# Optical Design with Zemax

### Transcript of Optical Design with Zemax

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Optical Design with Zemax

Lecture 5: Aberrations II

2012-11-20

Herbert Gross

Winter term 2012

2 5 Aberrations II

Time schedule

1 16.10. Introduction

Introduction, Zemax interface, menues, file handling, preferences, Editors, updates, windows, Coordinate systems and notations, System description, Component reversal, system insertion, scaling, 3D geometry, aperture, field, wavelength

2 23.10. Properties of optical systems I Diameters, stop and pupil, vignetting, Layouts

3 30.10. Properties of optical systems II Materials, Glass catalogs, Raytrace, Ray fans and sampling, Footprints, Types of surfaces, Aspheres

4 06.11. Properties of optical systems III Gratings and diffractive surfaces, Gradient media, Cardinal elements, Lens properties, Imaging, magnification, paraxial approximation and modelling

5 13.11. Aberrations I Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams, Aberration expansions, Primary aberrations,

6 20.11. Aberrations II Wave aberrations, Zernike polynomials, Point spread function, Optical transfer function

7 27.11. Advanced handling

Telecentricity, infinity object distance and afocal image, Local/global coordinates, Add fold mirror, Vignetting, Diameter types, Ray aiming, slider, multiconfiguration, universal plot, IO of data, Lens catalogs

8 04.12. Optimization I Principles of nonlinear optimization, Optimization in optical design, Global optimization methods, Solves and pickups, variables, Sensitivity of variables in optical systems

9 11.12. Optimization II Systematic methods and optimization process, Starting points, Optimization in Zemax

10 18.12 Imaging Fundamentals of Fourier optics, Physical optical image formation, Imaging in Zemax

11 08.01. Illumination Introduction in illumination, Simple photometry of optical systems, Non-sequential raytrace, Illumination in Zemax

12 15.01. Correction I Symmetry principle, Lens bending, Correcting spherical aberration, Coma, stop position, Astigmatism, Field flattening, Chromatical correction, Retrofocus and telephoto setup, Design method

13 22.01. Correction II Field lenses, Stop position influence, Aspheres and higher orders, Principles of glass selection, Sensitivity of a system correction, Microscopic objective lens, Zoom system

14 29.01. Physical optical modelling I Gaussian beams, POP propagation, polarization raytrace, polarization transmission, polarization aberrations

15 05.02. Physical optical modelling II coatings, representations, transmission and phase effects, ghost imaging, general straylight with BRDF

5 Aberrations II

Rays and Wavefronts

Rays and Wavefront forms an orthotomic system

Any closed path integral has zero value

Corresponds to law of Malus and Fermats principle

Ref: W. Singer

3

5 Aberrations II

Wave Aberration in Optical Systems

Definition of optical path length in an optical system:

Reference sphere around the ideal object point through the center of the pupil

Chief ray serves as reference

Difference of OPL : optical path difference OPD

Practical calculation: discrete sampling of the pupil area,

real wave surface represented as matrix

Exit plane

ExP

Image plane

Ip

Entrance pupil

EnP

Object plane

Op

chief

ray

w'

reference

sphere

wave

front

W

y yp y'p y'

z

chief

ray

wave

aberration

optical

systemupper

coma ray

lower coma

ray

image

point

object

point

4

AP

OE

OPL rdnl

)0,0(),(),( OPLOPLOPD lyxlyx

R

y

WR

y

y

W

p

''

p

pp

pp y

yxW

y

R

u

yy

y

Rs

),(

'sin

'''

2

5 Aberrations II

Relationships

Concrete calculation of wave aberration:

addition of discrete optical path lengths

(OPL)

Reference on chief ray and reference

sphere (optical path difference)

Relation to transverse aberrations

Conversion between longitudinal

transverse and wave aberrations

Scaling of the phase / wave aberration:

1. Phase angle in radiant

2. Light path (OPL) in mm

3. Light path scaled in l

)(2

)(

)(

)()(

)()(

)()(

xWi

xki

xi

exAxE

exAxE

exAxE

OPD

5

5 Aberrations II

Wave Aberration

Definition of the peak valley value

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

6

5 Aberrations II

Wave Aberrations

Mean quadratic wave deviation ( WRms , root mean square )

with pupil area

Peak valley value Wpv : largest difference

General case with apodization:

weighting of local phase errors with intensity, relevance for psf formation

dydxAExP

ppppmeanpp

ExP

rms dydxyxWyxWA

WWW222 ,,

1

pppppv yxWyxWW ,,max minmax

pppp

w

meanppppExPw

ExP

rms dydxyxWyxWyxIA

W2)(

)(,,,

1

7

0),(1

),( dydxyxWF

yxWExP

5 Aberrations II

Wave Aberrations

x

z

s' < 0

W > 0

reference sphere

ideal ray

real ray

Wave front

R

C

y'

reference

plane

(paraxial)

U'

Wave aberration: relative to reference sphere

Choice of offset value: vanishing mean

Sign of W :

- W > 0 : stronger

convergence

intersection : s < 0

- W < 0 : stronger

divergence

intersection : s < 0

8

y

z

W < 0

Wave aberrationy'p

y'

Reference

sphere

Wave front

Transverse

aberration

Pupil

plane

Image

plane

Tilt angle

y

W

p

'Re

yR

yW

f

p

tilt

Change of reference sphere:

tilt by angle

linear in yp

Wave aberration

due to transverse

aberration y‘

ptilt ynW

5 Aberrations II

Tilt of Wavefront

9

uznzR

rnW

ref

p

Def

2

2

2

sin'2

1'

2

Paraxial defocussing by z:

Change of wavefront

y

z

W > 0

Wave aberrationy'p

z'Reference sphere

Wave front

Pupil

planeImage

plane

Defocus

5 Aberrations II

Defocussing of Wavefront

10

01

0cos

0sin

)(),(

mfür

mfürm

mfürm

rRrZ m

n

m

n

''

0'*

'

1

0

2

0)1(2

1),(),( mmnn

mm

n

m

nn

drrdrZrZ

n

n

nm

m

nnm rZcrW ),(),(

1

0

*

2

00

),(),(1

)1(2drrdrZrW

nc m

n

m

nm

5 Aberrations II

Zernike Polynomials

Expansion of the wave aberration on a circular area

Zernike polynomials in cylindrical coordinates:

Radial function R(r), index n

Azimuthal function , index m

Orthonormality

Advantages:

1. Minimal properties due to Wrms

2. Decoupling, fast computation

3. Direct relation to primary aberrations for low orders

Problems:

1. Computation oin discrete grids

2. Non circular pupils

3. Different conventions concerning indeces, scaling, coordinate system ,

11

1. Fringe - representation

- CodeV, Zemax, interferometric test of surfaces

- Standardization of the boundary to ±1

- no additional prefactors in the polynomial

- Indexing accordint to m (Azimuth), quadratic number terms have circular symmetry

- coordinate system invariant in azimuth

2. Standard - representation

- CodeV, Zemax, Born / Wolf

- Standardization of rms-value on ±1 (with prefactors), easy to calculate Strehl ratio

- coordinate system invariant in azimuth

3. Original - Nijboer - representation

- Expansion:

- Standardization of rms-value on ±1

- coordinate system rotates in azimuth according to field point

k

n

n

gerademn

m

m

nnm

k

n

n

gerademn

m

m

nnm

k

n

nn mRbmRaRaarW0 10 10

0

000 )sin()cos(2

1),(

5 Aberrations II

Zernike Polynomials: Different Nomenclatures

12

5 Aberrations II

Zernike Polynomials

+ 6

+ 7

- 8

m = + 8

0 5 8764321n =

cos

sin

+ 5

+ 4

+ 3

+ 2

+ 1

0

- 1

- 2

- 3

- 4

- 5

- 6

- 7

Zernike polynomials orders by indices:

n : radial

m : azimuthal, sin/cos

Orthonormal function on unit circle

Expansion of wave aberration surface

Direct relation to primary aberration types

Direct measurement by interferometry

Orthogonality perturbed:

1. apodization

2. discretization

3. real non-circular boundary

n

n

nm

m

nnm rZcrW ),(),(

01

0cos

0sin

)(),(

mfür

mfürm

mfürm

rRrZ m

n

m

n

13

drrdrZrWc jj

),(*),(1

1

0

2

0

min)(

2

1 1

i

ijj

N

j

i rZcW

WZZZcTT 1

5 Aberrations II

Calculation of Zernike Polynomials

Assumptions:

1. Pupil circular

2. Illumination homogeneous

3. Neglectible discretization effects /sampling, boundary)

Direct computation by double integral:

1. Time consuming

2. Errors due to discrete boundary shape

3. Wrong for non circular areas

4. Independent coefficients

LSQ-fit computation:

1. Fast, all coefficients cj simultaneously

2. Better total approximation

3. Non stable for different numbers of coefficients,

if number too low

Stable for non circular shape of pupil

14

5 Aberrations II

Zernike Polynomials: Explicite Formulas

n m Polar coordinates

Interpretation

0 0 1 1 piston

1 1 r sin x

Four sheet 22.5°

1 - 1 r cos y

2 2 r 2

2 sin 2 xy

2 0 2 1 2

r 2 2 1 2 2

x y

2 - 2 r 2

2 cos y x 2 2

3 3 r 3

3 sin 3 2 3

xy x

3 1 3 2 3

r r sin 3 2 3 3 2

x x xy

3 - 1 3 2 3

r r cos 3 2 3 3 2

y y x y

3 - 3 r 3

3 cos y x y 3 2

3

4 4 r 4

4 sin 4 4 3 3

xy x y

4 2 4 3 2 4 2

r r sin 8 8 6 3 3

xy x y xy

4 0 6 6 1 4 2

r r 6 6 12 6 6 1 4 4 2 2 2 2 x y x y x y

4 - 2 4 3 2 4 2

r r cos 4 4 3 3 4 4 4 2 2 2 2

y x x y x y

4 - 4 r 4

4 cos y x x y 4 4 2 2

6

Cartesian coordinates

tilt in y

tilt in x

Astigmatism 45°

defocussing

Astigmatism 0°

trefoil 30°

trefoil 0°

coma x

coma y

Secondary astigmatism

Secondary astigmatism

Spherical aberration

Four sheet 0°

15

Deviation in the radius of normalization of the pupil size:

1. wrong coefficients

2. mixing of lower orders during fit-calculation, symmetry-dependent

Example primary spherical aberration:

polynomial:

Stretching factor of the radius

New Zernike expansion on basis of r

166)( 24

9 Z

r

14

24

44

2

949

23

)(13

)(1

Z

rZrZZ

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

c4

c1

c9 / c

9

5 Aberrations II

Zernike Coefficients for Wrong Normalization

16

5 Aberrations II

Zernike Expansion of Local Deviations

Small Gaussian bump in

the topology of a surface

Spectrum of coefficients

for the last case

model

error

N = 36 N = 64 N = 100 N = 144 N = 225 N = 324 N = 625

original

Rms = 0.0237 0.0193 0.0149 0.0109 0.00624 0.00322 0.00047

PV = 0.378 0.307 0.235 0.170 0.0954 0.0475 0.0063

0 100 200 300 400 500 6000

0.01

0.02

0.03

0.04

17

Orthogonalization of Zernike

Polynomilas for ring shaped

pupil area

Basis function depends on

obsuration parameter e:

no easy comparisons

possible

5 Aberrations II

Tatian Polynomials for Ring Pupils

41 2 3 5 6

3431 32 33 35 36

107 8 9 11 12

1613 14 15 17 18

2219 20 21 23 24

2825 26 27 29 30

18

5 Aberrations II

Polynomial for Rectangular Pupil Areas

Systems with rectangular pupil:

Use of Legendre polynomials Pn(x)

1. Factorized representation

Problem: zero-crossing lines

2. Definition of 2D area-orthogonal

Legendre functions

General shape of the pupil area:

Gram-Schmidt-orthogonalization

drawback:

1. Individual function for every pupil shape

2. no intuitive interpretation

3. no comparability between different systems possible

W x y A P x P ynm n m

mn

( , ) ( ) ( )

mnwenn

n

mnwenn

dxxPxP mn

12

2

0

)()(

1

1

19

2D-Legendre polynomials

for rectangular areas

Application:

Spectrometer slit aperture

5 Aberrations II

Legendre Polynomials

y

x

20

5 Aberrations II

Testing with Twyman-Green Interferometer

detector

objective

lens

beam

splitter 1. mode:

lens tested in transmission

auxiliary mirror for auto-

collimation

2. mode:

surface tested in reflection

auxiliary lens to generate

convergent beam

reference mirror

collimated

laser beam

stop

Short common path,

sensible setup

Two different operation

modes for reflection or

transmission

Always factor of 2 between

detected wave and

component under test

21

5 Aberrations II

Interferograms of Primary Aberrations

Spherical aberration 1 l

-1 -0.5 0 +0.5 +1

Defocussing in l

Astigmatism 1 l

Coma 1 l

22

5 Aberrations II

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

23

5 Aberrations II

Diffraction at the System Aperture

Self luminous points: emission of spherical waves

Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave

results in a finite angle light cone

In the image space: uncomplete constructive interference of partial waves, the image point

is spreaded

The optical systems works as a low pass filter

object

point

spherical

wave

truncated

spherical

wave

image

plane

x = 1.22 l / NA

point

spread

function

object plane

5 Aberrations II

Fraunhofer Point Spread Function

Rayleigh-Sommerfeld diffraction integral,

Mathematical formulation of the Huygens-principle

Fraunhofer approximation in the far field

for large Fresnel number

Optical systems: numerical aperture NA in image space

Pupil amplitude/transmission/illumination T(xp,yp)

Wave aberration W(xp,yp)

complex pupil function A(xp,yp)

Transition from exit pupil to

image plane

Point spread function (PSF): Fourier transform of the complex pupil

function

1

2

z

rN

p

Fl

),(2),(),( pp yxWi

pppp eyxTyxA

pp

yyxxR

i

yxiW

pp

AP

dydxeeyxTyxEpp

APpp

''2

,2,)','(

l

''cos'

)'()('

dydxrr

erE

irE d

rrki

I

l

0

2

12,0 I

v

vJvI

0

2

4/

4/sin0, I

u

uuI

-25 -20 -15 -10 -5 0 5 10 15 20 250,0

0,2

0,4

0,6

0,8

1,0

vertical

lateral

inte

nsity

u / v

Circular homogeneous illuminated

Aperture: intensity distribution

transversal: Airy

scale:

axial: sinc

scale

Resolution transversal better

than axial: x < z

Ref: M. Kempe

Scaled coordinates according to Wolf :

axial : u = 2 z n / l NA2

transversal : v = 2 x / l NA

5 Aberrations II

Perfect Point Spread Function

NADAiry

l

22.1

2NA

nRE

l

5 Aberrations II

Ideal Psf

r

z

I(r,z)

lateral

Airy

axial

sinc2

aperture

cone image

plane

optical

axis

focal point

spread spot

5 Aberrations II

Abbe Resolution and Assumptions

Assumption Resolution enhancement

1 Circular pupil ring pupil, dipol, quadrupole

2 Perfect correction complex pupil masks

3 homogeneous illumination dipol, quadrupole

4 Illumination incoherent partial coherent illumination

5 no polarization special radiale polarization

6 Scalar approximation

7 stationary in time scanning, moving gratings

8 quasi monochromatic

9 circular symmetry oblique illumination

10 far field conditions near field conditions

11 linear emission/excitation non linear methods

Abbe resolution with scaling to l/NA:

Assumptions for this estimation and possible changes

A resolution beyond the Abbe limit is only possible with violating of certain

assumptions

I(r)

DAiry / 2

r0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 4 6 8 10 12 14 16 18 20

Airy function :

Perfect point spread function for

several assumptions

Distribution of intensity:

Normalized transverse coordinate

Airy diameter: distance between the

two zero points,

diameter of first dark ring 'sin'

21976.1

unDAiry

l

2

1

2

22

)(

NAr

NAr

J

rI

l

l

'sin'sin2

l

ak

R

akrukr

R

arx

5 Aberrations II

Perfect Lateral Point Spread Function: Airy

log I(r)

r0 5 10 15 20 25 30

10

10

10

10

10

10

10

-6

-5

-4

-3

-2

-1

0

Airy distribution:

Gray scale picture

Zeros non-equidistant

Logarithmic scale

Encircled energy

5 Aberrations II

Perfect Lateral Point Spread Function: Airy

DAiry

r / rAiry

Ecirc

(r)

0

1

2 3 4 5

1.831 2.655 3.477

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2. ring 2.79%

3. ring 1.48%

1. ring 7.26%

peak 83.8%

Axial distribution of intensity

Corresponds to defocus

Normalized axial coordinate

Scale for depth of focus :

Rayleigh length

Zero crossing points:

equidistant and symmetric,

Distance zeros around image plane 4RE

22

04/

4/sinsin)(

u

uI

z

zIzI o

42

2 uz

NAz

l

'sin' 22 unRE

l

5 Aberrations II

Perfect Axial Point Spread Function

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I(z)

z/

RE

4RE

z = 2RE

5 Aberrations II

Defocussed Perfect Psf

Perfect point spread function with defocus

Representation with constant energy: extreme large dynamic changes

z = -2RE z = +2REz = -1RE z = +1RE

normalized

intensity

constant

energy

focus

Imax = 5.1% Imax = 42%Imax = 9.8%

Spherical aberration Astigmatism Coma

c = 0.2

c = 0.3

c = 0.7

c = 0.5

c = 1.0

5 Aberrations II

Psf with Aberrations

Zernike coefficients c in l

Spherical aberration,

Circular symmetry

Astigmatism,

Split of two azimuths

Coma,

Asymmetric

5 Aberrations II

Comparison Geometrical Spot – Wave-Optical Psf

aberrations

spot

diameter

DAiry

exact

wave-optic

geometric-optic

approximated

diffraction limited,

failure of the

geometrical model

Fourier transform

ill conditioned

Large aberrations:

Waveoptical calculation shows bad conditioning

Wave aberrations small: diffraction limited,

geometrical spot too small and

wrong

Approximation for the

intermediate range:

22

GeoAirySpot DDD

0,0

0,0)(

)(

ideal

PSF

real

PSFS

I

ID

2

2),(2

),(

),(

dydxyxA

dydxeyxAD

yxWi

S

Important citerion for diffraction limited systems:

Strehl ratio (Strehl definition)

Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity

DS takes values between 0...1

DS = 1 is perfect

Critical in use: the complete

information is reduced to only one

number

The criterion is useful for 'good'

systems with values Ds > 0.5

5 Aberrations II

Strehl Ratio

r

1

peak reduced

Strehl ratio

distribution

broadened

ideal , without

aberrations

real with

aberrations

I ( x )

35

Approximation of

Marechal:

( useful for Ds > 0.5 )

but negative values possible

Bi-quadratic approximation

Exponential approach

Computation of the Marechal

approximation with the

coefficients of Zernike

2

241

l rms

s

WD

N

n

n

m

nmN

n

ns

n

c

n

cD

1 0

2

1

2

0

2

12

1

1

21

l

5 Aberrations II

Approximations for the Strehl Ratio

22

221

l rms

s

WD

2

24

l

rmsW

s eD

defocusDS

c20

exac t

Marechal

exponential

biquadratic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

36

pppp

pp

vyvxi

pp

yxOTF

dydxyxg

dydxeyxg

vvH

ypxp

2

22

),(

),(

),(

),(ˆ),( yxIFvvH PSFyxOTF

pppp

pp

y

px

p

y

px

p

yxOTF

dydxyxP

dydxvf

yvf

xPvf

yvf

xP

vvH

2

*

),(

)2

,2

()2

,2

(

),(

llll

5 Aberrations II Optical Transfer Function: Definition

Normalized optical transfer function

(OTF) in frequency space

Fourier transform of the Psf-

intensity

OTF: Autocorrelation of shifted pupil function, Duffieux-integral

Absolute value of OTF: modulation transfer function (MTF)

MTF is numerically identical to contrast of the image of a sine grating at the

corresponding spatial frequency

I Imax V

0.010 0.990 0.980

0.020 0.980 0.961

0.050 0.950 0.905

0.100 0.900 0.818

0.111 0.889 0.800

0.150 0.850 0.739

0.200 0.800 0.667

0.300 0.700 0.538

5 Aberrations II

Contrast / Visibility

The MTF-value corresponds to the intensity contrast of an imaged sin grating

Visibility

The maximum value of the intensity

is not identical to the contrast value

since the minimal value is finite too

Concrete values:

minmax

minmax

II

IIV

I(x)

-2 -1.5 -1 -0.5 0 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Imax

Imin

object

image

peak

decreased

slope

decreased

minima

increased

5 Aberrations II

Optical Transfer Function of a Perfect System

Aberration free circular pupil:

Reference frequency

Cut-off frequency:

Analytical representation

Separation of the complex OTF function into:

- absolute value: modulation transfer MTF

- phase value: phase transfer function PTF

ll

'sinu

f

avo

ll

'sin222 0

un

f

navvG

2

000 21

22arccos

2)(

v

v

v

v

v

vvHMTF

),(),(),( yxPTF vvHi

yxMTFyxOTF evvHvvH

/ max

00

1

0.5 1

0.5

gMTF

Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on the

azimuthal orientation of the object structure

Generally, two MTF curves are considered for sagittal/tangential oriented object structures

5 Aberrations II

Sagittal and Tangential MTF

y

tangential

plane

tangential sagittal

arbitrary

rotated

x sagittal

plane

tangential

sagittal

gMTF

tangential

ideal

sagittal

1

0

0.5

00.5 1

/ max