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Imaging and Aberration Theory

Lecture 11: Wave aberrations

2019-01-11

Herbert Gross

Winter term 2018

2

Schedule - Imaging and aberration theory 2018

1 19.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems

2 26.10.Pupils, Fourier optics, Hamiltonian coordinates

pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates

3 02.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media

4 09.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas

5 16.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations

6 23.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders

7 30.11. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options

8 07.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options

9 14.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum

10 21.12.Sine condition, aplanatism and isoplanatism

Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics

11 11.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations

12 18.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement

13 25.01. Point spread function ideal psf, psf with aberrations, Strehl ratio

14 01.02. Transfer function transfer function, resolution and contrast

15 08.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability

1. Definition of wave aberrations

2. Performance criteria

3. Primary aberrations

4. Order expansions

5. Non-circular pupil shapes

6. Statistical aberrations

7. Measurement of wave aberrations

3

Contents

Law of Malus-Dupin:

- equivalence of rays and wavefronts

- both are orthonormal

- identical information

Condition:

No caustic of rays

Mathematical:

Rotation of Eikonal

vanish

Optical system:

Rays and spherical

waves orthonormal

wave fronts

rays

Law of Malus-Dupin

object

plane

image

plane

z0 z1

y1

y0

phase

L = const

L = const

srays

rot n s

0

4

R

y

WR

y

y

W

p

''

Relationship to Transverse Aberration

Relation between wave and transverse aberration

Approximation for small aberrations and small aperture angles u

Ideal wavefront, reference sphere: Wideal

Real wavefront: Wreal

Finite difference

Angle difference

Transverse aberration

Limiting representation

5

yp

z

real ray

wave front W(yp)

R, ideal ray

C

reference

plane

y'

reference sphere

q

u

idealreal WWWW

py

W

tan

Ry'

Wave Aberration

Exact relation between wave aberration and ray deviation

General expression from geometry

a describes the lateral aberration

Substitution of angle by scalar

product

Exact relation is quadratic in R

Approximation for large R

with

6

qcos1

'1

'

22

0

R

asan

ss

assssnWW

r

rr

2

2222

41

2'

R

asa

R

asanWW

'2

''

22

zR

yx

yR

yx

R

xW

pp

pp

'

'

'

z

y

x

a

real ray

C

R

s

sr

reference

sphere

q

a

A

W

wave

front

z

yp

center of

reference

sphere

real

intersection

point

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

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

7

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

8

Pupil Sampling

y'p

x'p

yp

xp x'

y'

z

yo

xo

object plane

point

entrance pupil

equidistant grid

exit pupil

transferred grid

image plane

spot diagramoptical

system

All rays start in one point in the object plane

The entrance pupil is sampled equidistant

In the exit pupil, the transferred grid

may be distorted

In the image plane a spreaded spot

diagram is generated

9

Wave Aberration

Definition of the peak valley value WPV

Reference sphere corresponds to perfect imaging

Rms-value is more relevant for performance evaluation

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

10

Wave Aberration Criteria

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

11

0),(1

),( dydxyxWF

yxWExP

Wave Aberrations – Sign and Reference

x

z

s' < 0

W > 0

reference sphere

ideal ray

real ray

wave front

R

C

reference

plane

(paraxial)

U'

y' < 0

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

12

y

z

W < 0

wave aberrationy'p

y'

q

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 q

linear in yp

Wave aberration

due to transverse

aberration y‘

Is the usual description

of distortion

q ptilt ynW

Tilt of Wavefront

13

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

Defocussing of Wavefront

14

Representation of primary aberrations

Seidel terms

Surface in pupil plane

Special case of chromatical aberrations

Primary Aberrations

Spherical aberration Coma

Primary monochromatic wave aberrations

Astigmatism Field curvature (sagittal)

Distortion

ypxp

ypxp

ypxp

ypxp

ypxp

4

1

222

1 rcyxcW pp cos3

2

22

2 yrcyxyycW ppp

222

3

22

3 cosrycyycW p 22

4

222

4 rycyxycW pp

cos3

5

3

5 rycyycW p

Axial color Lateral color

Primary chromatic wave aberrations

2

1

22

1

~~rbyxbW pp

ypxp

ypxp

cos~~

22 yrbyybW p

Ref: H. Zügge

15

Relation :

wave / geometrical aberration

Primary Aberrations

Type

ypxp

ypxp

ypxp

ypxp

ypxp

4

1 rcW

cos3

2 yrcW

222

3 cosrycW

22

4 rycW

cos3

5 rycW

sin' 3

1 rcx

cos' 3

1 rcy

2sin' 2

2 yrcx

0'x

cos' 2

3 rycy

sin' 2

4 rycx

cos' 2

4 rycy

0'x

3

5' ycy

Spherical

aberration

Symmetry to

Periodicity

axis

constant

point

1 period

Wave aberration Geometrical spot

Coma

Astigmatism

Field

curvature

(sagittal)

Distortion

Symmetry to

Periodicity

Symmetry to

Periodicity

Symmetry to

Periodicity

Symmetry to

Periodicity

one plane

1 period

two planes

2 period

one plane

1 period

axis

constant

point

1 period

one straight line

constant

one straight line

2 periods

two straight lines

1 period

)2cos2('2

2 yrcy

Ref: H. Zügge

16

Relation :

wave / geometrical aberration

Primary Aberrations

ypxp

ypxp

Axial color

Lateral color

Symmetry to

Periodicity

Symmetry to

Periodicity

one plane

1 period

axis

constant

point

1 period

one straight line

constant

Type Wave aberration Geometrical spot

2

1

~rbW

cos~

2 yrbW

sin~

' 1 rbx

cos~

' 1 rby

0'x

yby 2

~'

Ref: H. Zügge

17

Special Cases of Wave Aberrations

Wave aberrations are usually given as reduced

aberrations:

- wave front for only 1 field point

- field dependence represented by discrete

cases

Special case of aberrations:

1. axial color and field curvature:

represented as defocussing term,

Zernike c4

2. distortion and lateral color:

represented as tilt term, Zernike c2, c3

18

yp

z

y'

Reference sphere

wave front

pupil

plane

ideal

image

plane

axial

chromatic

yp

z

y'

reference

sphere

wave front

pupil

plane

ideal

image

plane

defocus

due to

field

curvaturechief ray

curved

image

shell

yp

z

y'

reference

sphere

wave front

pupil

plane

ideal

image

plane

image

height

differencechief ray

Special Cases of Wave Aberrations

3. afocal system

- exit pupil in infinity

- plane wave as reference

4. telecentric system

chief ray parallel to axis

19

yp

z

reference

plane

wave front

image in

infinity

yp

z

y'

reference

sphere

wave front

pupil

planeideal

image

plane

axial

chromatic

Expansion of the Wave Aberration

Table as function of field and aperture

Selection rules:

checkerboard filling of the matrix

Field y

Spherical

y0 y 1 y 2 y3 y 4 y 5

Distortion

r

1

y

y3

primary

y5

secondary

r

2

r 2

Defocus

Aper-

ture

r

r

3

r3y Coma primary

r 4

r 4 Spherical

primary

r

5

r

6

r 6 Spherical

secondary

DistortionDistortionTilt

Coma Astigmatism

Image

location

Primary

aberrations /

Seidel

Astig./Curvat.

r cosq

cosq

cos2q

r cosq

Secondary

aberrations

r cosq

r 4y 2 cos

2q

Coma

secondary

r5y cosq

r3y

3 cos

3q

r3y

3 cosq

r2 y

4

r2 y

4 cos

2q

r 4y 2

r22

yr

22 y

20

mlk

ml

p

k

klmp ryWryW,,

cos'),,'( qq

isum Ni number of terms

Type of aberration

2 2 image location

4 5 primary aberrations, 4th order

6 9 secondary aberrations, 6th order

8 14 8th order

Polynomial Expansion of Wave Aberrations

Taylor expansion of the wavefront:

y' Image height index k

rp Pupil height index l

q Pupil azimuth angle index m

Symmetry invariance:

1. Image height

2. Pupil height

3. Scalar product between image and pupil vector

Number of terms

sum of indices in the

exponent isum

y r y rp p' ' cos q

pr

'y

21

type of aberration wave aberration transverse aberration longitudinal aberration

u w u w u w

spherical 4 0 3 0 2 0

coma 3 1 2 1

astigmatism 2 2 1 2 0 2

Petzval curvature 2 2 1 2 0 2

distortion 1 3 0 3

axial chromatical 2 0 1 0 0 0

chromatical magnif. 1 1 0 1

The exponents of the Taylor expansion on the aperture depends on the kind of

representation of the aberrations.

The exponent grows by 1 in the sequence longitudinal-transversal-wave aberrations

The Seidel term ‚3rd order‘ is valid only for transverse aberrations

Dependence on aperture and fiels size for the primary aberrations:

Taylor - Expansion

22

pdpppappcpspp yyAryAyyAyryArAyryW 3222224 ''''),,'(

Taylor Expansion of the Primary Aberrations

Expansion of the monochromatic aberrations

First real aberration: primary aberrations, 4th order as wave deviation

Coefficients of the primary aberrations:

AS : Spherical Aberration

AC : Coma

AA : Astigmatism

AP : Petzval curvature

AD : Distortion

Alternatively: expansion in polar coordinates:

Zernike basis expansion, usually only for one field point, orthogonalized

23

4

lPVW

Rayleigh Criterion

The Rayleigh criterion

gives individual maximum aberrations

coefficients,

depends on the form of the wave

Examples:aberration type coefficient

defocus Seidel 25.020 a

defocus Zernike 125.020 c

spherical aberration

Seidel 25.040 a

spherical aberration

Zernike 167.040 c

astigmatism Seidel 25.022 a

astigmatism Zernike 125.022 c

coma Seidel 125.031 a

coma Zernike 125.031 c

24

a) optimal constructive interference

b) reduced constructive interference

due to phase aberrations

c) reduced effect of phase error

by apodization and lower

energetic weighting

d) start of destructive interference

for 90° or l/4 phase aberration

begin of negative z-component

Rayleigh criterion:

1. maximum of wave aberration: Wpv < l/4

2. beginning of destructive interference of partial waves

3. limit for being diffraction limited (definition)

4. as a PV-criterion rather conservative: maximum value only in 1 point of the pupil

5. different limiting values for aberration shapes and definitions (Seidel, Zernike,...)

Marechal criterion:

1. Rayleigh crierion corresponds to Wrms < l/14 in case of defocus

2. generalization of Wrms < l/14 for all shapes of wave fronts

3. corresponds to Strehl ratio Ds > 0.80 (in case of defocus)

4. more useful as PV-criterion of Rayleigh

Criteria of Rayleigh and Marechal

14856.13192

lllRayleigh

rmsW

25

PV and Wrms-Values

PV and Wrms values for

different definitions and

shapes of the aberrated

wavefront

Due to mixing of lower

orders in the definition

of the Zernikes, the Wrms

usually is smaller in

comparison to the

corresponding Seidel

definition

26

Typical Variation of Wave Aberrations

Microscopic objective lens

Changes of rms value of wave aberration with

1. wavelength

2. field position

Common practice:

1. diffraction limited on axis for main

part of the spectrum

2. Requirements relaxed in the outer

field region

3. Requirement relaxed at the blue

edge of the spectrum

Achroplan 40x0.65

on axis

field zone

field edge

diffraction limit

l [m]0.480 0.6440.562

0

Wrms [l]

0.12

0.06

0.18

0.24

0.30

27

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

Expansion of wave aberration surface

Zernike polynomials orders by indices:

n : radial

m : azimuthal, sin/cos

Orthonormal function on unit circle

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

0)(cos

0)(sin

)(),(

mfor

mform

mform

rRrZ m

n

m

n

28

Orthogonalization of Zernike

Polynomilas for ring shaped

pupil area

Basis function depends on

obsuration parameter e:

no easy comparisons

possible

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

29

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

( , ) ( ) ( )

mnif

n

mnif

dxxPxP mn

12

2

0

)()(

1

1

30

Orthogonalization of Zernike polynomials on a unit square

Gram-Schmidt-Orthogonalization procedure

2-Dimensional Legendre Polynomials

j

m

j

mjm ZcQ 1

kjjk dydxQQ

1

1

1

1

mmk

m

k

km ZaQaQ

1

1

lj

j

j

l

l

jlmm yxdQ

max max

0 0

,

1

1,...3,2,1m

kj

jkjmk mkcac

1

1

1

1 1

2 2

1

m

k

m

k

k

j

mjkjkmkmmm

mmm

PcTTP

ca

31

2D-Legendre polynomials

for rectangular areas

Application:

Spectrometer slit aperture

First few polynomials:

quite similar to Zernikes

Legendre Polynomials

y

x

32

At any surface the wavefront can be compared with the ideal one before and after the

refraction/reflection

Due to the additivity of the phase, at any surface the contribution can be calculated

Practical problems:

- collimated intermediate ray paths

- change of normalization radii and grid distortion

- change of wavefront/Zernikes during propagation

- choice of reference surface not trivial

- parabasal ray calculation inaccurate

33

Wave Aberration Generated at a Surface

real

ray

z

paraxial

ray

intermediate ideal

object plane

intermediate ideal

image plane

spherical image wave

surface

spherical object wave

real waves

n‘n

W. Welford, Aberrations of optical systems, Hilger 1986

W. Welford, Opt Acta 19 (1972) p.719, A new total aberration formula

Complete system:

Additivity of phase delay at every surface is obvious

Practical problems:

- change of normalization radii

- grid distortion

- huge amount of information, systematic analysis complicated

- analytical representation not possible

34

Wave Aberration Additivity

P

arbitrary ray

y

1 2 3

y'

P'

surfaces exit pupil

total Wtot

W1

W2 W3

surface contributions

ray pencil

Real distortion of the grid in at the various mirror positions

35

Grid Distortion2

20

mm

M1

11

0 m

m

26

0 m

m

M2 M3

Complex field with statistical phase

Correlation of the phase:

structural function

Coherence function

For gaussian statistics

Auto covariance function

PSD, power spectral density

)()()( ri

o erErE

)()()(

2

1

2112 ririrD

ee

)(2

1

2112

12

)()()(rD

oo erErEz

2

2

12)( 2 ca

r

erD

2

22

2),(),(),( ca

yx

eyyxxyxyxC

222222),(ˆ),( yxc vva

cyx eayxCFvvS

Statistical Wave Aberrations

36

Description:

1. in the spatial domain: topology of the rough surface

2. in the spatial frequency domain

Statistical Wave Aberrations

h(x) C(x)

x

PSD()A()

Fourier

transform

| |2

< h1h

2 >

correlation

square

height,

topologycorrelation

function

amplitude

spectrumpower spectral

density

Fourier

transform

x

37

Spatial Frequency of Surface Perturbations

Power spectral density of the perturbation

Three typical frequency ranges, scaled by diameter D

limiting line

slope m = -1.5...-2.5

log A2Four

long range

low frequency

figure

Zernike

mid

frequencymicro

roughness

1/l

oscillation of

the polishing

machine

12/D1/D 40/D

38

Atmospheric turbulence:

statistical phase screen

Scale below 1 cm:

Tatarski regime, viscosity

PSD

Scale from 1 cm to 5 m:

Kolmogorov regime

PSD

Greater length scales:

Karman regime

PSD

2

ik

k

Ta eb

3

11

kbTa

6

11

2

22 4

o

KaL

kb

Atmospheric Turbulence

const

Karman

large scale

Kolmogorov

inertial subrange

viscose

dissipation

Tatarski

k

= b

Tae-k

2

ko outer scale ki

inner scale

LogCn

2

(k)

=bKo

k 311

39

Measurement of Wave Aberrations

Wave aberrations are measurable directly

Good connection between simulation/optical design and realization/metrology

Direct phase measuring techniques:

1. Interferometry

2. Hartmann-Shack

3. Hartmann sensor

4. Special: Moire, Holography, phase-space analyzer

Indirect measurement by inversion of the wave equation:

1. Phase retrieval of PSF z-stack

2. Retrieval of edge or line images

Indirect measurement by analyzing the imaging conditions:

from general image degradation

Accuracy:

1. l/1000 possible, l/100 standard for rms-value

2. Rms vs. individual Zernike coefficients

40

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

41

Interferograms of Primary Aberrations

Spherical aberration 1 l

-1 -0.5 0 +0.5 +1

Defocussing in l

Astigmatism 1 l

Coma 1 l

42

Real Measured Interferogram

Problems in real world measurement:

Edge effects

Definition of boundary

Perturbation by coherent

stray light

Local surface error are not

well described by Zernike

expansion

Convolution with motion blur

Ref: B. Dörband

43

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

44

Wave front determines local direction of propagation

Propagation over distance z : change of transverse intensity distribution

Intensity propagation contains phase information

Principle of Psf Phase Retrieval

z

intensity caustic

I(x,y,z)

wave front : W(x,y)

local Poynting vector

direction of propagation

propagation

45

Transport of intensity equation couples phase and intensity

Solution with z-variation of the intensity delivers start phase at z = 0

Determine phase from intensity distribution.

- Inverse propagation problem : ill posed

- Boundary condition : measured z-stack I(x,y,z)

Algorithm for numerical solution

- IFTA / Gerchberg Saxton ( error reduction )

- Acceleration ( conjugate gradients, Fienup,...)

- Modal non least square-methods

- Extended Zernike method

Applications:

- Calculation of diffractive components for given illumination distribution

- Wave front reconstruction

- Phase microscopy

),(),,(),,(

yxWzyxIz

zyxIk

Propagation of Intensity

46

Principle of phase retrieval for metrology of optical systems

Measurement of intensity caustic z-stack

Reconstruction of the phase in the exit pupil

Phase Retrieval

z

x

y

opticalsystem

exit

pupil

image plane

z

x

3D- intensity image

stack I (x,y,z)

W( ,y)

wave front

x

47

Gerchberg-Saxton-Algorithm

Iterative reconstruction of the

pupil phase with back-and-forth

calculation between image and

pupil:

IFTA / Gerchberg-Saxton

Substitution of known intensity

Problems with convergence:

Twin-image degeneration

Modified algorithms:

1. Fienup-acceleration

2. Non-least-square

3. Use of pupil intensity

48

Phase Space Interpretation

Known measurement of intensity in defocussed planes:

- Several rotated planes in phase space

- Information in and near the spatial domain

Calculation of distribution in the Fourier plane

Wave equation is valid

Principle : Tomography

xp

minimal

defocus plane

pupil plane

maximal

defocus plane

desired

distribution

measuring

planes

phase

space

angle

image

plane

x'

49

Example Phase Retrieval

Evaluation of real data psf-stack

model phasemeasurement

50

Retrieval

without / with

Apodization

Correlation

over z

Phase Retrieval with Apodization

M

R no A

Dif

R wi A

Dif

51

Object Space Defocussing

a) defocussing in

image space

z

object

plane

several

detector planes

intensity caustic I(x,y,z)

pupil

one

wavefront

lens

b) defocussing in

object space

z

several

object

planes

one fixed

detector plane

intensity caustic I(x,y,zobj)

pupil

several

wavefronts

lens

52

Comparion phase

retriev vs Hartmann

test

Case of coma

Measurements

Zernike-

coefficients

0

0.0

1

0.0

2

0.0

3

0.0

4

0.05

0.0

6

0.0

7Coma

a13

a15

| A13 |

| A15 |HartmannRetrieval

+field- field -zone + zoneaxis

Ref: B. Möller

53