www.iap.uni-jena.de

Design and Correction of Optical

Systems

Lecture 5: Wave aberrations

2017-05-12

Herbert Gross

Summer term 2017

2

Preliminary Schedule - DCS 2017

1 07.04. Basics Law of refraction, Fresnel formulas, optical system model, raytrace, calculation

approaches

2 14.04. Materials and Components Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,

aspheres, diffractive elements

3 21.04. Paraxial Optics Paraxial approximation, basic notations, imaging equation, multi-component

systems, matrix calculation, Lagrange invariant, phase space visualization

4 28.04. Optical Systems Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,

photometry

5 05.05. Geometrical Aberrations Longitudinal and transverse aberrations, spot diagram, polynomial expansion,

primary aberrations, chromatical aberrations, Seidels surface contributions

6 12.05. Wave Aberrations Fermat principle and Eikonal, wave aberrations, expansion and higher orders,

Zernike polynomials, measurement of system quality

7 19.05. PSF and Transfer function Diffraction, point spread function, PSF with aberrations, optical transfer function,

Fourier imaging model

8 26.05. Further Performance Criteria Rayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based

criteria, further options

9 02.06. Optimization and Correction Principles of optimization, initial setups, constraints, sensitivity, optimization of

optical systems, global approaches

10 09.06. Correction Principles I Symmetry, lens bending, lens splitting, special options for spherical aberration,

astigmatism, coma and distortion, aspheres

11 16.06. Correction Principles II Field flattening and Petzval theorem, chromatical correction, achromate,

apochromate, sensitivity analysis, diffractive elements

12 23.06. Optical System Classification Overview, photographic lenses, microscopic objectives, lithographic systems,

eyepieces, scan systems, telescopes, endoscopes

13 30.06. Special System Examples Zoom systems, confocal systems

14 07.07. Further Topics New system developments, modern aberration theory,...

1. Rays and wavefronts

2. Wave aberrations

3. Expansion of the wave aberrations

4. Zernike polynomials

5. Performance criteria

6. Non-circular pupil areas

7. Measurement of wave aberrations

Contents

3

Rays and waves carry the same information

Wave surface is perpendicular on the rays

Wave is purely geometrical and has no diffraction properties

4

Ray-Wave Equivalent

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

5

Fermat principle:

the light takes the ray path, which corresponds

to the shortest time of arrival

The realized path is a minimum and therefore

the first derivatives vanish

Several realized ray pathes have the same

optical path length

The principle is valid for

smooth and discrete

index distributions

Fermat Principle

P1

x

z

P1

Cds

2

1

0),,(

P

P

dszyxnL

2

1

.

P

P

constrdsnL

P'

Pn d1 1

n d2 2

n d3 3

n d4 4

n d5 5

6

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

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

8

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 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

9

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

10

11

Wave Aberrations

Quality assessment:

- peak valley value (PV)

- rms value, area average

- Zernike decomposition for detailed

analysis

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

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

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

15

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

16

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

17

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

18

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

19

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

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 ,

20

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 into elementary functions / shapes

Zernike functions are defined in circular coordinates r,

Ordering of the Zernike polynomials by indices:

n : radial

m : azimuthal, sin/cos

Mathematically orthonormal function on unit circle for a constant weighting function

Direct relation to primary aberration types

n

n

nm

m

nnm rZcrW ),(),(

01

0)(cos

0)(sin

)(),(

mfor

mform

mform

rRrZ m

n

m

n

21

Azimuthal Dependence of Zernike Polynomials

Azimuthal spatial

frequency

r1

r3

r2

r1r3

r2

31

2 4

22

Zernike Polynomials: Meaning of Lower Orders

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°

23

Zernike Fringe vs Zernike Standard Polynomials

Fringe coefficients:

Z 4,9,16 = n² spherical

azimutal order f grows if

number increases

Standard coefficients - different term numbers

ZN = 0.05

RMS = 0.05l

SR ~ 1 – 40 ZN² = 0.9

Elimination of tilt

No Elimination of defocus @ Zemax

In radial symmetric system

for y-field (meridional)

sinus-terms vanish

2

4

24

W

rp1

4th order

4th and 2nd order

4th, 2nd and 0th order

4

1

-1

3

2

0

166)( 24 ppp rrrW

Balance of Lower Orders by Zernike Polynomials

Mixing of lower orders to get the minimal Wrms

Example spherical aberration:

1. Spherical 4th order according to

Seidel

2. Additional quadratic expression:

Optimal defocussing for edge

correction

3. Additional absolute term

Minimale value of Wrms

25

Zernike Polynomials

Advantages of the Zernike polynomials:

- de-coupling due to orthogonality

- stable numerical computation

- direct relation of lower orders to classical aberrations

- optimale balancing of lower orders (e.g. best defocus for spherical aberration)

- fast calculation of Wrms and Strehl ratio in approximation of Marechal

Necessary requirements for orthogonality:

- pupil shape circular

- uniform illumination of pupil (corresponds to constant weighting)

- no discretization effects (finite number of points, boundary)

Different standardizations used concerning indexing, scaling, sign of coordinates

(orientation for off-axis field points):

- Fringe representation: peak value 1, normalized

- Standard representation Wrms normalized

- Original representation according to Nijbor-Zernike

Norm ISO 10110 allows Fringe and Standard representation

26

drrdrZrWc jj

),(*),(1

1

0

2

0

min)(

2

1 1

i

ijj

N

j

i rZcW

WZZZcTT 1

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

27

Performance Description by Zernike Expansion

Vector of cj

linear sequence with runnin g index

Sorting by symmetry

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

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

circular

symmetric

m = 0cos terms

m > 0

sin terms

m < 0

cj

m

28

Different standardizations used concerning:

1. indexing

2. scaling / normalization

3. sign of coordinates (orientation for off-axis field points)

Fringe - representation

1. CodeV, Zemax, interferometric test of surfaces

2. Standardization of the boundary to ±1

3. no additional prefactors in the polynomial

4. Indexing according to m (Azimuth), quadratic number terms have circular symmetry

5. coordinate system invariant in azimuth

Standard - representation

- CodeV, Zemax, Born / Wolf

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

- coordinate system invariant in azimuth

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),(

Zernikepolynomials: Different Conventions

29

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

30

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

31

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

32

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

33

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

34

Rms of the wavefront changes

with field position

scaled in l

Only one scalar number per field

point:

fine structure suppressed

35

Rms of Wavefront as Function of the Field

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

36

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

37

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

38

Interferograms of Primary Aberrations

Spherical aberration 1 l

-1 -0.5 0 +0.5 +1

Defocussing in l

Astigmatism 1 l

Coma 1 l

39

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

40

Hartmann Shack Wavefront Sensor

detector

xDmeas

u

hf

array

spot

offset

Darray

Dsub

refractive

index n

wave-

front

Lenslet array divides the wavefront into subapertures

Every lenslet generates a simgle spot in the focal plane

The averaged local tilt produces a transverse offset of the spot center

Integration of the derivative matrix delivers the wave front W(x,y)

41

Typical setup for component testing

Lenslet array

Hartmann Shack Wavefront Sensor

subaperture

point spread

function

2-dimensional

lenslet array

a) b)

test surface

telescope for adjust-

ment of the diameter

detector

collimator

lenslet

array

fiber

illumination

beam-

splitter

42