Stray light analysis and control

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Library of Congress Cataloging-in-Publication Data

Fest, Eric C.Stray light analysis and control / Eric Fest.

pages cmIncludes bibliographical references and index.ISBN 978-0-8194-9325-5

1. Optical instruments–Design and construction. 2. Light–Scattering. I. Title.QC372.2.D4F47 2013621.36–dc23

2012049924

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SPIE—The International Society for Optical EngineeringP.O. Box 10Bellingham, Washington 98227-0010 USAPhone: +1 360.676.3290Fax: +1 360.647.1445Email: [email protected]: http://spie.org

Copyright C© 2013 Society of Photo-Optical Instrumentation Engineers (SPIE)

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

The content of this book reflects the work and thoughts of the author(s). Every effort has been madeto publish reliable and accurate information herein, but the publisher is not responsible for thevalidity of the information or for any outcomes resulting from reliance thereon.

Printed in the United States of America.First printing



ContentsPreface xi

Acknowledgments xv

Chapter 1 Introduction and Terminology 1

1.1 Book Prerequities 41.2 Book Organization 41.3 Stray Light Terminology 6

1.3.1 Stray light paths 61.3.2 Specular and scatter stray light mechanisms 71.3.3 Critical and illuminated surfaces 81.3.4 In-field and out-of-field stray light 81.3.5 Internal and external stray light 91.3.6 “Move it or Block it or Paint/coat it or Clean it” 9

1.4 Summary 10

Chapter 2 Basic Radiometry for Stray Light Analysis 13

2.1 Radiometric Terms 132.1.1 Flux, or power, and radiometric versus

photometric units 142.1.2 Reflectance, transmittance, and absorption 162.1.3 Solid angle and projected solid angle 162.1.4 Radiance 182.1.5 Blackbody radiance 182.1.6 Throughput 222.1.7 Intensity 232.1.8 Exitance 232.1.9 Irradiance 242.1.10 Bidirectional scattering distribution function 25

2.2 Radiative Transfer 292.2.1 Point source transmittance 312.2.2 Detector field of view 322.2.3 Veiling glare index 32

v



vi Contents

2.2.4 Exclusion angle 322.2.5 Estimation of stray light using basic radiative transfer 332.2.6 Uncertainty of stray light estimates 36

2.3 Detector Responsivity 362.3.1 Noise equivalent irradiance 362.3.2 Noise equivalent delta temperature 37

2.4 Summary 38

Chapter 3 Basic Ray Tracing for Stray Light Analysis 41

3.1 Building the Stray Light Model 413.1.1 Defining optical and mechanical geometry 413.1.2 Defining optical properties 43

3.2 Ray Tracing 433.2.1 Using ray statistics to quantify speed of convergence 433.2.2 Aiming scattered rays to increase the speed

of convergence 453.2.3 Backward ray tracing 483.2.4 Finding stray light paths using detector FOV 493.2.5 Determining critical and illuminated surfaces 503.2.6 Performing internal stray light calculations 513.2.7 Controlling ray ancestry to increase speed

of convergence 553.2.8 Using Monte Carlo ray splitting to increase

speed of convergence 553.2.9 Calculating the effect of stray light on

modulation transfer function 563.3 Summary 58

Chapter 4 Scattering from Optical SurfaceRoughness and Coatings 61

4.1 Scattering from Uncoated Optical Surface Roughness 624.1.1 BSDF from RMS surface roughness 684.1.2 BSDF from PSD 704.1.3 BSDF from empirical fits to measured data 714.1.4 Artifacts from roughness scatter 72

4.2 Scattering from Coated Optical Surface Roughness 734.3 Scattering from Scratches and Digs 754.4 Summary 75

Chapter 5 Scattering from Particulate Contaminants 77

5.1 Scattering from Spherical Particles (Mie Scatter Theory) 785.2 Particle Density Function Models 80

5.2.1 The IEST CC1246D cleanliness standard 815.2.2 Measured (tabulated) distribution 87



Contents vii

5.2.3 Determining the particle density function using typicalcleanliness levels, fallout rates, or direct measurement 875.2.3.1 Use of typical cleanliness levels 895.2.3.2 Use of fallout rates (uncleaned surfaces only) 895.2.3.3 Use of a measured (tabulated) density function 90

5.3 BSDF Models 915.3.1 BSDF from PAC 915.3.2 BSDF from Mie scatter calculations 925.3.3 BSDF from empirical fits to measured data 925.3.4 Determining the uncertainty in BSDF from the

uncertainty in particle density function 925.3.5 Artifacts from contamination scatter 93

5.4 Comparison of Scatter from Contaminants and Scatterfrom Surface Roughness 95

5.5 Scattering from Inclusions in Bulk Media 955.6 Molecular Contamination 985.7 Summary 98

Chapter 6 Scattering from Black Surface Treatments 101

6.1 Physics of Scattering from Black Surface Treatments 1026.1.1 BRDF from empirical fits to measured data 1046.1.2 Using published BRDF data 1096.1.3 Artifacts from black surface treatment scatter 111

6.2 Selection Criteria for Black Surface Treatments 1126.2.1 Absorption in the sensor waveband 1136.2.2 Specularity at high AOIs 1136.2.3 Particulate contamination 1146.2.4 Molecular contamination 1146.2.5 Conductivity 114

6.3 Types of Black Surface Treatments 1146.3.1 Appliques 1156.3.2 Treatments that reduce surface thickness 1156.3.3 Treatments that increase surface thickness 116

6.3.3.1 Painting 1166.3.3.2 Fused powders 1166.3.3.3 Black oxide coatings 1196.3.3.4 Anodize 119

6.4 Survey of Widely Used Black Surface Treatments 1206.5 Summary 120

Chapter 7 Ghost Reflections, Aperture Diffraction, andDiffraction from Diffractive Optical Elements 123

7.1 Ghost Reflections 1237.1.1 Reflectance of uncoated and coated surfaces 124



viii Contents

7.1.1.1 Uncoated surfaces 1247.1.1.2 Coated surfaces 125

7.1.2 Reflectance from typical values 1267.1.3 Reflectance from the stack definition or

predicted performance data 1287.1.4 Reflectance from measured data 1287.1.5 Artifacts from ghost reflections 1287.1.6 “Reflective” ghosts 131

7.2 Aperture Diffraction 1327.2.1 Aperture diffraction theory 1327.2.2 Calculation of aperture diffraction in stray light

analysis programs 1337.2.3 Artifacts from aperture diffraction 1347.2.4 Expressions for wide-angle diffraction calculations 135

7.3 Diffraction from Diffractive Optical Elements 1377.3.1 DOE diffraction theory 1387.3.2 Artifacts from DOE diffraction 1407.3.3 Scattering from DOE transition regions 140

7.4 Summary 142

Chapter 8 Optical Design for Stray Light Control 145

8.1 Use a Field Stop 1458.2 Use an Unobscured Optical Design 1478.3 Minimize the Number of Optical Elements between the

Aperture Stop and the Focal Plane 1488.4 Use a Lyot Stop 150

8.4.1 Calculating Lyot stop diameter from analyticexpressions 151

8.4.2 Calculating Lyot stop diameter from coherentbeam analysis 152

8.5 Use a Pupil Mask to Block Diffraction and Scatteringfrom Struts and Other Obscurations 153

8.6 Minimize Illumination of the Aperture Stop 1548.7 Minimize the Number of Optical Elements, Especially

Refractive Elements 1548.8 Avoid Optical Elements at Intermediate Images 1558.9 Avoid Ghosts Focused at the Focal Plane 1558.10 Minimize Vignetting, Including the Projected Solid Angle

of Struts 1568.11 Use Temporal, Spectral, or Polarization Filters 1578.12 Use Nonuniformity Compensation and Reflective Warm

Shields in IR Systems 1578.13 Summary 160



Contents ix

Chapter 9 Baffle and Cold Shield Design 163

9.1 Design of the Main Baffles and Cold Shields 1649.2 Design of Vanes for Main Baffles and Cold Shields 167

9.2.1 Optimal aperture diameter, depth, and spacingfor baffle vanes 168

9.2.2 Edge radius, bevel angle, and angle for baffle vanes 1729.2.3 Groove-shaped baffle vanes 172

9.3 Design of Baffles for Cassegrain-Type Systems 1749.4 Design of Reflective Baffle Vanes 1789.5 Design of Masks 1819.6 Summary 181

Chapter 10 Measurement of BSDF, TIS, and SystemStray Light 183

10.1 Measurement of BSDF (Scatterometers) 18310.2 Measurement of TIS 18610.3 Measurement of System Stray Light 188

10.3.1 Sensor radiometric calibration 18810.3.2 Collimated source test 18910.3.3 Extended source test 19010.3.4 Solar tests 191

10.3.4.1 Using direct sunlight 19110.3.4.2 Using a heliostat 192

10.4 Internal Stray Light Testing 19310.5 Summary 193

Chapter 11 Stray Light Engineering Process 195

11.1 Define Stray Light Requirements 19511.1.1 Maximum allowed image plane irradiance and

exclusion angle 19611.1.2 Inheritance of stray light requirements from

comparable systems 19811.2 Design Optics, Pick Surface Roughness, Contamination

Levels, and Coatings 19811.3 Build Stray Light Model, Add Baffles and Black Surface

Treatments 19811.4 Compute Stray Light Performance 19911.5 Build and Test 20011.6 Process Completion 20211.7 Summary 20211.8 Guidelines and Rules of Thumb 202

Index 205



PrefaceIn 1741, the great Swiss mathematician Leonhard Euler was asked by King Fred-erick the Great of Prussia to write a tutorial on natural philosophy and sciencefor his niece, the Princess of Anhalt-Dessau. Euler agreed and began writing thetutorial as a series of letters to the Princess, about one a week, for nearly 250 weeks.These letters were eventually published as a collection and became some of thefirst popular science writing.1

Portrait of Leonhard Euler, by Johann Georg Brucker (1756).

In a letter entitled “Precautions to be observed in the Construction ofTelescopes”2 (shown in the second figure), Euler recommends that the Princess

xi



xii Preface

“. . . (enclose the telescope) in a tube, that no other rays, except those whichare transmitted through the objective, may reach the other lenses. . . If by anyaccident the tube shall be perforated ever so slightly, the extraneous light wouldconfound the representation of the object.”

Excerpts from Leonhard Euler’s tutorial. The figures show the telescope before and afterthe addition of field stops, which were added for stray light control.

He also suggests that she “[. . . ] blacken, throughout, the inside of the telescope,of the deepest black possible, as it is well known that this colour reflects not therays of light, be they ever so powerful”.

Though he calls them “diaphragms” and not field stops, Euler goes on to suggesttheir use as a further means of “diminishing the unpleasant effect of which I havebeen speaking.” This unpleasant effect is, of course, what we now call stray light,and this letter shows that it was identified as a problem hundreds of years ago. It isremarkable that the methods Euler discussed to control it (i.e., the use black surfacetreatments, field stops, and baffles) are still some of the primary methods used tocontrol it today (see Chapters 6, 8, and 9, respectively). Of course, some thingshave changed; Euler and the Princess didn’t have the massive computing powerwe have today, and therefore were unable to predict the stray light performanceof a telescope to the accuracy that is now possible. In addition, the occurrence ofstray light in their telescope was an “unpleasant effect” and was not as serious a



Preface xiii

problem as, say, the loss of scientific data due to stray light in a multi-billion-dollarspace-based telescope.

However, the letter shows that the problem and many of its solutions remainthe same. The goal of Euler’s letter and of this book are similar: to provide opticalengineers with the information and analytical tools necessary to design and buildoptical systems with sufficient stray light control. In addition to Euler’s letter,there have been hundreds of papers published on the subject, and it is impossibleto include the content of all of them here. Therefore, only the content that ismost applicable to the task of optical system engineering is discussed. This isan important distinction, as many previous publications deal with the science ofoptical scattering and stray light, but fewer address the application of this sciencein engineering practice. This book summarizes the important scientific results,providing references for more detailed study, and then applies these theories tothe engineering of optical systems. This book also considers the economics ofperforming stray light analysis, which is a dimension that is also lacking in thecurrent literature. Sometimes the engineer tasked with performing a stray lightanalysis has months of time and a large budget, and other times has 15 minutes andno budget. This book provides tools and solutions for a spectrum of budgets, andquantifies the accuracy associated with each approach.

Eric FestTucson, AZ

February [email protected]

1. T. McGew (Ed.), Discussion of Euler’s “Letters to a German Princess”,http://homepages.wmich.edu/∼mcgrew/euler.htm.

2. L. Euler and N. de Condorcet, Letters of Euler to a German Princess, on DifferentSubjects in Physics and Philosophy, Volume 2, H. Hunter, Trans., translated fromthe French and published by Murray and Highley (1802).



AcknowledgmentsMany people helped me write this book, and I’d like to take a moment to thankthem.

I’d like to thank Dave Rock, who gave me my first job in optics and, to thisday, serves as my role model. Much of the content of this book I learned from him,and I will always be grateful for all he taught me and for the helpful feedback heprovided for this book.

I’d like to thank my co-workers, including Chad Martin, John McCloy, DaveMarkason, and Dave Jenkins, from whom I’ve learned a tremendous amount aboutstray light analysis. Special thanks goes to Mike Schaub, who helped me set up theZemax model of the Maksutov–Cassegrain telescope used throughout this book. I’dalso like to thank Scott Sparrold at Edmund Optics, Margy Green at Raytheon, andMichael Dittman at Ball Aerospace for many fruitful discussions and for providingme with some of the material in this book. I’d also like to thank Chris Staats atSchmitt Measurement Systems for teaching me the intricacies of measuring BSDF.

This book probably would not have happened without the help of Rich Pfistererof Photon Engineering LLC, who encouraged me to write it and provided anexcellent model for it in his Stray Light Short Course Notes. Rich also spent manyhours reviewing it, and I thank him for his tireless efforts.

I was very fortunate to have the help of Bob Breault of Breault ResearchOrganization, who is one of the founders of the science of stray light analysis andwho provided me with many comments and suggestions that greatly improved thisbook. For the many hours he spent reviewing and discussing it with me, I thankhim.

I also owe thanks to the other reviewers of this book, who gave selflessly of theirtime and by doing so greatly improved it: Scott Ellis, Paul Spyak, Rick Juergens,and Matt Jenkins. I’d also like to thank the people at SPIE Press who made thisbook a reality, especially the book’s editor, Scott McNeill, who provided invaluablefeedback and who was very understanding when I asked for schedule extensions.

Last, but certainly not least, I’d like to thank my wife, Gina, who accommodatedmy writing schedule with incredible patience. I am extremely fortunate to bemarried to her.

This book is dedicated to my daughters, Fiona and Marlena.

xv


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Chapter 1

Introduction and TerminologyStray light is defined as unwanted light that reaches the focal plane of an opticalsystem. Figure 1.1 illustrates an example of noticeable stray light. This photographof clear, featureless sky was taken with a digital camera and zoom lens, with the sunjust outside the field of view (FOV). The bright spots (called artifacts) are causedby light from the sun that strikes the camera lens and reaches the focal plane byphysical mechanisms such as scattering from surface roughness, scattering fromparticulate contamination, and ghost reflections. These mechanisms are discussedin Chapters 4, 5, and 7, respectively.

Another example of stray light is shown in Fig. 1.2. This picture, also of clear,featureless sky, was taken with a Maksutov–Cassegrain telescope whose baffleswere shortened to admit stray light. Light from the sun, which is located justoutside the FOV, bypasses the primary and secondary mirror and passes directlythrough the hole in the primary to shine directly on the image plane. This type ofstray light path (called a zeroth-order path) is discussed in Chapters 3 and 9.

The effect of aperture diffraction is shown in Fig. 1.3. Light from the streetlamp inside the FOV diffracts from the iris of the camera and results in the radialstreaks seen in this image. Aperture diffraction is discussed in Chapter 7.

In Figs. 1.1–1.3, stray light in the optical system resulted in unwanted lightin the final image. The artifacts were not intended by the designer of the opticalsystem; they occur because it is not possible to perfectly control the path of lightbetween the scene and the focal plane of the system. The best that the designer cando is to use stray light control techniques to reduce it to a level that ensures properfunctioning of the system. Stray light control is important in all optical systems butespecially in the following scenarios:13

� Observing faint objects in the presence of the sun or other bright sources. Thisscenario is very common, and occurs in a wide variety of optical systems,from consumer cameras to space-based telescopes.

� Making high-accuracy radiometric measurements, such as the type madeby sensors on weather satellites. The presence of stray light in these mea-surements will lower their accuracy, especially if the stray light is not wellcharacterized.

� Projecting and displaying high-contrast images. Stray light in the displaysystem will reduce the contrast ratio of these systems.

1



2 Chapter 1

Scatter from surface roughness and particulate contamination

Ghost reflections

Figure 1.1 Stray light from the sun, which is just outside the lower-left corner of field of view.

� Making images using infrared camera systems, which can be sensitive tostray light from self-emission of the camera system itself.

� Making spectroscopic and other multi-band measurements, because straylight can often result in crosstalk between wavelength bands.

It is important for designers of optical systems to understand the consequencesof stray light on system performance and to take appropriate steps early in thedesign process to control it. Because it is not possible to present a stray lightanalysis of every possible type of optical system, this book describes analytictools (such as mathematical models of optical scatter) and engineering practicesthat can be applied to any system. Examples of the application of these toolsto a variety of systems are provided; an often-used example is the commercialMaksutov–Cassegrain telescope shown in Fig. 1.4.

Direct sunlight

Figure 1.2 Stray light from the sun just outside the lower edge of FOV, shining throughthe hole in the primary mirror of a Maksutov–Cassegrain telescope whose baffles areundersized.



Introduction and Terminology 3

Aperture diffraction

Figure 1.3 Aperture diffraction from a bright source (a street lamp) inside the FOV.

Figure 1.4 An Orion Apex 90 Maksutov–Cassegrain telescope with a Nikon D90 camera,mounted to an equatorial mount. This system is used throughout this book to collect imagesof stray light.



4 Chapter 1

1.1 Book Prerequities

It is assumed that the reader has knowledge of basic radiometry, optical designprinciples, and error analysis. For those not familiar with these topics, there are anumber of good references.1−4 Knowledge of error analysis is especially importantin stray light analysis, as it uses many approximations. In addition to these topics,knowledge of Fourier optics5,6 and Mie scattering theory7 will provide a deeperunderstanding of the material.

Though not required to obtain an understanding of the basic concepts of straylight analysis and design, knowledge of a stray light analysis program is needed toperform all but the most basic analysis. This book considers a stray light analysisprogram as one with the following features (chapter numbers that discuss eachfeature are given for reference):

� The ability to model optical sources of many types, including point andextended, monochromatic and polychromatic, unpolarized and polarized, andincoherent and coherent (Chapter 3).

� The ability to model complex optical and mechanical surfaces (Chapters 8and 9).

� The ability to model the specular reflectance and transmittance of opticalcoatings as a function of angle of incidence (AOI), wavelength, and polar-ization (Chapter 7).

� The ability to trace rays nonsequentially, split them, and aim scattered rays(Chapter 3).

� A variety of scatter models, including the Harvey or ABg model (Chapter4), a contamination scatter model (Chapter 5), and a model suitable formodeling scattering from non-shift-invariant black surface treatments, suchas the general polynominal, tabulated, or scripted BSDF model (Chapter 6).Of these, the most important is the tabulated BSDF model because it can beused to mimic the behavior of any other model.

� The ability to track ray paths, including the ability to isolate them so thatthey can be plotted and analyzed individually (Chapter 3).

� The ability to compute and plot irradiance and intensity distributions(Chapter 3).

� The ability to perform optimization (Chapter 9).� A scripting language that allows repetitive tasks to be automated.

Three such programs available at the time of this writing are FRED,8 ASAP,9

and TracePro.10 This book (in particular, Chapter 3) will suggest ways in whichthese programs should be used for maximum efficiency; however, it is not a tutorialon the details of using each program. For this information, please consult thesoftware documentation. Familiarity with an optical design program such as CODEV12 or Zemax11 is also useful.

1.2 Book Organization

This book is divided into two parts. The first part, which consists of Chapters 1–7,provides the basic principles necessary to model the stray light performance of




optical systems. These principles include basic stray light terminology, radiometry,and the physics of stray light mechanisms. Because this book is intended primarilyfor optical system engineers, the scientific foundations of these principles arepresented in summary form, with references provided for more detailed study. Thefirst part builds familiarity with those mathematical techniques and models thatare available in most commercial stray light analysis programs, which is importantbecause they are the primary tools used by engineers to analyze and control straylight.

The second part of the book, which consists of Chapters 8–11, demonstrateshow the basic principles can be applied in the design, fabrication, and testing phasesof optical system development to ensure that the system has adequate stray lightperformance. Included in this part is a review of the relationship between opticaldesign form and stray light performance, basic baffle design, and stray light testingmethods. Chapter 11 presents a process to design and build optical systems thathave adequate stray light performance. A key component of this process is thedevelopment of system stray light requirements, without which it is difficult tojudge the adequacy of any design.

Throughout the book, the most important equations or concepts are highlightedwith a box around them. In addition, each chapter ends with a summary that containsits most important points.

The Orion Apex 90 Maksutov–Cassegrain telescope shown in Fig. 1.4 isused throughout the book to illustrate stray light artifacts. This telescope has a90-mm entrance pupil diameter and a 1250-mm focal length ( f /13.89). A NikonD90 digital SLR camera with a DX-format focal plane array (23.6 mm ×15.8 mm) is mounted to it to capture images. The system has a FOV of 1.08 deg ×0.72 deg. The approximate optical prescription of the telescope was determinedthrough measurements and entered into the Zemax11 optical design program andFRED; the resulting model is shown in Fig. 1.5. The system, referred to as the“baseline” Maksutov–Cassegrain system throughout the book, is on an equatorialmount that allows its line of sight (LOS) to be easily adjusted to follow the sun.

Corrector lens

Secondary mirror baffle

Secondary mirror

Primary mirror (stop)

y

z

Primary mirror baffle

Main baffle

FPA (NikonDX format)

x (intopage)

Figure 1.5 Approximate optical model in FRED of the Orion Apex 90 Maksutov–Cassegraintelescope system shown in Figure 1.4.



6 Chapter 1

1.3 Stray Light Terminology

1.3.1 Stray light paths

A light path is a unique sequence of events experienced by a beam of light, endingat the image plane. Most optical systems have only one intended light path; however,stray light mechanisms such as reflections from refractive optics (also called Fresnelor ghost reflections) or scattering from surface roughness result in a multitude ofunintended light paths that do not follow the intended one. An example of such astray light path is described as follows: “Light leaves the sun, transmits throughthe first surface of the lens, ghost reflects off of second surface of lens, ghostreflects off of first surface of lens, and transmits through the second surface ofthe lens to the focal plane.” This path is illustrated in Fig. 1.6. As mentionedearlier, it is not possible to eliminate all of these paths; it is only possible toreduce their magnitude, in this case through the use of antireflection (AR) coatings(see Chapter 7).

Paths are often categorized by their order, which refers to the number of straylight mechanisms (or events) that occur in the path. For instance, the path describedabove is a second-order path because it contains two ghost reflection events. Non-stray light events in the path (such as “transmits through lens 1”) are not countedin the order. As will be discussed in Chapters 3 and 9, it is possible for a stray lightpath to have zero order, such as direct illumination of the focal plane. Such a pathis shown in Fig. 3.8 for the baseline Maksutov–Cassegrain telescope. The presenceof a zeroth-order path (also called a “sneak” path) usually indicates that the opticalsystem is inadequately baffled (baffle design is discussed in Chapter 9).

The ratio of the magnitude of light at the end of the path to the magnitude oflight at the beginning of the path is sometimes called the path transmittance andgenerally decreases as an , where a is some number less than 1, and n is the pathorder. Therefore, the lower the order of the path, the more light it produces on thefocal plane, and the more of a concern it is in the design and analysis of the system.The process of designing a system with adequate stray light control essentiallyconsists of identifying its stray light paths and then reducing or eliminating them,beginning with the zeroth-order paths and then paths of each successive order,until the stray light requirement for the system is met. This process is discussed ingreater detail in Chapter 11.

Sun

Lens

Ghost reflection

Ghost reflection

Focal plane

Figure 1.6 Example of a second-order ghost reflection path.




1.3.2 Specular and scatter stray light mechanisms

Stray light mechanisms decrease the path transmittance of the intended optical pathand increase the path transmittance of unintended paths, and generally fall into oneof two categories: specular or scatter. They differ in that light from a specularmechanism is deterministic, obeying either Snell’s laws of reflection and refractionor the grating equation. Both Snell’s laws and the grating equation will be discussedin this chapter.

Snell’s law of reflection states that the angle of the reflected ray relative to thesurface normal of the reflecting surface is equal to the AOI �i , and Snell’s law ofrefraction states that the angle of transmission �′

t relative to the surface normal ofthe refracting surface of all refracted rays is given by

ni sin (�i ) = nt sin (�t ), (1.1)

where ni and nt are the refractive indices of the incident and transmitting media,as shown in Fig. 1.7. The polarization directions labeled in this figure refer tothe plane of oscillation of the electric field of the incident wave: a wave that ispolarized perpendicular to the plane of incidence (which is the plane that containsthe incident ray and the surface normal) is called s-polarized, and a wave that ispolarized parallel to the plane of incidence is called p-polarized.2 As will be shownin Chapter 7, the orientation of polarization is often important when computingthe magnitude of the light specularly reflected or transmitted from an interface. Aconsequence of Snell’s law of refraction is that if ni > nt , then light incident on theinterface at angles greater than or equal to the critical angle �c, computed as

�c = sin−1(

nt

ni

), (1.2)

will undergo total internal reflection (TIR), in which 100% of the light incident onthe interface will be reflected. Ghost reflections, such as those shown in Fig. 1.6,are examples of specular mechanisms that follow Snell’s laws.

Incident ray Reflected ray

Transmitted ray

ni

s-polarization direction (out of page)

p-polarization direction

nt

θi θi

θt

Figure 1.7 Specular reflection and refraction at an index boundary.



8 Chapter 1

Incident rayZeroth-orderray

θi θ0

θ+1

+1 orderray

d

Figure 1.8 Diffraction from a grating.

The grating equation is used to predict the direction of rays diffracted from agrating, and is given by

m�

d= |sin �m − sin �i | , (1.3)

where m is the order of the diffracted ray, � is the wavelength of the incident beam,d is the period of the grating, and �m is the angle that the mth diffracted ordermakes with the surface normal, as shown in Fig. 1.8.

By contrast, scatter mechanisms do not obey Snell’s laws or the grating equa-tion, and the angle of the scattered ray with respect to the surface normal cantake on any value. As an example, see the illustration of scattering from opticalsurface roughness in Fig. 4.2. Light never undergoes a perfect specular reflectionor transmission though a surface, even a highly polished optical surface; there isalways some small amount of scatter. This scatter usually has an impact on straylight performance and is therefore usually modeled. Techniques to model surfacescatter are discussed in Chapters 4–6.

1.3.3 Critical and illuminated surfaces

A critical surface is one that can be seen by the detector (which can be an electronicdetector, a piece of film, the human eye, or some other device), and an illuminatedsurface is one that is illuminated by a stray light source. All optical surfaces in anoptical system are usually critical. In order for first-order stray light to reach thefocal plane, there generally must be at least one surface that is both critical andilluminated (an exception to this rule is internal stray light from self-emission ininfrared optical systems, see Section 1.3.5). This concept, illustrated in Fig. 1.9, iscentral to the process of stray light analysis and design. First-order stray light pathsmust occur on a surface that is both critical and illuminated in order to have anyeffect on the stray light performance of the system.

1.3.4 In-field and out-of-field stray light

Sources of stray light can be either inside or outside the nominal FOV of thesystem, and the stray light that results from these sources is referred to as “in-field”or “out-of-field” stray light, respectively. In-field stray light often manifests itself




Light from source

Surface not critical but illuminated

Surface critical and illuminated

Detector

Surface critical but not illuminated

Baffle vane

Optical system

Detector lines-of-sight

Figure 1.9 Critical and illuminated surfaces.

as a “halo” around a point source in the FOV, demonstrated in Fig. 1.3. Out-of-fieldstray light often manifests itself as irradiance distribution that varies across theFOV, as demonstrated in Fig. 1.1. Ghost reflections result in bright spots (suchas the ones in Fig. 1.1) in either case, though, in the out-of-field case these spotsusually occur only for sources very close to the edge of the FOV.

1.3.5 Internal and external stray light

Sources of stray light can be either internal or external to the optical system itself.Internal stray light sources are usually a concern only in infrared optical systems,in which the self-emission of the sensor itself can result in stray light at the focalplane. Internal stray light is also called thermal background, thermal self-emission(TSE), nearfield background, or nearfield stray light. External stray light is usuallya concern in all optical systems, regardless of the waveband they operate in. Acommon source of external stray light is the sun.

1.3.6 “Move it or Block it or Paint/coat it or Clean it”13

This phrase (or ones similar to it14) summarizes the methods used to control straylight:

� “Move it” usually refers to moving an object so that it is not critical orilluminated, or both (see Chapter 9).

� “Block it” usually refers to inserting a baffle so that an object is not criticalor illuminated, or both (see Chapter 9).

� “Paint/coat it” usually refers to making a surface black to reduce its scatter(see Chapter 6), putting an AR coating on a lens surface to reduce its re-flectance (see Chapter 7), or making a surface smoother to reduce its scatter(see Chapter 4).

� “Clean it” usually refers to reducing particulate contamination on a surfaceto lower its scatter (see Chapter 5).

The effect of each of these methodologies on the stray light performance of asystem is discussed in Section 2.3.6 using a simple analytic model of system straylight.



10 Chapter 1

1.4 Summary

Stray light is defined as unwanted light that reaches the focal plane of an opticalsystem. This book assumes that the reader is familiar with basic radiometry andoptics, and is familiar with one or more stray light analysis programs, such as FRED,ASAP, or TracePro. A stray light path is a unique sequence of events experiencedby a beam of light, ending at the image plane; these events are typically eithertransmission, reflection, diffraction, or scatter from a surface. The order of thestray light path is given by the number of stray light mechanisms (such as ghostreflections, diffraction from a grating, or scatter) that occur in the path, and themagnitude of stray light on the focal plane typically decreases exponentially withthe path order. Stray light mechanisms are either specular or scatter: specularmechanisms (such as ghost reflections and grating diffraction) are deterministicin that the direction rays take after the mechanism are determined by Snell’s lawsof reflection or refraction, or the grating equation; scatter mechanisms (such assurface roughness and contamination scatter) are nondeterministic in that rays cango in any direction after the mechanism. A critical surface is one that can be seenby the detector, and an illuminated surface is one that can be illuminated by straylight source: first-order scatter paths usually require at least one surface that is bothcritical and illuminated. In-field stray light is generated from sources inside theFOV, whereas out-of-field stray light is due to sources outside the FOV. Internalstray light is generated by objects inside the sensor, and is usually only a problemin infrared sensors. External stray light is generated by objects outside the sensor,such as the sun. The phrase “Move It or Block It or Paint/Coat It or Clean It”summarizes the methods used to control stray light.

References

1. W. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York,(2008).

2. E. Hecht, Optics, 4th Ed., Addison-Wesley, Reading, MA (2001).3. J. Palmer and B. Grant, The Art of Radiometry, SPIE Press, Bellingham, WA

(2009) [doi: 10.1117/3.798237].4. J. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in

Physical Measurements, 2nd Ed., University Science Books, Sausalito, CA(1997).

5. J. Gaskill, Linear Systems, Fourier Transforms, and Optics, John Wiley & Sons,New York (1978).

6. J. Goodman, Introduction to Fourier Optics, 2nd Ed., McGraw-Hill, New York(1996).

7. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Par-ticles, Wiley, New York (1998).

8. FRED Software, Photon Engineering, LLC, http://www.photonengr.com.9. ASAP Software, Breault Research Organization, http://www.breault.com.




10. TracePro Software, Lambda Research Corporation, http://www.lambdares.com.

11. Zemax Software, Radiant Zemax, LLC, http://www.radiantzemax.com.12. CODE V Software, Synopsys, Inc., http://www.synopsys.com.13. “Stray Light Short Course Notes,” Photon Engineering, LLC (2011), used by

permission.14. R. Breault, “Control of Stray Light,” in The Handbook Of Optics, Vol IV,

3rd Ed., M. Bass, G. Li, and E. Van Stryland, Eds., pp. 7–11, McGraw-Hill,New York (2010).



Chapter 2

Basic Radiometry for StrayLight AnalysisRadiometry is the science of detecting and measuring optical radiation,2 and is nec-essary to perform stray light analysis. Optical radiation is an electromagnetic wavewhose wavelength � is between 1 × 10−2 and 1 × 103�m. As shown in Fig. 2.1and Table 2.1, optical radiation includes the ultraviolet, visible, and infrared por-tions of the electromagnetic spectrum.

This chapter presents a brief review of the radiometric concepts needed toperform basic stray light analysis. One of the most important results of this chapteris the first-order stray light model of an optical system given in Eq. 2.47, becauseit can be used to obtain an estimate of the stray light performance of a system.It can also be used to validate some of the results obtained using stray lightanalysis software. Doing so is crucial, as setting up such an analysis in softwarecan be complicated and difficult to do without errors. A comprehensive review ofradiometry is beyond the scope of this book; however, there are a number of goodreferences.1−3

2.1 Radiometric Terms

This chapter introduces a number of radiometric quantities, many of which aredefined in their differential form. For instance, exitance M is defined in Section2.1.8 as

M = d�

dA, (2.1)

where d� is the differential flux emitted by the source, and dA is the differentialarea of the source. It is important to note that, as with all differential equations, thisequation reduces to

M = �

A(2.2)

if � is constant over the area defined by A.

13



14 Chapter 2

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

UV VIS IR

EUV UVC

UVA

UVB NIR SWIR MWIR LWIR VLWIR

Wavelength (μm)

Figure 2.1 Optical radiation wavebands in the electromagnetic spectrum.

2.1.1 Flux, or power, and radiometric versus photometric units

The flux, or power, of a source is equal the number of photons/second (ph/s) itemits. This quantity is represented by the Greek symbol �. The flux of a sourcecan be a function of many variables, such as position, direction, wavelength, andpolarization. This quantity is also expressed in watts, which can be computed fromph/s:

watts =n∑

i=1

hc

�i, (2.3)

Table 2.1 Optical radiation wavebands in the electromagnetic spectrum.

Band Sub-band Min �(�m) Max �(�m)

X-ray 1.00E-05 1.00E-02

Extreme Ultaviolet (EUV) 1.00E-02 0.121Ultraviolet C (UVC) 0.1 0.28

Ultraviolet (UV) Ultraviolet B (UVB) 0.28 0.315Ultraviolet A (UVA) 0.315 0.4

Visible (VIS) 0.4 0.7

Near Infrared (NIR) 0.7 1Short-Wave Infrared (SWIR) 1 3

Infrared (IR) Mid-Wave Infrared (MWIR) 3 5Long-Wave Infrared (LWIR) 8 12Very Long-Wave Infrared (VLWIR) 12 30

1.00E + 03

Microwave 1 .00E + 03 1.00E + 06



Basic Radiometry for Stray Light Analysis 15

where n is the number of photons, h is Planck’s constant (6.626 × 10−34 Joule·s),c is the speed of light (3 × 108 meters/s), and �i is the wavelength (in meters) ofthe i th photon. Flux is commonly expressed in either watts (W) or ph/s, thoughthe latter may be preferred for systems using solid-state detectors (which includeCMOS detectors used in consumer cameras) because it is easier to compute thephotocurrent (in amperes) in such detectors from the flux in ph/s (i.e., photocurrent=quantum efficiency × flux, see Section 2.3). Conversely, watts may be preferablefor systems using uncooled microbolometers, as the output of these detectors isdirectly proportional to the incident flux in watts.

Both photons/s and watts are radiometric units, and as such can be used toquantify the flux in any optical system, regardless of its waveband of operation.In optical systems operating in the visible waveband (roughly 0.4–0.7 �m), flux isoften quantified in photometric units such as lumens (lm), which can be computedfrom ph/s as

lumens = (680.002)

0.7�m∫0.4�m

n∑i=1

hc

�ip (�i ) d�, (2.4)

where p(�) is the photopic luminosity function, which quantifies the response of astandard human eye to light. Figure 2.2 plots p(�), which is also available in tabularform from a variety of online sources.6 Lumens are used to quantify the magni-tude of the flux as perceived by the human eye and are not useful outside of thevisible waveband. All of the radiometric units discussed in the following sectionshave photometric equivalents in which the flux is quantified using lumens. Radiom-etry performed using photometric units is called photometry.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.38 0.48 0.58 0.68 0.78

Phot

opic

lum

inos

ity

func

tion

Wavelength (μm)

Figure 2.2 The photopic luminosity function4 p(�).


SRBK005-02 SRBK005 March 13, 2013 19:10 Char Count= 0

16 Chapter 2

2.1.2 Reflectance, transmittance, and absorption

Reflectance is the amount of flux reflected by a surface, normalized by theamount of flux incident on it. Transmittance is the amount of flux transmitted bya surface, normalized by the amount of flux incident on it. Any flux not reflectedor transmitted is absorbed ( ). Conservation of energy requires that

+ + = 1. (2.5)

By Kirchoff’s radiation law,22 the flux emitted by a hot object must be equal to theamount absorbed by it; therefore, the emittance ε of an object must be equal to .

2.1.3 Solid angle and projected solid angle

In the spherical coordinate system shown in Fig. 2.3, the solid angle of an objectas viewed from a particular point in space is equal to

=2

1

2

1

sin( ) d , (2.6)

where 1 and 2 define the extent of the object in the azimuthal coordinate, and 1and 2 define the extent of the object in the elevation coordinate. The units of solidangle are steradians (sr).

A geometry often encountered in radiometry, called a right circular cone, isshown in Fig. 2.4, in which 1 = 0, 1 = 0, and 2 = 2 . Its solid angle is equal to

= 2 [1 − cos ( 2)] . (2.7)

Using 2 = 90 deg in Eq. (2.7) gives the solid angle of hemisphere (2 ).The definition of projected solid angle is similar to the definition of solid angle,

except for the addition of a cosine term:

=2

1

2

1

sin ( ) cos ( ) (2.8)

Figure 2.3 Solid angle geometry.

dθ

d ,dθ




Figure 2.4 Solid angle of a right circular cone.

This geometry is illustrated in Fig. 2.5. The units of projected solid angle aresteradians, just as for solid angle.

There are a number of common cases for which the value of the projected solidangle is simple to compute. The first of these is the right circular cone (shown inFig. 2.4), which is equal to

� = sin2(2) . (2.9)

Using 2 = 90 deg in Eq. (2.9) gives the solid angle of a hemisphere (). Theprojected solid angle divided by is often called the geometric configurationfactor (GCF).

The projected solid angle of an optical system can be computed from itsworking f -number ( f /#′), which is equal to

f/

#′ = f

DEP(1 + m), (2.10)

where f is the effective focal length of the system, DEP is equal to the diameter ofthe entrance pupil (computed as twice the height of the marginal ray), and m is themagnification of the system (which is equal to the image distance divided by theobject distance). The projected solid angle of an optical system is given by7

� f/# =

4(

f/#′)2 . (2.11)

Figure 2.5 Projected solid angle geometry.



18 Chapter 2

dA

dAcos(θ)

θ

dω

n

d2Φ

Figure 2.6 Quantities used in the definition of radiance.

2.1.4 Radiance

The radiance of a source L is equal to

L = d2�

dA cos () d�, (2.12)

where d� is the differential power emitted by the differential projected area of thesource dA cos () into the differential solid angle d�, as shown in Fig. 2.6.

The units are ph/s-unit area/sr, or in photometric units as candela/m2 (alsocalled “nits”). Radiance is used to quantify the amount of light or “brightness”of a surface: the more flux a surface emits per unit area or the more flux it emitsper projected solid angle, the greater its radiance. It is an elemental radiometricquantity, and other quantities, such as intensity or exitance (discussed later), arederived by integrating it over solid angle or area, respectively. If absorption lossesare neglected, radiance is conserved through an optical system, and thus the radianceof an image is the same as the radiance of the exit pupil and of the scene (it is said tobe “invariant”). A surface whose radiance is constant with respect to the emittanceangle is said to be Lambertian. Though treating a surface as Lambertian is oftena useful approximation, in practice no surface is perfectly Lambertian.

2.1.5 Blackbody radiance

The Planck blackbody equation can be used to compute the spectral radiance L�

(in ph/s-cm2-sr-�m) of an extended source from its temperature:

L� (�,T ) = C1

�4[exp

(C2

/�T

) − 1] , (2.13)

where C1 = 5.99584 × 1022 photons-�m5/s-cm2-sr, C2 = 14387.9 �m-K, and Tis the temperature of the source in kelvin. This function is plotted in Fig. 2.7 as afunction of wavelength for several temperatures.

Equation (2.13) does not account for variations in radiance versus wavelengthdue to changes in emissivity of the extended source. These variations, which aredetermined by the chemical composition of the source and which all sources have,can result in an error in the radiance predicted by the Planck equation, and thereforemay need to be considered in the calculation. For example, the spectral radianceof the sun is shown in Fig. 2.8 along with the radiance predicted by the Planckequation for an ideal blackbody at 5800 K. Chemical species in the sun result inabsorption bands that are not predicted by the Planck equation.




0

1E+21

2E+21

3E+21

4E+21

5E+21

6E+21

7E+21

8E+21

9E+21

0 1 2 3 4

Radi

ance

(ph/

s-cm

2 -μm

)

Wavelength (μm)

4000 K 5000 K 6000 K

Figure 2.7 Blackbody radiance versus wavelength.

0.E+00

1.E+21

2.E+21

3.E+21

4.E+21

5.E+21

6.E+21

7.E+21

8.E+21

9.E+21

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Radi

ance

(ph/

s-cm

2 -str

-)

Wavelength (μm)

5800 -K Blackbody Measured Solar

μm

Figure 2.8 The spectral radiance of an ideal 5800-K blackbody and the measured exo-atmospheric spectral radiance of the sun.8



20 Chapter 2

Table 2.2 Equivalent solar blackbody temperatures for typical sensor wavebands.

WavebandEquivalent Solar BB Error in Band-Integrated

Min �(�m) Max �(�m) Name Temperature (K) Radiance

0.4 0.7 VIS 5848 0.436%

0.7 1 MIR 5761 −0.109%

1 3 SWIR 5986 −2.333%

3 5 MWIR 5656 1.448%

8 12 LWIR 4983 0.758%

One of the ways to reduce the magnitude of the error resulting from the useof the Planck equation is to determine the blackbody temperature of the sourcethat minimizes the chi-squared difference between the actual spectrum and theblackbody spectrum in the waveband of interest. This was done for the spectrumshown in Fig. 2.8 (i.e., 5800 K is best-fit), and in Table 2.2 for the visible andIR sensor wavebands. The resulting error in the band-integrated radiance is alsogiven in Table 2.2. Error analysis must be performed to determine the error in anyquantity dependent on this radiance.

Figure 2.9 shows the apparent exo-atmospheric radiance of the sun, and assuch it does not account for the reduction in apparent radiance due to atmospheric

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

Tran

smit

tanc

e

Wavelength (μm)

Figure 2.9 Transmittance from ground to space versus wavelength of the atmosphere,looking straight up from the ground.




Table 2.3 Band-averaged transmittance from ground to space of the atmosphere, lookingstraight up from the ground.

WavebandAverage

Min �(�m) Max �(�m) Name Transmittance

0.4 0.7 VIS 0.598

0.7 1 NIR 0.722

1 3 SWIR 0.581

3 5 MWIR 0.490

8 12 LWIR 0.726

absorption and scattering (extinction), which can significantly reduce it. Atmo-spheric extinction is a complicated phenomenon that is a function of many variables,including wavelength, the geometry of the path of light through the atmosphere,and weather conditions, to name a few. The software MODTRAN9 was createdto deal with this complexity—it can predict atmospheric extinction as a functionof these variables and more. Figure 2.9 shows the transmittance from ground tospace of the default MODTRAN atmospheric model, looking straight up from theground, which is the maximum transmittance possible from the ground to space.Table 2.3 gives the same transmittance averaged over typical sensor wavebands;this table quantifies the error in apparent radiance of the sun that results fromneglecting atmospheric extinction.

The total radiance L in a waveband can be calculated by integrating Eq. (2.13)over the waveband of interest:

L =�2∫

�1

L�d�, (2.14)

where �1 and �2 are the minimum and maximum wavelengths of the waveband.The wavelength �peak corresponding to the peak radiance can be computed from

the temperature of the blackbody T using Wien’s displacement law for photons:

�peak = 3670

T, (2.15)

where �peak is in �m, and T is in kelvin. Integrating Eq. (2.13) over all wavelengthsgives the total radiance emitted by a blackbody Ltotal, which can be expressedusing the Stefan–Boltzman law for photons:

Ltotal =(

�p

)T 3, (2.16)

where �p is the Stefan–Boltzman constant (1.5204 × 1013 ph/s-cm2-K3). The term in Eq. (2.16) converts the units to radiance (ph/s-cm2-sr).



22 Chapter 2

dAsθs

dΩc

ns ncθc

dΩs dAc

d

Figure 2.10 Quantities used in the definition of throughput.

2.1.6 Throughput

The throughput G of an optical system (also called its etendue or “A� product”)is defined as

G=∫As

∫Ac

dAs cos(s) dAc cos(c)

d2=

∫As

∫�c

dAs cos(s) d�c =∫Ac

∫�s

dAc cos(c) d�s,

(2.17)where dAs is the differential area of the source, s is the angle between the normal ofthe source surface ns and the vector between the center of the source and collectorsurfaces, dAc is the differential area of the collector, and c is the angle betweenthe normal of the collector surface nc and the vector between the center of thesource and collector surfaces, as shown in Fig. 2.10. The units of throughput areunit-area-steradian.

As shown in Eq. (2.17), this quantity can also be expressed as an integral overthe projected solid angle of the collector (�c) or source (�s). If the source andcollector are normal to each other (i.e., s = c = 0) and As and Ac are muchgreater than d2, then Eq. (2.17) reduces to

G ≈ As Ac

d2= As�c = Ac�s . (2.18)

A common mistake when computing G is to calculate it as either As�s or Ac�c.Calculating G in this way is known as the “ice-cream cone” mistake7, showngraphically in Fig. 2.11.

Ωs

Ac

Ωs

As

Figure 2.11 Calculation of throughput using the correct geometric factors (left) and incorrectfactors (right).




G can be expressed in terms of the projected solid angle of an optical system� f/# as

G ≈ Ac� f/#, (2.19)

where Ac is the area of the focal plane for which the G is being computed (often,the area of pixel in a digital imaging system). Throughput is a geometric quantitythat is invariant at any point between the source of light in a scene and the imageplane of the optical system; it is conserved across a refractive index boundary as

n21G1 = n2

2G2, (2.20)

where n1 and G1 are the refractive index and throughput on one side of the boundary,and n2 and G2 are the refractive index and throughput on the other side. Notice thatthroughput also appears in the definition of radiance [Eq. (2.12)] and, neglectingabsorption losses, is invariant through an optical system.

2.1.7 Intensity

The intensity I of a point source is given by

I = d�

d�, (2.21)

where d� is the differential flux emitted by the source, and d� is the differentialsolid angle into which the point source is emitting, as shown in Fig. 2.12. Intensitycan only be defined for point sources, that is, sources that have an infinitely smallextent. Though no real-world sources exactly meet these criteria, this way of defin-ing the brightness of a source is often useful. Intensity is specified in ph/s-sr, or, inphotometric units, in candela (lm/sr).

2.1.8 Exitance

Exitance M is equal to the flux per unit area emitted by a source, defined as

M = d�

dA, (2.22)

where d� is the differential flux emitted by the source, and dA is the differentialarea of the source, as shown in Fig. 2.13. Exitance is specified in ph/s-unit area, or,in photometric units, in lux (lm/m2). For Lambertian surfaces, the exitance of the

dωdΦ

Figure 2.12 Quantities used in the definition of intensity.



24 Chapter 2

dA

dΦ

Figure 2.13 Quantities used in the definition of exitance.

surface is related to its radiance L as

M = L . (2.23)

2.1.9 Irradiance

The irradiance E incident on a surface is equal to

E = d�

dA, (2.24)

where d� is the differential flux incident on the surface, and dA is the differentialarea of the surface, as shown in Fig. 2.14. Irradiance is specified in ph/s-unit area,or, in photometric units, in lux (lm/m2). The only difference between exitance andirradiance is the direction of propagation of light.

The irradiance at a distance d from a point source with intensity I is given bythe cosine-cubed law as

E = I cos3()

d2, (2.25)

where is the angle between the vector to the source and the surface normal, asshown in Fig. 2.15.

The irradiance at a distance d from a surface source with radiance L and areaA (shown in Fig. 2.16) is given by the cosine-to-the-fourth law as

E = LA cos4()

d2. (2.26)

dA

dΦ

Figure 2.14 Quantities used in the definition of irradiance.




θI

d

E

Figure 2.15 Relationship between point source intensity I and irradiance E.

2.1.10 Bidirectional scattering distribution function

The bidirectional scattering distribution function (BSDF) is the radiance of ascattering surface, normalized by the irradiance incident of the surface:

BSDF (i , �i , s, �s) = dL (i , �i , s, �s)

dE (i , �i ), (2.27)

where i and �i are the elevation and azimuth angles of the incident ray, s and �s

are the elevation and azimuth angles of the scattered ray (as shown in Fig. 2.17),dL is the differential radiance of the scattering surface, and dE is the differentialincident irradiance.10

Equation (2.32) can also be written in terms of the differential scattered flux perdifferential projected solid angle d�s/d�s, normalized by the differential incidentflux d�i as

BSDF = d�s/

d�s

(d�i ) cos s. (2.28)

The units of BSDF are 1/sr. Another (much less-widely used) quantity is the cosine-corrected BSDF (also called the scatter function), which is equal to BSDF × cos s .In some publications, BSDF and cosine-corrected BSDF are used interchangeably;because of this, it is important to verify which definition is being used. In general,BSDF is used instead of cosine-corrected BSDF. It is often referred to as either thebidirectional reflectance distribution function (BRDF) or bidirectional transmit-tance distribution function (BTDF), depending on the direction the scattered lightis propagating relative to the scattering surface. A less-often used term is the bidi-rectional diffraction distribution function (BDDF), which is used to compute theeffects of aperture diffraction (see Section 7.2). Surfaces whose BSDF at normal

θL,A

d

E

Figure 2.16 Relationship between surface source radiance L and irradiance E.



26 Chapter 2

z

x

y

θi

φi

θs

φs

Incident beam

Scattered beam

(αi , βi)

(αs , βs )

Figure 2.17 Angles used in the definition of the bidirectional scattering distribution function.

incidence varies only with the elevation angle s are called isotropic scatterers, andthose that also vary with the azimuthal angle �s are called anisotropic scatterers.Most surfaces whose scatter is modeled in stray light analysis (such as polishedoptical surfaces and black surface treatments) are isotropic scatterers whose BSDFis a weak function of wavelength. The BSDF of any real surface is always greaterthan zero and can be greater than 1.

It is sometimes more convenient to express the independent variables of theBSDF using direction-cosine space coordinates instead of angles. The coordinatesof the incident ray (�i , �i ) and scattered ray (�s , �s) are related to the angles as

�i,s = cos �i,s sin i,s (2.29)

�i,s = sin �i,s sin i,s, (2.30)

as shown in Fig. 2.17.For reasons that will become more apparent in Chapter 4, BSDF is often plotted

as a function of |sin s – sin i |, as shown in Fig. 2.18. This quantity is also oftenreferred to as |� – �0|, where � = sin s and �0 = sin i . Because BSDF is oftenplotted in this way, it is important to be able to properly interpret this plot, whichcan look strange in the very common case of i �= 0 and �s – �i = 0 deg or 180deg (i.e., non-normal incidence, in-plane scatter). In this case, the BSDF curvesplits in two (see Fig. 2.18). One of these curves represents scatter in the “forward”direction (away from the incident beam, as shown in Fig. 2.19), and the other in the“backward” direction.




1.E-03

1.E-02

1.E-01

1.E+00

1.E-02 1.E-01 1.E+00 1.E+01

BSD

F (1

/sr)

|sin s− sinθ i|

Forward scatter

Backward scatter

θ

Figure 2.18 Measured BSDF of Aeroglaze Z306 black paint11 at 0.6328 �m (i = 45 deg),plotted versus |sin s – sin i |. More details from this set of data are shown in Fig. 6.3.

θiθi

Incident raySpecularly

Backward- scattered

ray

Forward- scattered

ray

reflected ray

Figure 2.19 Scattering geometry for the BSDF data presented in Fig. 2.18.



28 Chapter 2

θ1

θ2

θ1

θ2

Figure 2.20 Reciprocity of BSDF. The BSDF must be the same if the incident and scatteredrays are switched, i.e., BSDF(1, 2) = BSDF(2, 1).

Conservation of energy demands that the BSDF obey reciprocity, that is:

BSDF (i , �i , s, �s) = BSDF (s, �s, i , �i ) , (2.31)

which means that the BSDF must be the same if the incident and scattered rays arereversed, as shown in Fig. 2.20.

The ratio of total power scattered by a surface in the reflected or transmitteddirection to the power incident on it is called the total integrated scatter (TIS),which is equal to the integral of the BSDF over the projected solid angle of thehemisphere:

TIS =2∫0

/2∫0

BSDF sin (s) cos (s) dsd�. (2.32)

In order for energy to be conserved, the TIS of any BSDF (whether measuredor modeled) must be ≤ 1. This quantity has a variety of names, including totalhemispherical reflectance (THR), diffuse hemispherical reflectance (DHR), andalbedo. For consistency, this book will always refer to this quantity as TIS. Somereferences12 define TIS as above, normalized by the specular reflectance of thesurface. Most stray light analysis programs, however, use the above definition. Formost BSDF functions used in stray light analysis, a closed-form solution of theirTIS integral does not exist, and therefore it must be evaluated numerically. All ofthe stray light programs mentioned in Section 1.1 can do this calculation.

As with emitting surfaces, the BSDF of a scattering surface that does notvary as a function of incident or scatter angle is called Lambertian. The TIS of aLambertian scatterer is related to its BSDF by the equation

BSDF = TIS

. (2.33)

In practice, no surface is perfectly Lambertian. One of the closest is Spectralon,13

which is composed of pressed polytetrafluoroethylene (also known as Teflon R©)powder, and is highly reflective (>99%). Spectralon is often used for radiometriccalibration because its radiance L is predictable (i.e., L ∼ incident irradiance/across a wide range of i and s).

While the basic radiometric definition of BSDF is straightforward, measuringand modeling it accurately can be difficult, as variation of BSDF with angle of




incidence, scatter angle, and position is often complicated for real surfaces. How-ever, modeling BSDF accurately is important because the stray light performanceof an optical system is usually a strong function of the BSDF of its components.(That is why Chapters 4–6 are devoted to the development of BSDF models foroptical surfaces, particulate-contaminated surfaces, and black surface treatments,and to the details of directly measuring BSDF and TIS.) Despite the sophisticationof these techniques, it is often difficult to predict the BSDF of a particular surface towithin a factor of two, especially if the time and money available to develop a modelare very limited. This may be surprising to optical engineers used to working withinterferometric data of surface figure, as such data is often much more accuratethan BSDF measurements or models. Indeed, even estimating the uncertainty ofBSDF measurements or models can be difficult because doing so often requiresmany more measurements than the available time or budget allow for. For instance,determining the true spatial variation in BSDF over a particular surface may requirethat the BSDF be measured at many points, which may be too time consuming orexpensive to perform. This problem is exacerbated by the fact that little data hasbeen published that establishes the expected spatial or angular variation in BSDFfor surfaces typically found in most optical systems. This book partially resolvesthis problem by providing estimates of the accuracy of BSDF measurements andmodels. However, many of the estimates presented are “best case” in that theycan determine the minimum uncertainty but not the maximum, or they are oftenbased on “best guesses” that are generally accepted by the optics industry but notsupported by rigorous studies. It is important to be aware of these limitations inaccuracy and use the results of BSDF predictions accordingly. A simple analysisof the effect of uncertainties on predicted stray light performance is shown inSection 2.3.6.

2.2 Radiative Transfer

Referring to Fig. 2.10, the differential flux d� on the collection surface due tosource surface radiance Ls is given by

d2� = LsdAs cos (s) dAc cos (c)

d2, (2.34)

and therefore the flux is given by

� =∫As

∫Ac

[Ls cos (s) cos (c)

d2

]dAsdAc. (2.35)

This equation makes no assumptions about the variation in Ls over the angularextent of the collection area Ac or about the relative sizes of d2, As , and Ac. If thesource is Lambertian, the equation simplifies to

� = Ls

∫As

∫Ac

[cos (s) cos (c)

d2

]dAsdAc = Ms As F, (2.36)



30 Chapter 2

Rs

Rc

H

Figure 2.21 Dimensions used in a disk-to-disk configuration factor equation.

where Ms is the exitance of the source, and F is the configuration factor (alsocalled the form factor), which is defined as

F = 1

As

∫As

∫Ac

[cos (s) cos (c)

d2

]dAsdAc. (2.37)

The value of F is a function of the shape, size, orientation, and distance betweenthe source and collection surfaces; closed-form solutions for F have been derivedfor a variety of common geometries.15 One very common geometry, illustrated inFig. 2.21, occurs when the source and collection surfaces are both disks that faceeach other.

The configuration factor F in this case is equal to

F = 1

2

⎡⎣X −

(X2 − 4

S22

S21

)1/2⎤⎦ , (2.38)

where S = R/H , and X = 1 + (1 + S22 )/S2

1 .A further simplification to Eq. (2.36) can be made if the source and collection

surfaces face each other (s = c = 0), d2 >> As , and d2 >> Ac. In this case,Eq. (2.36) simplifies to the well-known formula

� = LAs Ac

d2= LAc�s, (2.39)

where �s is as shown in Fig. 2.11. In many cases, the assumptions used in thederivation of Eq. (2.39) are valid, and therefore this equation is often used for basicradiometric transfer calculations. Equation (2.39) can be rewritten as

Ec = L�s, (2.40)

where Ec is the irradiance on the collector surface. Substituting the expression forBSDF into this equation, the relationship between the source power and irradiance




on the collector due to scattering can be written as

Ec = Ei (BSDF) �, (2.41)

where Ei is the irradiance incident on the scatterer from the source, and � isthe projected solid angle between the scatterer and the collector. This equationdemonstrates that way in which scattered light propagates through a system: BSDFtransforms irradiance into radiance, and solid angle transforms radiance intoirradiance. This equation reveals the fundamental contributors to the stray lightlevel. Often, the only term in this equation that can be reduced to zero is theprojected solid angle �,24 which is often done in the design of optical systems byusing baffles (see Chapter 9). This fact supports the “Move It or Block It” designphilosophy discussed in Section 1.3.6.

2.2.1 Point source transmittance

A transfer function commonly used to describe the stray light performance of anoptical system is its point source transmittance (PST), which is equal to the amountof stray light on the focal plane of an optical system divided by the amount of lightincident at the entrance aperture of the system. Multiple definitions of PST havebeen used, usually either as the ratio of fluxes or as the ratio of irradiances. Thisbook uses the latter definition:

PST = ESL

Einc, (2.42)

where ESL is the irradiance on the focal plane due to stray light, and Einc is theirradiance from a point source at infinity (collimated) incident on a plane normalto the incident beam. This plane is usually at the entrance aperture of the system,though in some systems this aperture may not be well defined; in such a system, theplane is usually defined at the first optical element of the system. This definitionof PST is sometimes called the point source normalized irradiance transmittance(PSNIT) or the normalized detector irradiance (NDI). ESL is typically definedas the irradiance averaged over the entire focal plane, and as such it does notcontain any information about the spatial distribution of the irradiance at the focalplane. However, it can be defined for one or more smaller regions on the focalplane, which can be used to describe the spatial distribution of irradiance. For mostoptical systems, PST it is usually a strong function of the elevation angle betweenthe stray light source and the optical axis of the system. (This angle is shown assun later on in Fig. 2.24.)

PST can be used to calculate the irradiance on the focal plane due to an extendedsource by integrating it over the source’s projected solid angle:

ESL =�2∫

�1

2∫1

EincPST (, �) sin dd�. (2.43)



32 Chapter 2

A similar transfer function also commonly used is solar source transmittance(SST), which is defined the same as PST, except that the incident irradiance is fromthe sun, which is not a point source but, as seen from earth, subtends an angle ofabout 32 arcsec (0.5333 deg). The resulting irradiance Einc can be computed asLsun�sun�atm, where Lsun is the apparent radiance of the sun in the sensor waveband,�sun is the projected solid angle of the sun (6.8052 × 10−5 sr), and �atm is the spectraltransmittance of any atmosphere between the sensor and the sun. The variation inSST with elevation angle is often quite different than the variation in PST of thesame system, especially for a system whose FOV is similar to the angular extent ofthe sun.

2.2.2 Detector field of view

Another transfer function commonly used to describe stray light performance isdetector field of view. Detector FOV is computed by putting an extended source atthe focal plane, propagating light backwards through the system, and calculating theintensity I as a function of elevation angle and azimuth angle � that results fromstray light in the system. This calculation is often performed in stray light analysissoftware in order to identify low-order stray light paths, as the detector FOV will bevery high at values of (,�) that correspond to low-order paths. Propagating lightbackwards through the system is a common technique used in stray light analysis,discussed further in Section 3.2.3.

2.2.3 Veiling glare index

Optical systems operating at visible wavelengths often use veiling glare index(VGI)14 to quantify stray light performance, which is equal to

VGI = Eout

Eout + Ein, (2.44)

where Eout is the irradiance on the focal plane due to stray light from Lambertianradiance outside the FOV, and Ein is the irradiance on the focal plane due to thesame Lambertian radiance inside the FOV. VGI is often determined using a veilingglare test, as pictured in Fig. 2.22. A white screen (as Lambertian and broad aspossible) is illuminated with a bright light, and a black region is placed in the centerof it that exactly subtends the FOV. In this configuration, the irradiance on the focalplane is Eout. The black region is then removed from the screen, and the irradianceat the focal plane is measured again, which determines Eout + Ein. Section 10.5discusses VGI tests.

2.2.4 Exclusion angle

System stray light requirements often specify an exclusion angle, which is usuallydefined as the minimum angle at which the stray light requirement (such as themaximum allowable irradiance at the focal plane due to stray light) must be met.




Black region that exactly fills FOV

White, Lambertian screen

Optical system Focal plane

Stray light

Light source

Light reflected from screen

Figure 2.22 A veiling glare test.

Because it can be difficult or impossible to reduce stray light for source anglesat or near the edge of the FOV, the exclusion angle is often greater than FOV/2,as shown in Fig. 2.23.

2.2.5 Estimation of stray light using basic radiative transfer

Equation (2.39) together with some of the terms defined previously, will now beused to predict the amount of out-of-field stray light in a simple optical system.The geometry of this system is shown in Fig. 2.24. The sun illuminates an opticalsystem from an off-axis angle, and the system optics scatter light to the systemfocal plane.

The terms in this analysis are as follows:� Lsun is the apparent radiance of the sun in the sensor waveband. Depending on

the accuracy required, this can be modeled using the solar radiance spectrumshown in Fig. 2.8, or (less accurately) by integrating the Planck blackbody

FOV/2

Optical system Focal plane

Exclusion angle

Figure 2.23 Geometry of the exclusion angle.



34 Chapter 2

Lsun

Ωsun , τatm

θsun

BSDFoptics

Ωoptics,τoptics

ESL

Figure 2.24 Geometry for simple solar-stray-light analysis.

equation [Eq. (2.13)] over the sensor waveband, using the equivalent solarblackbody temperature for the waveband (Table 2.2).

� �sun is the projected solid angle of the sun. From earth, the sun subtendsabout 32 arc sec, so, using Eq. (2.9) for a right circular cone, �sun = 6.8 ×10–5 sr.

� �atm is the spectral transmittance of the atmosphere. If this data is available,then it should be multiplied by the blackbody radiance of the sun and thenused in the integrand to compute Lsun. If it is not available, then the band-averaged values shown in Table 2.3 can be used.

� sun is the off-axis angle of the sun, as measured from the center of the FOVof the sensor.

� BSDFoptics approximates the scattering of the entire optical system. This valueis usually dependent on sun, and is often approximated by the surface rough-ness and contamination scattering of the first element in the system, such asthe objective lens in a refractive system or the primary mirror in a reflective.Surface roughness scattering is discussed in Chapter 4, and contaminationscattering in Chapter 5. Both chapters discuss analytic models (such as theHarvey model) for this term. As a worst case scenario, the term can be cal-culated as the sum of the scattering from all of the optics in the system. Theuse of a single term to model the scattering of the entire optical system isone of the largest approximations made in this analysis, and therefore canbe a significant source of error. Obviously, a better analysis would use a raytracing model of the system.

� �optics is the solid angle of the optical system, as defined by Eq. (2.11).� �optics is the transmittance of the optical system, which can be computed as the

product of the transmittance or reflectance of every element in the system.� ESL is the desired value of the computation and is the irradiance on the focal

plane due to stray light that results from solar illumination of the optics.

ESL can be computed by applying the equation of radiometric transfer� = LAc�s twice—once to compute the flux on the optics from the sun, and




again to compute the flux on the focal plane from the scattering of the optics. First,the flux �inc from the sun on the optics can be computed as

�inc = Lsun�atm Aoptics�sun cos (sun) , (2.45)

where Aoptics is the illuminated area of the optics. The cosine factor must be addedto account for the fact that the sun is illuminating the optics at an angle. For thepurposes of this analysis, it is desirable to compute the irradiance incident on theoptics Einc rather than the flux, so both sides must be divided by Aoptics:

Einc = Lsun�atm�sun cos (sun) . (2.46)

Now ESL can be computed by computing the radiance of the scattering from theoptics from BSDFoptics and multiplying by the solid angle and transmittance of thesystem �optics and �optics:

ESL = Lsun�atm�sun cos (sun) BSDF optics (sun) �optics�optics, (2.47)

or, in terms of SST,

SST (sun) = BSDFoptics (sun) �optics�optics. (2.48)

This simple model of the stray light performance of an optical system neglectsmany factors, such as the radii of curvature of the optics and the effect of ghostreflections; however, it is a quick way to get a rough estimate of the amount ofstray light in the system. Indeed, prior to widespread availability of computers,calculations such as these were the only way to compute the stray light level. Someresearch has shown that this equation can be used to obtain a good estimate of ESL

using only the BSDF of only the first illuminated optic in the system.25

A similar equation has been derived26 for estimating the irradiance at the focalplane due to in-field stray light EIFSL as a function of position from the center ofthe focal plane r :

EIFSL = EincAinc

AjBSDFj

[r

2(

f/

#)

aj

]�optics�optics, (2.49)

where Ainc is the area of the beam incident on the system, Aj is the area of thebeam at the j th element, and BSDFj is the BSDF of the j th element. Studies haveshown that predictions made with this equation agree well with predictions madeusing more detailed ray tracing models.27

Using a derivation similar to the one for Eq. (2.47), the detector FOV I()can also be computed by assuming a source of radiance Lfp and area Afp at thefocal plane, propagating the light backwards through the system, and computingthe resulting intensity, which results in the equation

I () = Lfp Afp�optics�opticsBSDFoptics () �atm. (2.50)



36 Chapter 2

2.2.6 Uncertainty of stray light estimates

Equation (2.47) also allows for a simple error analysis to be performed. Assumingthat the uncertainty in any of the values (e.g., Lsun and BSDFoptics) is independentand random, the uncertainty in the focal plane irradiance �ESL can be computedas17

�ESL =√√√√(

ESL

Lsun�Lsun

)2

+(

ESL

BSDF optics�BSDF optics

)2

(2.51)

=�sun cos (sun) � f/#�atm�optics

√(BSDFoptics�Lsun

)2 + (Lsun�BSDFoptics

)2,

where �Lsun is the uncertainty in Lsun, and �BSDFoptics is the uncertainty inBSDFoptics. Calculations of this type are often used to determine the validity of astray light analysis, which is important because the value of �BSDFoptics can belarge, as discussed in Sections 4.4 and 5.7.

2.3 Detector Responsivity

Most modern optical systems use solid state detectors to convert light at the focalplane to an electronic signal, and it is important to consider the effects of thisconversion when computing system stray light performance. A comprehensivediscussion of these effects is beyond the scope of this book;18 however, someimportant effects include the following:

� All solid state detectors have noise that limits sensitivity, and this noiseis proportional to the temperature of the detector material. For detectorsoperating in the MWIR (3–5 �m) and longer wavelengths, the detectormaterial generally must be cryogenically cooled in order for the detectornoise to be low enough for the detector to be useful.

� The number of electrons generated per incident photon is called the quantumefficiency of the detector and is a function of wavelength, as shown for severaltypical detector materials in Fig. 2.25. When performing stray light analysis, itis often important to consider the quantum efficiency of the detector, becausestray light that occurs in a spectral region of low quantum efficiency is notas important as stray light that occurs in a region of high quantum efficiency.This effect is discussed in more detail in Section 8.11.

2.3.1 Noise equivalent irradiance

All solid state detectors have electronic noise (such a Johnson or shot noise18)that limits their sensitivity. Its magnitude is often quantified by noise equivalentirradiance (NEI), which is the minimum irradiance that can be detected in thepresence of the noise. NEI is often used as a unit of measure when specifying theirradiance due to stray light, e.g., “the irradiance on the detector due to stray lightwas 10× NEI.”




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16

Qua

ntum

Effi

cien

cy

Wavelength (μm)

Si InGaAs InSb HgCdTe

Figure 2.25 Quantum efficiency versus wavelength of typical detector materials.19,20 Inaddition to the detector material itself, quantum efficiency is also a function of the ARcoating on the detector.

2.3.2 Noise equivalent delta temperature

Detector noise in infrared systems is often quantified as noise equivalent deltatemperature (NEDT), also called noise equivalent temperature difference. In an IRsystem, the perceived temperature of the scene Tscene is often computed from theirradiance on the detector from the scene Escene as

Tscene = Tref + Escene − Eref( L

/ T

)�optics�optics

, (2.52)

where Tref is the temperature of the reference source (usually a laboratory black-body) used to calibrate the sensor; Eref is the irradiance on the detector from thereference source; and dL/dT is the derivative of the Planck equation as a function oftemperature, integrated over the waveband of interest, computed at Tref for photonsas:

L

T=

�2∫�1

−C1C2 exp(C2

/�Tref

)�5Tref

[exp

(C2

/�Tref

) − 1]2 . (2.53)

NEDT can be computed from NEI by plugging it into Eq. (2.52):

NEDT = NEI( L

/ T

)�optics�optics

. (2.54)



38 Chapter 2

2.4 Summary

Radiometry is necessary to perform stray light analysis. A central concept inradiometry is radiance (usually symbolized by the letter L), which defines theamount of optical flux (either watts or, equivalently, ph/s) from an object perarea per solid angle. Radiance is a precise definition of the familiar concept ofbrightness. The irradiance on a surface (E , in ph/s-mm2) due to radiance L froman object is equal to L�, where � is the projected solid angle between the objectand the surface. The Planck blackbody equation can be used to predict the emittedradiance of an object based on its temperature and on the waveband of interest.The blackbody equation can be fit to the emission spectra of natural sources suchas the sun. Due to the presence of sharp emission and absorption peaks in thesolar spectrum, the temperature of the best blackbody fit varies as a function ofwaveband, from about 5800 K in the visible to about 5000 K in the LWIR.

BSDF quantifies the amount an object scatters, and is equal to L/E , where L isthe radiance of the object due to scatter, and E is the irradiance incident on it. Ingeneral, BSDF is a function of angle of incidence, scatter angle, wavelength, andposition, and as such can be difficult to model. Scattered flux propagates through asystem via two laws: E = L� and L = BSDF × E .

In many cases, the only way to prevent the propagation of flux scattered froma surface is to prevent other surfaces from seeing it (� →0), which motivates theuse of baffles in optical systems (for instance, to block the sun).

A common way of quantifying the stray light performance of a system is pointsource transmittance (PST), which is equal to the irradiance on the focal plane ofan optical system due to stray light divided by the amount of light incident at theentrance aperture of the system. The PST of a system can be modeled analyticallyusing Eq. (2.47). This equation neglects many important stray light mechanisms,such as ghost reflections; however, it can be used to quickly estimate PST, to validatethe results of some calculations in stray light analysis software, and estimate thesensitivity of the system stray light performance as a function of uncertainties inquantities such as BSDF, which can often be uncertain by a factor of 2× or more.

References

1. W. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York(2008).

2. J. Palmer and B. Grant, The Art of Radiometry, SPIE Press, Bellingham, WA(2009) [doi: 10.1117/3.798237].

3. W. Wolfe, Introduction to Radiometry, SPIE Press, Bellingham, WA (1998)[doi: 10.1117/3.287476].

4. M. Born and E. Wolf, Principles of Optics, 7th Ed., Cambridge University Press(1999).

5. D. Goldstein, Polarized Light, 3rd Ed., CRC Press, Boca Raton, FL (2010).6. Commission International De L’Eclairage (CIE), http://www.cie.co.at/.




7. P. Spyak, “Seven deadly radiometry mistakes,” Proc. SPIE 8483, 848302[doi:10.1117/12.929939].

8. ASTM International, “ASTM E490 – 00a(2006) Standard Solar Constant andZero Air Mass Solar Irradiance Tables” (2006), http://www.astm.org.

9. MODTRAN software, Ontar Corporation, www.ontar.com.10. F. Nicodemus, J. Richmond, J. Hsia, I. Ginsberg, and T. Limperis, Geometric

Considerations and Nomenclature for Reflectance, U. S. Dept. of Commerce,NBS Monograph 160 (1977).

11. W. Viehmann and R. Predmore, “Ultraviolet and visible BRDF data on space-craft thermal control and optical baffle materials,” Proc. SPIE 675, 67–72(1986) [doi: 10.1117/12.939484].

12. J. Stover, Optical Scattering: Measurement and Analysis, 3rd Ed., SPIE Press,Bellingham, WA (2012) [doi: 10.1117/3.975276].

13. LabSphere, Inc., http://www.labsphere.com.14. International Standards Organization (ISO), ISO 9358: Optics and optical in-

struments – veiling glare of image forming systems – definitions and methodsof measurements (1994).

15. J. Howell, “A Catalog of Radiation Heat Transfer Configuration Factors,”http://www.engr.uky.edu/rtl/Catalog/.

16. J. Goodman, Introduction to Fourier Optics, 2nd Ed., McGraw-Hill, New York(1996).

17. J. Taylor, An Introduction to Error Analysis: The Study of Uncertainties inPhysical Measurements, 2nd Ed., University Science Books, Sausalito, CA(1997).

18. E. Dereniak and G. Boreman, Infrared Detectors and Systems, John Wiley &Sons, New York (1996).

19. The Raytheon Vision Systems Infrared Wall Chart, www.raytheon.com.20. Next Generation Space Telescope website, astro.berkeley.edu/∼jrg/ngst/.21. International Standards Organization (ISO), “ISO 9358 - Optics and optical

instruments – Veiling glare of image forming systems – Definitions and methodsof measurement,” http://www.iso.org (1994).

22. E. Hecht, Optics, 4th Ed., pp. 582, Addison-Wesley, Reading, MA (2001).23. E. Freniere, R. Stern, and J. Howard, “SOAR: a program for rapid calcula-

tion of stray light on the IBM PC,” Proc. SPIE 1331, 107–117 (1990) [doi:10.1117/12.22654].

24. R. Breault, “Control of Stray Light,” in The Handbook Of Optics, Vol IV,3rd Ed.,M. Bass, G. Li, and E. Van Stryland, Eds., McGraw-Hill, New York (2010).

25. A. Greynolds, “Formulas for estimating stray light levels in well-baffled opticalsystems,” Proc. SPIE 257, 39–49 (1980) [doi: 10.1117/12.959600].

26. G. Peterson, “Analytic expressions for in-field scattered light distributions,”Proc. SPIE 5178 0277-786X (2004) [doi: 10.1117/12.509120].

27. J. Harvey, N. Choi, A. Krywonos, G. Peterson, and M. Bruner, “Image degrada-tion due to scattering effects in two-mirror telescopes,” Opt. Eng. 49(6), 063202(2010) [doi: 10.1117/1.3454382].



Chapter 3

Basic Ray Tracing for StrayLight AnalysisAs shown in Section 2.3.5, developing analytic models of system stray light per-formance (even ones with very limited fidelity) results in complicated equationswith many terms. For this reason, most nontrivial stray light problems today areanalyzed by ray tracing in stray light analysis software. A list of these programsis given in Section 1.1. These programs have many features and can be difficultto use. This chapter provides guidance in the most efficient ways of using them toperform stray light analysis and design. As mentioned previously, this chapter willnot go into the details of how to use a given software package; for this information,please contact the software vendor.

3.1 Building the Stray Light Model

Building a model of an optical system in stray light analysis software usuallyconsists of two steps: defining the optical and mechanical geometry, and thendefining its optical properties.

3.1.1 Defining optical and mechanical geometry

The geometric descriptions of the optical surfaces (such as the radii of curvatureof the lens and mirror surfaces) in an optical system are usually determined byimporting them from an image quality optimization program such as CODE V orZemax. This operation is illustrated in Fig. 3.1, which shows the Zemax model ofthe baseline Maksutov–Cassegrain telescope and the model obtained by importingit into FRED. Stray light control features such as baffles are usually not present inthe original model because they are not necessary for image quality analysis.

To make sure no errors occurred during import, the image quality of the straylight model should be compared to the original model. One of the simplest waysto do this is to compare the RMS spot size at the defined field points. If the twomodels are the same and the same rays are used to calculate both (i.e., same number,wavelengths, pupil positions, etc.), the RMS spot sizes should agree very closely,

41



42 Chapter 3

Figure 3.1 Zemax (top) and FRED (bottom) models of the baseline Maksutov–Cassegraintelescope.

to within a percent or two. If they do not, it is usually because of some differencebetween the two programs:

� Different refractive index models, especially for IR materials.� Different aperture dimensions. If the aperture dimensions were not explicitly

defined in the original optical model, then they have to be computed, and thestray light analysis program may have computed them differently.

� Different number of surfaces. This sometimes occurs because the stray lightanalysis program generates extra surfaces in optical elements with sharedsurfaces, such as refractive doublets.

If, after checking for the common errors listed above, the two programs still donot agree, then it may be necessary to compare the positions and direction cosinesof individual rays on a surface-by-surface basis in order to determine the source ofthe discrepancy. Also, be aware that optical surfaces imported from a mechanicalCAD program (such as Pro-Engineer1 or CATIA2) through an IGES or STEP fileare generally not represented accurately enough and will not ray trace correctly.

The next step in building the stray light model is to define its mechanical ge-ometry, such as its struts and baffles. These structures (such as baffles) are oftendesigned in the stray light analysis program, and other times they are importedfrom a mechanical CAD program through an IGES or STEP file. Be aware that themathematical representation (such as a nonuniform rational B-spline or NURBS)of a piece of mechanical geometry in a stray light analysis program may make itmore difficult to position because it uses an inconvenient coordinate system, andmay (because it is a high-order polynomial) make it slower to ray trace than simplersurfaces such as planes. In addition, CAD files often contain more detail than isnecessary for performing a stray light analysis; for instance, the file may containgeometry that is on the inside of a bulkhead and therefore can never be illuminatedor critical. The presence of this geometry in the stray light model slows the ray tracebecause the software has to check for ray intersections with it. For these reasons,



Basic Ray Tracing for Stray Light Analysis 43

it is usually best to use NURBS or other complex representations of mechanicalgeometry only when necessary. For instance, it is not necessary to use a NURBS sur-face to model a simple plane or cylinder; a simple (low-order) surface will suffice.Such surfaces are used to model the baffles in the stray light model of the baselineMaksutov–Cassegrain telescope shown in Fig. 1.5. If it is necessary to import CADgeometry, then it is best to import only those surfaces that can affect the stray lightperformance of the sensor, such as those that are both critical and illuminated.

3.1.2 Defining optical properties

Stray light analysis typically requires (at a minimum) that the BSDF of surfacesthat are both critical and illuminated be defined. BSDF is typically not definedin the image quality analysis model or in the mechanical CAD model; therefore,after importing geometry from these models, it must be defined in the stray lightanalysis program. Defining BSDF is nontrivial [for more details, see Chapters 4and 5 for defining the BSDF of optical surfaces, and Chapter 6 for mechanicalsurfaces (such as black-painted surfaces)]. It is also usually necessary to modelthe specular reflectance and transmittance of the optical surfaces. If the opticalgeometry was imported from an image quality analysis model, this informationmay be automatically imported (it is often not and must be manually copied intothe stray light analysis program). If this information is not available, see Chapter 7for information on modeling it.

Once the model’s geometry and optical properties have been defined, theyshould be used to perform some basic radiometric calculations whose results canbe confirmed with closed-form solutions. For instance, the irradiance on the focalplane due to a unit radiance source in the FOV can be computed and compared to theestimate using Eq. (2.40) (using �c = �/[4 * ( f /#)2]), or the irradiance on the focalplane due to scattering from the first optical surface in the system can be computedin the stray light analysis program and estimated using Eq. (2.47). It is important todo this because there are many opportunities to err when setting up the stray lightmodel. A single error in setting up the source flux and geometry, the system geome-try and optical properties, or the ray trace controls will result in an erroneous result.

3.2 Ray Tracing

Performing a ray trace in a stray light analysis program can be complicated andrequires a detailed understanding of its features and operation. This chapter dis-cusses optimal ways to set up the ray trace to quickly obtain reliable results, findstray light paths, and perform internal stray light analysis.

3.2.1 Using ray statistics to quantify speed of convergence

Stray light analysis programs often use a Monte Carlo (pseudo-random) ray propa-gation algorithm to pick the starting locations, direction cosines, wavelengths, andother parameters of the rays it generates. Therefore, the results of the ray trace are afunction of (among many things) the random number seed used to start it and of the



44 Chapter 3

Scattering surface

Incident ray

DetectorScattered rays

Raytrace 1 Raytrace 2

Figure 3.2 Monte Carlo ray tracing of the same system with different starting randomnumber seed. The scattered rays are at different angles for each raytrace.

number of rays traced. The random number seed is used to initialize the sequenceof random numbers generated; a ray trace performed twice with the same seed usedto initialize it will produce the same results. Typically, when a stray light analysisprogram is first opened, it is seeded with the same number, and therefore openingthe program twice and running the same ray trace will produce the same results.Once the program is open, it can be set to start every ray trace with the same seed(which means the program will generate the same result if a ray trace is run again)or not. An example of a ray trace that was repeated with a different seed is shownin Fig. 3.2. In this ray trace, rays are incident on a surface and rays are scatteredfrom it, some of which hit a detector surface. The rays are scattered at differentangles because of the different random seed.

Because the rays are generated at different angles for each ray trace, the numberof rays (and thus the total flux) that hits the detector changes. Generally, the morerays that hit the detector, the less the flux changes, and thus the faster the conver-gence. One way to quantify the variation in flux (or in any value computed in aMonte Carlo simulation, such as irradiance or intensity) with the number of rays isto divide the ray trace into n ray traces and collect statistics after each one.3 The vari-ation is computed as the relative error4 �of the quantity, which can be estimated as3

� = �

〈x〉 , (3.1)

where 〈x〉 is the mean value of the quantity x ,

〈x〉 = 1

n

n∑i=1

xi , (3.2)

and � is the standard deviation of the quantity x , computed as

� =

√√√√√√n∑

i=1

(xi − 〈x〉)2

n − 1. (3.3)




Table 3.1 Variation in flux on the detector and the relativeerror of the flux � as a function of the number of ray traces nperformed for the geometry shown in Fig. 3.2.

n � (W) �

10 0.02130 0.32161

100 0.02135 0.05716

1000 0.02132 0.0067

As the number of ray traces (and hence the number of rays) increases, the relativeerror of the quantity x will decrease. These calculations were performed for the raytrace shown in Fig. 3.2, in which a Lambertian scatterer is illuminated by a single1-W ray, and scattered rays are traced to the detector. The flux on the detector �

and its relative error � are shown in Table 3.1 as a function of the number of raytraces n (and hence the number of rays). As expected, � decreases with the numberof rays traced, the flux on the detector converges to its true value (0.02131 W),and the true value of the flux is always within the estimated error �(1 ± �).

Many stray light analysis programs do not have a built-in way to compute �,and therefore it must be added using the scripting language or in another programto post-process the data. By computing the relative error of the quantity of interest,the number of rays and/or ray traces necessary to obtain a given relative error canbe determined.

Now that a means of calculating the efficiency of a ray tracing simulation hasbeen established, methods of improving this efficiency will be discussed.

3.2.2 Aiming scattered rays to increase the speed of convergence

A technique often used to increase convergence speed is scattered ray aiming,which is illustrated in Fig. 3.3. This figure shows rays incident on a surface and

Scattered rays aimed into hemisphere

Scattered rays aimed at detector

(a) (b)

Figure 3.3 Scattered rays aimed (a) into a hemisphere and (b) to the detector.



46 Chapter 3

(a) Backwards ray trace from point source at focal plane

(b) Backwards ray trace from extended source at focal plane

(c) Forward ray trace, scatter aimed at virtual image

Back (inside) surface of corrector

Figure 3.4 Steps in computing the virtual image of the detector for the back surface of thecorrector.

scattering; on the left side, the scattered rays are traced into a hemisphere, and onthe right, they are aimed at the detector surface. Obviously, most of the rays aimedinto the hemisphere are wasted, as very few go to the detector. By contrast, noneof the rays aimed at the detector are wasted, and therefore this ray trace results ina much lower relative error. All of the stray light analysis programs discussed inSection 1.1 allow rays to be aimed in this way.

In Fig. 3.3, rays from the scattering surface can be aimed directly at the detectorbecause there are no optics between them. This is often not the case, as shown bythe example in Fig. 3.4(c). In order for rays scattered from the back (inside) surfaceof the corrector to reach the detector, they must first reflect from the primary andsecondary mirrors. This is done for the ray trace in Fig. 3.4(c) by first determiningthe virtual image of the detector as seen by the back surface of the corrector. Thiscan be achieved via the following steps:

1. Define a point source at the focal plane and trace it backward through thesystem, making sure that the rays fill the pupil, which usually means tracinginto a cone whose full divergence angle is equal to 2*sin–1[1/(2* f /#)] (see




Section 3.2.3 for more information on backward ray tracing). This step isshown in Fig. 3.4(a).

2. Stop (absorb) those rays on the surface whose scatter is to be aimed.3. Use a best-focus calculation to determine where these rays come to a focus.

This is the location of the virtual image.4. Define an extended source at the focal plane that is the same size as the

image, and trace it backward through the system, again making sure to fillthe pupil. This step is shown in Fig. 3.4(b).

5. Stop (absorb) those rays on the surface whose scatter is to be aimed.6. Propagate these rays along their direction cosines to the virtual image loca-

tion and determine the spatial extent of the ray cloud. This extent is the sizeof the virtual image. This image may be highly aberrated, in which case theshape of its maximum extent may not be a rectangle, though typically it iswell approximated by one.

7. Define a surface or curve at the virtual image of the detector, and aim raysfrom the scattering surface at it. Most or all of these rays will hit the detector.This step is illustrated is Fig. 3.4(c).

Calculation of the virtual image of the detector should be computed for everysurface in the system whose scatter is to be analyzed, as doing so will greatlyincrease the efficiency of the ray trace. The location and size of the virtual imageof the focal plane is different in each optical space of the system. Optical spacesare defined by the intended (nominal) path that light takes through the opticalsystem, and change after the light reflects or refracts through a surface. The opticalspaces in the baseline Maksutov–Cassegrain telescope are illustrated in Fig. 3.5.Optical spaces #3 and #4 refer to the same physical space; however, rays from thescene pass through optical space #3 prior to reflection from the primary mirror, andrays pass through optical space #4 after reflection from the primary. The virtualimage calculation described above was performed for optical space #3, for whichthe apparent location and size of the focal plane as seen through the primary andsecondary mirrors was computed. Provided that it can see the virtual image of the

1 2

5

3,4

Figure 3.5 Optical spaces in the baseline Maksutov–Cassegrain telescope.



48 Chapter 3

focal plane, any scattering surface in this optical space (such as the primary mirrorbaffle shown in Fig. 1.5) can be aimed at the focal plane using this virtual image. Theprimary mirror baffle can also be aimed at the focal plane from both optical spaces#4 and #5; rays aimed from optical space #4 will be aimed at the virtual imageof the focal plane as seen in reflection from the secondary, and rays aimed fromoptical space #5 will be aimed directly at the focal plane. Because the virtual imagecalculation must be performed for each optical space, it is recommended that thesteps to compute it be automated using a script, which is generally straightforward.

Stray light analysis programs generally allow scatter to be aimed in a variety ofways, including into a cone of a specified angular divergence. This method may bemore appropriate for imaging rays at virtual images that are at or close to infinity,such as the virtual image for optical space #1. In this optical space, the angularextent of the focal plane is equal to the FOV of the system, and therefore aimingscattered rays into a cone whose divergence is equal to the FOV will ensure that allof the rays hit the focal plane.

3.2.3 Backward ray tracing

The previous chapter illustrated one use of backward ray tracing, in which raysare traced backward from the focal plane into the optical system. Though it mayseem counterintuitive at first, any radiometric (and therefore stray light) analysiscan be performed using a backward ray trace from the focal plane to the source.This is due to the invariance of throughput (Section 2.2.5), which states that thethroughput from the source to the focal plane must be the same as the throughputfrom the focal plane to the source; therefore, the flux transferred from the focalplane with radiance L to the source is the same as the flux transferred in the otherdirection. For example, the flux � transferred from the sun to the focal plane in thesimple analytic stray light model developed in Section 2.3.6 can be computed bymultiplying both sides of Eq. (2.50) by the area of the focal plane Afp:

� = Lsun�atm�sun cos (�sun) BSDF optics (�sun) �optics�optics Afp. (3.4)

This equation is the same, even if the solar radiance Lsun is propagated fromthe focal plane to the sun. Therefore, the irradiance at the detector due to straylight ESL can be computed by placing a source of radiance Lsun, tracking backward,and computing the power � from scattering and other stray light mechanisms on asurface that has the same angular extent as the sun. ESL is then given by �/Afp.

The same results can be obtained tracing rays in either direction; however,one method may be more efficient than the other. It is typically more efficient totrace rays from the object with smaller angular extent to the object with largerangular extent. Therefore, in a solar stray light analysis of the baseline Maksutov–Cassegrain telescope, it is more efficient to do a forward ray trace. That is, to tracerays from the sun (angular extent =0.5333 deg) through the optical system and tothe detector (angular extent =0.72 deg, which is its minimum FOV). Because thesun has a small angular extent, solar stray light analysis is usually performed usingforward ray tracing. Conversely, if the same system were used to observe the earth




(a) Forward ray trace

(b) Backward ray trace

Back (inside) surface of corrector

Figure 3.6 (a) Forward ray trace from a source of small angular extent (the sun), and(b) backward ray trace to a source of large angular extent (the earth from an altitude of824 km).

from orbit, a backward ray trace would usually be more efficient, as the angularsubtense of the earth from a typical orbital altitude (824 km) is 86.3 deg. Forwardand backward ray tracing for these cases (sun and earth, respectively) are illustratedin Fig. 3.6. In both Figs. 3.6(a) and (b), scatter from the back (inside) surface of thecorrector is being analyzed; however, the ray aiming for this surface is different foreach ray trace. In Fig. 3.6(a), rays are aimed at the virtual image of the detector,which is the most efficient way to perform this analysis, as described earlier. InFig 3.6(b), rays are aimed in the other direction and at the angular extent of the earth,which is the most efficient way to aim rays for this ray trace. This figure illustratesthe need to define different aim regions for different ray tracing directions.

More methods of improving efficiency are discussed in the remaining sectionsof this chapter. Many of these methods utilize backward ray tracing as a way ofimproving efficiency.

3.2.4 Finding stray light paths using detector FOV

Section 2.3.2 discussed the concept of detector FOV, in which the intensity as afunction of azimuth and elevation angles I (�, �) due to a source at the focal planeof the system is calculated. Detector FOV is the primary method of identifyinglow-order stray light paths, and should be performed in nearly every straylight analysis. It is performed in stray light analysis software by simply backwardray tracing from the focal plane and collecting the rays on a surface in front of thesystem [similar to the ray trace shown in Fig. 3.6(b)], and then binning these rays



50 Chapter 3

250

0

125

187.5

62.5

Intensity (W/sr)

Angle with Y Axis (deg)Angle with Y Axis (deg)Angle with Y Axis (deg)Angle with Y Axis (deg)

Ang

le w

ith X

Axi

s (d

eg)

Ang

le w

ith X

Axi

s (d

eg)

Ang

le w

ith X

Axi

s (d

eg)

Ang

le w

ith X

Axi

s (d

eg)

Figure 3.7 Detector FOV of the modified baseline Maksutov–Cassegrain system. The rect-angle in the center corresponds to the nominal FOV, and the ring around it corresponds tothe zeroth-order stray light path through the hole in the primary mirror. Axes are defined inFig. 1.5. The maximum grayscale value was truncated to show detail.

according to their angular distribution. Thus, in this plot, all details of the spatialdistribution of the rays are lost. In order to identify all possible stray light paths,it is recommended to trace backward into a hemisphere. The detector FOV of themodified baseline Maksutov–Cassegrain system is shown in Fig. 3.7. The systemwas modified by removing the primary and secondary mirror baffles. This figurewas computed using a unit radiance source (1 W/mm2-sr) over the entire extent ofthe FPA (23.6 mm × 15.8 mm). No stray light mechanisms higher than zeroth-orderwere turned on for this ray trace. The region of high intensity in the center of theplot corresponds to the FOV of the system (1.08 deg × 0.72 deg). The magnitudeof this intensity is given by

I (0, 0) = LAfp�optics�optics

�FOV, (3.5)

where L is the radiance of extended source at the focal plane (= 1 for Fig. 3.7),and �FOV is the projected solid angle of the FOV (= Afp/EFL2).

In a system with no stray light (which, in practice, does not exist), Fig 3.7would contain only the small bright square in the center, which corresponds to thenominal FOV. However, the presence of the zeroth-order stray light path throughthe hole in the primary mirror results in the annular region of intensity that extendsto about ± 12 deg. A ray trace of this zeroth-order path is shown in Fig. 3.8. Thisexample illustrates the benefit of using the detector FOV to identify such paths.

3.2.5 Determining critical and illuminated surfaces

First-order stray light paths must come from surfaces that are both illuminated andcritical, and therefore identifying such surfaces is important. An efficient way to




Figure 3.8 Zeroth-order path through the hole in the primary mirror in the baselineMaksutov–Cassegrain system.

do this is to first perform a backward ray trace (which, as described in the previouschapter, should be performed anyway to identify zeroth-order paths) and make alist of the surfaces that receive flux from the focal plane. Then, perform a forwardray trace from the external source and make a list of the surfaces that receive flux.Surfaces on both lists are both illuminated and critical, and will therefore contributeto stray light at the focal plane, as shown in Fig. 3.9. The backward ray trace needsto be performed only once, but the forward ray trace may need to be performedmultiple times, once for each angle of the external source.

3.2.6 Performing internal stray light calculations

An efficient way to calculate internal stray light from self-emission in IR systems(such as Fig. 3.10) is as follows:

1. Define a Lambertian source at the detector whose area A is that over whichthe internal stray light is to be computed. The size of a detector pixel is

Rays to a critical surface

Rays to an illuminated surface

Main baffle is critical and illuminated

Figure 3.9 A critical and illuminated surface in the Maksutov–Cassegrain telescope. Dueto overviewing, the main baffle is both critical and illuminated. Overviewing is discussedfurther in Section 8.3.



52 Chapter 3

Figure 3.10 LWIR camera, 7.5–10.5 mm, f /1.67, FOV = 15.3 deg. Based on US Patent#5,909,308.

usually the smallest area that is used. Make the radiance of this source 1/A,and aim it into a hemisphere so that all surfaces surrounding the detectorcan, if possible, be hit by rays from the source.

2. Trace rays backward from the source, and output a list of the flux hittingeach surface. Because the radiance of the source is equal to 1/A, the flux onthe i th surface is equal to [by Eq. (2.39)] the projected solid angle × pathtransmittance of the surface (�i ��) as seen by the source on the detector.This step is illustrated in Fig. 3.11.

3. Move the source to all the points on the detector at which the internal straylight is to be calculated, and repeat step 2.

4. Load the resulting solid angles into a spreadsheet, as shown in Table 3.2. Theirradiance ESL at each point on the detector from which rays were traced isequal to

ESL =n∑

i=1

Li εi�i �i , (3.6)

where Li and εi are the in-band blackbody radiance and emissivity of the i thsurface, respectively. This calculation can be performed for every source location

Lens E1 Lens E2

Lens E3

Field stop

Lens E4

FilterDetector

Housing coneCold

shieldAperture

stop (cold)

Scene

Figure 3.11 Backward ray tracing in a LWIR camera system to calculate internal stray light.




Table 3.2 Spreadsheet used to compute irradiance at the detector from internal stray lightin the LWIR camera shown in Fig. 3.11.

Detector Location x = 0, y = 0

Radiance Projected DetectorTemperature (ph/s- Solid Irradiance

Surface (K) Emissivity mm2-sr) Angle∗� (sr) (ph/s-mm2)

El 300 0.0218 2.8605E+13 0.2037 5.8282E+12

E2 300 0.0114 1.4970E+13 0.2121 3.1759E+12

Field Stop 300 1 1.3092E+15 0 0.0000E+00

E3 300 0.0012 1.5701E+12 0.2209 3.4684E+11

E4 300 0.0159 2.0802E+13 0.2300 4.7844E+12

Filter 300 0.0030 3.9218E+12 0.2608 1.0227E+12

Aperture Stop 77 1 6.1015E+08 0.6012 3.6680E+08

Housing Cone 300 1 1.3092E+15 0 0.0000E+00

Cold Shield 77 1 6.1015E+08 2.2797 1.3909E+09

Scene 300 1 1.3092E+15 0.1947 2.5488E+14

Total 2.7003E+14

Detector Location x = 0, y = 1,5875 mm

Radiance Projected DetectorTemperature (ph/s- Solid Irradiance

Surface (K) Emissivity mm2-sr) Angle∗� (sr) (ph/s-mm2)

El 300 0.0000 0.0000E+00 0.1901 0.0000E+00

E2 300 0.0114 1.4970E+13 0.1830 2.7394E+12

on the detector, and a map of the irradiance over the detector can be computed, asshown in Table 3.2. The table shows that the primary contributors to internal straylight in the center of the detector (location 0,0) are the warm aperture stop andthe housing cylinder around the detector, which indicates the need to cryogenicallycool these surfaces. This topic is discussed more in Section 8.12.

Because the values of Li are not a function of the ray trace, the temperatureof any surface in the sensor can be set to any value and the resulting detectorirradiance map (Fig. 3.12) instantly recomputed. As discussed later in Section7.1.2., the reflectance of the AR coatings on the lenses was assumed to be 0.02,and the reflectance of the bandpass filter coating was assumed to be 0.1. In order to



54 Chapter 3

Irradiance (ph/s-mm2)

2.70E14

2.62E14

2.66E14

2.64E14

2.68E14

Figure 3.12 Irradiance distribution at the detector from internal stray light in the LWIRcamera shown in Fig. 3.11.

make the values in Table 3.2 easier to verify by hand calculations, ghost reflectedrays were not traced during the raytrace simulation; therefore, the contributionto internal stray light due to ghost reflections is not captured. However, ghostreflections from refractive optics in IR systems generally reflect the FOV of thedetector to warm lens barrels and other mechanical structures (such as the “housingcone” in Fig. 3.11) in the sensor, and thus they typically increase the irradianceat the detector due to internal stray light by Li j�i �i , where Li , �i , and �i arethe radiance, projected solid angle, and path transmittance of the i th mechanicalstructure surface, respectively, and j is the reflectance of the j th refractive surface.The similarity between this equation and Eq. (3.6) illustrates that the reflectanceof refractive optics acts to increase internal stray light in a manner similar totheir emissivity. Section 8.7 notes that the increase can be reduced by placing therefractive optic close to or inside the cold shield, so that the detector views coldgeometry in reflection from it. The design of the cold shield for this system isdiscussed in Section 9.2.2.




ni nt

Ancestry = 0

Ancestry = 1 Ancestry = 0

Ancestry = 2Ancestry = 1

Figure 3.13 Ancestry levels of split rays in a ghost reflection ray trace.

3.2.7 Controlling ray ancestry to increase speed of convergence

The number of times that a ray has been split is called its ancestry, and it isimportant to control in order to increase the speed of convergence. Rays that havebeen split once (either through ghost reflections or scattering) have an ancestry of1, as shown in the ghost reflection ray trace in Fig. 3.13. Rays with an ancestry of1 model first-order stray light, rays with an ancestry of 2 model second-order straylight, etc. The flux of each ray generally decreases as roughly (0.01)n , where n is theancestry (or order) of the ray. In addition, the number of rays at each ancestry levelincreases as 2n for ghost rays and Nn for scattered rays, where N is the numberof scattered rays per incident ray. Therefore, with each increase of ancestry level,the stray light analysis program is tracing exponentially more rays of exponentiallyless flux, causing it to use all available memory and dramatically slow down theray trace. For this reason, the ray trace controls should be set so that the maximumancestry level of the rays generated is equal to the maximum stray light path orderto be analyzed. This is determined by the stray light requirements of the system(see Section 11.1). In most cases, it is not necessary to analyze path orders greaterthan 2; however, as detectors become more sensitive, this will less often be thecase. Case studies of this issue have been published.5

3.2.8 Using Monte Carlo ray splitting to increase speedof convergence

When performing a ghost reflection analysis, the default behavior in most straylight analysis programs is to split the incident ray into two rays at the refractivesurface, one in the reflected direction, the other in the transmitted direction, asshown in Fig. 3.13. As discussed in Chapter 2, this may result in a large number ofrays generated during the ray trace, especially if there are a large number of surfacesfrom which ghost reflections are being traced (which could use all available RAMin the computer and, therefore, slow down the ray trace). If n rays are incident onthe refractive surface, then n rays will be propagated in the transmitted direction,



56 Chapter 3

each with a flux equal to �i (1 – R), where �i is the flux of the incident ray, and Ris the reflectance of the surface. In addition, n rays will be traced in the reflecteddirection, each with a flux of �i R, for a total of 2n rays.

One way to reduce the number of rays traced is to use Monte Carlo ray splitting,in which only the reflected ray is traced if r ≤ R, where r is a uniform randomvariable between 0 and 1 (inclusive), and R is the reflectance of the surface. Ifthis condition is not met, then only the transmitted ray is traced. The flux of thetransmitted or reflected rays is the same as that of the incident ray �i . If n raysare incident on the surface, then, as n → ∞, (1 – R)n rays will be traced in thetransmitted direction, and Rn in the reflected, for a total of n rays. Thus, the ray tracewill not generate as many rays during the ray trace, which can reduce the amountof memory used by the ray trace and thus improve the speed of the simulation.However, this may result in fewer rays reaching the focal plane and thereforeincrease the relative error of the irradiance on it. The only way to decrease therelative error would be to trace more rays, which will result in a longer ray trace.Therefore, Monte Carlo ray splitting may reduce the amount of memory requiredfor a ray trace at the cost of longer ray tracing time; however, the time to completethe raytrace may still be less than the time required to perform it without usingMonte Carlo ray splitting.

3.2.9 Calculating the effect of stray light on modulationtransfer function

Stray light can reduce system image quality by lowering its modulation transferfunction (MTF), which can be used to quantify the image quality of a system. Thiscalculation can be computationally intensive and therefore often must be done ina stray light analysis program. This section discusses one method of performingthis calculation. A comprehensive review of MTF, which is a definition that comesout of Fourier optics theory, is beyond the scope of this book; however, there are anumber of good references.6,7

MTF quantifies the ability of the optical system to reproduce the spatial mod-ulation of the scene, and is equal to

MTF ( f ) = Emax( f ) − Emin( f )

Emax( f ) + Emin( f ), (3.7)

where Emax( f ) is the irradiance at the focal plane due to the maximum radiance of apattern in the scene of spatial frequency f , and Emin( f ) is the irradiance due to theminimum radiance of the same pattern, as shown in Fig. 3.14(a). A perfect opticalsystem will have Emin = 0 and therefore MTF = 1. In the simplest approximation,stray light degrades the MTF by increasing Emin to Emin + ESL, as shown in Fig3.14(b).

MTF is also equal to the normalized Fourier transform of the point spreadfunction (PSF)7, which is the irradiance distribution at the focal plane E f (x , y) due




Emin Emax Emin + ESL

(a) (b)

Figure 3.14 Image of an MTF bar target (a) without stray light and (b) with stray light.

to a collimated input beam:

MTF( fX , fY ) = � [E f (x, y)

]∞∫

−∞

∞∫−∞

E f (x, y) dxdy

. (3.8)

Therefore, MTF can be computed in stray light analysis software by computingthe irradiance at the focal plane due to a collimated input beam and taking its Fouriertransform. This calculation can often be performed in the stray light analysisprogram itself, as most of them include a fast Fourier transform (FFT) algorithm.If stray light mechanisms (such as surface roughness scattering) have been turnedon in the stray light model, then the MTF computed from E f (x , y) will quantifythe effect of stray light. Typically, E f (x , y) is computed using either coherentbeam analysis (for a diffraction-limited or nearly diffraction-limited systems) orgeometric ray tracing analysis (for geometric aberration-limited system). Doingthis calculation in stray light analysis software can be complicated, and the bestsource of information for performing it is usually the software documentation.(This book does not discuss this calculation in detail, although some details ofdiffraction modeling are given in Section 7.2.)

Examples of these calculations are shown in Figs. 3.15 and 3.16. The formershows a ray trace of the optical system used to demonstrate the calculations. Thissystem is diffraction limited on-axis at 0.6328 m. Coherent beam analysis was

Figure 3.15 The optical system used to perform the PSF calculations shown in Fig. 3.16.This system is f /10, has an EPD of 10.2 mm, and is diffraction limited for the on-axis beamat 0.6328 m.



58 Chapter 3

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

0 0.1 0.2 0.3

Irra

dian

ce (W

/mm

2 )

Y Position (mm)

No scatterw/1000 Angstrom Surface Roughness

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200M

TFSpatial Frequency (1/mm)

No scatterw/1000 Angstrom Surface Roughness

Scatter reduces peaks

Scatter increases valleys

(a) (b)

Figure 3.16 The (a) PSF and (b) MTF of the diffraction-limited system shown in Fig 3.15,with and without scattering from the back surface of the lens. The oscillation in the minimumvalues of the PSF curve is an aliasing artifact.

used in FRED to compute the PSF of this beam without the effects of stray light[Fig 3.16(a)]. This PSF matches the theoretical PSF predicted for a diffraction-limited system with a circular pupil (namely, the J1 Bessel function, as describedin Section 7.2.1).7 The MTF of the system was computed by taking the Fouriertransform of the PSF, and the resulting function matches the theoretical MTFpredicted for this system.

Stray light was added to the system by modeling the back surface of the lenswith a BTDF function for 1000-

′A RMS surface roughness (see Section 4.1.1 for

defining this model). For any practical optical system, this level of roughness (whichhas a TIS of 24.6%) is unacceptably high. However, it is used here to make theeffect of scatter on PSF and MTF more noticeable when plotted. The ray tracewas repeated, and the resulting PSF and MTF are plotted in Fig. 3.16. In the PSFplot [Fig. 3.16(a)], scattering has redistributed the irradiance from the center of thePSF to its edges, resulting in a decrease in the peak PSF and an increase in theminimum PSF at Y positions ≥ about 0.15 mm. This “washing out” of the PSFcan also be seen in the MTF [Fig. 3.16 (b)], which has decreased for all spatialfrequencies except the DC term. Such broadening of the PSF and reduction of theMTF is typically the effect of stray light on image quality.

3.3 Summary

The most accurate way to predict the stray light performance of a system is typi-cally by using a stray light analysis program. The model of the system is usu-ally created in these programs by first importing the optical prescription from an




image quality optimization program like CODE V or Zemax. There are often bugsin this import process, and therefore the RMS spot size or other image qualitymetric should be computed in the stray light analysis program and compared toits value in the original program to make sure the import occurred without error.In addition, the model should be used to perform basic radiometric calculationsthat can be confirmed with closed-form solutions, because when constructing thestray light model, there are many opportunities to make an error. Geometry canalso be imported from a CAD program such as Pro/ENGINEER (now called CreoElements/Pro) or SolidWorks via an IGES or STEP file, though this should be donesparingly because CAD geometry can slow down the ray trace.

Stray light analysis programs often use a Monte Carlo algorithm to generaterays, which can take a long time to converge if not set up properly. The speed ofconvergence is dramatically improved if scattered rays are aimed at the focal plane.The relative error of the quantity of interest (such as flux on the focal plane) shouldbe calculated to determine if it is accurate enough for purposes of the simulation.Limiting the ancestry of split rays and/or using Monte Carlo ray splitting can alsoincrease the speed of convergence.

Backward ray tracing, in which rays are propagated from the focal plane tothe entrance aperture of the system, is an important part of stray light analysisand should be performed on most systems. It is used to generate the detector FOV,which is a 2D map of the power transferred to the detector as a function of fieldangle, and is a function of all of the stray light mechanisms in the system. Thismap can be used to quickly identify low-order stray light paths and is thereforean important part of the analysis. The backward ray trace is also used to identifycritical surfaces and perform internal stray light analysis in IR systems. The effectof stray light on MTF can be computed by taking the Fourier transform of thestray-light-broadened PSF.

References

1. Pro-Engineer Software, Parametric Technology Corporation (PTC), http://www.ptc.com.

2. CATIA Software, Dassault Systemes, http://www.3ds.com.3. D. Jenkins and E. Fest, “Robust error estimation in optical analysis software

using subdivided and recombined ray traces,” Proc. SPIE 6289, 62890O (2006)[doi: 10.1117/12.681187].

4. B. Frieden, Probability, Statistical Optics, and Data Testing, Springer, New York(1983).

5. P. Le Houiller and E. Freniere, “To split or not to split: Case studies on MonteCarlo analysis of illumination ray tracing concerning the usefulness of ray-splitting,” Proc. SPIE 6338, 633803 (2006) [doi: 10.1117/12.680883].

6. J. Goodman, Introduction to Fourier Optics, 3rd Ed., Roberts & Company,Englewood, CO (2005).




Chapter 4

Scattering from Optical SurfaceRoughness and CoatingsOptical surfaces are the surfaces of the lenses and mirrors that form the imagein an optical system. Though these surfaces are typically very smooth, none areperfectly smooth, and their residual roughness will scatter light. An example of thetype of artifact that results from surface roughness scattering is shown in Fig. 1.1.Surface roughness of the camera’s optics has scattered sunlight from outside theFOV into its FOV, thus increasing its out-of-field stray light. Scattering from par-ticulate contaminants (discussed in Chapter 5) results in artifacts similar to thosefrom surface roughness and also contributes to this artifact. Roughness of a typicalground and polished fused silica surface is shown in Fig. 4.1. The roughness profileof optical surfaces varies greatly with the substrate material and the fabrication andfinishing processes; the roughness profile shown in Fig. 4.1 is not representative ofall optical surfaces.

There are a number of ways to model surface roughness scatter; in general,the more fidelity the model has, the more measurements (and therefore the moretime and money) required to develop it.2 The relationship between the accuracy ofthe BSDF model and the accuracy of the predicted stray light performance of thesystem depends on the location of the optic in the system; optics that are illuminatedinfluence the system performance more than those that are not. In general, theless the scatter from a particular surface contributes to stray light at the focalplane, the less accurately its scatter needs to be modeled. A few models of surfaceroughness scatter are presented in this chapter, each requiring different inputs. Inorder to understand these models, it is necessary to review the relationship betweenthe surface roughness profile, the wavelength, and the BSDF. A comprehensivereview of this theory is beyond the scope of this book, so an abbreviated versionis presented here (there are a number of good sources for further reference3−5).After a review of the basic physics, a number of models are presented that makeuse of this theory to develop a scatter model of the surface. This chapter alsodiscusses scattering from scratches and digs. Because scattering from any realsurface is always the sum of scattering from its roughness and scattering fromparticulate contaminants on it (there will always be some), the models discussedin this chapter may not accurately describe scatter from real surfaces, especially if

61



62 Chapter 4

Figure 4.1 Atomic-force microscope image of a ground and polished fused silica surface.1

The portion of the surface shown is about 10 �m across, and its RMS roughness is about5 A. The streaks are from the grinding process.

they are heavily contaminated. Scattering from particulates is discussed in Chapter5, which includes a comparison between scattering from surface roughness andscattering from particulates.

4.1 Scattering from Uncoated Optical Surface Roughness

The sag of an optical surface is generally described as the sum of two profiles:its optical figure profile and its surface roughness profile, as shown in Fig. 4.2.The optical figure of a surface determines its image-forming properties, and assuch it determines the direction of the specularly reflected or transmitted rays. Inmost cases, the figure is described by the general conic equation, plus any asphericterms.6 The surface roughness determines the magnitude and angular distributionof light scattered from it. Light scattered at angles near the specular beam is

Incident beam Specularly reflected beam

Near angle scatter

Large angle scatter

Surface roughness

Surface optical figure

Figure 4.2 Scattering from surface roughness.



Scattering from Optical Surface Roughness and Coatings 63

generally referred to as near specular or near angle scatter, and light scattered atlarge angles from the specular beam is called large angle scatter. The roughnessfunction is determined by the process used to finish the surface and is generallymuch more difficult to model a priori than the figure function. As a result, mostdescriptions of the surface roughness rely on measured data. A common way toobtain this data is with a surface profilometer or a white-light interferometer.7

An important quantity that can be computed from this profile z(x , y) is itspower spectral density (PSD). The 2D PSD S2 of this profile is defined as

S2( fx , fy) = limL→∞

1

L2

∣∣∣∣∣∣∣L/2∫

−L/2

L/2∫−L/2

z (x, y) exp[−2�i( fx x + fy y)]dxdy

∣∣∣∣∣∣∣

2

, (4.1)

where fx and fy are the spatial frequencies (in inverse length units), and L is thelength of the measured profile. Thus, the PSD is the modulus squared of the Fouriertransform of the surface.

Another important quantity that can be computed from the roughness profileis the RMS roughness � of the surface (sometimes called Rq), and is given by

� =⎧⎪⎨⎪⎩ lim

L→∞1

L2

L/2∫−L/2

L/2∫−L/2

[z (x, y)]2 dxdy

⎫⎪⎬⎪⎭

1/2

. (4.2)

RMS roughness is probably the most common way of specifying optical finish andas such is commonly called out on optical component drawings. When measuring �for the purposes of predicting scatter, it is important to consider the waveband of thesensor in which the optic will be used, as the effective value of � (sometimes calledthe total effective surface roughness or ��) changes with the sensor waveband.8

This topic is discussed more fully in Section 4.1.1.The measured PSD of a typical mirror surface with �� = 13.1 A is shown in

Fig. 4.3. Due to the randomness of the finishing processes such as polishing, mostmeasured PSDs will exhibit a significant amount of noise.

In general, the PSD of most optical surfaces (in which the RMS roughness� is much less than the wavelength of light �) are well approximated using theK-correlation model8 (also called the ABC model) whose functional form is givenby

S( f ) = A[1 + (Bf )2]−C/2, (4.3)

where f = ( f 2x + f 2

y )1/2, A is the magnitude of the PSD at low frequencies (asshown in Fig. 4.3), 1/B is the spatial frequency at which the “roll-off” occurs and isproportional to surface spatial wavelength (also called the autocorrelation length)of the surface profile, and C is the slope of the PSD at frequencies above 1/B.



64 Chapter 4

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

Pow

er S

pect

ral D

ensi

ty (μ

m4 )

Spatial Frequency (1/μm)

5X 10X 50X-1 50X-2 K correlation fit

A

B

slope=-C

Figure 4.3 Measured PSD S2 versus spatial frequency f for a typical mirror surface with� = 13.1 A.9 The measured data (digitized from the original reference) is labeled by themagnification of the objective used in the white-light interferometer to obtain it. Data fromtwo interferometers with 50× objectives is shown. The K-correlation model parameters areA = 4.64 × 10–3�m4, 1/B = 1 × 10–3�m–1, and C = 1.55.

The relationship between PSD and BSDF can be determined by the Rayleigh–Rice perturbation theory (also called the “Golden Rule”)2 for � << � as

BSDF (�i ; �s, �s) = 4�2�n2

�4cos �i cos �s QS2( fx , fy), (4.4)

where �i is the incident angle, �s and �s are the elevation and azimuth scatterangles (as shown in Fig. 2.18), �n is the difference in refractive index across theboundary of the scattering surface (=2 for mirrors), Q is the “polarization factor”2

that is a complicated function of the dielectric constant of substrate, �i , �s , �s , thepolarization state of the incident beam, and the polarization state of interest of thescattered light. However, in the case of in-plane scatter (�s= 0 or 180 deg) for asurface whose reflectance or transmittance is not a strong function of the angleof incidence or polarization (such as high-reflectance mirror), Q is approximatelyequal to the specular reflectance or transmittance of the surface. S2 is the 2D PSD(as defined above), and

fx = sin �s cos �s − sin �i

�, (4.5)




fy = sin �s sin �s

�. (4.6)

Note the similarity between these equations and the general grating equation, whichis given by

f = |sin �s − sin �i |m�

, (4.7)

where m is the order of the diffracted beam. Thus, the Rayleigh–Rice perturbationtheory treats the optical surface as a collection of diffraction gratings of differentfrequencies. Surfaces whose scattering varies as a function of | sin �s − sin �i | aresaid to be shift invariant. In general, the BSDF predicted by Eq. (4.4) correlates wellwith measured data, though models with different obliquity factors [cos �i × cos �s

in Eq. (4.4)] have been proposed.10

Because most optical surfaces scatter isotropically and the roughness of mostoptical surfaces is well described by the K-correlation model, Eq. (4.4) can bewritten as

BSDF (|sin �s − sin �i |)

= 4�2�n2

�4cos �s cos �i Q

{A

[1 +

(B |sin �s − sin �i |

�

)2]}−C/2

. (4.8)

Fig. 4.4 shows the BSDF that is predicted by Eq. (4.8) using the measured and K-correlation model PSDs shown in Fig. 4.4 [= 0.6328 �m, �n = 2 (i.e., a mirror),and Q = 1]. Neglecting the Q cos �s factor, the functional form of BSDF versus|sin �s – sin �i | is the same as the function form of PSD versus f .

Equation (4.8) is commonly parameterized as the 3-parameter Harvey model18

(also called the Harvey–Shack model):

BSDF (|sin �s − sin �i |) = b0

[1 +

( |sin �s − sin �i |l

)2]s/2

, (4.9)

where

b0 = 4�2�n2 Q A

�4,

l = �

B,

s = −C.

A closed-form solution for the TIS of a Harvey scatterer at �i=0 can be computedby plugging Eq. (4.9) into Eq. (2.32). For s �= –2,

TIS = 2�b0

ls(s + 2)

[(1 + l2)

s+22 − (l2)

s+22

], (4.10)



66 Chapter 4

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

BRD

F (1

/sr)

|sinθθs–sinθθi|

5X 10X 50X (1) 50X (2) K-correlation fit Measured BRDF

Figure 4.4 BRDF versus |sin �i – sin �s| for the 13.1-A RMS roughness mirror whose PSDis given in Fig. 4.3 (� = 0.6328 �m, and �n = 2).

and for s =2,

TIS = �b0l2 ln(

1 + 1

l2

). (4.11)

At �i �= 0, the TIS must be evaluated numerically. This calculation can be performedby most stray light analysis programs.

The Harvey equation neglects the cos �s × cos �i obliquity term in Eq. (4.8).As will be discussed in Chapter 5, this term is usually neglected because its effectcannot be measured for surfaces with particulate contamination (which all surfaceshave). The Harvey model is a very common way of modeling the BSDF ofan optical surface and is supported by most stray light software. The Harveymodel derived from the K-correlation model parameters discussed in this sectionis plotted in Fig. 4.5, along with the K-correlation data for reference. The Harveycoefficients are b0 = 4.5665 sr–1, l = 6.3280 × 10–4, and s = –1.55. The TIS ofthis model is 6.7675 × 10–4. If the substrate is set to Schott N-BAK2 (n = 1.5381@ 0.6328 �m), then the b0 coefficient drops to 0.3305 sr–1, and the TIS drops to4.8981 × 10–5.

Also plotted in Fig. 4.5 is the ABg model, which is very similar to the Harveymodel and also commonly used. The function form of the ABg model is

BSDF = A

B + |sin �s − sin �i |g . (4.12)




1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

BRD

F (1

/sr)

|sinθs–sinθi|

K-correlation fit Harvey Abg

Figure 4.5 K-correlation, Harvey, and ABg models for the 13.1-A RMS roughness mirrorwhose PSD is given in Fig. 4.3 (� = 0.6328 �m).

The three coefficients of the ABg model can be computed from the Harveymodel coefficients as

B = l−s,

g = −s,

A = b0 B.

Using the above conversion of coefficients, the ABg model and the Harveymodel predict nearly equal values of BSDF, especially for very small and very largevalues of | sin �s − sin �i |. However, the two differ slightly, especially in the regionof | sin �s − sin �i | = l. The difference between the two is generally considered tobe negligible. The TIS of the ABg function is easier to compute analytically byfirst converting the coefficients to Harvey model coefficients and using Eq. (4.10),since the ABg model is difficult to integrate analytically.

The Harvey model BSDF can be computed for any incident ray and scatteredray direction. Figure 4.6 shows the BRDF versus scatter angle of the Harvey modeldiscussed above, computed for �i = 0 deg and �i = 70 deg. The scatter angle isexpressed as a direction cosine pair, with the x-direction cosine equal to sin �s ×cos �s , and the y-direction cosine equal to sin �s ×sin �s .

Now that the relationship between surface roughness and BSDF has beenderived, methods of using this relationship to model the BSDF of optical surfaceswill be discussed, roughly ascending in order of complexity and fidelity. In each



68 Chapter 4

Figure 4.6 BRDF versus scatter angle of the Harvey model of the 13.1-A RMS roughnessmirror whose PSD is given in Fig. 4.3 [� = 0.6328 �m, �i = 0 deg (left), and �i = 70 deg(right)]. Calculations are performed in FRED.

of these methods, the three parameters of the Harvey model are determined bymeasurements of the surface roughness or scattering.

4.1.1 BSDF from RMS surface roughness

As mentioned earlier, RMS roughness is commonly specified and measured foroptical components, and in the absence of measured PSD or BSDF data, it can beused to model the BSDF of an optical surface. When using this technique, the totaleffective surface roughness7�� must be used, which is defined as

�� =1/�∫

1/d

S2 ( f ) d f , (4.13)

where � is the minimum wavelength in the spectral waveband of interest, and dis the measurement length (e.g., the spatial extent of the FOV of the white-lightinterferometer used to measure S2). This definition of roughness must be usedbecause spatial frequency components greater than 1/� do not generate opticalscatter, as Eq. (4.5) shows (the numerator of this equation cannot be greater thantwo and is typically not greater than 1). Thus, when using the roughness valueto determine the amount of optical scatter from the surface, it is importantto specify the maximum spatial frequency of interest as 1/�. Otherwise, theroughness may be measured at spatial frequencies that do not result in opticalscatter, and the value of the roughness may suggest that the surface scatters morethan it does. It is particularly important to do this when calling out roughnessfor optical component fabricators, as band-limiting the roughness measurementmay make it easier for the fabricator to satisfy the roughness requirement. Themaximum frequency is usually defined as an input into the analysis software of




Table 4.1 Typical values of RMS roughness �.

Component Type Typical RMS Roughness (Angstroms)

Glass (Fused silica, Schott N-BK7) 10

Post-polished nickel-plated mirror 10

Silicon Carbide mirror 10

Germanium 40

Diamond-turned beryllium mirror 100

Sapphire 100

the device used to measure the roughness, such as a white-light interferometer orsurface profilometer.

Ideally, the measured �� of the as-fabricated surface or similar surface wouldbe available; however, because stray light analysis should be performed prior tofabrication, it is often the case that �� is not available. Therefore, values of � (not��) for typical optical surfaces are provided in Table 4.1. Care should be takenwhen using these values, as the � of the actual surface to be modeled may be verydifferent from these values, depending on the fabrication method used. Also, thesevalues are for �, not ��, and therefore are the worst case, as the �� values will beband-limited and thus lower.

Once the �� of the surface is known, it can be used to compute the TIS as11

TIS =[

2��n�� cos �i

�

]2

. (4.14)

By setting this equation equal to the TIS obtained by integrating the BSDF of thesurface [Eq. (10)] over the projected solid angle of a hemisphere, the value of theHarvey scatter model parameter b0 can be solved as

b0 = 2��n2 B2

�4

�2� (−C − 2)

1 − (1 + B2/

�2)(2+C)/2(4.15)

for C �= –2, and

b0 = 2��n2 B2

�4

�2�

ln (1 + B2/

�2)(4.16)

for C = −2. Because this model does not require a PSD or BSDF measurement, thevalues of K-correlation model parameters B and C are unknown. However, typicalvalues for these parameters can be used: B is typically equal to 200 �m for opticalsurfaces, and C is typically equal to 1.5. From these values and the wavelength, thevalue of b0 can be computed, as well as the Harvey model parameters l and s—thus,a complete model can be obtained. Of course, the fidelity of this model may below: B can vary from 100–500 �m, and C can vary from –2.5 to –1.5 for typical



70 Chapter 4

optical surfaces. These variations in B and C result in values of b0 that vary from0.15–45.71 for a mirror at � = 0.6328 �m. Thus, b0 can vary by about two ordersof magnitude for typical optical surfaces; therefore, by using this model, the BSDFof the surface at or near the specular direction can be in error by this amount. It isalso rare to work with a surface that does not have at least one coating layer on it: aswill be discussed in Section 4.2, the BSDF of a coated surface can be very differentfrom the BSDF of an uncoated surface (though, in general, the difference increaseswith the number of layers, so the scattering from surface with just one or two layersis usually similar to the scattering from just the substrate). If the surface is coated,then the measured roughness is of the coating, not the substrate, which can havedifferent roughness (though, again, the fewer layers there are in the coating, themore similar the roughness of the coating is to the substrate). Nevertheless, in theabsence of any other data, this model may be the most accurate approximation thatcan be made.

4.1.2 BSDF from PSD

As shown earlier in this chapter, BSDF can be computed from the measured surfacePSD. However, PSD values measured from white-light interferometers (such asthose shown in Fig. 4.3) and other devices can contain a number of errors thatreduce the accuracy of the predicted BSDF. To reduce these errors, it is suggestedthat the following process be used to convert PSD values to BSDF:8

1. Window the PSD data, and remove dead and/or noisy pixels.2. Remove the base surface figure (especially when using a low-magnification

objective).3. Correct the data for instrumental effects such as field curvature.4. Perform an azimuthal average to remove noise.5. Combine the PSD data from several magnifications to cover all required

spatial frequencies.6. Calculate the band-limited surface roughness, and verify that it is reasonable

for the type of surface being modeled.7. Fit to the K-correlation model, and verify that the B and C parameters are

physically reasonable.8. Compute the BSDF.

Many stray light analysis programs allow the measured PSD to be input directlyinto them, making it possible to skip steps 6 and 7. One way to validate the accuracyof the measured PSD data is to compute its effective surface roughness usingEq. (4.13) (this calculation is often implemented in the analysis software of thedevice used to measure the PSD) and compare it to its expected value given thefabrication and polishing process used to fabricate the optic. Disagreement betweenthe two may indicate a problem with the PSD measurement technique, with thefabrication process, or both.

One of the benefits of using PSD is that it can be used to compute the BSDFof the surface at any wavelength. This feature makes it attractive for predictingscattering at ultraviolet wavelengths, as direct measurement of the BSDF at these




wavelengths can be difficult. Another benefit to using PSD data is that it can veryaccurately predict BSDF, especially for uncoated surfaces. As Fig. 4.4 shows, theBSDF predicted from a PSD measurement matches the directly measured BSDFto within about a factor of two.

There are, however, a number of disadvantages to using PSD data. One is thatusing it requires the lengthy data acquisition and reduction process described earlier.Another is that PSD data neglects the effect of any coatings and contaminants onthe surface which, as will be shown in Section 4.2 and Chapter 5, can be non-negligible. Given the nature of these disadvantages, it is often the case that directmeasurement of the BSDF in a scatterometer (as discussed in Section 10.1) willyield more accurate results for the same or less effort, provided that it is easy tomeasure the BSDF at the wavelength of interest.

4.1.3 BSDF from empirical fits to measured data

Arguably the most accurate way to represent the BSDF of a surface is to measure itdirectly at the wavelength of interest. Such a measurement often takes just as muchtime and effort as measuring the PSD but has none of the issues associated with con-verting PSD to BSDF (such as those discussed in the previous section), and it hasthe added benefit of accounting for the effect of any surface coatings and particulatecontaminants. BSDF measurements are usually performed using a scatterometer,which is discussed in detail in Section 10.1. BSDF measurements must be per-formed at a finite number of angles of incidence and scatter angles; in order to pre-dict it at arbitrary angles, an interpolation function must be fit to the data. For opticalsurfaces, one of the best models for doing this is the Harvey model because its func-tional form accurately describes scatter from optical surfaces. The Harvey modelparameters (b0, l, and s) are determined by a fitting algorithm that seeks to minimizethe residual between the measured BSDF values and the model, computed as√√√√ N∑

i=1

[log10 (BSDFi,measured) − log10(BSDFi,predicted)

]2, (4.17)

where N is the number of points. This method of computing the residual uses thelogarithm of the BSDF to ensure a good fit at all scatter angles, even those whoseBSDF values are different by many orders of magnitude, as is often the case.

An example of such a fit is shown in Fig. 4.4. As is often the case, the measuredBSDF does not exhibit the decrease at large scatter angles that results from thecos �s term in the PSD model [Eq. (4.4)]. The usual reason for this is that opticalsurfaces always have some particulate contamination, and scattering from theseparticulates increases the BSDF at large scatter angles.8 Thus, the Harvey model(which does not have a cos �s term) often does a better job of fitting measuredBSDF data than the PSD model, though not all of the scatter it models is due tosurface roughness scatter.

In practice, the Harvey model can be used to provide a good mathematical fitto the BSDF of a variety of surface types other than uncoated optical surfaces, such



72 Chapter 4

as coated optical surfaces and even some nonoptical surfaces such as anodizedaluminum and some specular black paints. Fitting to the Harvey model also allowsfor easy determination of the TIS of the surface through Eq. (4.10). If the totaleffective surface roughness of the surface is known or can be estimated, then theTIS determined from the Harvey model can be computed to the TIS computedfrom the roughness [Eq. (4.14)] as a means of validating the accuracy of the BSDFmeasurement.

Once the fit is performed using the residual given in Eq. (4.17), the uncertaintyin the BSDF predicted by the Harvey model can be estimated as

1

N

N∑i=1

∣∣BSDFi,measured − BSDFi,predicted

∣∣BSDFi,measured

. (4.18)

The minimum uncertainty in measured BSDF is about 5%,12 though round-robin tests have indicated that, in practice, the average uncertainty can be as muchas 50%.19 When in doubt, the more conservative 50% value should be used.

4.1.4 Artifacts from roughness scatter

Figure 4.7 shows the simulated artifacts that result from the application ofthe Harvey model to the back (inside) surface of the corrector lens in the

20 Angstroms40 Angstroms

10 Angstroms 5 Angstroms

log10(irradiance in ph/s-mm2)

11.12

8.47

9.79

9.13

10.46

Figure 4.7 Irradiance distribution at the focal plane due to surface roughness scatter fromthe back (inside) surface of the corrector of the Maksutov–Cassegrain system. Simulationdone in FRED.




Maksutov–Cassegrain telescope. The Harvey model b0 coefficient was computedusing Eq. (4.15), assuming n = 1.5381 (Schott N-BAK7 at 0.6328 �m), s = –1.5,and l = 0.01 radians. These artifacts were generated by modeling the sun justoutside the bottom-left corner of the FOV. As predicted, the irradiance at the focalplane (which is a function of the BSDF) scales as �2

� . Although the optical systemsare different, the measured spatial distribution of irradiance shown in Fig. 1.1 issimilar to the predicted distributions shown in Fig. 4.7. (Closer correlation betweenpredicted and measured irradiance distributions is shown later in Fig. 11.3.)

4.2 Scattering from Coated Optical Surface Roughness

Equation (4.4) establishes the relationship between the BSDF of a particle-free,uncoated optical surface and its surface roughness. However, most optical surfaceshave some sort of coating, such as an antireflection, protective, or bandpass coating(or some combination thereof), and these coatings will affect the BSDF of thesurface. In addition to increasing the roughness (and therefore the TIS) of thesurface, coating layers can change the angular distribution of the surface’s BSDF.In general, the more layers the coating has, the greater the difference between itsBSDF and that of the uncoated surface. A model has been developed that can beused to predict the BSDF of a coated surface given the complex refractive indexand PSD of the substrate, and the thickness and real index of refraction of everylayer of the dielectric coating (often referred to as the stack definition).14 Thismodel also requires that the correlation between the roughness of each layer bespecified; coatings in which the roughness profiles of every layer (including thesubstrate) are identical have correlated roughness, and those in which the profilesare statistically independent have uncorrelated roughness. Examples of both areshown in Fig. 4.8. Because it is very difficult to measure the degree of correlationbetween the roughness of every layer, this calculation is often performed for bothcases (correlated and uncorrelated), and the results averaged or the worst case used.

The predicted BTDF from a 28-layer bandpass coating14 (whose transmittanceversus wavelength is shown in Fig. 4.9) is shown in Fig. 4.10. The predictedBTDF has many peaks and valleys across the total range of scatter angles, whichis very different than the BTDF of a single-surface scatterer that has only one peakcentered on the specular direction. This demonstrates the importance of accounting

Substrate Coating

Figure 4.8 Correlated (top) and uncorrelated roughness (bottom).



74 Chapter 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.6 0.625 0.65 0.675 0.7

Tran

smit

tanc

e

Wavelength (μm)

Figure 4.9 Transmittance versus wavelength of a 28-layer bandpass coating.

for surface coatings when modeling surface roughness scatter, especially bandpasscoatings, which typically have many (>2) layers.

In practice, using this model to predict the scattering of a coated surfaceis difficult. To date, this theory has not been incorporated into any commercialsoftware package, though source code to compute it is available.15 The calculationrequires the stack definition as input, which optical coating vendors typically treatas proprietary, and therefore this information is usually not available. And as

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90

BTD

F (1

/str

)

Scatter Angle from Normal (degrees)

Correlated

Uncorrelated

Figure 4.10 BTDF versus scatter angle of a 28-layer bandpass coating at AOI = 45 degand wavelength = 0.65 �m. The transmittance versus wavelength of the same coating isshown in Fig. 4.9.




mentioned previously, the degree of correlation between the roughness profiles ofthe different layers is difficult to measure, and the difference between the predictedBSDF of a stack with correlated and uncorrelated roughness can be significant(10×), as shown in Fig. 4.10. Given all these difficulties, the BSDF of a coatedsurface is usually directly measured and fit (as discussed previously).

4.3 Scattering from Scratches and Digs

Scratches and digs are cosmetic defects in optical surfaces, and their allowed num-ber and length per unit area of optical surface are specified using scratch and dignumbers, which are commonly used throughout the optical component fabricationindustry. The meaning of these numbers is defined in MIL-STD-13838;16 however,this definition is not specific enough for these numbers to be used in a meaningfulway to perform stray light analysis. For instance, there is no width associated with ascratch of a particular number, and therefore it is not possible to compute the percentarea coverage (PAC) of the scratch, which is necessary (but, arguably, not suffi-cient) to compute the BSDF of the surface. Studies have been performed in whichthe BSDF of a surface with particular scratch and dig numbers was measured;17

however, it is difficult to extrapolate these results to surfaces with different (or eventhe same) scratch and dig numbers. Therefore, the most accurate way to accountfor any scattering due to scratches and digs is direct BSDF measurement of thesurface.

4.4 Summary

Optical surfaces cannot be made perfectly smooth, and therefore all will scatterlight due to their surface roughness. The magnitude and angular distribution of thesurface BSDF is a function of the profile of the surface at the microscopic level. Themost common model used to compute the BSDF due to surface roughness scatteringis the Harvey model, which is available in most stray light analysis programs andis a function of three coefficients. These coefficients can be computed from thestatistics of the surface roughness; at a minimum, the RMS surface roughness� is required (Table 4.1 gives the roughness of typical optical elements). Thecoefficients can also be determined by fitting them to a set of measured BSDFdata. Given the difficulty of measuring the surface roughness statistics (especiallyfor surfaces with coatings), this is often the most accurate way of determining thecoefficients. Even using this most accurate method, the predicted BSDF can be inerror by 50% or more.

The total scatter from surface roughness typically varies as (�/�)2, where� is the wavelength. It is important when specifying a surface roughness call-out on a drawing to also specify the spatial frequency bandwidths of the RMSroughness.

It is difficult to predict the BSDF of a surface based on its scratch and dignumbers, because the standard used to define these numbers does not provideenough information for BSDF to be computed. Therefore, the best way to determinethe BSDF of a surface with a given scratch/dig is to measure it.



76 Chapter 4

References

1. B. Ma et al., “Evaluation and analysis of polished fused silica subsurface qualityby the nanoindenter technique,” Appl. Opt. 50(9), C281 (2011).

2. R. Pfisterer, “Approximated scatter models for stray light analysis,” Optics andPhotonics News, p.16 (Oct 2011).


4. T. Germer, “Predicting, Modeling, and Interpreting Light Scattered by Sur-faces,” SPIE Short Course SC492 notes (2004).

5. J. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering,2nd Ed., Optical Society of America, Washington, D.C. (1999).

6. W. Smith, Modern Optical Engineering, 3rd Ed., McGraw-Hill, New York(2000).

7. D. Malacara, Optical Shop Testing, 3rd Ed., John Wiley & Sons, New York(2007).

8. M. Dittman, “No such thing as � – flowdown and measurement of surfaceroughness requirements,” Proc. SPIE 6291, 62910P (2006) [doi: 10.1117/12.678314].

9. M. Dittman, “K-correlation power spectral density & surface scatter model,”Proc. SPIE 6291, 62910R (2006) [doi: 10.1117/12.678320].

10. J. Harvey and A. Krywonos, “Unified scatter model for rough surfaces at largeincident and scatter angles,” Proc. SPIE 6672, 66720C (2007) [doi:10.1117/12.739139].

11. A. Greynolds, “Relative micro-roughness scattering from the surfaces of atransmitting optical element,” Proc. SPIE 511, 35–37 (1984) [doi: 10.1117/12.945033].

12. Sales brochure for the Complete Angle Scatter Instrument (CASI), SchmittMeasurement Systems, http://www.schmitt-ind.com (2012).

13. D. Rock, “The OARDAS stray radiation analysis software,” Proc. SPIE 3780,138–147 (1999) [doi: 10.1117/12.363771].

14. J. Elson, “Multi-layer coated optics: guided-wave coupling and scattering bymeans of interface random roughness,” JOSA A12(4), 729–738 (1995).

15. T. Germer, The SCATMECH code library, National Institute of Standardsand Technology (NIST), http://physics.nist.gov/Divisions/Div844/facilities/scatmech/html/index.htm.

16. “Performance specification – optical components for fir control instruments;general specification governing the manufacture, assembly, and inspection of”,Military Standard (MIL STD) MIL-PRF-13830B (1963).

17. I. Lewis, A. Ledebuhr, and M. Bernt, “Stray light implications of scratch/digspecifications,” Proc. SPIE 1530, 22–34 (1991) [doi: 10.1117/12.50493].

18. Advanced Systems Analysis Program (ASAP) User’s Manual, Breault ResearchOrganization (2011).

19. T. Leonard and P. Rudolph, “BRDF Round Robin Test of ASTM E1392,”Proc. SPIE 1995, 285–293 (1993) [doi: 10.1117/12.162658].


SRBK005-05 SRBK005 February 28, 2013 18:41 Char Count=0

Chapter 5

Scattering from ParticulateContaminantsAll surfaces have some amount of particulate contamination (i.e., dust) that in-creases their BSDF above the level predicted by their surface roughness. An exam-ple of the type of artifact that results from scatter of similar contaminants is shownin Fig. 1.1. In this figure, particulates (such as the ones shown in Fig. 5.1) on thecamera’s optics have scattered sunlight from outside the FOV into its FOV, thusincreasing its out-of-field stray light. Scattering from surface roughness (discussedin Chapter 4) results in artifacts similar to those from particulate scatter and alsocontributes to this artifact. For a comparison of the magnitude of these two typesof scattering, see Section 5.6.

A number of models have been developed to compute the BSDF of contam-inated surfaces. As with surface roughness scatter models, the more fidelity themodel has, the more input data (and therefore effort to implement) it requires.As the next section demonstrates, the BSDF of a surface due to contaminationscatter is a strong function of the particle density function f (D), which is equalto the projected areal density (in units such as 1/mm2) of the particle distribu-tion as a function of particle diameter D, and the difference between the BSDFmodels presented in this chapter lies in the way they describe f (D). In the mostapproximate model, f (D) is represented using a simple equation. In the mostaccurate model, it is represented as a table of values determined by detailed in-spection of particles on the surface itself. Details of performing inspection inorder to determine the input parameters for either model are also presented in thischapter.

Computation of BSDF from f (D) is usually done using Mie scatter theory, andthus this chapter begins with a review of this theory. As with surface roughnessscatter theory, a comprehensive review of Mie scatter theory is beyond the scopeof this book; however, there are a number of good references.1−3 There are alsomany details related to the control and inspection of particulate contaminationthat are not covered in this chapter but are covered elsewhere.4 These referencesalso discuss the effects of molecular contamination, which are discussed brieflyhere.

77



78 Chapter 5

(a)

(b)

Figure 5.1 Typical dust particles on black acrylic: (a) shows an image 3.44 mm in diameter.The white rectangle in (a) is 0.69 mm in diameter, and is shown in (b) (image courtesy ofMargy Green).

5.1 Scattering from Spherical Particles (Mie Scatter Theory)

The angular distribution of intensity due to scattering by a spherical particle ofcomplex refractive index N and diameter D, due to illumination by a beam ofvacuum wavelength � = can be predicted using Mie scatter theory.1 This theorypredicts that the magnitude and angular distribution of scattered light is proportionalto the size parameter x , given by

x = �Re (N ) D

�(5.1)

and to the relative refractive index m, given by

m = N

Nm, (5.2)



Scattering from Particulate Contaminants 79

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

s-polarized

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

Scatter angle from undevi-ated beam

p-polarized

D = 0.010 μm,x = 0.078

D = 0.200 μm,x = 1.519

D = 1.000 μm,x = 7.596

Figure 5.2 Normalized intensity I versus scatter angle from an undeviated beam � for ans-polarized (left) and p-polarized (right) incident beam. D is equal to the sphere diameter,and x is the size factor [� = 0.6328 �m, and N = m = 1.53 + 0.0005i (typical dust)].

where Nm is the real refractive index of the media surrounding the particle. Thevariation in normalized intensity versus scatter angle for typical dust particles(m = 1.53 + 0.0005i for � = 0.6328 �m) of different diameters is shown in Fig. 5.2.It has been shown18 that it is accurate to model nonspherical dust particles (suchas the ones shown in Fig. 5.1) as spheres.

Particles that are small compared to the wavelength (x < ∼0.1) exhibit theRayleigh scattering pattern, which is isotropic for s-polarized incident light andsymmetric in the forward and backward directions for p-polarized incident light.For either polarization state, the magnitude of forward scatter relative to backwardscatter increases with the particle diameter. Mie scatter calculations are compu-tationally intensive; however, they have been implemented in some stray lightanalysis programs (see Section 1.1). Code to perform the calculations has alsobeen published.1,5 All such calculations require the complex refractive index ofdust, which is given for various wavelengths in Table 5.1.

Figure 5.3 shows a spherical particle on an optical surface. Light incident onthe surface at AOI �i is forward scattered at an angle of � f from the undeviated

Table 5.1 Complex index of refraction versus wavelength fordust.6

Wavelength (�m) Complex Index of Refraction

0.6328 1.53 + 0.00051

1.15 1.50 + 0.001)

3.39 1.50 + 0.02i

10.6 1.70 + 0.21



80 Chapter 5

Particle

Incident beam

Forward-scattered

beam

Forward- scattered beam (if

p-polarization vector

s-polarization vector (out of page)

Backward-scattered

beam

θi

θf

θb

θs

Surface

Figure 5.3 Geometry for Mie scattering of a particle on an optical surface.

beam, and backscattered at an angle �b. The backscattered beam makes an angle �s

with respect to the surface normal.Assuming that the incident beam is unpolarized, the BRDF of a surface con-

taminated with particulates of N diameters is computed as

BRDF (�s) = 1

(2�/

�)2 cos �s

N∑i=1

f (Di )

×[

RIs(Di , � f ) + RIp(Di , � f ) + Is (Di , �b) + Ip (Di , �b)

2

], (5.3)

where � is the wavelength of light, �s is the scatter angle with respect to the surfacenormal, f (Di ) is the density of particles of the i th diameter, Is(Di ,�) and Ip(Di ,�)are the s-polarized and p-polarized (respectively) intensity versus scatter angle ofparticles of diameter Di , and R is the surface reflectance. Similarly, the BTDF ofan unpolarized incident beam is equal to

BTDF (�s) = 1(2�

/�)2 cos �s

N∑i=1

f (Di )

[TIs

(Di , �f

) + TIp(Di , �f

)2

], (5.4)

where T is the transmittance of the surface. In general, the BSDF predicted bythese equations correlate well with measured data,3 though models with differentobliquity factors [the 1/cos �s in Eqs. (5.3) and (5.4)] have been proposed.7 Thisequation will be used in the next chapters to predict the BSDF given f (D).

5.2 Particle Density Function Models

There are a number of ways to model f (D); two of the most widely used will bediscussed in this section.




5.2.1 The IEST CC1246D cleanliness standard

The most widely used method of specifying f (D) is the Institute of EnvironmentalSciences and Technology (IEST) CC1246D standard.9 This standard, which is de-rived from the discontinued but nearly identical MIL-STD-1246C,10 can quantifythe number of particles on a surface using a single number, called the clean-liness level (CL) of the surface, though in some cases other input values arerequired. Once the CL for a surface is known, the BSDF of the surface due toparticulate scatter can be computed from Eq. (5.3) and from Mie scatter theory.IEST-STD-CC1246D defines the number of particles Np (per 0.1 m2) whose diam-eters are greater than or equal to D as

Np (S, C L , D) = 10|S|[log210(C L)−log2

10(D)], (5.5)

where S is the particle distribution slope, CL is the cleanliness level of the surface,and D is the particle diameter in �m. Figure 5.4 shows Np versus D for typicalCLs and the default slope value (–0.926). For these cases, the CL correspondsto the particle size whose density is 1/0.1m2. This standard is valid only forD ≥ 1 �m.

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

log 10

(num

ber o

f par

ticl

es p

er 0

.1 m

2 of

dia

met

er ≥

D)

log102(particle diameter D in m)

CL = 200 CL = 400 CL = 600

1 particle

1 μm 200 μm 400 μm 600 μm

μ

Figure 5.4 Number of particles of diameter ≥ D versus D given by IEST-STD-CC1246D(slope = −0.926) for CLs 200, 400, and 600. The x intercept of each line occurs at theparticle whose diameter is equal to CL �m.



82 Chapter 5

Particle slope is a function of the environment in which the particle contamina-tion occurred and the process used to clean the surface; generally, the more cleaningperformed on the surface, the lower the value of S, as cleaning removes larger par-ticles but leaves the smaller ones. The value for S defined in CC1246D is –0.926,which is representative of particle distributions on cleaned surfaces.11 CC1246Ddoes not provide a value for S for any other conditions; however, other studies havebeen conducted to determine its value.11−13 In particular, it was found that particlefallout distributions for clean room environments can be better approximated usinga slope of –0.383.11

The PAC of a particle distribution is the percentage of the surface area obscuredby particles and can be computed from Eq. (5.5) as

PAC = 10K+|S| log210(C L), (5.6)

where K = –7.245 if S = 0.926, and K = –5.683 if S = –0.383. The PAC ofthe distribution is importnant because (PAC/100) is approximately equal to thetotal integrated scatter (TIS) of the surface due to contamination scatter (moreon this later in this section). The constants assume a maximum particle diameterof 2000 �m, which is based on the assumption that any particles larger thanthis will have been removed in the cleaning process.8 The PAC computed in thisway will vary drastically as a function of S; for a typical value of CL (300), thePAC is 0.02736% for S = 0.926, and 0.00047% for S = −0.383. This variabilitymakes it difficult to compare the BSDF of particle distributions with the same CLbut different S. To eliminate this variability, the parameter Rs is introduced intoEq. (5.5):

Np (S,CL,D) = (Rs) 10|S|[log210(C L)−log2

10(D)], (5.7)

where

Rs = PAC (S = −0.926)

PAC (S). (5.8)

For a given CL, the PAC computed from Eq. (5.7) will always be equal to thatpredicted by Eq. (5.6) for S = −0.926. The Rs parameter does not appear in theIEST-STD-CC1246D standard, however, it is widely used throughout the opticsindustry (including many stray light analysis programs), and therefore Eq. (5.7) isused throughout this book to compute Np.

Because Eqs. (5.3) and (5.4) require the number of particles per 0.1 m2 ofdiameter D and not the number of particles per 0.1m2 of diameter greater than orequal to D, f (D) must be derived from Np(D) as

f (S,CL,D) = − d

dDNp (S,CL,D) , (5.9)

f (S,CL,D) =[

2 |S|D

log10 (D)]

(Rs) 10|S|[log210(C L)−log2

10(D)]. (5.10)




1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+00 1.E+01 1.E+02 1.E+03

Num

ber o

f Par

ticl

es p

er 0

.1 m

2

Particle Diameter (μm)

CL = 200, S = –0.926 CL = 400, S = –0.926 CL = 600, S = –0.926

CL = 200, S = –0.383 CL = 400, S = –0.383 CL = 600, S = –0.383

Figure 5.5 Number of particles of diameter D versus D given by IEST-STD-CC1246D(S = −0.926, and S = −0.383) for CLs 200, 400, and 600.

f versus D for typical CLs are shown in Fig. 5.5. For the same CL, the S =–0.383distribution has more large particles than the S = –0.926 because the cleaningprocess has removed them.

The BRDF due to contamination scattering on a perfect (reflectance =1) mirrorat 0.6328 �m is shown in Figs. 5.6 and 5.7, and for the same mirror at 10.6 �min Figs. 5.8 and 5.9. In each case, the BRDF was computed using Eq. (5.3). Theangular dependence of the BRDF due to particulate scatter is very similar to thatof surface roughness scatter (compare to Fig. 4.4); however, it is not perfectlyshift-invariant. The large peak at |sin �s – sin �i | = 1 is due to the 1/cos �s obliquityfactor in Eq. (5.3) and has been observed experimentally.3 The slope of the scatterdistribution (i.e., the specularity) is higher for the 0.6328-�m case than for 10.6-�mcase because the size parameters are higher for 0.6328 �m, and therefore theforward scatter is more peaked (as shown in Fig. 5.2).

As mentioned the total integrated scatter (TIS) of a contaminated mirror isapproximately equal to its PAC/100, and therefore Eq. (5.6) can be used to estimatethe TIS. However, this assumption neglects the effects of diffraction and absorp-tion, which can significantly affect the TIS. In particular, the extinction paradox1



84 Chapter 5

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E-03 1.E-02 1.E-01 1.E+00

BRD

F (1

/sr)

|sin θs− sin θ i|

CL = 200 CL = 400 CL = 600

Figure 5.6 BRDF versus |sin �s – sin �i | of a perfect mirror (reflectance = 1) due to contam-ination scatter (0.926 slope assumed, � = 0.6328 �m). Calculations performed in FRED.

Figure 5.7 BRDF versus scatter angle of a perfect mirror (reflectance = 1) due to contam-ination scatter (0.926 slope assumed, � = at 0.6328 �m) for �i = 0 deg and �i = 70 deg.The scatter angle is expressed as a direction cosine pair, with the x-direction cosine equalto sin �s × cos �s, and the y-direction cosine equal to sin �s × sin �s. Calculations performedin FRED.




1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E-03 1.E-02 1.E-01 1.E+00

BRD

F (1

/str

)

|sinθs − sinθ i|

CL = 200 CL = 400 CL = 600

Figure 5.8 BRDF versus |sin �s – sin �i | of a perfect mirror (reflectance = 1) due to contam-ination scatter (0.926 slope assumed, � = at 10.6 �m). Calculations performed in FRED.

Figure 5.9 BRDF versus scatter angle of a perfect mirror (reflectance = 1) due to contam-ination scatter (0.926 slope assumed, � = at 10.6 �m) for �i = 0 deg and �i = 70 deg. Thescatter angle is expressed as a direction cosine pair, with the x-direction cosine equal tosin �s × cos �s, and the y-direction cosine equal to sin �s × sin �s. Calculations performedin FRED.



86 Chapter 5

Table 5.2 CL versus PAC and TIS of a perfect mirror (reflectance = 1). TIS computed usingMie theory in FRED.

CL PAC (%) TIS, � = 0.6328(%) TIS, � = 10.6(%)

200 0.00455 0.00859 0.00556

400 0.10585 0.19980 0.12940

600 0.79796 1.50601 0.97530

(a diffraction effect) can significantly increase the TIS over PAC/100 by as much asa factor of 2. Table 5.2 compares the TIS computed using diffraction theory withthe PAC computed using Eq. (5.6). The TIS values were computed using the samecode (FRED) that computed the BRDF shown in Figs. 5.6–5.9. One of the reasonsthat the TIS values at 10.6 �m are lower than those at 0.6328 �m is because theparticles have more absorption at 10.6 �m, which can be seen in the complex indexof refraction of dust given in Table 5.1. Despite this, the extinction paradox stillmakes the TIS at 10.6 �m larger than the PAC. Therefore, when using the PAC toestimate TIS, be aware that it can be in error by a factor of 2.

In general, the angular distribution of scatter from an IEST-STD-CC1246Dparticulate distribution is similar to the scatter from surface roughness. This meansthat the Harvey model discussed in Section 4.1 can fit the predicted BSDF from par-ticulate contamination, which may be desirable when performing first-order straylight calculations using Eq. (2.47). The error associated with such an approximationcan be easily computed from the residuals of the Harvey fit to the contaminationBSDF. The Harvey model can also be used to fit the combined scatter from surfaceroughness and contaminants.

To summarize, the IEST-STD-CC1246D standard is the most widely usedmethod of specifying particulate contamination and is supported by most stray lightanalysis programs. In order to compute the BSDF of a surface due to contaminationscatter, these programs require the following input values:

� The complex refractive index of the particles at the wavelength of interest.Typical values for dust are given in Table 5.1.

� The reflectance and transmittance of the substrate.� The slope of the particle size distribution: S = –0.926 for freshly cleaned

optics, and S =–0.383 for exposed optics.� The maximum particle diameter.� The CL.

Because it requires only a few values as input, this model is very convenient touse, and for this reason most stray light analysis programs include it. However, insome cases, the IEST-STD-CC1246D particle distribution may not model the actualdistribution accurately enough. An example of such a case is a stray light analysis ofa radiometer with high radiometric accuracy; in order to determine the radiometric




error resulting from stray light, the analysis must very accurately determine theBSDF of one or more of the contaminated surfaces. The error associated with theuse of the IEST-STD-CC1246D distribution is discussed in Section 5.3.4. If thisdescription is not accurate enough, it may be necessary to use the actual measuredsize distribution of particulates on the optics to be modeled. This is the topic ofthe next chapter. (Note this model makes a number of simplifying assumptions,and therefore care must be taken when using it; otherwise, the predicted BSDF cangrossly misrepresent that of the actual surface.)

5.2.2 Measured (tabulated) distribution

Through detailed inspection of the particulates on a surface, a table of f (Di )values can be determined. Studies have been performed that show good correlationbetween predicted and measured scatter for such distributions.3 Most stray lightanalysis programs allow the user to define a tabulated f (Di ) for their particulatescatter model. Provided the particulate distribution remains constant followinginspection, this method is the most accurate way of modeling scattering fromparticulates.

5.2.3 Determining the particle density function using typicalcleanliness levels, fallout rates, or direct measurement

Two methods of describing f (D) have been discussed: using the IEST-STD-CC1246D CL and slope, and using measured tabulated data. Methods of deter-mining these parameters are discussed in this section. As with modeling scatterfrom surface roughness, the more time and effort spent in determining the parame-ters, generally the more accurate the model will be. Before discussing any of thesemethods, it is necessary to define the term cleanroom and discuss how its airborneparticulate levels are quantified.

A cleanroom is a space with distinct boundaries in which airborne particulatelevels are controlled through the use of protective clothing for personnel, filters toremove airborne particulates, positive air pressure to keep dirty air out, and othertechniques.4 ISO standard 14644-115 specifies the allowable particulate level ina cleanroom using cleanroom class numbers, which are given in Table 5.3. Thisstandard is similar to the discontinued Federal Standard 209E; equivalent FEDSTD 209E cleanroom classes are also given in Table 5.3. Both standards use largernumbers to define dirtier cleanrooms. Semiconductor fabrication can require aClass 1 cleanroom, aerospace applications typically do not require cleanrooms ofclass less than 5, and the air in a “typical” room is usually Class 9. Cleanroomsuse high-efficiency particulate air (HEPA) filters, which are 99.97% efficient atremoving 0.3-�m diameter particles. Typically, most particulates in cleanroomscome from the people working in them, and therefore the more heavily traffickedthe cleanroom, the more difficult it is to keep clean.4



88 Chapter 5

Tab

le5.

3IS

O14

644-

1cl

eanr

oom

clas

ses.

Max

imum

Par

ticu

late

s/m

3

FE

DST

D20

9EIS

OC

lass

≥0.1

�m

≥0.2

�m

≥0.3

�m

≥0.5

�m

≥1�

m≥5

�m

Cla

ssE

quiv

alen

t

110

2

210

024

104

31,

000

237

102

358

1

410

,000

2,37

01,

020

352

8310

510

0,00

023

,700

10,2

003,

520

832

2910

0

61,

000,

000

237,

000

102,

000

35,2

008,

320

293

1,00

0

735

2,00

083

,200

2,93

010

,000

83,

520,

000

832,

000

29,3

0010

0,00

0

935

,200

,000

8,32

0,00

029

3,00

0R

oom

air




Table 5.4 Typical IEST-STD-CC1246D CLs.

Surface Preparation CL S Comments

PAC = 0.0045%. Difficult toFreshly Precision Cleaned 200 −0.926 maintain outside of a cleanroom

Average Cleanliness Level for Sensorwith High Contamination Control

300 −0383 PAC = 0.01 20%

Average Cleanliness Level for Sensorwith Low Contamination Control

500 −0383 PAC = 03165%

5.2.3.1 Use of typical cleanliness levelsIt is often the case that inspection of particulate contaminants on the as-built hard-ware is not possible when scattering analysis is performed. This may be because theanalysis is being performed early in the design phase and the assembly environmenthas not yet been defined; because the assembly environment exists but there is notenough time or budget to inspect it; or because the surfaces are inaccessible. Inthese cases, it is necessary to use estimates of the CL, which are given in Table 5.4.A sensor with high contamination control is typically one that is assembled in aClass 5 or lower cleanroom, and operates in a vacuum or is housed in a sealedcavity that is purged to keep out particulates. An example of such a system is aspace-based telescope. A sensor with low contamination control is typically onethat is assembled in a less clean environment, such as a laminar flow bench in atypical lab, and is in a sealed cavity but is not purged. An example of such a sensoris a commercial camera lens.

The error in BSDF that results from using the typical CLs in Table 5.4 isapproximately equal to the uncertainty in the predicted PAC, which is typicallyabout a factor of 2. This is because the TIS = PAC/100 (as discussed in Section5.2.1), and the BSDF due to contamination scatter scales linearly with the TIS.

5.2.3.2 Use of fallout rates (uncleaned surfaces only)The CL of a surface that has been exposed to fallout (S = –0.383) can be estimatedby the cleanroom class of its assembly environment,14 the length of time it isexposed to this environment,4 and the spatial orientation in which it is exposed.4,15

An equation that relates the CL to these parameters is

CL = 10∧

√√√√√(

1

0.926

)log10

⎧⎨⎩Forient� t

[10NISOCLASS ·

(0.1

5

)2.08]0.773

⎫⎬⎭ − A,

(5.11)where

� A = [(–7.245 + 5.683) – 0.383 log210(5)] / 0.926 = –1.8889;

� Forient is the orientation factor (1 for upward-facing horizontal surfaces, 0.1for vertical surfaces, and 0.01 for downward-facing horizontal surfaces);



90 Chapter 5

1.E+01

1.E+02

1.E+03

1.E+04

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Surf

ace

Clea

nlin

ess L

evel

Exposure Time (days)

Class 4 (Horiz.) Class 5 (Horiz.) Class 6 (Horiz.)

Class 4 (Vert.) Class 5 (Vert.) Class 6 (Vert.)

Figure 5.10 Surface CL versus time of exposure for a conventional cleanroom environmentand horizontal orientation of the surface.

� � is a constant determined by the number of air changes per hour of thecleanroom (� = 81 for a cleanroom with fewer than 15–20 changes/hourand approaching still air, � = 8.1 for a conventional cleanroom with 15–20changes/hour, and � = 2.7 for a laminar flow cleanroom11); and

� NISOCLASS is the 14644-1 class of the cleanroom.

The variation in CL as a function of these parameters is shown in Figs. 5.10 and5.11.

Assuming the variables in Eq. (5.11) are independent and their measurementerror is randomly distributed, the resulting uncertainty in CL can be computed asthe RSS of the uncertainties in the independent variables.17 A similar calculationwas used to derive the uncertainty in predicted irradiance due to stray light givenin Eq. (2.50).

5.2.3.3 Use of a measured (tabulated) density functionThe particle density function can be determined by direct inspection of the surface.A discussion of inspection techniques is beyond the scope of this book; however, anumber of references exist.3,11




1.E+01

1.E+02

1.E+03

1.E+04

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Surf

ace

Clea

nlin

ess L

evel

Exposure Time (days)

Class 4 (Horiz.) Class 5 (Horiz.) Class 6 (Horiz.)

Class 4 (Vert.) Class 5 (Vert.) Class 6 (Vert.)

Figure 5.11 Surface CL versus time of exposure for a laminar flow cleanroom environmentand horizontal orientation of the surface.

5.3 BSDF Models

Once the PAC or particle density function has been determined, it can be used inone of the models discussed below to determine BSDF.

5.3.1 BSDF from PAC

The angular variation in BSDF from contaminants and from surface roughnessis similar. Therefore, the Harvey BSDF model (discussed in Section 4.1) can beused to approximate scatter from contaminants. The coefficients can be computedin much the same way they are computed in Section 4.1.1 from RMS surfaceroughness: pick typical values for s and l (–1.5 and 0.01 rad), compute the PAC,and then compute the b0 coefficient as

b0 =(PAC

/100

)ls (s + 2)

2�[(

1 + l2)(s+2)/2 − (

l2)(s+2)/2

] . (5.12)

The PAC can be computed from the CL using Eq. (5.6) or determined from inspec-tion. As with the RMS roughness method of predicting BSDF, this method can pre-dict BSDF that is in error by an order of magnitude or more. However, in the absenceof any other data or method, it may be the best approximation that can be made.



92 Chapter 5

5.3.2 BSDF from Mie scatter calculations

As mentioned previously, most stray light analysis programs have a methodto compute the BSDF of a contaminated surface using Mie scatter calculations. Thecontamination can usually be specified using the cleanliness level and slope. Thelargest source of error in this calculation is most likely due to the approximate na-ture of the IEST-STD-CC1246D particle size distribution. This error is discussedin Section 5.3.4.

5.3.3 BSDF from empirical fits to measured data

As with surface roughness scattering, an accurate method of modeling scatter-ing from particulates is to measure the BSDF of the contaminated surface. Thisworks well if the surface roughness of the substrate and the contamination level arethe same or similar to the surface being modeled. The Harvey model, which wasdiscussed in the previous chapter, can often be used with high accuracy to empiri-cally fit the measured BSDF from the contaminated surface.

5.3.4 Determining the uncertainty in BSDF from the uncertainty inparticle density function

As mentioned previously, the particle size distribution given by IEST-STD-CC1246D is idealized and real particle distributions are only approximated by it.Applications that require the scattering from particulates to be modeled with highfidelity may need to use the measured particle size distribution, especially if themeasured distribution is very different than the ideal distribution. There are a num-ber of inspection methods to determine particle density versus diameter, includingmanual inspection and counting with a microscope and the use of automated in-spection equipment.16

The uncertainty in BSDF (�BSDF) that results from uncertainty in f (Di) can beestimated by first assuming that the uncertainties in f (Di ) are independent and ran-dom. Then, using the equation for maximum uncertainty,17 �BSDF can be written as

�BSDF (�s) = 1(2�

/�)2 cos �s

√√√√ N∑i=1

[� f (Di ) Ii (�s)]2, (5.13)

where � f (Di ) is the uncertainty in f (Di ). Equation (5.13) is difficult to evaluatein most stray light analysis software because � f (Di )Ii (�s) must be evaluatedseparately for each i . Easier to compute is the worst-case estimate17 given by

�BSDF (�s) = 1(2�

/�)2 cos �s

N∑i=1

|� f (Di )| Ii (�s). (5.14)

All methods of determining the BSDF due to scattering from particulate contami-nants are based on Eq. (5.3), and therefore it can be used to estimate the uncertaintyassociated with any of them. An example of this calculation is presented in thissection, which can be used to estimate the minimum uncertainty of any methodthat uses CL to describe f (D).




1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+00 1.E+01 1.E+02

Num

ber o

f Par

ticl

es p

er 0

.1 m

2

Particle Diameter (μm)

3a (Measured) 3a (Fit, CL = 682)

3b (Measured) 3b (Fit, CL = 776)

Figure 5.12 Measured particle size distributions for samples 3a and 3b,17 and their best-fitIEST-STD-CC1246D distributions (–0.383 slope assumed for both samples).

As mentioned, the IEST-STD-CC1246D particle size distribution is an approx-imation to any real particle distribution and, as such, the BSDF predicted by it hasan uncertainty associated with it. This uncertainty can be estimated by comparingthe best-fit IEST-STD-CC1246D particle distributions to measured distributionsand then using Eq. (5.14) to predict the uncertainty in BSDF. Figure 5.12 shows thebest-fit IEST-STD-CC1246D f (D) to a set of published measurements of f (D).18

Because the contaminated samples were not cleaned prior to inspection, the modelused to fit the distribution has a slope of –0.383.

The values of |� f (Di )| were computed as the absolute value of the differencebetween the measured predicted values of f (Di ) and then used as inputs intoEq. (5.14). �BRDF(�s) was then computed by using |� f (Di )| instead of f (Di )as the tabulated input for a stray light analysis program. The resulting values of�BRDF(�s)/BRDF(�s) for samples 3a and 3b are shown in Figs. 5.13 and 5.14.The value of �BRDF(�s)/BRDF(�s) averaged over the points measured for bothsamples 3a and 3b is 0.46, which is an estimate of the minimum uncertainty inBSDF associated with using the IEST-STD-CC1246D distribution to model a real-world particle distribution. The maximum uncertainty is determined using valuesof |� f (Di )| associated with the method of determining f (D) and then determining�BSDF(�s) as described earlier. Typically, those most familiar with the inspectionmethod used to determine f (D) are best able to estimate the error in the method.

5.3.5 Artifacts from contamination scatter

Figure 5.15 shows the simulated artifacts that result from the application of theIEST-STD-CC1246D particulate contamination model to the back (inside) surface



94 Chapter 5

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-03 1.E-02 1.E-01 1.E+00

Resi

dual

BRD

F/BR

DF

BRD

F (1

/sr)

|sin θs– sinθI|

BRDF

Residual BRDF/BRDF

Average Residual BRDF/BRDF

Figure 5.13 BRDF predicted by particle counts from sample 3a, the residual BRDF fromthe BRDF model using the best-fit CL shown in Fig. 5.12, and the ratio of the two. Theaverage residual is 0.62. The steps in the curves are due to the small number (17) ofparticle diameters measured.

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-03 1.E-02 1.E-01 1.E+00

Res

idua

l B

RD

F/BR

DF

BR

DF

(1/s

r)

|sin θs– sin θ i|

BRDF

Residual BRDF/BRDF

Average Residual BRDF/BRDF

Figure 5.14 BRDF predicted by particle counts from sample 3b, the residual BRDF fromthe BRDF model using the best-fit CL shown in Fig. 5.12, and the ratio of the two. Theaverage residual is 0.42. The steps in the curves are due to the small number (19) ofparticle diameters measured.




CL500 CL400

CL300 CL200

log10(irradiancein ph/s-mm2)

13.08

9.79

11.44

10.61

12.26

Figure 5.15 Simulated irradiance distribution at the focal plane due to particulate contam-ination scatter from the back (inside) surface of the corrector of the Maksutov–Cassegraintelescope. Simulation done in FRED.

of the corrector lens in the baseline Maksutov–Cassegrain telescope. These artifactswere generated by placing the sun just outside the bottom-left corner of the FOV.Although the optical systems are different, the measured spatial distribution of irra-diance shown in Fig. 1.1 is similar to the predicted distribution shown in Fig. 5.15.Closer correlation between predicted and measured irradiance distributions isshown in Fig. 11.3.

5.4 Comparison of Scatter from Contaminants and Scatterfrom Surface Roughness

For typical particle density functions, the TIS from particulates is not a strong func-tion of wavelength , as shown in Fig. 5.16. However, the TIS from surface roughnessscatter generally varies as (1/wavelength)2. Therefore, scatter from surface rough-ness generally dominates at short wavelengths (≤2 �m for typical roughnessesand particle density functions) and contamination scatter at long (>2 �m). Figure5.16 also illustrates why it is difficult to scale visible BSDF measurements to theinfrared, and vice-versa: the physical processes are different.

5.5 Scattering from Inclusions in Bulk Media

Scattering from bulk inclusions or bubbles inside the bulk material of a refractiveoptic (also called “volume scattering”) can be predicted in much the same way as



96 Chapter 5

1.E-05

1.E-04

1.E-03

1.E-02

0.1 1 10 100

TIS

Wavelength ( m)

Contamination (Measured)

Contamination (Fit)

Roughness (29.5- Å RMS, Theoretical)

Roughness + Contamination (Measured)

Roughness (Theoretical) + Contamination (Fit)

μ

Figure 5.16 TIS versus wavelength of scattering from particulates and roughness.25 TISis measured at multiple points on a contaminated piece of polished flint glass with RMSroughness of 29.5 A. Measured contamination TIS is estimated as the difference betweenthe minimum TIS and the average TIS.

scattering from particulates on optical surfaces. As with scattering from surfaceparticulates, computing the scatter from bulk inclusions can be computationallyintensive, and therefore these calculations are usually performed using stray lightanalysis software.

ISO Standard 10110-319,20 defines “1/N × A” as the standard notation forquantifying the number and size of bulk inclusions, where N is the number ofmaximum-size defects allowed, and A is the “grade number,” which is the squareroot of the cross-sectional area of the maximum-size defect allowed (in mm). Thisstandard allows for defects with smaller grade numbers than A, provided that thenumbers of these defects obey the following relation:

n∑i=0

Ni

(2.5)i ≤ N , (5.15)

where n + 1 is the number of discrete defect sizes, Ni is the number of defectsof the i th particle size (with N0 being the number of defects of area A2, N1

being the number of defects of area [A/sqrt(2.5)]2, etc.). A typical callout for bulkdefects is “1/2 × 0.16”, which means 2 defects of diameter 0.16 mm (160 �m)are allowed, or a larger collection of smaller defects. Thus, the size of the defectsare similar to the sizes of particulate contaminants, and thus Mie scatter theorycan be used to determine the angular scatter distributions from these defects.Unfortunately, most stray light analysis programs do not support Mie calculations




for bulk inclusions, and instead use an approximate equation called the Henyey–Greenstein model.21 This model typically has two parameters as input: the first isthe extinction coefficient � , which is used to determine the probability of a rayscattering in a volume pscat as

pscat (t) = 1 − exp (−� t) , (5.16)

where t is the path length of the ray through though the volume, assuming noscattering. Thus, the larger � and t are, the higher the probability that the raywill be scattered by an inclusion. � can be computed as (volumetric density ofthe inclusions) × (cross-sectional area of inclusions), and therefore can be derivedfrom the ISO 10110-3 specification as

� =(

N

V

)A2, (5.17)

where V is the volume of the optical component. The other parameter is g, whichis used in probability density function of the scattered ray direction:

p (�) =(

1

4�

)1 − g2

[1 + g2 − 2g cos (�)

]3/2. (5.18)

Probability distribution functions such as this one are used in Monte Carlo simu-lations of volume scatter to reduce the number of scattered rays traced, which canotherwise quickly overflow the available memory. Variable g is the model param-eter between –1 and 1 that determines the angular distribution of scattered light,as shown in Fig. 5.17. When g<0, the backscattered intensity is larger relative to

0.5

1

1.5

2

30

210

60

240

90

270

120

300

150

330

180 0

θ

p

g = −0.5 g = 0 g = 0.72

Figure 5.17 Scattered intensity distributions determined by the Henyey–Greenstein func-tion [Eq. (18)].



98 Chapter 5

the forward; when g = 0, the intensity is isotropic; and when g>0, the forwardscatter is larger relative to the backscatter. The value of g for a particular inclusionsize can be chosen by computing the size parameter x for the inclusion [as givenin Eq. (5.1)] and picking the value of g that best approximates the Mie scatterdistribution for x (see Fig. 5.3). The Henyey–Greenstein model can be used in thisway to obtain a rough approximation (to within about an order of magnitude) ofthe magnitude of light from bulk scattering. The uncertainty associated with thismethod can be computed by using the standard error analysis techniques17 on theequations presented earlier.

5.6 Molecular Contamination

In addition to particulate contaminants, molecular contaminants can also degradeoptical system performance.4 Molecular contamination is formed from volatilesoutgassed from materials (such as paints, adhesives, and circuit boards) in ornear the optical system. These contaminants can form films that absorb light inthe sensor waveband. The amount of molecular contamination that a materialcan generate is quantified by its total mass loss (TML) and its percent collectedvolatile condensable materials (CVCM). A testing method (ASTM E 595-07)22

has been developed for quantifying these values, and data has been published forsome materials.23,24 In Chapter 6, TML and CVCM will be discussed as selectioncriteria for black surface treatments. The TML and CVCM can be reduced in somematerials using a bake-out process.4

Such films can also increase the BSDF of the optics on which they are deposited,and thus degrade the stray light performance of the system, though this effect ispoorly understood. As with any surface whose scattering properties are unknown,the best way to understand the scattering properties of a surface with molecularcontamination is to measure it.

5.7 Summary

Scattering from particulate contamination (dust) on a surface can increase itsBSDF above the level predicted by surface roughness scatter. A model based onMie scattering theory has been used to accurately predict scattering from particlesand is available in most stray light programs. This model requires the refractiveindex of the dust particles (see Table 5.1) and the particle density function, whichdescribes the projected area of the particles as a function of their diameter. Aninternational standard (IEST-STD-CC1246D) has been established to describe theparticle density function using two parameters, the cleanliness level CL and theparticle distribution slope S, where CL is equal to the particle diameter whosedensity is 1 per 0.1 m2, and S is the variation in particle density versus diameter.Research has indicated that S should be –0.926 for surfaces that have been recentlycleaned, and –0.383 for those that have not. CL can be determined a priori by usingtypical values (given in Table 5.5), cleanroom fallout parameters, or by inspection.Even if inspection data is used, the best accuracy that can be obtained in modeling




BSDF is about ± 50%. The TIS of particulate scatter is not a strong function ofwavelength, whereas roughness scatter varies as (1/l)2; therefore, roughness scatterusually dominates at shorter wavelengths (VIS and NIR), and contamination scatterdominates at longer wavelengths (MWIR and LWIR). The Henyey–Greensteinmodel is available in most stray light analysis programs and is often used to modelscattering from bulk inclusions.

References

1. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Par-ticles, John Wiley & Sons, New York (1998).

2. H. van de Hulst, Light Scattering by Small Particles, Dover, Mineola, NY(1981).

3. P. Spyak and W. Wolfe, “Scatter from particulate-contaminated mirrors. Part 1:theory and experiment for polystyrene sphere and �g 0.6328 mm,” Opt. Eng.31(8), 1746–1756 (1992) [doi: 10.1117/12.58708].

4. A. Tribble, Fundamentals of Contamination Control, SPIE Press, Bellingham,WA (2000) [doi: 10.1117/3.387881].

5. T. Germer, The SCATMECH code library, National Institute of Standardsand Technology (NIST), http://physics.nist.gov/Divisions/Div844/facilities/scatmech/html/index.htm.

6. P. Spyak and W. Wolfe, “Scatter from particulate-contaminated mirrors. Part4: properties of scatter from dust for visible to far-infrared wavelengths,” Opt.Eng. 31 (8), 1775–1784 (1992) [doi: 10.1117/12.58711].

7. D. Jenkins, E. Fest, R. Kremer, and P. Spyak, “Improved Mie scatter theorymodel for particulate contamination that conserves energy and obeys reci-procity,” Proc. SPIE 6291, 62910Q (2006) [doi: 10.1117/12.681173].

8. P. Ma, M. Fong, and A. Lee, “Surface particle obscuration and BRDF predic-tions,” Proc. SPIE 1165, 381–391 (1989) [doi: 10.1117/12.962866].

9. IEST-STD-CC1246D: Product cleanliness levels and contamination controlprogram, Institute of Environmental Sciences and Technology (2002).

10. “Product cleanliness levels and contamination control program,” Military Stan-dard (MIL STD) 1246C (1994).

11. O. Hamberg and E. Shon, “Particle size distribution on surfaces in clean rooms,”Aerospace Corporation report (1984).

12. R. Peterson, P. Magallanes, and D. Rock, “Tailored particle distributions de-rived from MIL-STD-1246,” Proc. SPIE 4774, 79–96 (2002) [doi: 10.1117/12.481665].

13. J. Fleming, B. Matheson, M. Dittman, F. Grochocki, and B. Firth, “Modelingparticle distributions for stray light analysis,” Proc. SPIE 6291, 62910T (2006)[doi: 10.1117/12.678321].

14. ISO 14644-1, International Organization for Standardization, “Cleanrooms andassociated controlled environments – Part1: Classification of air cleanliness,”ISO 14644-1:1999(E).



100 Chapter 5

15. J. Buch and M. Barsh, “Analysis of particulate contamination buildup on sur-faces,” Proc. SPIE 777, 43–54 (1987) [doi: 10.1117/12.967066].

16. Clemex Corp. website, http://www.clemex.com.17. J. Taylor, An Introduction to Error Analysis, University Science Books, Sausal-

ito, CA (1997).18. P. Spyak and W. Wolfe, “Scatter from particulate-contaminated mirrors. Part 2:

theory and experiment for dust at � = 0.6328 mm,” Opt. Eng. 31(8), 1757–1764(1992) [doi: 10.1117/12.58709].

19. International Standards Organization, ISO 10110-3: Material imperfections –bubbles and inclusions (1996).

20. D. Aikens, “The Truth About Scratch and Dig”, Optical Society of America,Washington, D.C. (2010).

21. L. Henyey and J. Greenstein, “Diffuse radiation in the galaxy,” AstrophysicalJournal 93, 70–83 (1941).

22. American Society for Testing and Materials (ASTM), ASTM E 595-07: Stan-dard Test method for Total Mass Loss and Collected Volatile CondensableMaterials form Outgassing in a Vacuum Environment (2007).

23. W. Campbell and J. Scialdone, “Outgassing Data for Selecting Spacecraft Ma-terials,” NASA Reference Publication 1134, Revision 2 (1990).

24. R. Predmore and E. Mielke, Materials Selection Guide, Revision A, NASAGoddard Space Flight Center (1990).

25. H. Bennett, “Reduction of Stray Light from Optical Components,” Proc. SPIE107, 24–33 (1977) [doi: 10.1117/12.964592].



Chapter 6

Scattering from Black SurfaceTreatmentsBlackening the mechanical housing surfaces in an optical system is one of theprimary methods of controlling stray light. Examples of such surfaces include theinner diameters of cylinders in which lenses and mirrors are mounted, struts tohold mirrors (usually the secondary mirror) in centrally-obscured systems, baffles,vanes, sunshades, cold shields, stops, and any other surface near the optical path.These surfaces may be critical, and therefore steps should be taken to ensure theyare not illuminated as well (see Chapter 9). An example of a black surface treat-ment (anodized aluminum) used in an optical system is shown in Fig. 6.1. Thisimage shows the inside of the barrel of the baseline Maksutov–Cassegraintelescope.

Although the use of black surface treatments can significantly reduce themagnitude of first-, second-, and higher-order stray light paths, the improvementgained in stray light performance by their use is often not enough to compen-sate for a system whose optical and/or baffle design is inadequate for its straylight requirement. Therefore, selection of black surface treatments should oc-cur at the appropriate time in the stray light design process, as discussed inChapter 11.

This chapter begins with a discussion of the physics of light scattering fromblack surface treatments and a discussion of the methods used to model itsBRDF (only BRDF models will be discussed, as the BTDF of a black surfacetreatment is usually zero). BRDF models of a couple of typical black surfacetreatments (Aeroglaze R© Z306 paint and anodized aluminum) are provided, as arereferences to publications that contain measured BRDF data. Caution should beused when using any set of published BRDF data (discussed in Section 6.1.2)because it is often difficult to know how similar the measured sample is tothe surface being modeled. The BRDF model data is followed by a discussionof the criteria that should be considered when selecting a black surface treat-ment, as there are usually many more criteria to consider than just its BRDF inthe sensor waveband. The chapter ends with a survey of popular black surfacetreatments.

101



102 Chapter 6

Corrector

Black anodize

Figure 6.1 Anodized aluminum used in the inner diameter of the barrel of the baselineMakstuov–Cassegrain telescope. The primary mirror has been removed so that the an-odization can be seen.

6.1 Physics of Scattering from Black Surface Treatments

Black surface treatments are engineered to minimize their TIS by using surfaceroughness profiles that are very different than those of optical surfaces, which areusually engineered to maximize their specular reflectance or transmittance. Blacksurface treatments usually seek to minimize their TIS and to be as Lambertian aspossible; some methods include the following:1,2

� The use of absorbing compounds, such as paints (e.g., Aeroglaze R© Z306) ordyes.

� The use of structures to trap light, such as dendrites or cavities (such as thosein anodized surfaces), as shown in Fig. 6.2.

Incident rays at θi = 0 deg Incident rays at θi = 65 deg

Figure 6.2 Dendrites in black surface treatments and their “shadowing” effect. The AOI �iis relative to the macroscopic surface normal. At small �i , more light gets trapped in thedendritic structures of the surface profile than at larger �i , as evidenced by the number ofrays reflected away from the surface. This is especially noticeable in the circled region.



Scattering from Black Surface Treatments 103

� The use of structures or particles to diffusely scatter light, such as those insandblasted surfaces.

� The use of interference coatings to minimize reflections, such as those usedin “dark mirrors,” that have low specular reflectance.

Most black surface treatments use more than one of the methods mentionedpreviously, and the use of these methods makes it difficult to predict their BRDFa priori from information such as their surface roughness profile, as can be donewith optical surfaces. For instance, the absorption of paint versus wavelength isa function of many independent variables, such as its chemistry, the method usedto apply it, and the environment in which it was applied. Likewise, the surfaceroughness profile of the dendrites or cavities is much more complex than that ofoptical surfaces, as shown in Figs. 6.9 and 6.10. The complexity of these surfaceprofiles makes it difficult to measure them and predict their scattering properties.For these reasons, the BRDF of black surface treatments is usually determinedthrough direct measurement.

Because of the structural differences between black surface treatments andoptical surfaces, the angular distribution of the BRDF of black surface treatmentsis often very different. Figure 6.3 shows the measured BRDF of Aeroglaze R© Z306paint on unpolished aluminum,3 which is a space-qualified paint widely used tocontrol stray light. The figure shows that as the AOI �i increases, the BRDF of thispaint becomes more specular (i.e., its slope is larger), and its TIS increases. This is aproperty of most black surface treatments (and, indeed, of most surfaces in general)and is called non-shift-invariance because its BRDF changes as a function of �i andnot just | sin �i – sin �s |. There are multiple reasons for this behavior; one is that thenear specular scatter is increasing due to the increase in specular reflectance with�i (as predicted by the Fresnel equations, see Fig. 2.3). Another reason is that thelight traps in the unpolished aluminum get “shadowed” as the �i increases, and thusthe effectiveness of these surfaces at trapping light decreases, as shown in Fig. 6.2.Compare this scatter from optical surfaces, whose TIS generally decreases with �i ,as predicted by diffraction theory in Eq. (4.14). As will be discussed in Chapter 9,the degradation in system stray light performance due to the high BRDF of blacksurface treatments at high AOIs can be mitigated through the use of proper baffledesign.

Some baffle designs call for the use of highly diffuse black surface treatments,and the surface roughness profiles described above are designed to achieve this(see Chapter 9). However, another type of baffle design uses specular baffles,which employ specular black surface treatments to control stray light. The methodsthat specular baffles use to control stray light include the use of light traps inwhich multiple reflections are used to absorb stray light, and the use of bafflegeometries that are designed to send stray light back out the entrance aperture ofthe optical system. Methods to model the BRDF of specular black surfaces andexamples of specular black surface treatments are included in the chapters thatfollow.



104 Chapter 6

1.E-03

1.E-02

1.E-01

1.E+00

1.E-02 1.E-01 1.E+00 1.E+01

BRD

F (1

/sr)

|sinθs − sinθ i|

θi = 5 deg (Measured) θi = 5 deg (Fit)



Figure 6.3 BRDF of Aeroglaze R© Z306 paint on unpolished aluminum at 0.6328 �m. Mea-sured data is digitized from the published data.3 Estimated TIS is 0.0592 at 5 deg, 0.0855at 45 deg, and 0.1114 at 60 deg.

6.1.1 BRDF from empirical fits to measured data

As discussed in the previous section, the most accurate way to model the BRDFof black surface treatments is usually to measure it and fit it to a model. Be-cause the BRDF for diffuse black surface treatments is not shift-invariant, andbecause the best method of determining the BRDF of these surfaces is to measureit, it is important that the BRDF measurements be made at multiple values of�i . To get a good fit, a minimum of three widely spaced values of �i are recom-mended, such as 5 deg, 45 deg, and 75 deg. Measuring the BRDF at or near thosevalues of �i at which it is illuminated may also increase the accuracy of thestray light prediction. Because the variation of BRDF versus �i for these surfacescan be very different from that of optical surfaces, typically models other than theHarvey model are used. One such model is the general polynomial (also called




Table 6.1 General polynomial cijk coefficients for the fit toAeroglaze R© Z306 data at 0.6328 �m shown in Fig. 6.3.

Cijk

k

i j 0 1

0 0 −1.7327 0.7579

1 0 0.3390 2.0569

1 1 −0.7254 −0.7427

2 0 −1.0807 1.0607

2 1 −3.2797 7.1209

2 2 −1.6929 2.1468

3 0 0.7937 −1.6387

3 1 5.9528 −9.3959

3 2 −1.1236 −0.8554

3 3 −0.7462 2.0412

the diffuse polynomial), whose form is given by

log10 (BSDF ) =n∑

k=0

⎡⎣ m∑

i=0

i∑j=0

cijk(Ui W j + U j W i )+

l∑i=l ′

cik log(1 + di T )

⎤⎦V k

2,

(6.1)

where U = �2s + �2

s , V = –�i �s – �i �s , W = �2i + �2

i , and T = U – 2V + W(variables shown in Fig. 2.17), and cijk, cik, and d are the model coefficients. Thismodel is used to fit the data in Fig. 6.3, and the coefficients of this fit are shown inTable 6.1. The advantages of this model are that it always obeys reciprocity, that itcan be used to fit data at multiple values of �i , and that the order of its polynomials(n, m, and l) can be increased to obtain a better fit to the measured data. However,it also has a number of disadvantages: its functional form is very complicated,and its coefficients do not correspond to easily described properties (such as theslope parameter s of the Harvey model). Its TIS must be computed numerically,and it is not guaranteed to be less than or equal to 1 for all coefficients. And aswith all higher-order polynomial functions, using high orders can obtain good fitsto measured data, but these polynomials tend to have nonphysical oscillations inthe BRDF at angles other than those that were measured. In practice, the generalpolynomial (which was used in the fit shown in Fig. 6.3) usually produces the most



106 Chapter 6

Figure 6.4 BRDF versus scatter angle of a general polynomial Aeroglaze R© Z306 model at0.6328 �m for �i = 0 deg and �i = 70 deg. The scatter angle is expressed as a direction-cosine pair, with the x-direction cosine equal to sin �s

∗ cos �s, and the y-direction cosineequal to sin �s × sin �s. Calculations are performed in FRED.

realistic fits when used at lower orders, such as n = 1 and m = 3. The Lorentziancomponent Eq. (6.1) (the cik coefficients) is typically used to fit BRDFs that arefairly specular and therefore was not used in Fig. 6.3 (l = 0). Direction-cosinespace plots of the same model are shown in Fig. 6.4.

The fit coefficients shown in Table 6.1 are obtained using the “Solver” utilityin Excel4 to minimize the residual as defined in Eq. (4.17). The coefficients arethen input into FRED, and its TIS computed at multiple values of �i to ensure thatenergy conservation is obeyed. If it is not, then an optimization routine may beneeded in which the merit function of the routine is the residual, with a penaltyfunction employed for those solutions that violate TIS ≥1. As when fitting anyset of measured data, obtaining good fits to BRDF data requires the use of theright fitting function for the data set and the right tools to perform the fit. Theuncertainty of the BRDF fit [as defined by Eq. (4.18)] shown in Fig 6.3 is 0.034,which can be used as an estimate of the minimum uncertainty in this BRDFmodel.

The same fitting process was used to compute the general polynomial modelcoefficients for Aeroglaze R© Z306 paint at 10.6 �m. The measured BRDF is shownin Fig. 6.5, the coefficients in Table 6.2, and the BRDF in direction-cosine spacein Fig. 6.6. The uncertainty of this fit is 0.20. The TIS at 10.6 �m is slightly lowerthan the TIS at 0.6328 �m (0.0198 versus 0.0592 at �i = 5 deg).

Another popular black surface treatment is anodized aluminum, whose mea-sured BRDF at 0.6328 �m is shown in Fig. 6.7, whose general polynomial fitcoefficients are shown in Table 6.3, and whose BRDF in direction-cosine space isshown in Fig. 6.8. The uncertainty of this BRDF fit is 0.1169. The coefficients inTable 6.3 are computed using the same process used to compute the coefficients




1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01

BRD

F (1

/sr)

|sin θs – sin θi|

θi = 5o (Measured) θi = 5o (Fit)


Figure 6.5 BRDF of Aeroglaze R© Z306 paint on unpolished aluminum at 10.6 �m. Measureddata is digitized from the published data.19 Estimated TIS is 0.0198 at 5 deg, 0.0941 at 80deg.

for Aeroglaze R© Z306 in Tables 6.1 and 6.2. This sample of anodized aluminum hashigher TIS than Aeroglaze R© Z306 at the same wavelength and is more specularat high angles of incidence, which requires the use of the Lorentzian component[i.e., the d value in Eq. (6.1)] in the general polynomial model to fit. This behav-ior is typical of many black anodized surfaces, which often have higher BRDFat high �i than black painted surfaces. In general, anodize treatments also showsome shift-invariance, as shown for larger values of |sin �s – sin �i | in Fig. 6.7. Theoptical properties of anodize treatments are often not as well-controlled as blackpaint and therefore can exhibit large lot-to-lot or vendor-to-vendor variation (seeSection 6.4.3).

The BRDF of specular black surface treatments is often shift-invariant, thoughthis is not always the case.3 When it is, the Harvey model can often be used toobtain a good fit to the nonspecular scattered light. Most black surface treatmentsused for stray light control (such as those whose BRDF data was shown previously)are very diffuse and therefore do not have a significant specular component thatneeds to be modeled. However, when working with an unknown surface treatment,



108 Chapter 6

Table 6.2 General polynomial cijk coefficients for the fit toAeroglaze R© Z306 at 10.6 �m data shown in Fig. 6.5.

Cijk

k

i j 0 1

0 0 −1.3751 4.5798

1 0 −10.2009 −3.8576

1 1 6.7630 −8.9005

2 0 17.4710 −2.2442

2 1 −2.9263 12.7094

2 2 −1.1703 3.5346

3 0 −10.7815 0.2042

3 1 −5.4900 3.0025

3 2 2.0500 −2.3326

3 3 3.9397 −4.9421

Figure 6.6 BRDF versus scatter angle of a general polynomial Aeroglaze R© Z306 model at10.6 �m for �i = 0 deg and �i = 70 deg. The scatter angle is expressed as a direction-cosinepair, with the x-direction cosine equal to sin �s × cos �s, and the y-direction cosine equal tosin �s × sin �s. Calculations performed in FRED.




1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E-02 1.E-01 1.E+00 1.E+01

BRD

F (1

/sr)

|sin θs – sinθ i|




Figure 6.7 BRDF anodized aluminum at 0.6328 �m. Measured data was provided courtesyof Schmitt Measurement Systems [ref Schmitt]. Estimated TIS is 0.1123 at 5 deg, 0.2991 at45 deg, and 0.5947 at 75 deg.

it is not advisable to assume that its specular component is zero; it is always bestto measure the specular component to determine if it is significant enough toinclude in the model.

6.1.2 Using published BRDF data

A great deal of BRDF data has been published for many black surfacetreatments.1,3,5,16 However, it is often difficult to use this data to construct a BRDFmodel of known accuracy for the following reasons:

� Many of these publications are many years old and may not represent theproperties of the surface treatment as it is being made today.

� The publications may report BRDF data from surfaces whose surface prepa-ration prior to the application of the black surface treatment is different thanthat of the surface of interest. For instance, bead-blasting a surface prior topainting it can lower its TIS and make its BRDF more Lambertian. Or the en-vironment under which the black surface treatment was applied (temperature,



110 Chapter 6

Table 6.3 General polynomial cijk coefficients for the fit toanodized aluminum at 0.6328 �m data shown in Fig. 6.7.

Cijk

k

i j 0 1

0 0 −0.4862 14.5971

1 0 −7.3626 −93.9341

1 1 31.7399 163.4810

2 0 11.2902 121.2815

2 1 −138.2943 −324.7588

2 2 181.6310 57.1636

3 0 −6.7127 −51.6019

3 1 84.5194 87.8442

3 2 −244.1509 99.3737

3 3 86.7686 −70.5166

Cijk (d = 508.0160)

k

i 0 1

0 −0.3234 −0.61661 −0.3231 −0.6166

humidity, tooling, etc.) may have been different than that of the surface ofinterest, which can also change its BRDF.

� Details of the scatterometer used to measure the BRDF and/or the experienceof its operator are often not discussed in the publications, and therefore it canbe difficult to know the accuracy of the data presented.

� The data presented in the publications is often at low resolution in AOI andscatter angle, making it difficult to obtain a good fit to the data.

� The data shown in these publications is difficult to obtain in electronic formand therefore must be digitized from the original, which will result in anadditional reduction in accuracy.

For these reasons and more, it is recommended that, if a BRDF model ofknown accuracy is desired, then a new sample of the black surface treatment beobtained and a BRDF measurement of it made. Doing so will ensure that the




Figure 6.8 BRDF versus scatter angle of a general-polynomial anodized aluminum modelat 0.6328 �m for �i = 0 deg and �i = 70 deg. The scatter angle is expressed as a directioncosine pair, with the x-direction cosine equal to sin �s × cos �s, and the y-direction cosineequal to sin �s × sin �s. Calculations performed in FRED.

surface is as representative (in as many ways as possible) of the final surface andthat sufficient data is collected to obtain a good fit (see Section 10.1 for a rec-ommendation of commercial facilities that provide such measurements). It is forthis reason that companies that make sensors whose stray light is well-controlled(such as Raytheon) maintain large internal databases of BRDF data that are notcommercially available. If accuracy to within a couple orders of magnitude is notimportant, then published data can be used, or the models of Aeroglaze R© Z306and anodized aluminum presented earlier in this chapter, as they are represen-tative of optical scatter of many black surface treatments (i.e., it has a typicalTIS and variation in BRDF versus AOI). If the TIS of the surface treatment tobe modeled is known, then the BRDF values predicted by the Aeroglaze R© Z306or anodized aluminum models can be scaled by the ratio of the TIS of the sur-face treatment to be modeled to the TIS of the Aeroglaze R© Z306 or anodizemodel.

6.1.3 Artifacts from black surface treatment scatter

A simulated artifact from anodized aluminum scatter in the baseline Maksutov–Cassegrain telescope are shown in Fig. 6.9. This artifact is due to scattering fromthe inner diameter of the primary mirror baffle, which is critical and is illuminatedby the sun at 15 deg. (The ray trace of this scatter path is shown in Fig. 9.16.) Theartifact due to scattering from Aeroglaze R© Z306 paint was also evaluated but is notshown in Fig. 6.9 because it is very similar. This is because the BRDF of the twotreatments are similar at the values of �i and �s that correspond to this scatter path.The spatial variation in irradiance of the artifact is a strong function of the angular



112 Chapter 6

3.320E13

2.185E13

2.753E13

1.618E13

irradiance in ph/s-mm2

1.050E13

Figure 6.9 Artifact due to scattering from the inside of the primary mirror baffle in thebaseline Maksutov–Cassegrain telescope. The sun is located 15 deg from the optical axis,along the diagonal, outside of the lower left corner. Simulation done in FRED.

variation in BRDF of the paint and of baffle geometry. The upper right corner ofthe detector is closest to the specular component of the reflection from the innerdiameter of the baffle, and is therefore the brightest region. The crescent-shapedshadow is due to the shape of the hole in the primary mirror.

6.2 Selection Criteria for Black Surface Treatments

For the purposes of controlling stray light, the most desirable property of a blacksurface treatment is low BRDF in the sensor waveband at the required range of AOIand scatter angle. There are also a number of obvious selection criteria, such as itsease of application, its ability to adhere to the substrate, its thermal characteristics,susceptibility to damage with handling, long-term durability, and its cost. However,there are many less obvious but still important factors to consider when selectinga black surface treatment, some of which will be discussed briefly here. Thesecriteria should be evaluated with respect to the position of the black surface inthe optical system.6 For instance, surfaces near the focal plane will probably havestricter requirements on the amount of molecular and particulate contaminationthat they are allowed to generate, as contaminants on the focal plane will havea greater impact on the performance on the system than contaminants on othersurfaces. Such an evaluation may result in the use of more than one type of blacksurface treatment in the optical system.




6.2.1 Absorption in the sensor waveband

Selection of a black surface treatment usually begins by identifying those surfacesthat have low TIS in the sensor waveband. This is usually done by comparingthe measured TIS [often reported as diffuse hemispherical reflectance (DHR) ortotal hemispherical reflectance (THR), see Section 10.2] versus wavelength of thecandidate treatments. It is especially important to do this when evaluating surfacesfor sensors that do not operate in the visible waveband because surfaces that areblack (i.e., have low TIS) in the visible are often not black at all in other wavebands.In particular, many treatments that have high absorption in the visible have muchlower absorption in the NIR, as shown for tungsten hexafluoride in Fig. 6.10.

For selecting surfaces for stray light control, TIS is a better measurement than(the more common) specular reflectance versus wavelength because it quantifiesthe magnitude of all of the light scattered from the surface, whereas specularreflectance quantifies only the specular component. This is true even for specularblack surfaces because low TIS is desirable in any surface that is used to controlstray light. TIS can be used to select surface treatments that are suitable for thesensor waveband; however, as such measurements are not angle-resolved, theygenerally are not used to predict the stray light performance of the system.

6.2.2 Specularity at high AOIs

This property is related to, but not the same as, high absorption in the waveband.As shown in Figs. 6.3 and 6.5, the TIS of most surface treatments increases withAOI, as does the specularity (i.e., the slope of the BRDF versus scatter angle).

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 2 4 6 8 10 12 14 16 18 20

Tota

l Int

egra

ted

Scat

ter (

Frac

tion

al)

Wavelength (μm)

Figure 6.10 TIS versus wavelength of tungsten hexafluoride.18



114 Chapter 6

High specularity can be very undesirable in a black surface treatment, as it can leadto stray light artifacts that are bright and narrow in angular extent. For instance,highly specular paint on the inner diameter of a lens tube can cause a “caustic” orfocus near the focal plane that can be very bright. As a general rule, the deeper thedendrites on the surface, the lower the specularity and TIS of the surface at highAOIs. Determining the specularity of a black surface treatment typically requiresa BRDF measurement.

6.2.3 Particulate contamination

Small pieces from the surface of a black surface treatment can break off, thusgenerating particulate contamination (see Chapter 5 for a discussion of the effectsof particulates on stray light performance). The amount of particulate generateddepends on many factors, such as the fragility of the surface, the vibrationalenvironment it is exposed to, and whether or not the surface rubs against another inthis environment. As discussed, some black surface treatments such as anodizationuse dendritic structures as light traps to lower the BRDF of the surface; in general,the deeper these structures are, the better they are at lowering the BRDF and themore fragile they are. Some of these surfaces (such as “Deep Sky Black,” see Table6.3) are so fragile that they cannot be touched without destroying their dendrites.In addition, these surfaces can trap other particulate contaminants and then releasethem under vibration.

6.2.4 Molecular contamination

As discussed in Section 5.7, molecular contamination refers to the volatiles thatoutgas from a material; this contamination can deposit on the sensor optics andreduce the system transmittance. Some black surface treatments (such as paint) canhave high TML and/or collected volatile condensable materials (CVCM), whichquantify the amount of molecular contamination generated by the material.

Materials in space-borne sensors are susceptible to the effects of atomic oxygen,which is present in low earth orbit and can attack the binder materials in paints.7

This effect increases the TML (due to ablation of the surface treatment) and canresult in the surface becoming more Lambertian.8

6.2.5 Conductivity

The conductivity of the black surface treatment may be important. For instance,high-conductivity surface treatments are often used on spacecraft to minimize thecharging effect that occurs when passing through radiation belts.9

6.3 Types of Black Surface Treatments

Comparing black surface treatments is made easier if they are categorized by type.They generally fall into one of three types.




6.3.1 Appliques

Appliques are freestanding surfaces that are attached to the substrate by adhe-sives or other means.10 Their surfaces often consist of numerous fibers that act aslight traps. They have a number of advantages: they may have lower BRDF thanblack paints or anodization, their BRDF does not vary greatly lot-to-lot becausethey are manufactured under controlled conditions (as opposed to paints, whoseproperties can vary depending on the application process), they can be durable tolight handling, they can be cheaper than paints or anodization, and the process ofadhering them to a surface can be more manufacturing-friendly than a paintingor anodization process. They also have a disadvantages: they can generate moremolecular and particulate contamination than other black surface treatments andcan have a lower laser-damage threshold. Perhaps the most well-known applique isblack flocking paper, which is available commercially.11 Appliques are often usedto control stray light in laboratory experiments.

6.3.2 Treatments that reduce surface thickness

These treatments make the surface more diffuse by changing its roughness pro-file. Examples include chemically etched electroless nickel plating, flame-sprayedaluminum, sandblasting, and beadblasting. In addition to changing the roughnessprofile, chemical etching of electroless nickel plating also increases its absorp-tion by changing the chemical composition of the surface coating.12 These surfacetreatments are highly durable, do not generate molecular contamination, and canincrease adhesion to paints. However, their BRDF can have large lot-to-lot variationbecause it is highly process-dependent. The treatments may change the variationin BRDF with AOI, scatter angle, and wavelength; and they may weaken thestrength of the substrate. An etched electroless nickel-plated surface is shown inFig. 6.11.

Figure 6.11 An etched electroless nickel-plated surface.13



116 Chapter 6

Figure 6.12 Scanning electron micrograph of a Ball IR Black Coating (image courtesy ofBall Aerospace and Technologies Corporation).

6.3.3 Treatments that increase surface thickness

Treatments that cover the surface are the most common type of black surfacetreatment, including paints and anodization (such as the coating shown in Fig.6.12).

6.3.3.1 PaintingThis is the most common way of blackening a surface. Some form of carbon isusually added to the paint to make it black. The paints available for stray light controlare usually applied by a high-pressure air sprayer, though some can be brushed on.The paint must be applied thickly enough to eliminate reflections from the substrate(generally1 greater than 0.003′′, especially in the far infrared14). The advantagesof paint are that there are many types available, its properties are well known, andit can be either sprayed or brushed on, which means it can be used for touch-up.Its disadvantages are that it increases the thickness of the surface to which it isbeen applied, it can generate a lot of molecular and particulate contamination, andit can be difficult to apply, especially if a masking-off process must be used. Someof the most popular diffuse paints used for stray light control include Aeroglaze R©

Z306, Akzo Nobel 463-3-8 (formerly known as Cat-A-Lac diffuse black), MH21,and MH2200. Specular black paints include Aeroglaze R© Z302 and Akzo Nobel443-3-8 (formerly known as Cat-A-Lac specular black). Contact information forthese paints is given in Table 6.4.

6.3.3.2 Fused powdersThese powders are usually ceramic, and they can be applied in a variety of methods(including spraying and brushing), but they require a cure at 500◦C. They may have




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118 Chapter 6

Tab

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high durability in space environments but may be easily damaged by handling.Perhaps the most well-known is CerablakTM (see Table 6.4).

6.3.3.3 Black oxide coatingsThese are usually made by reacting a conductive surface material (such as copper)with an oxidizing process. If the substrate is not conductive, then it may be coatedwith one first. These coatings often have very low TIS; however, they are also oftenvery fragile. Popular coatings for stray light control include Epner Laser BlackTM

and Ebonol R© C (see Table 6.4).

6.3.3.4 AnodizeThis is a very common process used to increase the natural oxide layer on ametal (typically aluminum or beryllium) surface by immersing the surface in anelectrolytic solution and passing a current through the solution. Dyes are thenoften applied to the surface. The most widely used specification for anodization isMIL-STD-8625F15, which defines three types:

� Type I is formed using chromic acid in the solution and is the oldest type ofanodization. It results in surfaces that are softer and thin (0.5–18 �m), andare harder to dye than Type II surfaces.

� Type II is formed using sulfuric acid in the solution and is the most commontype of anodization. It results in surfaces that are thicker than Type I surfaces(18–25 �m).

� Type III (also called “hardcoat”) is also formed using sulfuric acid but isthicker (> 25 �m) than Type II.

Note that MIL-STD-8625F does not specify any of the optical properties ofthe anodized surface. Anodization is a very common process available from manyvendors and is therefore often used as a black surface treatment for stray lightcontrol. The advantages of typical anodization are that it is durable, does notgenerate a lot of molecular and particulate contamination, and is widely available.One disadvantage is that its TIS can be a few percent higher than comparable blackpaints, especially at large AOIs (though this is less true of anodized berylliumthan anodized aluminum). In addition, its TIS can vary widely as a function ofwavelength and may be significantly higher (a few percent) at IR wavelengths(NIR-LWIR). Also, because anodize is often used only to protect a surface orfor cosmetic reasons, its vendors are often inexperienced at controlling its opticalproperties; as a result, its optical properties can vary widely lot-to-lot. For example,the vendor might not use a consistent substrate preparation process, which can resultin a large change in TIS. Therefore, sufficient metrology should be in place beforeaccepting anodized parts from such vendors.

In addition to the typical anodization that is available from most vendors,there are some specialized types (such as Deep Sky BlackTM and Pioneer OpticalBlackTM, see Table 6.4) that are engineered specifically to have low BRDF, andtherefore may be more desirable to use than typical anodization. Martin Black wasa similar treatment but is no longer available. However, these types of anodization



120 Chapter 6

may not work well in all wavebands, may be more fragile than typical anodization(due to the use of deeper dendrites), and may be more expensive. Contact theirvendors for more information.

6.4 Survey of Widely Used Black Surface Treatments

A number of papers have been published in which black surface treatments havebeen evaluated for particular applications,5,6 and some of the results of these studiesare summarized in Table 6.3. Some important points:

� The table refers to black surface treatment products that are commerciallyavailable at the time of this book’s publication; however, there is no guaranteethat these products will always be available or that they will always have thesame product name. For instance, Aeroglaze R© used to be called Chemglaze R©,and Akzo Nobel 463-3-8 used to be called Cat-A-Lac Diffuse Black.

� As discussed with respect to BRDF data in Section 6.1.2, the data presentedin the publications cited in the table is generally of unknown accuracy, andtherefore if the accuracy of the BRDF, TIS, CVCM, or any other data isimportant, then a new sample should be obtained from the vendor and ameasurement made of the quantity of interest.

� Sometimes the vendors listed in the table do not control the optical propertiesof the treatment well, especially in the UV and IR wavebands. Therefore, aslong as the visible reflectance of the coating remains roughly the same, theymay, without warning, change the formulation (and therefore the reflectance)of the product. For instance, the dye used in an anodization process may bechanged. The vendor needs to therefore be made aware of these issues relatedto changing the product’s formulation and, if necessary, either not change it ortake steps to control its optical properties. Preferably, some sort of metrology(such as TIS tests) would be performed on the surface treatment to verifythat its reflectance meets the requirement.

� Most of these treatments work well in the VIS-LWIR waveband but mayor may not work well in the UV or VLWIR. In particular, the absorption ofmany of these treatments is much less in the VLWIR. Studies of black surfacetreatments for UV and VLWIR have been published.5

� This list is not exhaustive. Longer lists have been published,1,5 although they,too, are not exhaustive.

6.5 Summary

Blackening the mechanical housing surfaces in an optical system is one of theprimary methods of controlling stray light. The microscope roughness profile ofthese surfaces often consists of dendrites that act as light traps; as a general rule,the deeper these dendrities, the lower the BRDF of the surface, and the morefragile it is. The most accurate way to model the BRDF of these surfaces is fitan interpolation function to a set of measured data. An interpolation functioncommonly used for this purpose is the general polynomial. The BRDF of these




surfaces can be much more specular at large AOIs than at normal incidence, andtherefore the data set should contain measurements across the full range of AOIs: 5deg, 45 deg, and 75 deg are recommended. For the purposes of stray light control,two of the most commonly used types of black surface treatment are black paintand anodize. Black paint can be used on nearly any type of substrate, can be usedacross a wide range of operational environments (temperature, pressure, etc.), andhas very repeatable BRDF. One of the most widely used paints for this purpose isAeroglaze R© Z306. Anodize is usually used on aluminum or beryllium, and can alsobe used across a wide range of operational environments. Because it is often usedfor cosmetic purposes, many anodize vendors do not control its optical propertieswell, and therefore sufficient metrology should be used to ensure consistent results.The BRDF of both painted and anodized surfaces is a function of the substrateroughness prior to blackening, of the chemicals used in the blackening treatment,and of the process used to apply it. For these reasons, BRDF measurements ofnew surface samples are almost always more accurate than BRDF measurementsfrom the published literature. A survey of widely used black surface treatments isprovided in Table 6.4.

References

1. M. Persky, “Review of black surfaces for space-borne infrared systems,” Reviewof Scientific Instruments 70(5) (1999).

2. S. Meier, “Methods to suppress stray light in black materials,” Proc. SPIE 5526,195–207 (2004) [doi: 10.1117/12.559812].

3. W. Viehmann and R. Predmore, “Ultraviolet and visible BRDF data on space-craft thermal control and optical baffle materials,” Proc. SPIE 675, 67–72(1987) [doi: 10.1117/12.939484].

4. Microsoft Excel Software, http://www.microsoft.com.5. S. Pompea and R. Breault, “Characterization and use of black surfaces for

optical systems,” in The Handbook Of Optics, Vol IV,3rd. Ed., M. Bass, G. Li,and E. Van Stryland, Eds., McGraw-Hill, New York (2010).

6. S. Pompea and S. McCall, “Outline of selection processes for black bafflesurfaces in optical systems,” Proc. SPIE 1753, 92–104 (1992) [doi: 10.1117/12.140712].

7. M. McCargo, R. Dammann, J. Robinson, and R. Milligan, “Erosion of DiamondFilms and Graphite in Oxygen Plasma,” Proceedings of the International Sym-posium on Environmental and Thermal Control Systems for Space Vehicles,pp. 1–5 (1983).

8. A. Whittaker, “Atomic Oxygen Effects on Materials,” STS-8 Paint Data Sum-mary, Marshall Space Flight Center (1984).

9. M. Birnbaum, E. Metzler, and E. Cleveland, “Electrically conductive blackoptical black paint,” Proc. SPIE 362, 60–70 (1982) [doi: 10.1117/12.934134].

10. K. Snail, D. Brown, J. Costantino, W. Shemano, C. Seaman, and T. Knowles,“Optical characterization of black appliques,” Proc. SPIE 2864, 465–474 (1991)[doi: 10.1117/12.258336].



122 Chapter 6

11. “Light Absorbing Black-Out Material”, available from Edmund Optics Inc.,http://www.edmundoptics.com.

12. R. Brown, P. Brewer, and M. Milton, “The physical and chemical propertiesof electroless nickel-phosphrous allows and low reflectance nickel-phosphorusblack surfaces,” J. Materials Chem. 12, 2749–2754 (2002).

13. S. Pompea, “Assessment of black and spectrally selective surfaces for straylight reduction in telescope systems,” Proc. SPIE 7739, 773921-1 (2010) [doi:10.1117/12.858219].

14. S. Smith and J. Fleming, “BRDF measurements of a new IR blackcoating with lower reflectance,” Proc. SPIE 3426, 333–343 (1998) [doi:10.1117/12.328473].

15. “Anodic coatings for aluminum and aluminum alloys,” Military Standard (MILSTD) 8625F (2003).

16. J. Miller, “Multispectral infrared BRDF forward-scatter measurements of com-mon black surface preparations and materials,” Opt. Eng. 45(5), 056401 (2006)[doi: 10.1117/1.2203635].

17. ASAP Reference Guide, Breault Research Organization (2012).18. R. Willey, Et. Al., “Total reflectance properties of certain black coatings (from

0.2 to 20.0 micrometers),” Proc. SPIE 384, 19–26 (1980) [doi: 10.1117/12.934933].

19. A. Ames, “Z306 black paint measurements,” Proc. SPIE 1331, 299–304 (1990)[doi: 10.1117/12.22669].



Chapter 7

Ghost Reflections, ApertureDiffraction, and Diffraction fromDiffractive Optical ElementsIn addition to the stray light mechanisms discussed in Chapters 4–6, there areothers that can significantly affect the stray light performance of an optical system:ghost reflections, aperture diffraction, and diffraction and scattering from diffrac-tive optical elements (DOEs, used in hybrid optics). This chapter discusses thephysics of these mechanisms and multiple methods for modeling them. As withthe modeling methods previously discussed, the more accurate methods requiremore effort to implement.

7.1 Ghost Reflections

Ghost reflections are a common stray light mechanism, responsible for the promi-nent stray light artifacts shown in Fig. 1.1. Ghost reflections are specular and occurat the interface of a refractive optic, as shown in Fig. 1.7. Any optical system withat least one refractive element (even a flat window) will have ghost reflections be-cause there will always be some light reflected at a boundary between two refractiveindices. Although they can occur at any field angle, they normally occur inside andnear the FOV.

The number of second-order (double-bounce) ghost reflection paths N is equalto

N = 1

2(n2 − n), (7.1)

where n is the number of refractive surfaces. This equation shows that the numberof double-bounce ghost reflection paths increases roughly quadratically with thenumber of refractive optical surfaces. This fact leads to the general optical designguideline that, to improve stray light performance, the number of refractive opticsshould be minimized (see Section 8.8).

123



124 Chapter 7

The path transmittance �p of a ghost reflection path is proportional to theproduct of the reflectance �i of each surface at which a ghost reflection occurs:

�p ∝n∏

i=1

�i , (7.2)

where n is the number of ghost reflections in the path. The values of �i are typicallysmall, and therefore even small variability or uncertainty in � can drasticallychange the path transmittance. For instance, the path transmittance of a second-order ghost reflection path with no other losses and �1 = �2 = 0.01 is equal to1 × 10−4. However, if �1 changes to 0.015, then the new path transmittance isequal to 1.5 × 10−4, an increase of 50%. Therefore, it is important to be aware ofthis high sensitivity in path transmittance to small variations and/or uncertaintiesin reflectance, because it can result in large variations in the irradiance of ghostreflection artifacts from unit-to-unit and can make it difficult to obtain agreementbetween measured and modeled artifact irradiance.

A surface of a refractive element can be either uncoated or coated with anAR or bandpass coating, and its specular reflectance determines the magnitude ofthe ghost reflection path. Methods of determining the reflectance of either type ofsurface are discussed in this chapter.

7.1.1 Reflectance of uncoated and coated surfaces

7.1.1.1 Uncoated surfacesThe s-polarized and p-polarized reflectance (�s and � p, respectively) of an uncoatedinterface between two media (such as air and glass) as a function of the polarizationstate of the incident beam is given by the Fresnel equations as

�s =∣∣∣∣ni cos �i − nt cos �t

ni cos �i + nt cos �t

∣∣∣∣2

, (7.3)

� p =∣∣∣∣ni cos �t − nt cos �i

ni cos �t + nt cos �i

∣∣∣∣2

. (7.4)

The variables in these equations are defined in Fig. 1.7; �s and � p are plotted for airon glass (Schott N-BK7, ni = 1.0029, nt = 1.5151 at wavelength = 0.6328 �m)in Fig. 7.1. The angle at which the reflectance of a p-polarized beam drops to zerois called Brewster’s angle �B , and is given by

�B = tan1(

ni

nt

). (7.5)

�B for this case is equal to 56.57 deg. Transmittance is also determined by theFresnel equations:

�s = nt cos �t

ni cos �i

∣∣∣∣ 2ni cos �i

ni cos �i + nt cos �t

∣∣∣∣2

, (7.6)



Ghost Reflections, Aperture Diffraction, and Diffraction from DOEs 125

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 15 30 45 60 75 90

ρor

τ

Angle of Incidence θ i (degrees)

ρs ρp τs τp

Figure 7.1 A plot of s- and p-polarized reflectance (�s and � p) and transmittance (�s and�p) of an air/N-BK7 interface at 0.6328 �m.

�p = nt cos �t

ni cos �i

∣∣∣∣ 2ni cos �i

ni cos �t + nt cos �i

∣∣∣∣2

. (7.7)

�s and �p are plotted in Fig. 7.1 for air on glass. For unpolarized light, � =(�s + � p)/2, and � = (�s + �p)/2.

For dielectric (nonabsorbing) materials such as air and N-BK7, ni and nt arereal values, and � = 1 − � for both s- and p-polarization; n is complex for absorbingmaterials (such as metals), and conservation of energy requires that � = 1 − � − � ,where � is the absorption of the material. Except in cases when the flux is veryhigh,4 � is usually equal to the emissivity ε of the material. ε is discussed in thecontext of blackbody emission in Section 2.1.5.

All stray light analysis programs listed in Section 1.1 can compute the � and �for uncoated surfaces.

7.1.1.2 Coated surfacesRefractive surfaces are usually coated with AR coatings. However, no such coatingsare perfect, and therefore ghost reflections will occur. In addition to the refractiveelements in the system, the front surface of a solid-state detector will also producea ghost reflection and should therefore be AR coated to increase transmittance andreduce reflectance.

AR and other coatings (such as bandpass coatings) usually consist of multiplelayers of material whose composition and thicknesses are determined using thin-film analysis programs such as Essential MacLeod,1 TFCalc,2 and FilmStar.3 Thedescription of the layer compositions and thicknesses is called the stack definition.



126 Chapter 7

Table 7.1 Stack definition of a seven-layer AR coating,5 optimized for the 0.4–0.7 �mwaveband. TiO2+ZrO2 is a commercial mixture.5 The refractive indices are given for0.55 �m.

Layer Thickness (�m) Material Refractive Index

1 0.0959 MgF2 1.39

2 0.0345 TiO2+ZrO2 2.12

3 0.0124 Al2O3 1.65

4 0.0665 TiO2+ZrO2 2.12

5 0.0421 Al2O3 1.65

6 0.0088 TiO2+ZrO2 2.1

7 0.01 Al2O3 1.65

Substrate N-BK7 1.52

Most of these coatings operate on the principle of multiple-beam interference, andthere are a number of references that discuss their design.4 An example of a stackdefinition for a seven-layer AR interference coating5 for optics operating in thevisible waveband (0.4–0.7 �m) is shown in Table 7.1. The reflectance of this stackas a function of � and �i is shown in Fig. 7.2. This figure shows that the reflectanceof the AR coatings is a strong function of both.

In addition to interference coatings, another type of AR coating is a gradient-index (or graded-index) coating.6 In this type of coating, the refractive index ofthe coating increases, either continuously or discretely, with its proximity to thesubstrate. This variation in refractive index can be accomplished multiple ways,including by using a lithographic process to deposit subwavelength structures(“micro-” or “nanostructures”) whose fill factors are varied to change the effectiveindex.7 Nanostructured materials can be used to obtain a low refractive index inone or more of the layers in a coating stack, which can improve its performance.5

Most stray light analysis programs provide a variety of methods in which thisreflectance can be calculated, the most common of which is discussed next. As inChapters 4–6, these methods are presented in order of increasing complexity and(consequently) fidelity. It is usually not necessary to use any of these methods tocompute the reflectance of an uncoated surface, as this can be done simply by usingthe Fresnel equations (see Section 7.1.1) and the refractive index of the substrate;most stray light analysis programs can perform this calculation.

7.1.2 Reflectance from typical values

Sometimes there is little or no information available about a particular coating.This may be because it is early in the design phase and the coating has not been




0

0.005

0.01

0.015

0.02

0.025

0.03

0.4 0.45 0.5 0.55 0.6 0.65 0.7

Reec

tanc

e

Wavelength (μm)

ρs and ρp ( i = 0 deg) ρs (θi = 45 deg) ρp (θi = 45 deg)θ

(ρ)

(ρ)

Figure 7.2 Reflectance of the seven-layer AR coating whose stack definition is given inTable 7.1.

designed yet. Or perhaps no information about the coating (stack definition, orpredicted or measured reflectance) is available from the vendor. In this case, it maybe that the only way to model its reflectance is to make an educated guess basedon the reflectances of similar coatings. For this purpose, a list of typical coatingreflectances is given in Table 7.2. This table gives values for both low-performanceand high-performance coatings; the difference between them is usually that high-performance coatings use more layers and cost more. These reflectances are typicalof most standard coatings, averaged over a typical waveband (e.g., visible, MWIR)and averaged over a typical range of �i s (0–30 deg). Most coatings are designed fornormal incidence (�i = 0), and therefore their reflectance usually increases with�i . These values should be increased by at least a percent or two for �i > 30. The

Table 7.2 Typical coating reflectances.

Low HighCoating Type Performance Performance

AR 0.02 0.005

Bandpass (in-band) 0.1 0.05



128 Chapter 7

uncertainty that results from using these values can be estimated as the differencein reflectance between the value in the table and that of the “high-performance” ARcoating shown Fig. 7.2 (averaged over the waveband, and over three AOIs between0 deg and 30 deg). This difference is about 0.003, which is most likely the minimumerror associated with using values in Table 7.2.

7.1.3 Reflectance from the stack definition or predictedperformance data

Most stray light analysis programs will compute the reflectance of a surface basedon its stack definition; this is always a more accurate method of modeling reflectancethan using a constant reflectance value. However, most coating vendors treat thestack definition as proprietary, and therefore it can be difficult to obtain. Some thinfilm and optical design programs (such as FilmStar and Zemax, respectively) haveimplemented a method by which an encrypted coating stack description can betransferred in an encrypted form; however, at the time of publication, this methodis not available in any stray light analysis programs. Easier to obtain is tabulatedreflectance data for the coating (reflectance versus � versus �i ), which can be outputfrom the thin film design program and input into most stray light analysis programs.The disadvantage of using either the stack definition or the tabulated reflectancedata generated from it is that the manufacturing errors that occur in the fabricationof the coating are not captured in the calculation. The magnitude of this uncertaintyvaries with coating type, though for AR coatings it can be as much as 25% of thenominal reflectance value. A better way may be to use the measured reflectance ofthe as-built coating, which is discussed in the next chapter.

7.1.4 Reflectance from measured data

Just as it is possible to input tabulated reflectance data generated by a thin filmdesign program into a stray light analysis program, it is also possible to inputtabulated measured data from a spectrometer. Because it is generally not possibleto make this measurement on powered (i.e., curved) optics, the measurement isusually performed on a flat witness sample that was coated at the same time as thepowered optic. Although the coatings on flat and powered optics are not identical,the use of measured data from a flat will accurately represent coating reflectanceto within 0.005, which is accurate enough for most applications. Manufacturers ofwidely-used spectrometers include Bruker8 and Thermo Fisher Scientific.9

7.1.5 Artifacts from ghost reflections

Ghost reflection artifacts were simulated in the zoom lens system shown in Fig. 7.3.This system was imported into FRED from Zemax. Two AR coatings have beenapplied to the surfaces: one is a typical AR coating with a reflectance of 0.02,and the other is the 7-layer stack definition given in Table 7.1. This system has15 surfaces (including the detector), and therefore according to Eq. (7.1) has




Figure 7.3 Zoom lens used in ghost reflection simulation. EFL = 28.9774 mm, EPD =5.795 mm (f/5), and FOV = 22.637 deg × 15.214 deg. US Patent #4,936,661. Prescriptioncourtesy of Radiant Zemax LLC.

105 two-bounce ghost reflection paths, some of which are shown in Fig. 7.4. Theresulting artifacts for the sun just outside the FOV at 14 deg from its center areshown in Fig. 7.5.

As expected, the magnitude of the ghost reflection artifacts is lower using the7-layer AR coating than the coating with the reflectance of 0.02, as the maximumreflectance of the 7-layer coating is about 0.005 at low AOIs. This figure demon-strates the high variability in ghost artifact irradiance with changes in reflectancediscussed earlier. The bright artifact in the center of the FOV is due to the straylight path shown in Fig. 7.6.

Ghost reflections usually occur at field angles inside or close to the nominalFOV, as shown in the PST curve of the baseline Maksutov–Cassegrain system inFig. 9.14. Their artifact irradiance can be reduced by using AR coatings with lowerreflectance, or by redesigning the system so that the ghost reflection is not as well-focused on the focal plane. Most optical design programs have a feature that allowsthe irradiance on the focal plane due to a ghost reflection path to be computed,and these calculations can be incorporated into the optimization of the system toreduce the irradiance of ghost reflections.

Ghost paths can sometimes contain one or more TIR events. Because TIR doesnot reduce the flux of the beam, it results in a stray light path with low order andhigh flux. TIR paths can sometimes occur from the flat and unused portions oflenses (sometime called “bevels”) that are used to mount the lens and are locatedaround the lens’ clear aperture. In this case, baffles should be added to the system toprevent the flat from being illuminated, or the flat should be ground and/or paintedblack. TIR paths can also exist in a lens with a surface that has a small radius ofcurvature or in a lens of high index, such as germanium. These paths usually occur

Figure 7.4 Ghost reflections in the FRED model of the zoom lens shown in Fig. 7.3.



130 Chapter 7

(a)

Irradiance (ph/s-mm2)

3.00E12

0.00

1.50E12

2.25E12

0.75E12

(b)

Figure 7.5 Ghost reflection artifacts in the zoom lens shown in Fig. 7.4. The AR coatingused for artifacts in (a) is a typical AR coating with a reflectance of 0.02, and the coatingused for artifacts in (b) is the 7-layer stack shown in Table 7.1. The sun was located 14 degfrom the center of the FOV, just outside its lower left corner. The solar irradiance incidenton the system is 1.52 × 1015 ph/s-mm2. Simulation performed in FRED.

for sources outside the FOV. If such a path occurs, the lens should be shielded fromoff-axis illumination or redesigned to eliminate the path.

In IR systems, ghost reflections can increase internal stray light by reflectingemission from warm geometry back to the detector. This issue is discussed furtherin Chapter 8.7.




Figure 7.6 Ghost reflection path responsible for the bright artifact in the center of the FOVshown in Fig. 7.5.

7.1.6 “Reflective” ghosts

Ghost reflections, as they are defined here, occur only in systems with refractiveelements. However, multi-bounce paths in all-reflective systems are often called“ghosts” because they produce similar artifacts and occur at similar off-axis angles(i.e., near the FOV). An example of such a multi-bounce path is shown in Fig. 7.7.This path does not exist in the nominal baseline Maksutov–Cassegrain telescopeprescription, but it can be created by removing the baffles and decreasing the diam-eter of the hole in the primary from 28 mm to 10 mm. In this path, light from insidethe FOV undergoes a double reflection between the primary and secondary mirrors(primary-secondary-primary-secondary), and then goes to the focal plane. Becausethis path involves no stray light mechanisms, it is a zeroth-order path and thereforevery bright. Such paths can be detected using a detector FOV plot (see Chapter 3)and can eliminated by redesigning the system (see Chapter 8) or often by usingbaffles around the primary and/or secondary mirrors (see Chapter 9).

Figure 7.7 Reflective ghost path in a modified Maksutov–Cassegrain system. The diameterof the hole in the primary mirror was reduced from its nominal diameter of 28 mm to adiameter of 10 mm.



132 Chapter 7

7.2 Aperture Diffraction

Aperture diffraction causes artifacts in the final image that are not present in thescene. These artifacts are generally unwanted and are thus stray light. Aperturediffraction often appears as streaks of light near the image of bright sources ofnarrow angular extent, such as the photograph of a street light at night shown inFig. 1.3; aperture diffraction is responsible for the 14 streaks that radiate out fromthe image of the street light. This image was taken with a digital SLR camera witha 7-bladed iris. Section 7.2.1 explains how the number of streaks is related to thenumber of straight edges (or sides) of the aperture.

Aperture diffraction is well described by diffraction theory, which relates theelectric field at the focal plane of an optical system to the electric field at theentrance aperture. A comprehensive review of this theory is beyond the scope ofthis book; however, there are a number of good references.10,11 A brief reviewof this theory is presented here, along with a discussion of how this theory isused in stray light analysis programs to simulate the effects of aperture diffraction.This discussion is similar to that presented in Section 3.2.9, in which the effect ofstray light on MTF is analyzed. Computing the magnitude of aperture diffractionfor large AOIs or large diffraction angles is often very computationally intensiveto perform using the typical coherent beam propagation algorithms available inmost stray light analysis programs, and therefore asymptotic methods based onasymptotic expressions are discussed in Section 7.2.4.

7.2.1 Aperture diffraction theory

Scalar diffraction theory predicts that the irradiance at the image plane E f (u, v) ofa diffraction-limited system from an on-axis unit-irradiance monochromatic planewave is equal to the Fourier transform of the pupil transmittance function tA(x , y):

Ef (u, v) = 1

�2 f 2|� [tA (x, y)]|2fX =u/� f , fY =v/� f , (7.8)

where � is the Fourier transform. Figure 7.8 illustrates the geometry used in Eq.(7.8). E f (u, v) is often called the point spread function (PSF) of the system.

In general, tA(x , y) is a complex function. Its modulus squared is intensitytransmittance of the pupil (it is always zero outside of the clear aperture), andits complex phase is the wavefront error of the optical system. In a diffraction-limited system with 100% transmittance, tA is equal to 1 everywhere inside theclear aperture. For example, the PSF of a system with a circular pupil of diameterD illuminated with an on-axis plane wave whose irradiance is 1 W/mm2 is10

E f (r ) =(

�D2

4� f

)2 [2

J1(�Dr/

� f )

�Dr/

� f

], (7.9)

where r = (u2 + v2)1/2

, f is the focal length of the optical system, and J1 is aBessel function. An example of this function is plotted in Fig. 7.9.




y

x

u

v

f

tA(x,y)

Incident beam

Optical system

Image plane

Figure 7.8 Geometry used in aperture diffraction analysis.

Now that the relationship has been established between the pupil transmittancefunction and the irradiance distribution on the focal plane, methods of computingare discussed next.

7.2.2 Calculation of aperture diffraction in stray lightanalysis programs

As mentioned previously, Eq. (7.8) is useful to gain understanding of the relation-ship between the shape of the aperture stop and the resulting pattern of aperturediffraction. However, because the equation for tA can be very complicated for someoptical surfaces (such as aspheres), aperture diffraction is usually computed in

0.0E+00

5.0E+04

1.0E+05

1.5E+05

2.0E+05

2.5E+05

-0.01 -0.005 0 0.005 0.01

Irra

dian

ce (W

/mm

2 )

Position (mm)

Figure 7.9 The PSF of a system with a circular pupil. The wavelength of the collimatedincident beam is 0.6328 �m, its irradiance is 1 W/mm2, the circular aperture is 2 mm indiameter, and the focal length of the ideal optic is 10 mm (f/5).



134 Chapter 7

stray light analysis programs using coherent beam propagation algorithms. Typi-cally, this calculation is performed by representing the electric field as the sum ofa number of Gaussian beams, then tracing these beams through the optical systemand coherently summing the resulting set of beams at the focal plane. This method(sometimes called Gaussian beam decomposition12) can accurately model all ofthe effects described above, and converges to the result predicted by Eq. (7.8) fordiffraction-limited systems. The details of performing this type of coherent beamanalysis are beyond the scope of this book; the best source of information on howto perform these calculations is generally the software documentation.

7.2.3 Artifacts from aperture diffraction

Stray light analysis software was used to generate the images shown in Fig. 7.10. Theleft side of this figure shows three commonly used aperture stop shapes: 7-sided iris,8-sided iris, and circular. Multi-bladed irises are especially common in commercialSLR cameras. The right side of the figure shows the image plane irradiance thatresults from diffraction from these apertures, computed using Fourier transformtechniques. All of these irradiance patterns are plotted on the same grayscale, thecenters of which have been truncated to show the detail in the diffraction pattern.

Some points of interest in these images:� The irradiance of the diffraction streaks is about three orders of magnitude

less than the peak irradiance. This suggests that the streaks are noticeableonly near sources of high radiance that have a low-radiance background [thiscondition applies to street lights at night (see Fig. 1.3)].

� The diffraction pattern from the 7-bladed iris (top pattern) has 14 streaks init, which match the number and spatial distribution of the streaks seen in Fig.1.3 (the picture was taken by a camera with a 7-sided iris). The diffractionpattern has 2× more streaks than the aperture stop has sides because of thelack of symmetry in an aperture stop with an odd number of sides.

� The diffraction pattern from the 8-bladed iris (middle pattern) has eightstreaks, which have much higher irradiance than the streaks from the 7-bladed iris. The diffraction pattern has 8 streaks because the symmetry ofaperture stops with an even number of sides results in diffraction patternswith the same number of streaks. These streaks also have higher irradiancethan those of the 7-bladed iris because of the constructive interference thatoccurs in the diffraction from aperture stops with an even number of sides.

� The diffraction pattern from the circular aperture, which has no straightedges, has no streaks.

� The irradiance of the diffraction artifacts drops off rapidly as a function ofradial distance from the center of the image.

These results demonstrate that aperture diffraction theory can be used to ac-curately predict the irradiance distribution in an optical system, that the irradianceof the streaks due to aperture diffraction can be reduced by using an aperture stopwith an odd number of sides, that the streaks can be eliminated by using a circular




Aperture Stop Point-Spread FunctionIrradiance (W/mm2)

100

0

50

4 mm 0.1 mm

25

75

Figure 7.10 Common aperture shapes (top to bottom: 7-sided iris, 8-sided iris, and circular)and their resulting far-field irradiance distributions. The wavelength of the collimated incidentbeam is 0.6328 �m, its irradiance is 1 W/mm2, the circular aperture is 2 mm in diameter,the focal length of the ideal optic is 10 mm (f/5), and the focal plane is 0.1 mm in diameter.The same grayscale is used for all diffraction patterns; it has been truncated from its peakvalue (about 2.5 × 105 W/mm2) to show more detail. The plot in the lower right corner isshown in cross-section in Fig. 7.9.

aperture stop, and that aperture diffraction is primarily an in-field stray light effect,as the magnitude of the artifacts drops off rapidly with distance from the image.

7.2.4 Expressions for wide-angle diffraction calculations

As mentioned earlier, it is difficult to calculate the irradiance on the focal plane dueto diffraction from a beam that is either (a) incident on the aperture at an angle ofincidence much greater than the angular subtense of the Airy disk (1.22�/D, whereD is the diameter of the entrance pupil) or (b) at a large diffraction angle fromthe normal of the aperture. This is because the size of the image plane over whichthe diffraction calculation must be performed in these cases can be very large



136 Chapter 7

θi

θdD

Aperture

Figure 7.11 Quantities used in the definition of BDDF.

compared to the wavelength and therefore requires (depending on the coherentbeam propagation algorithm used) either a large number of points in the FFT gridor a large number of Gaussian beams to sample the plane adequately. In these cases,it is often easier to model diffraction from the aperture as a scattering phenomenonby using the bidirectional diffraction distribution function (BDDF).15 The BDDFof an unobscured circular aperture can be computed by normalizing the far-fielddiffraction pattern of the aperture by the irradiance incident on it:17

BDDF = D2

�2

[� J1(�D |sin �d − sin �i |

/�)

�D |sin �d − sin �i |/

�

]2

, (7.10)

where �d is the angle between the diffracted ray and the surface normal, and �i isthe AOI, as shown in Fig. 7.11.

Using the BDDF, diffraction from the aperture can be modeled in the same wayas any other BSDF and can therefore be plugged into a first-order stray light modelof the system [such as Eq. (2.47)] or into a stray light analysis program. Becausethis is a far-field approximation, it is valid only when computing the diffractedirradiance on a surface that is very far (>> 2D2/�) from the aperture or on the focalplane of a system where the diffracting aperture is or is near the entrance pupil.

Equation (7.10) is plotted in Fig. 7.12. This function is shift-invariant and istherefore a straight line when plotted vs. | sin �d − sin �i |. This function oscillatesat high frequency with | sin �d − sin �i | due to the Bessel function (J1), and theseoscillations (especially at large values of | sin �d − sin �i |) are often not observed inpractice due to averaging over the waveband and over detector pixel area. Therefore,it is often easier to model this function using the asymptotic expression15

BDDF = �

�3 D |sin �d − sin �i |3. (7.11)

This function is also plotted in Fig. 7.12. The slope of this function is –3, whichis generally much less (i.e., much steeper) than the slope of a BSDF function forother types of scatter, such as surface roughness scatter (which is usually between–2.5 and –1.5; see Section 4.1.1), indicating that diffraction can be the dominantstray light mechanism at scatter angles near the specular beam but is usually not atangles far from specular. It is possible to block wide-angle diffraction of this typewith a Lyot stop (see Section 8.4).




1.E-10

1.E-08

1.E-06

1.E-04

1.E-02

1.E+00

1.E+02

1.E+04

1.E+06

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00

BDD

F (1

/sr)

|sin θd – sin θ i|

PSF Based Asympto c

Figure 7.12 BDDF due to diffraction by a circular pupil, assuming a pupil diameter of 10.2mm and a wavelength of 0.6328 �m.

Figure 7.13 A germanium binary optic (image courtesy of Edmund Optics, Inc.).

Transition zone width w (typical ~0.005”) Zone height h

Figure 7.14 Cross-section of a DOE.



138 Chapter 7

7.3 Diffraction from Diffractive Optical Elements

DOEs and hybrid optics use a diffraction grating to focus light. DOEs are purelydiffractive elements, such as a circular grating that acts as a lens. Hybrid optics aretraditional lenses or mirrors with a DOE added to it, usually to perform color cor-rection. A picture of a hybrid germanium lens is shown in Fig. 7.13, and the crosssection of a kinoform DOE is shown in Fig. 7.14. In a kinoform DOE, thephase grating varies smoothly as a function of position. In a binary DOE, the phasegrating varies in discrete steps. Binary DOEs will not be discussed, though thereare good references.13

A detailed description of the design of diffractive optics is beyond the scopeof this book; however, there are a number of good references.13,14

7.3.1 DOE diffraction theory

The efficiency m of a DOE is the fraction of incident flux that gets diffracted into themth order. Different orders get diffracted at different angles, as shown in Fig. 7.15.

DOEs are usually designed to operate at a particular wavelength, AOI, andorder (usually +1), and they have a theoretical efficiency of 1 at this condition.However, light incident on them at any other � or �i will result in a nonzero valueof at other orders; these orders can couple stray light into the FOV. For a DOEoptimized for the +1 order, m is given by13

m =

⎧⎪⎪⎨⎪⎪⎩

sin{

�

(�0 cos �i0

� cos �i− m

)}

�

(�0 cos �i0

� cos �i− m

)⎫⎪⎪⎬⎪⎪⎭

2

, (7.12)

where �0 and �i0 are the wavelength and AOI that the DOE is optimized for, and �and �i are the wavelength and AOI of the incident beam. The height h of the DOE(as shown in Fig. 7.14) for 100% efficiency diffraction into the +1 order is equal to�0(cos �i0)/2. m for a DOE with �0 = 0.5876 �m and �i0 = 0 is plotted in Fig. 7.16.The figure shows that m drops rapidly as the difference between the order number

θiθ-1

θ0

θ1

Incident ray

Diffracted rays

Figure 7.15 Diffracted orders from a DOE.




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4 0.45 0.5 0.55 0.6 0.65 0.7

η

Wavelength ( m)

m = 1, θi = 0 m = 0, i = 0

m = -1, θi = 0 m = 1, θi = 25

θ

μ

Figure 7.16 Diffraction efficiency versus order m, wavelength, and AOI �i for a DOE de-signed for m = +1, � = 0.5876 �m, and �i = 0.

m and the design order (+1) increases. It also shows that, for a particular order,that the peak of the efficiency curve shifts � and �i . Equation (7.12) can be usedto generate a table of m versus � and �i , which can be input into most stray lightanalysis programs. Some programs allow only a table of m versus � to be defined;in this case, m should be computed as the average over the expected range of �i .

An example of a binary singlet lens is shown in Fig. 7.17. The glass in lensis Schott N-BK7, and it has a focal length of about 129 mm. A DOE, optimizedfor 0.5876 �m for +1 order, is on the back surface of the lens. A point from thecenter of the object at a finite conjugate on-axis source whose wavelengths spanthe visible waveband is traced through the lens, and the –1, 0, and +1 orders aretraced to the image plane.

Figure 7.17 Diffraction of the –1, 0, and +1 (design) orders in an achromatic singlet. Theinput ray bundle is collimated and monochromatic at 0.4861 �m. The diffractive is on theback (inside) surface of the lens (prescription courtesy of Radiant Zemax LLC).



140 Chapter 7

8 mm

Irradiance (W/mm2)

4.0

-1 order-1 order

0 order0 order

+1 order+1 order

3.0

2.0

1.0

0.0

Figure 7.18 Artifacts from DOE diffraction. The input beam is as shown in Fig. 7.17 andhas unit irradiance. Image generated in FRED.

7.3.2 Artifacts from DOE diffraction

Figure 7.18 shows the simulated artifacts from the ray trace of the achromaticsinglet shown on Fig. 7.17. As expected, the closer the order is to the design order(+1), the more focused it is and the higher its efficiency. As this figure suggests,diffraction from DOEs has a significant effect on the stray light performance of thesystem only at angles inside or near (usually within a few degrees of) the nominalFOV.

7.3.3 Scattering from DOE transition regions

A DOE consists of a number of zones with sharp transition regions between eachof them, as shown in Fig. 7.14. Because it is not possible to manufacture transitionzones that are infinitely thin, they will always have some finite width, typicallyabout 0.005′′ wide.

These regions are difficult to polish; they will generally have high surfaceroughness and will therefore scatter light and increase the BSDF of the DOEelement as a whole. An estimate of the increase in BSDF of the element due to




w

r1

r2

r

w

Figure 7.19 Geometry of DOE transition zones.

scattering from the transition zones BSDFtz can be computed as

BSDFtz = TIStz

�= (1 − PACtz )

�= 1

�

n∑i=1

r2i − (ri − w)2

r2, (7.13)

where TIStz is the total integrated scatter of the transition zones, which, as shown,is equal to one minus the percent area coverage of the zones. n is the number ofzones, ri is the radius of the i th zone, w is the width of the transition zone, andr is the radius of the optic, as shown in Fig. 7.19. This estimate assumes that thetransition zones are perfect Lambertian scatterers (TIS = 1 in the transition zoneregion), which, given that the transition zones are difficult to polish, is a reasonableassumption. This equation illustrates that, for the purpose of stray light control,the fewer and narrower the transition zones are (i.e., the smaller n and w are,respectively), the lower the BSDF of the surface.

Equation (7.13) can be used in a stray light analysis program to define theBSDF of a DOE due to transition zone scattering. This equation neglects thespatial distribution of the transition zones, which may be important in cases suchas when the DOE is only partially illuminated due to shadowing by a baffle or otherstructure. In this case, some of the transition zones are illuminated and some arenot, which can significantly affect the amount of light scattered by them. This effectcan be accounted for by modeling the transition zones as flat annular surfaces onthe DOE (as shown in Fig. 7.19) and making these surfaces Lambertian scatterers.The TIS of the DOE (averaged over its aperture) modeled in this way will be thesame as that predicted by Eq. (7.13).



142 Chapter 7

7.4 Summary

Ghost reflections (also called “lens flare” in photography) occur in optical systemswith one or more refractive elements. They typically occur for sources inside ornear the nominal FOV. The number of second-order ghost reflection paths increasesquadratically with the number of refractive surfaces. Ghost reflections can be re-duced by using AR coatings or by redesigning the optical system so that ghostreflections are not well focused on the focal plane. There are many ways to modelthe reflectance of these coatings; typical values are given in Table 7.2, though thesevalues neglect variations in wavelength and AOI. More-accurate models can be con-structed using the thin film stack definition or, if not available, tabulated reflectancedata. The irradiance of ghost reflection artifacts is very sensitive to small uncer-tainties or variations in AR coating reflectance. A detector FOV calculation shouldbe performed for reflective systems to identify “reflective ghosts,” which are multi-bounce paths from reflective optics and can result in artifacts with high irradiance.

Diffraction from the aperture stop of an optical system can result in streaksif the stop has straight edges. Aperture diffraction from sources outside the FOVcan be modeled as a BDDF and can contribute to stray light at the focal plane,especially for sources near the edge of the FOV. A Lyot stop can be used to reducesuch diffraction (see Section 8.4).

Diffractive optical elements (DOEs) are often used in optical systems to correctchromatic aberration. Theoretically, they have 100% diffraction efficiency at onlyone combination of wavelength and AOI; at other combinations they will diffractlight into multiple orders, which can create stray light at the focal plane. Theefficiency of these orders can be calculated using the diffraction efficiency equationand can be modeled in stray light analysis software. In addition, scattering can occurfrom the rounded edges between the zones of the grating and increase the BSDF ofthe DOE. As a result, reducing the number of zones in the DOE lowers the BSDFof the DOE surface.

References

1. Thin Film Center Inc., Essential Macleod software, http://thinfilmcenter.com.2. Software Spectra Inc., TFCalc software, http://sspectra.com.3. FTG Software Associates, FilmStar software, http://ftgsoftware.com.4. A. MacLeod, Thin-Film Optical Filters, 3rd Ed., John Wiley & Sons, New York

(2001).5. T. Murata, H. Ishizawa, and A. Tanaka, “High-performance antireflective coat-

ings with a porous nanoparticle layer for visible wavelengths,” Appl. Opt. 50(9),C403-C407 (2011).

6. W. Southwell, “Gradient-index antireflection coatings,” Opt. Let. 8(11), 584–586 (1983).

7. D. Hobbs and B. Macleod, “Design, fabrication, and measured performance ofanti-reflecting surface textures in infrared transmitting materials,” Proc. SPIE5786, 349–364 (2005) [doi: 10.1117/12.604532].



Ghost Reflections, Aperture Diffraction, and Diffraction from Diffractive Optical. . . 143

8. Bruker Corporation, http://www.bruker.com.9. Thermo Fisher Scientific, http://www.thermoscientific.com.

10. J. Goodman, Introduction to Fourier Optics, 3rd Ed., Roberts & Company,Englewood, CO (2005).


12. A. Greynolds, “Propagation of generally sstigmatic gaussian beams along skewray paths,” Proc. SPIE 560, 33–50 (1986) [doi: 10.1117/12.949614].

13. D. O’Shea, T. Suleski, A. Kathman, and D. Prather, Diffractive Optics:Design, Fabrication, and Test, SPIE Press, Bellingham, WA (2003) [doi:10.1117/3.527861].

14. S. Sparrold and G. Forman, “Hybrid optical components deliver benefits forsystem design,” Laser Focus World Magazine, (Dec 2011).

15. E. Freniere, R. Stern, and J. Howard, “SOAR: a program for rapid calcula-tion of stray light on the IBM PC,” Proc. SPIE 1331, 107–117 (1990) [doi:10.1117/12.22654].

16. M. Caldwell and P. Gray, “Application of a generalized diffraction analysis tothe design on nonstandard Lyot-stop systems for earth limb radiometers,” Opt.Eng. 36(10), 2793–2808 (1997) [doi: 10.1117/1.601506].

17. “Stray Light Short Course Notes,” Photon Engineering LLC (2011), used withpermission.



Chapter 8

Optical Design for StrayLight ControlTypically, the goal of the optical design is to meet the image quality, size, weight, andcost requirements over the desired waveband, aperture, and FOV. It is also very im-portant to consider the system stray light requirements and choose an optical designform that will meet them. This chapter shows that not all design forms have equalstray light performance, and some stray light requirements simply cannot be metwith some optical design forms, regardless of how many black surface treatmentsand baffles are added to the system. This chapter discusses methods for controllingstray light in the optical design form, and the pros and cons associated with each.Because it is often difficult to say whether or not one method is better than another(all of them involve trade-offs), they are presented here in no particular order. Meth-ods used to control external stray light are also useful in controlling internal straylight, and therefore the applicability of each method to both external and internalstray light control is discussed. Nonuniformity compensation and reflective warmshields, which are used to internal stray light only, are discussed in Section 8.14.Not all of the methods discussed are appropriate for all systems; for instance, spaceconstraints may make it impossible to add a field stop. Determining which methodsare appropriate requires balancing the system stray light requirement with all otherrequirements.

8.1 Use a Field Stop

An intermediate field stop is an aperture in the optical system at an intermediateimage that prevents light from outside the FOV from reaching the focal plane.In general, the field stop should be placed as far forward in the optical train aspossible, which will minimize the number of illuminated surfaces in the system.An example of a system with a field stop is the modified Maksutov–Caessegraintelescope shown in Fig. 8.1. This system has been modified from the baseline byreplacing the detector with a field stop and reimaging it with a singlet lens. Themodification makes the system longer but allows the baffles to be eliminated, asthey are no longer needed to block the zeroth-order path through the hole in theprimary mirror. This improves the stray light performance of the system because

145



146 Chapter 8

DetectorField stop Reimaging lensAperture Stop

Figure 8.1 The Maksutov–Caessegrain telescope with a field stop and reimaging lensadded.

the baffles were both illuminated and critical. The improvement can be seen in thesolar source transmittance (SST) curves for the system with and without a fieldstop, as shown in Fig. 8.2. Adding a field stop may not completely eliminate theneed for baffles in all systems; for instance, the main baffle in Fig. 8.1 may still benecessary to prevent illumination of the primary mirror.

The effectiveness of the field stop is partially determined by how closely itsaperture fits the intermediate image. In general, this means that the aberrations(especially field curvature) at the intermediate image must be well controlled.

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

0 10 20 30 40 50 60 70 80

SST

Angle of sun from center of FOV (deg)

without field stop with field stop

Figure 8.2 SST for the Maksutov–Cassegrain system with and without a field stop (Figs.8.1 and 1.5, respectively). The peak in the SST at about 25 deg for the system without afield stop is due to scattering from the inside of the primary mirror baffle.



Optical Design for Stray Light Control 147

Also, the tighter the manufacturing and alignment tolerances are on the field stop,the better it can fit the intermediate image, and the more effective it is. Typicalmanufacturing and alignment tolerances require that the field stop aperture beoversized by 0.005′′ (0.127 mm) in order to not vignette the beam. The rules forsizing the edge of the field stop aperture are the same as those for baffle vanes,which are discussed in Section 9.2. In systems that require very good stray lightcontrol, undersizing the field stop so that it vignettes the edges of the field but isguaranteed to block all zeroth-order paths is an acceptable trade-off.

A field stop is typically not the most effective means of reducing internalstray light in an IR system (a cold stop is usually more effective, see Section 8.4);however, it can be useful if it prevents critical surfaces or the detector from beingilluminated by self-emission from internal structures.

The use of a field stop is one of the most effective ways to reduce externalstray light in an optical system, but it requires that the system form an intermediateimage, which therefore increases the number of optical elements in the systemover one that has no field stop. The resulting system will be longer and have ahigher cost. Systems that have very strict stray light requirements, such as manymilitary, astronomical, or space-borne systems,1 typically require field stops. Moreinformation on the use of field stops can be found in Smith (2008).2 The lensadded to the system in Fig. 8.1 also reimages the primary mirror (which wasthe aperture stop) to a new aperture stop surface that is immediately in front ofthe detector. This is called a reimaged pupil (sometimes called a relayed pupil)design, and it also improves its stray light performance, as discussed in Sections8.3–8.5.

8.2 Use an Unobscured Optical Design

A central obscuration, such as the secondary mirror in a Cassegrain telescope,often increases both external and internal stray light in the optical system be-cause it increases the number of critical surfaces in the system. For instance, thesecondary in a Cassegrain system usually requires struts to support it, and someportion of these struts will always be critical. In addition, such designs also oftenrequire the use of baffles to block low-order stray light paths, such as those in thebaseline Maksutov–Cassegrain system (see Fig. 1.5), and at least some portion ofthese baffles is usually critical. For this reason, unobscured optical designs (such asthe off-axis three-mirror anastigmat shown in Fig. 8.3) typically have better straylight performance than obscured designs. If an obscured design is necessary forimage quality, packaging, or other requirements, there are techniques that can beused to improve its stray light performance:

� If struts are used, add a bevel to their sides so the sides cannot be seen by thedetector (see Section 8.11).

� Use a main baffle to prevent critical areas of the obscuration or struts frombeing illuminated (see Section 9.1).

� Use a mask placed at an image of the central obscuration or struts to blockdiffraction, scatter, and emission from them (see Section 8.5).



148 Chapter 8

Field stop

Secondary

Primary

Tertiary

Aperture stop

Focal plane

Figure 8.3 TMA design with the aperture stop directly in front of the image plane (US Patent#4,265,510).

8.3 Minimize the Number of Optical Elements between theAperture Stop and the Focal Plane

Minimizing the number of optical elements between the aperture stop or its con-jugate and the focal plane reduces the number of critical surfaces in the systemand can therefore improve stray light performance. All objects preceding the stopin the optical path will not be critical unless they are imaging elements, centralobscurations, or objects that vignette the FOV.

For example, the first-order scatter path from the inner diameter of the primarymirror baffle in the baseline Maksutov–Cassegrain telescope (shown in Fig. 9.16)would be blocked if the stop were moved from the primary mirror to directly infront of the focal plane. Another example of a path that would be blocked is theoverviewing path shown in Fig. 8.4. Overviewing occurs when a portion of thedetector corresponding to a particular field point views an object via a portion ofan optical element that corresponds to a different field point. This phenomenonis illustrated in Fig. 8.4. Rays are traced forward and backward at the edge of thefield, and some of the rays that are traced backward reflect off of the portion ofthe secondary (the lower portion in Fig. 8.4) that is intended to be used by therays at the opposite edge of the field. These rays miss the primary mirror and hitthe inner diameter of the main baffle, making it critical. Moving the stop fromthe primary mirror to another surface further down the intended optical path will




Backwards ray trace fromsame field point (black rays)Overviewing occurs from the

portion of the secondary notused at this field

Forward ray trace atedge of field (grey rays)

Primary mirroris the stop

Figure 8.4 Overviewing in the baseline Maksutov–Cassegrain telescope.

eliminate this overviewing path. For instance, if the secondary mirror were the stop,the beam footprints for all field points would coincide there, and this path wouldbe eliminated.

Moving the aperture stop closer to the focal plane can increase the size andcomplexity (and therefore the cost) of the optical system. For instance, moving thestop from the primary mirror to the secondary mirror in the baseline Maksutov–Cassegrain telescope requires that the diameter of the primary mirror increase sothat it does not vignette. One way to avoid this problem is to use a reimaged pupildesign, such as the one shown in Fig. 8.1. In this design, a lens reimages theprimary mirror to the aperture stop, which is located in front of the focal plane.This improves the stray light performance of the system while keeping the diameterof the primary as small as possible. The TMA design shown in Fig. 8.3 is anotherexample of such a design. Reimaged pupil designs are commonly used in systemsthat require good stray light control (see Lyot stops in Section 8.4 and pupil masksin Section 8.5).

Moving the stop closer to the focal plane also prohibits the use of stop-symmetryto improve image quality and therefore may increase the number of optical ele-ments required. For instance, the classical double-Gauss optical design is roughlysymmetric about the aperture stop, and this symmetry improves its image quality.If the stop were moved closer to the focal plane, this symmetry would be broken,and optical elements would need to be added to maintain its image quality. Movingthe stop can also increase the magnification between the entrance and exit pupils,thus making the system more sensitive to alignment tolerances.

Making the aperture stop the last element in the optical path before the detectorand keeping it cold by incorporating it into the cryogenically cooled dewar assemblyis one of the primary methods to reduce internal stray light. In many systems it



150 Chapter 8

is necessary to do this in order to lower the internal stray light enough so thatthe light from the scene can be detected (the LWIR camera shown in Fig. 3.11is an example of such a system). An aperture stop used in this way is called acold stop, and it is typically the entrance aperture into a cryogenically cooled baffleassembly called the cold shield.6,7 The cold shield prevents the detector from seeingthe warm geometry that surrounds it and usually has vanes to prevent first-orderscatter from reaching the detector. (The LWIR camera shown in Fig. 3.11 has acold shield whose design is discussed in Section 9.2.2.) Lyot stops (discussed inthe next section) are also often used in a similar way, the difference being that theaperture of the Lyot stop is sized precisely to block diffraction from the entrancepupil.

8.4 Use a Lyot Stop

Diffraction from the entrance pupil can increase stray light on the focal plane due toout-of-field sources. As shown in Section 7.2.4, diffraction increases dramaticallyas the field angle gets smaller. If it is necessary for the optical system to have goodstray light performance at these angles (for instance, if the exclusion angle is verysmall), then it may be necessary to block the diffraction with a Lyot stop (alsosometimes called a glare stop).8 Lyot stops are placed at an image of the pupil andare slightly undersized relative to the nominal size of the pupil image, as shown inFig. 8.5. In this figure, light is incident on the entrance pupil at an angle �i that isoutside the nominal FOV of the system and is therefore blocked by the field stop.8

However, diffraction from the entrance pupil can propagate through the field stopand reach the detector. At a subsequent pupil plane, this diffracted light is refocusedto produce a bright ring at the geometrical image of the entrance pupil. A Lyot stopcan be placed at this plane and, if slightly undersized relative to the image of the en-trance pupil, can block some of this diffracted light. Not all of the diffracted light isblocked by the Lyot stop; diffraction occurs again at the field and Lyot stops (second-order diffraction) but can be reduced further by adding another “stage” consisting

Entrance pupil

Primary imaging optical system

Field stop

Reimaging optical system

Lyot stop

Detector

Diffraction pattern

Ring-shaped diffraction pattern

θi

Diffracted raysUndiffracted rays

Figure 8.5 Lyot stop geometry.8 The incident angle �i is outside the system FOV.




of a slightly undersized field stop and Lyot stop.9 The stop at the back of the systemin the telescope with the reimaged pupil shown in Fig. 8.1 could also be a Lyot stop.Two methods are discussed later in this chapter for computing the improvement instray light performance due to the addition of a Lyot stop: the first uses an analyticapproximation, and the second uses coherent beam propagation in stray lightanalysis software.

As with a field stop, the use of a Lyot stop requires that the optical systemform an intermediate image (of the entrance pupil rather than the focal plane) thatis accessible, and therefore a system with a Lyot stop may be longer and havemore elements than one without. Also, because the Lyot stop is smaller than thenominal stop, adding a Lyot stop increases the system f/# slightly. Therefore, inorder to keep the f/# constant, the nominal stop must be oversized slightly so thatthe desired f/# is obtained when the Lyot stop is added.

If the apparent radiance of the Lyot stop is lower than that of the nominalaperture stop, then it can reduce internal stray light in IR systems by blockingradiance from the stop. Lowering the apparent radiance of the Lyot stop can be donein a number of ways, typically either by making it highly emissive (black) and coldor by making it highly reflective (often by gold-coating it) and positioning it so thatit reflects cold geometry back to the detector. Lyot stops used in these configurationsare called (in the former case) cold stops or (in the latter) reflective warm stops;they are often placed close to the detector, either as part of the cryogenically cooleddewar/cold shield assembly or close to it to provide reflection. Such stops are oftenused in reflective telescopes in which the (warm) primary mirror is the nominalstop of the system. Using a Lyot stop in this way is similar to putting the nominalaperture stop close to the detector and making it cold, as described in Section 8.3.

8.4.1 Calculating Lyot stop diameter from analytic expressions

An approximate expression for the ratio R of diffracted flux that reaches the focalplane with the Lyot stop present to the flux with no Lyot stop present has beenderived for a system with a circular stop as4

R = 2

[�kr�(1 − �2)]2, (8.1)

where k = 2�/�, r is the nominal radius of the Lyot stop (i.e., the radius of thepupil image), � is equal to sin (FOV /2), and � = 1 − (a/r ), where a is the amountby which the radius of the Lyot stop is undersized. This equation assumes thatthe aperture stop is circular, that pupil aberrations are small, that the aperture stopis being illuminated at an angle �i >> FOV/2 (which ensures that the angularvariation in diffracted flux inside the FOV is negligible), that the central lobe of theAiry disk (1.22�/D) is much greater than �, and that a/r << 1. These dimensionsare shown in Fig. 8.6. R is inversely dependent on �, and therefore Lyot stops aremore effective at longer wavelengths. The optimal value of R is typically whena/r < 0.05. R versus a/r for a typical system is shown in Fig. 8.7. Because thediameter of the Airy disk increases with wavelength, the effectiveness of the Lyot



152 Chapter 8

Nominal stop radius r

Optical axis

Lyot stop

Undersize amount aRing-shaped

diffraction pattern (only one side shown)

Figure 8.6 Lyot stop geometry.

stop also increases with wavelength. Other expressions for R have been derived10

that do not assume �i >> FOV/2.

8.4.2 Calculating Lyot stop diameter from coherent beam analysis

The analytic method of sizing the Lyot stop discussed previously requires manyassumptions, such as that the system has a circular aperture. Of course, theseassumptions are not valid for all systems, and in these cases it may be necessaryto use other methods. A much more general method is to use a stray light analysisprogram to optimize the size of the Lyot stop, which can be done using the aperturediffraction analysis techniques discussed in Section 7.2.

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

1.0E-03

1.2E-03

1.4E-03

0 0.02 0.04 0.06 0.08 0.1

R

a/r

0.6328 μm 1.06 μm

Figure 8.7 The ratio R of diffracted flux at the focal plane for a system with a Lyot stop tothe diffracted flux at the focal plane for the same system without a Lyot stop, as a functionof the ratio of the Lyot stop undersize a to the nominal stop radius r (see Fig. 8.6). The FOVof this system is 1 deg, and the nominal stop radius is 12 mm.




8.5 Use a Pupil Mask to Block Diffraction and Scattering fromStruts and Other Obscurations

Just as the Lyot stop can be used at an image of the aperture stop to block diffractionand scattering from it, so can a mask be used at the image of the struts or otherobscuration to block stray light. For instance, a circular mask can be placed at theaperture stop of a telescope with a reimaged pupil to block scattering from thesecondary obscuration and reflections from the unused portion of the secondary;such a mask is shown in Fig. 8.8. The mask shown is painted onto the back of awindow at the aperture stop. Notice that the mask exactly fills the hole that the raybundle created by the secondary mirror, which prevents the detector from seeingthe unused portion of the secondary mirror. In practice, a system with such a maskmay be difficult to assemble because the mask must be precisely aligned with theobscuration. In systems with struts, similar masks can be used to block scatteringand diffraction. The mask shown in Fig. 8.8 is at the pupil; however, the image ofthe struts and other obscurations may not be exactly at this plane, and therefore themask may need to be oversized, which would reduce the collection efficiency of thesystem. The mask also may need to be oversized in order to account for fabricationand alignment tolerances.

If the mask has lower apparent radiance than the struts or other obscura-tions, then it can significantly reduce internal stray light in IR systems. Theapparent radiance of the mask can be reduced using the methods discussedin Section 8.4. These masks are commonly used in Cassegrain-type telescopesto block emitted and reflected radiance from the secondary obscuration; such

Field stop Reimaging lens DetectorAperture Stop

w/Mask

Aperture Stop

Mask

Figure 8.8 A mask used to block stray light from the obscuration in the Maksutov-Cassegrain system with a reimaged pupil. This mask is created by applying paint to theback of the window.



154 Chapter 8

masks are often implemented using a gold coating and, as such, are called golddots.

8.6 Minimize Illumination of the Aperture Stop

The aperture stop is always a critical surface, and therefore it is important tominimize illumination of it. In some systems, such as the TMA system shown inFig. 8.3, this is not a problem, as the back surface of the aperture stop is criticalbut this surface cannot be easily illuminated. However, it may be a problem in thebaseline Maksutov–Cassegrain system because the primary mirror is the aperturestop, and therefore the region around it (such as the outside edge of the primary andthe inner diameter of the barrel near it) is critical due to overviewing (see Section8.3). If this region is illuminated, then it will scatter directly to the focal plane.This problem can be avoided by making the aperture stop a surface that is moredifficult to illuminate, such as the secondary mirror. However, this solution requiresan increase in the diameter of the primary mirror to avoid vignetting, which wouldincrease the size and cost of the system.

8.7 Minimize the Number of Optical Elements, EspeciallyRefractive Elements

The more optical elements in the system, the more stray light mechanisms that canoccur that could adversely affect stray light performance. As mentioned in Section7.1, the number of second-order stray light paths due to any pair of stray lightmechanisms (such as ghost–ghost paths or roughness scatter–roughness scatterpaths) is equal to 0.5(n2 − n), where n is the number of surfaces. Therefore, thestray light performance of a system degrades roughly as the square of the number ofoptical surfaces it has. Of course, in well-designed systems, increasing the numberof optical elements improves image quality, and therefore a trade-off may needto be made between image quality and stray light performance. It may also benecessary to make a trade-off between the cost of the AR coatings and stray lightperformance because the more layers the coating has, the lower its reflectance canbe made, but the higher its cost.

In general, refractive elements create more external stray light than reflectivebecause they have two stray light mechanisms that reflective optics do not: ghostreflections and bulk scatter (although bulk scatter is usually very low in mostmodern refractive optics). Refractive elements can also increase internal stray lightin IR systems because they may have higher emission than reflective elements andbecause they usually reflect emission from warm geometry back to the detector,even if they are AR coated (methods to compute internal stray light due to ghostreflections are discussed in Section 3.2.6). It may be possible to mitigate theseproblems by placing refractive optics close to or inside the dewar/cold shieldassembly, which can reduce emission from them. This action can also lower internalstray light by causing the detector to see cold geometry (such as itself) in reflectionfrom the optic instead of warm geometry, which is the reason to place bandpass




filters (which can have high reflectance, on the order of 10%) close to or insidethe cold shield, as was done in the LWIR camera design shown in Fig. 3.11. Thisdesign option is discussed further in Section 8.12. However, since it is generallyimpractical to cool all of the refractive optics in the system and/or place themnext to the cold shield, reflective optics are generally preferred over refractive forthe purpose of controlling stray light. Because refractive systems are often morecompact and have larger FOVs than reflective, it may be necessary to balance thestray light performance of the system with its size and/or image quality.

8.8 Avoid Optical Elements at Intermediate Images

Surface imperfections such as scratches, digs, and particulates on an optical elementat or near an intermediate image will get reimaged at the focal plane. Theseimperfections can increase the magnitude of stray light on the focal plane andcreate persistent and undesirable image artifacts, and therefore optical elements(such as windows) should not be located at intermediate images.

8.9 Avoid Ghosts Focused at the Focal Plane

Ghost reflections that are focused at the focal plane result in artifacts with highirradiance and thus can be problematic. An example of such a ghost path is shownin Fig. 8.9. Light at the corner of the field ghost reflects off of the solid state detectorand then again off of a lens surface whose radius of curvature is concentric withit, which focuses it on the opposite side of the detector. Thus, this path results inwell-focused ghost images of point sources and can therefor be problematic.

It is possible for ghost reflection paths from surfaces not concentric with thefocal plane to still be focused on it, and therefore the focus positions of all ghostpaths should be evaluated. If they occur near or at the focal plane such that the stray

Surface with radius of curvature centered on detector

Detector

Figure 8.9 Telephoto system with a surface concentric to the detector. Light from the edgeof the field (upper ray fan) ghost reflects from the front surface of the detector, then ghostreflects from the back surface of the second lens and is refocused on the detector (lowerray fan).



156 Chapter 8

light requirement is violated (see Section 11.1), the system should be redesignedso that such paths do not occur. Most optical design programs like CODE V andZemax can calculate the irradiance at the focal plane of in-field ghost paths basedon a paraxial raytrace; this feature should be used as early in the design process aspossible to identify such ghost paths and, if necessary, redesign the system.

Paths such as these are responsible for narcissus in IR systems and are gener-ally not desirable, though they may sometimes be used to lower the backgroundirradiance. This issue is discussed in more detail in Section 8.12.

8.10 Minimize Vignetting, Including the Projected SolidAngle of Struts

Surfaces that vignette are critical surfaces, are often illuminated, and have highemissivity; therefore, reducing or eliminating vignetting can decrease stray light.

Many classical telescope designs require the use of struts to hold the secondarymirror in front of the primary mirror, as shown for the Cassegrain system inFig. 8.10. Because the struts are in the FOV, they always have some surfaces thatare critical and can therefore degrade the stray light performance of the system.There are a number of ways to mitigate this problem; one is to prevent the strutsfrom ever being illuminated. However, this is often difficult or impractical becauseit may require a very long main baffle to do so (see Section 9.1). Another techniqueis to minimize the projected solid angle of the struts as seen by the focal plane bymaking them as thin as possible and by beveling their sides9 at an angle of FOV/2(plus slightly more for tolerances), as shown in Fig. 8.10.

Struts are usually warm, have high emissivity, and are in the FOV. Therefore,they can be significant contributors to internal stray light. Minimizing their

Secondary mirror

Strut

y

z

xy (out of page)

Rectangular (not ideal)

x

Beveled (ideal)

Sides

z (optic axis)

z

Angle=FOV/2

x (into page)

Primarymirror Strut Profiles

Figure 8.10 Beveling the sides of struts in a Cassegrain system to prevent them from beingcritical.




projected solid angle, as shown in Fig. 8.10, can reduce the internal stray lightfrom struts.

8.11 Use Temporal, Spectral, or Polarization Filters

In some cases, it may be possible to exploit some difference between the light fromthe intended object or scene and the light from a stray light source to filter out thestray light. Some examples include the following:

� Temporal filtering: LIDAR systems actively illuminate their targets withcoherent light. This light can be pulsed temporally, and a lock-in amplifiercan then be used to separate the desired signal from a stray light signal thathas different temporal characteristics, such as light from the sun, which iscontinuous.

� Spectral filtering: Some MWIR systems use InSb detectors, which are sen-sitive from 0.6–5.5 �m. These systems are often designed to look at objectsin the 3–5 �m band, which corresponds to a spectral region of the atmo-sphere with high transmittance. Therefore, light from 0.6–3 �m is generallynot useful. However, the sun has high flux in this band; by using a bandpassfilter to block it, the stray light performance of the system can be improved.

� Polarization filtering: LIDAR systems illuminate their targets with coherentlight, which is often strongly linearly polarized. In this case, a polarizer canbe used at the detector to pass the light from the illuminator but block someor all of the light from stray light sources that are polarized differently, suchas light from the sun, which is unpolarized.

As with external stray light, internal stray light can be reduced if it has a tempo-ral, spectral, or polarization characteristic that is different from the desired signal.Internal stray light is usually continuous (not pulsed), broadband, and unpolarized.If the desired signal is different in any way, then perhaps some of the internal straylight can be filtered. In practice, the most common filtering that is used to reduceinternal stray light is spectral filtering, in which a bandpass filter is placed in frontof the detector to transmit only the waveband of interest, thus blocking light frombroadband external stray light sources (such as the sun) and internal sources.

8.12 Use Nonuniformity Compensation and Reflective WarmShields in IR Systems

Systems working in the IR must be designed to control internal stray light, whichresults from in-band emission of the sensor itself.6,7 If the irradiance at the focalplane due to internal stray light is a significant fraction of (or larger than) theirradiance from the scene, then it can reduce sensitivity to or completely obscurethe scene irradiance. As discussed in Section 8.3, most IR systems use a cold stop tolower the irradiance from internal stray light below the level of the scene irradiance.However, this does not eliminate internal stray light; at the very least, emission fromwarm optical surfaces typically remains, and this emission often results in a pattern



158 Chapter 8

of irradiance on the focal plane that is spatially nonuniform. This pattern candistract from or obscure the scene and is therefore not desirable. Most modern IRsystems compensate for this pattern using a nonuniformity compensation (NUC)algorithm, in which the image of internal stray light is recorded by looking at auniform scene and then digitally subtracting this image from all subsequent imagesof the scene. This works well if none of the components of the sensor moves orchanges temperature. However, it is impossible to eliminate temperature changescompletely, and thus the irradiance on the focal plane due to internal stray lightwill change after recording the NUC image—subtracting the NUC image may notremove the effects of internal stray light. This means that the NUC operation mayneed to be performed again, which can interfere with the operation of the sensor; forinstance, performing the NUC algorithm can temporarily blind the sensor, whichmay be unacceptable for some applications.

The dominant contributor to nonuniformity is often narcissus, in which thecold detector sees itself or other cryogenically cooled components via stray lightmechanisms (such as a direct reflection or ghost reflections) from one or more op-tical components. Narcissus often results in a large variation in uniformity becauseone portion of the detector may see mostly cold geometry (sometimes cryogeni-cally cooled to 77 K) in reflection, whereas another portion may see mostly warmgeometry (usually at ambient temperature ∼300 K). As described earlier, this pat-tern can be removed using a NUC algorithm; however, temperature changes inthe sensor and the movement of any components for such purposes as scanning,zooming, or focusing can cause the irradiance pattern to change. Thus, quantifyingand reducing the magnitude of the narcissus effect can be important in the design ofthe system. Narcissus can be reduced as a part of the optical design process, usuallyby reducing the concentricity of refractive optical surfaces with the focal plane.

An example of a narcissus path is shown in the ray trace in Fig. 8.11 (which isthe same camera shown in Fig. 3.11). Emitted light from the detector propagatesthrough the system and ghost reflects off of the front (outside) surface of theobjective lens and goes back to the detector. These reflections typically occur fromsurfaces that are perpendicular to the optical axis, such as flat windows or theportion of lens and mirror surfaces near the optical axis. Many such paths exist

Ghost reflection

Field stop

Cold stop

Detector

Figure 8.11 A narcissus path in the LWIR camera shown in Fig. 3.11.




Irradiance Difference (ph/s-mm2)

-5.69E5

-7.80E5

-6.75E5

-7.27E5

-6.22E5

Figure 8.12 Change in irradiance due to narcissus and a change in detector temperaturefrom 77 K to 76 K.

in the sensor shown in Fig. 3.11, and the resulting change in detector irradiancedue to a 1-K change in the detector temperature (from 77 K to 76 K) is shown inFig. 8.12. The pattern is lower at the center than at the edges because the detectorhas a larger projected solid angle to itself in its center; it is therefore more affectedby the change in temperature. The magnitude of this nonuniformity pattern can bereduced by applying better AR coatings to the optics (the system shown in Fig.3.11 has coatings with a reflectance of 0.02).

In some cases, narcissus is intentionally created in order to prevent the detectorfrom seeing warm, highly emissive geometry. Such is the case when a bandpass isplaced close to or inside the cold shield (see Section 8.7). Another case is whena reflective warm shield is used, which is a mirror that reflects the view of thedetector back at itself.10 A reflective warm shield is often a spherical mirror whoseradius of curvature is equal to the distance between the vertex of the warm shieldand the detector. This configuration reflects the view of the detector back at itself,though the warm shield may require aspheric terms in order to ensure all potionsof the detector see cold geometry in reflection. Because of their residual emissivity(which is often about 0.01 or more), reflective warm shields typically do not lower



160 Chapter 8

Dimple

Rays from the detector retro-reflected by the dimple

Detector

Figure 8.13 Rays from the center of the detector, retro-reflected by a dimple added to theunused portion at the center of the secondary mirror in the Maksutov–Cassegrain telescope.

internal stray light as much as cryogenically cooled cold stops. Note that the use ofa reflective warm stop may result in new external-stray-light paths.

An example of a reflective warm shield is a secondary mirror dimple, whichis located in the unused central portion of the secondary mirror. The radius ofcurvature of the dimple is equal to the distance from its vertex to the detector; itensures that the detector does not see warm geometry in reflection. An example ofa secondary mirror dimple is shown in Fig. 8.13.

8.13 Summary

It is important to establish system stray light requirements prior to designingthe optical system and to consider these requirements during system design. Notall optical designs have the same stray light performance, and some stray lightrequirements simply cannot be met with certain optical design forms, regardless ofhow many black surface treatments and baffles are added. There are many designrules that can be used to improve the stray light performance of a system; mostof them increase the complexity and/or size of the system. Rules used to controlexternal stray light are also often useful for controlling internal stray light. Theserules include using a field stop (and putting it as far forward in the system aspossible), using a Lyot stop, putting the aperture stop as close to the detector aspossible, reducing the size of or eliminating central obscurations, reducing thenumber of optical elements (especially refractive elements), and using temporal,spectral, or polarization filters to block stray light.

Cold stops and cold shields are often used in IR systems as the primary meansof reducing internal stray light. Another means involves placing refractive optics(especially those with high reflectance, such as bandpass filters) close to or insidethe cold shield, where they reflect the FOV of the detector to cold geometry. Evenif these methods are used, internal stray light is typically still present in the systemand can result in a spatially nonuniform pattern of irradiance at the detector; thispattern can be removed using a nonuniformity compensation (NUC) algorithm.Internal stray light can also be reduced by using reflective warm shields.




An example of a system that follows most of the rules described above is theoff-axis TMA system shown in Fig. 8.3. With the exception of the use of a Lyotstop and filters, all of the design rules described earlier for minimizing externaland internal stray light are followed in this system. Of course, this system may notwork for every application, especially those requiring larger FOVs or more compactdesigns. However, its good stray light performance is one reason why the off-axisTMA has been the design of choice for many remote sensing and military systemsin which stray light control is critical.1

References

1. E. Fest, “VIIRS polarization sensitivity testing and analysis,” Proc. SPIE 7461,746102 (2009) [doi: 10.1117/12.828154].

2. W. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill, New York(2008).

3. R. Noll, “Reduction of diffraction of a use of a Lyot stop,” JOSA 63(11),1399–1402 (1973).

4. B. Johnson, “Analysis of diffraction reduction by use of a Lyot stop,” JOSA A4 (8), 1376–1384 (1987).

5. W. Wolfe, Introduction to Infrared System Design, SPIE Press, Bellingham,WA (1996) [doi: 10.1117/3.226006].

6. E. Dereniak, Infrared Detectors and Systems, John Wiley & Sons, New York(1996).

7. M. Caldwell and P. Gray, “Application of a generalized diffraction analysis tothe design of nonstandard Lyot-stop systems for earth limb radiometers,” Opt.Eng. 36 (10), 2793–2808 (1997) [doi: 10.1117/1.601506].

8. T. Birge, “Stray light analysis of the cryogenic limb array etalon spectrometer,”Proc. SPIE 675, 152–159 (1987) [doi: 10.1117/12.939493].

9. R. Breault, “Control of stray light,” in The Handbook Of Optics, Vol IV, 3rd Ed.,

M. Bass, G. Li, and E. Van Stryland, Eds., McGraw-Hill, New York (2010).10. Y. Shaham, M. Umbricht, and S. Rudin, “Cold shield effectiveness in MWIR

cameras,” Proc. SPIE 2269, 438–449 (1994) [doi: 10.1117/12.188675].



Chapter 9

Baffle and Cold Shield DesignBaffles and vanes are usually used to block low-order stray light paths and areoften the primary means of controlling stray light in an optical system. Baffles arecylindrical or conical shaped tubes used to enclose a system or block zeroth-orderstray light paths, and vanes are structures that go on baffles to block scatteringfrom them. Baffles are generally used to block light from sources well outside thenominal FOV of the system and should be designed to not vignette.9 They canbe difficult to fabricate and add cost and weight to the system; however, they areessential in some systems to ensure proper functioning. An example of such asystem is the baseline Maksutov–Cassegrain telescope (shown in Fig. 1.5), whichhas three baffles:

� A large cylindrical baffle around the primary mirror to prevent direct illumi-nation of the primary. Such a baffle is often called the main baffle.10

� A baffle in the center of the primary mirror and another around the secondarymirror. These baffles block the zeroth-order external stray path through thehole in the primary mirror. A ray trace of this path is shown in Fig. 3.8.

Another example of a baffle is the cylindrical cold shield around the detector inthe LWIR camera shown in Fig. 3.11. This baffle (which was discussed in Chapters8.3 and 8.4) blocks the zeroth-order internal stray light path from self-emission ofthe housing around the detector.

The design of any of these baffles should be performed as early in the opticaldesign process as possible, as their size and placement can significantly affectboth the optical performance (e.g., vignetting and stray light) and the mechanicalcharacteristics (e.g., size and weight) of the system. This chapter discusses howbaffles and vanes can be used to improve system stray light performance, and isdivided into the following sections:

� Section 9.1 discusses a method to determine the optimal length and diameterof the main baffle.

� Section 9.2 discusses methods to determine the optimal size and placementof baffle vanes.

� Section 9.3 discusses the design of the primary and secondary mirror bafflesin the baseline Maksutov–Cassegrain system.

163



164 Chapter 9

� Section 9.4 discusses some nontraditional vane designs to utilize highly spec-ular, nonplanar surfaces. These baffles are usually more difficult to fabricatethan traditional planar baffles but can perform better in some applications.

� Section 9.5 discusses the design of masks and dimples to prevent stray lightpaths from unused portions of the optics, such as the unused portion of thesecondary mirror in Cassegrain-type systems.

The methodologies presented in this chapter can be applied to the design of bafflesof any type of optical system, even those not specifically considered here.

9.1 Design of the Main Baffles and Cold Shields

Main baffles, such as the one shown in Fig. 9.1, are common in optics and areused to shadow (i.e., prevent direct illumination of) an optical element or the focalplane. The baseline Maksutov–Cassegrain telescope has a main baffle (which, whenused on a telescope, is sometimes called the telescope barrel) that also serves as amounting structure for the corrector lens and is often used in a similar way for othercentrally obscured systems. Commercial camera lenses (such as the one shown inFig. 9.2) also often have main baffles, which are sometimes called lens hoods. Theelement that is shadowed is referred to generically in this book as the collector. Inthe baseline Maksutov–Cassegrain telescope, the collector is the primary mirror.In the camera in Fig. 9.2, the collector is the first element of the zoom lens.

The length L and diameter D of the main baffle, and the diameter d of thecollector determine the minimum angle of a collimated off-axis source �min at which

Optical axis

θmax

L

D

Collector

d

θmin

Figure 9.1 Main baffle length and diameter.



Baffle and Cold Shield Design 165

Main baffle (lens hood)

Collector (zoom lens)

Figure 9.2 A main baffle (lens hood) and collector (zoom lens) on a commercial cameralens. The lens hood has notches in it to prevent vignetting at the corners of the FOV.

the collector is no longer directly illuminated. �min is given by

�min = tan−1(

D − d

2L

), (9.1)

and �max is given by

�max = tan−1(

D + d

2L

). (9.2)

These quantities are shown in Fig. 9.1.To the first order, the effect of the main baffle on the PST of this system can

be determined by computing the percent overlap in the projected area of the mainbaffle entrance aperture and the collector, as shown in Fig. 9.3. This PST is oftencalled the shadow function of the baffle because it determines the amount that thecollector is shadowed. The shadow function is equal to unity (i.e., no shadowing)

D

d

Ltanθ

Overlap region

Figure 9.3 Geometry used for baffle shadow function calculation. The dimensions refer tothose shown in Fig. 9.1.



166 Chapter 9

1.E-03

1.E-02

1.E-01

1.E+00

0 2 4 6 8 10 12

PST

θ (deg)

L/D = 5, D/d = 1 (θmin = 0 deg, θmax = 11.3 deg)

L/D = 5, D/d = 1.5 (θmin = 1.9 deg, θmax = 9.5 deg)

L/D = 10, D/d = 1 (θmin = 0 deg, θmax = 5.7 deg)L/D = 10, D/d = 1.5 (θmin = 1.0 deg, θmax = 4. deg)8

Figure 9.4 PST at the collector versus angle of a collimated source from optical axis � fordifferent baffle geometries. The dimensions refer to those in Fig. 9.1.

for source angles less than or equal to �min, and the baffle is often designed suchthat �min is equal to half of the system FOV, as it is usually not desirable to shadow(or vignette) the FOV. In systems with small FOVs, setting �min equal to half theFOV may not be practical, as it may result in a very long main baffle. The shadowfunction is equal to zero (i.e., totally shadowed) for source angles greater than orequal to �max, and the baffle is often designed such that �max is equal to the exclusionangle, which is often a part of the stray light requirement for the system anddefines the minimum angle at which the stray light requirement must be met (seeSection 2.2.3). �min, �max, and PST for different values of L/D and D/d are shownin Fig. 9.4. In the case where the D = d, the shadow function PST is given by10

PST = 1 − 1

4

∣∣∣∣5(

1 −∣∣∣∣ �

�max

∣∣∣∣)

− 1 +(

�

�max

)∣∣∣∣ (9.3)

if |�/�max| ≤ 1, and zero otherwise.Figure 9.4 shows that the larger the value of D/d, the larger the value of �min,

which suggests that systems with large FOVs must have a main baffle much widerthan the primary (i.e., a large D) to prevent vignetting. In practice, this increase insize may be prohibitive, and therefore some systems have D/d = 1 and accept thevignetting that occurs.

Figure 9.4 also shows that the larger the value of L/D, the smaller the valueof �max, which suggests that longer main baffles can block light from off-axissources closer to the edge of the FOV. Although it is desirable, from a stray light




Sugar-scoop baffleSugar-scoop baffle

Figure 9.5 Sugar-scoop baffle on the Infrared Astronomical Satellite.12

performance standpoint, to make �max as close to the edge of the FOV as possible,in practice this is often impractical because it results in a very long and heavymain baffle. In some cases, a portion of the main baffle might not be illuminated,either as a natural consequence of the illumination geometry or by controlling theorientation of the sensor relative to the illumination. In these cases, the weight ofthe baffle can be reduced by removing the portion of it that is never illuminated.Such a baffle is sometimes called a sugar-scoop baffle and was used on the InfraredAstronomical Satellite (IRAS), as shown in Fig. 9.5.

The simplistic calculation of the PST from the shadow function shown in Fig.9.4 neglects the effect of scatter from the inner diameter (ID) of the baffle, whichis discussed in the next chapter.

9.2 Design of Vanes for Main Baffles and Cold Shields

The ID of the main baffle is often both illuminated and critical as seen from thecollector, and will therefore scatter directly to it. The radius of the main baffle cancreate a caustic in the reflected beam and thus create regions of high irradiance

Depth Spacing

Diameter

Angle

Edge radius

Bevel angle

Figure 9.6 Illustration of the baffle vane parameters in Table 9.1.



168 Chapter 9

Table 9.1 Design parameters for baffle vanes.

Vane Parameter Definition

Aperture Diameter of the hole in the vane

Depth Distance from the tip of the vane to the ID of the main baffle

Spacing Distance between adjacent vanes

Edge radius Radius of curvature of edge of vane aperture

Bevel angle Angle of bevel relative to vane surface

Angle Angle relative to main baffle

Coating Type of paint or treatment

on the collector, especially if the surface treatment of the ID is specular. Thesereflections can be blocked using vanes (apertures within the main baffle). Thedesign variables for vanes are given in Table 9.1.

The goal in baffle vane design is typically to block all first-order stray lightpaths off of the ID of the main baffle and to not vignette the FOV of the collector.

9.2.1 Optimal aperture diameter, depth, and spacing for baffle vanes

The goal in baffle vane design is typically to block all first-order stray light pathsoff of the ID of the main baffle and to not vignette the FOV of the collector. Anexample of a vane design (for the cold shield for the LWIR camera shown in Fig.3.11) is shown in Fig. 9.7.

The initial design consists of only the entrance aperture, main baffle, andcollector. The entrance aperture diameter is usually determined by the FOV and/orthe f /# of the system. The diameter of the collector used in baffle vane design isalways its maximum diameter; in most cases, this diameter will be the diagonal ofthe primary mirror (i.e. the critical portion) or of the detector.

The dotted lines in Fig. 9.7(a) are construction lines used to determine thebaffle vane depth, spacing, and bevel angles. Approximate solutions to these linescan be done graphically using a drawing program or exactly using linear equations,which is the recommended method and is used here. The numbers in Fig. 9.7 referto the steps used in constructing the design:

1. Draw a ray between the +y edge of the entrance aperture and the +y edgeof the collector. This ray defines a “keep-out” zone that prevents the FOV ofcollector from being vignetted.

2. Draw a ray from the −y edge of the critical portion of the collector to the+y corner of the main baffle.

3. Place a baffle at the intersection point between the rays from step 1 and step 2.




Collector

Main baffle Entrance aperture

1

2 3

4 5

6

Illuminated areas

Critical areas

y

z

Bevel guide lines

y

zIncident

rays

Remaining third-order scatter path

Remaining second-order scatter path

(a) (b)

Figure 9.7 (a) Steps in a diffuse baffle vane design for a cold shield, and (b) examples ofremaining higher-order paths. Figures are not to scale.

4. Draw a ray from the −y edge of the entrance aperture to the +y edge of thesecond baffle vane aperture.

5. Draw a ray from the intersection point between the ray from step 4 and themain baffle to the −y edge of the critical portion of the collector.

6. Place a baffle at the intersection point between the ray from step 1 and theray from step 5.

Repeat steps 4–6 as many times as necessary, until the end of the main baffle isreached. Figure 9.7(a) shows that this baffle design works by preventing adjacentportions of the ID of the main baffle from being both critical and illuminated,as seen from the collector. The only paths that remain are higher-order ones, asillustrated in Fig. 9.7(b). The number of baffle vanes should not be allowed toget so large that the magnitude of scattering from the edges of the vanes exceedsthat of the scattering from the ID of the main baffle without vanes. The maximumdiameter of the baffle vane apertures are computed using the maximum dimensionof the critical portion of the collector; however, the vane apertures may need to besmaller in other cross-sections because the “keep-out” zone is smaller. In general,increasing the depth of the vanes beyond the depth computed using the algorithmshown in Fig. 9.7 does little to improve the stray light performance of the system.If knife-edges and bevels are to be used on the vane tips, the bevel angles shouldbe chosen to prevent the bevels from being both illuminated and critical, as shownin Fig. 9.7. This issue is discussed in more detail in Section 9.2.2.

Figure 9.8 shows an out-of-plane view of the cold shield designed in Fig. 9.7.Because the entrance aperture is circular and the collector is square, the optimal



170 Chapter 9

y

x z

Figure 9.8 Out-of-plane view of the cold shield designed in Fig. 9.7. The diagonal of thedetector is along the y axis. The model was created in FRED.

aperture shapes are “racetracks,” or squares with rounded corners (also called“squircles”). The radii of these corners are approximately equal to (D/2)×(1 − z/L),where D is the diameter of the circular entrance aperture, z is the axial distancebetween the vane and the entrance aperture, and L is the distance between theentrance aperture and the collector. Table 9.2 gives the dimensions of baffle andvanes.

One way of quantifying the performance of this design is to compute its coldshield efficiency �, which is defined as

� = Edirect

Edirect + Eindirect, (9.4)

where Edirect is the irradiance on the detector that comes directly from a uniform,Lambertian scene; and Eindirect is the irradiance on the detector that comes fromscattering and all other stray light mechanisms within the cold shield cavity. Thequantity is similar to the VGI defined in Eq. (2.44). The efficiency of this cold shieldis computed by entering the design into FRED and performing a backwards raytrace to the entrance (cold stop) aperture, and allowing rays from the ID and from thevanes to scatter directly or indirectly to the aperture. The Aeroglaze R© Z306 BRDF

Table 9.2 Dimensions of the apertures in the cold shield design shown in Fig. 9.8. The IDof the main baffle cylinder is 12 mm.

Racetrack CornerAperture z Location (mm) Semi-Width (mm) Radius (mm)

Cold stop 0.0000 2.9289 2.9289

Vane #1 3.2666 2.6094 1.7120

Vane #2 6.5341 2.3708 0.8029

Detector 9.7649 2.1596 0.0000




Table 9.3 Cold shield efficiency with and without baffle vanes. The cold shieldgeometry is shown in Fig. 9.8. The inside of the cold shield is painted withAeroglaze R© Z306, and the analysis is performed at 10.6 �m.

Configuration Efficiency (fractional)

w/o baffle vanes 0.9758

w/ baffle vanes 0.9999

model at 10.6 �m given in Chapter 6 is used to model the scattering properties ofthe black surface treatment. The results are shown in Table 9.3. The addition ofthe baffle vanes reduces all first-order scatter paths and raises the efficiency about0.02, giving it near-perfect efficiency. This analysis neglects scattering from thevane edges, which will decrease efficiency.

Cold shield efficiency is usually the best way to quantify the effectiveness ofthe baffle vane design for cold shield. However, for other applications, such ascontrolling solar stray light in the main baffle of a telescope, it is useful to showthe PST of the baffle with and without vanes (Fig. 9.9). The PST is computed asthe irradiance on the collector divided by the irradiance at the entrance aperture.As expected, adding the vanes significantly reduces the PST, by as much as a factorof 1900. This analysis also neglects the effect of reflections from the edges of thevane apertures.

For some vane geometries, specular black paints may yield better stray lightperformance.11

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

25 40 55 70 85

PST

θ (deg)

No Vanes With Groove Vanes

With Straight Vanes

Figure 9.9 PST of the baffle design shown in Fig. 9.8 for three configurations: with novanes, with groove vanes, and with straight vanes (see Section 9.2.3).


SRBK005-09 SRBK005 March 14, 2013 10:37 Char Count= 0

172 Chapter 9

9.2.2 Edge radius, bevel angle, and angle for baffle vanes

Because the edges of the baffle vanes can scatter directly to the collector, their edgeradii should be made as small as possible. This is usually done by adding a bevelto the baffle vanes and by making the edge a knife edge, as shown in Fig. 9.6. Thesmallest edge radius that can be easily made is typically about 0.005 (0.127 mm).Care must be used when applying paint to these edges because it can increase theirradius, which increases their projected solid angle and the amount of light theyscatter to the collector. Paint can also be easily chipped off of these edges, whichcan increase their BRDF. The effect of edge scatter on stray light performance canbe evaluated in stray light analysis software by adding the edges to the model.This analysis is especially important to do in baffle designs with many vanes, asscattering from the edges may result in a system with more stray light than a systemwith fewer vanes.

The bevel angle for each vane should be chosen so that the bevel is not bothcritical and illuminated, as shown in Fig. 9.7(a). Because the bevel on the entranceaperture is likely to be illuminated, it should face outward so that it is not critical.The bevels on the other vanes should face inward so that they are not illuminated,as doing so increases the angle between the scattered and specular rays in the firstreflection of higher-order scatter paths to the collector, such as the ones shownin Fig. 9.7(b). Increasing this angle generally decreases the BRDF and thereforereduces the flux in these paths.

The bevels shown in Fig. 9.7 were designed assuming that the bevel angle can bedifferent for each vane. However, the vanes may be easier to fabricate if the bevel an-gle is constant. In this case, it is possible to prevent the bevels from being both criti-cal and illuminated by orienting them as shown in Fig. 9.10.2 In order to prevent theirbevels from being critical, vanes near the entrance aperture of the main baffle shouldbe oriented so that their bevels face outward. The opposite should be done for vanesnear the collector: their bevels should be oriented facing inward, to prevent themfrom being illuminated. The distance z0 from the entrance aperture at which the ori-entation of the bevels should switch equal to [− D × L + L × √(d ∗ D)]/ (d − D),where d, D, and L are those variables shown in Fig. 9.1. This design requires thatthe bevel angle be less than arctan{z0 / [D + (d − D) × z0 / (2 × L)]}.

is a function of the parameters of the main baffle (length, diameter, and entranceaperture diameter), of the diameter of the collector, and of the bevel angle.

Using a nonzero baffle vane angle (such as the one shown in Fig. 9.6) typicallydoes little to improve stray light performance.2 Such baffles are much more difficultto fabricate and are therefore rarely used.

9.2.3 Groove-shaped baffle vanes

In some cases, groove-shaped vanes (such as the ones shown in Fig. 9.11) maybe easier to fabricate than the straight vanes shown in Fig. 9.7. For instance,groove-shaped vanes may be easier to lathe into the ID of a lens barrel becausethey are a continuous surface, whereas straight vanes are not. This is especiallytrue in systems with tight space constraints and with a large number of vanes.




Illumination rays Critical rays

Bevels face outward to prevent them from being critical

Bevels face inward to prevent them from being illuminated

Collector

Main baffle

Figure 9.10 Optimal vane bevel orientation for vanes with constant bevel angles.

The rules for designing groove-shaped vanes are the same as those for designingstraight vanes: the design should prevent overlap between critical and illuminatedareas. This rule was used to design optimal groove-shaped vanes for the coldshield considered earlier; the resulting design is shown in Fig. 9.11. The stepsused to generate this design are similar to the steps used to generate the straight-vane design. A tighter space constraint was used on the groove-vane design; the

Collector

Main baffle Entrance aperture y

z

Groove angle

Figure 9.11 Groove-shaped vane design for the cold shield shown in Fig. 9.7. Figure is notto scale.



174 Chapter 9

minimum ID of the vanes was larger than that of the straight vane design. Thiswas done so that the performance of groove vanes in a system with tight spaceconstraints could be demonstrated. The PST of the groove vane design is shown inFig. 9.9. This PST calculation was performed assuming that same paint applied tothe straight vanes (Aeroglaze R© Z306 at 10.6 �m) was also applied to the groovevanes.

As the figure shows, the groove-vane design does not perform as well asthe straight-vane design, which is because the vane cavities are shallower in thegrooved design (because there are more of them) and because groove vanes are notas efficient at trapping light as straight vanes; as a result, there is more flux in thesecond-order stray light paths in the grooved design than in the straight design. Forthis reason, groove-vane designs generally do not perform as well as straight-vanedesigns with the same depth and spacing. Groove vanes may also be fabricatedwith constant groove angles, which may make them easier to fabricate but will alsodegrade their performance relative to optimal groove vane designs, such as the oneshown in Fig. 9.11.

9.3 Design of Baffles for Cassegrain-Type Systems

Optimal baffle designs for Cassegrain-type systems block the zeroth-order path tothe focal plane (shown in Fig. 3.8) while minimizing vignetting by the primary andsecondary mirror baffles. An example of an artifact that results from this path isshown in Fig. 1.2. This artifact is created by shortening the primary mirror bafflein the baseline Maksutov–Cassegrain telescope and then taking a photograph of afeatureless scene with the sun just outside the FOV. The sun directly illuminates thedetector via the path shown in the ray trace in Fig. 3.8. The primary mirror baffleis shortened in the stray light model, and the artifact is reproduced in Fig. 9.12.

Figure 9.12 The stray light artifact due to the zeroth-order path in the modified Maksutov–Cassegrain telescope. The telescope was modified by shortening the primary mirror baffleby about 1′′. The sun is at 10 deg from the optical axis, just outside the lower edge of the FOV.This artifact is very similar to the artifact observed in the as-built system shown in Fig. 1.2.




Primary mirror baffle entrance aperture (z1, y1)

y

z

Secondary mirror baffle entrance aperture (z2, y2)

Limit ray

Figure 9.13 The baffles in the baseline Maksutov–Cassegrain system. The limit ray goesthrough the tips of the primary and secondary mirror baffles to the edge of the detector.

The size of the baffles for Cassegrain systems with spherical primaries andsecondaries can be computed using closed-form solutions.3 For other systems, thebaffle sizes can be computed using the optimizer in a stray light analysis program.Figure 9.13 shows the four variables that must be optimized: the axial locationsand semidiameters of the primary and secondary mirror baffle entrance apertures(z1, y1 and z2, y2, respectively).

The merit function should be computed as the weighted sum of the followingquantities:

1. The flux that leaves the system through the entrance aperture via the zeroth-order path from a backward ray tracing source on the focal plane. The sourcemust fill the image and should be collected on a surface at the telescopeentrance aperture. The merit function should be heavily weighted by thisflux, as the goal is to eliminate this flux completely.

2. The flux vignetted by the baffles from forward ray tracing sources at thecenter and corner field points.

Optimizing the baffles in this way eliminates the zeroth-order path and minimizesthe amount of vignetting. This method is used to compute the size of the baffles forthe baseline Makstuov–Cassegrain system, and the resulting sizes closely matchthose in the as-built system. The baffles are shown in Fig. 9.13. In this design, themain baffle does not block the zeroth order path to the detector, because it is tooshort and the detector is too small. However, if the main baffle is long enough, itcan help block the path, and the optimal primary and secondary mirror baffles willvignette less than they would without it. If the �min of the main baffle [as definedin Eq. (9.2)] is equal to half of the system FOV, then it will completely block thezeroth-order path, and no primary and secondary mirror baffles are necessary; thisis usually impractical, as it requires the main baffle to be very long.

In optimal (or nearly optimal) baffle designs, the tip of the secondary mirrorbaffle, the tip of the primary mirror baffle, and the edge focal plane all lie on the



176 Chapter 9

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 10 20 30 40 50 60 70 80

SST


zeroth-order path

scattering from optics

ghosts from the corrector

scattering from primary mirror baffle

scattering from secondary mirror baffle

scattering from main baffle

Illumination of the inside of the primary mirror baffle

Illumination of the inside of the main baffle via reflection from the

secondary mirror

Figure 9.14 SST of elements in the baseline Maksutov–Cassegrain system, as predictedby the FRED model.

same limit ray, as shown in Fig. 9.13. This design ensures that the baffles are assmall as possible while ensuring that the zeroth-order path is blocked.

The SST predicted by the FRED model from each element in the system isshown in Fig. 9.14, and the total SST of the system with and without baffles isshown in Fig. 9.15.

A number of conclusions can be drawn from Fig. 9.15:� As expected, blocking the zeroth-order path with the baffles greatly reduces

the SST at small angles.� Adding baffles increases the SST at larger angles because they are illuminated

and can scatter directly to the detector. The peak at about 15 deg in the SSTof the primary mirror baffle is due to scattering from the inner diameterof the primary mirror baffle, as shown in Fig. 9.16. This path raises theSST significantly because the scattering from the inside of the baffle isnear specular. This path can be reduced by adding vanes to the inside of




1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 10 20 30 40 50 60 70 80

SST


without baffles with baffles

Figure 9.15 SST of the entire baseline Maksutov–Cassegrain system, with and withoutbaffles, as predicted by the FRED model.

the baffle, by lowering its BRDF, or by shadowing the baffle with a largersecondary baffle and/or longer main baffle. It can also be eliminated bymaking the aperture stop the last element in the system before the focalplane, as discussed in Section 8.3.

� The overviewing path off of the ID of the main baffle that occurs at 25 deg(illustrated in Fig. 3.9) is not a concern in this system because its flux is much

Figure 9.16 Near-specular scattering from the inside of the primary mirror baffle due toillumination by the sun at 15 deg from the center of the FOV.



178 Chapter 9

less than the flux of the path off of the ID of the primary mirror baffle. Thisis not the case in a system without a primary mirror baffle, as shown in Fig.9.15. The magnitude of this path could be reduced by adding vanes to the IDof the main baffle or by lowering its BRDF. It could be eliminated by makingthe main baffle longer or by moving the aperture stop from the primary to anelement deeper in the system, as discussed in Section 8.3.

� Ghosts from the corrector occur only at small angles, which is typical ofghost reflection paths.

� Most stray light at high angles is due to scattering from the corrector. Thiscan be reduced by shadowing the corrector with a baffle or by reducing thecorrector’s surface roughness or particulate contaminants.

Comparison between the predicted and measured SST is presented in Section11.4; the predicted matches the measured to within a factor of 5. Agreement towithin a factor of two is considered very good, as the agreement between thepredicted and measured BSDF of a single optical surface is often not better thanthis (as discussed in Sections 4.1.3 and 5.3.4). As expected, the first-order modeldoes not predict scattering from the baffles.

9.4 Design of Reflective Baffle Vanes

A number of baffle vane designs have been developed that reflect incident lightback out the entrance aperture of system.4 Unlike the vanes discussed previously inthis chapter, these vanes have high reflectance, on the order of 0.9 or higher. Thesevanes have a number of advantages: they have low emissivity (and therefore mayhave better internal stray light performance than highly absorbing baffles and donot heat up as much), can generate less particulate and molecular contamination,and may perform better over systems with large wavebands. However, they alsohave a number of drawbacks, including that they can have worse external stray lightperformance than comparable absorbing baffles, they can be difficult and expensiveto fabricate, and can be heavier than standard vanes; therefore, they may be suitablefor some but not all applications.

An ellipsoidal baffle vane design5 is shown in Fig. 9.17. Each vane is a sectionof an ellipse that has one focus at the edge of the vane in front of it and another at theedge of the entrance aperture. This arrangement ensures that all of the rays that liein the plane of the ellipse will be rejected out the entrance aperture. Unfortunately,some skew rays (about 10%) will not be rejected, and therefore the back sides ofthe baffle vanes need a black surface treatment. The vanes must be positioned sothat no entering rays can strike the inner diameter of the main baffle.

Another design, patented by Lockheed,7 is designed using alternating confocalellipses and hyperbolas, as shown in Fig. 9.18. The focusing properties of theseconic sections are such that any in-plane ray that enters between the two foci must,after one or more reflections, be rejected between the foci. Analysis indicates thatthe same is true for all skew rays.




Cross-section

Reflected rays Oblique view

F1

F2

Figure 9.17 Elliptical baffle vanes. All in-plane rays are rejected by the baffle vanes, whichhave one focus at the edge of the entrance aperture (point F1) and another at the edge ofthe vane in front of them (point F2).

Reflected rays Oblique view

Cross-section

Figure 9.18 Lockheed–Stavroudis baffle vanes, which are composed of alternating confocalellipses and hyperbolas.



180 Chapter 9

-6

-5

-4

-3

-2

10 30 50 70 90

log 10

(PST

)

Off-Axis Angle (deg)

Lockheed-Stavroudis Design

Absorbing Design

Figure 9.19 PST of the Lockheed–Stavroudis specular baffle vane design and a compa-rable absorbing baffle vane design. The characteristics of the vane designs are given inTable 9.4.

Figure 9.19 compares the PST of the Lockheed–Stavroudis design shown inFig. 9.18 to a comparable absorbing baffle design.4 The characteristics of eachdesign are given in Table 9.4. The Harvey model (discussed in Chapter 4) wasused to model the scattering properties of the baffle vanes in each design. In theLockheed–Stavroudis design, the Harvey model is typical of mirror surfaces, andwith a high slope and low TIS (about 0.01). In the absorbing design, the Harveymodel is Lambertian with a TIS of 0.06. The PST of the Lockheed-Stavroudis

Table 9.4 Characteristics of the Lockheed–Stavroudis and absorbing baffle vane designsevaluated in Fig. 9.19

Lockheed-Stavroudis AbsorbingVane Design Vane Design

Diameter of entrance aperture (in) 4 4

Baffle length (in) 11.6 11.6

Edge radii (in) 0.0039 0.0039

Number of vanes 8 8

Vane depth (in) 0.5 1

Specular reflectance 0.9 0

Slope of Harvey scatter model −1.5 0

TIS of Harvey scatter model 0.01 0.06




Unused portion of secondary

Figure 9.20 The unused portion of the secondary mirror due to the secondary obscurationin the baseline Maksutov–Cassegrain design.

design is higher because it has first-order scatter paths to the detector, whereas theabsorbing baffle design does not. This plot demonstrates that the benefits of usinga specular baffle vane design must be weighed against poorer external stray lightperformance.

9.5 Design of Masks

In many optical systems, portions of one or more of the optics are unused, and theseunused portions can couple stray light to the focal plane. For instance, the pupil inCassegrain systems has a hole in its center due to the secondary obscuration, andtherefore the center of the secondary mirror is unused, as shown in Fig. 9.20.

External stray light paths that can use this portion of the secondary includereflective ghost paths, such as the one shown in Fig. 7.7. This and other paths canbe blocked by applying a black surface treatment (such as paint) to mask off theunused portion of the secondary mirror. Internal stray light paths off of unusedareas such as this can be mitigated by using a dimple, as discussed in Section 8.12.

9.6 Summary

Baffles and vanes are usually used to block low-order stray light paths, and as suchare often the primary means of controlling stray light in an optical system. Bafflesare cylindrical or conical tubes used to enclose a system or block zeroth-orderstray light paths, and vanes are structures that go on baffles to block scatteringfrom them. The main baffle is a cylindrical baffle usually designed to prevent someelement (such as the primary mirror or detector) from being illuminated.

The algorithm shown in Section 9.2.1 can be used to determine the optimal sizeand position of vanes within a main baffle. The configuration is optimal becauseit completely prevents the overlap between critical and illuminated areas usingthe smallest number of vanes. Scattering from the edge of the baffle vanes can bereduced by making the edge radii as small as possible (i.e., using knife edges). Edgescatter can never be eliminated completely, and therefore adding more vanes to thesystem can sometimes be detrimental to system stray light performance. Bevels onknife-edged vanes should be designed such that they are not both illuminated and



182 Chapter 9

critical. Groove-shaped vanes are often used on the inside diameter of main bafflesand lens tubes as a means of reducing stray light; they may be easier to fabricate(but may not perform as well) as straight baffle vanes.

The zeroth-order path through the hole in the primary mirror in Cassegrain-typesystems can be blocked using baffles on the primary and secondary mirror. Thesebaffles can be designed for minimum vignetting using the optimization algorithmin stray light analysis software.

Highly reflective baffle designs exist that reflect much of the light incident onthe inside of the main baffle back out the entrance aperture. These designs can beadvantageous because they may have better internal stray light performance thancomparable absorbing baffle vane designs and may not generate as much particulateand molecular contamination. However, they can be difficult to fabricate and mayhave worse external stray light performance.

Masks can be used on unused areas of optical elements (such as in the centerof the secondary mirror) to block stray light paths that use these areas.

References

1. E. Freniere, “First-order design of optical baffles,”Proc. SPIE 257, 19–28 (1980)[doi: 10.1117/12.959598].

2. R. Breault, “Vane structure design trade-off and performance analysis,” Proc.SPIE 967, 90–117 (1988) [doi: 10.1117/12.948095].

3. W. Hales, “Optimum Cassegrain Baffle Systems,” Appl. Opt. 31(25), 5341–5344 (1992).

4. G. Peterson, S. Johnston, and J. Thomas, “Specular baffles,” Proc. SPIE 1753,65–76 (1992) [doi: 10.1117/12.140692].

5. J. Bremer, “Baffle design for earth radiation rejection in the cryogenic limb-scanning interferometer/radiometer,” Proc. SPIE 245, 54–62 (1980) [doi:10.1117/12.959333].

6. W. Linlor, “Baffle System Employing Reflective Surfaces”, NASA TechnicalMemorandum 84406 (1983).

7. O. Stravroudis and L. Foo, “System of reflective telescope baffles,” Opt. Eng.33(3), 675–680 (1994) [doi: 10.1117/12.159338].

8. R. Breault, “Control of Stray Light,” in The Handbook Of Optics, Vol IV, 3rd Ed.,M. Bass, G. Li, and E. Van Stryland, Eds., pp. 7–10, McGraw-Hill, New York(2010).

9. “Stray Light Short Course Notes,” Photon Engineering LLC, used with per-mission (2011).

10. A. Greynolds, “Formulas for estimating stray light levels in well-baffled opticalsystems,” Proc. SPIE 257, 39–49 (1980) [doi: 10.1117/12.959600].

11. E. Freniere, “Use of specular black coatings in well-baffled optical systems,”Proc. SPIE 675, 126–133 (1986) [doi: 10.1117/12.939490].



Chapter 10

Measurement of BSDF, TIS,and System Stray LightDespite the great deal of work that has gone into the development of BSDF andsystem stray light models, the most accurate way to determine these quantities isoften to measure them directly. Such measurements are a vital part of developingthe stray light model and of validating it. This chapter discusses the techniquesand equipment used to measure BSDF, TIS, and system stray light. Referencesto companies that manufacture such equipment, as well as those that providemeasurement services, are provided.

10.1 Measurement of BSDF (Scatterometers)

BSDF is typically measured in a device called a scatterometer.1 Scatterometersmeasure the BSDF as a function of (usually) AOI and scatter angle. (Figure 2.17illustrates the angle-naming convention used in this chapter.)

Although there are a number of designs for scatterometers, most use the samebasic design discussed in this chapter. However, before discussing this design,the concept of instrument signature (also called instrument profile) needs to beintroduced. The instrument signature of a scatterometer is the BSDF that theinstrument measures in the absence of a sample, and is caused by scattering anddiffraction (stray light) occurring within the instrument itself.2,3 All scatterometershave an instrument signature, which increases as |sin �s − sin �i | decreases. Themagnitude of the instrument signature relative to the BSDF of the sample undertest determines the minimum scatter angle from specular that can be measured;therefore, the better the scatterometer, the lower its instrument signature and thecloser to the specular beam the BSDF of the sample can be measured. It is importantto characterize the instrument signature prior to measurement, so that it can bedistinguished from the scatter of the sample. For flat samples, the instrumentsignature can be characterized simply by performing a measurement with no samplein place; however, for curved samples, the optical power of the sample changes theinstrument signature, and a more complicated data reduction process must be used.3

A number of techniques exist by which the instrument signature of a scatterometercan be reduced.1

183



184 Chapter 10

LaserPinhole

MirrorSample

Detector aperture

Detector optics

Detector

Chopper

Figure 10.1 Optical schematic of a typical scatterometer. The size and separation of theelements is not to scale.

An optical schematic of a typical scatterometer is shown in Fig. 10.1. Thisfigure shows the instrument configuration for �i = 0 deg and transmitted scattermeasurement (BTDF); however, most scatterometers can be configured for different�i and reflected scatter (BRDF) measurement. Light from a laser is passed througha spatial filter (which usually consists of an objective lens and a pinhole) and achopper wheel before being collected by the mirror and focused onto the aperture(i.e., the entrance pupil) of the detector assembly. Between the mirror and theaperture is the sample, which is illuminated by the beam. The optics in frontof the detector focus the illuminated sample spot onto the detector (which hasa field stop to limit the FOV of the detector to reduce stray light). This systemis essentially a projection-condenser system (also called a specular or Kohlerillumination system), in which the mirror acts as the condenser and the detectoroptics as the projector. These systems provide uniform illumination of the sampleas well as high collection efficiency of light scattered from the sample. The detectoris mounted on a goniometer so that it can move around the sample (both in thetransmitted direction for BTDF measurement and in the reflected direction forBRDF measurement) and measure scatter as a function of angle. The apertureof the detector optics is usually variable, which allows the angular resolution ofthe instrument to be changed. Using a smaller aperture will increase the angularresolution of the measurement but will also lower the flux on (and therefore the SNRof) the detector, so this is a trade-off that must be made. Typically, a smaller detectoraperture is used closer to the specular beam where the BSDF (i.e., the signal) ishigh and where it also changes rapidly with scatter angle, and larger apertures areused at larger angles from specular where the BSDF is lower. The chopper is usedin conjunction with a lock-in amplifier to isolate the signal from laser from thesignal that results from sources of stray light, such as (for visible wavelengths)room lighting. The cleaner and smoother the optics used in the scatterometer are,the lower its instrument signature. Provided that the laser, pinhole, detector optics,and detector are chosen appropriately, this scatterometer design can be used to



Measurement of BSDF, TIS, and System Stray Light 185

Laser sourcesSampleDetector

Illuminated spot

Figure 10.2 The SMS CASI scatterometer (image courtesy of SMS).4

measure scatter across a wide range of wavelengths. Figure 10.1 does not show anypolarization elements in the scatterometer; however, a variety of elements can beinserted in order to characterize the polarized scatter properties of the sample.

A widely used scatterometer of this type is the Schmitt Measurement Systems(SMS) complete angle scatter instrument (CASI),4 shown in Fig. 10.2. This instru-ment has been sold commercially for many years and has a wide variety of features.Specifications for this instrument are provided in Table 10.1.

Although using these types of scatterometers is generally straightforward, thereare some difficulties to be aware of:

� A laser provides a high SNR but limits the wavelengths at which the BSDFcan be measured. This can mean that the BSDF cannot be measured in thewaveband of interest or that the BSDF that is highly wavelength dependentacross a typical sensor waveband (which is not true for most optics and blacksurface treatments) cannot be accurately measured. In these cases, it may bemore appropriate to use a scatterometer with a broadband blackbody source,5

even though its SNR may be lower.� Elongation of the spot at the sample and aligning the FOV to this elongated

spot can make measurement of scatter at high values of �i (>∼ 80 deg)difficult.

Table 10.1 Specification of the CASI scatterometer.

Parameter Value

Wavelengths (typical) 0.325, 0.6328, 1.06, 3.39, and 10.6 �m

Goniometer arm length 50 cm

Angular extent of detector apertures 0.12◦, 0.29◦, 0.47◦, 1.59◦

Minimum angle from specular (typical) 0.1◦

Total system accuracy 5%

Total system linearity 2%

Noise equivalent BSDF (typical) 5E-8 1/sr



186 Chapter 10

� Measurements at UV and IR wavelengths can be more difficult than mea-surements at visible wavelengths because it is more difficult for the operatorto align an invisible beam. For this reason, some scatterometers (includingthe CASI) have a visible laser beam co-aligned with the nonvisible beam.

Though scatterometers of this type can be used to obtain very accurate mea-surements of BSDF, they are not easily portable and therefore are difficult to use forin situ measurements. SMS, as well as Surface Optics Corporation (SOC),5 makehandheld scatterometers that can be used for such measurements, though they aregenerally of much lower accuracy and angular resolution. In addition, both SMSand SOC offer scatter measurement services in which samples can be sent to themfor measurement.

10.2 Measurement of TIS

TIS can be measured in a scatterometer by measuring the BSDF, fitting it to a model(as described in Chapters 3–5), and computing the integral of the BSDF over theprojected solid angle of the hemisphere. However, errors in the measured value ofthe BSDF along with (probably more significantly) the residual between the modelfit and the measured BSDF can result in a significant error in the predicted TIS.A more accurate way of measuring TIS is to directly measure it using either aCoblentz or integrating sphere, as shown in Figs. 10.3 and 10.4, respectively. Ina Coblentz sphere, the sample is illuminated at a spot at or near the center of thehemisphere, and scatter from it is reflected by the highly reflective, highly specularmaterial (such as polished aluminum) on the inside of the hemisphere to a detectorthat is also near the center of the hemisphere.6 In an integrating sphere, the sampleis placed in a port on the side of the integrating sphere opposite the opening for theilluminating beam, and scatter from it is reflected by the highly reflective, highlyLambertian material (such as Spectralon for visible wavelengths, or diffuse goldfor IR) to a detector on the side of the sphere.

Coblentz spheres are usually purchased as a part of a larger instrument, such asa spectrometer.7,8 Integrating spheres can be purchased individually;9 a commercialintegrating sphere is shown in Fig. 10.5.


reflected ray

Sample

Coblentz sphere

Detector

Scattered rays

Figure 10.3 Measurement of TIS using a Coblentz sphere.





reflected ray

Sample

Detector

Scattered rays

Integrating sphere

Figure 10.4 Measurement of TIS using an integrating sphere.

Figure 10.5 A commercial integrating sphere (image courtesy of LabSphere, Inc.).



188 Chapter 10

10.3 Measurement of System Stray Light

The last step in the stray light engineering process is the test of the system straylight performance (see Chapter 11). The purpose of this test is usually to determineif the system meets its stray light requirements and to validate, as much as possible,the results of the stray light analysis. For systems with very low stray light levels,validating the model may be difficult or impossible because the testing proceduremay not be sensitive enough. There are a number of methods of testing stray light,and some are better suited to testing a particular type of stray light requirementthan others. Two basic types of stray light test geometries are discussed here: acollimated source test and a broad source test. Because either test may requireradiometric calibration of the sensor, it is necessary to discuss this topic first.

10.3.1 Sensor radiometric calibration

Stray light tests typically quantify the flux on the focal plane of the sensor due tostray light, and in order to do so it may be necessary to perform absolute radiometriccalibration of the sensor prior to performing the test. Cases in which it is necessaryto perform this calibration include the following:

� When the results of the test are to be reported in absolute radiometric units,such as ph/s-cm2. It is often necessary to do this when comparing the testresults to stray light model results.

� When one or more of the quantities necessary to report the test results cannotbe measured directly. For instance, it may be necessary to do calibration fora PST test (see Chapter 11) because even though PST is a relative measure-ment, it may not be possible for the sensor to measure the incident irradiancedirectly. For instance, the incident irradiance may cause the detector to sat-urate (which is typical in PST tests), as a high incident irradiance is usuallyrequired to keep the stray light flux on the detector above the detector noiselevel.

Calibration typically determines the ratio between the sensor output (e.g.,digital counts) and the flux (e.g., ph/s-cm2) incident on the sensor focal plane, andis usually performed by imaging a source of known radiance using the sensor undertest. Such sources include:

� For visible sensors: calibrated lamps,10 integrating spheres, and Spectralondiffuser plates,9 and

� For IR sensors: blackbodies11 (for IR sensors) and gold diffusers.9

The details of performing these tests vary greatly, depending on the source andsensor used, and therefore are not discussed here. Of course, not all tests requirecalibration; for instance, it is often possible to measure VGI without calibratingthe sensor because VGI is a ratio whose numerator and denominator can often bemeasured without saturating the detector (as discussed in Section 2.2.3). Whetheror not it is necessary to calibrate the sensor prior to testing is determined by thedetails of the source and sensor, and by the type of test data desired.




10.3.2 Collimated source test

The purpose of a collimated source test is to measure the amount of stray light onthe focal plane from illumination by a source of narrow angular extent, such as apoint source or sun-like source. Such a test directly measures PST or SST, and isvery common.13 Of course, it is not possible to make a real source whose spatialextent is infinitely small, and therefore any PST test will be an approximation. Theoptical schematic for a typical collimated source test is shown in Fig. 10.6. A sourceaperture is placed at the focus of a collimator, which is used to illuminate an opticalsystem on a rotation stage. The type of source used depends on the waveband ofthe optical system (see Chapter 9). A chopper is often placed between the sourceand the collimator to modulate the signal, and the chopper and sensor output arepassed through a lock-in amplifier (LIA) to isolate the signal from the source.The collimator usually consists of an off-axis parabola (OAP) or other collimatingoptics. The off-axis illumination angle of the system off-axis is selected by rotatingthe optical system relative to the incoming beam.

The size of the source aperture is determined by the type of test performed,the radiance of the source, and the level of stray light and noise characteristicsof the sensor. For instance, in a PST test, the source aperture is usually made assmall as possible in order to closely simulate a point source. However, the sourceaperture must also be large enough to generate enough photons on the sensor toproduce stray light that is measureable. The relationship between the in-band sourceradiance Ls , the source aperture area As , the collimator focal length f , and theirradiance incident on the sensor Ein is given by

Ein = Ls

(As

f 2

)(10.1)

In order for the stray light generated by the collimator to be detectable, it must begreater than the detector noise, e.g.,

Ein[PST(�off-axis)] > NEI (10.2)

Collimating optics, focal length f

Source, radiance Ls and aperture area As

Incident irradiance, Ein

Rotation stage

Optical system under test

θoff-axis

Lock-In Amplifier

Chopper

Figure 10.6 Optical schematic of a collimated source test.



190 Chapter 10

In a PST test, the source aperture is usually made just big enough so that thecondition defined in Eq. (10.2) is satisfied. In general, it is also necessary to use acollimator whose aperture is big enough to fill the entrance aperture of the opticalsystem under test. However, be aware that overfilling the entrance aperture willresult in illumination of the geometry surrounding the optical system under test,which could in turn scatter light back into the sensor and reduce the accuracy ofthe test.

The collimated source test is also often used to simulate SST. The collimatorand source are typically chosen such that Ein is equal to the in-band solar irradiance.Because As is usually the easiest parameter to change in this test, it is useful to beable to compute it based on the solar radiance, which can be done thusly:

As =(

Lsun�sun

Ls

)f 2 (10.3)

where �sun = 6.805 × 10−5 sr, and Lsun is the in-band solar radiance (see Section2.1.5 for computing this). Because Ls is almost always less than Lsun (i.e., theradiance of a ∼5800-K blackbody), the solid angle of the test source is almostalways larger than �sun, which means that the SST test will not exactly simulate thesun’s angular extent (which is nominally 0.53 deg). The apparent angular extent ofthe simulated solar source �source will be roughly equal to

�source = tan−1

(√A

f

)(10.4)

This may mean that the irradiance at the focal plane due to stray light will varydifferently as a function of �off-axis, especially near the edge of the FOV; because�source is bigger than �sun, the PST will vary more slowly with the �off-axis. Whetheror not this is important to the test results needs to be considered when planning thetest; if it is, then test methods that use the sun (discussed in Section 10.4) may bemore appropriate.

10.3.3 Extended source test

Extended source tests usually use a broad diffuser or integrating sphere to illuminatethe sensor from a wide range of off-axis illumination angles. An example of anextended source test is the veiling glare test, first discussed in Section 2.2.3. Thistest is usually done only for systems whose waveband is in the visible, and isusually performed using a broad, white diffuse reflector (such as a white wall orscreen) with a small, removable black region in the center (as shown in Fig. 2.22).The reflector is illuminated using a light source that is large enough to uniformlyilluminate it. A broader diffuse reflector produces a more accurate test. The blackregion is sized to exactly subtend the sensor FOV (which is rectangular for mostdigital camera systems). The measurement is performed by pointing the sensorat the diffuse source without the black region and sampling the sensor output (indigital cameras, this is just taking an image), and then positioning the black region




so that it just fills the FOV and taking another sample. The values of Eout and Ein inEq. (2.44) can then be computed by summing the magnitude of each sensor outputsample over the FOV for each sample taken (Eout for the first, Ein for the second).For digital cameras, this corresponds to summing the grayscale values of all ofthe pixels in each image. This test is a relative test, and therefore no calibrationis necessary, although care should be taken not to saturate the sensor output. It isalso possible to perform this test with an integrating sphere,14 using the exit portof the sphere as the extended, Lambertian source. This is possible only if a blackplug that exactly fills the FOV can be placed on the side of the integrating spherethat is opposite the exit port.

10.3.4 Solar tests

The sun is a common source of stray light, and therefore solar stray light require-ments and tests are very common. A method to test SST using a collimated sourcetest is described in Section 10.3.2; this section discusses two other testing methodsthat use the sun as the light source in the test. These methods have the advantageof using the actual angular extent and radiance of the sun, and therefore can yieldmore-accurate results than the collimated source test. However, these tests alsosuffer from a number of drawbacks, one of which is that the solar radiance isa strong function of weather conditions. This introduces variability into the test;the radiance of the sun can vary minute-to-minute, which can make determiningthe observed solar radiance and comparing the results of different tests difficult.Another drawback is that solar tests usually must be performed outside, and thusthe system and its test equipment may be exposed to dust and other contaminants.Whether or not these drawbacks are significant depends on the details of the sensorbeing tested (a smaller FOV usually results in a less accurate collimated sourcetest) and the required accuracy of the test results.

10.3.4.1 Using direct sunlightThe simplest way to perform this test is to use direct sunlight to illuminate thesensor. Because the goal of the test is usually to characterize stray light from thesun as a function of its angular location relative to the edge of the FOV, the edge ofthe FOV needs to be set at a known angle from the sun. Perhaps the simplest wayto do this is to use an equatorial mount (such as the one shown in Fig. 1.4). Thesemounts are commonly used for astronomical telescopes; they are inexpensive andwidely available. If the equatorial mount is properly leveled and aligned with thecelestial pole, then the sun can be set at a given angle with respect to the centerof the FOV by first centering it on the sun (taking care not to damage the focalplane by using an ND filter or other attenuator, if necessary) and then rotating thetelescope along the right ascension or declination axes to an orientation at a knownangle from the sun. Using an equatorial mount also allows the sun to be easilytracked across the sky. The output of the sensor at each solar angle can then bescaled using the scale factor S to determine the stray light irradiance on the focalplane, which can then be compared to the requirement and to the model results.



192 Chapter 10

There are a number of difficulties in using direct sunlight to perform the test:the first is that the range of solar angles is limited by the location, time of day, andthe ability to precisely orient the optical axis of the sensor. Another is that the suncan illuminate airborne particulates such as water vapor or dust, that can change theapparent angular extent of the sun and make it difficult to repeat the test results. Yetanother is that the sun moves, which may be a problem in testing systems requiringlong exposures. This may be solved by using an equatorial mount with a motor drive.

10.3.4.2 Using a heliostatA solution to the problem of movement of the sun described in the previous sectionis to use a heliostat,12 which is a device designed to reflect the sun is such a wayas to keep its angle of incidence on the sensor constant. Most heliostats consist offlat mirrors, one of which is usually mounted to an equatorial mount with a motordrive. A typical heliostat is shown in Fig. 10.7. The use of a heliostat eliminates

Figure 10.7 A commercial 1-m diameter heliostat (image courtesy of Heliotrack.com).




the problem of tracking the sun but does not eliminate the problem of variabilityof solar radiance over time.

10.4 Internal Stray Light Testing

Testing of internal stray light in IR systems can be performed by attaching thermo-couples to the internal housing components and recording their temperature alongwith the sensor output. These temperatures can then be used in stray light modelssuch as those discussed in Section 3.2.6 to validate the model with the sensoroutput. If necessary, the sensor temperature can be changed by putting it outside orby using heating elements.

10.5 Summary

Despite the great deal of work that has gone into the development of BSDF andsystem stray light models, the most accurate way to determine these quantities isoften to measure them directly. All of these tests have some source of error. BSDFis measured in a device called a scatterometer. Scatterometers use blackbodies orlaser sources, and usually use a goniometer or multiple detectors to measure BSDFat multiple scatter angles. TIS can be measured using an integrating sphere, andis available as an option for some spectrometers. The stray light performance ofa system can be measured using a source and collimator to simulate PST or in aveiling glare test. The sun can also be used to test the performance of the system,though it is often easier to do so using a heliostat, which will reflect the sun ata constant angle relative to the sensor FOV. Variability in weather conditions canmake solar test repeatability difficult. When possible, it is important to correlatethe stray light predicted from the model with the measured stray light; this issue isdiscussed further in Chapter 11.

References


2. K. Klicker, J. Stover, and D. Wilson, “Near specular scatter measurementtechniques for curved samples,” Proc. SPIE 967, 255–263 (1988) [doi:10.1117/12.948110].

3. E. Fest, “Data reduction of BSDF measurements from curved surfaces,” Proc.SPIE 7069, 70690K (2008) [doi: 10.1117/12.792570].

4. Schmitt Measurement Systems (SMS), http://www.schmitt-ind.com.5. Surface Optics Corporation (SOC), http://www.surfaceoptics.com.6. T. Lindstrom and A. Roos, “Reflectance and transmittance measurements of

anisotropically scattering samples in focusing Coblentz spheres,” Review ofScientific Instruments 71(6), 2270–2278 (2000).

7. Thermo Fisher Scientific Inc., http://www.thermoscientific.com.8. Bruker Corporation, http://www.bruker.com.



194 Chapter 10

9. LabSphere, Inc., http://www.labsphere.com.10. Gamma Scientific, http://www.gamma-sci.com.11. Infrared Systems Development Corporation, http://www.infraredsystems.com.12. Heliotrack.com, http://www.heliotrack.com.13. J. Fleming, F. Grochocki, T. Finch, S. Willis, and P. Kaptchen, “New stray

light test facility and initial results,” Proc. SPIE 7069, 70690O (2008) [doi:10.1117/12.798920].

14. J. Jablonski, C. Durell, and G. McKee, “Design and characterization of uniformradiance source systems for veiling glare testing of optical systems via theintegral method,” Proc. SPIE 8014, 801412-1 (2011) [doi: 10.1117/12.883880].



Chapter 11

Stray Light EngineeringProcessThe goal of this book is to present tools and information necessary to design opticalsystems with stray light performance adequate to their purpose. This goal can beachieved using the process shown in Fig. 11.1. References to chapters relevant toeach step in the process are presented in the figure. This process is generic and canbe applied to many aspects of system performance; requirements for the systemare established, the initial system is designed, and its performance is evaluated rel-ative to the requirements. If performance does not meet the requirements, changesare made and its performance is re-evaluated; this process is repeated until therequirements are met. Tests are then performed on as-built hardware to verify thatthe design meets its requirements. The process is flexible and can be adapted tobetter fit the constraints of a given project; for instance, the project may involvemodifying as-built hardware, and therefore many of the design decisions (such asthe choice of surface roughness) may have already been made and are thereforeinflexible. In such cases, the process may need to change or steps may need to beomitted. This chapter discusses each of the steps in more detail.

11.1 Define Stray Light Requirements

A stray light requirement defines what stray light performance is acceptable forthe optical system, and is determined by evaluating its purpose and the manner inwhich it is to be used. This is often one of the most difficult steps in the designprocess, because it requires an in-depth understanding of the purpose of the system,and because there is almost never a “perfect” set of requirements. Establishingrequirements often involves balancing performance and ease-of-use with size,cost, and complexity; the stricter the stray light requirement is, the more difficultit will be to achieve, and thus the more complicated and expensive the systemmust be, and the more difficult it will be to test. A “zero stray light” requirementis not realistic: all optical surfaces have some roughness and contamination thatscatter, and all refractive surfaces will produce ghost reflections, even if they areAR coated, and these mechanisms will produce stray light. In the same way, a veryloose stray light requirement (or none at all) is also usually not realistic; there are

195



196 Chapter 11

1. Define Stray Light Requirements from System Requirements (11)

2. Design Optics (8), Pick Surface Roughness (4), Contamination

Levels (5), and Coatings (7)

3. Build Stray Light Model (3), Add Baffles (9) and Black

Surface Treatments (6)

Model Agrees With Test Results?

Requirements Met?

5. Build and Test (10)

6a. Done 6b. Partially Done, Add Stray Light to Risk Register

Requirements Met?

Yes

Yes Yes

No

No

No

4. Compute Stray Light Performance (Detector FOV, PST, etc.) (2,3)

Figure 11.1 Stray light engineering process flowchart. Numbers in parentheses refer to thechapters that address the topic.

few systems that can tolerate zeroth-order stray light paths. It is impossible to listhere all of the requirements that any conceivable optical system might require; thismust be done by the optical system designers (often the person performing thestray light analysis) and by people familiar with the purpose of the system and themanner in which it is to be used. However, a few typical stray light requirementswill be discussed here.

11.1.1 Maximum allowed image plane irradiance and exclusion angle

The maximum allowed image plane irradiance defines the maximum amount oflight that can be on the image plane of an optical system from stray light sources.It can be defined in absolute units (e.g., ph/s-mm2) or in relative units (e.g., PST),and can be defined for a single area of the image plane (such as the entire imageplane) or for multiple areas of the image plane (the latter may be referred toas an “image irradiance distribution” requirement). For systems with a detector,the maximum allowed image plane irradiance is often set equal to the detector’s



Stray Light Engineering Process 197

minimum detectable irradiance. Obviously, if the stray light is not detectable, thenit is not a problem. This minimum detectable irradiance is usually a function ofdetector noise and is given by the detector manufacturer. It can be referred to bya number of different terms: minimum optical flux (i.e., ph/s) per pixel, minimumdetectable irradiance (MDI), noise equivalent irradiance (NEI), noise equivalenttemperature difference (NEDT, used in IR systems only), and others. For moderncamera systems, these values are often very small, and thus it is rare for a system notto have some detectable stray light. Because the amount of stray light irradiance onthe detector usually increases as the angle of the stray light source (i.e., sun, streetlights) to the center of the FOV decreases, and because it is impossible to reduceall of this near-FOV stray light, the maximum allowed irradiance requirement isoften accompanied by an exclusion angle requirement. The exclusion angle definesthe minimum angle of the stray light source at which the maximum allowed imageplane irradiance requirement is met. (This geometry is illustrated in Fig. 2.25.) Theexclusion angle is determined by setting the maximum allowed irradiance (eitherfrom the minimum detectable irradiance or from another method), performing astray light analysis of the system, and then determining the source angle at whichthe irradiance requirement is first met. The exclusion angle requirement warns theuser of the system that sources near the FOV may result in a high level of stray lightand thus may change the way they use the system in order to avoid this condition.

As mentioned earlier, some systems also have a requirement on the allowedspatial distribution of stray light irradiance at the focal plane. Typically, theserequirements specify a maximum allowed variation in irradiance over the focalplane; this is often intended to eliminate sharp edges in the stray light irradiancethat could cause problems with image processing algorithms. Such systems mayneed very good control over zeroth-order paths and ghost reflections, as thesemechanisms often result in artifacts with sharp edges.

A less stringent way of specifying the maximum allowed irradiance ESL due tostray light is to set it at a level equivalent to a single grayscale bit:

ESL = Escene

2b(11.1)

where Escene is the irradiance at the detector due to a typical scene, and b isthe number of grayscale bits. This method will be used to define the stray lightrequirement for the baseline Maksutov–Cassegrain system. Because this system isintended for use in astronomy, Escene will be computed using a typical astronomicalobject (the planet Mars): assuming an 8-bit grayscale output (which is widely usedin digital image formats such as JPEG), ESL can be computed as

ESL = Lsun�sun Mars(�/�)� f/#

256(11.2)

where Lsun is the in-band (visible) radiance of the sun (2.1 × 1019 ph/s-mm2-sr for a 5800-K blackbody integrated from 0.4–0.7 �m), �sun Mars is the projectedsolid angle of the sun as seen from Mars [�(1.38 × 109 km/2)2/ (228 × 109 km)2 =2.9 × 10−5 sr], � is Mars albedo (0.15), and � f/# is the solid angle of the Maksutov–



198 Chapter 11

Cassegrain system [�/4(13.9)2 = 4.1 × 10−3 sr]. Adding an additional factor offive for margin, this yields a stray light requirement of ESL = 8.46 × 107 ph/s-mm2.

11.1.2 Inheritance of stray light requirements fromcomparable systems

This is not a stray light requirement in itself but a method of defining requirements,such as maximum allowed image plane irradiance or veiling glare. This methodoften works well for consumer camera systems because answering the question“What performance does the user expect?” is often an easier question to answerthan the question “What performance does the user need?”, and it assumes thatthe stray light performance of the comparable system is known or can be analyzed.Most modern consumer cameras, including digital SLR and digital cell phonecameras, control stray light by using AR coatings on their optical surfaces andby roughening and/or blackening any mechanical surfaces that are near the mainoptical path. Although these features reduce stray light, it is still detectable inthese systems, especially in the lighting scenarios such as the sun just outside theFOV and street lights in the FOV at night, and especially when long exposures areused. However, this level of stray light control is acceptable for most consumerapplications, and therefore the corresponding levels of stray light in these systemscan be used to define the stray light requirement for new systems that will be usedin a similar way.

11.2 Design Optics, Pick Surface Roughness, ContaminationLevels, and Coatings

Once the requirements have been established, the optical system is designed, andthe surface scatter and reflectance of the optical surfaces are chosen. This designprocess often starts with a comparison of the requirements to the approximatestray light performance of the system given by equations such as Eq. (2.50). Thiscomparison can provide a starting point for a design. For instance, if the approximatemodel of the scattering from the optics [BSDFoptics in Eq. (2.50)] indicates that thereis little or no margin to the stray light requirement, then the system may need afield stop, as scattering from baffles and other structures will increase the stray lightbeyond that from just optics scatter, as demonstrated in the Maksutov–Cassegrainsystem (see Figs. 8.1 and 9.13). This analysis can also inform the choice of thenumber of surfaces and their required roughness and cleanliness. The parametersused for the baseline system are given in Table 11.1. Typical values are used for allparameters; none are measured or obtained from the vendor.

11.3 Build Stray Light Model, Add Baffles andBlack Surface Treatments

This step involves the use of the techniques presented throughout the book, inparticular Chapter 3 (the construction of the stray light model) and Chapter 9




Table 11.1 Surface properties assumed for the baseline Maksutov–Cassegrain model.

RMS RoughnessSurface (Angstroms) Cleanliness Level Reflectance

Corrector (Front) 13.1∗ 500 ∼0.005∗∗

Corrector (Back) 13.1∗ 500 ∼0.005∗∗

Primary Mirror 13.1∗ 500 0.99

Secondary Mirror 13.1∗ 500 0.99

∗The roughness scatter model is based on the measured PSD data from Section 4.1.2.∗∗The 7-layer AR coating model from Section 7.1.3 is used.

(the design of baffles). The baffles in the as-built Maksutov–Cassegrain telescopeclosely match the optimal baffle configuration, as discussed in Section 9.3. Ituses anodized aluminum as its black surface treatment, and therefore the generalpolynomial BRDF model shown in Fig. 6.7 is applied to all baffles.

11.4 Compute Stray Light Performance

Once the model is constructed, the analysis usually starts with a detector FOV raytrace to identify zeroth-order paths and critical surfaces, and a forward raytraceto identify the illuminated surfaces. If, after blocking low-order paths, the systemmeets the requirement, then the process can proceed to the next step, which is tobuild and test. If not, more design work is necessary. If the predicted stray lightperformance exceeds the requirement by a large margin, another optical designform (such as one with a field stop) may need to be considered.

Using the moon as the stray light source, the irradiance on the focal plane dueto stray light in the baseline Maksutov–Cassegrain system is computed in FREDand plotted in Fig 11.2. The irradiance on the telescope due to the moon Emoon isequal to

Emoon = Lsun�sun(�/�)�moon (11.3)

where �sun is the projected solid angle of the sun as seen from the earth or moon(6.80 × 10−5 sr), a is the moon albedo in the visible (0.12), and �moon is theprojected solid angle of the moon as seen from the earth (which is nearly equalto �sun). The resulting value of Emoon is 3.65 × 109 ph/s-mm2, which is used in afirst-order radiometric model and in the FRED model to predict the irradiance onthe detector due to stray light. The results are shown in Fig. 11.2. The first-orderradiometric model is Eq. (2.50), with BSDFoptics equal to the sum of the Harveyfunctions for roughness scatter from the corrector and the primary and secondarymirrors. This model neglects contamination scatter and the effects of the baffles(main, primary, and secondary). Nevertheless, it matches the prediction of theFRED model fairly well, except for the large peak at about 20 deg due to scatteringfrom the ID of the main baffle (as shown in Fig. 9.17). This agreement is somewhat



200 Chapter 11

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

0 10 20 30 40 50 60 70 80

Irra

dian

ce (p

h/s-

mm

2 )

Angle of moon from center of FOV (deg)

Predicted (First order, roughness only)

Predicted (FRED)

Measured

Requirement

Figure 11.2 Predicted and measured irradiance due to the stray light at the detector of thebaseline Maksutov–Cassegrain system.

fortuitous; in the FRED model, the decrease in scatter from the optics due to theshadow function of the main baffle is offset by the increase in scattering due toillumination of the baffles. The result is that the FRED model appears to match thefirst-order model better than would be expected. Both models meet the requirementwith margin, as shown in Fig. 11.2.

11.5 Build and Test

Once the analysis of the system indicates that it will meet its stray light require-ments, a prototype system should be built and tested, if feasible. The prototypeshould have all of the same characteristics as those used in the analysis, and (ifpossible) it should not be a scaled version of the final system, as manufacturingvariables are likely to be different. The uncertainty analysis presented in Chapters2–6 indicate the importance of doing stray light testing, as it is often difficult topredict the SST or another stray light metric to better than a factor of 2 of theas-built hardware. (Methods for stray light testing are discussed in Chapter 10.)

Measured data from a solar test using the actual system hardware (with baffles)is also shown in Fig. 11.2. The sun is used instead of the moon because it has nearlythe same angular extent and because it is much brighter and therefore provides more




signal. The system is calibrated by photographing a Spectralon panel illuminatedby the sun and then computing the calibration constant as (grayscale value ofSpectralon panel in image)/(predicted irradiance at detector due to solar-illuminatedSpectralon panel), as described in Chapter 10. The system is positioned at knownangles relative to the sun (using the equatorial mount, also described in Chapter10), and photographs (bitmaps) are then taken, scaled using the calibration constantand the ratio of the lunar irradiance at the earth to solar irradiance; the averageirradiance over the bitmap is then computed. The measured irradiance due to straylight matches the predicted to within a factor of 4, which is good considering thatnone of the BSDF models used in the prediction are based on measurements of theactual hardware. A comparison of the predicted and measured stray light artifactsis shown in Fig. 11.3. The measured artifact was generated using the sun as thestray light source and then scaling accordingly for the moon. At this source angle(15 deg), scattering from the ID of the primary mirror baffle is the most significantcontributor to irradiance at the focal plane, as illustrated in Fig.11.2. Therefore, theartifact shown in Fig. 11.3 is very similar to the artifact shown in in Fig. 6.9, whichis due solely to scatter from the ID of the primary mirror baffle. The measuredand predicted artifacts agree closely, which demonstrates the validity of the modelsmethods presented in this book.

(a)

(b)

8.623E7 (3.320E13)

5.740E7 (2.210E13)

7.182E7 (2.765E13)

4.299E7 (1.655E13)

irradiance in ph/s-mm2

2.857E7 (1.100E13)

Figure 11.3 (a) Measured and (b) predicted stray light artifact in the baseline system for themoon at 15 deg from the center of the FOV. The moon is just outside the lower left cornerof the FOV. Irradiance values due to the moon and (in parentheses) the sun are given.



202 Chapter 11

11.6 Process Completion

If the model correlates to the test data to within the uncertainties of the modelingtechnique and the requirement is met, then the process is finished. If the uncertain-ties are not well quantified (as is often difficult to do), the amount of stray lightin the as-built system may be less than predicted, and therefore the requirementmay be met but the model not correlated. This result should be considered, at best,a “conditional pass,” as it may indicate an unknown deviation between the modeland the as-built hardware. An example would be a baffle that is oversized relativeto its specification or drawing, thus blocking more stray light than predicted. Thepresence of this deviation increases the risk to the system, as the mistake could beresolved in the future, resulting in a sudden decrease in stray light performance. Inaddition, having a model (especially one that is well correlated with measurements)allows problems in the hardware to be quickly and accurately evaluated for impactto stray light performance. For instance, a part of the system may be received fromthe vendor missing part of its anodization. Rather than immediately rejecting thepart, the stray light model can be used to evaluate the effect of the missing an-odization and determine whether or not the part needs to be sent back. Therefore,whenever possible, the model should be correlated with the measurements.

11.7 Summary

The stray light performance of an optical system can be designed to meet the needsof the end user by using the stray light engineering process flow chart shown inFig. 11.1 to guide the design phase. In order to apply this process effectively, itis necessary to understand basic radiometry, the mechanisms by which stray lightcan reach the focal plane, and the methods by which these mechanisms can bemodeled. Before designing the system, it is important to establish its stray lightrequirements so that the appropriate features can be added to the design, as itis often difficult or impossible to introduce them once the system is built. Thereare many commercially available stray light analysis programs that can be usedto accurately predict the stray light performance of the system and evaluate theeffect of stray light control mechanisms, though it is important to also performstray light testing of the system to verify that the requirements are met. Stray lighttesting of the as-built system and correlation of the test results with predictions isrecommended, as this makes it possible to use the model to determine the effect ofusing noncompliant parts on system performance.

11.8 Guidelines and Rules of Thumb

There are a number of useful guidelines and rules of thumb that can be used todesign optical systems that have sufficient stray light control. These guidelines arepresented here.

� Stray light analysis often follows Pareto’s law, which means that it takes 20%of the time to identify 80% of the obvious stray light paths, and the other80% of the time to identify the last 20%.7




� Unlike most users of optical instruments, the stray light designer’s primaryconcern is often not the nominal field but rather all of the interior surfacesthat scatter light. It is necessary to look beyond the radii of the imagingapertures to find the sources of unwanted flux. Removing these sources fromthe field of the detector is a real possibility and will result in a significantimprovement in the system.6 This is the motivation behind backward raytracing and detector FOV (see Section 3.2.3).

� Modifying surface properties is often one of the least-effective methods ofreducing out-of-field stray light.7 This is because only incremental improve-ments in a surface’s BSDF are possible. For instance, the scattering of anoptical surface can be improved by lowering its surface roughness, but itcannot be eliminated completely. The same is true of contamination scatter.Therefore, a more-effective method is often to prevent the surface from beingeither illuminated and/or critical. This reduces the projected solid angle termin Eq. (2.41) to zero.

� Modifying surface properties is often the only way to reduce in-field straylight because baffles cannot be used to block it.8

� The closer a surface is to the nominal optical path, the more likely it is toreflect light to the detector, so pay close attention to these surfaces. Examplesinclude the lens barrel IDs and lens edges.

� As stated in Chapter 4, the relationship between the accuracy of the BSDFmodel and the accuracy of the predicted stray light performance of the systemdepends on the location of the optic in the system; optics that are illuminatedinfluence the system performance more than those that are not. In general,the less that scatter from a particular surface contributes to stray light at thefocal plane, the less accurately its scatter needs to be modeled.

� If overlap between illuminated and critical regions is unavoidable, then thesurfaces on which this occurs (such as optical surfaces) should have as low aBSDF as possible.

References

1. S. Pompea, R. Pfisterer, and J. Morgan, “Stray light analysis of the Apache PointObservatory 3.5-meter telescope system,” Proc. SPIE 4842, 128–138 (2003)[doi: 10.1117/12.459471].

2. G. Baudin et al., “Medium resolution imaging spectrometer (MERIS) stray lightdesign,” Proc. SPIE 2864, 313–321 (1996) [doi: 10.1117/12.258322].

3. J. Miller, R. English, J. Schweyen, and G. Peterson, “National Ignition Facilitymain laser stray light analysis and control,” Proc. SPIE 3492, 300–305 (1999)[doi: 10.1117/12.354141].

4. K. Ellis, “Stray light characteristics of the Large Synoptic Survey Telescope(LSST),” Proc. SPIE 7427, 742708 (2009) [doi: 10.1117/12.830599].

5. A. Lowman and J. Stauder, “Stray light lessons learned from the Mars Reconnais-sance Orbiter’s optical navigation camera,” Proc. SPIE 5526, 240–248 (2004)[doi: 10.1117/12.566080].



204 Chapter 11

6. R. Breault, “Control of Stray Light.” in The Handbook Of Optics, Vol IV, 3rd Ed.,M. Bass, G. Li, and E. Van Stryland, Eds., pp. 7–11, McGraw-Hill, New York(2010).

7. “Stray Light Short Course Notes,” used with permission, Photon EngineeringLLC (2011).


SRBK005-ind SRBK005 March 14, 2013 10:13 Char Count=0

IndexAABC model, 63, See K-correlation modelABg model, 4, 66, See surface roughness

scatterrelationship to the Harvey model, 67

absorption, 16Aeroglaze R© Z302 paint, 116Aeroglaze R© Z306 paint, 101–103, 111,

116, 170BRDF at 0.6328 �m, 104–106BRDF at 10.6 �m, 107, 108

Akzo Nobel 463-3-8 paint, 116albedo, 28, See total integrated scatter (TIS)anodized aluminum, 101, 106, 111

BRDF at 0.6328 �m, 109–111image of, 102

antireflection (AR) coatings, 6stack prescription of, 126

aperture diffraction, 1, 132artifacts, 135calculation in stray light analysis

programs, 133theory of, 132wide-angle approximations, 135

aperture stopsreducing stray light by moving closer to

focal plane, 148reducing stray light by preventing

illumination of, 154

Bbaffle vanes, 163, 168

algorithm to determine optical size andspacing of, 168

design parameters, 168determining edge radii, 172determining optimal aperture

diameters, depth, and spacing, 168effect of vanes angle, 172effect on point source transmittance,

171ellipsoidal reflective, 178groove vanes, 172optimal orientation of bevels, 172reflective, 178

baffles, 163design of optimal baffles for

Cassegrain-type systems, 174main, 163

bidirectional scattering distributionfunction (BSDF), 25

anisotropic, 26bidirectional diffraction distribution

function (BDDF), 25bidirectional reflectance distribution

function (BRDF), 25bidirectional transmittance distribution

function (BTDF), 25cosine-corrected, 25isotropic, 26reciprocity, 28and TIS, 28

black surface treatmentsanodize, 119appliques, 115Ball IR Black, 116beadblasting, 115black oxide coatings, 119

205



206 Index

black surface treatments (Contd.)dendritic structures and, 102electroless nickel plating, 115flame-sprayed aluminum, 115fused powders, 116painting, 116sandblasting, 115selection criteria, 112survey, 117, 118treatments that increase surface

thickness, 116treatments that reduce surface

thickness, 115

CCat-A-Lac black paint, 116, See Akzo

Nobel 463-3-8 paintCerablakTM, 119cleanrooms, 87

FED STD 209E classes, 87Coblentz sphere, 186cold shields, 54, 163

efficiency, 170example design, 170

collector, 164collimated source test, 189configuration factors, 30, See radiative

transfer equationcosine-cubed law, 24cosine-to-the-fourth law, 24critical surfaces, 8

determining by raytracing, 50

DDeep Sky BlackTM, 119detector field of view, 32, 49detector responsivity, 36

of typical materials, 37diffraction from diffractive optical

elements, 137, 138artifacts, 140efficiency as a function of AOI and

wavelength, 138scattering from DOE transition zones,

140

diffractive optical elementskinoforms, 137transition zones, 138, 140

diffuse hemispherical reflectance (DHR),28, See total integrated scatter (TIS)

diffuse polynomial model, 105, Seegeneral polynomial model

direction-cosine space coordinates, 26

EEbonol R© C, 119emissivity, 18Epner Laser BlackTM, 119etendue, 22, See throughputexclusion angle, 32, 197exitance, 23extended source test, 190

Ffield stops

reducing stray light by using, 145, 147tolerances and their effect on stray

light, 147filters

reducing stray light by using temporal,spectral, or polarization, 157

flux, 14f -number, 17

Ggeneral polynomial model, 104

fitting measured BSDF data to, 106Lorentzian component, 107total integrated scatter (TIS) of, 105

general polynominal model, 4, See blacksurface treatment scatter

geometric configuration factor (GCF), 17ghost reflections, 1, 6, 7, 123

in all-reflective systems (reflectiveghosts), 131

artifacts, 130model for uncoated surfaces using

refractive index (Fresnel equations),124

modeled using measured data, 128



Index 207

modeled using stack prescription, 125modeled using tabulated performance

data, 128modeled using typical values, 127path transmittance of, 124reducing stray light by preventing focus

at focal plane, 155relationship between number of

surfaces and number of paths, 123and total internal reflection (TIS), 129

golden rule, 64, See power spectraldensity (PSD): Rayleigh-Riceperturbation theory

grating equation, 8, 65

HHarvey model, 4, 65, See surface

roughness scattercoefficients for a typical mirror, 66relationship to the K-correlation

model, 66total integrated scatter (TIS) of, 65

heliostat, 192Henyey-Greenstein scatter model, 97hybrid optics, 137

IIEST-CC1246D cleanliness standard,

81accuracy of, 86, 93cleanliness level, 81particle distribution slope, 81particle distribution slope for freshly

cleaned and other surfaces, 82relationship between cleanliness level

and fallout rate, 89typical cleanliness levels, 89

illuminated surfaces, 8determining by raytracing, 50

Infrared Astronomical Satellite (IRAS),167

integrating sphere, 186, 191intensity, 23iris diffraction patterns, 135irradiance, 24

KK-correlation model, 63, 65

relationship to the Harvey model, 66

LLabSphere, Incorporated, 187lens hoods, 164, See main bafflesLWIR camera, 52Lyot stops

reducing stray light by using, 150sizing, 151, 152

Mmain baffle

computing maximum source angle ofcollector shadow, 165

computing minimum source angle ofcollector shadow, 165

effect on point source transmittance,165

lightweighting, 167point source transmittance with and

without vanes, 171sizing, 164

Makstuov–Cassegrain system, 129optimal baffles for, 175

Maksutov–Cassegrain telescope, 1, 5, 73,101, 111, 145, 146, 163, 164, 174,199

measured stray light performance of,200

Martin Black, 119mechanical design software, 42

importing from, 42MH21 paint, 116MH2200 paint, 116Mie scatter theory, 78

Rayleigh scatter, 79relationship to scattering from

particulate contaminants, 80MIL 1246C cleanliness standard, 81,

See IEST-CC1246D cleanlinessstandard

MIL-STD-13838, 75, See scattering fromscratches and digs



208 Index

minimum detectable irradiance (MDI),197

MODTRAN, 21modulation transfer function (MTF)

effect of stray light on, 56molecular contaminants, 98

collected volatile condensablematerials (CVCM), 98

total mass loss (TML), 98moon as a stray light source, 199“Move It or Block It or Paint/Coat It or

Clean It”, 9

Nnarcissus, 158

nonuniformity pattern of, 159raytrace, 158reducing by changing optical design, 158

near specular scatter, 63near-angle scatter, 63, See near specular

scatternoise equivalent delta temperature

(NEDT), 37noise equivalent detector irradiance

(NEDT), 197noise equivalent irradiance (NEI), 36, 197nonuniformity compensation, 157normalized detector irradiance (NDI), 31,

See point-source transmittance (PST)

Oobscurations

reducing stray light by making smalleror eliminating, 147

reducing stray light by reducing oreliminating, 156

optical design softwareCODE V, 4, 41, 156Zemax, 4, 41, 156

optical elementsreducing stray light by eliminating at

intermediate image, 155reducing stray light by reducing

number of, 154optical finish, 63, See surface roughness

optical radiation, 13overviewing, 51, 148, 149, 154, 177

Pparticle density function, 77

measured, 87particulate contaminants

image of, 78percent area coverage (PAC), 82refractive index of, 79

particulate contamination scatter, 1photocurrent, 15photopic luminosity function, 15Pioneer Optical BlackTM, 119Planck blackbody equation, 18point source normalized irradiance

transmittance (PSNIT), 31, Seepoint-source transmittance (PST)

point source transmittance (PST), 31measurement using a collimator, 189

point spread function (PSF), 56, 132of circular aperture, 132

polarization, 7p-polarized, 7s-polarized, 7

power, 14, See fluxpower spectral density (PSD), 63

of a typical mirror, 63and Rayleigh–Rice perturbation theory,

64relationship to surface roughness

scattering, 64projected solid angle, 16

of a right circular cone, 17of an optical system, 17units, 16

pupil masksreducing stray light by using, 153

Rractrack apertures, 170radiance, 18

blackbody, 18Lambertian, 18solar, 18



Index 209

radiative transfer equation, 29radiometric calibration, 188, 201radiometry, 13

photometric units, 15photometry, 15radiometric units, 15

raytracing, 43backward, 48, 51, 52computing detector FOV, 49determining critical and illuminated

surfaces, 50improving speed of, 45, 55Monte Carlo sampling algorithm, 43performing internal stray light analysis,

51ray aiming in, 45ray ancestry, 55reducing memory use during, 55statistical stability of, 43, 44using Monte Carlo ray splitting in, 55using to calculate MTF, 56

reflectance, 16RMS spot size, 41rules of thumb, 202

Sscatter from black surface treatments, 101

absorption in the sensor waveband, 113AOI dependence of, 103artifacts, 112conductivity and, 114modeled using measured BRDF data,

104modeled using published BRDF data,

109molecular contaminants and, 114particulate contaminants and, 114specularity at high AOIs, 113

scatter from inclusions in bulk media, 95Henyey-Greenstein model, 97ISO 10110-3 standard, 96

scatter from particulate contaminantscomparison to scatter from surface

roughness, 95wavelength dependence of, 95

scatter from surface roughnesscomparison to scatter from particulate

contaminants, 95scattering from particulate contaminants,

77, 78, 83accuracy of scatter models, 92artifacts from, 93modeled using measured scatter data, 92modeled using percent area coverage, 91relationship between percent area

coverage and total integrated scatter,83

relationship to Mie scatter theory, 80wavelength dependence of, 83

scattering from scratches and-digs, 75scattering from surface roughness, 61

artifacts, 72coated surfaces, 73modeled using measured BSDF data,

71modeled using RMS roughness, 68polarization dependence of, 64relationship between total integrated

scatter and RMS roughness, 69relationship to power spectral density,

64uncoated surfaces, 62

scatterometer, 110, 183Complete Angle Scatter instrument

(CASI), 185optical schematic of, 184

Schmitt Measurement Systems (SMS),185

shadow function, 165, See mainbaffles:effect on point sourcetransmittance

Snell’s lawof reflection, 7of refraction, 7

solar source transmittance (SST), 32breakdown for Maksutov-Cassegrain

system, 176measurement using a heliostat, 192reducing stray light using field stop,

146



210 Index

solid angle, 16of a right circular cone, 16units, 16

source source transmittance (SST)measurement using a collimator,

189measurement using direct sunlight,

191sources

extended, 18Spectralon, 28, 201squircles, 170, See racetrack aperturesStefan–Boltzman constant, 21stray light

artifacts, 1–3equation for first-order estimation of,

35external, 9in-field, 8internal, 9out-of-field, 8uncertainty in estimate, 36

stray light analysis software, 4, 28, 66ASAP, 4building models in, 41FRED, 4, 41, 106, 108, 111, 170,

176TracePro, 4

stray light artifactsfrom aperture diffraction, 135from black surface treatment

scattering, 112of ghost reflections, 130from higher-order DOE diffraction,

140from particulate contamination scatter,

93from surface roughness scattering,

72stray light engineering process,

195flowchart, 196

stray light mechanisms, 7scatter, 8specular, 7

stray light paths, 6first-order, 168path transmittance, 6second-order, 6zeroth-order, 1, 6, 51, 163, 174, 175

stray light requirements, 195inheritance from comparable systems,

198maximum allowed image plane

irradiance, 196stray light testing

external, 32, 189, 190internal, 193

strutsreducing stray light by beveling sides,

156Surface Optics Corporation (SOC),

186surface profilometer, 63surface roughness

image of, 62measurement of, 63relationship to power spectral density

(PSD), 63, 70RMS surface roughness, 63total effective, 63typical values, 69

surface roughness scatter, 1system stray light

measurement of, 188

Tthree-mirror anastigmat (TMA), 147throughput, 22

ice-cream cone mistake, 22invariance, 23

total hemispherical reflectance (THR),28, See total integrated scatter(TIS)

total integrated scatter (TIS), 28of a Lambertian scatterer, 28measurement of, 186

total internal reflection (TIR), 7transmittance, 16

atmospheric, 21



Index 211

Uuncooled microbolometers, 15

Vveiling glare

index, 170index (VGI), 32test, 33, 190

vignetting, 147, 148, 163, 174, 175reducing stray light by reducing or

elimination, 156

volume scattering, 95, See scatter frominclusions in bulk media

Wwhite-light interferometer,

63Wien’s displacement law, 21working f -number, 17

Zzoom lens, 128, 165


Eric Fest is an optical engineer at an aerospace company in Tucson, AZ, where he develops visible and infrared electro-optical systems. He has 20 years of experience performing stray light and polarimetric analysis. He has a Ph.D. in Optics from the University of Arizona, has published several papers, and holds 3 patents.


Stray light analysis and control

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