081948332X

Bellingham, Washington USA

AppliedPrismatic and

Refl ective Optics

Dennis F. Vanderwerf

Library of Congress Cataloging-in-Publication Data

Vanderwerf, Dennis.Applied prismatic and reflective optics / Dennis Vanderwerf.

p. cm. – (Press monograph ; 200)Includes bibliographical references and index.ISBN 978-0-8194-8332-41. Lenses–Design and construction–Mathematics. 2. Mirrors–Design and

construction–Mathematics. 3. Prisms–Design and construction–Mathematics. 4.Fermat’s theorem. 5. Refraction. 6. Reflection (Optics) I. Title.

QC385.2.D47V36 2010681 ′.423–dc22

2010021193

Published by

SPIEP.O. Box 10Bellingham, Washington 98227-0010 USAPhone: +1 360.676.3290Fax: +1 360.647.1445Email: [email protected]: http://spie.org

Copyright c© 2010 Society of Photo-Optical Instrumentation Engineers

All rights reserved. No part of this publication may be reproduced or distributed inany form or by any means without written permission of the publisher.

The content of this book reflects the work and thought of the author(s). Everyeffort has been made to publish reliable and accurate information herein, but thepublisher is not responsible for the validity of the information or for any outcomesresulting from reliance thereon.

Printed in the United States of America.

About the cover: The image on the cover shows linear Fresnel solar concentrationat work. The solar concentrator powers an air-conditioning system at South WestGas Corporation’s facilities in Phoenix, Arizona. The system was designed byHelioDynamics (photograph courtesy of Lee Langan).

Contents

Preface............................................................................................................... xiii

Chapter 1 Introduction and Background ........................................................ 1

1.1 Snell’s Law of Refraction.............................................................................. 1

1.2 Optical Dielectric Materials ......................................................................... 2

1.3 Fresnel Reflection at a Dielectric Surface................................................ 4

1.4 External Reflection at an Optical Surface ................................................ 5

1.5 Internal Reflection at an Optical Surface ................................................. 6

1.6 Reflection Phase Shifts at a Planar Interface .......................................... 7

1.7 Antireflection and Reflection Coatings..................................................... 9

1.8 Effective f /# of a Converging Light Beam............................................. 9

1.9 Refraction and Translation of Skew Rays at Planar Surfaces ........... 10

1.10 Convergent Beam through a Tilted Plate ................................................. 13

1.11 Reflection and Translation of Skew Rays at Planar Surfaces............ 17

1.12 Reflection Matrix ............................................................................................. 18

1.13 Orientation of Viewed Images through Prisms ...................................... 18

1.14 Intersection Coordinate Matrix ................................................................... 19

1.15 Three-Mirror Beam-Displacing Prism ...................................................... 21

1.16 Refraction Matrix ............................................................................................. 24

1.17 Four-Mirror Beam-Displacing Prism ........................................................ 25

1.18 90-deg Beam-Deviating Prism..................................................................... 28

References ....................................................................................................................... 32

Chapter 2 General Prisms and Reflectors ...................................................... 33

2.1 Equilateral Prism .............................................................................................. 33

2.2 Abbe Dispersing Prism .................................................................................. 35

2.3 Pellin–Broca Dispersing Prism ................................................................... 36

2.4 Penta Prism......................................................................................................... 38

2.5 Right-Angle Prism ........................................................................................... 39

2.6 Porro Prism......................................................................................................... 40

2.7 Dove Prism ......................................................................................................... 42

2.8 Brewster Laser-Dispersing Prism ............................................................... 44

2.9 Littrow Prism ..................................................................................................... 46

v

vi Contents

2.10 Schmidt Prism ................................................................................................... 49

2.11 Pechan Prism ..................................................................................................... 53

2.12 Schmidt–Pechan Prism................................................................................... 54

2.13 Cube-Corner Retroreflector .......................................................................... 56

References ....................................................................................................................... 60

Chapter 3 Polarization Properties of Prisms and Reflectors ......................... 61

3.1 Prisms Producing Polarized Light .............................................................. 61

3.1.1 Uniaxial double-refracting crystals ........................................... 61

3.1.2 Nicol polarizing prism ................................................................... 61

3.1.3 Glan–Foucault polarizing prism ................................................. 63

3.1.4 Glan–Thompson polarizing prism ............................................. 64

3.1.5 Glan–Taylor polarizing prism...................................................... 64

3.1.6 Beam-displacing polarizing prism ............................................. 65

3.1.7 Wollaston polarizing prism .......................................................... 66

3.1.8 Nomarski polarizing prism ........................................................... 67

3.1.9 Rochon polarizing prism ............................................................... 67

3.1.10 MacNeille polarizing beamsplitter cube .................................. 68

3.1.11 Birefringent multilayer reflective polarizing film ................. 70

3.1.12 Polarizing beamsplitter elements using birefringentpolarizing film................................................................................... 71

3.1.13 Wire-grid polarizing beamsplitter .............................................. 72

3.1.14 Polarizing beamsplitter using frustrated total internalreflection ............................................................................................. 73

3.1.15 Polarizing beamsplitter prism with common polariza-tion output .......................................................................................... 74

3.2 Prisms Controlling the Polarization of Light .......................................... 75

3.2.1 Fresnel rhomb retarders ................................................................. 75

3.2.2 Total-internal-reflecting cube-corner retarders ...................... 78

3.2.3 Phase-coated total-internal-reflecting right-angle prismretarders............................................................................................... 80

3.3 Polarization Preservation in Prisms and Reflectors .............................. 82

3.3.1 Polarization-preserving total-internal-reflecting prism....... 82

3.3.2 Polarization-preserving two-piece reflective axicon............ 87

3.3.3 Polarization-preserving total-internal-reflecting cube-corner prism....................................................................................... 89

3.3.4 Stokes parameters ............................................................................ 89

3.3.5 Depolarizing cube-corner prism ................................................. 90

3.4 Plane of Polarization Rotation Using Total-Internal-ReflectingPrisms and Reflectors ..................................................................................... 92

3.4.1 90-deg polarization-rotating prism with coaxial beamoutput ................................................................................................... 92

3.4.2 90-deg polarization-rotating prism with retroreflectedbeam output ....................................................................................... 92

Contents vii

3.4.3 90-deg polarization-rotating prism with orthogonalbeam output ....................................................................................... 92

3.4.4 Double Fresnel rhomb polarization rotator withcollinear beam output ..................................................................... 93

3.4.5 Four-mirror 90-deg polarization rotator with collinearbeam output ....................................................................................... 94

References ....................................................................................................................... 95

Chapter 4 Specialized Prism Types ................................................................ 97

4.1 Dispersing Prism .............................................................................................. 97

4.1.1 Refracting direct-vision prism..................................................... 97

4.1.2 Reflective dispersing prisms with collinear output .............. 98

4.1.3 Direct-vision prisms with wavelength tuning ........................ 99

4.1.4 Total-internal-reflecting dispersing prism ............................... 99

4.1.5 Multiprism negative dispersion................................................... 101

4.2 Refracting Achromatic Compound Prism................................................ 101

4.3 Anamorphic Prisms for Beam Compression and Expansion ............ 103

4.3.1 Beam expander with orthogonal output ................................... 104

4.3.2 Beam compressor with coaxial output ..................................... 105

4.3.3 Beam expander with collinear output ....................................... 106

4.3.4 Wedge prism beam compressor/expander ............................... 107

4.3.5 Anamorphic prism pair with coaxial output ........................... 108

4.3.6 Multiprism dispersive compressors and expanders .............. 109

4.4 Achromatic Anamorphic Prism................................................................... 111

4.4.1 Air-spaced prism pair with coaxial output .............................. 111

4.4.2 Compound prisms with orthogonal output.............................. 113

4.4.3 Refracting/total-internal-reflecting prism pair with or-thogonal output................................................................................. 113

4.5 A Misalignment-Tolerant Beam-Splitting Prism................................... 116

4.6 Axicon Prism ..................................................................................................... 116

4.7 A Variable Phase-Shifting Prism ................................................................ 116

References ....................................................................................................................... 119

Chapter 5 Prism and Mirror System Design, Analysis, and Fabrication ....... 121

5.1 Prism Design and Analysis ........................................................................... 121

5.1.1 Sectional element approach for prism design ........................ 122

5.1.2 Right-angle prism sections ........................................................... 124

5.1.3 Experiential design of multiple reflectors ............................... 124

5.1.4 Matrix methods for design and analysis .................................. 125

5.1.5 Evolutionary prism design using a genetic algorithm ......... 126

5.1.6 A three-mirror tabletop lectern projector................................. 127

5.1.7 Prism aberrations ............................................................................. 128

viii Contents

5.2 Prism Quality Specifications ........................................................................ 130

5.2.1 Surface quality and flatness specifications .............................. 130

5.2.2 Optical material properties ........................................................... 130

5.2.3 Specifying angular accuracies ..................................................... 131

5.2.4 Tolerancing a Dove prism ............................................................. 131

5.2.5 Techniques for prism angle measurement ............................... 131

5.3 Survey of Fabrication Methods ................................................................... 135

5.3.1 Ground and polished glass prism ............................................... 135

5.3.2 Fabrication of a Penta prism by measurement of theangular deviation error ................................................................... 135

5.3.3 Molded, pressed, and fire-polished prisms.............................. 137

5.3.4 Fabrication of large prisms ........................................................... 137

5.4 Some prism-mounting methods .................................................................. 137

References ....................................................................................................................... 138

Chapter 6 A Selection of Prism Applications ................................................. 141

6.1 Laser Scanning.................................................................................................. 141

6.1.1 Reflective scanning prism ............................................................. 141

6.1.2 Refractive prism-beam scanning and steering ....................... 141

6.1.2.1 Single-wedge prism ....................................................... 141

6.1.2.2 Wedge prism pairs .......................................................... 143

6.1.2.3 LADAR guidance system using prism pairs.......... 145

6.1.2.4 Rotating square-plate linear scanner ........................ 146

6.2 Interferometry and Spectroscopy................................................................ 149

6.2.1 Laser interferometer with prism polarization rotator .......... 149

6.2.2 Polarization interferometer using a Wollaston prism .......... 149

6.2.3 Multipass optical cell for laser interferometer ....................... 150

6.2.4 Nomarski polarized-light interferometer ................................. 151

6.2.5 Aplanatic prism spectrograph...................................................... 152

6.3 Prismatic Optical Devices ............................................................................. 153

6.3.1 Prism switch for fiber-optic connections ................................. 153

6.3.2 Laser gyro readouts......................................................................... 153

6.3.3 Reflecting wedge prism for optical reader .............................. 155

6.3.4 Total-internal-reflecting touch switch using a Dove prism 157

6.3.5 Inspection device for window surfaces .................................... 158

6.4 Viewing, Display, and Illumination Systems .......................................... 159

6.4.1 Direct-view system for a microdisplay..................................... 159

6.4.2 Binocular surgical loupe with flare reduction ........................ 160

6.4.3 Inversion prism for range finders................................................ 161

6.4.4 Prism transforming transmitted intensity profile .................. 161

References ....................................................................................................................... 163

Contents ix

Chapter 7 Projection Displays ........................................................................ 165

7.1 Color-Separating and Color-Combining Prisms.................................... 1657.1.1 Three-channel Philips RGB separating prism........................ 1657.1.2 Philips prisms in reflective LCD projection displays .......... 1667.1.3 Crossed dichroic x-cube prisms for projection displays..... 1687.1.4 Prisms for digital light processing projection ........................ 1707.1.5 Other types of color-separating prisms for projectors......... 173

7.2 Polarizing Beamsplitters for Projection Displays ................................. 1757.2.1 MacNeille polarizing beamsplitters........................................... 1757.2.2 Cartesian polarizing beamsplitters ............................................. 1767.2.3 Wire-grid polarizing beamsplitters in projection displays . 177

7.3 Illuminators for Projection Displays.......................................................... 1797.3.1 Hollow tunnel integrators.............................................................. 1797.3.2 Solid light pipes................................................................................ 1807.3.3 Effect of light-pipe cross section on uniformity.................... 1807.3.4 Solid microprismatic light homogenizer.................................. 1817.3.5 Tapered-tunnel illuminator for projection displays .............. 183

References ....................................................................................................................... 184

Chapter 8 Microprismatic Arrays .................................................................... 187

8.1 Roof Prism Linear Array ............................................................................... 1878.2 Square Prismatic Hollow Light Guide ...................................................... 1958.3 Circular Prismatic Hollow Light Guide.................................................... 1978.4 Luminaire with Contoured Prismatic Extractor ..................................... 1998.5 Elliptical Light Guide with Directional Output ..................................... 1998.6 Prismatic Backlighting Devices .................................................................. 2028.7 Brightness Enhancement for Liquid Crystal Displays ........................ 2098.8 Polarizing Prismatic Sheet ............................................................................ 2168.9 Prismatic Reflective Polarizer Film ........................................................... 2178.10 LCD Backlights Producing Polarized Light ........................................... 2178.11 Prismatic Array Beamsplitters and Combiners ...................................... 2228.12 Polarization Converters Using Prismatic Arrays ................................... 2268.13 Cube-Corner Arrays ........................................................................................ 2288.14 Dove Prism Arrays .......................................................................................... 231References ....................................................................................................................... 234Some commercial nonsequential ray-tracing programs:.................................. 235

Chapter 9 Fresnel Lenses ............................................................................... 237

9.1 Basic Refractive Fresnel Lens Design ...................................................... 2379.1.1 Design example: Fresnel lens collimator/searchlight .......... 240

9.2 High-Transmission Fresnel Lens Doublet ............................................... 2429.3 Reflective Fresnel Lenses .............................................................................. 245

9.3.1 First-surface reflector design parameters................................. 2459.3.2 Second-surface reflector design parameters ........................... 247

x Contents

9.4 Refractive Planar Circular Fresnel Lens Solar Applications ............. 248

9.4.1 Multilens solar furnace .................................................................. 248

9.4.2 Multilens-array solar simulator................................................... 248

9.5 Refractive Meniscus Fresnel Lenses.......................................................... 248

9.6 Reflective Planar Linear-Focus Solar Concentrators ........................... 250

9.6.1 Tilted linear-focus reflective solar concentrator .................... 250

9.6.2 Linear-focus concentrator using a linear Fresnel lensand a crossed linear total-internal-reflecting array ............... 250

9.6.3 Planar reflective spot-focus concentrator using orthogo-nal refractive and reflective linear Fresnel lenses ................. 253

9.7 Curved Linear Fresnel Lens Solar Concentrators ................................. 255

9.8 Flexible Fresnel Lens Solar Concentrators.............................................. 260

9.8.1 Sectional planar solar concentrators.......................................... 260

9.8.2 Inflatable curved solar concentrators ........................................ 260

9.9 Fresnel Lenses Using Total Internal Reflection ..................................... 261

9.9.1 Low-profile overhead projector................................................... 262

9.9.2 Curved catadioptric Fresnel lenses ............................................ 262

9.9.3 Photovoltaic solar concentrator using total internalreflection ............................................................................................. 264

9.10 Fresnel Lenses for Rear-Projection Screens ........................................... 264

9.11 Fresnel Lens Manufacture............................................................................. 265

9.12 Achromatic Fresnel Lenses........................................................................... 265

9.12.1 Combination of high- and low-dispersion materials............ 267

9.12.2 Achromatic catadioptric Fresnel lenses ................................... 267

9.12.3 Dispersion-compensated achromatic Fresnel lens................ 271

9.12.4 Design example: achromatic dual-grooved Fresnel lensfor overhead projector .................................................................... 273

9.12.5 Achromatic zone plate using a Fresnel lens ........................... 274

9.13 Diffraction and Coherence Effects in Fresnel Lenses.......................... 276

9.13.1 Diffraction compensation in a Fresnel lens reflector ........... 276

9.13.2 Phase-optimized Fresnel lens ...................................................... 277

9.13.3 Phase-optimized Fresnel lens for use in an IR intrusiondetector ................................................................................................ 278

9.14 Design of a Fresnel Lens Illuminator Using Genetic Algorithms ... 278

References ....................................................................................................................... 281

Afterword........................................................................................................... 285

Index................................................................................................................... 287

Preface

This text deals primarily with the optics of refracting and reflecting planar surfacesin the form of prismatic refracting and reflecting components, and the design,analysis, and applications of these components. Optical prisms consist of multipleplanar surfaces, constructed to a specified geometry and formed from opticalglass or plastic. The surfaces may have thin-film coatings that contribute to theirfunctionality. Optical prismatic elements can be classified into two general types:those that are used in imaging systems, such as binoculars or projectors, andthose used in nonimaging systems, such as spectrometers, illuminators, and solarconcentrators. In addition to well-known prism systems, new applications of prismsare being introduced in the fields of electro-optics, metrology, prismatic films andarrays, projection displays, and others.

Chapter 1 introduces and reviews the optical concepts that are useful forthe topics developed in the succeeding chapters. In Chapter 2, some better-known prism types are discussed, along with the essential ray-trace equations thatdefine their specific properties. This includes both single and compound prisms,along with cube-corner retroreflectors. Birefringent prisms and polarizing beam-splitting prisms that produce polarized light are discussed in Chapter 3, includingprisms that affect the polarization state of light, such as polarization-preservingprisms and prisms that rotate the plane of polarization. Prisms with collinear andcoaxial dispersion properties, achromatic multiprisms, and anamorphic designsfor beam expansion and compression are examined in Chapter 4. In Chapter 5,several methods of prism design are reviewed, including some of the more recentmethodologies. This chapter also covers prism fabrication, tolerancing, choiceof optical material, and some mounting methods. Specific uses of prisms inoptical systems, such as scanning, beam steering, spectroscopy, interferometry,light coupling and switching, and viewing and illumination are presented inChapter 6. Chapter 7 covers the use of prisms as dichroic color beamsplitters andcombiners, polarizers, and light-beam homogenizers and integrators in projectiondisplays. Microprism arrays are very useful for light guides, luminaires, brightness-enhancement sheets, backlight displays, and sheet polarizers. These applicationsare detailed in Chapter 8. Last, Chapter 9 covers Fresnel lens optics and the useof both refractive and reflective lenses in illumination, solar concentration, anddirect-view displays. Several design methods for producing achromatic and phase-corrected Fresnel lenses are also presented.

References and examples are drawn from specialized texts, journal articles,conference proceedings, trade publications, and patent literature. I wish toacknowledge the editorial assistance of Gwen Weerts of SPIE Press for hersuggestions and contributions during the composition of this book.

Dennis F. VanderwerfAustin, Texas

xi

Chapter 1Introduction and Background

1.1 Snell’s Law of Refraction

One of the most important laws in the analysis and design of prisms, and opticalsystems in general, is Snell’s law of refraction, named for Willebrord Snell. Itrelates the angles of incidence and refraction at the boundary of two materials withdiffering refractive index (sometimes called the index of refraction). The refractiveindex n is defined as the ratio of the velocity of light in a vacuum c to the velocityof light in the material vmat:

n =c

vmat. (1.1)

Since the velocity of light is reduced when traveling through optical materials, nis greater than unity. For the special case of air, which has a refractive index ofapproximately 1.0003, we assume the refractive index of air to be unity for mostoptical calculations.

Snell’s law can be derived geometrically or from Fermat’s principle, named forPierre de Fermat.1,2 It is usually stated in the following form:

n sin I = n′ sin I′, (1.2)

where n is the refractive index of the incident medium, and n′ is the refractiveindex of the transmitting medium. I is the angle of incidence, measured relativeto the boundary surface normal, and I′ is the angle of refraction at the boundarysurface of the second medium (see Fig. 1.1). Snell’s law is applicable to plane orcurved surfaces, and both rays lie in a common plane called the plane of incidence.

A related law for reflecting surfaces is the law of reflection. It can also be derivedgeometrically or by using Fermat’s principle. It is stated in the following form:

I = I′, (1.3)

where I is the angle of incidence, and I′ is the angle of reflection, as illustrated inFig. 1.2. Since both incident and reflected rays are in the same medium, refractiveindex is not a factor in the directional change, and both rays lie in the commonplane of incidence.

1

2 Chapter 1

Figure 1.1 Snell’s law of refraction.

Figure 1.2 Law of reflection.

1.2 Optical Dielectric Materials

When applying Snell’s law, the refractive indices depend on the wavelength λ ofthe incident light (other than air). Refractive indices of optical materials for variouswavelengths are usually obtained from data tables from the manufacturer, or inoptical material reference books, and are determined by careful measurement ofthe indices for various spectral lines. The refractive index n is usually specifiedwith a subscript indicating the spectral line used for measuring the index—forexample, nC at λ = 656.3 nm, nd at λ = 587.3 nm, and nF at λ = 486.1 nm.There are also various formulas known as dispersion equations that approximatethe continuous change in refractive index with wavelength. The dispersion of anoptical material is defined as the change in refractive index with wavelength andis an important consideration for most prism designs. For visible light, the Abbenumber, or ν-number, gives a measure of the material dispersion and is defined as

ν =nd − 1

nF − nC. (1.4)

By comparing the ν-number between various glasses, relative dispersions can becompared, since a lower ν-number indicates a material of higher dispersion. This is

Introduction and Background 3

important in the choice of glass type for prism design, especially where dispersionreduction (e.g., achromatic prisms) is a requirement. In Fig. 1.3, a ray of visiblelight is refracted and dispersed at a planar air–glass surface.

There are many optically transparent dielectric optical materials, in the formof glasses and plastics, suitable for use in optical element fabrication. Althoughthere are hundreds of types of optical glass available for use by a lens designer,as listed in the Schott, Ohara, or Hoya glass catalogs, a more limited number isusually employed for optical prisms. Table 1.1 gives some representative typesused in commercially available prisms. The most important factors to be consideredin choosing a material for prism construction are the availability, cost, intendedspectral range, stability, transmission quality, and ability to be accurately machinedand polished, or in some cases, molded.

Figure 1.3 Dispersion of refracted light for crown glass.

Table 1.1 Typical prism materials.

Description Code or nC nd nF ν-number

brand name λ = 656.3 nm λ = 587.3 nm λ = 486.1 nm

Crown glass BK7 1.5143 1.5168 1.5228 64.2

Optical crown glass B270 1.5202 1.5229 1.5291 58.5

Flint glass F2 1.6150 1.6200 1.6331 36.4

Flint glass F5 1.5988 1.6034 1.6146 38.0

Extradense flint glass SF10 1.7209 1.7283 1.7464 28.5

Acrylic plastic(PPMA)

Lucite,Plexiglas

1.4892 1.4918 1.4978 57.5

Polycarbonate Lexan 1.5799 1.5855 1.5994 30.0

Fused silica UV grade HerasilSuprasil

1.4565 1.4586 1.4632 67.6

Fused quartz IR grade InfrasilVitreosilIR

1.4565 1.4586 1.4632 67.6

4 Chapter 1

1.3 Fresnel Reflection at a Dielectric Surface

When light is refracted at a dielectric optical surface, a portion of the incident lightis reflected, so all of the light is not transmitted. The amount of reflected light isderived from the Fresnel amplitude coefficients for reflection, named for AugustinJean Fresnel:

rp =n′ cos I − n cos I′

n cos I′ + n′ cos I, (1.5)

or

rp =tan(I − I′)tan(I + I′)

, (1.6)

rs =n cos I − n′ cos I′

n cos I + n′ cos I′, (1.7)

or

rs =−sin (I − I′)sin (I + I′)

, (1.8)

where rp is the reflection coefficient for p-polarized light (electric field vectorparallel to the plane of incidence), and rs is the reflection coefficient for s-polarizedlight (electric field vector perpendicular to the plane of incidence). Figures 1.4(a)and (b) show the angles of incidence, refraction, and reflection I, I′, I′′; the electricfield vectors E‖, E⊥; and the propagation vectors k, k′, and k′′, which define thedirection of the incident, refracted, and reflected rays. The orthogonal magneticfield vectors are shown as dotted, since the magnetic induction for dielectricmaterials is negligible.

The Fresnel reflections Rp and Rs are given by

Rp = r2p =

tan2(I − I′)tan2(I + I′)

, (1.9)

Rs = r2s =

sin2(I − I′)sin2(I + I′)

. (1.10)

The surface transmissions are

Tp = 1 − Rp, (1.11)

Ts = 1 − Rs. (1.12)

For unpolarized light, the reflection R is given by the average of Rp and Rs:

R =12

[tan2(I − I′)tan2(I + I′)

+sin2(I − I′)sin2(I + I′)

]. (1.13)


Figure 1.4 (a) Fresnel reflection vectors for s-polarized light (n′ > n). (b) Fresnel reflectionvectors for p-polarized light (n′ > n).

For normal incidence of light (I = I′ = 0), the reflection for unpolarized light R isgiven simply by

R =(n′ − n)2

(n′ + n)2 . (1.14)

The boundary transmission T for unpolarized light is then given by

T = 1 − R. (1.15)

1.4 External Reflection at an Optical Surface

The reflection parameters of light depend on whether the ray is incident on amedium of higher or lower refractive index. We will first consider the case wherethe medium is of a higher refractive index, or external reflection. Such an exampleis an air–glass interface. Figure 1.5 shows the Fresnel reflection coefficients Rs and

6 Chapter 1

Figure 1.5 External reflection at surface n′ > n.

Rp at an air–glass interface as a function of the angle of incidence I for BK7 opticalglass. The average value for unpolarized light is also shown. At a specific angle ofincidence IBrew, known as Brewster’s angle for David Brewster, the p-polarizedlight is completely refracted, with IBrew + I′ = 90 deg. The reflected light is thencompletely s-polarized, and from Snell’s law it follows that

IBrew = arctan(n′

n

). (1.16)

For an air–BK7 glass interface, with n′ = 1.5229 and n = 1.0, IBrew = 56.71 deg.

1.5 Internal Reflection at an Optical Surface

In the case where the light ray is incident on a medium of lower refractive index,we have internal reflection. Figure 1.6 illustrates the reflection coefficients for aBK7 glass–air interface as the angle of incidence varies. It is apparent that thereis also an incidence angle for which Rp = 0. Using Eq. (1.16), IBrew = 33.29 deg,which is the complement of the Brewster angle for external reflection. There is alsoan angle of incidence for which both Rp and Rs approach unity, such that the lightis completely reflected for all angles of incidence greater than this critical angle.This is called total internal reflection (TIR) and can provide 100% reflectance at


Figure 1.6 Internal reflection at surface n > n′.

all wavelengths of interest. No applied reflective coating can exactly achieve thisreflectance value.

The critical angle Icrit for total internal reflection is calculated from

Icrit = arcsin(n′

n

). (1.17)

For a BK7–air interface, with n′ = 1.0 and n = 1.5229, Icrit = 41.04 deg. BothBrewster reflection and total internal reflection are important considerations inmany prism designs.

1.6 Reflection Phase Shifts at a Planar Interface

When light is reflected from a planar surface, the component of the electric vectorE parallel to the plane of incidence E‖ and the component perpendicular to theplane of incidence E⊥ undergo a phase shift ∆ϕ. The phase shift for externallyreflected light varies as a function of the angle of incidence I for an interfacewhere n1 (incident medium) > n2 (transmitting medium). Assume that n1 = 1.0and n2 = 1.51. The E⊥ phase shift ∆ϕ⊥ remains constant at 180 deg for all I valuesbetween 0 and 90 deg. The E phase shift ∆ϕ‖ = 0 deg, for 0 deg ≤ I < IBrew, thenchanges abruptly to 180 deg for IBrew ≤ I ≤ 90 deg.

For internally reflected light (n1 > n2), ∆ϕ⊥ = 0 deg for 0 deg ≤ I ≤ Icrit, and∆ϕ⊥ varies continuously from 0 to 180 deg for Icrit < I ≤ 90 deg. This continuous

8 Chapter 1

variation can be calculated3,4 from the following:

∆ϕ⊥ = 2 arctan

−√

sin2 I − n∗2

cos I

, (1.18)

where n∗ ≡ n2/n1, and n∗ < 1.0, ∆ϕ‖ = 180 deg for 0 deg ≤ I < IBrew, ∆ϕ‖ = 0 degfor IBrew ≤ I ≤ Icrit, and ∆ϕ‖ varies continuously for Icrit ≤ I ≤ 90 deg (TIRregion). This continuous variation can be calculated from the following:

∆ϕ‖ = 2 arctan

−√

sin2 I − n∗2

n∗2 cos I

. (1.19)

The relative phase shift is defined as δ = ∆ϕ⊥ − ∆ϕ‖ and is calculated from

δ = 2 arctan

cos I√

sin2 I − n∗2

sin2 I

. (1.20)

The maximum value δmax is calculated from

δmax = 2 arctan[(1 − n∗2)

2n∗

], (1.21)

δmax occurs at the incident angle I (δmax), where

I(δmax) = arctan

√2n2

2

n21 − n2

2

. (1.22)

For n1 = 1.51 and n2 = 1.0, Icrit = 41.47 deg, IBrew = 33.51 deg, δmax = 45.94 deg,and I (δmax) = 51.34 deg.

Table 1.2 shows some calculated values for selected parameters, where thesmallest I value is slightly larger than Icrit. There are two values of I for whichδ = 45.0 deg, because there will be two values of I for every value of δ.

In Fig. 1.7, the δ variation is plotted as I varies from normal to a grazingincidence angle.

Table 1.2 Phase changes for total internal reflection, in degrees.

I 41.472 45 48.63 51.34 54.62 75 90∆Φ⊥ −0.2676 −38.63 −56.21 −67.03 −78.81 −139.6 −180.0∆Φ‖ −0.6102 −77.26 −101.2 −112.9 −123.81 −161.7 −180.0δ 0.3426 38.63 45 45.94 45 22.08 0


Figure 1.7 Phase shift for internal reflection.

1.7 Antireflection and Reflection Coatings

By the use of optical thin-film coatings, the reflectance of optical surfaces canbe significantly reduced. These antireflection (AR) interference coatings can bea single-layer film or a multilayer film of various thicknesses and materials. Thereflectivity is wavelength dependent. By judicious choice of coating materials, itis possible to reduce the reflectance to zero for a single wavelength using twolayers. By using more layers and different coating materials, the reflectance can belowered over an extended wavelength range.5 The reflectance of a common glasssurface can be reduced from about 4% to less than 0.5% over the visible spectrumby a cost-effective multilayer coating.6

High-reflectance surfaces can be obtained from metallic coatings, such asaluminum or silver. These metallic coatings can be further overcoated withmultilayer thin films to enhance the reflectance or make them more durable.For example, over the visible spectrum, the reflectance of an aluminum coatingcan be increased from about 92% to about 98% using four alternating layersof silicon dioxide and titanium dioxide.7 It is also possible to produce a high-reflectance (>99%) multilayer coating using alternating high- and low-indexdielectric coatings, with select wavelength band reflectance. Multilayer coatingscan be designed having other unique optical characteristics.

1.8 Effective f/# of a Converging Light Beam

It is often convenient to describe a converging beam of light in terms of aneffective f -number, designated by f /#. Consider a convergent beam of light withray angle θ′, produced by collimated light incident on an aplanatic lens, as shown

10 Chapter 1

in Fig. 1.8(a). Then, the effective f /# of the lens is defined by

f /# =n

2n′ sin θ′, (1.23)

where n is the refractive index on the object side of the lens, and n′ is the refractiveindex on the image side.8 For an aplanatic lens in air, n = n0 = 1.0, n′ = n0

′ = 1.0,and the minimum allowable f /# = f /0.5. Without considering the actual lens, wecan describe a convergent light beam with half-angle θ′ as having this effectivef /#. For example, if θ′ = 15 deg, the beam f /# = f /1.9; if θ′ = 30 deg, the beamf /# = f /1.0; and if θ′ = 60 deg, the beam f /# = f /0.58.

Consider a convergent f /1.0 beam (θ = 30 deg) that is incident on the frontsurface of a glass cube (n′ = n1

′ = 1.517) in air and is focused on the back surfaceof the cube by movement of the cube away from the lens [see Fig. 1.8(b)]. Aresultant focal shift ∆S′ occurs. The internal ray angle θ′ in the cube is reducedto 19.2 deg, yielding an effective beam f /# of f /0.12 at the focus, using a valueof n′ = 1.517. However, if the cube is moved toward the lens, such that the focusfalls outside the cube in air, then the effective beam f /# at the focus is returned tof /1.0, with a resultant focal shift ∆S′ as shown in Fig. 1.8(c).

1.9 Refraction and Translation of Skew Rays at Planar Surfaces

Up to now, we have been considering rays that lie only in the meridional(tangential) plane. Although this type of ray trace is useful, it is often necessary totrace rays that are incident on an optical surface in an arbitrary plane of incidence.These are called skew or oblique rays. Figure 1.9(a) shows the refraction of ageneral skew ray. The law of refraction can be written in vector form as thefollowing cross-product:

n(K × k) = n′(K′ × k). (1.24)

Here, n is the refractive index of the incident medium, n′ is the refractive index ofthe refracting medium, K is the incident ray vector, K′ is the refracted ray vector,and k is the vector normal to the refracting boundary.

The following derived equations can be used in a Cartesian coordinate system tocalculate the angles of refraction and intersection coordinates at the next occurringsurface:

Ki′ =

( nn′

)Ki + ki

[cos I′ −

( nn′

)cos I

], (1.25)

where

cos I =∑

Kiki, where (i = x, y, z), (1.26)


and

cos I′ =∑

Ki′ki, where (i = x, y, z), (1.27)

where I is the angle of incidence, I′ is the angle of refraction, ki is the directioncosine of the normal for the planar boundary in the forward direction, Ki is thedirection cosine of the incident ray, and Ki

′ is the direction cosine of the refractedray. The summation is over the x, y, and z coordinates. The Kx

′, Ky′, and Kz

′

values then define the direction of the ray incident on the next surface. I′ is usuallycalculated directly from I using Snell’s law [see Fig. 1.9(b)].

Figure 1.8 (a) f /# of lens in air, (b) f /# in glass cube, and (c) f /# through glass cube.

12 Chapter 1

Figure 1.9 (a) Refraction of a skew ray, vector representation. (b) Refraction of a skew ray,Cartesian coordinates.

The translation of the ray to the next planar surface is calculated from theintersection of the ray with the next planar surface. Since prism design deals mainlywith planar surfaces, we do not have the added complication of calculating the rayintersection at a curved surface, as for a lens. The normal form of the planar surfaceis given by

xkx + yky + zkz = P, (1.28)

where P is the perpendicular distance from the plane to the origin.The line representing the ray is described by the following set of equations:

x − xn−1

Kx=

y − yn−1

Ky=

z − zn−1

Kz= d, (1.29)

where (xn−1, yn−1, zn−1) are the intersection coordinates from the previous surface,and d is the ray distance between the two surfaces.


Figure 1.10 Translation of light ray between planar surfaces.

The intersection coordinates (x, y, z) are then calculated from the simultaneoussolution of Eq. (1.28) and Eq. (1.29), and these become the initial values for thetranslation to the next surface (see Fig. 1.10).

1.10 Convergent Beam through a Tilted Plate

We now use Eq. (1.25) through Eq. (1.29) to trace an f /1.9 convergent beam oflight through a tilted glass plate (or slab or cube) of thickness I, with a nominalrefractive index n′ = 1.5168. The traces can be performed in the tangential plane,an oblique plane, or the sagittal plane. In Fig. 1.11, a convergent f /1.9 beamof light (maximum half-angle θ = 15 deg) originates from a circular referencesurface having a radius R0. This could be the exit pupil of a positive lens thatfocuses the beam to a point at a distance z f from the reference surface. Any rayρ originating from this reference plane from a point (x0, y0, z0) is specified by theangle ϕmeasured from the positive x axis and the radial distance r from the origin,where ρ = r/ sin θ, x0 = r cosϕ, y0 = r sinϕ, and z0 = 0. The direction cosines ofthese rays are then calculated from the following equations:

Kx1 =x0

ρ= −cosϕ sin θ (1.30)

Ky1 =−y0

ρ= −sinϕ sin θ (1.31)

Kz1 =z f

ρ. (1.32)

14 Chapter 1

Figure 1.11 Glass plate in a convergent beam.

If a glass plate of thickness T is inserted in the beam normal to the optical axis,there is a longitudinal displacement of the focus from the original focus.

Now, the glass plate is positioned a distance d01 from the reference surface andtilted around the y axis by angle ω, as in Fig. 1.12. The direction cosines of theplate entrance surface 1 and the plate exit surface 2 are then

kx1 = kx2 = −sinω, (1.33)

ky1 = ky2 = 0, (1.34)

kz1 = kz2 = cosω. (1.35)

Figure 1.12 Astigmatism from a tilted plate in a convergent beam—tangential plane.


The angle of incidence I1 and angle of refraction I1′ at plate surface 1 are then

calculated from Eq. (1.26) and Snell’s law:

I1 = arccos(Kx1kx1 + Ky1ky1 + Kz1kz1), (1.36)

I1′ = arcsin

[( nn′

)sin I1

]. (1.37)

The direction cosines of the refracted ray at plate surface 1 are then calculated fromEq. (1.25):

Kx1′ =

( nn′

)(Kx1 − kx1 cos I1) + kx1 cos I1

′, (1.38)

Ky1′ =

( nn′

)(Ky1 − ky1 cos I1) + ky1 cos I1

′, (1.39)

Kz1′ =

( nn′

)(Kz1 − kz1 cos I1) + kz1 cos I1

′. (1.40)

These values are then used for the ray incident on the exit plate surface 2. Theintersection coordinates (x1, y1, z1) at the plate surface 1 are calculated usingEq. (1.28) and Eq. (1.29), where P01 = d01 cosω. Then,

x1 =

kx1 + ky1

(Ky1

Kx1

)+ kz1

(Kz1

Kx1

)P01 − ky1

[y0 −

(Ky1

Kx1

)x0

]− kz1

[z0 −

(Kz1

Kx1

)x0

] , (1.41)

y1 = y0 +

(Ky1

Kx1

)(x1 − x0), (1.42)

z1 = z0 +

(Kz1

Kx1

)(x1 − x0). (1.43)

For surface 2 of the plate, P02 = P01+T . Then, the direction cosines of the refractedray (Kx2

′,Ky2′,Kz2

′) and the intersection coordinates (x2, y2, z2) at surface 2 arecalculated by the reapplication of Eq. (1.38) through Eq. (1.43). The intersectioncoordinates (x3, y3) of the refracted ray at the original focal plane (z3 = z f ) can becalculated from

x3 = x2 +

(Kx2

′

Kz2′

)(z f − z2), (1.44)

y3 = y2 +

(Ky2

′

Kz2′

)(z f − z2). (1.45)

It is useful to calculate the focal-point coordinates (x3, y3, z3) in both thetangential and sagittal planes for various ray angles in the convergent beam. Forthe nonsymmetric tangential plane (ϕ = 0 deg and 180 deg, and y3 = 0), the z3

16 Chapter 1

coordinate is calculated from the intersection of the upper ray U with the lower rayL. At this intersection, xL

3 = xU3 and xU

3 = xL3 . Then,

z3 =

xL2 − xU

2 +

(Kx2

′U

Kz2′U

)zU

2 −

(Kx2

′L

Kz2′L

)zL

2

Kx2′U

Kz2′U −

Kx2′L

Kz2′L

, (1.46)

x3 = xU2 +

(Kx2

′U

Kz2′U

)(z3 − zU

2 ). (1.47)

For the symmetric sagittal plane (ϕ = 90 deg and 270 deg, and x3 = 0), the z3focus coordinate is calculated at the position where the ray intersects the z axis, orwhere y3 = 0. It follows that

z3 = z2 −

(Kz2′

Ky2′

)y2. (1.48)

We now apply the previous equations to trace through a 45-deg tilted BK7 glassplate (or slab or cube) with a thickness of 5 mm. Plates of this tilt angle appearin many optical applications. The beam half-angle of the boundary ray is 15 deg,and the beam radius at the reference plane is 25 mm. The calculated focal pointof the undeviated beam is z f = 93.30 mm. With the plate inserted, the focalpoint in the tangential plane for the boundary rays moves outward and downwardwith coordinates x3 = −1.853, y3 = 0, and z3 = 96.40. In the sagittal plane, thefocal point moves outward along the z axis with coordinates x3 = 0, y3 = 0, andz3 = 95.020. The paraxial foci are calculated from a very small cone (θ ≈ 0.5 deg)centered on the optical axis, where zt

3 (paraxial) in the tangential plane = 96.322,and zs

3 (paraxial) in the sagittal plane = 94.974.The resultant astigmatism is then calculated from the difference of these paraxial

foci:

Astigmatism = zs3 (paraxial) − zt

3 (paraxial) = −1.348. (1.49)

This value is significant and needs to be considered, especially for imaging systemsin which a tilted plate or slab is an optical component. The longitudinal sphericalaberration (LSA) is calculated from the following equation:

LSA = z3 (θ) − z3 (paraxial). (1.50)

Since the tilted plate is nonrotationally symmetric around the optical axis, the LSAwill vary with both the ray angle θ and the planar angle ϕ of the ray fan. Figure 1.13plots the LSA as a function of θ in the tangential and sagittal planes. Color andcoma are also introduced by a tilted plate in a convergent light beam. There are


Figure 1.13 LSA as a function of θ in tangential and sagittal planes.

concise formulas available for the calculation of astigmatism, chromatic aberration,spherical aberration, and coma for the tilted plate.9

1.11 Reflection and Translation of Skew Rays at Planar Surfaces

For a reflective surface, the law of reflection can be described by the followingvector equation:

K′ = K − 2k cos I, (1.51)

where K is the incident ray vector, K′ is the reflected ray vector, k is the upwardvector normal at the reflecting surface, and I is the angle of incidence [seeFig. 1.14(a)].

For a Cartesian coordinate system, the following equations result:

Ki′ = Ki − 2ρki, (1.52)

where

ρ =∑

Kiki = cos I, where (i = x, y, z). (1.53)

Here, Ki is the direction cosine of the incident ray, ki is the direction cosine of thereflecting surface normal pointing into the mirror, and I is the angle of incidence.Equation (1.53) is interpreted as follows: the arc cosine of ρ yields the angle ofincidence I [see Fig. 1.14(b)].

18 Chapter 1

Figure 1.14 (a) Reflection of a skew ray, vector representation. (b) Reflection of a skewray, Cartesian coordinates.

1.12 Reflection Matrix

Equations (1.51) and (1.52) can be combined and recast in matrix form to assist inmultiple sequential reflector computations. For a review of matrix and determinantmathematics, see, for example, Kreyszig,10 or for optical applications, Kloos.11

The following matrix equation results:Kx′

Ky′

Kz′

= R

KxKyKz

, (1.54)

where R is the reflection matrix:

R =

1 − 2k2x −2kxky −2kxkz

−2kxky 1 − 2k2y −2kykz

−2kxkz −2kykz 1 − 2k2z

. (1.55)

This defines the direction of the reflected ray. For a series of multiple reflectingsurfaces, the reflection matrix can be multiplied to calculate the direction of thefinal ray:

R = (Rn) (Rn−1) (. . .) (R1), (1.56)

where the matrix multiplication is performed in the opposite direction in whichthe light strikes the mirrors. In general, matrix multiplication is noncommutative,unless both matrices are diagonal and of the same order.

1.13 Orientation of Viewed Images through Prisms

The terminology to describe the image orientation of the object viewed througha mirror system or a reflecting prism can be confusing. We follow closely thedefinitions of Malacara and Malacara12 for inversion (geometric reflection about


a horizontal axis) and reversion (geometric reflection about a vertical axis) todescribe the image orientation of an object as viewed through the mirror system orprism, not being concerned for the present with how the orientation was produced.If the viewed image can be read normally, then it is called a right-handed image orreadable image. Readable images can be rotated over any rotation angle and remainreadable. If the viewed image is unreadable, then the image has been transformedfrom a right-handed to a left-handed coordinate system (sometimes called a changein parity). An unreadable image will remain unreadable at any rotation angle. If theunreadable image is erect, it is said to be reverted. A reverted image is producedby the reflection from a vertical wall-mounted mirror. If the unreadable image isupside down, the image is said to be inverted. An inverted image is produced, forexample, by the reflection from a horizontal body of water. By these definitions,both inverted and reverted images are unreadable. The sequential operations ofinversion and reversion will produce a readable image that is rotated 180 deg.

By these definitions, a prism that produces a readable upside-down image (i.e.,rotates the image by 180 deg) is not an inverting prism. (See also the use of theterms inversion and reversion, as used by Levi.13) However, prisms that rotate animage 180 deg, regardless of readability, are often called inverting prisms in theliterature. Figures 1.15(a) to (d) illustrate several of these cases, where the dottedbox represents a mirror system or reflecting prism of unspecified design. The exitimage represents how the object appears when viewed, and is not a “projected”image. When the determinant of R, |R| = −1, an image viewed through theprism will be unreadable. It does not, however, tell us anything about rotation ordisplacement of the viewed image.

To determine the actual orientation of the viewed image, we consider howthe object coordinate system (x0, y0, z0) is transformed or rotated into a viewedcoordinate system (x′, y′, z′). This can be determined by use of the following matrixmultiplication, where RT is the transpose of the reflection matrix R.14

Then, x′

y′

z′

= RT

cosαx cosαy cosαzcos βx cos βy cos βzcos γx cos γy cos γz

x0y0z0

, (1.57)

where RT represents a rotation of the viewed coordinate system relative tothe object coordinate system, (αx, βx, γx) are the direction angles of the rotatedcoordinate system relative to the x0 axis, (αy, βy, γx) are the direction angles relativeto the y0 axis, and (αz, βz, γz) are the direction angles relative to the z0 axis.

1.14 Intersection Coordinate Matrix

The intersection coordinates (xi, yi, zi) at the next reflecting surface can becalculated by the same procedure as for plane refracting surfaces, as in Eq. (1.28)and Eq. (1.29), ensuring that the next reflective surface is in a position to interceptthe incident ray. Again, there are advantages to calculating the intersection

20 Chapter 1

(a) (b)

(c) (d)

Figure 1.15 (a) Viewed image is undeviated, erect, and unreadable. (b) Viewed image isdisplaced, rotated 180 deg, and readable. (c) Viewed image is displaced, deviated 90 deg,erect, and readable. (d) Viewed image is deviated by angles α and β, erect, and readable.

coordinates using matrices. This calculation can also be defined in matrix formas follows:13,14

1xnynzn

= C

1

xn−1yn−1zn−1

, (1.58)

where C is the intersection-coordinate matrix:

C =

1 0 0 0

PKx(i−1)/ρ 1 − kxKx(i−1)/ρ −kyKx(i−1)/ρ −kzKx(i−1)/ρPKy(i−1)/ρ −kxKy(i−1)/ρ 1 − kyKy(i−1)/ρ −kzKy(i−1)/ρPKz(i−1)/ρ −kxKz(i−1)/ρ −kyKz(i−1)/ρ 1 − kzKz(i−1)/ρ

, (1.59)

where i = 1, 2, 3, . . . , P is the perpendicular distance from surface i to the origin,and ρ is defined as in Eq. (1.53). Computation of the coordinate matrix C for asurface requires knowledge of the preceding surface coordinates and the directioncosines of the incident ray. For a series of multiple reflecting surfaces, the mirror


coordinate matrices can be multiplied to calculate the coordinates of the final rayat a defined surface:

C = (Cn) (Cn−1) (. . .) (C1). (1.60)

1.15 Three-Mirror Beam-Displacing Prism

In general, an odd number of reflecting surfaces produces an unreadable image,and the image direction and rotation are determined by the mirrored surfaceorientations. An even number of mirrored surfaces will produce a readable image.Consider the three-mirror reflecting prism in Figs. 1.16(a) and (b). Reflection frommirror surfaces M1 and M2 occurs by total internal reflection for most opticalglasses. Reflecting surface M3 must be coated with a reflective material. This is alimited-aperture prism, with the square entrance and exit apertures Ap×Ap definedas shown. θ = 60 deg, ϕ = 120 deg, ψ = 150 deg, input face length = 2Ap,M1 length = 2Ap, M2 length = 3Ap, and M3 length = 2Ap/ sinα. If we considerentrance rays normal to the planar entrance aperture, then the exit ray will emergenormal to the exit aperture, and since there is no ray deviation due to refraction,we treat this as a pure three-mirror system. Place the (x, y, z) coordinate systemorigin along a central ray entering the center of the aperture. The distance of theorigin from the entrance facet is somewhat arbitrary, but we place it at the positionwhere the extension of the M3 facet crosses the z axis. The direction cosines of theincident ray are

Kx01 = 0 Ky01 = 0 Kz01 = +1, (1.61)

and for the reflecting surfaces, the direction cosines of each mirror normal are

kx1 = cos (30 deg) ky1 = 0 kz1 = cos (60 deg), (1.62a)

kx2 = 0 ky2 = 0 kz2 = 1, (1.62b)

kx3 = −cos (60 deg) ky3 = 0 kz3 = −cos (30 deg), (1.62c)

From Eq. (1.55), the reflection matrices for each reflecting surface are

R1 =

−0.50 0 −0.8660 1 0

−0.866 0 0.50

, (1.63a)

R2 =

1 0 00 1 00 0 −1

, (1.63b)

R3 =

0.750 −0.50 −0.433−0.50 0 −0.866−0.433 −0.866 0.250

, (1.63c)

22 Chapter 1

Figure 1.16 (a) A three-mirror reflecting prism; viewed image is displaced and inverted.(b) A three-mirror reflecting prism, perspective view.


and from Eq. (1.56),

R = (R3)(R2)(R1) =

−1 0 00 1 00 0 1

. (1.64)

The determinant |R| = −1. Therefore, the image is unreadable.Then,

Kx′

Ky′

Kz′

= R

Kx0

Ky0

Kz0

= R

001

=

001

. (1.65)

This simply states that the exiting ray, the ray reflected from M3, is parallel to theentrance ray. However, it is not collinear, being displaced by a defined distance.The transposed reflection matrix RT is

RT =

−1 0 00 1 00 0 1

, (1.66)

which in this case is identical to R, and the matrices are symmetric. For the x′

axis, αx = 180 deg, αy = 90 deg, and αz = 90 deg; for the y′ axis, βx = 90 deg,βy = 0 deg, and βz = 90 deg; and for the z′ axis, γx = 90 deg, γy = 90 deg, andγz = 0 deg. The direction of the x′ axis is reversed and the directions of the y′ andz′ axes remain unchanged, and therefore the viewed image is inverted.

The intersection coordinates at each surface can be calculated using Eq. (1.58).The P values for the reflecting planes are as follows:

P1 = (Ap/2) sin (60 deg), (1.67a)

P2 = 2Ap sin (60 deg), (1.67b)

P3 = 1.5Ap cos (60 deg). (1.67c)

Assume for these calculations that Ap = 1.0. Using Eq. (1.53), ρ1 = 0.50 andI1 = 60 deg, ρ2 = 0.50 and I2 = 60 deg, and ρ3 = 0.8660 and I3 = 30 deg.

Equation (1.59) then yields the coordinate matrices C1, C2, and C3:

C1 =

1 0 0 00 0 1 00 0 1 0

0.866 1.732 0 0

, (1.68a)

24 Chapter 1

C2 =

1 0 0 0−3.0 0 1 1.732

0 0 1 01.732 0 0 0

, (1.68b)

C3 =

1 0 0 0

−0.750 0.50 1 −0.8660 0 1 0

−0.433 −0.289 0 0.50

. (1.68c)

Then, 1x3y3z3

= C

1x0y0z0

, (1.69)

where

C = (C3)(C2)(C1) =

1 0 0 0−3.0 1 0 1.732

0 0 1 00.866 0.577 0 0.50

, (1.70)

yielding x3 = −3.0, y3 = 0, and z3 = 0.866. Thus, the exiting central ray throughprism surface 2 is displaced by a distance 3Ap from the incident central ray.

1.16 Refraction Matrix

As we defined a reflection matrix for a mirror, we can define a refraction matrixfor refraction at a boundary between materials having different refractive indices.Using Eqs. (1.38), (1.39), and (1.40), the direction cosines of the refracted ray are

Kx′ =

( nn′

)Kx + kx

[ρ′ −

( nn′

)ρ

], (1.71a)

Ky′ =

( nn′

)Ky + ky

[ρ′ −

( nn′

)ρ

], (1.71b)

Kz′ =

( nn′

)Kz + kz

[ρ′ −

( nn′

)ρ

], (1.71c)

and

ρ = cos I =∑

Kiki, where i = (x, y, z), (1.72)

ρ′ = cos I′ =

√1 −

[( nn′

)sin I

]2. (1.73)


Equations (1.71a), (1.71b), and (1.71c) can be recast in matrix form as1

Kx′

Ky′

Kz′

= R

1

KxKyKz

, (1.74)

where the refraction matrix R is defined as

R =

1 0 0 0

kx[ρ′ − (n/n′)ρ] n/n′ 0 0ky[ρ′ − (n/n′)ρ] 0 n/n′ 0kz[ρ′ − (n/n′)ρ] 0 0 n/n′

. (1.75)

For a general prism in which both refraction and reflection occur, we cannotdirectly multiply refraction matrices R by reflection matrices R of the form inEq. (1.55), since they are of a different order. To obtain a system matrix S by matrixmultiplication, we introduce a modified fourth-order matrix R′ for the reflectionmatrix, such that

R′ =

1 0 0 0

−2kxρ 1 0 0−2kyρ 0 1 0−2kzρ 0 0 1

, (1.76)

where ρ is defined as in Eq. (1.72). We call R′ the fourth-order reflection matrix,and it yields the direction cosines of the reflected ray as R. Then,

1Kx′

Ky′

Kz′

= R′

1

KxKyKz

. (1.77)

The use of the third-order matrix R is preferable for pure mirror systems,because it requires only the direction cosines of the reflective surface normals. BothR and R′ will yield the same direction cosines of the reflected ray. The coordinatematrix C for refractive surfaces is obtained from Eq. (1.59), where ρ is defined asin Eq. (1.72).

1.17 Four-Mirror Beam-Displacing Prism

To produce a readable image, we can replace the single inclined M3 reflector ofthe three-mirror reflecting prism example with two reflective contiguous mirrorsM3 and M4, having a 90-deg included angle [Fig. 1.17(a)]. This is often called aroof mirror, which by itself produces an erect and readable image. In this prism,

26 Chapter 1

the roof vertex line is inclined at the same angle β = 30 deg as the original singlemirror. The reflections from M3 and M4 now occur by total internal reflection.Then,

kx3 = −cos (45 deg) sin (30 deg)ky3 = cos (45 deg)kz3 = −cos (45 deg) cos (30 deg),

and

kx4 = −cos (45 deg) sin (30 deg)ky4 = −cos (45 deg)kz4 = −cos (45 deg) cos (30 deg),

where M3 and M4 could be reversed in computational sequence with no resultantdifference.

Then, the corresponding reflection matrices are

R3 =

0.750 −0.50 −0.433−0.50 0 −0.866−0.433 −0.866 0.250

, (1.78a)

R4 =

0.750 0.50 −0.4330.50 0 0.866−0.433 0.866 0.250

. (1.78b)

R1 and R2 remain as defined in Eq. (1.63a) and Eq. (1.63b).Then,

R = (R4)(R3)(R2)(R1) =

−1 0 00 −1 00 0 1

. (1.79)

The determinant |R| = +1, indicating that the image is readable.Then,

Kx′

Ky′

Kz′

= R

Kx0

Ky0

Kz0

= R

001

=

001

. (1.80)

The exiting ray remains parallel to the entrance ray.The transposed reflection matrix RT is

RT =

−1 0 00 −1 00 0 1

. (1.81)


Thus, both the x′ axis and the y′ axis are reversed, and the z′ axis is unchanged.The viewed image is therefore readable and rotated 180 deg. This prism, as inFigs. 1.17(a) and (b), has a circular area of nonessential glass removed and is calleda Leman prism (sometimes called a Sprenger-Leman prism or Leman-Sprengerprism). Reflection at M3 and M4 occurs by TIR, each having an angle of incidenceof about 52 deg, eliminating the need for any reflective coatings. The Leman prismis used in several optical system applications, such as a monocular spotting scope.

Figure 1.17 (a) A four-mirror reflecting prism with roof prism face; viewed image is rotated180 deg and readable. (b) A four-mirror reflecting prism with roof prism face, perspectiveview.

28 Chapter 1

1.18 90-deg Beam-Deviating Prism

The four-surface single prism in Fig. 1.18(a) is useful for producing an erect andreadable viewed image with a deviation angle δ = 90 deg. The square inputaperture of side Ap is inclined at 45 deg to the z axis. Figure 1.18(b) shows thedesign parameters for BK7 glass (nd = 1.5168). There are two refractions and tworeflections. Total internal reflection occurs at surface 2 (BC), while surface 3 (CD)must be coated with a reflective material. Surface AD is nonworking. The objectplane 0 and the viewing plane 5 are in the positions shown, with the coordinatesystem origin on the central ray at the object plane.

The sequential ray-tracing equations are

I1 = 45 deg, (1.82a)

I1′ = arcsin

(sin I1

n

), (1.82b)

I2 = I1 − I1′ + 45 deg = I2

′, (1.82c)

I3 = I2′ − 45 deg = I3

′, (1.82d)

I4 = 45 deg − I3′ = I1

′, (1.82e)

I4′ = arcsin(n sin I4) = I1, (1.82f)

δ = 45 deg + I4′ = 90 deg. (1.82g)

The prism dimensions are determined by the input aperture Ap and the intersectionof the lower refracted ray AC at the far corner of the prism. The prism dimensionsare then calculated from

AB =Ap

sin (45 deg), (1.83a)

BC = AB tan (θ − I1′), (1.83b)

CD =AB

sinψ, and (1.83c)

AD = CD[sin (ψ − I1

′)sin I1

′

]. (1.83d)

Let Ap = 25 mm. Then, AB = 35.4 mm, BC = 67.1 mm, CD = 50.1 mm, andAD = 31.8 mm.

The system matrix S is given by

S = (R4) (R3′) (R2

′) (R1), (1.84)


(a)

(b)

Figure 1.18 (a) A 90-deg beam-deviating prism, perspective view. (b) A 90-deg beam-deviating prism, design parameters.

30 Chapter 1

and 1

Kx45Ky45

Kz45

= S

1

Kx01Ky01

Kz01

, (1.85)

where

S =

1 0 0 01 1 0 00 0 1 0−1 0 0 1

. (1.86)

Then, Kx45 = 1, Ky45 = 0, Kz45 = 0, and the ray is deviated 90 deg in the +xdirection.

The fourth-order determinant |S| = +1, and therefore, the image is readable.Alternatively, since the two planar refracting surfaces do not change the readabilityof a viewed image, it could be directly inferred that the even number of mirrorsin this prism will produce a readable image. The ray intersection coordinates ateach surface are calculated using Eq. (1.58) and Eq. (1.59) for each surface, in theorder in which the ray hits the surfaces. At the viewing plane 5, the deviated ray isdisplaced 2.4460 units in the z direction.

To calculate the ray intersection coordinates at each surface, we construct thecoordinate matrix Ci at each surface using Eq. (1.59). Then,

C = (C5)(C4)(C3)(C2)(C1) =

1 0 0 1

12.50 0 0 00 0 1 0

61.15 −1 0 0

. (1.87)

Then, 1x5y5z5

= C

1x0y0z0

. (1.88)

If x0 = +12.5 mm, then z5 = 48.65 mm, and if x0 = −12.5 mm, then z5 =

73.65 mm, such that the exit-beam width equals the input aperture height, andthere is no anamorphic expansion or compression of the exit beam. Table 1.3 givesthe relevant ray-trace data.

A useful construction for many prism types is the tunnel diagram, shown forthis 90-deg deviating prism (Fig. 1.19). Here, the prism is unfolded about eachreflecting surface, which shows the direct path of a ray. The tunnel diagram shows


Table 1.3 90-deg deviating prism ray-trace data.

Surfacenumber

Intersectioncoordinates(mm)

ρ, I, I′ Exit-raydirectioncosines

Surfacenormaldirectioncosines

Perpendicularsurfacedistance toorigin (mm)

0 Origin surface x0 = 0 ρ0 = 0 Kx01 = 0 kx0 = 0 P0 = 0y0 = 0 I0 = 90 deg Ky01 = 0 ky0 = 0z0 = 0 I0

′ = 90 deg Kz01 = 1.0 kz0 = 1.0

1 Refractor x1 = 0 ρ1 = 0.7071 Kx12 = −0.2959 kx1 = −0.7071 P1 = 8.84y1 = 0 I1 = 45 deg Ky12 = 0 ky1 = 0z1 = 12.50 I1

′ = 27.787 deg Kz12 = 0.9552 kz1 = 0.7071

2 Reflector x2 = −11.22 ρ2 = 0.4662 Kx23 = −0.9552 kx2 = 0.7071 P2 = 26.5y2 = 0 I2 = 62.213 deg Ky23 = 0 ky2 = 0z2 = 48.72 I2

′ = 62.213 deg Kz23 = 0.2959 kz2 = 0.7071

3 Reflector x3 = −37.50 ρ3 = 0.9552 Kx34 = 0.9552 kx3 = −1.0 P3 = 37.5y3 = 0 I3 = 17.215 deg Ky34 = 0 ky3 = 0z3 = 56.86 I3

′ = 17.215 deg Kz34 = 0.2959 kz3 = 0

4 Refractor x4 = −23.65 ρ4 = 0.8847 Kx45 = 1.0 kx4 = 0.7071 P4 = 26.5y4 = 0 I4 = 27.868 deg Ky45 = 0 ky4 = 0z4 = 61.15 I4

′ = 45 deg Kz45 = 0 kz4 = 0.7071

5 Viewing plane x5 = 12.50 ρ5 = 0 — kx5 = 1.0 P5 = 12.5y5 = 0 I5 = 0 deg ky5 = 0z5 = 61.15 I5

′ = 0 deg kz5 = 0

Figure 1.19 Tunnel diagram for the 90-deg deviating prism.

32 Chapter 1

that the input and exit surfaces are optically parallel, and there will be no dispersionwhen collimated light enters the prism as shown. However, the prism will exhibitthe aberrations of a tilted plate when used in convergent or divergent light beams.Therefore, unless there are additional elements to correct these aberrations, thisparticular prism is recommended for use only in collimated light.

References

1. E. Hecht, “The propagation of light,” Chapter 4 in Optics, 2nd ed., 87–92,Addison-Wesley, Reading, MA (1987).

2. T. V. Higgins, “All rays lead to geometrical optics,” Laser Focus World 30(4),89–97 (1994).

3. M. Born and E. Wolf, Principles of Optics, 4th ed., 49–50, Pergamon Press,London (1970).

4. J. Lekner, Theory of Reflection: of Electromagnetic and Particle Waves,194–195, Springer, New York (1987).

5. T. V. Higgins, “Reflections on surfaces, coatings, and thin films,” Laser FocusWorld 30(9), 61–67 (1994).

6. CERAK, Inc., “Coatings: selecting thin-film materials,” in The PhotonicsDesign and Applications Handbook, 46th ed., Book 3, 88–98, LaurinPublishing, Pittsfield, MA (2000).

7. C. K. Carniglia, “Mirrors: coating choices make a difference,” in ThePhotonics Design and Applications Handbook, 46th ed., Book 3, 307–310,Laurin Publishing, Pittsfield, MA (2000).

8. M. R. Hatch and D. E. Stoltzmann, “The f-stops here,” Optical Spectra, 88–91(June, 1980).

9. W. J. Smith, Modern Optical Engineering, 2nd ed., 99, McGraw-Hill, NewYork (1990).

10. E. Kreyszig, Advanced Engineering Mathematics, 5th ed., 289–299, JohnWiley, New York (1983).

11. G. Kloos, “Optical components,” Chapter 2 in Matrix Methods for OpticalLayout, SPIE Press, Bellingham, WA (2007) [doi:10.1117/3.737850].

12. D. Malacara and Z. Malacara, “Prisms,” Chapter 18 in Handbook of LensDesign, Marcel Dekker, New York (1994).

13. L. Levi, “Plane surfaces, mirrors, and prisms,” Chapter 8 in Applied Optics,Vol. 1, John Wiley, New York (1980).

14. R. E. Hopkins, “Mirror and prism systems,” Chapter 7 in Applied Optics andOptical Engineering, Vol. 3, R. Kingslake, Ed., Academic Press, New York(1968).

Chapter 2General Prisms and Reflectors

2.1 Equilateral Prism

The equilateral prism is one of the most available and most widely known of theprism types. It is normally used as a dispersing prism, separating white light intoits component visible colors. The prism has three planar surfaces at equal 60-degangles (Fig. 2.1) and is commercially available in optical glass or plastic, withspecified surface and angular accuracies.

A multispectral light ray entering surface 1 at angle of incidence I1 is refracted atangle I1

′, incident on surface 2 at angle I2, and refracted at angle I2′. The refraction

angles vary for each wavelength λ in the light ray according to Snell’s law, and theexiting beam is dispersed into the familiar visible spectrum. The resulting deviationangle δ(λ) is the change in direction of the exit beam from the entrance beam,where

I1′ = arcsin

(sin I1

nλ

), (2.1a)

I2 = 60 deg − I1′, (2.1b)

I2′ = arcsin(nλ sin I2), (2.1c)

δ(λ) = I1 + I2′ − 60 deg. (2.1d)

There is an allowable range of incident angles for which the rays refracted atsurface 1 hit surface 2 and are refracted into a dispersed beam. In particular, if

Figure 2.1 Equilateral dispersing prism.

33

34 Chapter 2

I2 exceeds the critical angle at surface 2, then the ray undergoes total internalreflection (TIR). For BK7 glass with nd = 1.5168 (λ = 587.3 nm), the criticalvalue I2crit = arcsin(1/nd) = 41.25 deg. The resultant minimum allowable valueof I1 = 29.19 deg is calculated using Eqs. (2.1a) to (2.1c). Rays that undergoTIR at surface 2 will exit through surface 3 as a nondispersed ray. The maximumallowable value of I1 is 90 deg, or close to grazing. Also, some of the internallyrefracted rays might directly hit surface 3 and undergo TIR. These rays arerefracted by surface 2 and are not dispersed. These specific cases are well describedby Southall.1

If we consider only rays that are directly refracted by surface 2, then we find thatthe deviation angle δ acquires a minimum value for a particular angle of incidence.This occurs when the rays pass through the prism symmetrically—that is, I1

′ = I2,or I1 = I2

′. For this case, all internal rays are parallel to surface 3 of the prism andtherefore pass directly to surface 2, and I1

′ = I2 = 30 deg. For BK7 crown glass atλ = 587.3 nm, then I1 = 49.32 deg and δmin = 38.65 deg. When the prism producesminimum deviation, then the Fresnel surface-reflection losses for unpolarized lightare minimized, and the prism transmission is maximized (see Fig. 2.2).

Since the equilateral prism is available in crown glass, extradense flint glass, andacrylic plastic, the question arises as to how much the dispersion is increased byusing a dispersive flint glass.

We define the angular dispersion α(δ) in the visible spectrum as the differencein the deviations for the blue F (486.1 nm) and red C (656.3 nm) wavelengths, asin Eq. (2.2):

α(δ) = δmin (486.1 nm) − δmin (656.3 nm). (2.2)

Table 2.1 shows the minimum deviation angles, the angle of incidence atminimum deviation, and the angular dispersion at minimum deviation for threedifferent visible wavelengths using three different glass types. It is seen that theangular dispersion is increased about fourfold by choosing an extradense flint glassover a crown glass.

Figure 2.2 Equilateral dispersing prism at minimum deviation.

General Prisms and Reflectors 35

Table 2.1 Minimum deviation angles for different glass types.

Glass type λ (nm) δmin (deg) I1 (deg) α (deg)

BK7 crown glass, ν = 64.2 486.1 38.43 49.21 0.748587.3 38.65 49.32656.3 39.18 49.59

Acrylic plastic, ν = 57.5 486.1 36.25 48.12 0.741587.3 36.47 48.24656.3 36.99 48.50

SF10 Extradense flint glass, ν = 28.5 486.1 58.73 59.37 2.93587.3 59.57 59.79656.3 61.67 60.83

The equilateral dispersing prism can be used as a component in a prismspectroscope.2 Measurement of the angle of minimum deviation δ(λ)min for specificwavelengths can be used to accurately calculate the refractive index n(λ) of theprism as a function of wavelength by use of the following equation:

n(λ) =

{sin[δ(λ)min + 60 deg]

2

}sin(30 deg)

. (2.3)

For this type of spectroscope, the angles of minimum deviation are alwaysnoninteger, and the observer or detector must rotate at a different rate than theprism rotation.

2.2 Abbe Dispersing Prism

Another type of dispersing prism is the Abbe prism, named for Ernst Karl Abbe.This is one of several types of “Abbe” prisms named after this pioneer in the fieldof optics. This 60/90/30-deg prism is illustrated in Fig. 2.3. A multispectral rayenters surface AB at angle of incidence I1, is refracted at angle I1

′, undergoes TIRat surface BC at angle of incidence I2 and angle of reflection I2

′, and is refractedat surface AC at angle of incidence I3 and angle of refraction I3

′. The resultingdeviation angle δ(λ) is calculated from the following ray-trace equations:

I1′ = arcsin

(sin I1

nλ

), (2.4a)

I2 = 90 deg − I1′ = I2

′, (2.4b)

I3 = I2 − 30 deg, (2.4c)

I3′ = arcsin(nλ sin I3), (2.4d)

δ(λ) = 60 deg + I1 − I3′. (2.4e)

If the ray passes through both refracting surfaces symmetrically, where I1 = I3′,

then δ(λ)const.dev = 60 deg, and the internal ray is parallel to surface 3, where

36 Chapter 2

Figure 2.3 Abbe dispersing prism with 60-deg constant deviation.

δ(λ)const.dev is equal to the angle α between the refracting surfaces. Then, I1′ =

30 deg, and the deviation angle can be held at the integral value 60 deg for allwavelengths by proper rotation of the prism to change the angle of incidence onsurface 1. This type of prism is referred to as a wavelength-dependent 60-degconstant deviation dispersing prism. This is not the minimum deviation angle.Table 2.2 shows some relevant values for a prism constructed of BK7 glass.

Table 2.2 Sample parameters for Abbe constant deviation dispersing prism.

Glass type λ (nm) I1 (deg) I′1 (deg) δ(λ)const.dev

BK7 crown glass, ν = 64.2 656.3 (nC = 1.5143) 49.21 30.0 60.0587.3 (nd = 1.5168) 49.32 30.0 60.0486.1 (nF = 1.5228) 49.59 30.0 60.0

2.3 Pellin–Broca Dispersing Prism

The Pellin–Broca prism, named for Phillippe Pellin and André Broca, is anotherwavelength-dependent constant deviation dispersing prism, where δ(λ)const.dev =

90 deg. As in Fig. 2.4, it is a four-sided prism ABCD, where α = 75 deg, andβ = 90 deg. The CD face is inactive, typically with 60 deg ≤ γ ≤ 90 deg. In thiscase, γ = 60 deg, yielding 135 deg for the fourth angle.

The basic ray-trace equations to calculate the deviation angle are

I1′ = arcsin

(sin I1

nλ

), (2.5a)

I2 = α − I1′ = I2

′, (2.5b)

I3 = I2 − 90 deg + α, (2.5c)

I3′ = arcsin(nλ sin I3), (2.5d)

δλ = 90 deg − I1 + I3′. (2.5e)


Figure 2.4 Pellin–Broca dispersing prism with 90-deg constant deviation.

When I1 = I3′, then δλ = 90 deg, I1

′ = 30 deg, and I2 = I2′ = 45 deg.

If the critical angle at surface BC is less than 45 deg, then TIR occurs, and noreflective coating is required on this face. By varying the angle of incidence I1, thewavelength that produces the constant deviation angle δλ

const.dev can be isolated.Table 2.3 shows the values for three visible wavelengths for a Pellin–Broca prismconstructed of BK7 glass. Using Eq. (2.5), the angular dispersion α(λ) ≈ 0.75 degat δ(λ) = 90 deg. The I1 values are the same as for the similar Abbe prism describedpreviously. Moreover, if the prism is rotated around a point on face BC, two-thirdsthe distance from B to C, then the input and output beams can be kept stationary.3

A variation of this standard Pellin–Broca prism is the Brewster’s-anglePellin–Broca prism (Fig. 2.5). The incident ray I1Brew, enters surface 1 at Brewster’sangle for a specific refractive index of the prism. For example, if nd = 1.5168 (BK7glass, λ = 587.3 nm), then I1Brew = 56.60 deg. For δλ = 90 deg, α = 78.40 deg.This results in very high transmission of p-polarized light at this wavelength.Brewster’s-angle Pellin–Broca prisms are available commercially in BK7, UV-grade fused silica, and crystal quartz for high-power laser pulses. Since we canachieve an exact 90-deg deviation angle only for a specific Brewster’s angleat a specific wavelength, δλ = 90 deg for other wavelengths requires that theincident angle I1 deviate slightly from Brewster’s angle. The angular dispersionα(λ) ≈ 0.98 deg at δλ = 90 deg. Table 2.4 gives typical parameters for a Brewster’s-angle Pellin–Broca dispersing prism.

Table 2.3 Sample parameters for Pellin–Broca dispersing prism.

Glass type λ (nm) I1 (deg) I′1 (deg) δconst.devλ (deg)

BK7 crown glass, ν = 64.2 656.3 (nC = 1.5143) 49.21 30.0 90.0587.3 (nd = 1.5168) 49.32 30.0 90.0486.1 (nF = 1.5228) 49.59 30.0 90.0

38 Chapter 2

Figure 2.5 Brewster’s-angle Pellin–Broca prism.

Table 2.4 Sample parameters for a Brewster’s-angle Pellin–Broca dispersing prism.

Glass type λ (nm) I1 (deg) I′1 (deg) δconst.devλ (deg)

BK7 crown glass, ν = 64.2 656.3 (nC = 1.5143) 56.46 33.40 90.0587.3 (nd = 1.5168) 56.60 (Brewster’s angle) 33.39 90.0486.1 (nF = 1.5228) 56.95 33.40 90.0

2.4 Penta Prism

The Penta prism is a solid prism having five sides with one included angle of 90 degand three included angles of 112.5 deg each, as shown in Fig. 2.6. One surface is notused optically. Planar surfaces AB and CD are usually reflection coated to ensureoperation over a wider range of input angles at surface 1. Although the Penta prismcannot discriminate wavelengths as the Pella–Broca prism can, it has the usefulproperty of producing a constant 90-deg deviation angle δ, independent of wave-length and the angle of incidence at surface 1. The basic ray-trace equations are

I1′ = arcsin

(sin I1

n

), (2.6a)

I2 = I1′ + 22.5 deg, (2.6b)

I3 = 45 deg − I2′, (2.6c)

I3′ = I3, (2.6d)

I4 = 22.5 deg − I3′, (2.6e)

I4′ = arcsin(n sin I4), (2.6f)

δ = 90 deg + I1 − I4′. (2.6g)

Since I1′ = I4, the Penta prism can be used to maintain a 90-deg deviation

angle without the need for precise prism alignment perpendicular to the plane ofincidence. Table 2.5 shows some sample data illustrating this property.


Figure 2.6 Penta prism with 90-deg constant deviation.

Table 2.5 Sample parameters for a Penta prism.

Glass type λ (nm) I1 (deg) I′1 = I4 (deg) δconst.devλ (deg)

BK7 crown glass, ν = 64.2 656.3 (nC = 1.5143) 0 0 90.0+10 +6.585 90.0−10 −6.585 90.0±20 ±13.05 90.0

587.3 (nd = 1.5168) 0 0 90.0+10 +6.574 90.0−10 −6.574 90.0±20 ±13.03 90.0

486.1 (nF = 1.5228) 0 0 90.0+10 +6.548 90.0−10 −6.548 90.0±20 ±12.98 90.0

2.5 Right-Angle Prism

The right-angle prism in Fig. 2.7 consists of three surfaces, two refracting andone reflecting, with included angles 45/90/45 deg. If the incident light is close tonormal to a side face, then reflection from the hypotenuse face can occur by TIR.For a wider angular range of incident angles, the hypotenuse should be reflectioncoated. The right-angle prism is nondispersive, and the deviation angle δ varieswith angle of incidence ±I1, being simply defined as

δ = 90 deg − 2I1. (2.7)

The right-angle prism can therefore be used as a variable achromatic beam deviatorby rotation of the prism. When used in an imaging application, the direct-viewed

40 Chapter 2

Figure 2.7 Right-angle reflecting prism.

image of a distance object is reverted. If the reflecting hypotenuse face is convertedto a 90-deg roof, then the prism is called an Amici roof prism (for Giovanni Amici).Reflection is then accomplished by TIR, and the direct-viewed image is readable(see Fig. 2.8).

2.6 Porro Prism

The Porro prism, named for Ignazio Porro, is a special type of right-angle prismwhere the incident light enters the hypotenuse face at near-normal incidence.The light is retroreflected from the right-angled faces by TIR and exits at thehypotenuse face. A Porro prism system can be constructed from two identical

Figure 2.8 Amici roof prism.


air-spaced right-angle prisms, oriented orthogonally, as shown in Fig. 2.9(a). Foran entrance aperture of dimensions A × A, the direct-viewed image is displaced inboth the x and y directions by a distance A, being readable and rotated 180 deg.Figure 2.9(b) illustrates a double Porro prism with shaped corners and bevelededges to reduce the weight and increase compactness. Shaping and beveling, alongwith tolerancing, is an integral part of prism design and fabrication. This shapedand beveled form of the Porro prism is used in binoculars to erect the image formedby the objective, where it is called a Type I Porro.

Another type of double Porro prism system is shown in Fig. 2.10, which is amodification of the Type I system by Abbe. Here, it is shown as a constructionfrom three right-angle prisms. It is often called a Porro–Abbe prism, or simplya Type II Porro. In this prism system, the direct-viewed image is displaced by adistance A in the y direction only. The Type II Porro can be made slightly morecompact than the Type I Porro.

Figure 2.9 (a) Double Type I Porro prism system, original design. (b) Shaped and beveledType I Porro prism system.

42 Chapter 2

Figure 2.10 Double Type II Porro prism system, Abbe modification.

Figure 2.11 Brewster’s-angle Porro prism.4 Adapted with permission from the OpticalSociety of America.

A single-element Brewster’s angle Porro prism can be constructed.4 The prismcan be considered to be a lower-half Porro prism (ADE), and an upper-halfBrewster’s angle wedge (ABCD), as in Fig. 2.11. The corner angles are α =78.4 deg, β = 101.6 deg, and γ = ε = 90 deg for nd = 1.5168 (BK7 glass).The prism is identical in construction to the Brewster’s-angle Pellin–Broca prismof Fig. 2.5 but is used differently. Light enters face AB at Brewster’s angle, whereIBrew = 56.60 deg, and is retroreflected at surfaces AE and DE by TIR. It then exitssurface AB at Brewster’s angle. This prism can be used as a 1D retroreflector withvery low loss for p-polarized light, less than 10−5 loss between 0.4 and 1.1 µm.

2.7 Dove Prism

A Dove prism, named for Heinrich Wilhelm Dove, sometimes called aHarting–Dove prism, is useful as an image rotator (Fig. 2.12). As shown, a readable


Figure 2.12 Dove prism design parameters.

erect object is viewed as inverted. Since it is often used in an optical system to erectan upside-down image, the Dove prism is sometimes called a derotator. When usedin collimated light (the only recommended mode), there is no dispersion, as can beeasily seen from a tunnel diagram. Incident light rays at the top of the input faceare directed to the bottom of the exit face, and vice versa, with TIR occurring atthe base face. When the Dove prism is rotated around the optical axis by an angleω, the viewed image is rotated by an angle 2ω.

The design parameters are as follows:

ϕ = arcsin(A

B

), (2.8a)

I = 90 deg − ϕ, (2.8b)

I′ = arcsin(sin I

n

), (2.8c)

θ = 90 deg − ϕ − I′, (2.8d)

L = A(

1tan θ

+1

tanϕ

), (2.8e)

where A= aperture height, B= length of input face, ϕ= base angle of input face, I=angle of incidence at input face, I′= angle of refraction at input face, n= refractiveindex of the prism, θ= angle of internal rays with base face, and L= length of theprism. Table 2.6 shows sample parameters for A = 1.0, varying B, with refractiveindices nd = 1.5168 (BK7) and nd = 1.4586 (UV fused quartz), with an aspectratio γ = L/A.

The Dove prism is not a very compact device, since the γ value is usuallybetween 4 and 6. Since the upper and lower incident rays have a longer internaloptical path than a central incident ray, the Dove prism can be used as a phaseshifter between adjacent input rays. Thus, the Dove prism can be used as a delayline between separated narrow beams.

A roof Dove prism substitutes a 90-deg roof for the base face of the conventionalDove prism. The reflection matrix R for the roof Dove prism is

R =

1 0 00 −1 01 0 −1

. (2.9)

44 Chapter 2

Table 2.6 Sample design parameters for a Dove prism, where A = 1.0.

nd B Φ (deg) I (deg) Θ (deg) L = γ

1.5164 1.20 56.44 33.56 12.18 5.2951.30 50.28 39.72 14.80 4.6151.4142 45.0 45.0 17.21 4.2281.50 41.81 48.19 18.76 4.063

1.4586 1.20 56.44 33.56 11.29 5.6741.30 50.28 39.72 13.73 4.9221.4142 45.0 45.0 16.0 4.4871.50 41.81 48.19 17.46 4.298

The determinant of R is +1, and the viewed image is readable, but still rotated180 deg.

A compound version of the single Dove prism is the double Dove prism(Fig. 2.13). Two identical single dove prisms are positioned base to base. Thisfunctions similarly to a single Dove prism, but the aperture is doubled, or the L/Aratio is halved. The base of one prism can be reflectorized and the faces cementedtogether, or a small air gap can be mechanically held for TIR. Each half of theprism produces an inverted image, but if the full aperture is used as a direct-visionprism, the viewed image is not continuous from top to bottom. Rotating doubleDove prisms are often used in scanning systems. Figures 2.14(a) to 2.14(c) showperspective views of the Dove prism, the roof Dove prism, and the double Doveprism.

2.8 Brewster Laser-Dispersing Prism

Figure 2.15 illustrates a laser-dispersing prism where the entrance and exit anglesare at a nominal design Brewster’s angle. For a prism material of UV-grade fusedsilica (nd = 1.4586), the Brewster’s angle IBrew = 56.57 deg. This type of prismis often used for p-polarized visible lasers where very low surface reflection lossis desired. The prism apex angle α is chosen such that the refracted internal rayis parallel to the prism base at the Brewster’s angle, where α is twice the value of

Figure 2.13 Double Dove prism design parameters.


Figure 2.14 (a) Dove prism producing an inverted image. (b) Roof Dove prism producing areadable image rotated 180 deg. (c) Double Dove prism producing an inverted discontinuousimage.

Figure 2.15 Brewster laser-dispersing prism.

46 Chapter 2

each refracted internal angle Iint. Since IBrew = 90 deg + Iint, the prism is isosceleswith base angle β = γ ≈ IBrew.

2.9 Littrow Prism

The basic Littrow prism, named for Joseph Johann Littrow, is a dispersing30/60/90-deg uncoated prism (Fig. 2.16). An incident ray I1 is refracted at the ABface, undergoes TIR at the BC face, and is refracted outward at the AC face, pro-ducing a wavelength-dependent deviation δ(λ). The basic ray-trace equations are

I1′ = arcsin

(sin I

n

), (2.10a)

I2 = 60 deg − I1′ = I2

′, (2.10b)

I3 = I2′ − 30 deg, (2.10c)

I3′ = arcsin(n sin I3), (2.10d)

δ(λ) = 90 deg − I1 + I3′, (2.10e)

where n is the refractive index at wavelength λ. For example, a BK7 prism withI1 = 20 deg would yield δ(486.1 nm) = 96.471 deg, δ(587.3 nm) = 96.274 deg,and δ(656.3 nm) = 96.192 deg.

If a high-reflectance coating is applied to the AC face, the prism functions asa wavelength-independent (nondispersing) prism with a constant deviation angleδλ

const.dev = 60 deg. This prism is often called a 60-deg Bauernfeind prism [seeFigs. 2.17(a) and 2.17(b)]. The basic ray-trace equations are

I1′ = arcsin

(sin I1

n

), (2.11a)

I2 = 60 deg − I1′ = I2

′, (2.11b)

Figure 2.16 30/60/90-deg Littrow dispersing prism.


Figure 2.17 (a) 30/60/90-deg Littrow 60-deg deviation reflecting prism. (b) Perspectiveview of 30/60/90-deg Littrow 60-deg deviation reflecting prism.

I3 = I2′ − 30 deg = I3

′, (2.11c)

I4 = 30 deg − I3′, (2.11d)

I4′ = arcsin(n sin I4), (2.11e)

δλconst.dev = 60 deg + I4

′ − I1. (2.11f)

Table 2.7 shows sample parameters for a Littrow 30/60/90-deg reflecting prism.For a direct-view system, the image is erect, readable, and deviated 60 deg.

If the prism of Fig. 2.16 is split through the apex angle, and the face ACopposite the hypotenuse AB is coated with a very high-reflectance (>99%)multilayer dielectric coating, we obtain a Littrow laser-dispersion prism, as shownin Fig. 2.18. If a ray is incident on face AB at the Brewster’s angle, and the vertexangle α = 90 deg − I1Brew, then the ray is retroreflected. A p-polarized laser beamwill be transmitted with very low loss. For example, using ultraviolet-grade fusedsilica (UVFS) glass (nd = 1.4586 at λ = 587.3 nm), I1Brew = 55.567 deg, andα = 34.433 deg = I1Brew. For a multispectral incident laser beam, the reflected rays

48 Chapter 2

Table 2.7 Sample parameters for a Littrow 30/60/90-deg reflecting prism.

Glass type λ (nm) I1 (deg) I′4 = I1 (deg) δconst.devλ (deg)

BK7 crown glass, ν = 64.2 656.3 (nC = 1.5143) 0 0 60.0

+20 +20 60.0

−20 −20 60.0

587.3 (nd = 1.5168) 0 0 60.0

+20 +20 60.0

−20 −20 60.0

486.1 (nF = 1.5228) 0 0 60.0

+20 +20 60.0

−20 −20 60.0

Figure 2.18 Brewster–Littrow laser-dispersing prism.

will be dispersed. The basic ray-trace equations are

I1 = I1Brew, (2.12a)

I1′ = arcsin

(sin I1

n

), (2.12b)

I2 = I1′ − I1 = I2

′, (2.12c)

I3 = α − I2′, (2.12d)


δ(λ) = I1 − I3′, (2.12f)

α(δ) = δ(656.3 nm) − δ(486.1 nm). (2.12g)

For this prism, the calculated angular dispersion α(δ) ≈ 0.77 deg. This Littrowdispersing prism can be used as a component in a tunable laser. By rotation of theprism, the retroreflected wavelength can be selected, especially for gas lasers thatoperate at discrete wavelengths.5


2.10 Schmidt Prism

The Schmidt prism, named for Bernhardt Woldemar Schmidt, is a 22.5/67.5/67.5-deg reflecting roof prism as shown in Figs. 2.19(a) and 2.19(b). A ray incidentnormal to the entrance face exits the exit face at a 45-deg deviation angle. Thedeviation angle δ and the readability and rotation of a direct-view image can bepredicted from the reflection matrix, assuming that there are no refractions.

For the coordinate system shown, the direction normals for the surfaces are:

kx1 = −cos (67.5 deg), ky1 = 0, kz1 = cos (22.5 deg), (2.13a)

kx2 = cos (67.5 deg), ky2 = 0, kz2 = cos (22.5 deg), (2.13b)

kx3 = −cos (45 deg), ky3 = −cos (45 deg), kz3 = 0, (2.13c)

kx4 = −cos (45 deg), ky4 = cos (45 deg), kz4 = 0, (2.13d)

kx5 = cos (67.5 deg), ky5 = 0, kz5 = −cos (22.5 deg), (2.13e)

kx6 = cos (67.5 deg), ky6 = 0, kz6 = cos (22.5 deg). (2.13f)

Considering no refraction at surface 1 or 2, Eq. (1.55) gives

R2 =

0.7071 0 −0.70710 1 0

−0.7071 0 −0.7071

, (2.14a)

R3 =

0 −1 0−1 0 00 0 1

, (2.14b)

R4 =

0 1 01 0 00 0 1

, (2.14c)

R5 =

0.7071 0 0.70710 1 0

0.7071 0 −0.7071

. (2.14d)

The prism reflection matrix R is then

R = (R5)(R4)(R3)(R2) =

−1 0 00 −1 00 0 1

. (2.15)

The direction cosines of the incident ray are Kx1 = −cos (7.5 deg), Ky1 = 0 deg,and Kz1 = cos(22.5 deg). Then, the direction cosines of the exit ray Kx6, Ky6, Kz6are Kx6

Ky6Kz6

= R

Kx1Ky1Kz1

. (2.16)

50 Chapter 2

Then, Kx6 = 0.3827 = cos(67.5 deg), Ky6 = 0, Kz6 = 0.9239 = cos(22.5 deg),and the beam deviation δ = 2 × 22.5 deg = 45 deg. The determinant |R| = +1,indicating that the image is readable. For this symmetric matrix, the transposeRT = R, and the rotated coordinates (x′, y′, z′) of the exit beam relative to (x, y, z)are x′

y′

z′

= RT

xyz

. (2.17)

Then, x′ = −x, y′ = −y, z′ = z, and the direct-view image is rotated 180 deg. TheSchmidt prism finds use in eyepieces and viewing systems, providing a 45-degdeviated and readable erect image of an upside-down object. Since a light beam issplit at the roof, the 90-deg vertex angle is held to an accuracy of 1 to 5 arcsec, ordown to 0.25 arcsec for very critical applications.

The question arises whether the roof surfaces can operate by TIR or whether areflective coating should be applied. Schmidt prisms are offered commercially withand without this reflective coating. Referring to Fig. 2.19(c), the direction cosinesof the ray reflected from roof surface 3 to roof surface 4 are calculated from

Kx34Ky34Kz34

= R3

Kx23Ky23Kz23

. (2.18)

Since Kx23 = −cos (22.5 deg), Ky23 = 0, and Kz23 = −cos (67.5 deg), thenKx34 = 0, Ky34 = cos(22.5 deg), and Kz34 = −cos (67.5 deg). Then, usingEq. (1.53), the angle of incidence at both surfaces 3 and 4 is 49.21 deg. For aSchmidt prism of BK7 glass, the critical angle I2crit is 41.25 deg at λ = 587.3 nm.For collimated light incident perpendicular to the entrance surface, uncoated roofsurfaces could function by TIR.

Figure 2.19(d) shows a convergent light beam incident on the Schmidt prism,where the focus is outside the prism. The minimum f /# of this beam is determinedby any TIR failure at surface 2. In the tangential (x-z) plane shown, the maximumI1 value is given by

I1max arcsin

[n sin(45 deg − I2crit)

], (2.19)

where I1max is the angle of incidence of the critical lower ray shown. This would

be the same as for a TIR right-angle prism, and its value is I1max = 5.7 deg for

BK7 glass (nd = 1.5168). By the use of SF10 glass (nd = 1.7283), I1max can be

dramatically increased to 16.8 deg, yielding an f /1.7 beam.Using the x′, y′, z′ coordinate system (z′ axis perpendicular to entrance

surface 1), the direction cosines of the critical lower tangential ray A1 are given


Figure 2.19 (a) Schmidt prism, side view layout. (b) Schmidt prism, end view layout. (c)Schmidt prism, perspective view. (d) Schmidt prism with convergent f /5 incident beam(I1 = 5.7 deg for n = 1.5168).

52 Chapter 2

simply by

Kx01′ = −cos (90 deg − I1

max) = −0.09932, (2.20a)

Ky01′ = 0, (2.20b)

Kz01′ = cos(I1

max) = 0.9951. (2.20c)

For the upper tangential ray A2:

Kx01′ = cos(90 deg − I1

max) = 0.09932, (2.21a)

Ky01′ = 0, (2.21b)

Kz01′ = cos(I1

max) = 0.9951. (2.21c)

For the sagittal ray B1:

Kx01′ = 0, (2.22a)

Ky01′ = −cos (90 deg − I1

max) = −0.2890, (2.22b)

Kz01′ = cos(I1

max) = 0.9951, (2.22c)

For the sagittal ray B2:

Kx01′ = 0 (2.23a)

Ky01′ = cos(90 deg − I1

max) = 0.2890, (2.23b)

Kz01′ = cos(I1

max) = 0.9951. (2.23c)

By referencing the tangential and sagittal rays to the original (x, y, z) coordinatesystem, we can use the prism geometry defined by Eqs. (2.11a) through (2.11f).The recalculated direction cosines for the tangential ray A1 are

Kx01 = −cos (90 deg − 22.5 deg + I1max) = −0.2890, (2.24a)

Ky01 = 0, (2.24b)

Kz01 = cos(22.5 deg − I1max) = 0.9573. (2.24c)

The recalculated direction cosines for the tangential ray A2 are

Kx01′ = −cos (90 deg − 22.5 deg − I1

max) = −0.4726, (2.25a)

Ky01′ = 0, (2.25b)

Kz01′ = −cos (22.5 deg + I1

max) = 0.8813. (2.25c)

The direction angles (αx, βy, γz) of the sagittal ray B1 are αx = 67.62 deg, βy =

84.3 deg, and γz = 23.18 deg. The resultant direction cosines are Kx01 = −0.3808,Ky01 = −0.09932, and Kz01 = 0.9193. The direction cosines for the sagittal ray B2are correspondingly Kx01 = −0.3808, Ky01 = 0.09932, and Kz01 = 0.9193. Using


the refraction matrix R from Eq. (1.75), and the reflection matrices in Eqs. (2.14a)to (2.14d), the calculated internal angles of incidence for the boundary rays aresummarized in Table 2.8.

Using Eq. (1.23), the minimum f /# of this entrance beam in air is ≈ f /5 forhighest efficiency without encountering light leakage at any surface.

2.11 Pechan Prism

The Pechan prism is a two-element prism separated by an air space, with fivereflections. The basic structure is shown in Figs. 2.20(a) and (b). The lower inputhalf is a bisected Penta prism used as a 45-deg deviator (sometimes called a 45-deg Bauernfeind prism), and the upper output half is similar to a Schmidt prismwith the roof section replaced by a planar-coated reflector surface. The air spaceis mechanically held at ≈0.05 mm. The output rays are nearly coaxial with theinput rays, but with a considerable increase in optical path length. Neglecting anyrefraction, the prism reflection matrix R is given by

R = (R5)(R4)(R3)(R2)(R1) =

−1 0 00 1 00 0 1

, (2.26)

where R1, R2, R3, R4, and R5 are calculated using Eq. (1.55). Here Kx′ = −Kx,

Ky′ = Ky, and Kz

′ = Kz. A ray entering surface 1 normally (Kx = 0,Ky = 0,Kz = 1)will exit surface 3 with no angular deviation. Since |R| = −1, the image isunreadable, and using Eq. (2.17), x′ = −x, y′ = y, and z′ = z. Therefore, a direct-viewed object would be undeviated, unreadable, and rotated 180 deg. A Pechanprism could be used as a compact image rotator like the Dove prism but can beused in convergent or divergent light beams. The same f /# limitations apply as forthe Schmidt prism.

The Pechan prism has four transmitting surfaces and four reflecting surfaces.Two of the surfaces perform double duty, functioning in both TIR and transmissionmodes. To achieve high throughput, the coated reflector surfaces should have a veryhigh reflectance coating, such as the 64-layer dielectric OASIS coating producedby Optricon, with a reflectance exceeding 99%. In addition, the external entranceand exit surfaces should be antireflection coated. For the internal surfaces, addingan antireflection coating may affect the TIR property, resulting in some lightleakage due to frustrated total internal reflection (FTIR).6,7

Table 2.8 Schmidt prism internal incident angles for f /5 convergent entrance beam(n = 1.5168). Units are in degrees.

Plane Imax1 I2 I3 I4 I5 I6

Tangential A1 +5.70 41.25 50.64 50.64 41.25 3.75Tangential A2 −5.70 48.76 47.96 47.96 48.76 3.75Sagittal B1 +5.70 45.12 45.72 52.73 45.12 3.75Sagittal B2 −5.70 45.12 52.73 45.72 45.12 3.75

54 Chapter 2

Figure 2.20 (a) Pechan prism design layout. (b) Pechan prism, perspective view.

2.12 Schmidt–Pechan Prism

The Schmidt–Pechan prism is a compound prism that has similar properties to theroof Dove prism but is more compact and can be used in convergent or divergentlight beams. It is sometimes called a Pechan roof prism, or simply a roof prism inthe binocular trade. It consists of a Schmidt prism air spaced from the same sectionof the Penta prism as used in the Pechan prism, yielding six reflective surfaces[see Figs. 2.21(a) and (b)]. Neglecting refraction at surfaces 1 and 7, the reflectionmatrix R is:

R = (R6) . . . (R1) =

−1 0 00 −1 00 0 1

. (2.27)

The determinant |R| = +1, indicating that the image is readable. UsingEq. (2.17), the rotated coordinates (x′, y′, z′) of the exit beam, relative to (x, y, z) of


Figure 2.21 (a) Schmidt–Pechan prism design layout. (b) Schmidt–Pechan prism,perspective view.

the entrance beam, are x′ = −x, y′ = −y, z′ = z, and the direct-view image is rotated180 deg. In addition to antireflection coatings, a special phase-correction coating isoften applied to one or both of the roof surfaces. This phase-correction coatingwas first introduced in Zeiss roof binoculars to compensate for a polarizationphase shift from reflections at the roof surfaces. Contrast and resolution of theviewed image are claimed to be improved. A binocular viewing instrument witha specified 9-layer phase-correction reflective coating on the roof surfaces ofa Schmidt–Pechan prism is described by Ito and Noguchi.8 Cojocaru has alsodescribed phase-retarding thin films for totally reflecting prisms.9 When usedin quality binoculars, Schmidt–Pechan prisms are often constructed of BaK4glass (nd = 1.5688, ν = 55.98) to provide a larger acceptance angle without TIRleakage.

56 Chapter 2

2.13 Cube-Corner Retroreflector

The simplest form of a cube-corner reflector consists of three first-surface mirrors,each aligned at right angles to the others. The reflection matrix for the hollow cube-corner reflector of Fig. 2.22 is obtained from the reflective surface vectors kx, ky,kz, and

R1 =

−1 0 00 1 00 0 1

, R2 =

1 0 00 −1 00 0 −1

, R3 =

1 0 00 1 00 0 −1

. (2.28)

Then,

Rcc = (R3)(R2)(R1) =

−1 0 00 −1 00 0 −1

. (2.29)

If used as an imaging system, the determinant |Rcc| = −1, indicating anunreadable image. Also, the image is inverted. Since Kx

′ = −Kx, Ky′ = −Ky,

and Kz′ = −Kz

′, the cube-corner functions as a constant deviation reflectoror retroreflector. Retroreflection occurs only when an incident ray hits all threemirrors, and the cube-corner needs to be oriented correctly with respect to areference axis. There are several methods of achieving this, and one method is todefine the reference or optical axis as the line that trisects the cube base solid angle.The direction angles of the reference axis with respect to the original coordinatesystem are α = β = γ = 54.74 deg.

The aperture geometry of a cube-corner reflector is usually hexagonal ortriangular, as shown in Figs. 2.23(a) and 2.23(b). For the hexagonal case, threefull sides of the cube are exposed to incident light, and the leading corners define aplane of incidence. For rays normal to this plane of incidence, every incident ray isretroreflected, or the hexagonal aperture is the effective aperture, and the geometricefficiency is 100%.

Figure 2.22 Cube-corner reflector geometry.


Figure 2.23 (a) Cube-corner reflector with hexagonal aperture. (b) Cube-corner reflectorwith triangular aperture.

For a triangular aperture cube-corner, each side of the cube reflector is truncatedto form three 45-deg right triangles. Then, for rays entering the geometric aperture,some incident rays will not be retroreflected, since part of the full cube is missing.Eckhardt has defined an effective aperture that varies with the incident angle ofthe light.10 For the triangle cube-corner, there are two separated triangular planesdefined by the forward single-side corners of the original cube, and the backdouble-side corners of the truncated cube (Fig. 2.24). The effective aperture isthe hexagonal-shaped overlap region of these triangles, which ensures that allrays entering this aperture will be retroreflected. Consider incident light along thereference axis of the cube, the equilateral triangles having sides of length a, insidethe full cube hexagon having sides of length b, and the effective aperture hexagonhaving sides of length c. Then,

Area (triangle) =

√34

a2, (2.30a)

Area (cube hexagon) =(32

)b2 cot(30 deg), (2.30b)

Area (overlap hexagon) =(32

)c2 cot(30 deg), (2.30c)

where a2 = 2, b2 = [1 − cos(120 deg)], and c = a/3.Let a ≡ 1.0. Then, the area (triangle) = 1.299, and the effective aperture area

(overlap hexagon) = 0.866. The retroreflection efficiency of the triangular cube-corner for normal incidence light is

Efficiency =Area (triangle)

Area (overlap hexagon)= 0.666. (2.31)

58 Chapter 2

Figure 2.24 Effective aperture of a triangular cube-corner reflector.

Thus, the efficiency of the triangular aperture cube-corner is ≈67% that ofthe hexagonal-aperture cube-corner for normal incidence of light. The effectiveaperture for off-axis rays can be calculated by projection of the overlap region ontoa plane normal to the reference axis. The dropoff in efficiency for other angles ofincidence is plotted in Fig. 2.25.

The hollow retroreflector can be considered as a solid-glass prism with arefractive index n ≡ 1.0. The solid-glass cube-corner reflector prism is the type

Figure 2.25 Retroflection efficiency for hollow cube-corner reflectors.10 Adapted withpermission from the Optical Society of America.


that is usually available from commercial manufacturers. The entrance aperture isusually circular, since the cube-corner is formed at the end of a short glass cylinder,as shown in Fig. 2.26. These cube-corner prisms are commercially available witha typical aperture diameter d in the 15–50-mm range, and a total length l in the10–40 mm range, using BK7 glass. The cube-corners can reflect by TIR, or areflector coating can be applied to increase the acceptance angle and minimizepolarization effects. The corner angles are usually held to an accuracy of ±1arcsec. Figure 2.27 shows the variation in retroreflective efficiency for hexagonal,triangular, and circular apertures of solid cube-corner prisms for various angles ofincidence, having a refractive index n = 1.5.

Figure 2.26 Solid glass cube-corner reflector with circular aperture.

Figure 2.27 Retroreflection efficiency for solid glass cube-corner reflectors (n = 1.50).10

Adapted with permission from the Optical Society of America.

60 Chapter 2

References

1. J. P. C. Southall, Mirrors, Prisms and Lenses, 3rd ed., 113–132, Macmillan,New York (1946).

2. R. Kingslake, “Dispersing prisms,” Chapter 1 in Applied Optics and OpticalEngineering, R. Kingslake, Ed., Vol. 5, 1–15, Academic Press, New York(1969).

3. W. M. McClain, “How to mount a Pellin–Broca prism for laser work,” Appl.Opt. 12(1), 153 (1973).

4. H. Moosmüller, “Brewster’s angle porro prism: a different use for aPellin–Broca prism,” Appl. Opt. 37(34), 8140–8142 (1998).

5. “Tunable operation,” in Introduction to Laser Technology, Section 10, MellesGriot 2009 Technical Guide, p. 10.15 (2009).

6. H. Osterberg, “Coating of optical surfaces,” Section 21 in Military Standard-ization Handbook—Optical Design, MIL-HDBK-141, 27–29, Defense SupplyAgency, Washington, DC (1962).

7. L. Li, “The design of optical thin film coatings with total and frustrated totalinternal reflection,” Optics and Photonics News 14, 24–30 (2003).

8. T. Ito and M. Noguchi, “Viewing optical instrument having roof prism and aroof prism,” U.S. Patent No. 6,304,395 (2001).

9. E. Cojocaru, “Simple relations for thin-film coated, phase retarding totallyreflecting prisms,” Appl. Opt. 33(14), 2878–2681 (1994).

10. H. D. Eckhardt, “Simple model of corner reflector phenomena,” Appl. Opt.10(7), 1559–1566 (1971).

Chapter 3Polarization Properties of Prismsand Reflectors

3.1 Prisms Producing Polarized Light

3.1.1 Uniaxial double-refracting crystals

Certain types of crystals, such as calcite (Iceland spar or calcium carbonate) exhibitthe property of double refraction or birefringence, as first observed in calcite byErasmus Bartholinus in 1669. For the class of crystals called uniaxial, there is onlyone direction where all light rays travel along the same path at a constant velocity.This direction defines the optic axis or principal axis, and any plane that containsthe optic axis is called a principal plane (sometimes called a principal section).The optic axis is not a specific line, but indicates a direction in the crystal wherethere is no double refraction. For all rays not traveling along the optic axis, thevelocity is determined by a pair of refractive indices called the ordinary refractiveindex no and the extraordinary refractive index ne, and the path of an incident rayis split into two rays, the so-called o-rays and e-rays. Birefringence is specified bythe number (no − ne). Moreover, these o-rays and e-rays are polarized and vibratein mutually perpendicular planes. Only rays traveling parallel to the optic axis willnot be split, and no is therefore assigned to this direction. One way to representthis refractive index variation is by use of the indicatrix.1 Figure 3.1(a) shows apositive uniaxial indicatrix in the shape of an oblate spheroid, where ne > no, andFig. 3.1(b) shows a negative uniaxial indicatrix in the shape of a prolate spheroid,where no > ne. Both have circular symmetry in planes normal to the optic axis, andwhen the indicatrix has a spherical shape, ne = no, and the crystal is isotropic.

3.1.2 Nicol polarizing prism

One of the first prism polarizers to utilize a birefringent crystal was developed byWilliam Nicol in 1828 and is known as the Nicol prism. The Nicol prism shownin Fig. 3.2 is constructed from negative uniaxial calcite, where no = 1.6584 andne = 1.4864 for λ = 589.3 nm. Calcite is a widely used material because ofits clarity, stability, high spectral transmission range (200–5000 nm), and highbirefringence. Two triangular sections are optically coupled at the hypotenuse bya thin coating of optically clear cement such as Canadian balsam (ncement ≈ 1.54),

61

62 Chapter 3

Figure 3.1 (a) Positive uniaxial indicatrix (ne > no). (b) Negative uniaxial indicatrix (no > ne).

Figure 3.2 Nicol prism polarizer made of calcite, no = 1.6584, ne = 1.4864.

with the optic axis direction as shown. An incident unpolarized ray is split at theentrance surface, with both rays becoming linearly polarized. By controlling theincident angle of the rays at the interface, the o-ray can undergo total internalreflection (TIR), where Io

crit = arcsin(ncement/no) ≈ 68 deg. Since ncement > ne,the e-ray is always transmitted and exits the prism as linearly polarized light. Thisseparation of o-rays and e-rays by TIR is a useful technique that is used in othertypes of polarizing prisms. Although the exit ray is parallel to the incident ray, thereis a slight lateral displacement (noncollinear), the angular field is limited, and theinterface cement will suffer damage at high power levels.

Polarization Properties of Prisms and Reflectors 63

3.1.3 Glan–Foucault polarizing prism

One of several Glan-type polarizing prisms is the Glan–Foucault prism, shown inFig. 3.3(a). The calcite prisms are air spaced at the interface, and each optic axis isperpendicular to the plane of reflection. There is no separation of the ray paths inthe first prism section, but the o-ray moves slower in the first section and undergoesretardance with respect to the e-ray. Again, TIR is used to separate the o-ray fromthe e-ray, and s-polarized light is emitted from the exit face. The o-ray is usuallyabsorbed by blackening the side face. The field of view is determined by TIR failureof the o-ray at the glass–air interface 2, or TIR of the e-ray at this interface. Forcalcite, no = 1.6557 and ne = 1.4852 at λ = 630 nm. The corresponding criticalangles are Io

crit = 37.16 deg and Iecrit = 42.32 deg. As shown in Fig. 3.3(b), the

maximum angle of incidence I1o for the o-ray at entrance surface 1 is estimated

by I1o

max = arcsin[no sin(38.5 deg − Iocrit)] ≈ 2.4 deg. The maximum angle of

incidence I1e for the e-ray is estimated by I1

emax = arcsin[no sin(Io

crit−38.5 deg)] ≈6.3 deg. This results in a narrow asymmetric field of view about the central axisin the tangential plane. A nominal angular field is given as 6 deg at λ = 633 nmby a commercial supplier of the Glan–Foucault prism, United Crystals Company.

Figure 3.3 (a) Glan–Foucault prism polarizer made of calcite, Icrit(no) = 37.1 deg, Icrit(ne) =

42.3 deg. (b) Asymmetric field of view of Glan–Foucault prism polarizer.

64 Chapter 3

Other specifications are a damage threshold of 30 W/cm2 continuous wave (CW)or 300 W/cm2 pulsed laser radiation and transmittance of s-polarized light > 60%at λ = 633 nm.

3.1.4 Glan–Thompson polarizing prism

The Glan–Thompson prism shown in Fig. 3.4 uses two cemented calcite prismswith each optic axis perpendicular to the plane of reflection. Using TIR separationat the glass–optical cement interface, s-polarized light is transmitted, while thereflected p-polarized light is absorbed by a blackened side face. The transmission ofs-polarized light is > 90%, and the angular field is approximately doubled to about12 deg compared to the Glan–Foucault prism, but the Glan–Thompson prism canaccept only up to 8 W/cm2 CW or 100 W/cm2 pulsed radiation due to the lowerdamage threshold of the optical cement.

3.1.5 Glan–Taylor polarizing prism

The last of the Glan group to be described here is the Glan–Taylor prism, shown inFig. 3.5(a). Two air-spaced calcite prisms are oriented with both optic axes parallelto the plane of reflection and parallel to the entrance and exit faces. Using TIRseparation, p-polarized light is transmitted, while the reflected s-polarized light iseither absorbed by a blackened side face or emitted through a clear exit window.The transmission of p-polarized light is > 85%, and the angular field is about 6 deg.It can accommodate the highest radiation level of the Glan group—up to 30 W/cm2

CW or 500 W/cm2 pulsed radiation. A modified form of the Glan–Taylor prism,shown in Fig. 3.5(b), can produce orthogonal s-polarized and p-polarized outputbeams. In addition, if the angle of incidence at the interface is close to Brewster’sangle, there will be little reflection of p-polarized light. However, the intensity ofthe s-polarized reflected beam will be much less than the transmitted p-polarizedbeam.

Figure 3.4 Glan–Thompson prism polarizer made of calcite, Icrit(no) = 37.1 deg, Icrit(ne) =

42.3 deg.


Figure 3.5 (a) Glan–Taylor prism polarizer made of calcite. (b) Glan–Taylor prism polarizerhaving orthogonal outputs made of calcite.

3.1.6 Beam-displacing polarizing prism

Figure 3.6(a) shows a cleaved calcite rhomb where the optic axis is inclined in theprincipal section. Each side has corner angles α = 78.08 deg and β = 101.92 deg.The optic axis direction is determined by equally trisecting a β-β-β oblique cornerof the crystal. A ray enters at the edge of the principal section. The undeviateds-polarized o-ray vibrates perpendicular to the principal section, and the deviatedp-polarized e-ray vibrates in the principal section, where both rays lie in theprincipal section. The exiting p-polarized ray is displaced and parallel to the exitings-polarized ray. Typical exit ray separation is nominally 4 mm. This polarizingprism has the advantage that both exit beams are completely polarized and of equalintensity, although obviously the entrance beam diameter must be small. As inFig. 3.6(b), a calcite prism pair can produce a variable beam displacement betweenthe p-polarized and s-polarized rays.2 Here, one wedge prism is slid relative toanother wedge prism.

66 Chapter 3

Figure 3.6 (a) Beam-displacement prism polarizer. (b) Variable beam-displacement prismpolarizer.2

3.1.7 Wollaston polarizing prism

Another type of beam-splitting polarizing prism is the Wollaston prism, (forWilliam Hyde Wollaston), shown in Fig. 3.7. It usually consists of two calcite right-angle prisms optically cemented together at the hypotenuse. The optic axis of eachsection is orthogonal to that of the other section. An unpolarized ray traversing thefirst prism section is not split, but the o-ray is retarded with respect to the e-ray.The o-ray vibrates parallel to the optic axis and the e-ray perpendicular to the opticaxis. Upon entering the second section, the o-ray becomes the e-ray, and vice versa.The o-ray, now slower, is bent toward the interface normal, and the e-ray is bentaway from the interface normal. Prisms with a deviation angle δ from about 5 to45 deg between the exit beams can be obtained, depending on the right-angle prismbase angles. For very high-power applications, the prisms may not be cementedtogether, resulting in a reduction of transmission.


Figure 3.7 Wollaston prism polarizer (calcite).

3.1.8 Nomarski polarizing prism

The Nomarski prism, named for Georges Nomarski, is a modified Wollaston prism(Fig. 3.8). The optic axis of the first right-angle calcite prism is skewed as shown,while the optic axis of the second prism is oriented the same as for the Wollastonprism. This angled optic axis causes the ordinary and extraordinary rays to intersectoutside the prism, forming an interference plane. The resulting phase shifts canbe detected by an analyzer. The exact distance of this interference plane fromthe prism is determined by the angle of the skewed optic axis and is set by themanufacturer. Normarski prisms are used in differential interference contrast (DIC)microscopes.

3.1.9 Rochon polarizing prism

Related to the Wollaston polarizing prism, the Rochon prism (for Alexis MarieRochon) has the optic axis of the first calcite prism section in the direction of theincident ray (Fig. 3.9), and there is no distinction between the o-ray and the e-rayin this section. The split at the second section interface produces no deviation ofthe s-polarized o-ray, while the p-polarized e-ray is deviated from the interfacenormal. The first calcite section can be replaced by a more robust isotropic glasssection, choosing a glass with a refractive index and dispersion close to either of the

Figure 3.8 Nomarski prism polarizer (calcite).

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Figure 3.9 Rochon prism polarizer (calcite).

refractive index values of calcite.3 Glass-calcite Rochon prisms are commerciallyavailable using FK5 glass (nd = 1.4875, νd = 70.41), which is close to ne and thedispersion of calcite. Typical beam-deviation angles are 5, 10, and 15 deg.

Other double-refracting crystals used in commercially available polarizingprisms are crystal quartz (circular and low birefringence), alpha-BBO (ne =

1.6021, no = 1.6776 at λ = 552 nm), YVO4 (ne = 2.2154, no = 1.9929at λ = 630 nm), magnesium fluoride, and titanium dioxide. Polarizing prisms’surfaces are usually antireflection coated.

3.1.10 MacNeille polarizing beamsplitter cube

Another method to produce polarized light is by the deposition of alternating high-and low-index film layers, as described by MacNeille.4 Figure 3.10 illustratesa seven-layer transparent thin-film stack having refractive indices n1 and n2,deposited between transparent bulk material having a refractive index n3, wheren1 � n2, and n1 > n3. All internal rays within the layers hit the next layer atBrewster’s angle. A fraction of the reflected light at each layer interface is thereforecompletely s-polarized. If we choose n1 = 2.3 (zinc sulfide) and n2 = 1.38

Figure 3.10 Polarizing thin-film stack.


(magnesium fluoride), then Brewster’s angles α and β are given by

α = arctan(n2

n1

)= 31.0 deg, (3.1)

β = arctan(n1

n2

)= 59.0 deg, (3.2)

where α + β = 90 deg and Snell’s law is satisfied at each layer interface.A useful incident angle from the bulk material to the first layer is θ = 45 deg.

The required refractive index of the bulk material is then calculated from Snell’slaw:

n3 = n1

[sinα

sin(45 deg)

]≈ 1.67. (3.3)

Since θ between n3 and n2 (53.9 deg) is not Brewster’s angle, this incident ray isnot completely s-polarized on reflection.

To maximize the intensity of the reflected s-polarized ray at each layer, the layerthickness is controlled such that the ray reflected from the next layer is in phasewith the incident ray. To achieve this, the physical thicknesses t1 and t2 of the layersare controlled to be

t1 =λ

4√

(n12 + n2

2)/n12, (3.4)

t2 =λ

4√

(n12 + n2

2)/n22, (3.5)

where λ is the wavelength of the incident light, nominally 550 nm. For these sevenlayers, approximately 50% of the incident light is reflected as s-polarized, whilethe other half is transmitted as p-polarized light.

Figure 3.11 shows a 50R/50T polarizing beamsplitter (PBS) cube, where thedeposited layers lie on the hypotenuse of a right-angle prism, and another right-angle prism is coupled to the hypotenuse using a thin coating of optical cementhaving a refractive index close to n3. From Eq. (3.3), a suitable material for thecube would be SF5 glass (nd = 1.673). Both the reflected s-polarized light and thep-polarized transmitted light are at least 95% polarized over the visible spectrum,and the beamsplitter is usable for 40 deg ≤ θ ≤ 50 deg, or ±5 deg from the idealincident angle at the interface. The extinction ratio is the ratio of the transmitted orreflected primary polarization component to the opposite polarization component.It is possible to increase the angular field of MacNeille PBS cubes by modifying thebeam-splitting coating, albeit with a reduced usable wavelength range.5 Moderncommercial PBS cubes of a modified MacNeille design can achieve an input f /#down to ≈ f /2.5 and a transmission extinction ratio ≈ 1,000:1. The minimumworking f /# of a PBS cube for projection display applications, without noticeableloss of contrast, has been stated to be ≈ f /3.3.6

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Figure 3.11 MacNeille 50R/50T PBS cube in convergent beam of half-angle θ.

3.1.11 Birefringent multilayer reflective polarizing film

A type of reflecting polarizer film has been developed by 3M that uses a multilayerstack of biaxial birefringent polymer layers.7 It is designed to produce highreflectance for light with its plane of polarization parallel to one axis, and hightransmission for light with its plane of polarization parallel to a second axis, bothover a wide range of incident angles. A biaxial birefringent material, where therefractive indices differ along all three axes, can be produced by stretching themultilayer stack in one direction (uniaxial stretching). Figure 3.12 shows a singleinterface between layers for a biaxial birefringent film, and the associated refractiveindices. For light incident in the x-y stretch plane, n10 = n1x, n20 = n2x forp-polarized light, and n10 = n1y, n20 = n2y for s-polarized light. For light incident inthe y-z nonstretched plane, n10 = n1y, n20 = n2y for p-polarized light, and n10 = n1x,n20 = n2x for s-polarized light. The x direction is the extinction direction, and they direction is the transmission direction. Typical values are n1x = 1.88, n1y = 1.64,n1z = variable, n2x = 1.65, n2y = variable, and n2z = variable. For the large indexdifferential of 1.88−1.65 = 0.23 in the stretch direction, there is a high reflectanceof s-polarized light for a stack of hundreds of layers, and the angular transmissionof p-polarized light depends on the n1z/n2z index ratio.

Figure 3.12 Two-layer single interface biaxial birefringent film.


An example of a reflective polarizing film consists of 601 alternating layers ofpolyethylene (PEN) and a copolymer of 2,6 napthalene dicarboxylate, methyl ester,dimethyl isophthalate, and dimethyl terephtalate with ethylene glycol (coPEN),with controlled layer thicknesses. The multilayer film was heated and stretchedin the x direction, producing a film ≈ 0.5 mm thick. The transmission versuswavelength of the polarizing sheet is plotted in Fig. 3.13, where the “a” curvecorresponds to p-polarized light at I = 0 deg, the “b” curve corresponds top-polarized light at I = 60 deg, and the “c” curve corresponds to the s-polarizedlight extinction at I = 0 deg in the stretched x direction.

There is very high transmission of p-polarized light (80–100%) over a wideangular range, and a very high extinction (reflectance) of s-polarized light in thevisible 400–700-nm range. Adding an antireflection coating will further increasethe transmission of p-polarized light.

3.1.12 Polarizing beamsplitter elements using birefringent polarizingfilm

A PBS has been developed using a type of 3M birefringent polarizing film insteadof the MacNeille stack.8 The polarizing film can be encased in a cube split alongthe diagonal [Fig. 3.14(a)], or between thin, tilted glass plates [Fig. 3.14(b)].These types of PBS elements are characterized by fixed polarization x-y axesof the polarizing film that do not vary with the angle of incidence, as for theMacNeille prism. Unlike the MacNeille PBS cube, the polarization properties are

Figure 3.13 Transmission versus wavelength for birefringent reflective polarizing film.7

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Figure 3.14 (a) A birefringent film PBS cube. (b) A birefringent film PBS plate.

also independent of the refractive index of the cube material. Since the polarizationaxes are invariant with respect to the PBS, this type is especially useful inconvergent or divergent light beams, and is known as a wide-angle CartesianPBS. The extinction ratio (ratio of transmitted or reflected primary polarizationcomponent to the opposite polarization component) can exceed 10,000:1 for abirefringent PBS cube, for both transmitted and reflected rays.

For either the cube or plate configuration, it is important to use a nonbirefringentglass, such as PBH56 (n ≈ 1.85), to avoid visible polarization effects. Themultilayer reflective polarizing beamsplitter film typically has 892 layers at about0.15-mm thickness and a refractive index n ≈ 1.55. The preferred orientation forlarge cone angle, high-contrast optical systems is such that p-polarized light istransmitted along the x axis, and s-polarized light is transmitted along the y axis.

3.1.13 Wire-grid polarizing beamsplitter

A wire-grid polarizer is a thin planar element consisting of spaced fine parallelmetallic lines (normally aluminum or gold wires) that are deposited on a cleardielectric substrate, as shown in Fig. 3.15. The line width w is approximately thesame as the line spacing (period) d, and both must be small with respect to thelight wavelength λ. Incident unpolarized light is reflected by the metallic lines ass-polarized light and transmitted through the dielectric substrate as p-polarized


Figure 3.15 Wire-grid polarizer.

light, where d < λ/2. Wire-grid polarizers were first produced for use in themicrowave region, because a wider grid spacing is more easily fabricated. For useas a visible-light PBS, the line spacing must have dimensions ≤ 100 nm. Planarpolarizers were first produced by Moxtek, Inc., as the ProFlux wire-grid PBS.9,10

Visible-light (420–700 nm) wire-grid planar polarizers are also commerciallyavailable from suppliers such as Edmund Optics, with a clear aperture up to44 mm2.

A wire-grid PBS cube has been constructed by encasing a wire-grid polarizerbetween two diagonal halves of a glass cube.11 The wire-grid polarizer substrate isCorning 1737F glass, which is cemented to the BK7 glass cube with Norland 61cement. The wires are oriented perpendicular to the triangular edges of the prismhalves, providing reflection of s-polarized light and transmission of p-polarizedlight. This wire-grid polarizer has an incident angle ≈ 45 deg, which limits theuseful wavelength range to longer than mid-visible.

3.1.14 Polarizing beamsplitter using frustrated total internal reflection

A polarizing prismatic beamsplitter can be designed that uses a thin film withincident angles greater than the critical angle.12 It is based on frustrated totalinternal reflection (FTIR) and thin-film interference effects. The basic PBS isillustrated in Fig. 3.16, where p-polarized light is reflected from the film interface,and s-polarized light is transmitted. The thin-film structure consists of a stack ofhigh- and low-index layers. For the low-index layers, the internal angle of incidencemust be greater than the critical angle, and the reflection phase changes depend onthe refractive index ratios and the angle of incidence. If the thicknesses of the low-index layers are thin, then the evanescent wave can tunnel to the next high-indexlayer, FTIR occurs at the high-index/low-index boundaries, and p-polarized light

74 Chapter 3

Figure 3.16 Thin-film PBS using frustrated total internal reflection.12

is reflected as from a metal reflector. At the low-index/high-index interfaces, thephase is varied such that for many thin layers, the transmitted light is s-polarizedand independent of wavelength and angle of incidence. There is a minimum angleof incidence Imin, where Imin > Icrit, where the PBS will function. The theory ofthis PBS is detailed in Ref. 12. This PBS is in contrast to the conventional type ofPBS, which transmits p-polarized light and reflects s-polarized light.

As a cited example, a 45-layer film is encased in glass prism halves, where n0 =

1.75. For the high-index layer, n1 = 2.35, and for the low-index layer, n2 = 1.38.The layer thicknesses d are varied between 8.3–44.2 µm for the high-index layers,and between 30.5–107.9 µm for the low-index layers. The minimum angle ofincidence at the film is calculated to be 62.50 deg, and the calculated critical angleis 52.05 deg. The usable angular range at the film is 64–77 deg, correspondingto ±11.4 deg at the prism entrance face. The extinction ratios are ≈ 30,000:1 forreflected p-polarized light, and > 50,000:1 for s-polarized transmitted light.

3.1.15 Polarizing beamsplitter prism with common polarization output

It is sometimes desirable to produce a beam having a common polarization modeand output direction. The modified PBS of Fig. 3.17(a) adds two right-angleprisms, a rhomb prism, a half-wave retarder thin plate or birefringent polymerfilm, and a 50R/50T nonpolarizing beamsplitter film.13 Collimated light enteringthe PBS cube is split into transmitted p-polarized light and reflected s-polarizedlight. The s-polarized light is retroreflected and displaced by prism 1. A half-wave retarder film is embedded between prism 2 and prism 3, which convertsthe s-polarized light to p-polarized light. A nonpolarizing beam-splitting film isembedded between prism 1 and right-angle prism 2, and the reflected half isintegrated with the directly transmitted p-polarized light from the PBS cube. Thetransmitted light is reflected by the rhomb prism 3 to exit collinear to the lowerbeam with a common polarization mode. Figure 3.17(b) illustrates an alternative


Figure 3.17 (a) Prism polarization converter with λ/4 retarder.13 (b) Prism polarizationconverter with λ/2 retarder.13

arrangement with a quarter-wave retarder on the entire hypotenuse face of prism 1.In both cases, the output aperture is doubled in the x direction, and the outputsubbeams are not integrated. Devices of this type are often called polarizationconverters.

3.2 Prisms Controlling the Polarization of Light

3.2.1 Fresnel rhomb retarders

It can be generally stated that every prism that has reflecting surfaces will affectthe polarization state of incident light in some manner. Also, most light that isreflected from terrestrial objects viewed by the unaided eye or through a viewingdevice such as binoculars is partially polarized. That is why polarized sunglassesare useful in reducing horizontally reflected glare. When the polarization state of

76 Chapter 3

the input beam can be controlled and specified, certain prism types are usefulfor changing the state of polarization in a predictable manner. A useful way tospecify the polarization of a light beam is the polarization ellipse in Fig. 3.18.Since the most general state of polarization is elliptic, the polarization ellipse canrepresent an arbitrary polarization state. The azimuth angle Ψ is defined as theangle between the major semiaxis a and the x axis. The ellipticity angle X is definedas X = arctan(b/a), where b is the semiminor axis, while the ellipticity is definedas (b/a). Thus Ψ = 0 deg, X = 0 deg would indicate linearly polarized light alongthe x axis, and Ψ = 90 deg, X = 45 deg would indicate circularly polarized light.

A well-known prism type is the Fresnel rhomb, named for Augustin JeanFresnel. Figure 3.19 shows a single Fresnel rhomb. Through two total internalreflections, 45-deg linearly polarized input light is converted to circularly polarizedoutput, or it functions as a 90-deg or quarter-wave retarder. The 45-deg linearlypolarized input, with equal components along the x axis and y axis, allows the TIRphase changes to be calculated from the equations in Sec. 1.6. From Eq. (1.20), a45-deg phase shift occurs at incident angles of 47.87 deg and 55.22 deg for BK7glass (nd = 1.5168) in air. The larger incident angle is normally chosen, as it

Figure 3.18 Polarization ellipse.

Figure 3.19 Fresnel rhomb quarter-wave retarder.


provides less wavelength dependence of the retardance. A conventional BK7 glass(nd = 1.5168) Fresnel rhomb has opposite acute vertex angles α = 55.22 deg,a physical length L, and a beam displacement D. For a square aperture havingsides A ≡ 1.0 unit, L = A tanα[1 + sin(2α − 90 deg)] = 1.94 units, andD = A [1 + sin(2α − 90 deg)] = 1.35 units.

Figure 3.20(a) shows how a half-wave retarder can be constructed from twoidentical quarter-wave rhomb retarders. The two rhombs can be cemented together,brought into optical contact, or air spaced. The output beam remains collinearwith the input beam, but the retarder length is doubled. Both single and doubleFresnel rhombs are available commercially with aperture diameters in the 10–20-mm range, and at optimized wavelengths in fused quartz, FK5, and BK7 glass.For quarter- and half-wave rhomb retarders in BK7 glass (I = 55.22 deg at λ =

587.56 mm), and using Eq. (1.20) at each reflecting surface, the total retardance isobtained. These rhomb retarders are nondispersive prisms. However, they are oftencalled achromatic retarders because the change in retardance with wavelength ismuch less than for crystal quartz plate retarders.14

A quarter-wave double Fresnel rhomb can be designed by joining two eighth-wave Fresnel rhombs. It would be useful for producing circularly polarizedcollinear output from a linearly polarized input. Figure 3.20(b) shows the designparameters for BK7 glass. Each of the four internal angles of incidence I =

74.76 deg, yielding a relative phase shift δ = 22.5 deg at each reflection. IfA = 20 mm, α = I = 74.76 deg, then the total length 2L = 273 mm, with D = 0.This is not a very compact retarder. Table 3.1 gives the retardance variation overthe visible spectrum for these rhomb retarders.

Figure 3.20 (a) Double Fresnel rhomb half-wave retarder. (b) Double Fresnel rhombquarter-wave retarder.

78 Chapter 3

Table 3.1 Retardance variation with wavelength for single and double Fresnel rhombs.

Wavelength λ (nm) λ/4 single Fresnelrhomb (deg)

λ/2 double Fresnelrhomb (deg)

λ/4 double Fresnelrhomb (deg)

90-deg retardance 180-deg retardance 90-deg retardance

656.27 89.8 179.6 89.9587.56 90.0 180.0 90.0546.07 90.2 180.4 90.1486.13 90.6 181.2 90.3435.83 91.0 182.0 90.5404.66 91.2 182.4 90.7

The single Fresnel rhomb can be further achromatized by coating one of thereflecting surfaces with a 20-nm-thick layer of MgF2 and changing the acuterhomb angles to α = 51.5 deg.15 This results in a maximum phase retardation ofexactly 90 deg, which further reduces variation of the retardance with the internalangle of incidence. The resulting improvement in Fig. 3.21 provides about 0.4-degretardance variation over the wavelength range 334.1 nm to 546.1 nm.14

3.2.2 Total-internal-reflecting cube-corner retarders

A prismatic cube-corner reflector formed as a single prism is shown in Fig. 3.22,where the face reflections occur by TIR. The internal reflections introduce phasechanges that alter the azimuth angle and ellipticity of incident polarized light, suchthat incident linearly polarized light is both rotated and converted to ellipticallypolarized light. The polarization changes that occur during reflection for a TIRcube-corner can be shown in a polarization pupil map. For a BK7 glass cube-cornerin air, Fig. 3.23 shows a six-segment pattern of the various polarization states ofthe output beam for a linearly polarized input beam.16

Figure 3.21 Phase retardance versus wavelength for uncoated and coated Fresnelrhombs using BK7 glass. (Adapted from Ref. 14, with permission from the Optical Societyof America.)


Figure 3.22 TIR cube-corner reflector.

Figure 3.23 Polarization pupil map for perfect TIR cube-corner reflector (BK7/airinterface). (Reproduced from Ref. 16, courtesy of ZEMAX.)

Player has analyzed the polarization properties of uncoated TIR cube-cornerreflectors for changes in retardance δ and azimuth angle Ψ.17 For this retroreflector,with input light parallel to the optical axis, the angle of incidence at each of thethree surfaces is I = arcsin

√2/3 = 54.74 deg. Using Eq. (1.20), the relative phase

80 Chapter 3

shift δ for the cube-corner at a single face is then

δ = arctan√

2 − 3/n2, (3.6)

where n is the refractive index of the cube-corner substrate in air.The retardance δcc for three reflections on the cube-corner is

δcc = 2 arccos

cos

(3δ2

)4

+

3 cos(δ

3

)4

, (3.7)

and the azimuth angle Ψ is

Ψ = 0.5 arctan

2√

3 sin(δ

2

)sin

(3δ2

)+ sin

(δ

2

) . (3.8)

Player has noted that if the refractive index n is selectively chosen—e.g., SF14glass—then the cube-corner retardance δ ≈ 90 deg and operates as a quarter-waveretroreflecting retarder.17 Table 3.2 gives the retardance and azimuth angles forvarious cube-corner substrates. For SF14 glass, the variation in the cube-cornerretardance with wavelength is given in Table 3.3. Over the visible spectrum, thereis a low variation of δ and an even lower variation of Ψ.

3.2.3 Phase-coated total-internal-reflecting right-angle prism retarders

A single low-index-glass right-angle prism can function as a 90-deg retarder bya single internal reflection by coating the reflecting face with a single layer ofdielectric film.18 The film layer produces additional retardance to the inherentTIR retardance. Figure 3.24 illustrates the retarder prism with a refractive index

Table 3.2 Variation of retardance and azimuth angle of TIR cube-corner reflector basedon various glass types.

Glass type Single-faceretardance δ (deg)

Cube-cornerretardance δ (deg)

Azimuthangle Ψ (deg)

Ideal (nd = 1.7675) 54.03 90.00 16.9BK7 (nd = 1.5168) 45.29 76.35 16.32SF14 (nd = 1.7618) 53.89 89.79 16.89LASF9 (nd = 1.8503) 55.85 92.78 17.03


Table 3.3 Wavelength variation of retardance and azimuth angle of TIR cube-cornerreflector using SK14 glass.

Wavelength λ (nm) Cube-corner retardance δ (deg) Azimuth angle Ψ (deg)

706.52 89.3 16.87656.27 89.48 16.88587.56 89.79 16.89479.99 90.6 16.93435.83 91.15 16.95404.66 91.68 16.98

Figure 3.24 Right-angle TIR prism quarter-wave retarder. (Adapted from Ref. 18, withpermission from the Optical Society of America.)

nprism ≈ 1.51, and Fig. 3.25 plots the phase shift difference δ as a function ofnormalized thickness d/λ for various film refractive indices at an angle of incidenceα = 45 deg. Retardance of 90 deg can be achieved for dual d/λ values, and for filmindices between 2.2 and 2.4, with small variation of retardance with wavelength.Thus for d/λ ≈ 0.125, the required film thickness at λ = 546 nm would bed ≈ 68 nm. This prism retarder can be further achromatized by additional thindielectric layers. For example, a three-layer coating can produce δ = 90.0 deg,varying < 0.02 deg at α = 45 deg, over a ±15% bandwidth. For these multilayercoatings, both the refractive indices and thicknesses are varied.

Three conjoined right-angle prisms, with coated reflecting faces, can produceuseful 90-deg or 180-deg achromatic retardation with all internal angles ofincidence α = 45 deg.19 There are four internal reflections producing collinearinput and output rays. Figure 3.26(a) illustrates a BK7 glass (n ≈ 1.52) prismwith any one of the four surfaces coated with a thin dielectric film of high-indexZnS (n ≈ 2.39), overcoated by a thicker layer of MgF2 (n ≈ 1.38). A retardanceof 180 deg is produced with a standard deviation of 1.66 deg over the visiblewavelength range of 400–700 nm. Figure 3.26(b) shows a fused quartz (n ≈ 1.46)

82 Chapter 3

Figure 3.25 Right-angle TIR prism retarder—phase retardation versus normalized filmthickness. (Adapted from Ref. 18, with permission from the Optical Society of America.)

prism having all four surfaces coated, producing a retardance of 90 deg over thevisible range, with a standard deviation of 1.03 deg. The coatings also consist of athin film of ZnS, overcoated by a thicker coating of MgF2. Table 3.4 summarizesthe design parameters for these right-angle prism retarders.

3.3 Polarization Preservation in Prisms and Reflectors

3.3.1 Polarization-preserving total-internal-reflecting prism

A useful application of prisms is to deviate or displace a polarized input beamwithout significantly affecting the polarization of the output beam. Such prismsare called polarization-preserving prisms. In the case of total-internal-reflectingsurfaces, which normally produce relative phase shifts for oblique polarized light,these phase shifts are compensated by the prism geometry, input-beam polarizationdirection, and film coatings on the facets.

Table 3.4 Design parameters for 90-deg and 180-deg right-angle prism retarders.19

Prism glass Thickness offirst-layerZnS (nm)

Thickness ofsecond-layerMgF2 (nm)

Average phaseshift (deg)

Standarddeviation ofretardance (deg)

Number ofcoatedsurfaces

BK7 15.63 67.53 179.84 1.66 1Fused quartz 0.17 50.0 89.73 1.03 4


Figure 3.26 (a) Coated right-angle prism 180-deg phase retarder using BK7 glass.(b) Coated right-angle prism 90-deg phase retarder using BK7 glass. (Adapted from Ref. 19,with permission from the Optical Society of America.)

Figure 3.27 illustrates an uncoated prism pair that displaces and deviates theincident light by 180 deg and preserves the polarization.20 The input prismproduces three total internal reflections in the x-z vertical plane, and the outputright-angle prism produces two total internal reflections in the y-z horizontal plane.The three reflections I1 = +75 deg, I2 = −60 deg, and I3 = +75 deg in the firstprism are balanced by the two reflections of I4 = +45 deg and I5 = +45 deg in thesecond prism, where the sign is positive for counterclockwise beam reflection and

84 Chapter 3

Figure 3.27 Polarization-preserving prism pair. (Adapted from Ref. 20, with permissionfrom the Optical Society of America.)

negative for clockwise reflection. Then,

I1 + I2 + I3 = I4 + I5 = 90 deg . (3.9)

The phase shifts are equal in orthogonal planes, and the prism pair will thentransmit any mode of polarized light unchanged.

Figure 3.28 shows a polarization-preserving prism that uses four orthogonalinternal reflections and produces a parallel displaced output.21 As shown, theelectric field vector E of the input beam is oriented at 45 deg to the x axis andhas components Ex and Ey. The beam enters the prism normally and undergoesan orthogonal TIR reflection at each surface. The angle of incidence I at each

Figure 3.28 Polarization-preserving prism with coaxial output. (Adapted from Ref. 21, withpermission from the Optical Society of America.)


TIR surface is 45 deg, where I > Icrit (Icrit ≈ 41.2 deg for BK7 glass). TheEx component undergoes p-polarization TIR at surface 1, s-polarization TIR atsurfaces 2 and 3, and p-polarization TIR at surface 4. The Ey component undergoess-polarization TIR at surface 1, p-polarization TIR at surfaces 2 and 3, ands-polarization TIR at surface 4. The components are rotated as shown, and there isa relative phase shift at each reflection that produces internal elliptical polarizationstates. However, by the use of these four reflecting surfaces, the accumulated phaseshifts for both Ex and Ey are equal, and there is no relative phase shift δ betweenthe input and exit beams. The beam exits the prism parallel to the input beam witha displacement D, and with the same polarization state and orientation as the inputbeam. This polarization-preserving prism is achromatic.

Another method to preserve the polarization of more general types of reflectingprisms is by the coating of thin-film dielectric layers on the TIR surfaces.Polarization is preserved by designing the retardance δ to be close to zero at eachTIR surface, where

δ = ∆Φ⊥ − ∆Φ‖ ≈ 0, (3.10)

∆Φ⊥ and ∆Φ‖ are defined in Eqs. (1.18) and (1.19), and δ is defined fromEq. (1.20):

δ = 2 arctan

cos I√

sin2 I − n0/n1

sin2 I

, (3.11)

where the glass index ng > n0 and I > Icrit. Cojocaru has computed the refractiveindices for one-, two-, and three-layer coatings, using quarter- and half-wave thickcoatings, to achieve polarization preservation on a glass substrate.22 For a single-layer coating, the following are given by

n21 = n0ng, where d1 =

λ0

4, (3.12)

n2g =

2n20

sin2 Ig

− n20, where d1 =

λ0

2, (3.13)

where n1 is the coating index and n0 = 1.0 (air). The optical thickness d1 at designwavelength λ0, angle of incidence Ig, and physical coating thickness t1 is given by

d1 = t1√

n21 − n2

g sin2 Ig. (3.14)

From Eq. (3.14), the retardance for TIR at Ig = 45 deg and ng = 1.61 for anuncoated surface is δ = 51.1 deg. The computed retardance for an ideal quarter-wave coating (n1 = 1.27) on a glass substrate (ng = 1.61) as a function of λ/λ0for Ig = 45 deg is plotted in Fig. 3.29(a). Figure 3.29(b) plots the retardancedependence on Ig, for ng = 1.61, n1 = 1.27, and λ0 = 500 nm.

86 Chapter 3

Figure 3.29 (a) Wavelength dependence of retardance for single λ0/4 layer (n0 = 1, ng =

1.61, n1 = 1.27). (b) Retardance variation versus angle of incidence for single λ0/4 layer(n0 = 1, ng = 1.61, n1 = 1.27). (Adapted from Ref. 22, with permission from the Optical Societyof America.)

In general, the retardance for a single-layer coated polarization-preserving TIRprism depends on four factors:

• The prism glass index ng

• The layer index n1

• The optical thickness-to-wavelength ratio d1/λ

• The angle of incidence Ig.


For example, the dependence of the layer index n1 is illustrated in Fig. 3.30for several values of the prism glass index ng. Wang et al. have analyzed thesefactors theoretically and experimentally using a fused-quartz right-angle prismwith ng = 1.457, Ig = 45 deg, n1 = 1.23, and d1/λ = 0.5.23 In Fig. 3.30, thedependence of the retardance on the layer index n1 is illustrated for several valuesof the prism glass index ng.

3.3.2 Polarization-preserving two-piece reflective axicon

It is also possible to design polarization-preserving first-surface metallic reflectors.A two-piece biconical axicon reflector element is often used in laser applications(Fig. 3.31). It is possible to preserve the polarization of light after two reflectionsby the application of a single dielectric layer to the metallic axicon surfaces.24 Thedielectric material is the same for both reflecting surfaces, but the normalized filmthickness ζ (ratio of physical thickness to film-thickness period) has a value ζ1 onone reflecting surface and ζ2 on the other for a given angle of incidence. To achievepolarization-preservation of the system,(

Rp

Rs

)1

(Rp

Rs

)2

= f (I1, ζ1) f (I2, ζ2) ≡ 1, (3.15)

where Rp and Rs are the complex reflection coefficients for p-polarized ands-polarized light at each surface, and I1 and I2 are the angles of incidence at each

Figure 3.30 Retardance dependence on layer index for several glass prism indices ng

(d1/λ = 0.5, n1 = 1.27, Ig = 45 deg). (Adapted from Ref. 23, with permission from the OpticalSociety of America.)

88 Chapter 3

Figure 3.31 Polarization-preserving two-piece reflecting axicon. (Adapted from Ref. 24,with permission from the Optical Society of America.)

surface. Values of ζ1 and ζ2 pairs are calculated by an iterative process, restrictingsolutions such that 0 ≤ ζ1,2 < 1. For example, a polarization-preserving axiconwas designed for a single-layer MgF2 coating (n = 1.38) on an aluminum substrate(complex refractive index n = 1.212 − 6.924i), at the He-Ne laser wavelengthof 0.6828 µm. Figure 3.32 shows the computed ζ1,2 values versus the angles ofincidence, where I1 = I2. For this system, polarization preservation is not possiblewhen I1,2 ≤ 34 deg. Representative results are summarized in Table 3.5, includingthe overall reflectances for uncoated and coated (polarization-preserving) axicons.As a bonus, the MgF2 coating protects the aluminum mirror coating and alsoenhances the reflectance for p-polarized light at higher angles of incidence.

Figure 3.32 Normalized film thicknesses versus angle of incidence for polarization-preserving axicon. (Adapted from Ref. 24, with permission from the Optical Society ofAmerica.)


Table 3.5 Representative results for MgF2/Al polarization-preserving axicon.24

I1, I2 (deg) ζ1 (µm) ζ2 (µm) Uncoatedp-reflectance

Uncoateds-reflectance

Coated reflectance

35 0.48173 0.51081 0.7906 0.8546 0.704845 0.43021 0.564371 0.7627 0.8733 0.713360 0.38793 0.611417 0.6896 0.9089 0.7352

3.3.3 Polarization-preserving total-internal-reflecting cube-corner prism

It is possible to preserve the polarization state of retroreflected light by a TIR cube-corner prism with the use of phase-compensation coatings.25 Both azimuth angleand ellipticity changes are considered for a TIR cube-corner of BK7 glass. In onemethod, an interference stack on all three faces can reduce the normal TIR phasedifference from δ = 45.29 deg at each face to δ = 0 deg. Vertical linear s-polarizedlight is incident on the cube-corner and exits the cube-corner as shown in Fig. 3.33.Data for a four-layer stack is given in Table 3.6.

Layer 1 is the first layer on the BK7 substrate, and the optical thickness is aquarter-wave layer with a physical thickness t.

3.3.4 Stokes parameters

Another way to describe the polarization state of a light beam is by the useof the Stokes parameters, named for George Gabriel Stokes. The four Stokesparameters, S 0, S 1, S 2, and S 3, are related to the polarization ellipse. One methodto describe the Stokes parameters is by a set of four sequential transmitting filtersthat are irradiated by randomly polarized light. Then the Stokes parameters are

Figure 3.33 Polarization-preserving TIR cube-corner prism.25

Table 3.6 TIR cube-corner stack design producing δ = 0 deg at each face.25

Layer Material Refractive index n (λ = 633 nm) Optical thickness4nt (nm)

1 SiO2 1.46 8132 TiO2 2.45 10663 SiO2 1.46 10904 TiO2 2.45 1702

90 Chapter 3

operationally defined by the following set of equations:26

S 0 = 2H0, (3.16a)

S 1 = 2H1 − 2H0, (3.16b)

S 2 = 2H2 − 2H0, (3.16c)

S 3 = 2H3 − 2H0, (3.16d)

where S 0 is the incident irradiance. H0, H1, H2, and H3 are the measuredirradiances through each sequential filter. S 1, S 2, and S 3 represent differentpolarization states such that:

• S 1 > 0 represents horizontal linearly polarized light.• S 1 < 0 represents vertical linearly polarized light.• S 1 = 0 represents elliptically polarized light where Ψ = ±45 deg, circularly

polarized light, or unpolarized light.• S 2 > 0 represents linearly polarized light, where Ψ = +45 deg.• S 2 < 0 represents linearly polarized light, where Ψ = −45 deg.• S 2 = 0 represents linearly polarized light at other Ψ values.• S 3 > 0 represents right handedness.• S 3 < 0 represents left handedness.• S 3 = 0 represents no specific handedness.

The Stokes parameters are usually normalized such that S 0 = 1 represents anincident beam of unit intensity and the values of S 1, S 2, and S 3 are either 0 or 1.Thus, the Stokes parameter set [S 0 S 1 S 2 S 3] = [1 0 1 0] describes the outputas linearly polarized light oriented at Ψ = +45 deg. Moreover, if two incoherentbeams are superimposed, each having specified Stokes parameter sets, then theresultant polarization state is the sum of the components of each Stokes parameterset. For unpolarized light, S 0 is a positive quantity, and S 1 = S 2 = S 3 = 0. Forcompletely polarized light, S 0

2 = S 12 + S 2

2 + S 32, and for partially polarized light,

the degree of polarization V is

V =

√S 1

2 + S 22 + S 3

2

S 0. (3.17)

3.3.5 Depolarizing cube-corner prism

By control of the orientation and tilt of a solid-glass TIR cube-corner, it is possibleto produce unpolarized output.27 Figure 3.34 shows a solid-glass TIR prism witha circular aperture (previously shown in perspective in Fig. 2.26). The cube-corneris viewed normal to the circular aperture along the direction of the input lightbeam. The back corners of the cube are A, B, and C, and the cube apex is D. Theforward cube-corners are E, F, and G. The back corners are projected on the planarcircular aperture (x-y plane) as A′, B′, C′, and D′, such that the back edges B-D,A-D, and C-D are projected on the circular aperture as B′-D′, A′-D′, and C′-D′.


Figure 3.34 TIR cube-corner prism orientation angle θ.27

The angle θ of the projected line B′-D′ with the horizontal x axis orients thecube-corner. A major diagonal of the cube connects the front corner E with the backcorner B.

For a BK7 solid-glass (n ≈ 1.52) cube-corner in air, the cube-corner is rotatedaround the major diagonal E-B, forming an angle σwith the incoming light beam ofintensity S 0, and is uniformly irradiated. The incoming linearly polarized light hasa polarization orientation angle Ψ. Figure 3.35 plots the calculated absolute valuesof the spatially integrated nonnormalized Stokes parameters |S 1

′′|, |S 2′′|, |S 3

′′|,calculated from the six Stokes parameters of the incident and reflected light at eachof the three cube-corner surfaces. For details of the calculations, see Kalibjian.27

There are two triple null points for which |S 1′′| = |S 2

′′| = |S 3′′| ≈ 0. S 0 = 1.0. This

occurs, for example, at σ = 6.12 deg, θ = 30 deg or 150 deg, and Ψ = 11.6 degor 101.6 deg. The retroreflected light is therefore unpolarized, and the cube-cornerreflector functions as a depolarizer for linearly polarized input.

Figure 3.35 Stokes parameters S versus polarization orientation Ψ for TIR cube-corner(σ = 6.117 deg; θ = 30 deg, 150 deg; n0 = 1.5113).27

92 Chapter 3

3.4 Plane of Polarization Rotation Using Total-Internal-ReflectingPrisms and Reflectors

3.4.1 90-deg polarization-rotating prism with coaxial beam output

To rotate the plane of polarization of a linearly polarized input beam by 90 deg,prisms are again constructed where the total internal reflections are restrictedto orthogonal planes. Figure 3.36 shows a prism with three orthogonal TIRreflections, where the input linearly polarized light is oriented along the x axis,such that E = Ex.28 The internal beam remains linearly polarized and is rotatedduring the internal reflections as shown. The output beam is displaced and parallelto the input beam with the plane of polarization rotated 90 deg.

3.4.2 90-deg polarization-rotating prism with retroreflected beam output

In Fig. 3.37, a prism with three orthogonal TIR reflections produces a retroreflectedoutput beam with the polarization plane rotated 90 deg. The linearly polarizedinput beam is oriented along the x axis.28 Figure 3.38 shows another type ofretroreflecting prism that rotates the plane of polarization 90 deg for a linearlypolarized beam. Here, the incident linearly polarized light can be oriented atany angle in the x-y plane.29 The internal beam undergoes four orthogonal TIRreflections, where two are s-polarized, and two are p-polarized. The net relativephase shift is zero, the output is linearly polarized, and the prism is achromatic.

3.4.3 90-deg polarization-rotating prism with orthogonal beam output

Another prism system has six orthogonal TIR reflections for a linearly polarizedinput beam (Fig. 3.39).29 Three internal reflections produce s-polarized phaseshifts, and three produce p-polarized phase shifts. The linearly polarized outputbeam is perpendicular to the input beam, and the plane of polarization is rotated90 deg relative to the input beam.

Figure 3.36 90-deg polarization-rotating TIR prism with linearly polarized input and coaxialoutput. (Adapted from Ref. 28, with permission from the Optical Society of America.)


Figure 3.37 90-deg polarization-rotating TIR prism with linearly polarized input andretroreflected output. (Adapted from Ref. 28, with permission from the Optical Society ofAmerica.)

Figure 3.38 90-deg polarization-rotating TIR prism with retroreflected output (α = 90 deg,β = 45 deg).29

3.4.4 Double Fresnel rhomb polarization rotator with collinear beamoutput

As shown in Sec. 3.2.1, when the double Fresnel rhomb is designed as a half-waveretarder, the plane of polarization of a 45-deg-oriented linearly polarized inputbeam is rotated 90 deg, and the output beam is collinear with the input beam. Theretardation of a BK7-glass double Fresnel rhomb varies continuously from about91.0 deg (λ = 360 nm) to 89.6 deg (λ = 900 nm) and is considered achromatic inretardance. (See Table 3.1 for some calculated retardance values.) Rotation of thedouble Fresnel rhomb by angle ϕ about the optic axis rotates the linear polarizationof a light beam by 2ϕ.

94 Chapter 3

3.4.5 Four-mirror 90-deg polarization rotator with collinear beam output

Smith and Koch have analyzed the use of multiple first-surface metallic mirrors torotate the plane of polarization of a linearly polarized input beam and produce acollinear output.30 Reflections are not restricted to orthogonal planes, but can be offaxis. By geometric phase analysis, they have shown that a minimum of four mirrorsis required to produce a 90-deg polarization rotation and produce collinear output.Although there are a multitude of solutions, a sample configuration is shown inFig. 3.40.

Figure 3.39 90-deg polarization-rotating TIR prism with orthogonal output.29

Figure 3.40 Four-mirror 90-deg polarization rotator with nonorthogonal reflections andcollinear output.30


References

1. E. E. Wahlstrom, Optical Crystallography, 3rd ed., 54–55, John Wiley, NewYork (1948).

2. G. Brasen et al., “Polarizing beamsplitter,” U.S. Patent No. 7,230,763 (2007).

3. E. O. Ammann and G. A. Massey, “Less-expensive Rochon prisms,” NASAReport Number MFS-20554, National Technology Transfer Center, Wheeling,WV (1970).

4. S. M. MacNeille, “Beam splitter,” U.S. Patent No. 2,403,731 (1946).

5. J. Mouchart et al., “Modified MacNeille cube polarizer for a wide angularfield,” Appl. Opt. 28(10), 2847–2853 (1989).

6. A. E. Rosenbluth et al., “Contrast properties of liquid crystal light valves inprojection displays,” IBM J. Res. Develop. 42, 359–386 (1998).

7. J. M. Jonza et al., “Polarizing beam-splitting optical component,” U.S. PatentNo. 5,962,114 (1999).

8. C. L. Bruzzone et al., “Polarizing beam splitter,” U.S. Patent No. 6,721,096(2004).

9. E. Gardner and D. Hansen, “An image quality wire-grid polarizing beamsplitter,” SID Symp. Dig. 34, 62–63 (2003).

10. R. T. Perkins et al., “Broadband wire grid polarizer for the visible spectrum,”U.S. Patent No. 6,122,103 (2000).

11. T. Baur, “A new type of beam splitting polarizer cube,” Proc. SPIE 5158,135–141 (2003) [doi:10.1117/12.510767].

12. L. Li and J. A. Dobrowolski, “High-performance thin-film polarizing beamsplitter operating at angles greater than the critical angle,” Appl. Opt. 39(16),2754–2771 (2000). See also L. Li and J. A. Dobrowolski, “Thin film polarizingdevice,” U.S. Patent No. 5,912,762 (1999).

13. J. J. Lee, “Polarizing prism for panel type liquid crystal display front projectorand optical prism using the polarizing prism,” U.S. Patent No. 5,717,472(1998).

14. J. M. Bennett, “A critical evaluation of rhomb-type quarterwave retarders,”Appl. Opt. 9(9), 2123–2129 (1974).

15. R. J. King, “Quarter-wave retardation systems based on the Fresnel rhombprinciple,” J. Sci. Instr. 43, 617–622 (1966).

16. M. Nicholson, “How to model corner cube retroflectors,” ZEMAX ApplicationNote, Zemax Development Corp, Bellevue, WA (2007).

17. M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt.35(11), 1813–1820 (1988).

18. E. Spiller, “Totally reflecting thin-film phase retarders,” Appl. Opt. 33(20),3544–3549 (1984).

96 Chapter 3

19. I. Fillinski and T. Skettrup, “Achromatic phase retarders constructed fromright-angle prisms: design,” Appl. Opt. 23(16), 2747–2751 (1984).

20. W. H. Steel, “Polarization-preserving retroreflectors,” Appl. Opt. 24(21),3433–3434 (1992).

21. E. J. Galvez, “Achromatic polarization-preserving beam displacer,” Opt. Lett.26, 971–973 (2001).

22. E. Cojocaru, “Polarization-preserving totally reflecting prisms,” Appl. Opt.31(22), 4340–4342 (1992).

23. Z. P. Wang et al., “Polarization-preserving totally reflecting prisms with asingle medium layer,” Appl. Opt. 36, 2802–2807 (1997).

24. R. M. A. Azzam and M. Emdadur Rahman Khan, “Polarization-preservingsingle-layer-coated beam displacers and axicons,” Appl. Opt. 21(18),3314–3322 (1984).

25. L. H. Lee and J. J. Bockman, “Phase-compensated cube-corner in laserinterferometry,” U.S. Patent No. 7,165,850 (2007).

26. E. Hecht, “Polarization,” Chapter 8 in Optics, 2nd ed., 321–326, Addison-Wesley, Reading, MA (1987).

27. R. Kalibjian, “Cube-corner depolarizer,” U.S. Patent No. 7,254,288 (2007).

28. N. I. Petrov, “Achromatic polarization rotator,” Appl. Opt. 45(25), 6340–6343(2007).

29. W. A. Challener IV, “Achromatic polarization-rotating right-angle prismsystem,” U.S. Patent No. 5,751,482, (1998).

30. L. L. Smith and P. M. Koch, “Use of four mirrors to rotate linear polarizationbut preserve input–output collinearity,” J. Opt. Soc. Am. A 13, 2102–2105(1996).

Chapter 4Specialized Prism Types

4.1 Dispersing Prism

4.1.1 Refracting direct-vision prism

A direct-vision prism disperses the input light with no significant angular deviationof the output. It is often called a direct-view spectroscope. The simplest type ofdirect-vision prism is shown in Fig. 4.1, where the displaced and dispersed outputis coaxial with the input. This compound prism usually consists of crown- andflint-glass prisms cemented at the interface surface. Let the crown glass have arefractive index nd = 1.5159 and a ν-number = 70.0, and the flint glass have arefractive index nd = 1.9626 and a ν-number = 19.7. If we choose the slope angleα of the interface, then the vertex angle β of the first prism can be calculated from

tan β =(n2 − 1) sinα√

n21 − n2

2 sin 2α − cosα, (4.1)

where n1 is the design refractive index of the crown glass, and n2 is the designrefractive index of the flint glass.1 To produce a dispersed output that is collinearwith the input, two of the above prisms are combined to form a double Amici prism(Fig. 4.2). Using the same crown and flint glasses, if we choose α = 45.0 deg, thenβ = 98.123 deg.

Figure 4.1 A direct-vision prism with coaxial output.

97

98 Chapter 4

Figure 4.2 The double Amici direct-vision prism with collinear output.

4.1.2 Reflective dispersing prisms with collinear output

There are several types of reflecting dispersing prisms of a single glass type thatproduce collinear output (output beam on same axis as input beam). The Kesslerdirect-vision prism is a classic design that uses two refractions and two totalinternal reflections (Fig. 4.3). Figure 4.4 shows a Sherman-type prism that uses tworefractions and four reflections.2 Two reflections are from coated reflector surfaces,and two undergo total internal reflection (TIR). This prism is compact and producesa collinear output, but it requires a high-index glass to achieve the dispersion of a60-deg equilateral refracting prism (see Table 2.1). Figure 4.5 illustrates anothercompact dispersing prism that uses two refractions and two reflections from coatedsurfaces.3

Figure 4.3 The Kessler direct-vision prism with collinear output.

Figure 4.4 The Sherman-type direct-vision prism with collinear output.

Specialized Prism Types 99

Figure 4.5 A direct-vision prism with collinear output using two refractions and tworeflections. (Adapted from Ref. 3 with permission from the Optical Society of America.)

4.1.3 Direct-vision prisms with wavelength tuning

All of the direct-vision prisms described produce collinear output at a singlewavelength. It is possible to arrange a pair of the prism types in Fig. 4.5 to varythe wavelength of the collinear output. In Fig. 4.6, a pair of two reflection prismsare coupled to rotate by the same angle in opposite directions about the indicatedray positions.4 The ray path between the prisms is coaxial to the input and outputbeams but is not collinear to these beams. Another type of rotational coupling fora pair of identical dispersing prisms produces collinear output wavelength tuning(Fig. 4.7).5 Each prism has two refractions and two internal reflections, with adispersed coaxial output (output beam parallel to input beam). The rotation axispositions are as shown, and the rotations are in opposite directions.

4.1.4 Total-internal-reflecting dispersing prism

Where collinear or coaxial output is not a requirement, a single TIR prism can bedesigned to disperse light, or to be nondispersive. The TIR prism shown in Fig. 4.8

Figure 4.6 A coupled dispersing prism pair with wavelength tuning and collinear output.(Adapted from Ref. 4 with permission from Elsevier.)

100 Chapter 4

Figure 4.7 A coupled dispersing TIR prism pair with wavelength tuning and collinearoutput.5

Figure 4.8 A single TIR dispersing prism.

has planar entrance and exit refracting surfaces with an intermediate TIR surface.The sequential ray-tracing equations are as follows:

I1 = θ1 − α, (4.2a)

I1′ = arcsin

(sin I1

n

), (4.2b)

I2 = 180 deg − I1′ − α − β, (4.2c)

I2′ = I2, (4.2d)

I3 = β − I2′, (4.2e)


δ = θ1 + θ3, (4.2g)

where θ1 is the incident ray angle, α is the refracting groove angle, β is thereflecting groove angle, θ3 is the exit ray angle, and δ is the deviation angle. I2must exceed the critical angle at the reflecting surface. For example, if θ1 = 60 deg,α = 35 deg, β = 80 deg, and nd = 1.5168 (BK7 glass), then I2 = 48.82 deg,I1 = 25 deg, θ3 = 51.74 deg, and δ = 111.74 deg.


The TIR prism can be designed to have normal positive dispersion, negativedispersion, or no dispersion (e.g., achromatic). The conditions for each are asfollows:

• If (2β + α) > 180 deg, the prism has positive dispersion.• If (2β + α) < 180 deg the prism has negative dispersion.• If (2β + α) = 180 deg the prism is nondispersive.

Thus, a glass or plastic prism has positive dispersion if α = 40 deg and β = 75 deg,negative dispersion if α = 20 deg and β = 75 deg, and no dispersion if α = 30 degand β = 75 deg. These relationships are valid for any normal refractive index andangle of incidence I1, provided that TIR occurs at surface 2. This prism type couldbe used to compensate for other dispersions in an optical system.

4.1.5 Multiprism negative dispersion

Consider the pair of identical prisms oriented as shown in Fig. 4.9, where theincident face of prism 1 is parallel to the exit surface of prism 2, and the exitsurface of prism 1 is parallel to the entrance surface of prism 2. The prisms areconfigured such that all rays are incident at the Brewster’s angle, and all operateat minimum deviation.6 The color components of the exiting rays are spatiallyand temporally separated and are coaxial to the incident ray and to each other. Thistwo-prism system can generate negative group-velocity dispersion. By the additionof another identical prism pair with controlled orientation and separations, as inFig. 4.10, an in-line (collinear) output is obtained. The output of this four-prismarrangement also has negative group velocity dispersion. The amount of dispersioncan be adjusted, and the theory is developed by Fork et al.6

4.2 Refracting Achromatic Compound Prism

A well-known refracting achromatic prism is constructed from two bonded prismshaving different refractive indices and dispersions (Fig. 4.11). The first prism isusually of low-dispersion glass, and the second prism is of high-dispersion glass.

Figure 4.9 A prism pair with coaxial output and negative dispersion.6

102 Chapter 4

Figure 4.10 A four-prism system with collinear output and negative dispersion.6

Figure 4.11 An achromatic prism combination of high- and low-dispersion glass.

The initial values of the vertex angles for each section, α1 and α2, are calculatedfrom the paraxial approximation for a two-element achromatic prism:7

α1 =δν1

(n1 − 1)(ν1 − ν2), (4.3)

α2 =δν2

(n2 − 1)(ν2 − ν1), (4.4)

where δ is the deviation angle.


The sequential ray-tracing equations for an angle of incidence I1 are

I1′ = arcsin

(sin I1

n1

), (4.5a)

I2 = α1 − I1′, (4.5b)

I2′ = arcsin

[sin I2

(n1/n2)

], (4.5c)

I3 = I2′ − α2, (4.5d)

I3′ = arcsin(n2 sin I3), (4.5e)

δ = I1 + I3′ − α1 + α2, (4.5f)

where α1 and α2 are entered as positive numbers and are related by the following:

tanα2 =n2′ + sin I2

′ − sin(δ − I1 + α1)n2′ + cos I2

′ − cos(δ − I1 + α1), (4.6)

where n2′ is the refractive index value that produces the target δ value. For visible

light, the deviation angles are calculated from Eqs. (4.3a) to (4.3g) for threewavelengths, nC (λ = 656.3 nm), nd (λ = 587.3 nm), and nF (λ = 486.1 nm).A merit function MF(δ) can be defined as

MF(δ) =

√(δ − δc)2 + (δ − δd)2 + (δ − δ f )2. (4.7)

If the glasses have been selected, the prism angles α1 and α2 could then bevaried to reduce MF(δ) to an acceptable value over the visible spectrum. Othervariables to consider are I1 in Eq. (4.1) and n2

′ in Eq. (4.6), where n2C ≤ n2′ ≤

n2F . Design methods, including glass selection, for several compound achromaticprism systems are described by Mercado.8 Figure 4.12(a) shows a two-elementachromatic prism with a deviation angle δ = 6 deg. Here, α1 = 42.0239 deg,n1d = 1.52855, ν1 = 76.98, and α2 = 25.5872 deg; n2d = 1.65160 and ν2 = 58.40.Figure 4.12(b) plots the change in the deviation angle over the visible spectrum.

4.3 Anamorphic Prisms for Beam Compression and Expansion

Anamorphic prism systems change the height or width of an incident beam alongone dimension by a specified amount. For example, an elliptically shaped beamcan be converted to a circular beam, or the aspect ratio of a beam can be modified.In fact, most refracting prisms produce some magnification change between inputand output beams. The problem is to specify its magnitude and direction for theintended application. The most useful output directions are collinear, coaxial, andorthogonal (output beam perpendicular to input beam). These anamorphic prismscan be of the dispersing type, or in some cases, produce an achromatic output.

104 Chapter 4

Figure 4.12 (a) An achromatic compound prism with a deviation angle δ = 6 deg.8 (b)Deviation angle variance of an achromatic compound prism over the visible spectrum.8

4.3.1 Beam expander with orthogonal output

Figure 4.13 shows an anamorphic beam expander that produces an orthogonaloutput and is based on the a Littrow-type prism.9 Vertex angle β is cut at Brewster’sangle θBrew, and the angle of incidence I1 at surface 1 is also θBrew for hightransmission of p-polarized light. For n = 1.5 and θBrew = 56.3 deg, TIR occurs atsurface 2, and the dispersed output is orthogonal. The magnification (expansion) ofthe beam could be calculated from the ray intersection coordinates using Eq. (1.58)and Eq. (1.59). Alternatively, using the angles of incidence I and refraction I′

at each surface of the prism, the total anamorphic prism magnification MAG(or demagnification for beam compression) can be calculated from the following


Figure 4.13 An anamorphic beam expander with orthogonal output.9

product:

MAG =

k∏i=1

cos Ii′

cos Ii, (4.8)

where i is the refracting surface number and k is the number of refracting surfaces.For this example, the magnification is

MAG =cos I1

′

cos I1×

cos I3′

cos I3=

cos(33.69 deg)cos(56.31 deg)

×cos(0 deg)cos(0 deg)

= 1.5,

which is the same as the prism design refractive index n. The dispersed output isonly exactly orthogonal at the design wavelength, with a slight deviation for thedispersed rays.

4.3.2 Beam compressor with coaxial output

Figure 4.14 illustrates a beam-compressing prism with a coaxial output that hastwo refractions and two total internal reflections.10 The design parameters are thevertex angle α, the prism refractive index n, and the beam-compression factorMAG = A′/A. For coaxial output, the design refractive index and vertex angleare related by

n =cosα

cos 3α. (4.9)

The sequential ray-tracing equations are as follows:

I1′ = I1 = 0, (4.10a)

106 Chapter 4

Figure 4.14 A beam-compressing prism with coaxial output.10

I2′ = I2 = 90 deg − α, (4.10b)

I3′ = I3 = I2

′ − α, (4.10c)

I4 = I3′ − α, (4.10d)

I4′ = arcsin(n sin I4). (4.10e)

Using Eq. (4.7), if α = 17 deg, then n ≈ 1.52 and I4′ = 90 deg−α. Then, the beam

compression is

MAG =cos I1

′

cos I1×

cos I4′

cos I4=

1.01.0×

0.2920.777

= 0.376.

4.3.3 Beam expander with collinear output

A single prism with two refractions and one total internal reflection (Fig. 4.15)can produce an expanded collinear output with an anamorphic magnification of2×.11 The intended use is to change the elliptically shaped collimated beam froma laser diode to a circular beam. The prism material is BK7 glass with a publishedrefractive index n = 1.51119 at λ = 780 nm. The vertex angles are α = 27.912 degand β = 41.388 deg. Let the entrance surface 1 be tilted at an angle ϕ with respect

Figure 4.15 An anamorphic beam expander with collinear output at 2× magnification.11


to the horizontal optic axis. The sequential ray-trace equations are

I1 = 90 deg − ϕ, (4.11a)

I1′ = arcsin

(sin I1

n

), (4.11b)

I2 = I1′ + α = I2

′, (4.11c)

I3 = I2′ − β, (4.11d)


δ = I3′ − 90 deg + α + β − ϕ. (4.11f)

From Eq. (4.8), the magnification MAG is

MAG =cos(38.65 deg)cos(70.9 deg)

×cos(40.0 deg)

cos(25.17 deg)= 2.0.

Since the deviation angle δ ≈ 0 deg, the output is coaxial. The prism length is thenadjusted so that the exit ray is on the same axis as the central entrance ray, resultingin a collinear output.

4.3.4 Wedge prism beam compressor/expander

A wedge prism usually has a vertex angle α between 2 and 20 deg and is often usedfor beam steering (see Sec. 6.1.2). If one surface of the prism is reflectorized, a thinanamorphic beam expander can be created.12 The output beam is orthogonal to theinput beam. It is realized that any prism designed for anamorphic compressioncan be used as an expander by reversing the direction of the input and vice versa.Figure 4.16 shows a reflecting wedge prism anamorphic compressor that convertsan input beam with an aspect ratio of 2:1 to an output beam with an aspect ratioof 4:3.13 The ray-trace equations are

I1 = ϕ, (4.12a)

I1′ = arcsin

(sin I1

n

), (4.12b)

I2 = α + I1′ = I2

′, (4.12c)

I3 = I2 + I2′ + I1 − I1

′ − ϕ, (4.12d)


δ = ϕ + I3′. (4.12f)

Here, α � (I3 − I1′)/2 and ϕ is the tilt angle of surface 1 from the vertical. To

account for the varying thickness of the prism, ϕ and α are adjusted until thedesired compression ratio is obtained. For a prism of B270 optical crown glass(nd = 1.5229) with ϕ = 16.9 deg and α = 14.0 deg, an anamorphic compressionA′/A = MAG ≈ 0.375 can be obtained.

108 Chapter 4

Figure 4.16 An anamorphic reflecting wedge prism compressor.13

4.3.5 Anamorphic prism pair with coaxial output

A useful method to produce a variable magnification is by using a pair of identicalair-spaced prisms “a” and “b”. By varying the tilt angles ϕa and ϕb, the anamorphicmagnification can vary from about 2× to 6×. These prisms are sold commerciallyas separate components or as a mounted assembly by suppliers such as MellesGriot or Edmund Optics. They are usually antireflection coated. Figure 4.17 showsa pair of prisms with a vertex angle α = 29.43 deg and SF11 glass (n ≈ 1.765).

Figure 4.17 An air-spaced anamorphic prism pair beam expander.


The ray-trace equations are

I1 = α + ϕa, (4.13a)

I1′ = arcsin

(sin I1

n

), (4.13b)

I2 = I1 − I1′ − ϕa, (4.13c)


δa = I2′ + ϕa, (4.13e)

I3 = δa + α − ϕb, (4.13f)

I3′ = arcsin

(sin I3

n

), (4.13g)

I4 = α − I3′, (4.13h)

I4′ = arcsin(n sin I4), (4.13i)

δab = ϕb − I4′, (4.13j)

where ϕa and ϕb are positive for clockwise rotation and chosen such that I3 = I1and δab ≈ 0 for coaxial output. The anamorphic magnification of the pair iscalculated from Eq. (4.8). The vertical separation between the input and outputbeams is determined by the spacing between the prisms. Table 4.1 shows sampledata for magnifications of 2×, 3×, and 4×.

When operating at 3× magnification, the incident angle at each prism is closestto the Brewster angle for SF11 glass, where IBrew = 60.47 deg for n = 1.765.

4.3.6 Multiprism dispersive compressors and expanders

The anamorphic prism pair beam expander of Sec. 4.3.5 is of the compensating pairtype, sometimes called the up-down configuration. The dispersion is reduced, butnot eliminated, when compared with the additive pair configuration in Fig. 4.18,sometimes called the up-up configuration. Duarte and Piper have calculated thecumulative dispersion for a series of anamorphic beam prism pairs in bothconfigurations.14 For use as beam expanders in pulsed dye lasers, the angle ofincidence at each surface is set equal, and the exit angles are set close to zero. Theprisms are identical and designed for BK7 glass (n ≈ 1.515), with a vertex angleα = 41.5 deg and a glass dispersion dn/dλ = 0.71×10−4 nm at the laser wavelengthλ = 510 nm. Duarte and Piper calculated and compared the cumulative dispersions

Table 4.1 Sample data for an anamorphic prism pair, α = 29.43 deg, and SF11 glass.

ϕa 21.2 deg 30.7 deg 35.2 degϕb 6.1 deg 0.0 deg −2.4 degI1 50.63 deg 60.13 deg 64.63 degI3 50.63 deg 60.13 deg 64.63 degδab 0.00117 deg 0.00260 deg 0.00273 degMAG 2.0 3.0 4.0

110 Chapter 4

Figure 4.18 A two-prism up-up configuration.

for single-, double-, and four-prism beam expanders at magnifications of 5.34× and60×. In particular, the four-prism compensating pair arrangement at 5.35×, with theangles of incidence close to Brewster’s angle, yielded a 4.8 factor reduction in thecumulative dispersion compared to the additive pair arrangement.14

By control of the prism vertex angles and prism angles of incidence ona separated prism pair in the up-down configuration, the positively directeddispersion of the first prism can be compensated by the negatively directeddispersion of the second prism, resulting in a near-achromatic prism pair. Trebinohas analyzed arrangements for multiple-prism beam expanders, consideringthe impact on total transmission and magnification.15 A common four-prismachromatic down-up-up-down configuration is shown in Fig. 4.19, consistingof two achromatic pairs, with an achievable magnification MAG ≈ 40, and atransmission > 50%.

Figure 4.19 A four-prism down-up-up-down configuration with MAG ≈ 40. (Adapted fromRef. 15 with permission from the Optical Society of America.)


It was found that the total transmission for a given magnification can beoptimized by the up-up-up-down configuration of Fig. 4.20, where the negativelydirected dispersion of the fourth prism compensates for the additive positivelydirected dispersion of the first three prisms.

It was determined that for an achromatic N-prism beam expander, whenMAG ≈> [2−1/(2N−1−1)]N , that the transmission at this magnification is optimizedfor an up-up . . . up-down configuration.

The number of prisms is not restricted to an even number. A three-prism up-up-down configuration that is achromatic with optimal performance at MAG ≈ 20 isshown in Fig. 4.21.

4.4 Achromatic Anamorphic Prism

The achromatic anamorphic multiprisms described in Sec. 4.3.6 are restrictive tochoice of prism vertex angles, angles of incidence, and the steering direction of thebeam output. However, there are several types of anamorphic prisms that can bemade achromatic by other methods.

4.4.1 Air-spaced prism pair with coaxial output

By making an air-spaced anamorphic prism pair of a high-dispersive and a low-dispersive glass, this type of beam expander/compressor can be made achromatic.

Figure 4.20 A four-prism up-up-up-down configuration with MAG ≥ 10. (Adapted fromRef. 15 with permission from the Optical Society of America.)

112 Chapter 4

Figure 4.21 A three-prism up-up-down configuration with MAG ≥ 5. (Adapted from Ref. 15with permission from the Optical Society of America.)

The design principles are similar to the refractive achromatic compound prismsdescribed in Sec. 4.2. In Fig. 4.22(a), an achromatic anamorphic prism pair beamcompressor provides a coaxial beam output over a 20-nm range of the designwavelength.16 It converts the elliptically shaped output of a diode laser lightsource to a circularly shaped output beam, with a magnification MAG = 0.333,to compensate for small variations in the laser diode wavelength. The pair iscomposed of a first prism of KF9 glass (nd = 1.52346, νd = 51.54) and a secondprism of SF11 glass (nd = 1.78472, νd = 25.68). All prism surfaces are nonnormalto the incident and exit beams to prevent any reflection returning to the light source.The relevant design values at a laser-diode design wavelength λ = 800 nm are givenhere:

αa = 35.927 degαb = 27.384 degϕa = I1 = 1.0 degϕb = −29.217 degI2 = 36.587 degI3 = 1.5 degI4′ = 56.601 deg

δa = I2′ − αa − ϕa = 27.717 deg

δab = 0 degMAG = 0.333.

Figure 4.22(b) plots the change in the deviation angle δab in the wavelengthrange 500 nm ≤ λ ≤ 1000 nm.


Figure 4.22 (a) An air-spaced achromatic anamorphic prism pair compressor.16 (b)Deviation angle versus wavelength for an achromatic prism pair.16

4.4.2 Compound prisms with orthogonal output

Another type of compound achromatic anamorphic beam expander shown inFig. 4.23(a) has an orthogonal output.17 The first glass is low-dispersion BK7crown glass (n = 1.5112) and is optically bonded to the second high-dispersionglass SF11 (n = 1.7660). Both indices are referenced to light at λ = 780 nm. Thereflecting surface requires a reflective coating, and the prism has an anamorphicmagnification MAG ≈ 1.75. Another achromatic compound beam expander shownin Fig. 4.23(b) uses TIR at the reflecting surface and also provides coaxial outputwith a magnification MAG ≈ 1.75.17

4.4.3 Refracting/total-internal-reflecting prism pair with orthogonaloutput

By combining a refracting prism having positive dispersion with a catadioptricprism or TIR prism (a prism using both refraction and reflection) having negativedispersion (Sec. 4.1.4), an achromatic beam compressor with orthogonal outputcan be constructed.13 Figure 4.24 shows the configuration where the material forboth prisms “a” and “b” is BK7 glass (nd = 1.5168, νd = 64.17). The sequential

114 Chapter 4

Figure 4.23 (a) A compound achromatic anamorphic beam expander with orthogonaloutput.17 (b) A compound achromatic anamorphic beam expander with orthogonal outputusing TIR.17

Figure 4.24 An air-spaced achromatic anamorphic prism pair beam compressor using acommon material.13


ray-tracing equations are

I1 = ϕ1, (4.14a)

I1′ = arcsin

(sin I1

n

), (4.14b)

I2 = α1 + I1′, (4.14c)


δa = I2′ − ϕ1 − α1, (4.14e)

I3 = δa + 90 deg − α2 + ϕ2, (4.14f)

I3′ = arcsin

(sin I3

n

), (4.14g)

I4 = 180 deg − α3 − I3′ − α2 = I4

′, (4.14h)

I5 = I4′ − α3, (4.14i)

I5′ = arcsin(n sin I5), (4.14j)

δab = 90 deg − I5′ + ϕ2. (4.14k)

Some design values at the design wavelength λ = 587.6 nm are listed here:

α1 = 18.6 degα2 = 68.0 degα3 = 38.2 degϕ1 = 30.0 deg = I1

ϕ2 = 8.02 degI2 = 37.85 degI3 = 49.96 degI4 = 43.48 degI5 = 5.28 degI5′ = 8.02 deg

δa = 19.93 degδab = 90.0016 deg

and

δa (λ = 486.1 nm) − δa (λ = 656.3 nm) = 0.430 deg,δab (λ = 486.1 nm) − δab (λ = 656.3 nm) = −0.000568 deg .

The anamorphic magnification MAG = 0.375 is calculated from Eq. (4.8). Theprism pair can be used to convert a 2:1 aspect ratio input beam to a 4:3 aspect ratiooutput beam.

116 Chapter 4

4.5 A Misalignment-Tolerant Beam-Splitting Prism

Figure 4.25(a) illustrates a beam-splitting prism comprised of two bonded sectionswith a beam-splitting coating on the interface.18 The external surfaces areaccurately parallel to the opposing faces, and the reflecting surfaces utilize TIRor are coated with a high-reflectance film. The corner angles are α = 45 deg andβ = 135 deg, while the beam-splitting coating reflects 50% and transmits 50%.When a light beam is incident normal to the input aperture, the beam is split andthe output is displaced and parallel to the input beam, or the output is coaxial. Thedisplacement is determined by the length of the prism for the application.

In Fig. 4.25(b), the prism is rotated in a clockwise direction by angle ϕ.This could be uncontrolled misalignment or an intentional tilt to prevent lightbeing reflected back along the original path. For a rotated prism (clockwise orcounterclockwise), the output beams remain coaxial to the input beam. Thisrotation-tolerant prism is optically equivalent to a tilted plate (Sec. 1.10). Incontrast, for a cube beam-splitting prism, the output beams do not remainorthogonal if the prism is rotated, although both prism types can be used inconvergent or divergent beams.

4.6 Axicon Prism

An axicon prism has a conical-shaped surface. It is sometimes called an axicon lensor rotationally symmetric prism. The most common configuration is a refractingplano-convex axicon, as shown in Fig. 4.26. With collimated light incident on theplano side, a linear focus is formed on the optic axis and a ring focus is formedon an image plane. The angle θ, measured from the axicon surface to a planeperpendicular to the optic axis, describes the surface. The prism can be modeledusing the single design parameter θ.19 Axicon prisms are available commerciallyin BK7 glass and quartz.

A type of reflecting plano-concave axicon prism shown in Fig. 4.27 uses a TIRconical surface.20 Collimated light entering the plano surface undergoes TIR at theconical surface (θ = 45 deg) and forms an illuminated concentric ring 360 degaround the sides of the axicon prism.

4.7 A Variable Phase-Shifting Prism

The phase of a coherent beam of light can be varied by means of a shifting prism.21

As shown in Fig. 4.28, an isosceles prism with vertex angle α and refractive indexn1 has light incident at an angle of minimum deviation Imin. The surround has arefractive index n0 (normally air). For the prism in position 1, the optical pathdistance (OPD) between the points A and B is OPD1 = n0L + n1l′ + n0L. When theprism is shifted by a distance Y to position 2, the OPD between the same pointsis OPD2 = n0l + n1l′ + n0l, where Y , L, l′, and l are defined as shown. Then thedifference in the optical path distance ∆OPD12 is

∆OPD12 = 2(n1l − n0L). (4.15a)


Figure 4.25 (a) A beam-splitting compound prism with coaxial output.18 (b) A tilted beam-splitting compound prism with undeviated coaxial output.18

118 Chapter 4

Figure 4.26 A refracting plano-convex axicon prism.

Figure 4.27 A reflecting plano-concave axicon prism.20

Figure 4.28 A prismatic phase shifter.21


When the prism is operated at a minimum deviation angle of incidence Imin, then∆OPD12 is derived by Childers21 to be

∆OPD12 =2Y sin(α/2)

cos Imin

[n0 − n1 cos

(Imin −

α

2

)]. (4.15b)

The phase of the exit beam is then precisely controlled by varying Y , without anytranslation or deviation of the light beam.

References

1. J. P. C. Southall, Mirrors, Prisms and Lenses, 3rd ed., 493–499, Macmillan,New York (1946).

2. B. Sherman, “Dispersion prism with no deviation,” U.S. Patent No. 3,057,248(1962).

3. M. V. R. K. Murty and A. L. Narasimhan, “Some new direct vision dispersionprism systems,” Appl. Opt. 9(4), 859–862 (1970).

4. M. V. R. K. Murty, “In-line dispersion prisms and methods of tuning differentwavelengths,” Opt. Laser Technol. 16, 255–257 (1984).

5. R. D. Tewari et al., “Modified in-line dispersion prism,” Opt. Eng. 31(6),1340–1341 (1992) [doi:10.1117/12.57696].

6. R. L. Fork et al., “Negative dispersion using prism pairs,” Opt. Lett. 9(5),150–152 (1984).

7. W. J. Smith, Modern Optical Engineering, 2nd ed., 90–91, McGraw-Hill, NewYork (1990).

8. R. L. Mercado, “Color-corrected prism systems,” U.S. Patent No. 4,704,008(1987).

9. A. B. Marchant, “Method and apparatus for anamorphically shaping anddeflecting electromagnetic waves,” U.S. Patent No. 4,759,616 (1988).

10. S. D. Fantone, “Optical system with anamorphic compression,” U.S. PatentNo. 4,627,690 (1986).

11. J. F. Forkner, “Anamorphic prism for beam shaping,” U.S. PatentNo. 4,623,225 (1986).

12. K. Yoshifusa and T. Yokota, “Beam converting apparatus with a parallel lightbeam input and output from one prism plane,” U.S. Patent No. 5,007,713(1991).

13. D. F. Vanderwerf, “Polarized illumination system for LCD projector,” U.S.Patent No. 5,995,284 (1999).

14. F. J. Duarte and J. A. Piper, “Dispersion theory of multiple-prism beamexpander for pulsed dye lasers,” Opt. Commun. 43, 303–307 (1982).

120 Chapter 4

15. R. Trebino, “Achromatic N-prism beam expanders: optimal configurations,”Appl. Opt. 24(8), 1130–1138 (1985).

16. F. C. Leuke, “Achromatic anamorphic prism pair,” U.S. Patent No. 5,596,456(1997).

17. M. Sugiki, “Anamorphic prism,” U.S. Patent No. 4,750,819 (1988).

18. C. F. Buhrer, “Optical beam splitter prism,” U.S. Patent No. 4,671,613 (1987).

19. “Modeling axicons,” ZEMAX Application Note, Zemax Development Corp.,Bellevue, WA (1999).

20. A. R. Henderson, “Prisms,” UK Patent Application No. 2,001,775A (1978).

21. B. A. Childers, “Prismatic phase shifter,” NASA Tech Brief LAR-14637,NASA Langley Research Center, Hampton, VA (1999).

Chapter 5Prism and Mirror System Design,Analysis, and Fabrication

5.1 Prism Design and Analysis

More optical tools are available for the analysis of prism and mirror systems thanfor the design of new systems. Some questions to be considered when approachingthe design of a prism for an optical system are:

• Is the prism to be used in an imaging optical system, or as an illumination controlelement, or both?• For imaging applications, what is the required orientation of the viewed or

projected image? Does the image shape need to be preserved—e.g., no linearmagnification?• Can an existing design, preferably one available from a commercial supplier, be

used?• Can the design be fabricated economically?• Is the required prism a single or compound element?• Are compound prisms air-spaced or bonded?• Are surface coatings required—e.g., antireflection or beam-splitting?• Does the prism need to be achromatic?• Will the prism be used in convergent or divergent light?• What aperture size is required?• What is the wavelength range of interest?• Are there size and weight limitations?• What are the mounting tolerance requirements?• What are the prism material clarity requirements, angular accuracy of faces, and

surface smoothness?• What is the thermal environment?

There is an extensive library of prism designs in the literature that may providethe requirement of your optical system. Many times, these designs can be modifiedby a change of glass material or the addition of special coatings. Often two or moreexisting prisms can be combined in series.

121

122 Chapter 5

5.1.1 Sectional element approach for prism design

Often, sections of simpler prisms can be combined to form another prism typewith new properties. Rothstein has described the combination of single isoscelesTIR prisms.1 In Fig. 5.1, a single isosceles prism has vertex angle α and base anglesβ. It is tilted at angle ϕ and usually truncated to eliminate the nonworking lowerportion. The directions of the three working surfaces are specified by the anglesψ with a horizontal optical axis, where ψ1 = 90 deg − ϕ, ψ2 = α/2 + ϕ, andψ3 = 90 deg − α − ϕ. For a prism with refractive index n with an air surround, thesequential ray-tracing equations are

I1 = ϕ, (5.1a)

I1′ = arcsin

(sin I1

n

), (5.1b)

I2 = 90 deg − ψ2 + I1 − I1′ = I2

′, (5.1c)

I3 = ψ2 + ψ3 − I2′, (5.1d)

I3′ = arcsin(n sin I3). (5.1e)

The deviation angle δ = 90 deg − ψ3 + I3′. Table 5.1 gives several calculated

values for a BK7 glass prism in air (nd = 1.5168). Although there is no anamorphicmagnification, the prism is dispersing, and the deviation angle δ depends on thevalue of ϕ.

Figure 5.2 shows another TIR deviator composed of two identical isoscelesprisms folded along a side to form a single prism with two TIR reflections. Theprism is tilted at angle ϕ. The directions of the four working surfaces are specified

Figure 5.1 A single isosceles prism with vertex angle α and base angles β, tilted at angle ϕ.

Prism and Mirror System Design, Analysis, and Fabrication 123

Table 5.1 Sample data for isosceles TIR prism, nd = 1.5168.

α (deg) β (deg) ϕ (deg) I2 (deg) δ (deg)

20.0 80.0 15.0 70.18 50.020.0 80.0 5.0 76.71 30.020.0 80.0 0.0 80.0 20.020.0 80.0 −10.0 86.57 0.030.0 75.0 15.0 65.18 60.030.0 75.0 5.0 71.71 40.030.0 75.0 0.0 75.0 30.030.0 75.0 −10.0 81.57 10.045.0 67.50 15.0 57.68 75.045.0 67.50 5.0 64.21 55.045.0 67.50 0.0 67.50 45.045.0 67.50 −10.0 74.07 25.0

Figure 5.2 A double TIR prism consisting of two isosceles prisms. (Adapted from Ref. 1with permission from the Optical Society of America.)

by the angles ψ with the optic axis, where ψ1 = 90 deg − ϕ, ψ2 = 90 deg − β + ϕ,ψ3 = 270 deg− 3β+ϕ, and ψ4 = β−ψ3. The sequential ray-tracing equations are:

I1 = ϕ, (5.2a)

I1′ = arcsin

(sin I1

n

), (5.2b)

I2 = 90 deg − ψ2 + I1 − I1′ = I2

′, (5.2c)

I3 = 180 deg + ψ2 − ψ3 − I2′ = I3

′, (5.2d)

I4 = ψ3 + ψ4 − I3′, (5.2e)

I4′ = arcsin(n sin I4). (5.2f)

124 Chapter 5

The deviation angle δ = 90 deg − ψ4 + I4′. Several values of the parameters are

given in Table 5.2 for a BK7 glass prism.The deviation angle is constant for any tilt angle, provided that TIR occurs at

both reflecting surfaces. The magnification remains constant at MAG = 1.0, andthe prism is achromatic. Reflecting isosceles prisms are used as sections of thePechan prism [see Fig. 2.20(a)].

5.1.2 Right-angle prism sections

Several well-known single-prism types contain basic right-angle prism sectionsor modified right-angle prisms. For example, the wavelength-discriminatingPellin–Broca prism in Sec. 2.3 is a fusion of three right-angle prisms, two30/90/60-deg sections, and one 45/90/45-deg section (see Fig. 5.3). The reflectingLittrow prisms in Sec. 2.9 use a single right-angle prism. It is noted that a reflectingisosceles prism and a right-angle prism are sections of the Penta prism of Fig. 2.6.

5.1.3 Experiential design of multiple reflectors

Prism or multiple-mirror design by pure trial and error implies little previousknowledge of the principles used and neglects the experience of the designer.

Table 5.2 Sample data for double isosceles TIR prism, nd = 1.5168.

β (deg) ϕ (deg) I2 (deg) I3 (deg) δ (deg)

67.50 15.0 57.68 77.32 90.067.50 5.0 64.21 70.79 90.067.50 0.0 67.50 67.50 90.067.50 −10.0 74.07 60.93 90.075.0 15.0 65.18 84.82 60.075.0 5.0 71.71 78.29 60.075.0 0.0 75.0 75.0 60.075.0 −10.0 81.57 68.43 60.078.750 15.0 68.93 88.57 45.078.750 5.0 75.46 82.04 45.078.750 0.0 78.75 78.75 45.078.750 −10.0 85.32 72.18 45.0

Figure 5.3 Pellin-Broca prism composed of three right-angle sections.


Experiential knowledge often leads to choices that produce a better and quickersolution. Smith presents a methodology for designing mirror systems andcertain prism types that uses an iterative procedure combined with experientialknowledge.2 Smith describes the layout of a four-mirror system that projects acorrectly oriented image to a rear-view screen positioned orthogonal to the object.A solution is obtained using a minimal number of mirrors, and direction changesof each reflected ray are restricted to the same plane or to an orthogonal plane (seeFig. 5.4).

5.1.4 Matrix methods for design and analysis

Prism elements and mirrors of known geometry can be inserted into commercialoptical design programs. As components in multiple-element optical systems,they can be positioned, sized, toleranced, and corrected for certain aberrations.However, for the determination and control of prism image orientation, these ray-tracing programs are not as useful. To address this, several specialized techniqueshave been developed for the design and analysis of prisms and reflectors.

The design and analysis of mirror systems using fourth-order reflection matriceshas been described by Pegis and Rao.3 Procedural examples are given forthe design of a reversion prism with no deviation, a Wollaston prism, and aPenta prism. A matrix technique for determining the general orientation of aviewed image through a series of k planar reflectors was developed by Wallesand Hopkins.4 Here, a coordinate transformation matrix directly produces theorientation if the reflection matrices are multiplied in the same order as the lightray reflections occur.

A more general analysis of a series of planar reflective and refractive surfaceshas been described by Liao and Lin.5 The analysis considers skew rays, andby the use of fourth-order homogeneous transformation matrices, the location

Figure 5.4 A four-mirror system with reflections in orthogonal planes.2

126 Chapter 5

and orientation of a local coordinate system is calculated for each surface. Thetechnique is applied to a Pechan prism and a glass cube-corner prism. Tsai and Linhave addressed the actual design of a prism using a minimum number of reflectingand refracting surfaces by defining a merit function Γ based on the required changein image orientation.6 It is shown that the use of the following merit function:

Γ =

−1 0 00 −1 00 0 −1

, (5.3)

which specifies an image inversion along three axes, independent of prismalignment, leads directly to a solid-glass cube-corner reflector prism.

5.1.5 Evolutionary prism design using a genetic algorithm

A prototype methodology has been developed that uses a genetic algorithm (GA)for the initial conceptual design of a prism having undefined geometry. Bentleyand Wakefield have investigated GAs in the evolutionary design of optical prismswithout any prior knowledge of a specific prism type.7 Basically, a populationof possible solutions is evaluated according to a fitness criterion, and the fittestoffspring are reproduced to form an improved population. Designs that deviatefrom this criterion are penalized. The binary-coded parameter to be optimized is agenotype, and the decoded design modified by the GA is a phenotype.

In evolutionary prism design, the process often begins with a purely randominitial geometry. The light path through the prism is not specified. The directionand size of the input light and the position and direction of the desired outputare specified (see Fig. 5.5). To simplify ray tracing, the light is assumedmonochromatic, and surface reflections (other than TIR or coated surfaces) areneglected. The GA begins an iterative process of evaluation and produces newpopulations of improved designs. Design fitness is evaluated by deviations of theoutput light from the target direction and intersection at a specified plane. A basicrequirement for any evolved design is the setting of limitations on prism size.

Figure 5.5 Input and output target-ray vector directions for initial prism geometry.7


Designs that fall outside the set limits in size are penalized with poor fitness but arenot completely eliminated. The evolved prisms are composed of primitive sections,and a design where a section becomes detached from the group is called fragmentedand is given a high penalty.

Certain types of prism designs present difficulties in reaching an optimizedsolution due to the presence of easily reached local minima, called deceptiveattractors. For the design of a rhomboid prism, Fig. 5.6 shows three failed attempts,and Fig. 5.7 shows two deceptive attractors for this problem. Evolutionary designhas also been applied to right-angle, roof, and rotating Dove prisms, and to aPenta prism.7 In another approach, fixed right-angle evolved sections were furtherevolved to produce nearly perfect Abbe and Porro prisms with four internalreflections.

For the application of GAs in the design of a Fresnel lens illuminator, seeSec. 9.14.

5.1.6 A three-mirror tabletop lectern projector

For many applications, the use of a multiple-reflecting glass prism is preferable tothe use of a series of mounted planar mirrors, mainly because of the stability ofthe integral prism structure. However, for systems with larger optical beams andsizes, the use of mirrors is the only choice. The lectern projector in Fig. 5.8 hasa horizontal glass stage on which a 285 × 285-mm overhead-type transparency isprojected to an integral rear dual-focal-length projection lens.8 The screen has a

Figure 5.6 Failed attempts at rhomboid prism design using a genetic algorithm. (Adaptedfrom Ref. 7 with permission from Wiley.)

Figure 5.7 Deceptive attractors for evolutionary design of rhomboid prism. (Adapted fromRef. 7 with permission from Wiley.)

128 Chapter 5

Figure 5.8 A tabletop lectern rear-screen projector.8

size of about 450× 450 mm, and the entire lectern is approximately 1.5 m wide by1 m high and deep. A dual-focal-length projection lens projects at magnificationsof 1.6× or 2.4×. The ray path needs to be displaced and rotated by 90 deg. Sincethe information on the transparency is projected from the reverted back side, threefolding mirrors are required, taking into account the 180-deg image rotation bythe projection lens. Figure 5.9 illustrates the layout of the projection lens L1 andthe mirrors M1, M2, and M3. The mirrors can be sized by experimental or analyticmethods and are trapezoidal in shape.

5.1.7 Prism aberrations

Although prisms are composed of flat surfaces, they exhibit many of theaberrations that are usually associated with lenses, but the aberrations are relatedto plane symmetrical systems. Sasián has classified and calculated significantaberrations for a refracting prism with a plane of symmetry and a vertex angle α.9

Figures 5.10(a) and 5.10(b) show the basic ray paths. Some of these aberrationsare:

• Constant lateral chromatic (uniform over field of view)• Constant astigmatism (uniform over field of view)• Image anamorphism (varies over aperture)• Constant coma (uniform over field of view)• Linear astigmatism (varies linearly over field of view)• Field tilt (tilts the image plane)• Quadratic distortions (keystone and curvature of imaged lines).


Figure 5.9 Optical path of rear-projection tabletop lectern.8

Figure 5.10 (a) Side view of isosceles refracting prism showing optical ray axis. (b) Topview of isosceles refracting prism showing chief and marginal rays. (Adapted from Ref. 9with permission from the Optical Society of America.)

130 Chapter 5

These aberrations can be eliminated for certain specific ray paths through theprism. The mathematical description of these aberrations and their elimination isdescribed in Ref. 9.

5.2 Prism Quality Specifications

Commercial suppliers often classify prisms by grades such as student,demonstration, standard, high-tolerance, precision, and calibration quality. It isnecessary to examine the printed specifications of the supplier to determine theapplicability for its intended use.

5.2.1 Surface quality and flatness specifications

Surface quality is usually specified by the scratch and dig standard. For a scratchand dig of 60/40, the largest observable scratch width does not exceed 60hundredths of a millimeter, and the maximum measured diameter of a dig, pit, orbubble does not exceed 40 hundredths of a millimeter. Surface flatness (sometimescalled figure) is specified by the maximum wavelength change over the entiresurface at λ = 632.8 nm (HeNe laser) using a standard test plate where the spacebetween the test plate and the surface changes by λ/2 for every counted fringe. Forexample, normal surface quality and flatness for a “standard” right-angle prismcould be 60/40 scratch and dig, and λ/2 for BK7 glass. A “precision” or “high-tolerance” right-angle prism might have 20/10 scratch and dig surface quality andλ/4 surface flatness for BK7 glass or λ/10 for fused silica.

5.2.2 Optical material properties

Some of the optical requirements for prism glass are spectral transmission, clarity,color, bubbles, occlusions, stress, and striae. Stress in glass produces birefringence,and stress birefringence is usually expressed in nm/cm for a given wavelength oflight. Stress can be significantly reduced by the process of annealing, where theglass is heated to an annealing point and then slowly cooled to the strain pointtemperature, and then dropped to room temperature at a determined rate. Twobenefits of annealing glass are (1) removal of internal strains that produce stressbirefringence effects, and (2) normalizing the glass so that the refractive index isuniform throughout the material.10 The refractive index variation for glass can bereduced to about ±1 × 10−4 to ±2 × 10−5 by fine annealing. Striae are local abruptchanges in refractive index. To designate striae, glasses are often classified as A, B,C, or D grade.11 Prisms that transmit high-energy beams or beam-splitting prismsmust contain few small bubbles or inclusions. Examples of glass descriptionswould be “BK7 Grade A fine annealed glass,” or “UV-grade synthetic fused silica.”

Some of the glass physical properties that can affect optical performance andfabrication are hardness (Knoop test), grindability, viscosity, coefficient of linearexpansion, thermal conductivity, and chemical resistance.


5.2.3 Specifying angular accuracies

The required accuracy of the face angles for prisms depends on the type of prismand may vary between faces of the prism. For an Amici roof prism, angle tolerancebetween the refracting faces may be 30 arcsec, but the TIR roof angle tolerancemust be held to 90 deg ± 5 arcsec. The Penta prism is sometimes supplied with atighter tolerance < 10 arcsec for the 90-deg angle and a looser ±3-arcmin tolerancefor the other angles.

5.2.4 Tolerancing a Dove prism

The Dove prism is used as an optical component in several interferometricapplications. In particular, for a rotational shearing interferometer, a high-precisionrotating Dove prism is used to shear or rotate the wavefront. Since manycommercially available Dove prisms do not have the required accuracy forthis interferometer, Herrera and Strojnik have analyzed Dove prism tolerancerequirements for interferometric use.12 The prism geometric parameters are heightH, width W, length L, and base angle β. The glass is BK7 at λ = 633 nm, and foran effective square aperture of 19.75 × 19.75 mm, the effect of glass homogeneitywas considered to be negligible. Also, polarization changes for a Dove prism wereconsidered to be negligible. Surface flatness over the 60% surface area analyzedwas assumed to be λ/10. For this analysis, tolerance errors were considered forprism length (∆L), base angles (∆β1,∆β2), and pyramidal angles (θ1, θ2) of theentrance and exit faces [see Figs. 5.11(a) and (b)]. Base values H = W = 25.4 mmand L = 107.5 mm were chosen to duplicate dimensions of a commerciallyavailable prism.

Collimated light from a square object grid was traced through the prism, whilethe vertices defining the edges were varied to simulate manufacturing errors. Theimage of this object grid displayed the effect of these errors, where errors in βand θ, but not L, generate a critical optical path difference (OPD). MaximumOPD occurs when ∆β1 = −∆β2 or θ1 = θ2. It was determined that to achievewavefront deviations < λ/10, the base angle tolerance must be ±0.37 arcsec, andthe pyramidal angle tolerance must be held at ±0.52 arcsec. A commerciallyavailable Dove prism with an angular tolerance of ±2 arcmin would produce anOPD about twice as large as the requirement for this interferometer.

5.2.5 Techniques for prism angle measurement

Once a prism is fabricated with target angle tolerances, the prism angles shouldbe measured to verify the required accuracy. Commercial instruments such as thePrismMaster goniometer using an electronic autocollimator can measure prismangles in reflection or transmission from 0 to 360 deg with a resolution upto 0.036 arcsec.13 Various other techniques are available using angle gauges,autocollimators, and interferometers.

Rao has developed alternative noninterferometric prism angle-measurementtechniques for specific prisms.14,15 A method for measuring the error α in the 90-deg angle of a 45/90/45-deg prism is illustrated in Fig. 5.12. The diagonal face of

132 Chapter 5

Figure 5.11 (a) Dove prism showing base-angle errors ∆β1 and ∆β2 and length error∆L. (b) Dove prism showing pyramidal angle error (θ1 + θ2). (Adapted from Ref. 12 withpermission from Elsevier.)

Figure 5.12 Technique for the measurement of error in the 90-deg angle of a right-angleprism.14

the prism is first placed on a well-cleaned optical flat. An optical plate with its facesparallel to less than 0.5 arcsec is temporarily attached in an approximate verticalposition to the polished face of the test prism. Collimated light from a HeNe laserreflects from one of the parallel faces of the plate and is reflected to a verticalrear screen, where its position A is recorded with a micrometer eyepiece. The test


prism is then rotated 180 deg, and the position B of the reflected spot on the screenis measured. The error α in the right-angle is then calculated from

α =AB

4OC, (5.4)

where AB is the distance between the spots, C is the midpoint position, and OC isthe distance from the plate to the screen.

Figure 5.13 shows a procedure for measurement of the error β of the 45-degangles. The diagonal face now rests on three steel balls, and a horizontal laserbeam is incident on the test prism face. The reflected spot position A is measuredon a horizontal rear screen. The prism is then rotated 180 deg, and the reflectedspot position B is measured. The error C is then calculated from

β =AB

4OC, (5.5)

where a positive error in β for one of the 45-deg angles indicates an equal negativeerror in the other 45-deg angle. For OC = 5 m, and a measurement accuracy of0.01 mm for AB, the calculated accuracy of the prism angle measurements is about1 arcsec.

Interferometric techniques for prism angle measurements can provide anaccuracy of up to 0.1 arcsec. Nunez and Sanchez have described interferometrictechniques for measuring prism vertex angles and pyramidal error.16 An alignedTwyman–Green interferometer and test prism [Fig. 5.14(a)] produce fringe patterns

Figure 5.13 Technique for the measurement of error in the 45-deg angles of a right-angleprism.14

134 Chapter 5

that indicate the absence of pyramidal error (A, B, C) and the presence of pyramidalerror (D, E, F), as shown in Fig. 5.14(b).

The pyramidal error εp can be calculated from the following:

εp =λ

4p(n − 1), (5.6)

where

λ = wavelength of illumination,p = fringe period,n = prism refractive index.

Figure 5.14 (a) Interferometric technique for measurement of pyramidal error in a prism.(b) Interferometric fringe patterns indicating absence and presence of pyramidal error.16


The percentage error dεp of the measured pyramidal angle is the derivative ofEq. (5.6):

dεp =

[dλ

4p(n − 1)

]−

[λdp

4(n − 1)p2

]−

[λdn

4p(n − 1)2

]. (5.7)

Using stated values of n = 1.61996, dn = 0.001, λ = 632.99 nm, dλ = 2.6 × 10−6,p = 2.98 mm, and dp = 0.01 mm, the percentage error in the pyramidal angle is0.52%.15

5.3 Survey of Fabrication Methods

5.3.1 Ground and polished glass prism

The most accurate surface figure for flat surfaces is achieved by grinding withsubsequent block polishing. By this method, a surface flatness up to λ/20 can beachieved for precision applications and interferometric use.

5.3.2 Fabrication of a Penta prism by measurement of the angulardeviation error

Most prisms are fabricated and then checked for the accuracy of the prism angles.Proper application of the Penta prism requires a precise angular deviation δ =

90 deg. Chatterjee and Kumar have developed a technique for monitoring theaccuracy of the deviation angle of a Penta prism during the fabrication processso that surface adjustments can be made.17 Figure 5.15 illustrates the ray path ofa Penta prism with a collimated laser beam (HeNe at 633 nm) incident normal tothe entrance surface. The reversed ray path of a Fresnel surface-reflection ray fromthe exit surface is also shown, exaggerated for clarity. The angles adjacent to theright-angle corner angle are given by 112 deg+α1 and 112 deg+α2, the right-anglecorner angle is given by 90 deg − γ, and the angle between the reflecting faces isgiven by 45 deg + β, where the angle errors α1, α2, γ, and β can have plus or minusvalues. It is shown that

δ = 90 deg + (n − 1)γ − 2nβ, (5.8)

where n is the refractive index of the prism. The error ε in the deviation angle is

ε = δ − 90 deg = (n − 1)γ − 2nβ, (5.9)

and if n = 1.5, thenε = 0.5γ − 3β. This shows that for this index, an error in β hassix times the effect as an angle in γ. Using Snell’s law and assuming small angles,the angle of refraction I1back

′ of the back-reflected ray at the entrance surface is

I1back′ = nI1back = n(2γ − 4β), (5.10)

136 Chapter 5

Figure 5.15 Technique of monitoring the accuracy of pentaprism angles during thefabrication process. (Adapted from Ref. 17 with permission from the Optical Society ofAmerica.)

and

ε =I1back

′

2− γ. (5.11)

From Eq. (5.10), I1back′ is twice as sensitive to an error in β than an error in γ

for any refractive index, and using Eq. (5.11), the deviation angle error ε can becalculated from measured values of I1back

′ and γ. During fabrication, I1back′ and

γ are measured, and the resultant deviation ε is calculated from Eq. (5.11). Thegeneral procedure is as follows:

1. From the ground prism blank, the entrance face AB and the exit face AE areblock polished to λ/10 surface flatness.

2. Face DC is set at noncritical angles 112.5 deg ± 1.0 arcmin. Although DC is anonworking surface of the Penta prism, it is polished for measurement viewingof the prism right angle.

3. The right angle is finished to an angular accuracy of a few fractions of an arcsecond by standard techniques. The sign of the angular error γ is determinedusing interference fringe analysis of the split wavefront reflected from the rightangle (90 deg +γ) through face DC using a HeNe laser, a Fizeau interferometer,and an adjustable reference plate R. This right angle becomes a reference anglefor the correction of the error in the 90-deg deviation angle δ.

4. Face BC is block polished to good figure and reflectorized. Using a Fizeauinterferometer at the face AB, the magnitude and sign of I1back

′ is measuredfrom the two-beam interference pattern formed by the back reflections of lightfrom the entrance face AB and the exit face AE as the face ED is polished.


5. The angular deviation angle error ε is calculated. If acceptable, face ED isreflectorized. If not, the prism can be reworked and remeasured.

Measured values for a 25-mm-wide × 45.0-mm-long Penta prism were reportedat γ ≈ 0.25 arcsec and I1back

′ ≈ 2.5 arcsec, yielding a deviation angle errorε ≈ ±1.25 + 0.25 arcsec.17

5.3.3 Molded, pressed, and fire-polished prisms

Molded and pressed-glass optical prisms are being produced that provide high-angle accuracies and surface flatness. Special precision molding glasses (SchottP-series) are available having a low transformation temperature.18 Fire polishingof glass can melt the surface of certain pressed or molded prisms enough to producea smoother surface. Typical fire-polishing temperatures range from about 700 ◦Cto 760 ◦C, and the surface melt time is from 5 to 20 min.

Many mass-produced plastic prisms of smaller size are produced by injectionmolding. To suppress flow marks on the optical surfaces during filling of the cavity,the gate is usually positioned along an edge of the formed prism. Injection-moldedplastic Amici roof prisms are used as a component in range-finder cameras.19

5.3.4 Fabrication of large prisms

A very large isosceles prism has been fabricated for the High Efficiency andResolution Canterbury University Large Echelle Spectrograph (HERCULES).20

It was prepared from a 23-kg prism blank of Schott Grade A fine annealed BK7glass. The vertex angle was 49.5 deg, with a height of 276 mm, a base of 268 mm,and a length of 255 mm. A Twyman–Green interferometer was used to test glasshomogeneity and wavelength distortion. Surface figure was checked using a 200-mm-diameter Zerodur reference flat.

5.4 Some prism-mounting methods

Mounting methods for optical components are generally classified as one of threetypes: kinematic mount, semikinematic mount, or nonkinematic mount. Mountingmethods for prisms differ from that of mirrors and lenses, since prisms often haveirregular shape, may have inputs and outputs that are not coaxial, or may havereflecting faces (TIR and coated) that require protection.

A mounted prism has six possible degrees of freedom: three translational alongperpendicular axes, and three rotational about these axes. When the prism is heldin position at six points, each point uniquely assigned to one degree of freedom,the prism is independently constrained at each point and is a kinematic mount. Thecontact points should be outside the clear aperture of the prism. The prism canthen be held in position by pressure at a single point. If the prism mount is definedat more than six support points, it is overconstrained and is nonkinematic. A truekinematic mount is superior in repositioning a prism that has been removed forsome purpose such as cleaning or recoating but may induce stress birefringencedue to excessive force at the support points.

138 Chapter 5

The semikinematic mount spreads each of the six contact points over a smallarea. The area should be a raised pad lapped coplanar with respect to the prismsurface to minimize any point contact.21,22 This is the preferable type of mountfor most prism elements, since there is negligible stress induced in the prism bycontact force. The prism can be secured by a clamp at a single point.

Another type of nonkinematic mount consists of mounting the prism directlyto a planar substrate by thin adhesive layers, where the mounting face is not usedoptically. This type of mount is resistant to shock and vibration, but the element isnot easily removed or repositioned.

Positional mountings are used when the prism needs to be accurately rotatedor repositioned, especially in laboratory applications. Various types of gimbaledthree-axis prism mounts are available commercially. The design of these mountscan be mechanically complex. A rotary beamsplitter prism mount has beendesigned with three axes of rotation, where each axis operates independently ofthe other two and does not affect their positioning.23 A two-axis angular adjustmotorized mount for a right-angle prism having high sensitivity and stability hasbeen described for use in an autoboresight. The angular range is ±0.75 mrad witha resolution of 10 µrad.24

References

1. J. Rothstein, “Isosceles total internal reflectors as optical elements,” Appl. Opt.2(11), 1191–1194 (1963).

2. W. J. Smith, Modern Optical Engineering, 2nd ed., 113–115, McGraw-Hill,New York (1990).

3. E. J. Pegis and M. M. Rao, “Analysis and design of plane-mirror systems,”Appl. Opt. 2(12), 1271–1274 (1963).

4. S. Walles and R. E. Hopkins, “The orientation of the image formed by a seriesof plane mirrors,” Appl. Opt. 3(12), 1447–1452 (1964).

5. T.-T. Liao and P. D. Lin, “Analysis of optical elements with flat boundarysurfaces,” Appl. Opt. 42(7), 1191–1202 (2003).

6. C.-Y. Tsai and P. D. Lin, “Prism design based on changes in imageorientation,” Appl. Opt. 45(17), 3951–3959 (2006).

7. P. J. Bentley and J. P. Wakefield, “Conceptual evolutionary design by a geneticalgorithm,” Eng. Design Automation 2(3), 119–131 (1997).

8. D. F. Vanderwerf, “Dual-magnification rear-projection lectern,” U.S. PatentNo. 4,561,740 (1985).

9. J. M. Sasián, “Aberrations from a prism and a grating,” Appl. Opt. 39(1), 34–39(2000).

10. F. Twyman, Prism and Lens Making: A Textbook for Optical Glassworkers,505, CRC Press, Boca Raton, FL (1988).


11. “Glass, Optical,” MIL-G-174B, Defense Supply Agency, Washington, DC(1988).

12. E. G. Herrera and M. Strojnik, “Interferometric tolerance determination for aDove prism using exact ray trace,” Opt. Commun. 281, 897–905 (2008).

13. Trioptics GmbH, Wedel, Germany.

14. S. M. Rao, “Method for measurement of the angles of 90-, 45-, 45-deg and 60-, 30-, 90- deg prisms,” Opt. Eng. 36(1), 197–200 (1997)[doi:10.1117/1.601159].

15. S. M. Rao, “Methods for making prism with submultiple of half angles:applications to the measurement of the angles of Pechan and Pellin–Brocaprisms,” Opt. Eng. 41(11), 2945–2950 (2002) [doi:10.1117/1.1512660].

16. A. Jaramillo-Nunez and C. Robledo-Sanchez, “Measuring the angles andpyramidal error of high-precision prisms,” Opt. Eng. 36(10), 2868–2871(1997) [doi:10.1117/1.601516].

17. S. Chatterjee and Y. P. Kumar, “Simple technique for the fabrication of apenta prism with high-accuracy right-angle deviation,” Appl. Opt. 46(26),6520–6525 (2007).

18. Schott Glass Catalog, available online at www.schott.com.

19. K. Tanaka and I. Kasai, “Roof prism,” U.S. Patent No. 5,946,147 (1999).

20. J.B. Hearnshaw et al., “HERCULES: a high-resolution spectrograph for smallto medium-sized telescopes,” in IAU 8thAsia-Pacific Regional Meeting, 289,11–15 (2003).

21. P. R. Yoder Jr., “Attributes of the successful optic-to-mount interface,”Chapter 2 in Design and Mounting of Prisms and Small Mirrors in OpticalInstruments, SPIE Press, Bellingham, WA (1998).

22. P. R. Yoder Jr., “Optomechanical design in five easy lessons,” SPIE’soemagazine, 29–32 (February 2004).

23. D. F. Arnone and F. S. Lueke, “Rotary beamsplitter prism mount,” U.S. PatentNo. 5,694,257 (1997).

24. E. J. Stolfi, “Motorized- axis-angular fine adjustment prism mount,” U.S.Patent No. 4,722,592 (1988).

Chapter 6A Selection of PrismApplications

6.1 Laser Scanning

6.1.1 Reflective scanning prism

A rotating reflective prism in the shape of a polygon cylinder, or spinner, is a well-known scanning technique. Figure 6.1 shows a rotating polygon where a fixed-direction single laser beam partially illuminates a face. The reflected beam scanscontinuously until a facet corner is encountered, where there is a discontinuity orretrace interval between the end of the facet scan and the succeeding scan fromthe next facet. The pyramidal faceted scanner in Fig. 6.2 reduces or eliminates theretrace interval.1 A convergent beam illuminates at least two facets to producea continuous-scan interval between these facets having a circular focal tracewith constant angular velocity. Thus, two illuminated facets will produce onecontinuous-scan interval, three illuminated facets will produce two continuous-scan intervals, and illumination of an entire pyramid having n facets will producen − 1 continuous-scan intervals.

6.1.2 Refractive prism-beam scanning and steering

6.1.2.1 Single-wedge prism

A single-wedge prism with vertex angle α and deviation angle δ, when rotatedabout an axis normal to one of the surfaces, produces a circular scan on a screen

Figure 6.1 A rotating reflective polygonal scanner.

141

142 Chapter 6

Figure 6.2 A rotating reflective pyramidal faceted scanner. (Adapted from Ref. 1 withpermission from Elsevier.)

Figure 6.3 Wedge prism oriented with input face normal to incident ray (“A” orientation).

or generates a cone with half-angle δ. The exact deviation angle depends on thevertex angle, prism refractive index, and the direction of the prism with respect tothe incident beam. When the wedge prism is in “A” orientation as in Fig. 6.3, thedeviation angle δ can be calculated from

I1 = I1′ = 0, (6.1a)

I2 = α, (6.1b)

δA = I2′ − α = arcsin(n sinα) − α. (6.1c)

A Selection of Prism Applications 143

When the wedge prism is in “B” orientation as in Fig. 6.4, the deviation angle iscalculated from

I1 = α, (6.2a)

I1′ = arcsin

(sinα

n

), (6.2b)

I2 = I1 − I1′, (6.2c)

δB = I2′ = arcsin(n sin I2), (6.2d)

For analyzing wedge prisms as scanners, the small angle or paraxial approximationis often used. Then, Eq. (6.1c) reduces to δ = nα − α = (n − 1)α, and Eq. (6.2d)reduces to δ = nI2 = n(α−α/n) = (n− 1)α. Thus, the paraxial values δparax are thesame for both prism orientations. Table 6.1 lists the exact and paraxial δ values fora BK7 glass (nd = 1.5168) wedge prism using some sample vertex angles α.

6.1.2.2 Wedge prism pairs

By placing two wedge prisms with vertex angles α1 and α2 in series with acollimated laser beam incident on prism 1, the deviated ray from prism 1 is incidenton prism 2. By rotating each independently about the optical beam axis, the finalbeam deviation from prism 2 can be controlled (see Fig. 6.5). When the first prismrotates at an angular velocity ±ω1 and the second prism rotates at an angular

Figure 6.4 Wedge prism oriented with input face inclined to incident ray (“B” orientation).

Table 6.1 Exact and paraxial ray deviations for wedge prisms at sample vertex angles.

α (deg) δA (deg) δB (deg) δparax (deg)

2 1.0344 1.0339 1.0336

6 3.123 3.108 3.101

10 5.271 5.201 5.168

14 7.527 7.326 7.235

18 9.951 9.497 9.302

144 Chapter 6

Figure 6.5 Operation of the double-wedge Risley prism.

velocity ±ω2, a variety of scan patterns are produced. The prism pair is knownas a Risley prism.

Marshall has generated a series of Risley prism scan patterns using the followingprocedure:2 A refractive index n = 1.50, and the paraxial form of the ray deviationis used for each prism, such that δ1 = α1/2 and δ2 = α2/2, where δ1 and δ2are vector quantities. Then the vector addition δ = δ1 + δ2 represents the totaldeviation. The x and y components of δ are

δx = δ1 cosω1t + δ2 cos(ω2t − ϕ), (6.3a)

δy = δ1 sinω1t + δ2 sin(ω2t − ϕ), (6.3b)

where ϕ is the relative orientation (e.g., phase angle) between the two prisms. Scanpatterns were then generated for specified values of the ratios (ω2/ω1), (α2/α1),and ϕ, and plotting δx against δy. Risley prisms can produce many types of scanpatterns, including those with loops and cusps. In terms of application, probablythe most useful scan patterns are circles, lines, ellipses, and spirals for scanning anarea. Figures 6.6(a) to 6.6(d) illustrate several scan patterns.

The Risley prism as described earlier has chromatic dispersion when used inbroadband light. In many applications, especially in the infrared, it is necessaryto minimize the angular dispersion over a wide spectral range. One method is touse achromatic compound-wedge prisms (see Sec. 4.2). LiF/ZnS compound-wedgeRisley-type prisms have been designed to have minimum dispersion over the 2-to 5-µm IR region, with a maximum steering angle δmax = 45 deg.3 There aretwo possible orientations for these achromatic prism pairs. The “A” orientationhas nonparallel faces between prism 1 and prism 2, while the “B” orientationhas parallel faces between the two prisms. Both orientations produce a deviationangle δmax. If a deviation angle is desired, prism 2 can be rotated 180 deg aboutthe reference axis. However, the rotated “A” orientation of Fig. 6.7(a) producesa small angular deviation over the spectral range, or an on-axis blind spot, andexact steering to δ = 0 deg is not possible. This is unacceptable for broadband


Figure 6.6 Several types of Risley scan patterns.2

IR countermeasure steering devices. The rotated “B” orientation of Fig. 6.7(b)provides exact steering to δ = 0 deg, since both the interior faces and the incidentand exit faces are parallel, and this is the preferred orientation.3

6.1.2.3 LADAR guidance system using prism pairs

By the use of two-wedge prism pairs, where each prism is independently rotatedin one pair, and the orientation can be changed for either pair, a laser detectionand ranging system (LADAR) can be constructed.4 In Fig. 6.8(a), high-speed(≈10,000 rpm) rotating prisms 1 and 2 are oriented as shown, and fixed-positionprisms 3 and 4 are oriented as shown. A collimated laser beam is incident onprism 1. When the rotation velocityω2 of prism 2 is slightly higher than the rotationvelocity ω1 of prism 1, a spiral scanning pattern is produced. An alternative prismarrangement [Fig. 6.8(b)] has prism 2 oriented 180 deg with respect to prism 1,while prisms 3 and 4 remain fixed in position. Another spiral scanning pattern ofsmaller diameter is produced for ω2 > ω1. In Fig. 6.8(c), the orientation of prism 4can be changed to 180 deg by a rapid acceleration motor, producing a shift in thecenter of the spiral scan pattern. Thus, both the size and position of the scanningpattern can be changed.

When a missile is launched at a target aircraft, an initial large scan identifies thetarget. The scan pattern is then shifted to sweep and track the aircraft, and the scanpattern is reduced in size for more precise tracking.

146 Chapter 6

Figure 6.7 (a) Achromatic Risley prism “A” configuration with prism 1 rotated 180 deg,showing an on-axis blind spot (δ > 0 deg). (b) Achromatic Risley prism “B” configurationwith prism 1 rotated 180 deg and δ = 0 deg.

6.1.2.4 Rotating square-plate linear scanner

A simple rotating refracting square plate of refractive index n can function as alinear scanner. A varying displacement s is produced by the varying angle ofincidence I of a small-diameter collimated laser beam as the plate edges rotatearound its center. The required plate thickness in the direction of the rotation axis isdetermined by mechanical stability during rotation. In addition, the plate thicknessmust accommodate the diameter of the incident laser beam. In Fig. 6.9(a), a squareplate ABCD having sides of length d is rotated about its center. In this position, afixed laser beam passes through the center of AB and the axis of rotation. Whenthe plate rotates in a clockwise direction, the exit beam is displaced by a distances, where

s = d sin I

1 − cos I√n2 − sin2 I

. (6.4)

For the position shown in Fig. 6.9(b), the scan direction is reversed whenthe incident beam passes through corner B from face AB to face BC, and thetransmitted beam moves from one end of the scan line to the other. The cycle isrepeated as each face and corner passes through the incident beam. The maximum


(a)

(b)

(c)

Figure 6.8 (a) Scan pattern for rotating and fixed wedge-prism pairs. (b) Smaller-diameterscan pattern for reoriented rotating prism and fixed-wedge-prism pairs. (c) Shifted scanpattern for rotating and reoriented fixed-wedge-prism pairs.4

value of s occurs when I = 45 deg and ensures that the beam always exits theopposite face. The scan is repeated as the next corner C intersects the incidentbeam, as shown in Fig. 6.9(c). Then, the scan length SL = 2s and is given by

SL = 2d sin 45 deg

1 − cos(45 deg)√n2 − sin2(45 deg)

. (6.5)

148 Chapter 6

Figure 6.9 (a) Reference position of a rotating square-plate scanner. (b) Rotating square-plate scanner at bottom of scan, I = +45 deg. (c) Rotating square-plate scanner at top ofscan, I = −45 deg.


6.2 Interferometry and Spectroscopy

6.2.1 Laser interferometer with prism polarization rotator

A uniaxial birefringent prism of specified geometry can be used as a primarycomponent in a double-pass laser interferometer system. It is used in conjunctionwith a phase conjugate mirror (PCM) to produce an output beam with 90-degpolarization rotation.5 Figure 6.10 shows the basic optical arrangement and thetwo prisms used. Prism 1 is made from calcite with the principal axis oriented asshown, with angles α = 38.5 deg, β = 83.5 deg, and γ = 90 deg. Prism 2 is aright-angle retroreflecting prism.

Linearly polarized light oriented at 45 deg enters face AD of prism 1 normally,where it is separated into an o-ray and an e-ray. The calcite refractive index forthe o-ray is no = 1.6428 and ne = 1.4799 for the e-ray. The o-ray undergoes TIRat point a, and the e-ray is refracted toward prism 2. The internally reflected ray atface AB undergoes total internal retroreflection to point b on face AB and exits faceAD to the PCM, collinear with the entrance ray. The retroreflected ray from prism 2is directed to the same point b, recombines with internally reflected rays, and exitsface AD to the PCM. The reflected ray from the PCM passes again through thesystem, and the recombined rays at point a are reflected from the PBS as output.The distance between prism 1 and prism 2 is adjusted in the double pass such thatthe difference in path lengths between the o-rays and the e-rays produces a 90-degpolarization rotation in the output.

6.2.2 Polarization interferometer using a Wollaston prism

In laser interferometry, it is often necessary to resolve ambiguities in fringecounting by providing two outputs, one horizontally polarized and the othervertically polarized, each output 90 deg out-of-phase with the other (phasequadrature). Figure 6.11 illustrates a polarization interferometer that producestwo such interference pattern outputs.6 Linearly polarized light at 45 deg reflectedfrom a beam-splitting mirror and right-angle prism 1 is converted to a circularlypolarized beam by a quarter-wave 90-deg retarder. This beam combines with thereference beam reflected from right-angle prism 2 at the beam-splitting mirror.The resultant interference pattern has circularly and linearly polarized components.

Figure 6.10 A laser interferometer producing a 90-deg rotation of input polarized light.5

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Figure 6.11 Polarization interferometer using a Wollaston prism. (Adapted from Ref. 6 withpermission from Elsevier.)

A Wollaston prism separates the interference patterns of the vertical and horizontalcomponents, producing two interference patterns, one horizontally polarized andthe other vertically polarized, both of which are in phase quadrature.

6.2.3 Multipass optical cell for laser interferometer

A multipass optical cell increases the optical path length of a laser beam in acompact physical space. It is often used as a laser interferometer componentto increase the difference between interfering beams. The cube-corner reflector(CCR) can be used to advantage for this purpose. Figure 6.12(a) shows a pair ofoffset TIR CCRs that produce a series of five double passes of an input laser beamentering in the plane of the axes.7 One CCR has a flat reflectorized segment atthe vertex. The output beam exits collinear to the input beam, and the number ofreflections is limited by overlapping of beams at the flat reflectorized segment.

By the use of an additional TIR right-angle prism and two CCRs with offsetaxes, the number of passes can be increased considerably.8 Here, the input beamis not in the plane of the CCR axes. When the reflected beam leaves the CCR,the right-angle prism returns the beam to the CCR for another series of passes.


A perspective view of an arrangement where there are 16 single passes of the lightbeam is shown in Fig. 6.12(b). The output beam exits coaxial to the input beam.The number of passes will eventually be limited, even with the use of additionalprisms, when the laser beam diameters completely fill the CCR apertures.

6.2.4 Nomarski polarized-light interferometer

A Normarski polarized interferometer is useful for obtaining interferograms offast light pulses with a short coherence length. A configuration for a Nomarskipolarized interferometer (Fig. 6.13) consists of two identical Wollaston prisms.9

Linearly polarized light at a 45-deg angle enters the first Wollaston prism and isseparated (or sheared) into o-rays and e-rays at the diagonal boundary. Lens L1

Figure 6.12 (a) A pair of cube-corner reflectors providing five passes. (b) A pair of cube-corner reflectors and a right-angle prism providing 16 passes (only input, output, and right-angle prism rays are shown). (Adapted from Ref. 8 with permission from the Optical Societyof America.)

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Figure 6.13 A Nomarski-polarized interferometer using two Wollaston prisms.9

transmits collimated light through the sample S, and it is converged into the secondWollaston prism by lens L2. By control of the diagonal angle α in the prisms, theoutput light rays can be brought close to parallel. By adjustment of the distancesbetween the lenses and prisms, the output can be made convergent or divergent.

For calcite Wollaston prisms of thickness l1 = l2 = 8 mm, α = 79 deg, andcollimating lens focal lengths f1 = f2 = 50 mm, an output shear distance of ≈3.5 mm is obtained. When the two prisms are in line, changes in the values of α, l1,and l2 have negligible influence on the performance of the interferometer, wherethe optical path difference between the two output beams can be brought to lessthan the coherence length of a subnanosecond light pulse.

6.2.5 Aplanatic prism spectrograph

A double-pass aplanatic prism spectrograph is illustrated in Fig. 6.14.10 The basicconfiguration consists of a dispersing prism with spherical concave and convexsurfaces, backed by a spherical concave mirror. A series of aplanatic (no sphericalaberration or coma) refractions at the prism faces produces a series of virtual andreal images at the aplanatic conjugates for these faces. The dispersion of the prismmaterial causes small changes in the location of these secondary images, resultingin a spectrum at the focal plane close to perpendicular to the incident light. Themagnification is close to 1:1 with a throughput > 90% in the visible spectrum.

Figure 6.14 A double-pass aplanatic prism spectrograph.10


The focal plane is flat and aberration free. The detector placed at the focal planeis usually a digital camera CCD matrix array. It remains a matter of definitionwhether the dispersing element is considered to be a prism or a section of a convex-concave lens. This prism spectrograph is offered commercially as the PARRISImaging Prism Spectrometer and sold by LightForm, Inc.11

6.3 Prismatic Optical Devices

6.3.1 Prism switch for fiber-optic connections

A switching device can be constructed that uses an optical prism as a transferelement between the inputs and outputs of a series of optical fibers without aphysical disconnect and reconnect.12 In Fig. 6.15(a), there are two parallel light-beam channels, each between a set of input and output optical fibers. The beamsare formed by a molded lens element attached to the end of each fiber. Between oneset of fibers is a hexagonal prism element that does not affect the beam directionwhen positioned as shown. When the prism is translated to the position shownin Fig. 6.15(b), the input and output are switched between the two channels byrefraction and total internal reflection within the prism.

A top view of a group of eight fiber-optic channels symmetrically placed ona circle around a central fiber-optic channel is shown in Fig. 6.16. The fiber-opticlens elements are mounted in holes in two rigid plates. By means of stepper motors,the prism can be rotated and translated to align any of the peripheral outputs to thecentral input. Once calibrated, this switching can be performed remotely.

6.3.2 Laser gyro readouts

The ring-laser gyroscope (gyro) is used extensively in inertial guidance systems.The basic operation is illustrated in Fig. 6.17. A linear laser with cavity modesconsisting of two waves traveling in opposite directions is mounted in a triangularcavity configuration with three mirrors. The beam directions are designated asclockwise (CW) or counterclockwise (CCW), and the resonance condition occurswhen the cavity length equals an integral number of wavelengths. The rotation ofthe cavity produces a difference in the path length between the CW and CCWbeams. If these two beams are output through one of the mirrors, the resultantinterference pattern can provide information on the rotation rate.

The fringe patterns give the instantaneous phase difference between the CW andCCW beams, with a prism readout device that combines the beams (Fig. 6.18).13

One of the gyro mirrors transmits a small percentage of the light (<0.1%) fromthe cavity. A prism ABC of refractive index n has a vertex angle α close to, butnot exactly, 90 deg, such that α = 90 deg + α′. The divergence δ of the beamsexiting face AB after two internal reflections is δ = 2nα′. For example, whenδ = 15 arcsec and for a HeNe wavelength λ = 0.633 nm, there is a 3-mm fringespacing. By the use of a detector much smaller than this fringe spacing, the rotationrate and direction can be determined by the rate that fringe intensity maxima movepast the detector.

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Figure 6.15 (a) A two-channel optical-fiber prism switch in a neutral position.12 (b) A two-channel optical-fiber prism switch in a switched position.12

Figure 6.19 illustrates another type of prism-based readout.14 Two symmetricprisms ABCD and AB′C′D are optically bonded along AD with a 50T/50R(transmittance/reflectance) beamsplitter coating. This compound prism is opticallybonded to one of the partially transmitting mirrors of the laser gyro. Faces ABand AB′ also have partially reflecting coatings. The CW and CCW laser beamsfrom the laser gyro intersect at a common point P1 on the laser gyro mirror. Bycontrol of the prism geometry, the reflected CW and CCW beams from surfaces ABand AB′ intersect at the common point P2. The transmitted component of the CWbeam becomes collinear to the reflected component of the CCW beam, and bothare directed to detector D1. The reflected component of the CW beam becomescollinear to the transmitted component of the CCW beam, and both are directedto detector D2. Thus, each detector receives 50% CW + 50% CCW, forming an


Figure 6.16 An eight-channel optical-fiber prism switch showing a single switch position.12

Figure 6.17 Basic layout of a ring-laser gyroscope.

interference spot whose intensity is proportional to the phase difference betweenthe components. From the intensity variation at either detector, the rotation ratecan be determined. To determine the rotation direction, additional techniques arerequired.14

6.3.3 Reflecting wedge prism for optical reader

A wedge prism can be used to modify the aspect ratio of a light beam in an opticalsystem (see Sec. 4.3.4). Figure 6.20 shows the components of an optical disk readerthat uses a semiconductor laser light source.15 Elliptical cross-section light emittedfrom the semiconductor laser element (λ = 780 nm) is collimated by lens L1 and

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Figure 6.18 A type of optical prism readout device for a laser gyro. (Adapted from Ref. 13with permission from Elsevier.)

Figure 6.19 A symmetric prism readout device for a laser gyro.14

enters the top surface of reflectorized wedge prism represented in cross section byABC. Surface AB is antireflection coated, and surface AC has a high-reflectancecoating. The wedge prism of refractive index n is inclined at angle ϕ such thatthe elliptical cross-section input beam is expanded in one dimension to a circularcross-section output beam. The anamorphic expansion is W2/W1 ≈ 3:1 when theprism is inclined at an angle ϕ = 27.4 deg, n = 1.765, and α = 12.4 deg, and theoutput and input beams are orthogonal.

The circular profile beam is focused to the reading plane of an optical disk bytracking and focusing lens L2. The modulated light from the disk then passes backthrough the system, where the beam cross section is compressed by the wedgeprism. The beam is focused by lens L1 and diffracted by a holographic element toa signal detector.


Figure 6.20 An optical disk reader using a wedge prism in double-pass mode.15

6.3.4 Total-internal-reflecting touch switch using a Dove prism

The evanescent wave that penetrates a surface when TIR occurs can be the basisof a touch switch when frustrated total internal reflection (FTIR) occurs. A touchswitch that controls the on-off state of an electrical device is based on the use ofFTIR in a Dove prism.16 A series of multiple TIR reflections occur within theDove prism, where the light beam enters and exits parallel to the base of theprism [Fig. 6.21(a)]. A photodetector connected to an electronic control circuitdetermines the on-off state of the electrical device. When the top surface ofthe prism is touched by a finger, skin moisture and oils produce FTIR, and thetransmitted light intensity is reduced. Finger pressure, beam size, the number ofTIR points along the prism, and the sensitivity of the electronic control circuitaffect the on-off sensitivity.

The geometry for multipoint TIR is shown in Fig. 6.21(b). The minimum lengthL of the TIR surface of the Dove prism can be calculated from15

L =2(d − y1)

tanϕ+ 2N(d − y1) tan I + 2(n − 1)y1 tan I, (6.6)

where

d = Dove prism thickness,y1 = distance of incident beam from the bottom of the prism,ϕ = Dove prism slope angle,N = number of TIR points on the touch surface,I = internal angle of incidence at the TIR point.

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Figure 6.21 (a) Optical touch switch showing total internal reflections in a Dove prism. (b)Geometry for frustrated multipoint.16

The sensitivity and reliability of the touch switch can be increased for larger Nvalues. It can be shown from Eq. (6.6) that for a fixed value of L, N can be increasedby decreasing the thickness d of the prism, and also that for fixed L and d values,N can be increased by reducing the value of ϕ.

6.3.5 Inspection device for window surfaces

For the purpose of visual inspection and detection of surface and subsurface defectsin a pane of window glass, light from a fiber-optic light guide is coupled into thewindow pane using an attached light-guiding prism, as shown in Fig. 6.22.17 The

Figure 6.22 Light-coupling prism for window-defect inspection device.17


slope of the entrance face is ≈ 70 deg with respect to the window surface, andthe polished bottom face of the prism is coupled to the window surface using anindex-matching fluid. Light entering the glass plate undergoes TIR at both the topand bottom surfaces of the window, and the prism length is controlled to avoidreemergence of light reflected off the bottom window surface through the prism.

The presence of a window surface defect or interior defect changes the directionof the light as it escapes, with the defect appearing as a bright spot on a darkerbackground. The suppression of stray light entering the window glass allows anautomated-defect-inspection technique to be implemented.

6.4 Viewing, Display, and Illumination Systems

6.4.1 Direct-view system for a microdisplay

Compact direct-view display systems are a necessary component for binocularhead-mounted displays. A system can be designed for a reflective imager as inFig. 6.23.18 Light is collected from an LED RGB array light source by a reflector.The light then passes through a collimating Fresnel lens, rectangular aperture stop,high-transmission diffusing plate, and prepolarizer that transmits s-polarized light.

The display panel is preferably of the reflective LCOS type. An image of thedisplay panel is formed at the exit pupil by the eyepiece optics. The beam-controloptics consists of two low-birefringence glass prism sections, a 30/90/60-degbottom section and a smaller 30/90/60-deg top section to provide the light pathas shown. Faces BC and CD of the bottom prism are coated with a high-reflectancephase-optimized coating to reduce undesirable polarization phase shifts. Totalinternal reflection from prism face AB also produces skew ray depolarizationand has a phase-optimized coating. A planar polarizing beamsplitter (PBS)—preferably a wire-grid polarizer (WGP)—is placed along AD, and the two prismsections are optically bonded along AD.

Figure 6.23 Direct-view microdisplay and illumination system.18

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S-polarized light from the Fresnel lens is reflected from prism faces CD and BC,undergoes TIR at face AB, and is reflected to the LCOS (liquid crystal on silicon)imager from the WGP. When the LCOS imager is in the dark state, there is nochange in polarization state of the reflected ray, and the reflected light from thePBS does not reach the viewer. In the bright state, the reflected p-polarized lightis transmitted by the PBS and reaches the viewer, with intermediate polarizationstates providing gray scale.

6.4.2 Binocular surgical loupe with flare reduction

A binocular surgical loupe may be used for surgery or other related applications.It is worn as a headpiece with identical loupes to provide binocular vision overan illuminated field. Figure 6.24 illustrates the components of a single loupe thatprovides a well-illuminated area over the field of view, and by the use of circularlypolarized light, minimizes viewed flare from the beam-splitting optics.19

Light from a fiber-optic illuminator is collimated by lens system L1 and passesthrough a one-piece circular polarizer CP1 to a 50T/50R beam-splitting cube. Thetransmitted component is reflected from mirror M1 for overall illumination of theobject, while the reflected component is focused by the objective lens system L2for directed illumination. The viewing optics consists of the objective lens L2,the transmitting beam-splitting cube, a circular polarizer CP2, a Schmidt–Pechanprism, and an eyepiece L3. The Schmidt–Pechan prism provides an erect image andlengthens the optical path between L2 and L3 for increased image magnification.The circular polarizers suppress flare that occurs at the beam-splitting surface butdoes not affect the mostly depolarized viewed illumination reflected back from theobject.

Figure 6.24 Biological surgical loupe using beam-splitting cube and Schmidt–Pechanprism.19


6.4.3 Inversion prism for range finders

For telescopic laser range finders, it is necessary to view the object, illuminate theobject with a laser light beam, and read the distance-measurement information. Itis useful to perform these three functions along a common axis, where the readoutdisplay would be in the field of view of the eyepiece optics. In Fig. 6.25, an afocaltelescopic system provides three channels for these functions that lie close to acommon axis.20 Prism 1 is a right-angle prism that has a beam-splitting dichroiccoating along a section that is optically bonded to prism 2. A reflective coatingis applied to the uncoated section of prism 1, and prism 2 has a roof section forcorrect orientation of the image. Both the bonded section and the roof section areat about a 60-deg angle with respect to the optical axis.

In the illumination mode, light from the laser diode is reflected from three facesof prism 1 and the beam-splitting coating and does not reach the eyepiece. In theview mode, light from the objective lens is internally reflected twice by prism 1,transmitted through the beamsplitter, internally reflected three times by prism 2,and then exits along the optical axis to the eyepiece optics. In the readout mode,light from the display is reflected by the folding mirror, imaged by the display lens,reflected by the beamsplitter, internally reflected three times by prism 2, and thenexits along the optical axis to the eyepiece optics.

6.4.4 Prism transforming transmitted intensity profile

The two-section prism device in Fig. 6.26 is useful for smoothing the distributionof an incident light beam having higher intensity at the center than at the edge. Itconsists of trapezoidal prism ABCD optically bonded to right-angle prism CDEwith a 50T/50R beam-splitting coating at the interface.21 Consider a collimatedincident beam having a Gaussian distribution incident on face AE. The incident

Figure 6.25 Telescopic range-finder prism assembly with three light channels along theoptical axis.20

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Figure 6.26 Two-section prism assembly transforming intensity distribution of the inputbeam.21

distribution profile is preserved at the exit surface BE by TIR at surface AB andtransmission through the beam-splitting coating. Another component undergoesa symmetric inversion due to TIR at surface AB and reflection from the beam-splitting coating. These half-intensity components combine at the exit surface BEto form a more uniform output light beam. Figure 6.27(a) illustrates the incidentGaussian-light-beam profile, and Fig. 6.27(b) shows how the original profile andthe inverted profile combine to form a uniform output.

This procedure can be extended to a three-section prism, as in Fig. 6.28.The input is a collimated beam with a much sharper falloff. A rhomboid prismABCD is optically bonded to a trapezoidal prism CDEF, which is bonded toa right-angle prism EFG. At each bonded interface, there is a beam-splittingcoating whose reflectance and transmittance produce a nearly equal intensity ateach output section. To accomplish this, the beam-splitting coating on interfaceEF has a reflectance R = 0.5 and a transmittance T = 0.5, while the beam-splitting coating on interface DC has a reflectance R = 0.333 and a transmittanceT = 0.667. Figure 6.29 shows the incident-light-beam distribution, the internalprofile inversions, and the more uniform composite output.

Figure 6.27 Input and output light-intensity profiles for two-section prism assembly.21


Figure 6.28 Three-section prism assembly transforming intensity distribution of the inputbeam.21

Figure 6.29 Input and output light-intensity profiles for three-section prism assembly.21

References

1. L. Beiser, “Laser scanning systems,” in Laser Applications, M. Ross, Ed.,Vol. 2, 71–86, Academic Press, New York (1974).

2. G. F. Marshall, “Risley prism scan patterns,” Proc. SPIE 3787, 74–86 (1999)[doi:10.1117/12.351658].

3. B. D. Duncan et al., “Wide-angle achromatic prism beam steering forinfrared countermeasure applications,” Opt. Eng. 42, 1038–1047 (2003)[doi:10.1117/1.1556393].

4. B. Sallee and J. K. Vinson, “Optical system for ladar guidance application,”U.S. Patent No. 6,371,405 (2002).

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5. D. A. Rockwell, “Polarization rotator with frequency shifting phaseconjugate mirror and simplified interferometric output coupler,” U.S. PatentNo. 5,483,342 (1996).

6. J. C. Owens, “Laser applications in metrology and geodesy,” in LaserApplications, M. Ross, Ed., Vol. 1, 84–86, Academic Press, New York (1971).

7. V. Ya. Barash et al., “A laser displacement interferometer,” USSR InventionBrevet No. 1,275,205 (1984).

8. A. L. Vitushkin and L. F. Vitushkin, “Design of a multipass optical cell basedon the use of shifted corner cubes and right-angle prisms,” Appl. Opt. 37(1),162–165 (1998).

9. H. Yu and S. Meng, “Wollaston prism design and working parameters in theNomarski polarized light interferometer,” Opt. Eng. 35(8), 2310–2312 (1996)[doi:10.1117/1.600805].

10. D. W. Warren et al., “Compact prism spectrograph based on aplanaticprinciples,” Opt. Eng. 36(4), 1174–1182 (1997) [doi:10.1117/1.601237].

11. LightForm, Inc., 601 Route 206, Hillsborough, NJ 08844.

12. C. F. Buhrer, “Multi-position optical fiber rotary switch,” U.S. PatentNo. 5,115,481 (1992).

13. F. Aronowitz, “The laser gyro,” in Laser Applications, M. Ross, Ed., Vol. 1,139–141, Academic Press, New York (1971).

14. S. P. Callaghan, “Readout for a ring laser,” U.S. Patent No. 4,582,129 (1986).

15. K. Yoshifusa and T. Yokota, “Beam converting apparatus with a parallel lightbeam input and output from one prism plane,” U.S. Patent No. 5,007,713(1991).

16. S. Sumriddetchkajorn, “Optical touch switch based on total internalreflection,” Opt. Eng. 42(3), 787–791 (2003) [doi:10.1117/1.1544457].

17. H. Weidner et al., “Improved illumination device for inspecting windowsurfaces,” NASA Tech Brief KSC-12127, NASA John F. Kennedy SpaceCenter, FL (2000).

18. P. L. Gleckman and M. Schuck, “Compact near-eye illumination system,” U.S.Patent No. 6,542,307 (2003).

19. B. Clark, “Optical system which allows coincident viewing, illuminating andphotographing,” U.S. Patent No. 5,078,469 (1992).

20. A. Perger, “Prism system for image inversion in a visual observation beampath,” U.S. Patent No. 6,292,314 (2001).

21. K. Matsuaka et al., “Optical element having function of changing the cross-sectional intensity distribution of a light beam,” U.S. Patent No. 4,641,920(1987).

Chapter 7Projection Displays

7.1 Color-Separating and Color-Combining Prisms

7.1.1 Three-channel Philips RGB separating prism

One of the first commercial applications of trichroic separation prisms wasdeveloped by Philips Corporation for use in early color television cameras. Widelyknown as the Philips prism, it consists of three prism sections and is used today inthree-channel charge-coupled device (CCD) video camcorders. Figure 7.1 showsone configuration.1 Converging light from a camera objective lens is split intored, green, and blue components that are directed to separate photocathode orCCD receivers. The geometry must be such that the optical path lengths for eachcolor are equal and there are two reflections for each split ray to preserve imageorientation. Face 2 of prism A has a first dichroic layer that reflects green lightand transmits red and blue light. There is a small air-gap separation between face2 and face 3 of prism B. Reflected green light from face 2 is directed to the greensensor by TIR from entrance face 1. A second dichroic layer on face 4 of prismB reflects the red light to face 3, where it is directed to the red sensor by TIR.Prism C is optically bonded to prism B, and the transmitted blue light is directedto the blue sensor. The varying angle of incidence on each dichroic layer causes

Figure 7.1 A Philips prism for three-color separation.

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the transmission and reflection curves to be shifted toward shorter wavelengths.However, transmitted rays that are incident on the first dichroic layer at largerangles of incidence have smaller angles of incidence on the second dichroic layer,and vice versa. Hence, the wavelength shifts that occur in each dichroic compensateeach other. This application of the Philips prism uses a single-pass mode withunpolarized light.

7.1.2 Philips prisms in reflective LCD projection displays

The Philips prism can be used as a color-separating and combining prism inprojection displays using reflective LCD imager s. A typical projection system isshown in Fig. 7.2. The reflective light modulator is preferably of the liquid crystalon silicon (LCOS) type, which has fabrication advantages and high-resolutioncapability in a small-size imager. The pixels of LCOS spatial-light modulatorsprovide variable rotation of the polarization plane of incident polarized light, suchthat by the use of an analyzer, transmission states can be continuously variedfrom bright to dark. A PBS cube can be used as the polarizer/analyzer, andthe Philips prism provides the color separation and recombination. However, thes-p polarization rotational splitting at the dichroic surfaces of the Philips prismincreases with increasing angle of incidence on the coatings, resulting in a loss ofimage contrast (ratio of transmitted light in the bright state to the transmitted lightin the dark state) and color fidelity. This is often called skew-ray depolarization orcompound-angle depolarization.

One method to address this depolarization is by a modification the geometryof the conventional Philips prism to decrease the angles of incidence on thedichroic coatings while maintaining the TIR requirements. The need for TIR in aconventional Philips prism limits the minimum angle of incidence on the dichroicsurfaces to about 25–30 deg. Figure 7.3 illustrates a modified three-element

Figure 7.2 Typical projection system using a Philips prism.

Projection Displays 167

Figure 7.3 A modified three-element dichroic prism with a matched noncubic PBS.(Adapted from Ref. 2 with permission from the Optical Society of America.)

trichroic prism design with a matched noncubic PBS.2 For a prism refractiveindex n ≈ 1.52 and an f /4 beam, the minimum angle of incidence on thedichroic coatings is reduced to about 16 deg, resulting in improved optical systemperformance. The four-element trichroic prism in Fig. 7.4 reduces the angles ofincidence on the dichroic surfaces.3

The angle of incidence on the first dichroic surface is I1 ≈ 11 deg, and theangle of incidence at the second dichroic surface I2 ≈ 20 deg. This decreases thephase change between the s-polarized and p-polarized transmission curves, andperformance is improved. Another method is to insert wavelength-compensationplates between the input/output surfaces of a Philips prism and the reflective lightmodulators.4 These plates are positioned normal to the system’s optical axis, andthe waveplate thickness can vary from near zero up to λ/2. Polarization rotation and

Figure 7.4 A four-element trichroic prism with reduced angles of incidence on the dichroicsurfaces.3

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ellipticity are introduced at both the dichroic surfaces and the TIR surfaces, and thepolarization characteristics of each channel are analyzed by means of polarizationpupil maps. The retardation of each waveplate is then calculated to reduce thepolarization orientation and ellipticity to a minimum in the “off” or “dark” state foreach channel. By the deposition of phase-control coatings on the dichroic surfacesof the Philips prism, the depolarization of beams having finite numerical aperturecan be reduced.5,6 For a Philips prism used in double-pass projection mode, thecross-polarization leakage can be reduced to ≈ 1 × 10−3 by these phase-controlcoatings.

7.1.3 Crossed dichroic x-cube prisms for projection displays

An important type of three-channel color prism for projection displays is thecrossed dichroic prism or x-cube prism. It is more compact than an equivalent-aperture Philips prism and consists of four 45/90/45-deg sections. Section Ahas a blue reflecting (red/green transmitting) dichroic coating on face 1 and ared reflecting (blue/green transmitting) coating on face 2. Section B has a redreflecting (blue/green transmitting) dichroic coating on face 3, and section D hasa blue reflecting (red/green transmitting) coating on face 4. Section C is uncoated.The sections are bonded together using a clear optical adhesive to form a cube.Figure 7.5 shows the light paths when the x-cube prism functions as a colorintegrator. It can be used in reverse to separate white light into red, blue, and greencomponents.

The typical use of an x-cube prism and three PBS cubes is a projection systemusing LCOS light modulators, as shown in Fig. 7.6. Because of the 45-deg angleof incidence at the dichroic surfaces, there are strong polarization shifts at these

Figure 7.5 Light paths for an x-cube prism used as a color integrator.


Figure 7.6 Typical use of an x-cube prism and three PBS cubes for an LCOS projectionsystem.

surfaces that degrade projected image contrast and color fidelity. This effect can bereduced by phase-shift coating s on the dichroic surfaces or by external waveplates.Polarization-insensitive x-cube prisms are commercially available.

By the use of off-axis projection optics and the placement of the polarizers inthe light path behind the x-cube prism, depolarization effects in the x-cube prismcan be minimized (Fig. 7.7). Unpolarized light enters a beam-separation prism atan angle such that it is directed to an x-cube prism by TIR, and there are negligibledepolarization effects from the dichroic coatings of the x cube. Linear polarizersare positioned between the x-cube prism and each reflective LCD imager suchthat a dark state is achieved in the “off” position. When the imagers are activated,polarized light returns to the x-cube prism, where depolarization effects can occur.

Figure 7.7 An off-axis projection system with minimized x-cube depolarization effects.7

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The beam enters the beam-separation prism at an angle, where it is transmittedto the projection lens. The image retains a high contrast and brightness since theviewer perceives only the light intensity, not the polarization state. This off-axisarrangement can also be used to minimize the depolarization effects in a projectionsystem using a Philips prism.

7.1.4 Prisms for digital light processing projection

Digital light processing (DLP) projection systems make use of the digitalmicromirror device (DMD) from Texas Instruments. This device consists of ahigh-resolution array of hinged micromirrors with a separation ≈ 1 µm. Eachmicromirror can be electronically tilted about the mirror diagonal at an angle ofabout ±10 to ±12 deg at a rate of over 5000 times per second. By the use ofa light-directing TIR prism, reflected light can be either directed to a projectionlens (“on”state) or to a heat sink or light dump (“off” state). The concept of DLPprojection for a single micromirror is shown in Fig. 7.8. Gray scale is achievedby controlling the ratio of “on” time to “off” time for each micromirror. Light-deflecting TIR prisms for DLP projection are commercially available.

For a three-chip color DLP projection system, color-splitting and combiningprisms are used. Both Philips prisms and x-cube prisms can be used. Figure 7.9shows a section of a DLP projector using a Philips prism in double-pass mode asa color splitter and combiner. Figure 7.10 illustrates a section of a DLP projectorwhere the x-cube prism is used as a color combiner, and the color splitting is donein a separate module.8 Although DLP projectors do not require polarized light,polarization shifts between the s-polarized and p-polarized components of light atthe 45-deg incidence angle on the dichroic coatings of an x cube can degrade thebrightness and color temperature of the projected image. Chen et al. have modifiedthe x-cube dichroic coatings to reduce these polarization wavelength shifts andhave reported a projected brightness increase of about 33% and an increase in thecolor temperature of about 18% for a DLP projector.9

Figure 7.8 DLP projection scheme for a single micromirror.


Figure 7.9 A section of a DLP projector using a Philips prism as a color splitter andcombiner.

Figure 7.10 Section of a DLP projector with an x-cube as a color combiner.8

The modified crossed–dichroic color-combining prism of Fig. 7.11 consistsof four optically bonded sections: an isosceles prism section, two quadrangularsections, and a pentagonal prism section.10 The base angle β is such that 30 deg ≤β ≤ 45 deg, and the acute angle ϕ between AC and BD is such that 60 deg ≤ϕ ≤ 90 deg. Bonded faces AE and EC have a blue-reflecting dichroic coating,and bonded faces BE and ED have a red-reflecting dichroic coating. The angle ofincidence of the light beams on the dichroic surfaces can be reduced to ≈ 30 deg,reducing any depolarization effects.

There are several types of light-directing TIR prisms for DLP projection.In Fig. 7.12, a DLP projector uses a three-section crossed TIR light-directingprism. For a DMD that has three stable states—e.g., +10 deg, flat, −10 deg—the

172 Chapter 7

Figure 7.11 A modified crossed–dichroic color-combining prism.10

Figure 7.12 A DMD projector using a three-section crossed TIR light-directing prism.(Adapted from Ref. 11 with permission from The Society for Information Display.)

output of two synchronized pulsed sources can be combined.11 A three-sectionbeam-separating TIR prism for DLP projection can provide a large separationof the exiting on-state beam and the off- and flat-state beams (Fig. 7.13).12 Thisproduces minimal spillage of the off- and flat-state beams into the projectionpath of the on beam, yielding a high-contrast projected image. Light-directingTIR prisms are sold commercially, such as the LightGateTM from Unaxis or theVikuitiTM TIR prism from 3M.


Figure 7.13 A three-section TIR light-directing prism for high-contrast DLP projection.12

7.1.5 Other types of color-separating prisms for projectors

Figure 7.14 shows a projector with a color-separating and combining prismconsisting of three identical prism sections and usable with LCOS imagers.13 FaceAD of prism 1 has a dichroic coating that reflects red light and transmits blue andgreen light. Face BD has a dichroic coating that reflects green light and transmitsblue light. The three prisms are optically bonded at the contact surfaces. The LCOSimagers for red, blue, and green light are positioned as shown. The TIR surfaces ACand AB have multilayer polarization phase shift–compensating coatings to improvecontrast. Polarized light from a PBS enters the prism through surface AC, and thepolarization-modulated output is directed to a projection lens.

Figure 7.14 Projector with a color-separating/combining prism consisting of three identicalprism sections.13

174 Chapter 7

Another color-separating/combining prism in Fig. 7.15 consists of four sectionsand is usable with reflective LCD imagers.14 A blue reflecting dichroic coatingis applied to face AE of prism 1 and face CE of prism 3. A red reflecting dichroiccoating is applied to face BE of prism 2 and face DE of prism 1. The prism sectionsare optically bonded at the contact surfaces. Entrance face AD is a TIR surface forreflected blue and red rays. A blue imager is positioned opposite exit face CD, ared imager opposite exit face AB, and a green imager opposite the exit face FG ofprism 4.

A three-channel color-separating/combining system for small reflective LCDimagers (e.g., LCOS) can be designed as a compact module using QuadCubeTM

architecture.15 It consists of four PBS cubes and various spectrally selectivepolarization filters or coatings (ColorSelect R© filters).16 This quad architecture canbe configured in various arrangements to provide collinear or orthogonal outputto a projection lens. A configuration that provides collinear output is shown inFig. 7.16, where the components are separated for clarity.17 A randomly polarized

Figure 7.15 A four-section color-separating/combining prism for reflective LCD imagers.14

Figure 7.16 A quad-cube color-splitting/combining prism with polarization beam splitting.17


white-light source provides s-polarized light through a plate polarizer, which isthen incident on a green polarization filter 1 in front of PBS 1.

For the green component, selective filter 1 rotates the plane of polarization ofgreen light 90 deg, while the red and blue components are unaffected. The p-polarized green light is transmitted by the PBS 1 and passes through PBS 2 tothe green reflective LCD imager. For the bright state, the LCD imager rotates theplane of polarization 90 deg, and the s-polarized green light is reflected from PBS2 and PBS 4. A green polarization filter 4 at the exit face of PBS 4 rotates the planeof polarization 90 deg, and the p-polarized light enters the projection lens.

For the red component, the incident s-polarized light is reflected in PBS 1 andpasses through a red polarization filter 2 positioned between PBS 1 and PBS 3,where it is converted to p-polarized light and passes through PBS 3 to the redreflective LCD imager. The reflected s-polarized light from the imager is reflectedin PBS 3 and is converted to p-polarized light by a red polarization filter 3positioned between PBS 3 and PBS 4. It passes unaffected through PBS 4 andgreen polarization filter 4 to the projection lens.

For the blue component, the incident s-polarized light is reflected in PBS 1,passes through red filter B, and is reflected in PBS 3. It passes to the blue imager ass-polarized light. The reflected p-polarized light from the imager passes unaffectedthrough PBS 3, red polarization filter 3, PBS 4, and green polarization filter 4 tothe projection lens. In practice, the polarization filters can be applied to the PBSsurfaces as optical coatings, and the PBS cubes can then be optically bonded toform a single module.

7.2 Polarizing Beamsplitters for Projection Displays

7.2.1 MacNeille polarizing beamsplitters

All practical projection systems use convergent or divergent light beams,and for compact LCOS systems, low- f /# beams are required. This places aconstraint on the performance of components, especially polarizing beamsplitters.The conventional MacNeille PBS described in Sec. 3.1.10 produces skew-raydepolarization when used in the fast (≤ f /2.5) illumination beams of most LCOSprojectors. The rotation of the s-polarization and p-polarization axes produces aloss of contrast and uniformity in the projected image. One method to improvethis depolarization is by the insertion of a quarter-wave retarder between the PBSand the reflective imager. Accurate alignment of the retarder is required, and thismethod is applicable when the PBS functions both as a polarizer and an analyzer.Another compensation method has been described that inserts an oblique-aligneduniaxial material on a glass substrate (o-plate compensator) at an input surfaceof a conventional PBS.18 The material can be a birefringent liquid crystal polymer(LCP plate compensator). For an f /2.0 beam, it was calculated that < 0.1% of non-s-polarized light was transmitted over a 200-nm bandwidth for the compensatedPBS, as compared to 0.8% for an uncompensated PBS.

A three-panel LCOS projector that uses polarization-compensated MacNeille-type PBSs is described by Chen et al.19 For color management, retarded stack

176 Chapter 7

filters (RSFs) are used that orthogonally rotate the polarization over a specifiedspectral range. To minimize filter leakage from the skew-ray depolarization ofthe PBSs, they are designed to have reflection invariance, with ≈ 0.2% leakageover its color band for an f /2.5 beam. In Fig. 7.17, a projector uses four PBSs, ablue/yellow RSF (rotates polarization of transmitted first color, retains polarizationof transmitted second color), a green/magenta RSF, a magenta/green RSF, and threeLCOS imagers. There is also an optional clean-up polarizer, a half-wave plate at 0deg between PBS 1 and PBS 2 to minimize effects of polarization mixing betweenthese PBSs, and a polarization-rotating quarter-wave plate at 45 deg between PBS2 and PBS 4. It is stated that this architecture can achieve a contrast ratio exceeding1000:1.

7.2.2 Cartesian polarizing beamsplitters

The wide-angle Cartesian polarizing beamsplitter was discussed in Sec. 3.1.12.The Cartesian PBS can be used to advantage in projection systems using reflectingLCD imagers, where the input beam f /# can be reduced to ≤ f /2.5.20 Figure 7.18illustrates a three-channel LCOS projector that uses an x-cube color-combiningprism and three Cartesian polarization beam-splitting cubes. The x cube is of thetype shown in Fig. 7.5.

The multilayer polarizing beamsplitter (MLPB) film has a lower refractive indexthan the surrounding glass cube and functions as a 45-deg tilted low-index plateimmersed in a higher-index medium. Astigmatism is generated when the PBSis used in a divergent or convergent beam. This astigmatism can be reduced bymodifying the design of the PBS.21 One method is shown in Fig. 7.19(a). AnMLPB/adhesive layer having a low refractive index n1 and a glass layer havinga high refractive index n2 are sandwiched between the two halves of a glass cubehaving an intermediate refracting index n0 such that n2 > n0 > n1. The thickness d1

Figure 7.17 A three-channel LCOS projection system using compensated MacNeillepolarizing beamsplitters.19


Figure 7.18 A three-channel LCOS projection system using three Cartesian polarizingbeamsplitters.20

of the MLPB film and the thickness d2 of the high-index layer are adjusted such thatthe astigmatism caused by the high-index layer reduces the astigmatism introducedcaused by the MRPB/adhesive layer. Typical values are n0 ≈ 1.85 (PMH55 glass),n1 ≈ 1.56, n2 ≈ 1.92 (PBH71 glass), d1 = 225 µm, and d2 = 3.8 mm.

Another method of astigmatism reduction uses a transparent wedge layerbetween the 225-µm-thick MLPB film and one half of the glass cube [Fig. 7.19(b)].The wedge material is not critical and can be an optical adhesive. The wedgereduces the astigmatism caused by the MLPB film. For a PBS cube with a heightof h = 35 mm, the wedge angle α is such that 0.15 deg ≤ α ≤ 0.25 deg, and thewedge thickness w = 129 µm.

An optical engine module has been developed consisting of three astigmatism-compensated Cartesian polarizing beamsplitters and an x-cube color prism.22

When used in a prototype projector operating at f /2.0, contrast has been measuredat > 1500:1. This module is sold commercially as the 3M VikuitiTM LCOS OpticalCore.

7.2.3 Wire-grid polarizing beamsplitters in projection displays

The design principles of a wire-grid polarizer (WGP) for use in the visibleregion were discussed in Sec. 3.1.13. Planar wire-grid polarizers have foundwidespread use and are usually used at 0 or 45-deg incidence in reflective LCDprojection systems. Wire-grid polarizers do not exhibit the depolarization effectsof a conventional MacNeille PBS at f /#s approaching f /2.0. However, the wire-grid polarizer is more wavelength dependent, and astigmatism is introduced when

178 Chapter 7

Figure 7.19 (a) Astigmatism correction in a Cartesian PBS using a high-index glass plate.(b) Astigmatism correction in a Cartesian PBS using an optical wedge.21

used in the tilted mode.23 When used with an LCOS projection system at 45 degin an f /2.0 beam, polarization rotation generated in the WGP is compensatedby an opposite rotation in the WGP in the reflected beam, and no quarter-wave plate is required for improved contrast.24 Yu and Kwok have measured theoptical properties of the WGP for projection displays.25 They have measured lightefficiency of ≈ 80%, and for optimum contrast have recommended an optimal tiltangle of 35 deg instead of the conventional 45 deg.

A projection architecture using color-selective green/magenta and red/bluepolarization filters and four wire-grid polarizing beamsplitters (WGPBSs) is shownin Fig. 7.20.26 Wire-grid polarizers are components in the UltreX-3 light enginethat is used in a rear-projection system.27 A WGP at 0-deg incidence can also beused as a prepolarizer in other projectors using PBS cubes, such as the CartesianPBS system of Fig. 7.18.


Figure 7.20 A three-channel projection architecture using wire-grid polarizers.26

7.3 Illuminators for Projection Displays

7.3.1 Hollow tunnel integrators

The light pipe or tunnel integrator is often used in projection displays to provideuniform illumination from a light source to the imaging optics. One form is ahollow pipe having a constant square or rectangular cross-sectional area with areflective coating on the wall interiors. Light from a nonuniform light source (ora focused image of the source) positioned at the input aperture enters the lightpipe. Multiple kaleidoscopic reflections within the pipe produce an array of imagesof varying uniformity, as shown in Fig. 7.21.28,29 When viewed through the exitaperture, the superposition of these images becomes the effective light source,producing a homogenized output of improved uniformity at the exit aperture.

Hollow light pipes are capable of transporting light of high heat flux but mustbe assembled from four reflectorized sections. Figure 7.22 shows a constructionmethod for a hollow light pipe of quadrilateral cross section.30 The internal surfaces

Figure 7.21 Reflected-light-source images from a hollow light beam homogenizer.

180 Chapter 7

Figure 7.22 Construction of a hollow beam homogenizer with a quadrangular aperture.30

of the side and top pieces can be coated with reflecting glass or polished metal, andthe pieces can be held together by shrink-wrap tubing.

Several factors that determine the required length of a light pipe for a givenaperture or aspect ratio are (1) cross-sectional shape of the light pipe, (2) f /# ofthe incidence beam, (3) centering accuracy of the light source, (4) light-sourcedistribution pattern, and (5) light-source size. Cassarly has modeled several lightpipe configurations using the LightTools R© nonsequential ray-tracing program,taking into account these factors among others.31 In general, longer light pipesand lower f /# input beams increase the number of superposed images and providebetter uniformity.

7.3.2 Solid light pipes

Light pipes constructed from solid material have some advantages in manufactureand mechanical stability. From an optical standpoint, TIR at the interior surfaceprovides high and nondegradable reflection. However, high–heat-flux light sourcesmay affect the thermal stability, the wall may require some protection to maintainTIR, and the end faces may require an antireflection coating.

A rectangular solid light pipe has been analyzed by Chang et al.32 Scramblingwithin the light pipe produces an output to a liquid-crystal light valve with theapproximate cross-sectional shape of the light pipe. For a 4:3 aspect ratio light-pipe cross section, an optical efficiency of 57% and light uniformity of 70% wascalculated.

7.3.3 Effect of light-pipe cross section on uniformity

The cross section of a light pipe can have a number of shapes. It has been foundthat when the reflected images of the entrance aperture completely fill the reflectionplane, good uniformity can be achieved. The reflected cross-sectional images forthe light pipe of Fig. 7.21 fulfill this condition. Hexagonal and equilateral triangleshapes can also be close-packed to fill the reflection plane and will produce gooduniformity. However, round and pentagon-shaped cross sections do not completelyfill the reflection plane and will produce poor uniformity, which has been verifiedby ray tracing.31 In particular, the circular-cross-section light pipe produces acentered hot spot.


7.3.4 Solid microprismatic light homogenizer

Figure 7.23 illustrates a solid cylindric light-beam homogenizer that operates incollimated light.33 A series of circular microprismatic grooves in the shape ofisosceles triangles are formed on the input and output surfaces having a diameterD. The grooves are isosceles triangles with a base angle β. An incident collimatedbeam is refracted at internal deviation angles ±δ, which are totally internallyreflected at the circular walls. The cylinder length L is adjusted such that raysincident on the perimeter of the entrance surface are refracted to the center of theexit surface, and rays incident near the center of the entrance surface are refractedto the perimeter of the exit surface.

The angle of incidence I1 = β at the entrance surface, and δ = I1−I1′ = β−I1

′. Tomaximize δ and prevent a refracted ray from hitting an adjacent facet, the followingcondition is required:

β =

(90 deg + I1

′)

2. (7.1)

Equation (7.1) is solved iteratively for β, and the length of the homogenizeris calculated from L = D/(2 tan δ). The rays reflected off of the walls arecollimated at the exit aperture. The homogenizer is achromatic, since dispersiongenerated at the input surface is canceled by opposite dispersion at the outputsurface. Figure 7.24(a) models a distribution of an input beam, and Fig. 7.24(b)indicates the homogenized output. For BK7 glass material (nd = 1.5168),calculated parameters are β = 62.98 deg, δ = 27.01 deg, D = 25 mm, and L =24.52 mm.

Figure 7.23 A solid cylinder light-beam homogenizer with microprismatic grooves on inputand output apertures.33

182 Chapter 7

Figure 7.24 (a) Modeled light distribution of input beam for microprismatic light-beam homogenizer. (b) Light distribution of output beam for microprismatic light-beamhomogenizer.


7.3.5 Tapered-tunnel illuminator for projection displays

If the input aperture of a hollow light pipe is illuminated with a divergent orconvergent circular light beam having a maximum half-angle θ, then the opticalthroughput, or étendue E, is defined by the geometric quantity:

E = πA sin2θ =

πA

4 ( f /# )2 [mm2 -steradian], (7.2)

where A = area of aperture (mm2), and the f /# = 1/(2 sin θ) = 1/(2NA),where NA is the numerical aperture. Thus, for an f /2.0 beam, θ = 14.5 deg andNA = 0.25, and for an f /1.0 beam, θ = 30.0 deg and NA = 0.50.

A hollow illuminator for a projection system has been described where thereflecting walls have the shape of a compound parabolic reflector (CPR) withdefined input and output aperture dimensions.34 The maximum value of thedivergence half-angle θo at the output aperture can be related to the maximuminput divergence half-angle θi, where θi � θo, and the étendue can be preservedfrom the input aperture to the output aperture with maximum collection efficiencyby the following condition:

Ai sin2θi = Ao sin2

θo. (7.3)

In Fig. 7.25, the maximum angle output ray angle θo at the upper-left edgeoriginates at the bottom-right edge of the opposite face.

Figure 7.25 Hollow illuminator having the shape of a compound parabolic concentrator.34

184 Chapter 7

For rectangular apertures, the condition for étendue preservation can be furtherrefined by the following:

aibi sin θia sin θi

b = aobo sin θoa sin θo

b, (7.4)

where

ai, ao = heights of input and output rectangular apertures,bi, bo = widths of input and output rectangular apertures,θi

a, θoa = input and output half-angles along rectangle heights,

θib, θo

b = input and output half-angles along rectangle widths.

A CPR illuminator for square apertures is shown in Fig. 7.26. For an f /2.0 outputbeam, θo

a = θob ≈ ±15 deg.

Figure 7.26 A CPR illuminator for a projection display with a square output aperture.34

References

1. L. P. G. Verdijk and E. Tienkamp, “Color-separating prism arrangementof which some surfaces adjoin dichroic layers,” U.S. Patent No. 4,009,941(1977).

2. H.-S. Kwok et al., “Trichroic prism assembly for separating and recombiningcolors in a compact projection system,” Appl. Opt. 39(1), 168–172 (2000).

3. J. W. Bowron, “Four prism color management system for projection systems,”U.S. Patent No. 6,644,813 (2003).

4. B. Bryars, “Systems, methods and apparatus for improving the contrast ratio inreflective imaging systems utilizing color splitters,” U.S. Patent No. 5,986,815(1999).

5. M. R. Greenberg and B. J. Bryars, “Skew ray compensated color separationprism for projection display applications,” SID Digest 31(1), 88–91 (2000).


6. A. E. Rosenbluth et al., “Design of phase-controlled coatings to correctskew–ray depolarization in LCOS projectors,” Displays 23(3), 121–138(2002).

7. B. J. Bryars, “Off-axis projection display system,” U.S. Patent No. 6,398,364(2002).

8. J. Huang, “Digital light processing projector,” U.S. Patent No. 7,144,116(2006).

9. H.-C. Chen et al., “Improving the illumination efficiency and color temperaturefor a projection system by depositing thin-film coatings on an x-cube prism,”Opt. Eng. 45(11), 113801 (2006) [doi:10.1117/1.2393154].

10. F.-C. Ho and J.-J. Huang, “Optical prism assembly,” U.S. Patent No. 7,224,531(2007).

11. D. Maes, “Dual TIR prism, a way to boost the performance of a DLPTM

projector,” Barco, Society of Information Display MEC Spring Meeting, Jena,Germany (March, 2008).

12. S. M. Penn, “Prism for high contrast projection,” U.S. Patent No. 6,959,990(2005).

13. J. Huang, “Color separation prism assembly compensated for contrastenhancement and implemented as reflective imager,” U.S. Patent No.6,704,144 (2004).

14. A. L. Huang, “Color-separating prism utilizing reflective filters and totalinternal reflection,” U.S. Patent No. 6,517,209 (2003).

15. M. G. Robinson et al., “High contrast color splitting architecture using colorpolarization filters,” SID Symp. Digest 31, 92–95 (2000).

16. G. Sharp et al., “LCOS projection color management using retarder stacktechnology,” Displays 23, 139–144 (2002).

17. T. Suzuki, “Color-separating and recombining optical system,” U.S. Patent No.6,984,041 (2006).

18. M. G. Robinson et al., “Wide field of view compensation technique for cubepolarizing beam splitters,” SID Symp. Digest 34, 874–877 (2003).

19. J. Chen et al., “High contrast MacNeille PBS based LCOS projection systems,”Proc. SPIE 5740, 78–91 (2005).

20. C. L. Bruzzone et al., “Reflective LCD projection system using wide-angleCartesian polarizing beam splitter,” U.S. Patent No. 6,486,997 (2002).

21. D. J. Aastuen et al., “Projection system having low astigmatism,” U.S. PatentNo. 6,786,604 (2004).

22. C. L. Bruzzone et al., “High-performance LCOS optical engine usingCartesian polarizer technology,” SID Symp. Digest 34, 126–129 (2003).

186 Chapter 7

23. S. Arnold et al., “An improved polarizing beamsplitter LCOS projectiondisplay based on wire-grid polarizer,” SID Symp. Digest 32, 1282–1285(2001).

24. E. Gardner and D. Hansen, “An image quality wire-grid polarizing beamsplitter,” SID Symp. Digest 34, 62–63 (2003).

25. X.-J. Yu and H.-S. Kwok, “Application of wire-grid polarizers to projectiondisplays,” Appl. Opt. 42(31), 6335–6341 (2003).

26. A. J. S. M. De Vaan and S. C. McClain, “Projection device with wire gridpolarizers,” U.S. Patent No. 6,873,469 (2005).

27. C. Pentico et al., “Ultra high contrast color management system for projectiondisplays,” SID Symp. Digest 34, 130–133 (2003).

28. M. M. Chen et al., “The use of a kaleidoscope to obtain uniform flux over alarge area in a solar or arc imaging furnace,” Appl. Opt. 2(3), 265–271 (1963).

29. L. J. Krolak and D. J. Parker, “The optical tunnel—a versatile electroopticaltool,” J. SMPTE 72, 177–180 (1963).

30. B. Wagner, “Method of producing a hollow mixing rod, and a mixing rod,”U.S. Patent No. 6,625,380 (2003).

31. W. J. Cassarly, “Design of efficient illumination systems,” Optical ResearchAssociates, SPIE Short Course Notes, SPIE 44th Annual Meeting, Denver, CO(1999).

32. C.-M. Chang et al., “A uniform rectangular illuminating optical system forliquid crystal light valve projectors,” SID Euro Display ’96, Birmingham,England, 258–260 (1996).

33. D. F. Vanderwerf and A. J. Herbert, “Prismatic light beam homogenizer forprojection displays,” U.S. Patent No. 6,024,452 (2000).

34. A. L. Duwaer and J. F. Goldenberg, “Light valve projection system withimproved illumination,” U.S. Patent No. 5,146,248 (1992).

Chapter 8Microprismatic Arrays

A microprismatic array is a series of prismatic elements that are positioned nextto each other in a 2D pattern or, in some cases, a 3D pattern. These prismaticelements can be identical or vary in geometry and have smaller dimensions thanconventional single prisms. They are often replicated in optical plastic and can bea rigid sheet or a flexible film, with the sheet area much greater than the thickness.These prismatic sheets or films can subsequently be shaped into other forms, suchas a rectangle or cylinder, for specialty applications.

8.1 Roof Prism Linear Array

The linear array in Fig. 8.1 consists of a flat sheet of repeating isosceles roof prisms,having a vertex angle ϕ. The prismatic array has a refractive index n, surroundedby air with refractive index n0. For an individual prism, an incident light-ray vectorwith direction cosines Kx1, Ky1, Kz1 is incident and refracted at planar surface 1.The refraction matrix R1 is derived from Eq. (1.75), where the surface normals arekx1 = 0, ky1 = 0, and kz1 = −1, referenced to the coordinate system shown, where

R1 =

1 0 0 00 n0/n 0 00 0 n0/n 0

kz1[ρ1′ − (n0/n)ρ1] 0 0 n0/n

, (8.1)

where

ρ1 = Kx1kx1 + Ky1ky1 + Kz1kz1 = cos I,

and

ρ1′ =

√[1 − (n0/n) sin I]2 = cos I′.

The direction cosines of the surface normals for surface 2 are kx2 = 0, ky2 =

cos[(180 deg − ϕ)/2], kz2 = −cos[(180 deg − ϕ)/2], and for surface 3 are kx3 = 0,ky3 = −cos[(180 deg − ϕ)/2], kz3 = −cos[(180 deg − ϕ)/2].

A perspective view of an individual prism of an array, with apex angle ϕ,and several possible paths of an incident ray are shown in Fig. 8.2. Ray A′

187

188 Chapter 8

Figure 8.1 Section of a microprismatic array.

Figure 8.2 Possible light-ray paths in a microprismatic array.

enters planar surface 1, normally at a 0-deg incident angle, undergoes TIR atsurfaces 2 and 3, and is retroreflected through surface 1. Ray A enters planarsurface 1 at an oblique angle, undergoes TIR at surfaces 2 and 3, and is refractedupward at surface 1. Ray B is refracted at surface 1, undergoes TIR at surface 2,is refracted at surface 3, and is transmitted downward. Ray C is refracted atsurface 1, directly refracted outward at surface 3, and transmitted downward. BothB and C ray paths are possible up to near-grazing incidence angle I1 ≈ 90 deg.Alternatively, rays incident from opposite directions could be reflected at surface 3and refracted outward through surface 2, or directly refracted outward throughsurface 2. Therefore, to calculate the spread of possible exit angles for an incidentlight beam, both the refractive and reflective matrices are required for surfaces 2and 3.

Microprismatic Arrays 189

From Eq. (1.76), the fourth-order reflection matrices R2′ and R3

′ are:

R2′ =

1 0 0 00 1 0 0

−2ky2ρ2 0 1 0−2kz2ρ2 0 0 1

, (8.2)

R3′ =

1 0 0 00 1 0 0

−2ky3ρ3 0 1 0−2kz3ρ3 0 0 1

, (8.3)

and the refraction matrices R2, R3, and R4 are

R2 =

1 0 0 00 n/n0 0 0

ky3[ρ2′ − (n/n0)ρ2] 0 n/n0 0

kz3[ρ2′ − (n/n0)ρ2] 0 0 n/n0

, (8.4)

R3 =

1 0 0 00 n/n0 0 0

ky3[ρ3′ − (n/n0)ρ3] 0 n/n0 0

kz3[ρ3′ − (n/n0)ρ3] 0 0 n/n0

, (8.5)

R4 =

1 0 0 00 n/n0 0 00 0 n/n0 0

kz4[ρ4′ − (n/n0)ρ4] 0 0 n/n0

, (8.6)

The appropriate system matrices S are given by

Reflected rays A,A′ : SA = R4 R3′ R2

′R1, (8.7a)

Transmitted ray B: SB = R3 R2′R1, (8.7b)

Transmitted ray C: SC = R3 R1. (8.7c)

The direction cosines of the exiting rays, Kx′, Ky

′, and Kz′, can then be

calculated from the direction cosines of the incident rays, Kx, Ky, Kz, by a matrixmultiplication similar to Eq. (1.65), such that

1

Kx′

Ky′

Kz′

= S

1

KxKyKz

. (8.8)

190 Chapter 8

If the vertex angle ϕ exceeds a certain value, then incident normal rays canbe directly refracted through either surface 2 or surface 3. This happens whenthe angle of incidence I at surfaces 2 and 3 is less than the critical angle Icrit,as in Fig. 8.2. This occurs when ϕ > 180 deg − 2Icrit, e.g., ϕ ≥ 96 deg forn = 1.5. When either ρ2 > cos I2crit or ρ3 > cos I3crit, there is TIR failure, andlight is refracted through the array. The range of incident angles I1 can then becalculated that define the boundary regions in which reflection or transmissionoccur. Figure 8.3 illustrates a hemispheric model1,2 in which an oblique rayincident at point (x1, y1, z1) of planar surface 1 passes through the hemisphericsurface having a radius R0. The intersection coordinates at the hemispheric surface(x0, y0, z0) are given by x0 = R0 sin I1 sinω, y0 = R0 sin I1 cosω, and z0 = R0 cos I1,where R0, I1, and ω are defined as shown.

If x1 = y1 = z1 = 0, then the direction cosines for an incident oblique ray arecalculated from

Kx = (x1 − x0)/R0 = −sin I1 sinω, (8.9a)

Ky = (y1 − y0)/R0 = −sin I1 cosω, (8.9b)

Kz = (z1 − z0)/R0 = −cos I1. (8.9c)

If n = 1.5, then ρcrit = 1/n = 0.6667, or Icrit = 41.81 deg.The most frequent uses for a roof prism array have ϕ = 90 deg. The normal

direction cosines for the reflecting facets 2 and 3 are given by kx2 = 0, ky2 =

cos(45 deg), kz2 = −cos(45 deg), and kx3 = 0, ky3 = −cos(45 deg), kz3 =

−cos(45 deg). For the case where a ray undergoes TIR at both facets, the directioncosines of the exit ray at planar surface 4 (same as 1), are calculated directly fromEq. (8.8), using SA.

Figure 8.3 Transmission and reflection regions of microprismatic array using a modelhemisphere.1


Table 8.1 gives the resultant exit ray directions for various incident angles inthe reflection region. TIR occurs where I2 and I3 both exceed, or are close to, thecritical angle. Retroreflection occurs when I1 ≤ 4.7 deg and ω = 0 deg. Specular-type reflection occurs for any value of I1 whenω = 90 deg, I1 = I4, and the entranceand exit angles are in the same plane. Skew reflections occur at other I1 and ωvalues, where I1 = I4, but the exit plane is rotated relative to the incident plane.The resultant line of retroreflection and the boundaries between the reflection andtransmission regions are as shown in Fig. 8.4. The reflection/transmission boundaryis determined by the I1 and ω values for which I3 = Icrit. Rays with a high exitangle, as in ray path B of Fig. 8.2, may not clear the adjacent groove in the arrayand can be refracted out through the input surface. To fully evaluate these types ofrays in prismatic structures, the technique of nonsequential ray tracing is useful.Several commercial nonsequential ray-tracing optical programs are listed at theend of the references. For light that is directly transmitted by refraction, as in raypath C, the range of output angles I5, measured relative to the z axis, is reducedfrom the range of input angles over a large portion of the transmission region.This is referred to as normalized transmission, where the output angles are broughtcloser to the array normal. This occurs when I1 ≥ 14.8 deg for various values ofω. Table 8.2 gives some representative input and exit angles in the transmissionregion. The nonnormalized/normalized transmission boundary is determined forthe I1 and ω values for which I5 ≈ I1.

Figure 8.4 Square linear prismatic light guide with light source centered on input aperture.(Adapted from Refs. 3 and 4 with permission from the author and the Optical Society ofAmerica.)

192 Chapter 8

Tab

le8.

190

-deg

roof

-pris

mar

ray

ina

refle

ctio

nre

gion

(n=

1.5,

I cri

t=

41.8

1de

g,θ

max

=27.6

deg)

.

Ray

path

I 1E

ntra

nce

angl

e(d

eg)

ω θ(d

eg)

Kx

Ky

Kz

I 2(d

eg)

I 3(d

eg)

Kx′

Ky′

Kz′

I 5E

xit

angl

e(d

eg)

x o y o z o (R0=

100

mm

)A′

00

00

4545

00

00

090

00

00

00

Ret

rore

flect

ion

−1

01

010

00

A′

478

00

048

1841

82re

flect

ion/

tran

smis

sion

boun

dary

00

478

00

90−

008

330

0833

833

3R

etro

refle

ctio

n−

099

650

9965

9865

A6

9845

−0

0859

4837

4182

refle

ctio

n/tr

ansm

issi

onbo

unda

ry

−0

0859

698

859

385

1−

008

590

0859

859

3Sk

ewre

flect

ion

−0

9926

099

2698

26

A8

955

−0

1267

4859

4182

refle

ctio

n/tr

ansm

issi

onbo

unda

ry

−0

1267

89

1267

827

−0

0887

008

878

874

Skew

refle

ctio

n−

098

800

9881

9880

A13

565

−0

2116

4931

4184

refle

ctio

n/tr

ansm

issi

onbo

unda

ry

−0

2116

135

2116

778

−0

0987

009

878

866

Skew

refle

ctio

n−

097

240

9724

9724

A45

686

−0

6584

5987

4182

refle

ctio

n/tr

ansm

issi

onbo

unda

ry

−0

6584

4565

8448

8−

025

800

2580

2580

Skew

refle

ctio

n−

070

710

7071

7071

A63

865

−0

8132

6749

4181

refle

ctio

n/tr

ansm

issi

onbo

unda

ry

−0

8132

638

8132

356

−0

3792

037

9237

92Sk

ewre

flect

ion

−0

4415

044

1544

15

A88

962

45−

088

6672

041

83re

flect

ion/

tran

smis

sion

boun

dary

−0

8866

889

8866

276

−0

4625

046

2546

25Sk

ewre

flect

ion

atθ

max

−0

0018

000

180

1745

A45

90−

070

7151

4251

42−

070

7145

7071

450

00

00

0Sp

ecul

ar-t

ype

refle

ctio

n−

070

710

7071

7071


Tab

le8.

290

-deg

roof

-pris

mar

ray

ina

tran

smis

sion

regi

on(n

=1.

5,I c

rit=

41.8

1de

g,θ

max

=27.6

deg)

.

Ray

path

I 1 Ent

ranc

ean

gle

(deg

)

ω θ(d

eg)

Kx

Ky

Kz

I 2(d

eg)

I 3(d

eg)

Kx′

Ky′

Kz′

I 5 Exi

tang

lefr

omz-

axis

(deg

)

x o y o z o (R0=

100

mm

)B

100

00

5164

−38

360

0−

6644

tran

smis

sion

bloc

kage

00

90−

017

36T

IRtr

ansm

issi

on−

091

6717

36−

098

480

3997

9848

C10

00

0m

isse

s38

350

023

55no

nnor

mal

ized

tran

smis

sion

00

90−

017

360

3996

1736

−0

9848

−0

9167

9848

C14

80

00

mis

ses

3519

00

148

norm

aliz

edtr

ansm

issi

onbo

unda

ry

00

90−

025

540

2559

2554

−0

9668

−0

9668

9668

C21

845

−0

2626

mis

ses

360

−0

2626

218

norm

aliz

edtr

ansm

issi

onbo

unda

ry

2626

748

−0

2626

026

2126

26−

092

85−

092

8692

85

B45

00

073

13−

1688

00

−71

11tr

ansm

issi

onbl

ocka

ge

00

90−

070

71T

IR−

094

4470

71−

070

71−

032

8770

71

(con

tinue

don

next

page

)

194 Chapter 8

Tab

le8.

2(c

ontin

ued

)

Ray

path

I 1 Ent

ranc

ean

gle

(deg

)

ω θ(d

eg)

Kx

Ky

Kz

I 2(d

eg)

I 3(d

eg)

Kx′

Ky′

Kz′

I 5 Exi

tang

lefr

omz-

axis

(deg

)

x o y o z o (R0=

100

mm

)C

450

00

mis

ses

1688

00

1920

norm

aliz

edtr

ansm

issi

on

00

90−

070

71−

032

8770

71−

070

71−

094

4470

71C

4564

2−

063

66m

isse

s39

37−

063

6645 no

rmal

ized

boun

dary

6366

505

−0

3078

030

8230

78−

070

71−

070

6970

71B

850

00

8662

−3

380

0−

5008

tran

smis

sion

bloc

kage

00

90−

099

62T

IR0

6641

9962

−0

0872

074

768

716

C85

00

0m

isse

s3

380

038

92no

rmal

ized

tran

smis

sion

00

90−

099

62−

064

1799

62−

008

72−

076

698

716

B88

9gr

azin

g62

4−

088

6272

02−

4180

refle

ctio

n/tr

ansm

issi

onbo

unda

ry

−0

8862

−70

1488

6227

6−

046

33T

IR−

034

2746

33−

000

180

3121

017

45C

889

graz

ing

624

−0

8862

mis

ses

4180

refle

ctio

n/tr

ansm

issi

onbo

unda

ry

−0

8862

7014

norm

aliz

edtr

ansm

issi

on

8862

276

−0

4633

031

5046

33−

000

17−

033

970

1745


The (x0, y0, z0) values on the hemisphere defining these boundary regions arecalculated using Eqs. (8.9a) to (8.9c) and a reasonable value of R0. Figure 8.3shows the resulting regions on the model hemisphere. Here θmax indicates themaximum ray angle θ, relative to the groove direction (x axis), for which a raywill be reflected by double TIR for any angle of incidence I1 from 0 to 90 deg. Ingeneral, θ = arccos(sin I sinω). However, there are angles of incidence < 90 degfor which θ > θmax. Tables 8.1 and 8.2 give representative data for the reflectionand transmission regions. For an extended array, both the reflection region and thetransmission region are useful for several types of optical illumination devices.The prismatic array can be a rigid sheet (about 4–6 mm thick), where the prismatictriangular grooves are precisely replicated in optical plastic with a typical periodwidth of about 3 mm.

In another type of array, a flexible film (about 0.5 mm thick) can be producedwhere the prismatic triangular grooves have a typical period width of about0.25 mm. The optical plastic is usually acrylic, with nd = 1.4918, or polycarbonate,with nd = 1.5855. Polycarbonate is more impact resistant, but acrylic has a lowerdispersion. The flexible prismatic film is produced and sold by the 3M Companyas 3M optical lighting film (OLF) in several configurations.

8.2 Square Prismatic Hollow Light Guide

One of the first applications of a 90-deg prismatic array sheet in the reflectionregion was for a prismatic hollow light guide (also called a light conduit or lightpipe). For the configuration in Fig. 8.4, four identical rigid sections of the array arearranged to form a square hollow structure, with the grooves facing outward andrunning along the guide.3 The sections can be molded from acrylic plastic, typically1 to 10 mm thick. A light source with a controlled beam spread is positioned at oneend, and the light rays are transported down the guide by total internal reflectionfrom the prismatic surfaces. The maximum beam half-angle θmax, as calculated byWhitehead et al.3,4 is

θmax = arccos

√1 − n2 sin2(22.5 deg)

1 − sin2(22.5 deg). (8.10)

For n = 1.5, θmax = 27.6 deg, corresponding to an f /1.08 beam.Light guides of this type are normally used to efficiently transport light from

the input end of the guide to the exit aperture of the guide, or to provide extendedillumination through an area along the length of the guide. In the first case, highreflectance must be maintained with minimum light loss through the walls. In thesecond case, the light must be efficiently transmitted through the guide wall overthe desired illumination area. The transport efficiency of a light guide depends onseveral factors, such as

• Imperfections of the array surfaces,• Dielectric absorption loss in the array material,

196 Chapter 8

• Fidelity of the prism geometry—e.g., peak sharpness and surface flatness,• Dust and contamination on the optical surfaces.

Considering these factors, calculation of the expected light throughput of a lightguide is an approximation at best. In general, for maximum transport of light, thebase length of the microprisms should be much greater than the distance of themicroprism base to the inner planar surface, and the cross-sectional area of thefour walls should be much less than the light-guide cross section. Also, a practicallight intensity for transport through the light guide is on the order of 10 W/cm2.In Fig. 8.5, a light transporter with an input beam at one end of the guide producesoutput at the other end. The light guide can also be encased in a reflective sheath,such that any escaping light can be redirected back through the wall to increase thetransport efficiency.

The second case provides more practical applications for illumination devices.Normal light leakage through the walls causes the light guide to glow. To increasethe transmitted light over a specified area for use as an illuminator, an extractor isused, where the extractor is any element or modification that allows light to escapefrom the guide that would normally be reflectively contained inside. It can be assimple as a tapered cutout or an array of patterned holes along a single wall. Theproblem then becomes how to maximize the extraction of light along a selectedlinear area on the guide, with close to uniform intensity along the illuminatedlength. One method is to place diffusing screens on the wall interior. This scatters

Figure 8.5 A high-throughput prismatic light guide.


the light transmitted to the prismatic wall, and light is then transmitted. Severalother methods to extract light are rounding the peaks of the prismatic grooves,curving the prism facets, modifying the prism vertex angle, or increasing thedivergence half-angle θ of the light in the guide, such that θ > θmax. The divergenceangle θ can be increased by adjusting the light-source module angular output,adding interior diffusion screens perpendicular to the guide axis, or placing aconvex mirror at the end of the guide.5 Figure 8.6 illustrates a typical configurationfor a square light guide luminaire, with a controlled input light module at one end,a tilted reflector at the other end, planar diffuse reflectors surrounding three wallsto redirect transmitted light back into the guide, and a diffusing plate along thebottom wall to even out the extracted light.

8.3 Circular Prismatic Hollow Light Guide

Circular prismatic hollow light guides can also be constructed using flexible 3Moptical lighting film, usually formed from acrylic or polycarbonate material. Theselight guides (sometimes called light tubes) have certain advantages over the squareprismatic light guide. For the square light guide of Fig. 8.4, rays from a centeredpoint source are limited in reflection by the angular divergence θmax. For example,an oblique light ray hitting the wall at I1 = 45 deg, θ = 55 deg, ω = 54.2 deg(n = 1.5), at the position shown, will be transmitted through the prismatic wallby TIR failure. For the circular prismatic light guide in Fig. 8.7, all rays froma centered point source produce an angle of incidence I1 that lies in a planewith the surface normal. All reflected rays then remain in the reflection regionand undergo specular-type reflection, where ω = 90 deg and cos θ = sin I1.Since the angle of incidence can vary from I1 ≈ 0 deg at the input end of theguide to I1 ≈ 90 deg at the end of an infinitely long tube, θmax can approach 90deg at the input, with little light loss through the wall. The angles of incidence

Figure 8.6 A square prismatic light guide luminaire.

198 Chapter 8

Figure 8.7 A circular prismatic light guide with sample light-ray paths.6

remain constant for subsequent reflections, significantly increasing the transportefficiency of the guide. However, for a high-divergence-angle incident beam, thenumber of interior reflections increases over the length of the tube, producing moreattenuation through the dielectric material. In practice, the input light consists of afinite-size source with an auxiliary reflector, e.g., parabolic or elliptical, smoothor segmented. To ensure that the majority of rays are internally reflected, thedivergence half-angle is often controlled to be about 30 deg, even for a circularlight guide.

For a flexible prismatic film that is curled into a cylinder, there is a minimumpractical diameter Dmin that can be formed. For a film having a thicknessT measured from the smooth side to the prism vertex, this diameter can beapproximated as Dmin ≈ CT , where C is a constant related to the modulus ofelasticity. For example, a 0.4-mm-thick acrylic microprismatic film having about3 prisms per millimeter can be formed into a cylinder with a minimum diameter ofabout 80 mm. This film can be self-supporting up to a diameter of about 450 mm.Film curvature also affects the reflected angle from total internal reflection at theprisms. The change in the reflection angle, compared to a planar array, is estimatedto be about ±1.3 deg.6

If maximum light transport is desired, the prismatic grooves must have a precisegeometry, and the dielectric must be of the highest clarity to minimize internalscattering. The transported light flux depends on the type of light source and anyauxiliary reflector and on the material of the guide. Figure 8.8 plots the amountof light transported down tubes of varying length-to-diameter ratios (sometimescalled aspect ratio or tube length in diameter units) for a polycarbonate tube.7 Themaximum recommended aspect ratio is ≈ 60. Data points are calculated from ameasured transport factor TF = 23.2 diameter/dB, using a tungsten source with adivergence half-angle ≈ 30 deg.2 The light loss in decibels = 10 log (Φ/Φ0), whereΦ0 is the initial beam flux, and Φ is the flux at a specified distance down the tube.For example, for a tube with a 15-cm diameter, the transported flux will be ≈ 80%


Figure 8.8 Transported light intensity in circular microprismatic light guide.7

(−1-dB loss) for a tube ≈ 3.5 m long. When used for maximum light transport, thecurved microprismatic film is often directly encased in a diffusely reflecting rigidcylinder. This preserves the cylindrical shape of the film and directs any escapinglight from the guide back into the tube, where it can be further transported. Oncethe light is transported from one remote location to another, the light is often usedfor illumination. Having the light source in a remote location allows control oflamp changing and maintenance, heat buildup, and spectral properties of the light.

8.4 Luminaire with Contoured Prismatic Extractor

The light guide of Fig. 8.9 is used as a luminaire with a contoured prismaticextractor surface and an internal linear diffuse reflecting element.8 Ideally, thelight source emits light with ray angles 0 < θ < 50 deg relative to the guide xaxis. Figure 8.10 shows a cross section of the light guide, where the rays shownare the projections of the rays on the x-y plane. It has been determined throughmeasurement and computation analysis that light is transmitted most efficientlythrough the guide wall when projection of the light ray on a plane perpendicular tothe axis of the guide forms an angle γ ≈ 60 deg with the normal of the inner surfaceof the extractor surface. Table 8.3 gives typical rays that yield γ ≈ 60 deg, whereI, ω, and θ are as previously defined, and ϕ is the slope angle of the contouredsmooth inner surface. Here, IP is the projected angle of I in the y-z plane, given byIP = arctan(tan I cosω) and ϕ = γ − IP.

8.5 Elliptical Light Guide with Directional Output

Consider the elliptical reflective cylinder with major and minor axes a and b, as inFig. 8.11. The two foci are directed along the length of the cylinder, and the insideof the cylinder is either coated with a highly reflective coating or is formed from

200 Chapter 8

Figure 8.9 A microprismatic light guide luminaire with a contoured extractor.8

Figure 8.10 Cross section of contoured light guide luminaire.8

Table 8.3 Some optimum light guide wall-transmission ray angles (γ ≈ 60 deg). Units arein degrees.

I ω θ IP ϕ

60.0 85.0 30.4 8.58 51.4

65.0 85.0 25.5 10.6 49.4

60.0 75.0 33.2 24.1 35.9

70.0 80.0 22.3 25.5 34.5

60.0 60.0 41.4 40.9 18.1

75.0 75.0 21.1 44.0 16.0

85.0 85.0 7.1 44.9 15.1


Figure 8.11 Focal properties of an elliptical reflective cylinder.

a highly reflective film such as 3M Silverlux. The focus coordinates are given by±x f , where

x f =√

a2 − b2. (8.11)

For an oblique ray originating at either focus and intersecting the reflective ellipseat (x1, y1, z1), the slope ϕ of the surface is

ϕ = arctan

bx1

a2

√1 −

x12

a2

. (8.12)

Then, the direction cosines of the surface normal are

kx1 = sinϕ, (8.13a)

ky1 = cosϕ, (8.13b)

kz1 = 1. (8.13c)

The distance D f 1 from the focal point to the surface intersection is

D f 1 =

√(x1 − x f )2 + y1

2 + z12, (8.14)

and the direction cosines of the incident ray are

Kx f 1 =(x1 − x f )

D f 1, (8.15a)

Ky f 1 =y1

D f 1, (8.15b)

Kz f 1 =z1

D f 1. (8.15c)

202 Chapter 8

Then, the direction cosines of the reflected ray are

Kx12 = Kx f 1 − 2ρkx1, (8.16a)

Ky12 = Ky f 1 − 2ρky1, (8.16b)

Kz12 = Kz f 1 − 2ρkz1, (8.16c)

where ρ = Kx f 1kx1 + Ky f 1ky1 + Kz f 1kz1.The ray angle θ is given by θ = arccos Kz12, and the intersection coordinates

(x2, z2) of the reflected ray with the y-z plane (y2 = 0) are

x2 = x1 −

(Kx12

Kz12

)y1, (8.17a)

z2 = z1 −

(Kz12

Kx12

)(x2 − x1). (8.17b)

These calculations show that x2 = −x f , or that any skew reflected ray originatingat x f will always intersect the second focal line at a distance z2 from the origin.Several cases are shown in Table 8.4, where the linear dimensions are in arbitraryunits.

Table 8.4 Ray intersections for an elliptic reflective cylinder.

a b xf x1, y1, z1 θ (deg) x2, y2, z2

4 3 2.65 2, 2.60, 10 14.99 −2.65, 0, 28.88

4 3 2.65 3, 1.98, 5 21.96 −2.65, 0, 18.84

10 5 8.66 4, 4.58, 10 33.17 −8.66, 0, 30.60

10 5 8.66 8, 3, 5 31.56 −8.66, 0, 32.55

4 3 −2.65 2, 2.60, 10 28.03 2.65, 0, 15.02

4 3 −2.65 3, 1.98, 5 50.12 2.65, 0, 6.68

10 5 −8.66 4, 4.58, 10 53.40 8.66, 0, 14.85

10 5 −8.66 8, 3, 5 73.54 8.66, 0, 5.91

This result can be used for the design of a light-guide luminaire with directionaloutput.9 In Fig. 8.12, a half-elliptic reflective cylinder has linear diffusely reflectingextractors running along the focal lines. Light scattered from either extractor isreflected from the elliptical wall toward the other extractor, where it is againscattered. Scattered or reflected light rays that hit the linear prismatic array exitaperture are normalized on transmission. γ is the angle between normal to theprismatic array (n = 1.5) and the projected angle of the incident ray with a planeperpendicular to the axis of the luminaire.

8.6 Prismatic Backlighting Devices

Backlighting devices often use linear prismatic arrays to extract or transmit lightover a larger rectangular area in a low-profile enclosure. They have applications forback-illuminated advertising displays and liquid crystal display (LCD) illuminatorsfor computer screens and television. In Fig. 8.13(a), a low-profile light fixture


Figure 8.12 A linear elliptic luminaire with linear directional output from a microprismaticextractor.9

provides directional or normalized output using a linear 90-deg microprism planararray.1 A 90-deg linear microprismatic array element is positioned above a seriesof tubular tungsten filament or fluorescent lamps, with the grooves parallel to thelamp axis. An optional clear or diffusely transmissive element can be positionedabove the prism array. A planar diffuse or specular reflector is positioned belowthe lamps. As shown in the cross-sectional view of Fig. 8.13(b), forward ray C isdirectly transmitted by the prismatic array (I1 ≥ 15 deg for n ≈ 1.5) and broughtcloser to the array normal. Forward ray A is retroreflected (I1 ≤ 15 deg for n ≈ 1.5)and then reflected back to the prismatic array, where it is normally transmitted.Back rays B and D undergo planar reflection and are then transmitted normallythrough the prismatic array. A small percentage of rays, such as ray E, are notnormalized. An essentially directional output is thus achieved.

Another type of backlighting device uses a reflective linear 90-deg microprismplanar array.10 In a cross section of this microprism array [Fig. 8.14(a)], an incidentray having a ray angle θ1 is refracted at facet 1, total internally reflected at facet 2,reflected at the planar mirror surface 3, and refracted at facet 4 of the adjacentprism with exit angle θ4

′ = θ1. The ratio of element thickness t to prism heighth is t/h ≈ 3. The ray-deviation angle is δ = 90 deg. Figure 8.14(b) shows thesame prism array rotated by angle ϕ, at the same entrance-ray angle θ1, with thefollowing ray-trace equations:

I1 = 45 deg + θ1 − ϕ, (8.18a)

I1′ = arcsin

(sin I1

n

), (8.18b)

I2 = 90 deg − I1′ = I2

′, (8.18c)

I3 = I2′ − 45 deg = I3

′ (8.18d)

I4 = 45 deg − I3′, (8.18e)


θ4 = 45 deg − ϕ − I4′, (8.18g)

δ = 90 deg − θ1 + θ4. (8.18h)

204 Chapter 8

Figure 8.13 (a) A low-profile linear prismatic illuminator with normalized output.1 (b) Crosssection of a low-profile illuminator.

The deviation angle δ for the array remains fixed at 90 deg, and θ1 = θ4 for allθ1 and ϕ values. This also holds for any value of n that provides TIR at facet 2.This reflective linear microprismatic array is therefore a wavelength-independentconstant 90-deg deviation element over a defined acceptance angle. The backlightfixture in Fig. 8.15 uses a horizontal planar prismatic array with linear collimatedlight sources at both ends. A transmissive diffusing plate is placed at the exitwindow. Figure 8.16 shows a curved array with a single linear collimated lightsource. The curvature improves the light uniformity at the exit window, and thecurvature profile is not critical. The backlight of Fig. 8.17 has an additionalreflective prismatic film that spreads the light from a collimated point source toan extended beam directed to the curved reflective linear prismatic array.


Figure 8.14 (a) Linear reflecting microprismatic array with a 90-deg deviation angle. (b)Tilted linear reflecting microprismatic array with a 90-deg deviation angle.

Figure 8.15 Dual-lamp backlight with flat reflective microprismatic array.10

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Figure 8.16 Single-lamp backlight with curved reflective microprismatic array.10

Figure 8.17 Single-lamp backlight with dual-reflective microprismatic arrays.10

The backlight fixture in Fig. 8.18 uses a curved transmitting linear microprismextractor array to provide directional output over a large area.11 The prisms facedownward and utilize TIR to bend the light; this type of array is sometimes calleda turning film. The linear light source is substantially collimated and can be tiltedto produce an incident ray angle θ at the lower prismatic surface. The isoscelesmicroprisms have a vertex angle α and base angles β, and ϕ is the slope of thecurved array at any position along the curve. Figures 8.19 and 8.20 illustrate typicalray paths, and the basic ray-trace equations are

I1 = θ + ϕ − α/2, (8.19a)

I1′ = arcsin

(sin I1

n

), (8.19b)

I2 = α + I1′, (8.19c)

I2′ = I2, (8.19d)

I3 = β − I2′, (8.19e)


δ = 90 deg − I3′ + ϕ + θ. (8.19g)


Figure 8.18 Low-profile backlight using curved refractive microprismatic array.11

Figure 8.19 Section detail of curved linear microprismatic extractor for backlight.

For example, if θ = 10 deg, ϕ = 0 deg, and α = 70.61 deg, the output isessentially normal to the prismatic array, for nd = 1.5855 (polycarbonate), withI2crit = 38.10 deg. The microprismatic array is dispersive.

The high-aspect light box of Fig. 8.18 has a length L, width W, and heightH. A semicircular curve is drawn from the upper edge of the box near the lightsource to the lower edge of the box. The required radius of curvature R is givenby R = (W2 + H2)/2H, where the center of curvature is located on the x axis. IfL = W = 300 mm, and H = 60 mm (5:1 box aspect ratio), then R = 780 mm.A transmissive extractor film consisting of isosceles microprisms is bent alongthis curve such that all collimated light rays with θ = 0 deg from the sourceare intercepted. For light to emerge close to the normal of the exit window, thevertex angle of each microprism must be varied as a function of the curve slopeangle ϕ = arcsin(Y/R), such that δ = 90 deg, or I3

′ = −ϕ. The exit window can

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Figure 8.20 Halfway point of curved linear microprismatic extractor (Y = L/2).

Table 8.5 Sample design parameters for a curved transmissive extractor. Units are indegrees.

Y ϕ α β I1 I2, I2′ I3

′

0 0 75.06 52.47 37.53 52.46 0.0082

W/2 11.09 64.03 57.99 20.93 51.01 −11.10

W 22.62 52.08 63.96 3.42 48.92 −22.62

be a clear plate, a diffuse transmissive plate, printed graphics, or a transmissiveelectronic display. Some sample design parameters are shown in Table 8.5.

Another type of low-profile backlighting device is shown in Fig. 8.21.12 Asolid light guide of optical plastic has a series of linear microprismatic groovesmolded into the lower surface. A linear light source is placed at one end, anda planar reflector at the other. A highly reflective diffuse scattering surface ispositioned adjacent to the lower grooved surface. A diffuse transmitting elementis positioned adjacent to the guide upper surface. Light rays that undergo TIR atthe leading prism facet are directed to the upper surface where they are diffuselytransmitted. Rays that undergo TIR at the upper surface and are transmitted by theforward prism facet are scattered by the diffuse reflector and reenter the guide tobe transmitted at the upper planar surfaces. Rays that undergo TIR at the trailingprism facet can be further reflected by the upper surface of the end mirror until theycan be extracted by the prismatic array.

The backlighting device in Fig. 8.22 uses a solid rectangular light guide ofoptical plastic with trapezoidal input prisms, miniature (10- to 12-mm diameter)fluorescent lamps, and involute-shaped reflectors at each end.13 A series oftrapezoidal microprisms that function as TIR extractors are placed on the topplanar surface of the light guide. The input prisms and extracting microprisms areoptically cemented to the light guide or molded as an integral part of the guide.

The backlighting device in Fig. 8.23 has a tubular light source in a highlyreflective housing, with a horizontal prismatic array providing directional input intoa hollow rectangular light guide.14 The vertical sides of the guide have height H anddepth D, with an internal reflective surface chosen to be specularly reflective when


Figure 8.21 A solid light guide luminaire with linear microprismatic array.12

Figure 8.22 Solid light guide backlight with trapezoidal prismatic array extractor.13

the aspect ratio H/D > 10 and diffusely reflective when H/D < 10. The insideof the top side of the box has a specular, highly reflective surface. The rear faceof the light guide has a highly reflective (>85%) surface having a narrow scatterdispersion angle between 5 and 15 deg. A suitable reflective material is radiantlight film embossed VM2000, available from 3M. The forward surface is a verticalprismatic array that extracts the light from the light guide. A protective clear ordiffusely transmitting view plate is positioned over this prismatic array.

8.7 Brightness Enhancement for Liquid Crystal Displays

The design of backlights for transmissive direct-view liquid crystal displays hasbeen a major area of design research. One of the components in many of thesebacklights is a microprismatic array that enhances the viewing brightness in anangular range close to the display normal. Such a backlighting assembly uses aserpentine-shaped tubular fluorescent light source, a 90-deg linear prismatic film,and a back reflector to redirect stray radiation in the direction of the LCD.15

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Figure 8.23 Light box backlight with vertical and horizontal linear microprismatic arrays.14

The concept of brightness enhancement derives from the normalized prism arrayas shown in Fig. 8.13(b), where a means is provided to efficiently recycle lightthat is reflected downward from the microprismatic array. For a high-reflectance(>98%) back reflector, any direct or reflected light is redirected to the prismaticarray, usually at a different angle. The transmitted light then increases the forwardbrightness. Figure 8.24 shows a direct-view LCD display backlight using a seriesof cold-cathode fluorescent tubes (CCFTs) with high-reflectance-scattering lowerand side reflectors. A transmissive diffusing plate, a brightness-enhancement 90-deg microprismatic array film, and a transmissive LCD are positioned above thelight guide. The linear microprisms run in the same direction as the fluorescenttubes. Upward light is scattered by the diffusing plate, and a portion is transmittedby the microprismatic array. About 50% of the light hitting the array is reflecteddownward. The downward- or side-directed light is scattered and redirected tothe array. Light transmitted by the array is normalized, resulting in brightnessenhancement close to the normal axis of the display.

These prismatic arrays are sold commercially in the form of a flexible film by3M as VikuitiTM brightness-enhancement film (BEF). For handling and durability,the approximately 25-µm-thick UV-cured acrylic prism array is laminated to a 250-µm polyester film material, PET. In one type of BEF, designated BEFII 90/50, thereis a fixed 90-deg prism apex angle and a fixed peak-to-peak separation (often calledpitch) of 50 µm. The 90-deg BEF yields the maximum gain, typically about 1.6, fora view direction normal to the array, and about 1.2 at a view angle of ±30 deg, witha fairly sharp cutoff at wider view angles. Increasing the prism angle increases theview angle, but reduces the gain, while decreasing the prism angle reduces boththe view angle and the gain. Embossing a matte surface on the planar side of the


Figure 8.24 LCD backlight with a brightness-enhancement linear microprismatic array.

BEF also gives a wider view angle, but reduces gain. Adding another sheet of BEFnear and orthogonal to the first sheet enhances the brightness in crossed viewingdirections.

The BEF prismatic structure may be modified to yield a softer cutoff, asdescribed by O’Neill and Cobb.16 One type of structure is shown in Fig. 8.25.Here the 90-deg vertex angle is maintained, but the valley angles are varied suchthat alternate valley angles have values of 70 deg and 110 deg. This results inan alternating groove pitch. Figure 8.26 plots measured values of luminance forvarious view angles, for a typical backlight suitable for liquid crystal displays.Curves are shown for no BEF, 90/50 BEF, and a soft-cutoff BEF.

When two identical periodic structures are positioned close to each other, anundesirable moiré pattern can be observed if the structures are slightly displaced.This can occur in BEF, for example, from a reflection of the prismatic surface fromthe planar surface of the BEF, or from other planar surfaces if elements are closelystacked. It is possible to minimize this effect by varying the peak pitch P betweengroups of microprisms or making the structured surface nonperiodic.17 Figure 8.27shows one type of peak pitch variation where each group of prisms maintains aconstant peak pitch, and the fixed peak pitch P of any group can have a value

Figure 8.25 Groove structure of brightness-enhancement film with soft cutoff.16

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Figure 8.26 Measured luminance of backlights for various view angles.16

Figure 8.27 Groove structure of brightness-enhancement film for moiré patternreduction.17

between 20 and 60 µm. Moreover, the number of peaks per group varies between 2and 20. A typical configuration would include groups with peaks spaced at 50, 40,30, and 20 µm.

If all of the peaks of the BEF lie on a common plane and are in contactwith another planar surface, as in a closely stacked array, the area near thepeaks may be optically coupled to the planar surface, and the prism TIR isfrustrated in this region. This is sometimes called wet out and causes a noticeablemottling and streaking in the display. Campbell et al. describe a method ofmanufacturing a BEF that has a continuously varying peak height along thelength of a groove.18 This is accomplished by mounting the cutting tool on thediamond turning machine to a fast tool servo actuator, with white-noise input


Figure 8.28 Brightness-enhancement film with microprisms of varying height.18

filtered by a bandpass filter transmitting 4 to 5.6 kHz. Figure 8.28 illustrates a BEFstructure having this geometry where the valleys also vary, and the peak heightis preferably between 4% and 8% of the average peak height measured from theplanar side. The average period of the variations is preferably between 5 and 16times the nominal structure height. A 90-deg peak angle can be maintained toprovide maximum brightness gain. In addition to reducing the optical coupling,the varying-height prisms mask small cosmetic defects introduced during themanufacturing process. A brightness-enhancement film with a random prismaticstructure is available commercially from 3M, designated as VikuitiTM BEF III-10T. A typical LCD TV may contain from 4 to 40 CCFLs, depending on thescreen size. When BEF is used in this application, it is normally oriented with thegrooves in the vertical direction. For thermal stability at larger-screen TV sizes,the BEF thickness is increased over that used on smaller screens, such as computermonitors.

There are other types of brightness-enhancement arrays that use differentgeometries or have different applications. Figure 8.29 illustrates a brightness-enhancement film for a display that is front illuminated at a small angle θ from thedisplay plane, often called sidelighting.19 The film has spaced microprisms withbase angles α and β and a prism height h, with a distance d between prism peaks.For a pixilated display—e.g., LCD—there should be at least two prisms per pixel,and no blockage by the adjacent prism. Light is reflected by TIR at a prism face,is diffusely scattered by a planar diffuse reflector beneath the pixel, and emergesfrom the film at angles θ′ from 0 deg to about ±30 deg. Recommended ranges forthe design parameters are

0 deg ≤ θ ≤ 5 deg, (8.20a)

85 deg ≤ α ≤ 90 deg, (8.20b)

40 deg ≤ β ≤ 50 deg, (8.20c)

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5 µm ≤ h ≤ 20 µm, (8.20d)

10h ≤ d ≤ 29h, (8.20e)

where α ≈ 90 deg − θ.Another type of brightness-enhancement film for a backlit display uses a linear

array of prismatic structures in the x-z plane with curved faces in the y-z plane,forming a lenticular array (Fig. 8.30).20 This array is optically coupled to a solidlight guide such that the curved microprisms refract in two directions and providenormalized light output over a 2D range.

Figure 8.29 Brightness enhancement of front-illuminated LCD.19

Figure 8.30 Brightness-enhancement film using curved prismatic facets.20


Another brightness-enhancement film consists of a linear array of solid light-extracting structures having curved sides approximating a parabolic shape.21

These microstructures are based on the compound parabolic concentrator (CPC)developed for efficient collection of radiant energy. Figure 8.31(a) illustrates theoperation of an individual light extractor. Two opposite parabolic-shaped surfacesections 1 and 2 are canted with the focal points F1 and F2, defining the inputaperture boundaries. A specified fan of rays through the aperture edge F1, andat ray angles ϕ relative to the z axis, undergoes TIR at the parabolic surface andrefraction at the planar output aperture. This defines the maximum beam angle θmaxthat is incident on the output aperture, where typically 10 deg ≤ θmax ≤ 30 deg.If a ray is traced from the input aperture edge F1 to the opposite edge of theoutput aperture, the minimum value of a ray angle ϕmin that will be reflected bythe parabolic surface is determined. Smaller values of ϕ will be directly refractedat the output aperture at an angle of incidence ≤ θmax. This film array is opticallycoupled to a solid light guide, preferably of the same optical material [Fig. 8.31(b)].A linear lamp and curved reflector inputs light into one end of the guide, with a

Figure 8.31 (a) Detail of compound parabolic extractor element. (b) Solid light guidebacklight with compound parabolic linear array extractor.21

216 Chapter 8

planar reflector at the other end. Multiple reflections within the light guide resultin light entering the entrance apertures, with TIR at either parabolic-shaped sideand refraction at the output aperture. Typically, the pitch p is between 10 µm and200 µm, the height h is between 10 and 100 µm, and the ratio of output apertureto input aperture is between 1.5 and 10.0. Factors to consider in the design of thisstructure are the preservation of TIR at the parabolic surfaces, the desired value ofθ, and ease of manufacture.

8.8 Polarizing Prismatic Sheet

It is possible to produce partially polarized light for collimated light over anextended area by the use of two prismatic arrays 1 and 2, as shown in Fig. 8.32.22

The prismatic arrays have vertex angles α1 and α2 and are separated by an air gap.Unpolarized light rays entering the planar surface are incident on both prismaticstructures at Brewster’s angle, causing some of the s-polarized light to be reflectedto the side. The prismatic sheets have refractive indices n1 and n2, and n0 = 1.0 forthe air surround. Then,

I1brew = arctan(n0

n1

), (8.21a)

α1 = 180 deg − 2I1Brew, (8.21b)

I2brew = arctan(n1

n0

), (8.21c)

Figure 8.32 Production of partially polarized light by prismatic sheets.22


I2′ = arcsin

[(n1

n0

)sin I2brew

], (8.21d)

α2 = 180 deg − 2I2′. (8.21e)

If n1 = n2 = 1.53, then I1Brew = 33.17 deg, I2Brew = 56.83 deg, α1 = α2 =

113.7 deg, I2′ = I1Brew, and I1

′ = I2Brew. Since some of the s-polarized light isremoved at the two prismatic surfaces, the light transmitted through the upperplanar surface is partially p-polarized, and in the same direction as the incidentlight.

8.9 Prismatic Reflective Polarizer Film

By combining two 90-deg prismatic arrays and a repeating series of high andlow refractive index film pairs (often called a MacNeille pair) deposited on oneof the prismatic surfaces, a retroreflective polarizing film can be constructed.23

Figures 8.33(a) and (b) illustrate this construction. The composite film stackis captured between two prismatic arrays by optically cementing one array tothe stack. For broadband visible light, a suitable high-index material is titaniumdioxide (nH = 2.2 to 2.5), and a suitable low-index material is silicon dioxide(nL = 1.45). In practice, there may be three optical stacks, each containing sixMacNeille pairs, the first stack having a quarter-wave thickness centered at 400 nm,the second centered at 550 nm, and the third centered at 700 nm. A suitableprismatic-array substrate is polycarbonate (nd = 1.586), with a commercial opticalcement having an index nA = 1.56. Typical reflective polarizer film thicknessis 130 µm for smaller displays to 550 µm for large displays, such as LCD TV.Figure 8.34 gives typical performance of the polarizing film. The transmitted p-polarized light is ≈ 80%, and the reflected s-polarized light is ≈ 96% over thevisible spectrum. This reflective polarizer film is sold by 3M as VikuitiTM dual-brightness-enhancement film (DBEF).

8.10 LCD Backlights Producing Polarized Light

An LCD usually consists of a matrix of liquid crystal material captured betweenabsorptive polarizing sheets. If the incident light can be prepolarized in thepolarization direction of these absorptive sheets, the transmission of the LCD canbe significantly increased, resulting in increased viewing brightness. A reflectivepolarizer sheet can be utilized in a backlight to prepolarize almost all of theincident light entering an LCD display. Figure 8.35 shows an LCD backlightthat uses a bank of fluorescent lamps. Unpolarized light (designated as s + p)emitted through the diffuse transmitting window strikes the reflective polarizerwhere the p-polarized light is transmitted and the s-polarized light is reflected. Thep-component passes through the oriented lower LCD polarizer with no significantabsorption. The reflected s-polarized radiation is depolarized by the lowerdiffuser/reflector, and this unpolarized light again strikes the reflective polarizer.Unpolarized light that strikes the lower diffuser/reflector directly from the lamps is

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Figure 8.33 (a) Polarizing thin-film stack between 90-deg microprismatic sheets. (b)Retroreflective polarizing film. (Adapted from Ref. 22 with permission from The Society forInformation Display.)


Figure 8.34 Typical performance of retroreflective polarizing film.23

Figure 8.35 Backlight-producing polarized-light output using reflective polarizing film.

also directed toward the reflective polarizer. This continuous polarization recyclingand conversion to p-polarization yields a significant brightness increase in theviewed display, where the on-axis brightness is approximately doubled.

Another type of polarized backlight for LCD panels is shown in Fig. 8.36.24 Alinear light source is coupled to a series of solid light pipe slabs that are opticallyisolated from each other. TIR within each section tends to collimate the lightentering a solid light guide, where the light is extracted upward by TIR at an array

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Figure 8.36 Polarized backlight for LCD panel using solid light guide with TIR microprismarray.24

of linear prisms on the lower surface. A reflecting polarizing sheet is placed atthe top surface of this light guide. The reflective polarizer sheet is oriented withthe microprism direction at a 45-deg angle to the microprism direction of the lowerTIR array. If the apex angle of the TIR array ≈ 90 deg, then the reflected s-polarizedlight undergoes two 45-deg phase changes during TIR at the two prism faces,and the upward-reflected light is circularly polarized as it reenters the reflectivepolarizer sheet. The p-polarized light is again transmitted, and the s-polarizedlight is reflected back toward the lower TIR prismatic array. Thus, the light iscontinuously recycled such that the p-polarized light entering the lower polarizerof the LCD panel is strengthened. The lower polarizer of the LCD panel has itspolarization axis oriented parallel to that of the reflective polarizer sheet.

Several types of polarized backlights utilize microstructured anisotropic layers.Figures 8.37(a) and (b) illustrate a solid light guide with a machined array oflinear prismatic grooves on the top surface.25 The light guide is an acrylate, withnguide = 1.48. The grooves are filled with a liquid crystal birefringent polymerhaving an ordinary refractive index no = 1.51 and an extraordinary refractive indexne = 1.65, with directions as indicated. Unpolarized light striking prism surface 2at angles ≥ Ie

crit (≈64 deg) undergoes TIR and is extracted from the backlight ass-polarized light. The transmitted p-polarized light is essentially undeviated. Theprism apex angle α = 90 deg, and prism height = 10 µm, with a groove spacing of100 µm. The depth of the light guide is ≈ 2 mm. For this geometry, the s-polarizedlight is not normalized but has a maximum intensity about 30 deg from the normal.

Another type of polarized backlight uses a similar solid light guide but has adifferent prism extractor geometry and uses a commercially available birefringentfilm.26 The film is a uniaxial-aligned liquid crystal polymer (LCP) from Dejima


Figure 8.37 (a) Polarized backlight using birefringent polymer microprism array. (b) Detailof groove structure for polarizing extractor linear microprism array. (Adapted from Ref. 25with permission from The Society for Information Display.)

Optical Films. The microprisms can be formed onto the top surface of the lightguide by photoreplication from a master mold (sometimes called the 2P technique).The LCP film is then pressed into the microprism array by a lamination processas a relief structure, where the uniaxial alignment of the film is maintained.Figure 8.38 illustrates the near-normal TIR extraction of s-polarized light by therelief structure. The s-polarized light is estimated to be about 100 times more thanthe p-polarized light over a ±10-deg cone from the top surface normal, while mostof the p-polarized light can be recycled within the solid guide using side and endreflectors.

It is possible to produce a light guide emitting p-polarized light by the device inFig. 8.39(a).27 The backlight consists of a prismatic array of linear trapezoids thathave MacNeille-type polarization-separating thin-film stacks applied to the basesand optically coupled to a solid light guide by an optical adhesive. The optimumangle of incidence I at the film stack is about 64 deg at λ = 550 nm. Then, the

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Figure 8.38 Detail of relief structure of polarizing extractor linear microprism array.(Adapted from Ref. 26 with permission from The Society for Information Display.)

base angle β of the trapezoid is (180 deg − I)/2 = 58 deg for the p-polarizedlight to be reflected (TIR) from the side of the trapezoid and exit normal to the topsurface. In addition, the reflected s-polarized light from the stacks that strikes thebase and end of the light guide can be converted to p-polarized light by placing aquarter-wave retarder over reflectors on the base and end sections. This convertedp-polarized light can then exit through the linear trapezoid array. Figure 8.39(b)plots the transmitted p-polarized and s-polarized light as a function of the angle ofincidence I. In practice, the backlight can be efficiently used to produce emittedp-polarized light about ±10 deg from the exit surface normal.

8.11 Prismatic Array Beamsplitters and Combiners

There are a number of applications where a linear prismatic array can be usedas a beamsplitter or a beam combiner for collimated light beams. Figure 8.40(a)shows a prismatic array with vertex angle α used as a refractive beamsplitterwith deviation angles ±θ and 100% throughput. For certain input angles andsubstrate materials, it is possible to design a refractive beam combiner with highthroughput. As shown in Fig. 8.40(b), for α = 52.1 deg, θ = ±45 deg, andn ≈ 1.59 (polycarbonate), all input rays exit normal to the array. In the catadioptricbeamsplitter of Fig. 8.41(a), both refraction and total internal reflection deviate theincident beam. The relationship between the angle θ, the prism vertex angle α, andthe material refractive index n is given by

θ = 90 deg −{α

2+ arcsin

[n cos

(3α2

)]}. (8.22)

Some of the incident light is not deviated by ±θ but is misdirected within theprismatic array.

Figure 8.41(b) illustrates a catadioptric beam combiner. Table 8.6 gives thefraction of beam fill for various values of α and θ, for n = 1.492 (PPMA).


Figure 8.39 (a) Polarized backlight using thin-film-stack linear microprism array. (b)Transmission curves for polarized light through thin-film-stack linear microprism array.27

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Figure 8.40 (a) Refractive beam-splitting prismatic array. (b) Refractive beam-combiningprismatic array.

Table 8.6 Fraction of beam fill for a catadioptric beam combiner prismatic array.

α (deg) θ (deg) Beam-fill fraction

50.0 42.3 1.55

58.0 56.5 1.11

58.0 58.3 1.05

60.0 60.0 1.0

61 61.7 0.947

62.0 63.5 0.895

70.0 77.7 0.446


(a)

(b)

(c)

(d)

Figure 8.41 (a) 70-deg catadioptric beam-splitting prismatic array. (b) 50-deg catadioptricbeam-combining prismatic array. (c) 60-deg catadioptric beam-splitting prismatic array. (d)60-deg catadioptric beam-combining prismatic array.

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When the beam-fill fraction is greater than 1.0, some of the incident lightoverspills the reflecting facet and does not exit normal to the array. When thebeam-fill fraction is less than 1.0, the reflected facets are underfilled, and althoughall rays exit normal to the array, the exit beams from adjacent microprisms arespatially separated. For the special case where α = 60 deg and θ = 60 deg,the reflecting facets are exactly filled, and there is negligible refraction, as withthe 60-deg reflective beamsplitter shown in Fig. 8.41(c). For the 60-deg reflectivebeam combiner shown in Fig. 8.41(d), the spatial integration is continuous, withno separation between the individual transmitted microbeams.

Figure 8.42 shows an overhead projector that combines the light output from twolamps using a 60-deg reflective beam combiner.28 Fresnel lenses 1 and 2 collimatethe light from each lamp and are oriented such that the entrance angles to thereflective beam combiner is ±60 deg. The spatially integrated beams are focused tothe entrance pupil of the projection lens and directed to the screen by the foldingmirror. The projected screen illumination is approximately doubled from that of asingle lamp of equivalent wattage.

8.12 Polarization Converters Using Prismatic Arrays

A polarization converter is a device that separates an unpolarized light beaminto s-polarized and p-polarized beams, converts one of the beams to a commonpolarization, and spatially integrates the beams into a common direction withhigh throughput. Figure 8.43 illustrates. A polarization converter for illuminating

Figure 8.42 Overhead projector with dual lamps using 60-deg catadioptric beam-combining linear microprismatic array.28


Figure 8.43 Polarization-converting backlight using linear microprismatic arrays.29

an LCD panel uses beam-splitting and beam-combining linear prismatic arrays,reflective polarizing film, a quarter-wave retarder film, and planar mirrors.29

Collimated light is incident on an entrance refracting beamsplitter with vertex angleα ≈ 105 deg and refractive index n ≈ 1.58. The light is split into two beams atbeam angles θ ≈ ±24 deg. The bottom and end of the guide are planar reflectors. Aquarter-wave retarder is placed over the end reflector, and incident s-polarized lightbecomes reflected p-polarized light. The reflective polarizing film is placed beneatha top catadioptric beam combiner having a vertex angle α ≈ 63 deg and refractiveindex n ≈ 1.58. The reflective polarizer transmits p-polarized light and reflectss-polarized light, and the beam combiner normalizes and spatially integrates thetransmitted light through the top exit window. The dimensions of the box formedby these components must be such that only s-polarized light from the reflectingpolarizer strikes the retarder. Also, all light rays must pass through the exit window,with no light being returned to the entrance window. This condition is satisfiedwhen the height H and length L of the light guide satisfy the following:

tan θ = 2H/L, (8.23)

where the aspect ratio AR of the light guide is H/L. For θ ≈ 24 deg and H = 25mm, the required length is L = 112 mm, with AR = 4.5.

Figure 8.44 illustrates a large-area planar polarization converter that utilizeslinear prismatic arrays, reflective polarizing film, and quarter-wave retarder film.30

Collimated unpolarized light enters the lower element at an entrance angle θ =

45 deg. The lower linear prismatic array consists of two linear 90-deg vertex anglearray structures on opposite sides of a clear substrate. The microprisms of thelower surface have one face clear, while the opposite face has a reflective coating.Incident light enters through the clear faces and passes through a quarter-waveretarder sheet (e.g., Nitto-Denko–type NRF-QF03A) with principal axis direction

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Figure 8.44 Planar polarization converter using microprismatic arrays.30

as shown and strikes the reflecting polarizer sheet (e.g., VikuitiTM DBEF). Thetransmitted p-polarized light passes directly to a refractive beam combiner withvertex angle α = 52.1 deg, n = 1.58, and is emitted perpendicular to the planarconverter. The reflected s-polarized light passes through the retarder where it isconverted to elliptically polarized light. This elliptically polarized light is reflectedin the opposite direction by the reflective facets of the lower surface. After passingthrough the retarder, the elliptically polarized light is converted to p-polarized lightand spatially integrated with the directly transmitted p-polarized light. The groovepitches of the upper and lower element facets are typically 100 to 500 µm, whilethe groove pitch of the TIR microprisms is typically 10 to 50 µm. With all elementsin contact, typical thickness of this polarization converter is between 6 and 8 mm.

8.13 Cube-Corner Arrays

Instead of a large-aperture single cube-corner, it is often desirable to form a larger2D array of micro cube-corners, as shown in Fig. 8.45. In this configuration, eachcube-corner has an equilateral-triangle entrance aperture, with a typical cube depthH ≈ 120 µm and typical triangular aperture altitude h ≈ 255 µm. These entrancewindows are hexagonal close-packed. Once a master die is produced, cube-cornersheeting can be reproduced by microreplication. Figures 8.46(a) and (b) show thegeometry of an individual cube and the cross section of a line separating adjacentcube-corners.31 The master die can be formed by machining three sets of linearV-grooves, each set rotated by a 60-deg angle. This can be accomplished bymounting a flexible machinable sheet on a drum-type diamond-turning machine,with the sheet precisely rotated for each set of grooves. The included tool angleβ to produce the triangular cube-corner configuration can be calculated from the


Figure 8.45 Cube-corner array die section.31

following relationships:

sin(90 deg −

β

2

)= cos

(β

2

)=

Lh, (8.24)

where the length of a cube edge L = h√

6/3. Then the ideal tool angle β =

2 arccos(√

6/3) = 70.5288 deg = 70 deg, 31 min, 43.7 sec. If the cube-cornerarray is then replicated in an optical plastic sheet, the sheet can retroreflect by TIRor by coating the back prism surfaces with a highly reflecting thin film. Types offabrication errors that can affect the divergence of the retroreflected beam are errorsin the 90-deg dihedral cube angles, mounting errors of the master substrate duringthe 60-deg rotations, included tool angle error, and tool mounting error.

In some applications, such as traffic signs and warning displays, it is desirableto further increase the viewing angle of reflected light over a conventional cube-corner reflective array. This can be accomplished by a modification of the groovestructure of the master die.32 Figure 8.47(a) illustrates three sets of parallel V-grooves, where two of the sets (A, B) intersect at an angle of 70 deg, and a thirdset (C) intersects at an angle of 55 deg. The included tool angle for the (A, B)set is βA,B = 60 deg, 36 min, and for the (C) set is βC = 88 deg, 51 min. Thisproduces “matched pairs” of cube-corner elements where the optical axis is tiltedwith respect to the trisection of the internal base angle [see Fig. 8.47(b)]. Whenthe cube-corner array is replicated in an optical plastic of index n, the optimum tiltangle is approximated by:

ϕ = 54.736 deg − arcsin(1/n). (8.25)

230 Chapter 8

Figure 8.46 (a) Geometry of single micro cube-corner. (b) Cross section of adjacent microcube-corners and required cutting tool angle.

For n = 1.49, ϕ = 12.6 deg, and for most optical plastics, ϕ varies between 12and 13 deg. The resultant reflected angular half-brightness angle (50% falloff frommaximum) was found to be ≈ 40 deg in the x-axis plane and ≈ 35 deg in the y-axisplane, for an array replicated in optical acrylic, with intermediate angles in otherperpendicular planes.

An alternative geometry for a cube-corner array uses hexagonal entranceapertures instead of the triangular shapes described previously. This has theefficiency advantage of using three full sides of the cube, instead of the truncatedsides of the triangle cube-corner (see Sec. 2.4). The percentage of the reflectedlight can approach 100% of the incident light. Figure 8.48 shows a section of ahexagonal cube-corner retroreflecting array. However, the manufacture of a masterdie for a hexagonal cube-corner array presents challenges. One method describedby Brinksmeier et al. is a pin-building method where the top of each hexagonallyshaped pin has a precision-machined cube-corner.33 Pin alignment restricts the


Figure 8.47 (a) Top section of wide-angle retroreflective sheet.32 (b) Perspective view oftilted-axis micro cube-corner.

microcubes to a minimum size of about 500 µm. Another method is diamondmicrochiseling (DMC), also described by Brinksmeier et al. Using this technique,a cube microstructure of about 100 µm can be achieved.

8.14 Dove Prism Arrays

A type of 2D prismatic array that uses Dove prisms can be constructed to providea larger entrance aperture D relative to the length L. The single-roof Dove prismshown in Fig. 2.14(b) provides a directly viewed readable image rotated 180 deg.However, these prisms cannot be close stacked in an array, resulting in gaps andloss of efficiency over the full aperture. To provide a close-stacked array, a double-roof Dove prism having a square cross section can be constructed, as shown inFig. 8.49.34 The upper roof is a nonworking optical surface whose only purposeis to achieve a continuous-array aperture with no voids. A thin air separationmust be maintained between the individual prisms, or the working roof surfacesmust be reflectorized. The array shown in Fig. 8.50(a) consists of 88 double-roofDove prisms with an effective input aperture of radius D, each prism having a

232 Chapter 8

Figure 8.48 Hexagonal cube-corner reflective array. Leading corners are circledon sample microcube. (Adapted from Ref. 33 with kind permission of SpringerScience+Business Media.)

Figure 8.49 Double-roof Dove prism with square cross section.34

side of length b, with a height d. Let D = 10 cm and d = 1 cm, and assume alength/aperture ratio γ = 6 for each individual Dove prism. Then d = 1.414b, andthe length L of each prism is γ × d ≈ 8.5 cm. Thus, the aspect ratio of the arrayγarray ≈ 8.5/10 = 0.85.

Figure 8.50(b) shows a side cross section of the array. For imaging purposes, acontinuous integrated image is viewed for an object at a far distance and could beused for large-field scanning and tracking. These arrays can also be used in pairsto function as a variable beam deflector in transmission.


Figure 8.50 (a) Double-roof Dove prism array for circular entrance beam.34 (b) Crosssection of double-roof Dove prism array.34

234 Chapter 8

References

1. J. F. Dreyer, “Light fixture providing normalized output,” U.S. PatentNo. 4,791,540 (1988).

2. S. G. Saxe, “Prismatic film light guides: performance and recentdevelopments,” Solar Energy Mat. 19, 95–109 (1989).

3. L. A. Whitehead, “Prism light guide having surfaces which are in octature,”U.S. Patent 4,260,220 (1981).

4. L. A. Whitehead et al., “New efficient light guide for interior illumination,”Appl. Opt. 21(18), 2755–2757 (1982).

5. L. A. Whitehead, “Prism light guide luminaire,” U.S. Patent No. 4,615,579(1986); U.S. Patent No. 4,750,798 (1988).

6. S. Cobb, “Totally internally reflecting light conduit,” U.S. PatentNo. 4,805,984 (1989).

7. C. Sease, “Light piping: a new lighting system for museum cases,” J. Am. Inst.Conserv. 32(3), 279–290 (1993).

8. S. G. Saxe, “Light pipe having optimized cross-section,” U.S. PatentNo. 5,309,544 (1994).

9. L. A. Whitehead, “Prism light guide luminaire with efficient directionaloutput,” U.S. Patent No. 5,339,382 (1994).

10. K. A. Aho et al., “Back-lit display,” U.S. Patent No. 4,874,228 (1988).

11. R. A. Miller et al., “High aspect lighting element,” U.S. Patent No. 5,190,370(1993).

12. R. E. DuNah et al., “Flat, thin, uniform thickness large area light source,” U.S.Patent No. 5,420,761 (1995).

13. J. Kuper, “Light directing optical structure,” U.S. Patent No. 5,761,355 (1998).

14. J. C. Wright and M. C. Lea, “Light-guide lights suitable for use in illuminateddisplays,” U.S. Patent No. 7,164,836 (2007).

15. A. Abileah et al., “Lighting assembly for a backlit electronic display includingan integral image splitting and collimating means,” U.S. Patent No. 5,161,041(1992).

16. M. B. O’Neill and S. Cobb, “Brightness enhancement film with soft cutoff,”U.S. Patent No. 5,917,664 (1999).

17. S. Cobb et al., “Variable pitch structured optical film,” U.S. PatentNo. 5,919,551 (1999).

18. A. B. Campbell et al., “Optical film,” U.S. Patent No. 6,354,709 (2002).

19. K. E. Epstein and R. P. Wentz, “Front-lit liquid crystal display havingbrightness enhancement film with microridges which directs light through thedisplay to a reflector,” U.S. Patent No. 5,608,550 (1998).


20. S. C. Tang, “Brightness enhancement film,” U.S. Patent No. 6,277,471 (2001).

21. J. Lee and D. Kessler, “Brightness enhancement film using a linear array oflight concentrators,” U.S. Patent No. 7,160,017 (2007).

22. M. Suzuki, “Reflective polarizer sheet on the backlighting unit,” SID SIDSymp. Digest, pp. 813–816 (1997).

23. M. F. Weber, “Retroreflecting sheet polarizer,” U.S. Patent No. 5,559,634(1996).

24. C.-Y. Tai et al., “Backlighting assembly utilizing microprisms and especiallysuitable for use with a liquid crystal display,” U.S. Patent No. 5,390,276(1995).

25. S. M. P. Blom et al., “Towards a polarized light-emitting backlight micro-structured anisotropic layers,” J. SID 10(3), 209–213 (2002).

26. H. J. Cornelissen et al., “Polarized light LCD backlight based on liquidcrystalline polymer film: a new method of manufacture,” SID SymposiumDigest 35, 1178–1181 (2004).

27. S.-M. Huang et al., “Polarizing light guide plate unit and backlight unitand display device employing the same,” European Patent ApplicationNo. 1850156A1 (2006).

28. D. F. Vanderwerf, “Multiple lamp illumination system for projection displays,”Proc. SPIE 2650, 54–62 (1996); “Projector with multiple light source,” U.S.Patent No. 5,504,544 (1994) [doi:10.1117/12.237018].

29. R. J. Saccomanno, “Hollow cavity light guide for the distribution of collimatedlight to a liquid crystal display,” U.S. Patent No. 6,443,585 (2002).

30. D. F. Vanderwerf, “Planar polarizer for LCD projectors,” U.S. PatentNo. 5,940,149 (1998).

31. R. F. Stamm, “Retroreflective surface,” U.S. Patent No. 3,712,706 (1973).

32. T. L. Hoopman, “Cube-corner retroreflective articles having wide angularityin multiple viewing planes,” U.S. Patent No. 4,588,258 (1986).

33. E. Brinksmeier et al., “Manufacturing of molds for replication of micro cube-corner retroreflectors,” Prod. Eng. Res. Devel. 2, 33–38 (2008).

34. T. Lian and M.-W. Chang, “New types of reflecting prisms and reflecting prismassembly,” Opt. Eng. 35(12), 3427–3431 (1996) [doi:10.1117/1.601103].

Some commercial nonsequential ray-tracing programs:

• ASAP, from Breault Research Organization, Inc., Tucson, AZ• ZEMAX, from Zemax Development Corp., Bellevue, WA• TracePro, from Lambda Research Corp., Littleton, MA• CODE V, from Optical Research Associates, Pasadena, CA• LightTools, from Optical Research Associates, Pasadena, CA

Chapter 9Fresnel Lenses

9.1 Basic Refractive Fresnel Lens Design

The refractive Fresnel lens consists of a series of adjacent microprisms designed toprovide a varying deviation angle over the lens area. These angles are programmedto focus or collimate incident light. The most common Fresnel lens consists of aseries of concentric grooves replicated in optical plastic, with grooves on one sideand a planar surface on the other, and having positive power. It is often referred to asa positive aspheric Fresnel lens. Figure 9.1 shows how a continuous aspheric lenscan be collapsed to a Fresnel lens surface, eliminating much of the bulk material.The profile of the continuous aspheric surface can be described by the standard

Figure 9.1 Relationship between a continuous aspheric surface and a Fresnel lenssurface.

237

238 Chapter 9

equation of an aspheric surface, axially symmetric about the z axis:

z =cx2

1 +√

1 − (k + 1)c2x2+ a1x4 + a2x6 + a3x8 + a4x10, (9.1)

where z and x are the coordinates of the surface, c is the vertex curvature, k is theconic constant, and a1, a2, a3, and a4 are the aspheric coefficients.

The basic geometry for ray tracing through a positive aspheric Fresnel lens isshown in Fig. 9.2(a), and the refractive geometry at a single echelon is shown inFig. 9.2(b).1 Here, θ1 and θ2 are the incident and exit-ray angles, θ2

′ is the angle ofincidence at the exit surface, t is the lens thickness, and n is the refractive index ofthe lens material. From this geometry and the application of Snell’s law, the grooveangle α can be calculated from the equation

tanα =sin θ1 + n sin θ2

′

n cos θ2′ − cos θ1

. (9.2)

Consider a single-element Fresnel lens with collimated light incident normal tothe planar side. Here, δ = θ2 = I2

′ − α, as shown in Fig. 9.3(a). Figure 9.3(b)graphs the deviation angle δ as a function of the microprism groove angle α.The maximum deviation angle δmaxis attained when TIR occurs at the groovedsurface—e.g., I2 = αmax = arcsin(1/n) ≈ 42.12 deg for an acrylic plastic lenswith n = 1.491, and δmax = 90 deg − αmax ≈ 47.9 deg. For larger values of α, thelens is no longer refractive. When collimated light is incident on the grooved side[Fig. 9.4(a)], where I2 = α−I1

′ and δ = θ2 = I2′, the deviation angle δ as a function

of the microprism groove angle is as shown in Fig. 9.4(b). Since δ is now limitedby TIR at the planar surface, up to 90-deg deviation is possible at α ≈ 84.0 deg foran acrylic plastic lens.

The groove angle α′ of a Fresnel lens can also be calculated from the derivativeof Eq. (9.1), where c and k now refer to the grooved surface:

tanα′ =dzdx, (9.3a)

or

tanα′ =cx√

1 − (k + 1)c2x2+ 4a1x3 + 6a2x5 + 8a3x7 + 10a4x9. (9.3b)

This is a concise way of specifying the groove angles of a Fresnel lens for anygroove frequency, normally between two and eight grooves per millimeter. Froma table of calculated groove angles from Eq. (9.2), the angles can be least squaresfitted to Eq. (9.3b) to calculate the values of the variables c, k, a1, a2, a3, anda4. Since Eq. (9.3b) is not linear in the variables, a general technique is used that

Fresnel Lenses 239

minimizes the sum of the squares of the residuals by successive corrections to thesevariables.2 The residuals are defined as (tanα′ − tanα), and initial estimates of thevariables are required.

Figure 9.2 (a) Basic geometry for ray tracing through a positive aspheric Fresnel lens. (b)Refractive geometry at a single echelon.

240 Chapter 9

Figure 9.3 (a) Fresnel lens with collimated light incident on the plano side. (b) Deviationangle versus groove angle of a Fresnel lens with collimated light incident on the plano side.

9.1.1 Design example: Fresnel lens collimator/searchlight

Consider a circular refractive Fresnel lens collimator, as shown in Fig. 9.5. The lensdiameter Ap = 300 mm, the distance of the light source from the planar entrancesurface is f = 250 mm, the material is acrylic plastic, and the lens thicknesst = 3 mm. We use initial estimates of c = 1/[(n − 1) f ] = 0.0081 mm−1, k = −1(parabolic), and a1 = a2 = a3 = a4 = 0. A sample of equally spaced rays aretraced as shown. Each groove angle α is calculated from Eq. (9.2), and all groove

Fresnel Lenses 241

Figure 9.4 (a) Fresnel lens with collimated light incident on the grooved side. (b) Deviationangle versus groove angle of a Fresnel lens with collimated light incident on the groovedside.

242 Chapter 9

Figure 9.5 A circular refractive Fresnel lens collimator/searchlight.

angles are then least squares fitted to Eq. (9.3b), yielding α′. The calculated α andα′ values are equal to five significant figures, and the groove profile parameters aresummarized as follows:

c = 0.0080816 mm−1,

k = −0.91915,a1 = 6.1322 × 10−10,

a2 = −4.970 × 10−14,

a3 = 1.3967 × 10−19,

a4 = −1.0211 × 10−24,

Sum of squares (SS) of residuals = 6.5733 × 10−12.

The results are summarized in Table 9.1.

9.2 High-Transmission Fresnel Lens Doublet

In many applications of Fresnel lenses, the surface reflection losses in the outerregion of a single lens are excessive, and a dual-element Fresnel lens needs tobe used. This is the case, for example, in the condensor lens near the stage ofa conventional overhead projector. Consider a 300-mm-aperture lens with design

Fresnel Lenses 243

Table 9.1 Least-squares aspheric fit of groove angles of a Fresnel lens collimator.

x1 Planar surface (mm) x2 Grooved surface (mm) α (deg) α′ (deg) Residual tan(α′) − tan(α)0.4960 0.50 0.23152 0.23152 2.6873 × 10−9

10.416 10.50 4.8519 4.8519 −1.570 × 10−7

20.337 20.50 9.4183 9.4183 2.2310 × 10−7

30.258 30.5 13.883 13.883 3.8844 × 10−7

40.181 40.5 18.204 18.204 2.4518 × 10−7

50.105 50.5 22.348 22.348 −5.6978 × 10−7

60.030 60.5 26.291 26.291 −4.6390 × 10−7

69.958 70.5 30.016 30.016 6.8274 × 10−7

79.888 80.5 33.516 33.516 −3.6294 × 10−7

89.820 90.5 36.789 36.789 −2.80 × 10−7

99.754 100.5 39.838 39.838 8.5820 × 10−7

109.69 110.5 42.672 42.672 −2.3251 × 10−7

119.63 120.5 45.30 45.30 6.9366 × 10−7

129.57 130.5 47.734 47.734 −1.5539 × 10−6

139.52 140.5 49.986 49.986 1.1441 × 10−6

149.47 150.5 52.0697 52.0697 −2.7640 × 10−7

conjugates f = 250 mm (light source position) and f ′ = 400 mm (focus position).Figure 9.6 shows a section of the lens where θ1 > θ4, and the angle of refractionI2′ at each groove of the first element is set equal to the angle of incidence I3 at

each groove of the second element. One can consider that the “air prism” betweenthe prism facets is operating at minimum deviation. This results in a divergence ofthe rays exiting the first element. Let ϕ ≡ I2

′ = I3, and ϕ′ ≡ I2 = I3′. From the

figure:

α2 − ϕ = ϕ − α3, (9.4a)

α2 = I2 + I1′, (9.4b)

α3 = I3′ + I4

′, (9.4c)

where

ϕ = ϕ′ +

(I1′ + I4)2

. (9.4d)

Using Snell’s law to relate ϕ and ϕ′, it follows that

ϕ = arctan

n sin

[(I1′ + I4)2

]n cos

[(I1′ + I4)2

]− 1

. (9.4e)

244 Chapter 9

Figure 9.6 Ray-trace parameters for a high-transmission Fresnel lens doublet.

Then, the groove angles can be calculated from

α2 = arcsin(sinϕ

n

)+ I1

′, (9.4f)

α3 = arcsin(sinϕ

n

)+ I4. (9.4g)

Table 9.2 gives sample data for this Fresnel lens doublet. The groove width= 0.5 mm, and Y2 and Y3 are set near the midpoint of each groove. The lensmaterial is acrylic plastic (n = 1.492). The transmission Ti at each surface iscalculated using Eqs. (1.13) and (1.15), and the total transmission T = T1T2T3T4for each ray traced. Fresnel lenses are usually mass produced through replication,and antireflection coatings are rarely used.

Groove Profile for α2: Groove Profile for α3:

c = 0.0070568 mm−1 c = 0.0064347 mm−1

k = −1.0632 k = −0.61104a1 = 8.8277 × 10−9 a1 = −1.2115 × 10−8

a2 = −3.0594 × 10−14 a2 = −5.8711 × 10−14

a3 = −3.4088 × 10−20 a3 = −6.6858 × 10−19

a4 = 2.5417 × 10−24 a4 = −9.9916 × 10−24

SS residuals = 1.7063 × 10−10 SS residuals = 5.8336 × 10−10

Fresnel Lenses 245

Table 9.2 Representative data for a Fresnel lens doublet.

x2, x3 (mm) x1 (mm) x4 (mm) α2 (deg) α3 (deg) T = T1T2T3T4 Transmission

0.250 0.2480 0.2487 0.1011 0.0869 0.8530

11.75 11.65 11.69 4.742 4.075 0.8530

23.25 23.06 23.13 9.334 8.022 0.8529

34.75 34.47 34.57 13.833 11.889 0.8529

46.25 45.88 46.02 18.198 15.642 0.8527

57.75 57.29 57.46 22.396 19.254 0.8524

69.25 68.70 68.91 26.403 22.706 0.8518

80.75 80.12 80.35 30.201 25.983 0.8508

92.25 91.53 91.79 33.780 29.078 0.8494

103.75 102.95 103.24 37.136 31.991 0.8475

115.25 114.36 114.68 40.270 34.724 0.8449

126.75 125.78 126.13 43.187 37.281 0.8417

138.25 137.20 137.57 45.896 39.673 0.8378

149.75 148.62 149.02 48.407 41.907 0.8330

In addition to surface-reflection losses, there are also geometric effects causedby the risers of each element. One effect occurs when light is directly blocked bythe risers, resulting in reduced throughput, as illustrated in Fig. 9.7(a). Anothereffect is a dark banding in the output light, as shown in Fig. 9.7(b). Where theriser angle β and groove width w can be controlled, it is possible to eliminateboth of these effects, as shown for the Fresnel lens doublet in Fig. 9.7(c). Toaccomplish this, the pitch and riser angles of the elements are controlled such thatlight leaving the peaks and valleys of the first element hit the corresponding peaksand valleys of the second element. A defined separation of the elements needs tobe maintained. The design of such a lens has been described by Dudnikov et al.3

However, there are maximum β values that allow release from the mold duringreplication, and in the case of overhead projector condensers, there are minimumallowable β values to avoid stage glare from spurious reflections off of therisers.4

9.3 Reflective Fresnel Lenses

Reflective Fresnel lenses are useful components for several applications, includingilluminators, solar energy concentrators, and compact overhead projectors, whereboth the light source and projection lens are positioned over the stage.

9.3.1 First-surface reflector design parameters

Figure 9.8 shows the basic geometry for a first-surface reflective Fresnel lens, withaxial design conjugates f and f ′, where f ′ > f . The groove angles α1 can be

246 Chapter 9

Figure 9.7 (a) Reduced throughput of a Fresnel lens due to direct riser blockage. (b)Dark banding in output of a Fresnel lens. (c) High continuous throughput of a Fresnel lensdoublet.3

Fresnel Lenses 247

Figure 9.8 Basic geometry of a first-surface reflecting Fresnel lens.

calculated from the following set of equations:

θ = arctan(

yf

), (9.5a)

θ′ = arctan

(yf ′

), (9.5b)

α1 =(θ + θ′)

2. (9.5c)

9.3.2 Second-surface reflector design parameters

A detail of the refraction and reflection at a groove of a second-surface Fresnellens reflector is shown in Fig. 9.9. The groove angles α2 can be calculated from thefollowing set of equations, where I1 = θ, I3

′ = θ′, and the material has a refractiveindex n:

I1′ = arcsin

(sin I1

n

), (9.6a)

I3 = arcsin(sin I3

′

n

), (9.6b)

α2 =(I1′ + I3)2

. (9.6c)

248 Chapter 9

Figure 9.9 Groove detail for a second-surface reflecting Fresnel lens.

9.4 Refractive Planar Circular Fresnel Lens Solar Applications

There are few uses of planar circular Fresnel lenses for solar-energy-concentrationapplications. This is mainly due to size limitations in fabrication, structural stabilityof large lenses when mounted in a near-horizontal position, the need for precisesolar tracking, and off-axis optical aberrations that affect the achievable focusingaccuracy.

9.4.1 Multilens solar furnace

One application uses a spatially separated array of lenses to construct a solarfurnace.5 Figure 9.10 depicts a section of the solar furnace. The lenses are mountedon a support structure, precisely oriented in the direction of the sun, such that eachlens is focused at the spherical tip of a heat-conducting rod. These rods have highheat conduction and feed into boiler tanks though which a vaporizing fluid flows.The heated fluid can be utilized to generate steam and drive a turbine.

9.4.2 Multilens-array solar simulator

Another application of planar circular Fresnel lenses is the construction of alow-cost, large-area solar simulator.6 Solar simulators attempt to reproduce solarradiation with regard to intensity, spectrum, and uniformity over a defined area, forcontrolled testing of materials and structures. The simulator consists of 143 lensesarranged in a close-packed hexagonal array, each lens having a tungsten-halogenlamp positioned one focal length (≈300 mm) from the lens, producing a uniformcollimated beam over an area 1.2 m2 (see Fig. 9.11).

9.5 Refractive Meniscus Fresnel Lenses

A refractive Fresnel lens having a meniscus shape solves some of the problemsof the flat Fresnel lens. Of particular interest is the fact that the meniscus Fresnellens can be made aplanatic, free from spherical aberration and coma. The theory iscovered in several papers by Delano and Kleinhans.7–9 Erismann has described thedesign of a polyethylene Fresnel lens with a spherical shape for use as an infraredmotion sensor, where collimated incident light is focused to a point.10 However,

Fresnel Lenses 249

Figure 9.10 A multilens solar furnace.5

Figure 9.11 Fresnel lens array for a large-area solar simulator.6

250 Chapter 9

its use in large-diameter solar concentrators is uncommon, for some of the samereasons as for the flat Fresnel lens.

9.6 Reflective Planar Linear-Focus Solar Concentrators

Many of the applications of Fresnel lenses as solar concentrators involve the useof reflective linear Fresnel lens es producing a linear focus. When used as solarconcentrators, the first- or second-surface linear reflectors, shown in Figs 9.8 and9.9, and operating at ≤ f /1.0, produce significant blockage of light by the risers inthe lens outer region.

9.6.1 Tilted linear-focus reflective solar concentrator

In Fig. 9.12(a), a tilted flat first-surface linear Fresnel reflector produces no riserblockage.10 The design constraint that eliminates riser blockage is

ϕ ≥ α ≥ 0, (9.7a)

or

−ϕ ≤ α′ ≤ 0, (9.7b)

where ϕ is the minimum tilt angle of the reflector, α is the facet angle with respectto the x-y coordinate system, α′ is the facet angle with respect to a rotated x′-y′

coordinate system, and α′ = α − ϕ. For the V-configuration of Fig. 9.12(b), withfocal length f ′, half-acceptance angle γ, and minimum tilt angle ϕmin, the apertureAp is calculated from

Ap = 2(

sin γcosϕ

)f ′. (9.8)

For the second-surface configuration of Fig. 9.13, there is no riser blockage when

ϕ ≥ α > 0, (9.9a)

−ϕ ≤ α′ < 0, (9.9b)

where the refraction at the planar surface requires that α > 0.

9.6.2 Linear-focus concentrator using a linear Fresnel lens and acrossed linear total-internal-reflecting array

Another type of planar reflecting linear-focus solar concentrator [Fig. 9.14(a)] has alinear Fresnel structure replicated on one side of a plastic sheet and an orthogonal90-deg TIR prism array replicated on the other side. No reflective coatings arerequired. Figure 9.14(b) shows a detailed section with the direction vectors definingthe prism facets. Rays are traced through one quadrant of the concentrator usingEqs. (1.25), (1.26), and (1.27) for the refractions and Eqs. (1.52) and (1.53) for the

Fresnel Lenses 251

Figure 9.12 (a) A tilted first-surface reflective linear-focus solar concentrator.11

(b) V-configuration for tilted reflective linear-focus solar concentrator.11

Figure 9.13 A tilted second-surface reflective linear-focus solar concentrator.

252 Chapter 9

Figure 9.14 (a) A linear-focus linear Fresnel lens/crossed TIR linear array solarconcentrator. (b) Groove detail of linear-focus reflecting solar concentrator.

Fresnel Lenses 253

reflections, where

kx1 = −sinα ky1 = 0 kz1 = −cosαkx2 = 0 ky2 = cos 45 deg kz2 = −cos 45 degkx3 = 0 ky3 = −cos 45 deg kz3 = −cos 45 degkx4 = sinα ky4 = 0 kz4 = cosα

There is a varying displacement of the incident and exit rays for this system. Theintersection coordinates at each surface can be calculated from Eq. (1.59), where

xi = xi−1 +

(Kx(i−1)

ρi

)(Pi − kxix − kyiy − kziz), (9.10a)

yi = yi−1 +

(Ky(i−1)

ρi

)(Pi − kxix − kyiy − kziz), (9.10b)

zi = zi−1 +

(Kz(i−1)

ρi

)(Pi − kxix − kyiy − kziz), (9.10c)

where i = (1, 2, 3, 4).Consider a reflective concentrator of size 500 mm2 with a centered line focus

f ′ = 500 mm from the panel. The panel thickness t is 2 mm, the groove width wat each surface is 0.5 mm, and the material is acrylic plastic. The P values are asfollows: P1 = P4 = (w/2) sinα, and P2 = P3 = w(

√2/2) + (t − w) sin 45 deg,

relative to a local coordinate system (x0, y0, z0) at the top surface, and centeredwith each refracting groove and the corresponding reflecting groove vertex. Thedirection cosines and coordinates are calculated for a series of exit rays for anarbitrary y value, and the groove angles α are iteratively adjusted until x5 ≈ 0 atthe image plane, where

x5 = x4 +

(Kx4

Kz4

)f ′. (9.11)

For this linear concentrator, the important displacement is (x1 − x4) between thecollimated entrance ray and the focused exit ray, orthogonal to the focal line. Thisvaries from about 0.0027 mm near the panel center to about 0.58 mm at the edgeof the panel. Figure 9.15 plots the calculated groove angle α as a function of thedistance x0 from the panel center.

9.6.3 Planar reflective spot-focus concentrator using orthogonalrefractive and reflective linear Fresnel lenses

If a linear refracting Fresnel lens and a linear reflecting Fresnel lens arecrossed orthogonally, 2D convergence of normally incident solar radiation can beachieved.11 For the panel shown in Fig. 9.16, the incident surface is the planar sideof the refracting element. All rays incident along any two orthogonal lines on thepanel can be directed to a common focal point, while incident rays outside thesedirections will deviate from this focus. This lack of radial symmetry results in anextended spot focus.

254 Chapter 9

Figure 9.15 Groove angle versus distance from center for a linear-focus reflecting solarconcentrator.

Figure 9.16 A spot-focus reflective solar concentrator using crossed linear Fresnel lenselements.11

Fresnel Lenses 255

For a 305-mm2 panel with a focal length f ′ = 305 mm and n = 1.491,Fig. 9.17(a) illustrates a histogram of the distribution of intersection points withinannular rings at the focal plane, when all incident rays along the x axis and the yaxis are focused to a point (axial design). Figure 9.17(b) shows the distribution ofintersection points when rays incident along the diagonals are focused to a point(diagonal design). In each case, the 95% energy containment radius R95 is shown.This type of planar concentrator can be supported on the back side and enlargedby a mosaic of square or rectangular elements. It requires two-directional trackingand could be useful for circular targets of extended area.

9.7 Curved Linear Fresnel Lens Solar Concentrators

Refractive linear Fresnel lenses formed along a circular or an aspheric curveare very useful as solar concentrators (see Fig. 9.18). They can be fabricated inlarge sizes and have better structural stability than flat lenses. There are variousmethodologies for their design. One method starts at the edge and designs towardthe center. Each microprism is designed for minimum deviation of the incidentray, making sure that each succeeding prism does not block any exit rays fromthe one that preceded it.12 As shown in Fig. 9.19, α is the prism angle, and the(x, z,ϕ) coordinates specify the position and slope of the curved surface. To avoidinterference from the riser of the next groove, set β < β′, where β′ = I2

′+90 deg−α.The basic ray-trace equations for collimated incident light are as follows, where δis the required deviation angle for focusing:

I1 = ϕ, (9.12a)

α = arctansin(δ − I1) + sin I1√

n2 − sin2 I1 − cos(δ − I1), (9.12b)

I1′ = arcsin

(sin I1

n

), (9.12c)

I2 = α − I1′, (9.12d)


δ = I2′ − α + I1. (9.12f)

For minimum deviation of the refracted ray, each microprism is designed such thatI1 = I2

′. Then,

αmin = 2I1′, (9.13a)

δmin = 2(I1 − I1′). (9.13b)

This minimizes surface reflection losses, thereby maximizing the lenstransmission, and reduces the effect of prism slope errors. The overall transmissioncan exceed 90% with f /# < f /1.0. For any finite value of the lens refractive index,I2 will eventually reach the critical angle as ϕ increases. Therefore, it is not possibleto design a concentrator that operates over a full hemisphere.

256 Chapter 9

Figure 9.17 (a) Histogram of ray intersection points at the focal plane for a spot-focusconcentrator (axial design). (b) Histogram of ray intersection points at the focal plane for aspot-focus concentrator (diagonal design).

Fresnel Lenses 257

Figure 9.18 A curved linear refractive Fresnel lens.

Figure 9.19 Ray-trace detail for a curved linear Fresnel lens.

Another design method considers the collimation angle ±θ0 of solar radiation,where θ0 = 0.27 deg = 4.7 mrad. The maximum theoretical limit of the geometricconcentration ratio C(θ0) ≤ 1/ sin θ0, where C(θ0) is the ratio of the lens inputaperture A to the target width a.13 Thus, the ideal maximum value is C(θ0) ≈ 200.For the curved linear Fresnel lens of Fig. 9.20, the edge-ray principle is applied

258 Chapter 9

Figure 9.20 Edge-ray principle for a curved linear solar concentrator. (Adapted fromRef. 14 with permission from the Optical Society of America.)

to each microprism. By adjustment of (x, y,ϕ,α), solar rays of ray angle +θ0 arefocused at the right edge of the target of length a, while those of ray angle −θ0are focused to the left edge of the target. The lens is designed outward from thecenter.14 With precise 2D tracking, all solar rays will then fall within the target. Inpractice, chromatic dispersion reduces the concentration ratio to about 40.

Kritchman has described a “color-corrected” curved Fresnel lens that doublesthe concentration ratio to about 80.15 These design methods produce a noncircular-shaped cross-sectional profile for the lens, where normally α > ϕ. Another designmethod for a curved linear Fresnel lens solar concentrator has been describedwhere the smooth and grooved surfaces are formed with constant radius ofcurvature.16

Using the edge-ray principle, the 2θ0 solar collimation acceptance angle canbe enlarged to a general acceptance angle 2θ. This relaxes the precise trackingrequirements but reduces the concentration ratio.17 The limiting values of θ areusually determined by the occurrence of TIR at the prism facet. Figure 9.21 showshow the maximum acceptance angle θmax in the x-z plane can be determined whenI2 = α−I1

′ = I2crit and θmax = ϕ−I1max. For example, when α = 60 deg,ϕ = 45 deg,and n = 1.492, then θmax = 17.68 deg. Similarly, θmax for an incident ray in thetransverse y-z plane can be calculated using Eqs. (1.25) to (1.27). The smallestvalue of θmax in either plane near the edge of the lens determines a conservativevalue for the acceptance angle.

From a practical standpoint, it is useful to reduce the tracking requirementfrom full 2D movement to movement in a single polar direction. It has beenfound that diurnal rotation of angle ω about the linear axis of the concentratorminimizes shortening of the focal point position from the lens center. See Fig. 9.22,

Fresnel Lenses 259

Figure 9.21 Ray-trace detail for a curved linear Fresnel lens with acceptance angle 2θ.(Adapted from Ref. 17 with permission from the Optical Society of America.)

Figure 9.22 Diurnal rotational tracking for curved linear Fresnel solar concentrator.(Adapted from Ref. 18 with permission from the Optical Society of America.)

where rotation of the entire connected unit occurs about the y axis. Kritchman hasanalyzed the performance of a polar-tracking curved linear Fresnel lens for twoseasonal changes—namely, θy-z = 0 deg (equinox), and θy-z = 23.4 deg (solstice).18

A curved linear Fresnel lens concentrator with minimal tracking requirementshas been described by Leutz et al.19 The resulting geometric concentration ratio is

260 Chapter 9

on the order of 1.5 to 2.0 and is usable for photovoltaic applications. The edge-rayprinciple is applied such that light at two orthogonal acceptance angles is directedto a linear absorber. The prisms are designed at an angle of minimum deviation.The concentrator is oriented with the linear axis in the east-west direction with aseasonal tilt. Acceptance half-angles are approximately ±45 deg in the east-westdirection and ±30 deg in the orthogonal direction.

9.8 Flexible Fresnel Lens Solar Concentrators

9.8.1 Sectional planar solar concentrators

In Fig. 9.23, a sectional refractive lens solar concentrator uses thin, flexible, linear,refracting Fresnel lenses supported by a frame.20 The lenses have the smoothsurface toward the sun and the grooved surface toward the absorbing target, whichmay be a linear heat exchanger or a linear array of photovoltaic cells. The lensesare held under light tension and are made from a material such as 0.4-mm-thickpolymethylmethacrylate (PMMA). Sections A and A′ are inclined at ϕA = 45 deg,and sections B and B′ are inclined at ϕB = 13.75 deg. The refracting grooves aredesigned for minimum deviation according to Eqs. (9.13a) and (9.13b). The lensesare allowed to flex and bow under wind, gravity, or other environmental factors.Due to the increased groove-angle tolerance of minimum deviation, it is estimatedthat reasonable focus will be maintained when the lens bows up to ±5 deg, wherethe bow angle is measured from the tangent point of the curve at the edge of eachsection.

9.8.2 Inflatable curved solar concentrators

For some solar applications in space, a rigid curved Fresnel lens can be replacedby an inflatable flexible Fresnel lens.21 In Fig. 9.24, the inflated solar concentrator

Figure 9.23 A sectional flexible Fresnel lens solar concentrator.20

Fresnel Lenses 261

Figure 9.24 An inflatable linear Fresnel lens solar concentrator.21

has the shape of a cylinder. The cylinder consists of two curved flexible reflectingsections, such as aluminized polyester film, and a transparent Fresnel lens section.The transparent forward section consists of a flexible polymer with a series of linearFresnel grooves formed on its interior surface and a smooth outer surface. Thereflective flat side sections are the same material as the curved reflecting section.These sections are joined at the contiguous edges to a rigid back section that is astrong thermal conductor. The entire concentrator assembly is then inflated witha low-pressure gas, such as hydrogen, helium, or nitrogen, through a valve on theback section. When the Fresnel lens section is oriented toward the sun, a linearfocus is formed on the back section. When the focal length f is greater than thecenterline axis of the cylinder, the Fresnel lens microprisms can be designed forminimum deviations as per Eqs. (9.13a) and (9.13b), resulting in a lens that ismore tolerant to microprism slope errors. The linear receiver would have a 5- to10-cm width for a photovoltaic array, and the back plate would conduct and radiateexcess heat into space. For the deflated concentrator, the flexible sections could befolded against the rigid back plate and then deployed by inflation in space.

9.9 Fresnel Lenses Using Total Internal Reflection

As the entrance angles become larger, a conventional Fresnel lens may no longerfunction as a refracting lens [see Figs. 9.3(b) and 9.4(b)]. In this region, a newclass of Fresnel lens can be used that focuses by a combination of refraction andtotal internal reflection. These are often called catadioptric Fresnel lenses or TIRFresnel lenses. They are based on the TIR prism design described in Sec. 4.1.2, but

262 Chapter 9

the preferred prism material is a plastic such as acrylic. Figure 9.25 depicts severalgrooves of a single-element catadioptric Fresnel lens.

9.9.1 Low-profile overhead projector

A catadioptric Fresnel lens can be used with a conventional refracting Fresnellens to form a two-element condensing lens in a low-profile overhead projector.22

Figure 9.26 illustrates the optical layout, which uses an off-axis section of thecatadioptric Fresnel lens. This allows the lamp to be positioned outside the clearstage area for cooling purposes and avoids any transition area between refractingand catadioptric regions of the lower lens. Because the condenser lens is closeto the image stage in an overhead projector, it would be difficult to match thetransmission at the boundary of these regions, and a brightness difference wouldbe visible on the projected stage area. The catadioptric Fresnel lens is designed toprovide light with a smaller ray angle to the centered refractive upper Fresnel lens.The stage aperture is normally about 285 mm2, and the height of the projector baseis between 75 mm and 100 mm.

9.9.2 Curved catadioptric Fresnel lenses

Much of the earlier work on curved catadioptric Fresnel lenses was for use inautomobile taillights (Fig. 9.27).23 Where a catadioptric Fresnel lens is used asa compact illuminator or searchlight, it is possible to use a centered single-elementFresnel lens with both dioptric and catadioptric regions. Modern plastic replicationtechniques have generated new interest in this type of lens. Figure 9.28 shows aplastic curved TIR circular diverging Fresnel lens having a height-to-diameter ratio

Figure 9.25 Groove detail of a single-element catadioptric Fresnel lens.

Fresnel Lenses 263

Figure 9.26 Low-profile overhead projector using an off-axis catadioptric Fresnel lens.22

Figure 9.27 A curved catadioptric Fresnel lens used as an automobile taillight.23

264 Chapter 9

Figure 9.28 A plastic high-acceptance-angle curved catadioptric Fresnel lens.24

on the order of 0.25 or less.24 The lens is capable of collecting light over almost afull hemisphere, with a 90-deg deviation angle near the edge of the lens. The outputlight can also be designed to provide convergent or collimated light. This type oflens can be provided with curved facets and could also be used as a light collector.25

9.9.3 Photovoltaic solar concentrator using total internal reflection

An integral three-zone circular solar concentrator lens for a single photovoltaic cellhas been designed. The lens is composed of three adjacent zones or regions.26 Thecentral region is a continuous-refracting conic surface, described by Eq. (9.1), witha1 = a2 = a3 = a4 = 0, with an exit-ray-angle range between 0 and 15 deg. Theintermediate zone is a refractive Fresnel lens with an exit-ray-angle range fromabout 15 to 30 deg. The outer zone is a catadioptric or TIR Fresnel lens with anexit-ray-angle range from about 30 to 70 deg. The square entrance aperture has aside length of 120 mm, and a 5.5 mm2 photovoltaic cell is placed 61.56 mm belowthe top center of the lens, yielding a geometric concentration ratio ≈ 476×.

Each of the regions is designed by the edge-ray principle to focus a solar raywith acceptance half-angle of θ0 = ±0.84 deg to each edge of the photovoltaiccell. (See Sec. 9.7 on the use of the edge-ray principle for a solar concentrator.)The incident surfaces of the center- and intermediate-refracting regions lie on acommon horizontal plane, but the incident surface for each groove of the outerTIR lens region is angled. The lens has a staircase shape in this region. The opticallosses for this solar concentrator are ≈ 8% from surface reflection losses, ≈ 6%from misdirected rays, and ≈ 2% from blocked rays. For an acrylic (PMMA) lens,the chromatic dispersion was determined at λ = 300, 600, and 1200 nm, and wasestimated to contribute to about 2% additional loss in efficiency. By control of thefacet angles, an overall optical throughput efficiency of ≈ 81% can be achieved.

9.10 Fresnel Lenses for Rear-Projection Screens

Rear-projection displays are of two general types: those with the projection lensaxis centered with the projection screen normal, and those where the projectionlens is off axis with respect to the projection screen normal. The centeredsystem of Fig. 9.29(a) uses a circular Fresnel lens in conjunction with a bulk-diffusing or matte surface screen. The use of a collimating Fresnel lens produces

Fresnel Lenses 265

a more uniform distribution of light over the screen-viewing area, as shown inFig. 9.29(b).27 In some cases, the separate Fresnel lens and diffusing screen canbe replaced by a single element with the Fresnel grooves on the input side and asurface relief microstructure diffuser on the output side.28 This reduces cost andsuppresses ghosting due to spurious surface reflections.

Much of the recent research is in off-axis rear projection, driven by the goalof designing thinner rear-projection television systems. For this application, off-axis catadioptric Fresnel lenses are extremely useful. Figure 9.30 shows a basicarrangement for an off-axis system with the grooves facing the light source. Itis realized that off-axis projection produces a distorted image on the screen thatcannot be corrected by the Fresnel lens. The device that produces the projectedimage—e.g., a DMD or LCOS imager—must provide a predistorted image tocompensate for the off-axis projection. Another property of the catadioptric Fresnellens for rear projection is depicted in Fig. 9.31(a), where the order of inputted raysis reversed at the diffuser. This can cause some loss of resolution in the projectedimage. Figure 9.31(b) shows how the order can be preserved by an outward curvingof the TIR facets.29

Some Fresnel lenses for off-axis projection use a catadioptric region for thehigh entrance angles and a contiguous refractive region for the lower ray-entranceangles. Figure 9.32 depicts an arrangement where the input surface is planar and thegrooves face the diffusing sheet.30 A light transmission match is attempted at theboundary between these regions. Figure 9.33 illustrates a rear-projection Fresnellens having grooves on both sides, with catadioptric and refracting regions, and atransition region.31

9.11 Fresnel Lens Manufacture

Fresnel lenses are usually replicated in plastic from a master that is produced ona lathe using a precision diamond tool (diamond-turning machine). Compressionmolding produces the highest quality in terms of groove sharpness and fidelity.This is usually the technique used for individual circular lenses, but it is sizelimited and requires cycle times of several minutes. A newer high-precisionmolding (HPM) process reduces the cycle time to seconds, while maintainingthe quality of compression-molded lenses.28 Injection molding and castingare alternative techniques. For linear Fresnel lenses, embossing techniquescan produce continuous elements of good quality. Photoreplication is anothermanufacturing technique for certain types of Fresnel lenses. Standard surface-tolerance parameters used for glass prisms are not always relevant for plasticFresnel lenses. Since manufacturing methods are under continuous improvementand modification, practitioners in Fresnel lens manufacture and design should beconsulted on these topics.

9.12 Achromatic Fresnel Lenses

Achromatic Fresnel lenses are useful in several applications, and there are severalmethods to design and produce achromatic Fresnel lenses.

266 Chapter 9

Figure 9.29 (a) A rear-projection screen using a centered refractive Fresnel lens. (b)Transmitted luminance distribution for a centered Fresnel lens rear-projection screen.(Adpated from Ref. 27 with permission by The Society for Information Display.)

Fresnel Lenses 267

Figure 9.30 A rear-projection screen with off-axis projection and a catadioptric Fresnellens.

9.12.1 Combination of high- and low-dispersion materials

One method, used in a virtual image display, employs a sealed acrylic Fresnellens doublet with the grooves facing inward, with a high-dispersion liquid fillingthe space between.32 The intervening liquid lens is negative in power. Typically,the lower-dispersion acrylic lens material has a ν-number ≈ 57.4, and the higherdispersion liquid has a ν-number ≈ 32.5.

9.12.2 Achromatic catadioptric Fresnel lenses

Using the principles for a single dispersing prism described in Sec. 4.1.2, it ispossible to design a catadioptric Fresnel lens that has positive, negative, or nodispersion. The sample groove shown in Fig. 9.34 produces no dispersion andis therefore achromatic. An isosceles triangle is formed by the refracting grooveangle α, TIR groove angle β, and apex angle γ, where β = γ. The lens will then beachromatic for any incident ray angle θ1, where

α = θ1 − θ3, (9.14a)

β =(180 deg − α)

2. (9.14b)

Single Fresnel lenses of this type would be most useful as off-axis sections withhigh incidence angles θ1, and the geometric throughput losses may require somerelaxation of perfect achromaticity.33

By constructing a doublet consisting of two catadioptric Fresnel lenses, it ispossible to provide an achromatic lens that is usable at lower incident ray angles.34

268 Chapter 9

(a)

(b)

Figure 9.31 (a) Order reversal of inputted rays for a catadioptric Fresnel lens. (b)Preservation of ray order using curved TIR facets.29

Fresnel Lenses 269

Figure 9.32 Rear-projection screen with Fresnel lens grooves facing the diffuser, withrefractive and catadioptric regions.30

Figure 9.33 Rear-projection screen with a Fresnel lens having grooves on both sides, anda refractive/catadioptric transition region.31

Figure 9.35 shows a typical groove pair, where

α3 = θ4 − θ1, (9.15a)

β2 =(180 deg − α3)

2, (9.15b)

α4 = θ4 − θ6, (9.15c)

β5 =(180 deg − α4)

2. (9.15d)

270 Chapter 9

Figure 9.34 Catadioptric Fresnel lens producing no dispersion.

Figure 9.35 Achromatic catadioptric Fresnel lens doublet useable at low-incident rayangles.33

It is also possible to combine a catadioptric Fresnel lens element havingnegative dispersion with a positive refractive Fresnel lens element (having positivedispersion), as shown in Fig. 9.36. Both elements are the same material, suchas acrylic plastic (nd = 1.492). By proper choice of the groove angles, the netdispersion can be brought close to zero. Both elements are acrylic plastic, andrepresentative groove angles α3 groove pair are: α1 = 25.2 deg, β2 = 71.9 deg,

Fresnel Lenses 271

Figure 9.36 Achromatic Fresnel lens doublet using catadioptric and refractive elements.34

α3 = 12.2 deg, and θ1 = 63 deg, θ2 = 20 deg, θ3 = 26.5 deg. Also,

α1 < (θ1 − θ2), (9.16a)

β2 < (180 deg − α1), (9.16b)

θ2 < θ3. (9.16c)

Alternatively, a catadioptric Fresnel lens having positive dispersion can becombined with a negative-refracting Fresnel lens to provide an achromaticdoublet.35

9.12.3 Dispersion-compensated achromatic Fresnel lens

A positive-refracting Fresnel lens can be made achromatic by placing a diffractivestructure on the Fresnel lens grooves, since the dispersion of a grating is ofthe opposite sign to that of a refractive lens.36 Figure 9.37 illustrates a Fresnellens singlet with a diffractive structure on the individual refracting grooves. Thediffractive grooves are referenced to the base groove, which would normally bemachined prior to the machining of the diffractive grooves. The base groove angleβ and the blazed diffractive groove angle β′ are referenced to a common line, whereβ′ is the angle that gives the same exit-ray angle at the blaze wavelength λB. Theblaze angle α is referenced to the base groove angle, where

α = β′ − β, (9.17)

and the diffractive groove period Λ is measured along the base groove length.

272 Chapter 9

Figure 9.37 Dispersion-compensated Fresnel lens with diffractive structure.36

Figure 9.38 Detail of diffraction parameters on a refracting groove.

The refraction, diffraction, and deviation angle at the surface of a diffractinggroove is detailed in Fig. 9.38. The chromatic dispersion of the diffractive structureis of opposite sign to that of a purely refracting groove, and can be specifiedbetween two wavelengths, λ1 and λ2, by the first-order grating equation:

n′ sin θ′dif = n sin θdif +λ

Λ, (9.18)

where

n′ = refractive index of the exit medium at wavelength λ,θ′dif = angle of diffraction, referenced to the base groove (grating plane),

n = refractive index of the incidence medium at wavelength λ,θdif = angle of incidence, referenced to the base groove (grating plane).

Fresnel Lenses 273

When the exit medium is air, the angle of diffraction θ′dif at wavelength λ can becalculated from

θ′dif = arcsin

(n sin θdif +

λ

Λ

), (9.19)

where Λ = diffractive groove period.The grating is blazed when Snell’s law is applied to the grating facet:

n sin θ = n′ sin θ′, (9.20)

where

θ = θdif + α = angle of incidence, referenced to grating facet,θ′ = θ

′dif + α = angle of incidence, referenced to grating facet.

The blaze wavelength λB is calculated from Eq. (9.18):

λB = Λ[sin(θ′ − α) − n sin(θ − α)

]. (9.21)

The diffractive chromatic angular dispersion ψ at the surface is defined by

ψ = ϕ′(λ1) − ϕ′(λ2), (9.22)

where

ϕ′(λ1) = exit angle for light of wavelength λ1,

ϕ′(λ2) = exit angle for light of wavelength λ2.

The first-order diffraction efficiency η of the blazed grating at wavelength λ is

η = sinc2[π

(λB

λ− 1

)]. (9.23)

By adjustment of Λ, it is possible to calculate a value of ψ that offsets the refractivedispersion in a Fresnel lens system, resulting in a significant reduction in thechromatic dispersion of the lens.

9.12.4 Design example: achromatic dual-grooved Fresnel lens foroverhead projector

For some overhead-projector applications, it is possible to use a single-elementFresnel lens with grooves on both sides instead of the conventional Fresnel lensdoublet.37,38 See Fig. 9.39, where f = 189.7 mm, f ′ = 434.0 mm, and thelens diameter A = 400 mm. The lens material is acrylic plastic (nd = 1.492 at

274 Chapter 9

Figure 9.39 A dual-grooved Fresnel lens for use in an overhead projector.

λ = 0.5876 µm). The red and blue wavelengths chosen for the color correctionwere λ1 = 0.656 µm (red C line), and λ2 = 0.486 µm (blue F line). The refractivegroove width W is fixed at W ≈ 0.5 mm.

As shown in Fig. 9.40, the diffractive structure is placed on the exit surface ofthe lens, and the blazed diffraction grating equations are applied to this surface.Initial grating period and blaze angles are supplied to the last groove, and bysuccessive iterations, the values are recalculated for each adjacent inward grooveuntil the angular chromatic dispersion ψ is reduced to less than 0.001 deg over theentire lens. Simultaneously, the difference between the refracting design exit angle(yellow light) and the diffraction exit angle is held to less than 0.006 deg over theentire lens.

Figure 9.41(a) displays the calculated diffractive groove period Λ over the entirelens, from 0 ≤ x ≤ 200 mm. Figure 9.41(b) plots the variation in the region wherediffraction becomes a significant factor in color correction, chosen as Λ ≤ 25 µm.Figure 9.41(c) plots the corresponding blaze angle α variation, and Fig. 9.41(d)shows the calculated angular dispersion for the refractive lens and the primaryand secondary colors for the dispersion-compensated lens. Refractive color maydominate near the lens center, where the diffraction effect is small, but in any case,the dispersion is small in this region. The diffraction efficiency η is calculated to be≈ 0.94 over the entire lens, averaged between the three wavelengths λ, λ1, and λ2.

9.12.5 Achromatic zone plate using a Fresnel lens

A diffractive Fresnel zone plate can be combined with a refractive Fresnel lensto provide a color-corrected Fresnel zone plate. Here, it is desired to correct the

Fresnel Lenses 275

Figure 9.40 Dual-grooved Fresnel lens with diffractive structure on an exit surface.

(a) (b)

(c) (d)

Figure 9.41 (a) Diffractive groove period Λ over the entire lens. (b) Diffractive grooveperiod Λ in a region where Λ ≤ 25 µm. (c) Blaze angle α variation in region where Λ ≤ 25 µm.(d) Uncorrected refractive color and dispersion-compensated color over the entire lens.

276 Chapter 9

inherent chromatic aberration of the zone plate by use of the oppositely dispersingFresnel lens. The design of such an achromatic zone plate for use in the UV andx-ray regions has been described and is shown in Fig. 9.42.39 The material is silicon(refractive index n ≈ 2.37). The power of the Fresnel lens is much less than that ofthe zone plate, retaining the high resolution of the zone plate. For high throughput,the thickness of the Fresnel lens can be reduced, and any introduced phase errorsare cancelled by choice of the zone positions on the plate. The silicon zone plateand silicon Fresnel lens are fabricated on a single silicon substrate.

9.13 Diffraction and Coherence Effects in Fresnel Lenses

9.13.1 Diffraction compensation in a Fresnel lens reflector

Where the light is incoherent and the microprism size is on the order of 0.5 to0.1 mm, diffraction effects in Fresnel lenses are not usually of concern. However,when the groove pitch approaches the spatial coherence length of the light,diffraction effects are apparent. Distinct diffraction orders appear near the focalpoint of the lens or reflector. An analysis has been performed on a first-surfaceFresnel reflector that minimizes these diffraction orders by randomizing the groovewidth of the reflector.40 Both coherent laser light and partially coherent sunlightwere considered. Figure 9.43 represents a completely randomized reflecting groovepattern for a 1.25-cm-aperture Fresnel reflector with a focal length of 20 cm, wherethe groove geometry is specified in terms of the groove height and the groovewidth. The groove width was randomly varied between 50 and 100 µm over thelens aperture, with a mean width of 78 µm and a standard deviation of 9.217 µm.Compared with a 78-µm fixed-groove-width reflector of the same aperture andfocal length, the systematic diffraction orders at the focus were substantiallyreduced.

Figure 9.42 Achromatic zone plate with Fresnel lens dispersion compensation.39

Fresnel Lenses 277

Figure 9.43 Representation of a Fresnel lens reflector having random groove widths.40

9.13.2 Phase-optimized Fresnel lens

Figure 9.44 illustrates the refracted wavefronts through several adjacent groovesof a Fresnel lens where collimated light is incident on the grooved surface. If theheights of the grooves are controlled such that the refracted wavefronts align withthe refracted wavefronts of the adjacent grooves, then the emerging wavefrontswill be in phase across the lens, and all rays will superpose coherently at the focuspoint for a given wavelength λ.41 This is called a phase-optimized Fresnel lens, or“tuned” Fresnel. Here, I is the angle of incidence, I′ is the angle of refraction, α isthe groove angle, h is the groove height, and α = I′. From the geometry and Snell’slaw, it is seen that λ′ = λ/ cos(I − I′), and the wavefronts of adjacent grooves havea phase difference ∆ϕ, where

∆ϕ = 2πh(

1λ′−

1λ

). (9.24)

For the wavefronts of adjacent grooves to be in phase, the groove height h isadjusted such that

h = k(

λ

n cos(I − I′) − 1

), where (k = 1, 2, 3, . . .). (9.25)

For conventional Fresnel lenses at a visible wavelength, the groove depths needbe modified by only a small percentage. In fact, the height change is so small

278 Chapter 9

Figure 9.44 Phase-compensated Fresnel lens. (Adapted from Ref. 41 with permissionfrom the Optical Society of America.)

compared to the normal vertical groove height h ≈ w tanα that Vanucci describesa laser interferometric process to monitor the cutting depth of the diamond toolduring the master die fabrication.41

9.13.3 Phase-optimized Fresnel lens for use in an IR intrusion detector

Figure 9.45 illustrates another method for providing a phase-optimized Fresnellens.42 Collimated light is incident on the plano surface, and the rays are focusedto a detector for use as an IR intrusion device. A series of zones is composed ofrefracting facets, each having an aspheric curvature. For each zone, the distance dbetween the facet and the focal point is such that d = (i + k)λ, where i is an integer,k is a fraction between 0 and 1, and λ is the design wavelength. The value of i isconstant within a zone and varies between zones, while the value of k is constantover the entire lens. Each facet is adjusted to be in phase with the preceding facet byadjustment of the facet curvature and thickness. Shifts between zones occur whenthe facet thickness approaches a target value, where the I value for the next zoneis adjusted by an integral value. A preferred lens material is polyethylene, havinga refractive index n ≈ 1.51, with transmittance in the 0.7–1.4-µ m range.

9.14 Design of a Fresnel Lens Illuminator Using GeneticAlgorithms

In Fig. 9.46, a single Fresnel lens is used to illuminate a reading area from anarray of LED light sources.43 A symmetric array of five LED light sources is fixed

Fresnel Lenses 279

Figure 9.45 Phase-optimized Fresnel lens for intrusion detector.42

Figure 9.46 Multiple-LED illumination system using a Fresnel lens.43

in position 107.25 mm above a Fresnel lens, with the grooves facing the lightsources and having a diameter of 200 mm and a fixed groove width of 0.5 mm.A circular reading area is placed 56.5 mm below the Fresnel lens and is dividedinto Nr equal-area rings for measurement of the illumination uniformity. Conicalreflectors surrounding each LED light source and between the LED array and theFresnel lens ensure that all emitted rays are directed to the Fresnel lens.

280 Chapter 9

Figure 9.47 (a) The optimized integer groove angles of the evolved Fresnel lens with 1000rays emitted from each LED, compared with the groove angles of the conventional Fresnellens. (b) A cross section of the optimized Fresnel lens.43

The object is to calculate a set of groove angles for the Fresnel lens thatmaximizes the illumination on the reading surface with an acceptable degree ofuniformity. To accomplish this, genetic algorithms (GAs) are used as a search andoptimization technique. The goal is to maximize a performance index J, where

J = I − Ip, (9.26a)

and

I = 5N − 2(5N − Rt), (9.26b)

where I is the effective number of rays hitting the reading surface, Ip is a penaltyindex for less uniform distribution of light rays over the reading surface, N is thenumber of uniformly distributed rays emitted by each LED, and Rt is the totalnumber of rays incident on the reading surface. The GAs search for a set of nparameters that maximize J. The primary design parameters are the groove anglesof the Fresnel lens, with the initial population set being the groove angles of aconventional single-focus Fresnel lens. A parameter represented by a set of mbinary digits is called a gene, and the n genes representing the n parameters are

Fresnel Lenses 281

formed as a binary string, called a chromosome. These chromosomes evolve byan iterative process into generations. The chromosomes are evaluated by a fitnesscriterion, and the mutated chromosomes that are fitter pass their traits to the nextgeneration. By this process, the best chromosome is produced, which representsthe optimal parameter value.

For this analysis, the number of groove angles Ng = 20 and Nr = 4. Raysare traced using the TraceProTM optical design program.44 It was found that anoptimized Fresnel lens could be obtained that improved the illumination anduniformity at the reading surface. Figure 9.47(a) shows the optimized integergroove angles of the evolved Fresnel lens with 1000 rays emitted from each LED,compared with the groove angles of the conventional Fresnel lens. Figure 9.47(b)illustrates a cross section of the optimized Fresnel lens.

References

1. D. F. Vanderwerf, “Approximating the Fresnel lens,” Electro-Optical SystemsDesign 14, 47–51 (1982).

2. J. B. Scarborough, Numerical Mathematical Analysis, 6th ed., 545–551, JohnHopkins Press, Baltimore, MD, (1966).

3. Y. A. Dudnikov et al., “The design of a large-diameter Fresnel condenser fromFresnel lenses,” Sov. J. Opt. Technol. 42(5), 451–454 (1975).

4. D. F. Vanderwerf, “Ghost-image analysis of a Fresnel lens doublet,” Proc.SPIE 1331, 143–157 (1990) [doi:10.1117/12.22674].

5. R. F. Bard, “Solar furnace,” U.S. Patent No. 3,985,118 (1976).

6. K. Yass and H. B. Curtis, “Low-cost air-mass 2 solar simulator,” NASA-TM-X-3059, NASA John H. Glenn Research Center, Cleveland, OH (1974).

7. E. Delano, “Primary aberrations of meniscus Fresnel lenses,” J. Opt. Soc. Am.66(12), 1317–1320 (1976).

8. E. Delano, “Primary aberration contributions for curved Fresnel lenses,” J.Opt. Soc. Am. 68(10), 1306–1309 (1978).

9. W. A. Kleinhans, “Aberrations of curved zone plates and Fresnel lenses,” Appl.Opt. 16(6), 1701–1704 (1977).

10. F. Erismann, “Design of a plastic aspheric Fresnel lens with a spherical shape,”Opt. Eng. 36(4), 988–992 (1997) [doi:10.1117/1.601292].

11. D. F. Vanderwerf et al., “Linear refractor/reflector solar concentrators,” Proc.SPIE 161, 23–28 (1978).

12. M. J. O’Neill, “Solar concentrator and energy collection system,” U.S. PatentNo. 4,069,812 (1978).

13. W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators,Academic Press, New York (1978).

282 Chapter 9

14. E. M. Kritchman et al., “Highly concentrating Fresnel lenses,” Appl. Opt.18(15), 2688–2695 (1979).

15. E. M. Kritchman, “Color-corrected Fresnel lens for solar concentration,” Opt.Lett. 5, 35–37 (1980).

16. S. Shanmugam, “Design of a linear Fresnel lens system for solarphotovoltaic electrical power source,” Proc. SPIE 4572, 556–564 (2001)[doi:10.1117/12.444226].

17. E. Lorenzo and A. Luque, “Fresnel lens analysis for solar energy applications,”Appl. Opt. 20(17), 2941–2945 (1981).

18. E. M. Kritchman, “Linear Fresnel lens with polar tracking,” Appl. Opt. 20(7),1234–1239 (1981).

19. R. Leutz et al., “Design of a nonimaging Fresnel lens for solar concentrators,”Solar Energy 65(6), 379–388 (1999).

20. R. H. Appeldorn, “Refracting solar energy concentrator and thin flexibleFresnel lens,” U.S. Patent No. 4,848,319 (1989).

21. M. J. O’Neill and A. J. McDanal, “Inflatable Fresnel lenses as concentratorsfor solar power,” NASA Tech Brief, Lewis Research Center (1999). See alsoM. J. O’Neill, “Inflatable Fresnel lens solar concentrator for space power,”U.S. Patent No. 6,111,190 (2000).

22. J. C. Nelson and D. F. Vanderwerf, “Catadioptric Fresnel lens,” U.S. PatentNo. 5,446,594 (1995).

23. R. N. Falge, “Lens,” U.S. Patent No. 2,023,804 (1935).

24. W. A. Parkyn and D. G. Pelka, “Compact non-imaging lens withtotally internally reflecting facets,” Proc. SPIE 1528, 70–81 (1991)[doi:10.1117/12.49131].

25. W. A. Parkyn et al., “Faceted totally internally reflecting lens with individualcurved faces on facets,” U.S. Patent No. 5,404,869 (1995).

26. C. P. Liu et al., “Optical design of a new combo solar concentrator,” Proc.SPIE 7423, 74230X (2009) [doi:10.1117/12.832320].

27. A. Davis et al., “Fresnel lenses in rear projection displays,” SID Digest 32,934–937 (2001).

28. M. F. Foley, “Technical advances in microstructured plastic optics for displayapplications,” SID Digest 30, 1106–1109 (1999).

29. Y. Huang, “Total internal reflection Fresnel lens and devices,” U.S. PatentNo. 7,230,758 (2007).

30. F. R. Engstrom, “Total internal reflection Fresnel lens and optical system usingthe same,” U.S. Patent No. 7,350,925 (2008).

31. M. D. Peterson and J. A. Gohman, “Fresnel lens for use with rear projectiondisplay,” U.S. Patent No. 6,804,055 (2004).

Fresnel Lenses 283

32. A. Cox, “Application of Fresnel lenses to virtual image display,” Proc. SPIE162, 130–137 (1978).

33. D. F. Vanderwerf, “Achromatic catadioptric Fresnel lenses,” Proc. SPIE 2000,174–183 (1993) [doi:10.1117/12.163633].

34. D. F. Vanderwerf, “Catadioptric Fresnel lens,” U.S. Patent No. 5,446,594(1995).

35. D. F. Vanderwerf, “Overhead projector with catadioptric Fresnel lens,” U.S.Patent No. 5,317,349 (1994).

36. K. C. Johnson, “Dispersion-compensated Fresnel lens,” U.S. PatentNo. 5,161,057 (1992).

37. D. F. Vanderwerf, “Dual grooved Fresnel lens for overhead projection,” U.S.Patent No. 4,900,129 (1990).

38. S. K. Eckhardt, “Dual grooved Fresnel lens for overhead projection,” U.S.Patent No. 5,803,568 (1998).

39. W. Yun and Y. Wang, “Achromatic Fresnel optics for ultraviolet and x-rayradiation,” U.S. Patent No. 6,917,472 (2005).

40. D. A. Gregory and G. Peng, “Random facet Fresnel lenses and mirrors,” Opt.Eng. 40(5), 713–719 (2001).

41. G. Vannucci, “A ‘tuned’ Fresnel lens,” Appl. Opt. 25(16), 2831–2834 (1986).

42. I. K. Pasco, “Fresnel lens,” U.S. Patent No. 5,151,826 (1992).

43. W.-G. Chen and C.-M. Uang, “Better reading light system with light-emitting diodes using optimized Fresnel lens,” Opt. Eng. 45(6), 063001 (2006)[doi:10.1117/1.2210472].

44. TracePro, Lambda Research Corporation, Littleton, MA.

Afterword

From the early ground-and-polished prism facets of glass lighthouse Fresnel lensesto modern microreplicated flexible Fresnel lenses for solar concentrators, newapplications of prismatic optical components for light control are constantly beingdeveloped. In addition, innovations utilizing single and compound prismatic andreflective optical components are emerging in the fields of metrology, polarizationcontrol, projection systems, and illumination and display lighting, among others.Several significant applications which may indicate future trends are:

• In the field of electronic projection, handheld LCOS-, DMD-, and LCD-based “cell phone” projectors (picoprojectors) require compact optical systems.Large-screen projection displays are being developed using colored laser-diode sources. Complex monolithic optics (CMO) optical engines are beinginvestigated for digital cinema projectors.• For screen illumination of large flat-panel LED televisions, there are LED light-

guiding optics and new types of color-combining prisms for flat-panel OLED(organic light-emitting diode) displays.• In the field of direct-view microdisplays, LED backlighting optics for handheld

devices such as cell phones, digital cameras, and tablet-type personal computersis an area of continuous development. There are new designs for LCOS-basednear-to-eye (NTE) miniature displays.• Ranging from observation of the very large to the very small, there are giant

segmented mirror telescopes (GSMTs) and new high-contrast stereo prismmicroscopes.• Beam-shaping prisms are being designed for high-power lasers.• New-generation Fresnel lens–based solar concentrating photovoltaic systems

(CPVs) are being designed and sold by several manufacturers.• In the field of machine vision and image processing, multispectral prism-based

smart cameras are being developed.• Microstructured metamaterial prisms exhibiting negative refractive index are

being fabricated, and applications are being proposed.

The techniques and examples presented in this book were intended to providea good background to analyze, evaluate, and understand these types of opticalapplications, among others. It is further hoped that the material in this book mayhave inspired readers to create novel and useful devices utilizing prismatic andreflective optical components.

285

Index

2D tracking, 25845-deg Bauernfeind prism, 5360-deg Bauernfeind prism, 4690-deg

beam-deviating prism, 28polarization-rotating prism, 92total-internal-reflecting prism array,250

AAbbe

number, 2prism, 35

Abbe, Ernst Karl, 35accumulated phase shift, 85achromatic

compound beam expander, 113Fresnel lens, 265N-prism beam expander, 111retarder, 77

afocal telescopic system, 161air-spaced prism pair, 111Amici prism

double, 97roof, 40, 131, 137

Amici, Giovanni, 40anamorphic

beam expander, 107expansion, 30prism

magnification, 104pair, 108system, 103

angularaccuracies, 131dispersion, 34

annealing, 130antireflection coatings, 9aplanatic

prism spectrograph, 152refractions, 152

aspect ratio, 115, 198astigmatism, 16, 177

reduction, 177autocollimator, 131automated defect inspection, 159axicon prism, 116azimuth angle, 80, 89

Bbacklighting, 206, 208

device, 202Bartholinus, Erasmus, 61base-angle tolerance, 131Bauernfeind prism

45-deg, 5360-deg, 46

beamcombiner, 222

catadioptric, 222, 227compression factor, 105compressor, 105expander, 104, 106

achromatic compound, 113anamorphic, 107N-prism, 111

287

Index 288

beam, (continued)fill fraction, 226steering, 107

beam-deviating prism, 90-deg, 28beamsplitter, 222beam-splitting

coating, 162film, nonpolarizing, 74prism, 116

biaxial birefringent polymer layers, 70binocular

head-mounted display, 159surgical loupe, 160

birefringence, 61birefringent

film, 220polarizing film, 71

blazeangle, 271wavelength, 273

blind spot, on-axis, 144block polishing, 135Brewster’s angle, 6, 44, 64, 68, 104,

216Pellin–Broca prism, 37wedge, 42

Brewster, David, 6brightness enhancement, 210

film, 210, 213dual, 217

Broca, André, 36

Ccalcite, 61Cartesian polarizing beamsplitter, 176

wide-angle, 72catadioptric

beam combiner, 222, 227Fresnel lens, 261, 267, 270, 271prism, 113

change in parity, 19charge-coupled device matrix array,

153chromatic dispersion, 264, 272

chromosome, 281clean-up polarizer, 176close-packed hexagonal array, 248coaxial output, 99, 105cold-cathode fluorescent tube, 210collinear output, 98, 106

wavelength tuning, 99color-corrected Fresnel zone plate,

274coma, 16complex reflection coefficient, 87compound parabolic

concentrator, 215reflector, 183

illuminator, 184compound-wedge Risley-type prism,

144compression molding, 265concentration ratio, 258constant deviation

angle, 37dispersing prism, 36

critical angle, 7cube-corner

reflectorhollow, 56solid-glass, 58

retardance, 80cumulative dispersion, 109curved

catadioptric Fresnel lens, 262linear Fresnel lens, 255, 257

Ddeceptive attractors, 127degree

of freedom, 137of polarization, 90

depolarization effect, 169deviation angle, 33, 237, 238diamond-turning machine, 228dichroic layer, 165dielectric optical materials, 3differential interference contrast, 67

289 Index

diffractioncompensation, 276efficiency, 273

diffractivegroove period, 271structure, 271

digitallight processing, 170

projector, 170micromirror device, 170

directionangles, 19, 52cosine, 11, 187

directional output, 202, 203direct-view

display, 159system, 47

direct-vision prism, 97dispersing prism, 33, 152dispersion equations, 2divergence angle, 197double

Amici prism, 97Dove prism, 44Fresnel rhomb, 77Fresnel rhomb polarization rotator,93isosceles total-internal-reflectingprism, 124refraction, 61

double-passlaser interferometer, 149projection, 168

Dove prism, 42, 157double, 44roof, 43rotating, 131

Dove, Heinrich Wilhelm, 42dual-brightness-enhancement film,

217dual-element Fresnel lens, 242dual-grooved Fresnel lens, 273

Eechelle spectrograph, 137edge-ray principle, 257, 258, 260, 264effective

aperture, 56f /#, 9

electric field vectors, 4elliptical light guide, 199ellipticity, 89, 168entrance aperture

hexagonal, 230equilateral

prism, 33triangle-entrance aperture, 228

e-rays, 61étendue, 183

preservation, 184evolutionary prism design, 126exit pupil, 159experiential design, 124external reflection, 5extractor, 196, 202extraordinary refractive index, 61eyepiece, 159–161

Ff /#, 9fabrication

error, 229methods, 135of a Penta prism, 135

Fermat’s principle, 1field of view, 160figure, 130film

birefringent, 220polarizing, 71

brightness-enhancement, 210, 213dual-brightness-enhancement, 217multilayer polarizing beamsplitter,176multilayer thin, 9nonpolarizing beam-splitting, 74reflecting polarizer, 70turning, 206

Index 290

fire polishing, 137Fizeau interferometer, 136flare reduction, 160flatness, 130flexible linear refracting Fresnel lens,

260fluorescent tube, cold-cathode, 210four-mirror

90-deg polarization rotator, 94beam-displacing prism, 25

fourth-order reflection matrix, 25,125, 189

Fresnellens

achromatic, 265catadioptric, 261, 267, 270, 271collimator, 240curved catadioptric, 262curved linear, 255, 257doublet, 242, 244, 245dual-element, 242dual-grooved, 273flexible linear refracting, 260inflatable flexible, 260meniscus, 248phase-optimized, 277, 278planar circular, 248positive aspheric, 237reflective, 245reflective linear, 250reflector, 247refractive, 237, 264total-internal-reflecting, 261, 264

reflection, 4reflector, linear, 250rhomb, 76

double, 77quarter-wave double, 77

rhomb polarization rotatordouble, 93

zone plate, color-corrected, 274Fresnel, Augustin Jean, 4, 76frustrated total internal reflection, 53,

73, 157

GGaussian-light-beam profile, 162gene, 280genetic algorithm, 126, 280geometric

concentration ratio, 257, 259efficiency, 56

Glan–Foucault prism, 63Glan–Taylor prism, 64Glan–Thompson prism, 64glass cube-corner prism, 126glass plate, tilted, 13goniometer, 131gyroscope, ring-laser, 153

Hhalf-wave

plate, 176rhomb retarder, 77

Harting–Dove prism, 42hemispheric model, 190hexagonal

aperture, 58array, close-packed, 248entrance aperture, 230

high-reflectance surfaces, 9histogram, 255hollow

cube-corner reflector, 56light pipe, 179

homogeneous transformation matrix,125

Iimage contrast, 166, 169index-matching fluid, 159indicatrix, 61inflatable flexible Fresnel lens, 260injection molding, 137, 265interference, thin-film, 73interferometer

Fizeau, 136laser, 150Nomarski polarized, 151polarization, 149

291 Index

internal reflection, 6intersection coordinate, 13

matrix, 19inversion, 18inverting prism, 19irradiance, 90isosceles

roof prism, 187total-internal-reflecting prism, 122

isotropic, 61

KKessler direct-vision prism, 98kinematic mount, 137Knoop test, 130

Lladar guidance, 145laser

diode, 106interferometer, 150

double-pass, 149laser-dispersing prism, 44law of reflection, 17LCD projection displays, 166least-squares fit, 243LED light source, 278left handedness, 90Leman prism, 27Leman–Sprenger prism, 27light

pipe, 179, 195transporter, 196tube, 197

light-directing total-internal-reflectingprism, 171

light-guide luminaire, 202light-guiding prism, 158linear

Fresnel reflector, 250scanner, 146

linear-focus solar concentrator, 250liquid crystal on silicon

imager, 160, 173spatial-light modulator, 166

liquid crystal polymer platecompensator, 175

Littrow30/60/90-deg reflecting prism, 47laser-dispersion prism, 47prism, 46

Littrow, Joseph Johann, 46Littrow-type prism, 104longitudinal spherical aberration, 16loupe, binocular surgical, 160

MMacNeille

pair, 217polarizing beamsplitter, 175polarizing beamsplitter cube, 69stack, 71

master die, 228, 229material properties, 130matrix

array, charge-coupled device, 153fourth-order reflection, 25, 125, 189homogenous transformation, 125intersection coordinate, 19methods for design, 125system, 28transposed reflection, 23

meniscus Fresnel lens, 248meridional plane, 10merit function, 103micrometer eyepiece, 132micromirror, 170microprismatic

array, 187light homogenizer, 181

microreplication, 228microscope, 67microstructured anisotropic layer, 220minimum deviation, 34, 101, 243,

255, 260angle, 119

mirror, roof, 25moiré pattern, 211mounting tolerance requirements, 121

Index 292

multilayerpolarizing beamsplitter film, 176thin films, 9

multipass optical cell, 150multiprism dispersive compressors,

109

Nnegative

dispersion, 101group velocity dispersion, 101uniaxial calcite, 61

Nicol prism, 61Nicol, William, 61Nomarski

polarized interferometer, 151prism, 67

Nomarski, Georges, 67nonbirefringent glass, 72noncubic polarizing beamsplitter, 167nonkinematic mount, 137nonpolarizing beam-splitting film, 74nonsequential ray tracing, 180, 191normalized

output, 203transmission, 191

numerical aperture, 168, 183

OOASIS coating, 53oblique rays, 10occlusions, 130off-axis rear projection, 265on-axis blind spot, 144optical

cell, multipass, 150disk reader, 155lighting film, 195path distance, 116

o-rays, 61ordinary refractive index, 61orientation of viewed images, 18orthogonal output, 104overhead projector, 226, 242, 245,

262, 273

Pparaxial approximation, 143Pechan

prism, 53, 124, 126roof prism, 54

Pellin, Phillippe, 36Pellin–Broca prism, 36, 124Penta prism, 38, 124, 125, 131, 135phase

conjugate mirror, 149difference, 155quadrature, 149shifter, 43

phase-coated total-internal-reflectingretarders, 80

phase-compensation coating, 89phase-correction coating, 55phase-optimized

coating, 159Fresnel lens, 277, 278

phase-shiftcoating, 169compensating coatings, 173

phase-shifting prism, 116Philips prism, 165, 166photoreplication, 221, 265Pierre de Fermat, 1planar

circular Fresnel lens, 248polarization converter, 227

plane of incidence, 1polar tracking, 259polarization

beamsplitters, wire-grid, 178converter, 75, 226ellipse, 76interferometer, 149pupil map, 78recycling, 219rotating prism, 90-deg, 92rotation, 167rotator, four-mirror 90-deg, 94wavelength shift, 170

polarization-preserving prism, 82

293 Index

polarized backlight, 219polarizer, wire-grid, 72, 177polarizing beamsplitter, 159

Cartesian, 176cube, 68

wire-grid, 73film, multilayer, 176MacNeille, 175noncubic, 167wide-angle Cartesian, 72wire-grid, 178

Porro prism, 40Type I, 41Type II, 41

Porro, Ignazio, 40Porro–Abbe prism, 41positional mountings, 138positive

aspheric Fresnel lens, 237dispersion, 101

p-polarized light, 4, 64prepolarizer, 159, 178pressed-glass optical prism, 137primitive sections, 127principal

axis, 61plane, 61section, 65

prism45-deg Bauernfeind, 5360-deg Bauernfeind, 4690-deg beam-deviating, 2890-deg polarization-rotating, 92Amici

double, 97roof, 40, 131, 137

array, 90-degtotal-internal-reflecting, 250axicon, 116beam-splitting, 116Brewster’s-angle Pellin–Broca, 37catadioptric, 113design, evolutionary, 126direct-vision, 97

dispersing, 33, 152constant deviation, 36

double isoscelestotal-internal-reflecting, 124Dove, 42, 157

double, 44roof, 43rotating, 127, 131

equilateral, 33four-mirror beam-displacing, 25Glan–Foucault, 63Glan–Taylor, 64Glan–Thompson, 64glass cube-corner, 126Harting–Dove, 42isosceles

roof, 187total-internal-reflecting, 122

Kessler direct-vision, 98laser-dispersing, 44Leman, 27Leman–Sprenger, 27light-directingtotal-internal-reflecting, 171Littrow, 46

30/60/90-deg reflecting, 47laser-dispersion, 47reflecting, 124

Littrow-type, 104magnification, anamorphic, 104Nicol, 61Nomarski, 67pair

air-spaced, 111anamorphic, 108refracting/total-internal-reflecting,113

Pechan, 53, 124, 126roof, 54

Pellin–Broca, 36, 124Penta, 38, 125, 131, 135Philips, 165polarization-preserving, 82

Index 294

prism, (continued)Porro, 40

Type I, 41Type II, 41

Porro–Abbe, 41pressed-glass optical, 137quality, 130reflective dispersing, 98reflector,solid-glass cube-corner, 58reversion, 125rhomboid, 127right-angle, 39, 161Risley, 144Risley-type

compound-wedge, 144roof, 54rotationally symmetric, 116Schmidt, 49Schmidt–Pechan, 54sections, right-angle, 124spectograph, aplanatic, 152spectroscope, 35Sprenger–Leman, 27switch, 153system, anamorphic, 103three-mirror beam-displacing, 21total-internal-reflecting, 113

double isosceles, 124light directing, 171

trichroic separation, 165wedge, 107, 155

anamorphic compressor, 107as scanners, 143

Wollaston, 66, 125, 150, 151x-cube, 168

prismatichollow light guide, 195, 197sheets, 187

prism-based readout, 154pyramidal

angle tolerance, 131error, 133faceted scanner, 141

QQuadCubeTM architecture, 174quarter-wave

double Fresnel rhomb, 77retarder, 175, 227rhomb retarder, 77

Rrandomly polarized light, 89ray tracing, nonsequential, 191readable image, 19rear-projection displays, 264reflecting

Littrow prism, 124polarizer

film, 70, 217sheet, 220, 228

reflectionand translation of skew rays, 17coatings, 9coefficient, complex, 87phase shifts, 7

reflectiveaxicon, two-piece, 87dispersing prism, 98Fresnel lens, 245LCD imager, 166, 174linear Fresnel lens, 250

reflector, hollow cube-corner, 56refracting/total-internal-reflecting

prism pair, 113refraction

and translation of skew rays, 10matrix, 24, 187

refractiveFresnel lens, 237, 264index, 1

extraordinary, 61ordinary, 61

relative phase shift, 8, 85retarded stack filter, 176retarder, 75

achromatic, 77quarter-wave, 175, 227

295 Index

retrace interval, 141retroreflection, 191

efficiency, 57reversion, 19

prism, 125rhomb retarder

half-wave, 77quarter-wave, 77

rhomboid prism, 127right handedness, 90right-angle prism, 39, 161

sections, 124right-handed image, 19ring-laser gyroscope, 153Risley prism, 144

scan patterns, 144Risley-type prism

compound-wedge, 144Risley-type prism, compound-wedge,

144Rochon, Alexis Marie, 67roof

Dove prism, 43mirror, 25prism, 54

Amici, 40, 131, 137array, 190array, 90-deg, 192isosceles, 187

rotatingDove prism, 127, 131refracting square plate, 146

rotationally symmetric prism, 116

Ssagittal

plane, 16ray, 52

Schmidt prism, 49Schmidt, Bernhardt Woldemar, 49Schmidt–Pechan prism, 54, 160scratch and dig standard, 130sectional element, 122

semiconductor laser light source, 155semikinematic mount, 137Sherman-type prism, 98sidelighting, 213skew rays, 10

reflection and translation of, 17refraction and translation of, 10

skew-ray depolarization, 166, 175Snell, Willebrord, 1Snell’s law, 1solar

collimation acceptance angle, 258concentrator, 250

linear-focus, 250furnace, 248simulator, 248

solar-energy concentration, 248solid light pipe, 180solid-glass cube-corner reflector, 58spatial coherence length, 276spectograph, aplanatic prism, 152spherical aberration, longitudinal, 16spinner, 141s-polarized light, 4, 64spot-focus concentrator, 253Sprenger–Leman prism, 27square plate, rotating refracting, 146Stokes parameters, 89–91Stokes, George Gabriel, 89stress birefringence, 130striae, 130surface quality, 130system matrix, 28, 189

Ttabletop lectern projector, 127tangential

plane, 15ray, 50

telescopic laser range finders, 161test plate, 130thermal environment, 121thin-film interference, 73

Index 296

three-mirror beam-displacing prism,21

tilted glass plate, 13total internal reflection, 6

frustrated, 53, 157phase changes, 76

total-internal-reflectingdeviator, 122extractor, 208Fresnel lens, 261, 264prism, 113

array, 90-deg, 250double isosceles, 124isosceles, 122light-directing, 171

retarders, phase-coated, 80touch switch, 157transmissive extractor, 207transport

efficiency, 198factor, 198

transposed reflection matrix, 23trapezoidal

microprism, 208prism, 161

triangular aperture cube-corner, 57trichroic separation prism, 165tunnel

diagram, 30, 43integrator, 179

turning film, 206

two-piece reflective axicon, 87Twyman-Green interferometer, 133,

137Type I Porro prism, 41Type II Porro prism, 41

Uuniaxial stretching, 70

Vvariable achromatic beam deviator, 39virtual image display, 267

Wwavelength-compensation plates, 167wedge prism, 107, 155

anamorphic compressor, 107as scanners, 143

wet out, 212wide-angle Cartesian polarizing

beamsplitter, 72wire-grid

polarizer, 72, 159, 177polarizing beamsplitter, 178

cube, 73Wollaston prism, 66, 125, 150, 151Wollaston, William Hyde, 66

Xx-cube prism, 168ν-number, 2

Dennis F. Vanderwerf has been involved in the fields of opticsand optical engineering for over 35 years. He has held technicalpositions at the Roswell Park Cancer Institute, Buffalo, NewYork, the NASA John H. Glenn Research Center, Cleveland,Ohio, and the 3M Company in St. Paul, Minnesota and Austin,Texas. He has worked in the areas of crystallography, solarradiometry, flow visualization optics, optical solar concentratordesign, lens and projection systems optical design, new product

development, quality assurance, and intellectual property management. He hasreceived the NASA Apollo Achievement Award and the 3M Corporate Circle ofTechnical Excellence Award. He holds a BS in physics from Canisius College,Buffalo, New York, an MS in physics from Ohio State University, Columbus,Ohio, and an MBA from the University of St. Thomas, St. Paul, Minnesota.He has numerous optical journal, trade magazine, and conference proceedingspublications, and is a named or sole inventor on 29 U.S. patents in the fields ofoptics and optical design. His current interests lie in scientific writing, novel opticaltechnology applications, and science and math education. Dennis F. Vanderwerfresides in Austin, Texas.

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