www.iap.uni-jena.de

Imaging and Aberration Theory

Lecture 5: Aberrations representations

2018-11-15

Herbert Gross

Winter term 2019

2

Schedule - Imaging and aberration theory 2019

1 18.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems

2 25.10.Pupils, Fourier optics, Hamiltonian coordinates

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

3 01.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media

4 08.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas

5 15.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations

6 22.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders

7 29.11. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options

8 06.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options

9 13.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spectrum

10 20.12.Sine condition, aplanatism and isoplanatism

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

11 10.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations

12 17.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement

13 24.01. Point spread function ideal psf, psf with aberrations, Strehl ratio

14 31.01. Transfer function transfer function, resolution and contrast

15 07.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsicand induced aberrations, reversability

1. Stop shift formulas

2. Lens aberration contributions

3. Pupil aberrations

4. Representation of geometrical aberrations

5. Representation of wave aberrations

6. Miscellaneous

7. Aberration measurements

3

Contents

Refracting Surface VII: Eikonal

Eikonal of 4th order

Coefficients

1. Spherical aberration

2. Astigmatism

3. Field curvature

4. Distortion

5. Coma

pppppp

ppppA

yyxxyxp

pssCyyxxyx

p

pssDyyxxyx

p

pssP

yyxxp

pssAyx

p

sSyx

p

psKL

22

4

322

4

322

4

22

2

4

22222

4

4222

4

4)4(

2

)(

2

)(

4

)(

2

)(

88

)(

''

11)'(

''

11)(

''

11)'(

''

11)(

''

11)'(

''

11)'(

''

11)'(

'

1

'

1''

11)'(

3

2

3

3

2

3

2

3

2

2

2

23

snnsQQ

R

bnnC

snnsQQQ

snnsQ

R

bnnD

snnsQQQ

snnsQQ

R

bnnP

snnsQ

R

bnnA

snnsQ

R

bnnS

sRssn

sRsns

R

bnnK

p

pp

ppp

pp

pp

p

pp

4

Stop Shift Formulas

If the stop is moved, the chief ray takes a modified way through the system

Approach: expansion of the surface coefficient formulas for small changes in the

pupil position p, p‘

The stop shift formulas shows the change of the Seidel coefficients due to this

effect

Also possible:

set of formulas for object or image shift

applicable for curved objects

imageplane

ray

y'chief ray 1

chief ray 2

ExP 1ExP 2

p'1

rpmax

z

p'2

5

Stop Shift Formulas

Stop shift formulas excplicite with the help of the

moving parameter

sph

coma

ast

curv

dist

Mix of aberration types due to stop shift: induced aberrations

Examples:

1. spherical aberration induces coma

2. coma induces astigmatism

,

h

hhE oldnew

II SS

IIIII SESS

IIIIIIIII SESESS 2

IVIV SS

IIIIVIIIVV SESESSESS 3233

old

position

new

position

old

chief raynew

chief ray

oldhnewh

6

Lens Contributions of Seidel

In 3rd order (Seidel) :

Additive contributions of thin lenses (equal w) to the total aberration value

(stop at lens position)

Spherical aberration

X: lens bending

M: position parameter

Coma

Astigmatism

Field curvature

Distortion

2

22

23

3 2

)1(

2

)1(2

1

2

1)1(32

1M

n

nnM

n

nX

n

n

n

n

fnnSlens

MnX

n

n

fnsClens )12(

1

1

'4

12

2'2

1

sfAlens

2'4

1

snf

nPlens

0lensD

7

Lens Contributions of Seidel

Spherical aberration

Special impact on correction:

1. Special quadratic dependence on

bending X

Minimum at

2. No correction for small n and M

3. Correction for large

n: infrared materials

M: virtual imaging

Limiting value

2

22

23

3 2

)1(

2

)1(2

1

2

1)1(32

1M

n

nnM

n

nX

n

n

n

n

fnnSlens

sphW

X

M = 6M = - 6

M = - 3M = 0

M = 3

n = 1.5

X

n

nMsph min

2 1

2

2

M

n n

ns

0

2

2

2

1

8

Photographic lens

Incidence angles for chief and

marginal ray

Field dominant system

Quasi symmetry can be seen at the

surface contributions of field aberrations

Symmetry disturbed for spherical

aberration

incidence angle

chief

ray

Photographic lens

60

40

20

0

20

40

60

marginal

ray

1

1 2 3 4 5 6 7 8 9 10 11 12 13

2 3

4

7

6 8

1112 13

sum

spherical

coma

astigmatism

curvature

distortion

axial

chromatic

lateral

chromatic

-0.1

0

0.1

-0.1

0

0.1

-0.1

0

0.1

-0.2

0

0.2

-0.4-0.2

0

0.20.4

-0.02

0

0.02

1 2 3 4 5 6 7 8 9 10 11 12 13

-0.01

0

0.01

9

Microscopic Objective Lens

Incidence angles for chief and

marginal ray

Aperture dominant system

marginal ray

chief ray

incidence angle

0 5 10 15 20 2560

40

20

0

20

40

60

microscope objective lens

10

Microscopic Lens

Large distance system

Problems with large diameters

Aplanatic front group

Not corrected for curvature and

distortion

Astigmatic contributions of

cemented surfaces corrected by

rear group

Sign of lateral chromatic

aberration in front group

spherical

coma

astigmatism

curvature

distortion

axial

chromatic

lateral

chromatic

10 151 sum

3174

810

1316

20 23

29

-1

0

1

-0.02

0

0.02

-2

0

2

-2

0

2

-4-2024

-0.05

0

0.05

-1

0

1

5 20 25

11

Lithographic Lens

Large effect of mirror on field curvature

Typical bulge structure shows the correction

of field curvature according to the Petzval

theorem

-0.2

0

0.2

10 20 30 40 6050

1. bulge 2. bulge 3. bulge1. waist 2. waist

-1

0

1

concave

mirror1. intermediate

image2. intermediate

image

12

Primary Aberration Spot Shapes

Simplified set of Seidel formulas:

field point only in y‘ considered

Spherical aberration S:

circle

Coma:

shifted circle

Astigmatism:

focal line

Field curvature:

circle

Distortion:

shifted point

PpPppp

PpPppp

ryPryCrSx

yDryPAryCrSy

sin'''2sin'''sin'''

''cos'')''2()2cos2('''cos'''

223

3223

6222

33

''''

sin''',cos'''

p

pppp

rSyx

rSxrSy

422222

22

''''''2''

2sin'''' ,)2cos2(''''

pp

PpPp

ryCryCyx

ryCxryCy

0',cos'''2' 2 xryAy Pp

24222

22

'''''

sin'''',cos''''

p

PpPp

ryPyx

ryPxryPy

0',''' 3 xyDy

13

Primary Aberration Spot Shapes

Schematically:

14

Optical Image Formation

optical

system

object

plane

image

plane

transverse

aberrations

longitudinal

aberrations

wave

aberrations

Perfect optical image:

All rays coming from one object point intersect in one image point

Real system with aberrations:

1. tranverse aberrations in the image plane

2. longitudinal aberrations from the image plane

3. wave aberrations in the exit pupil

15

Longitudinal aberrations s

Transverse aberrations y

Representation of Geometrical Aberrations

Gaussian image

plane

ray

longitudinal

aberration

s'

optical axis

system

U'reference

point

reference

plane

reference ray

(real or ideal chief ray)

transverse

aberrationy'

optical axis

system

ray

U'

Gaussian

image

plane

reference ray

longitudinal aberration

projected on the axis

l'

optical axis

system

ray

l'o

logitudinal aberration

along the reference ray

16

Representation of Geometrical Aberrations

ideal reference ray angular aberrationU'

optical axis

system

real ray

x

z

s' < 0

W > 0

reference sphere

paraxial ray

real ray

wavefront

R

C

y'

Gaussian

reference

plane

U'

Angle aberrations u

Wave aberrations W

p

pp

pp y

yxW

y

R

u

yy

y

Rs

),(

'sin

'''

2

17

Angle Aberrations

Angle aberrations for a ray bundle:

deviation of every ray from common direction of the collimated ray bundle

Representation as a conventional spot diagram

Quantitative spreading of the collimated

bundle in mrad / °

perfect

collimated

real angle

spectrum

real beamu

z

Typical representation:

Longitudinal aberration as function of aperture

(pupil coordinate)

If correction at the edge: maximum residuum

at the zone

Typical: largest gradients at the edge

Correcting aspheres or high NA of

higher order:

oscillatory behavior

2/1

Aperture Dependence of Longitudinal Aberration

-0.04 0 0.04 0.08 0.12 0.16 0.2

rp

s'

l = 644 nm

l = 546 nm

l = 480 nm

rp

s'-3.00E-003-2.40E-003-1.80E-003-1.20E-003-6.00E-0040.00E+0006.00E-0041.20E-0031.80E-0032.40E-003

19

spherical aberration4 colors

coma in zone4 colors

coma in full field4 colors

image shells/astigmatism4 colors

distortionmain color

chromatical differencein magnification

secondarychromatic

Longitudinal Aberration Chart

20

Pupil Sampling

Ray plots

Spot

diagrams

sagittal ray fan

tangential ray fan

yp

y

tangential aberration

xp

y

xp

x

sagittal aberration

whole pupil area

Tangential / sagittal / skew rays

View along optical axis

22

Ray Selection Planes

x

ytangential

meridional

rays

x

ysagittal

rays

x

yskew

rays

Transverse Aberrations

Typical low order polynomial contributions for:

defocus, coma, spherical aberration, lateral color

This allows a quick classification of real curves

linear:

defocus

quadratic:

coma

cubic:

spherical

offset:

lateral color

pprSy cos''' 3

)2cos2('''' 2

PpryCy

pprKy cos'''

23

Combinations of basic shapes

24

Interpretation of Transverse Aberrations

defocus+spherical+coma

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

defocus+comaspherical+coma+distortion

spherical 3rd+spherical 5th

spherical+defocus

defocus+spherical 5th

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

yp

y'

yp

y'

yp

y'

yp

y'

yp

y'

yp

y'

Transverse Aberrations

y x

xp

yp

1

-1

1-1

5 m5 m

l= 486 nm

l= 588 nm

l= 656 nm

Dy

Dx

tangential sagittal

Classical aberration curves

Strong relation to spot diagram

Usually only linear sampling along the x-, y-axis

no information in the quadrant of the aperture

25

Best resolution:

- bright central peak in spot

- tangent at transverse aberration

curve

Best contrast:

- mean straight line over complete

pupil of transverse aberration curve

- smallest maximal deviation

Different criteria give slightly different

best image planes

Best Image plane

0

z

Wrms

0

z

Ds

u

y

best matching

in the

centre

minimal difference

ymaxy

min

ymax

26

Transverse Aberrations

Characteristic chart for the representation of transverse aberrations

axis field zone full field

meridional

deviation

sagittal

deviation

yp

y

x

yy

x

yp yp

xp xp

wavelengths:

365 nm

480 nm

546 nm

644 nm

27

Pupil Aberration

rp

'yp/yp

1%

2%

Characteristic chart for the representation of pupil aberration

Distortion of the pupil grid from the entrance to the exit pupil

Pupil aberration can be interpreted as the spherical aberration of the chief ray for the

pupil imaging

yp

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

xp

ideal

real

28

Sine condition not fulfilled:

- nonlinear scaling from entrance to exit pupil

- spatial filtering on warped grid, nonlinear sampling of spatial frequencies

- pupil size changes

- apodization due to distortion

- wave aberration could be calculated wrong

- quantitative mesaure of offence against the sine condition (OSC):

distortion of exit pupil grid

Sine Condition

xo xp

sphere distorted exit

pupil surface

object

plane

exit

pupil

optical

system

sx

u

xp

x'o

u'

x'p

x'p

image

plane

entrance

pupil

grid

distortion

1sin

unf

xD

ap

p

Photometric effect of pupil distortion:

illumination changes at pupil boundary

Effect induces apodization

Sign of distortion determines the effect:

outer zone of pupil brighter / darker

Additional effect: absolute diameter of

pupil changes

OSC and Apodization

focused +20 m +50 m-20 m-50 m

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-0.05 - barrel

+0.05 - pincushion

no

distortion

rp

intensity

Variation of Chromatical Aberrations

Representation of the image location

as a function of the wavelength

axial chromatical shift

Representation of the chromatical

magnification difference with field

height

lateral chromatical aberration

l

-100 -80 -60 -40 -20 0 20 40 60 80 100

500 nm

s' [m]400 nm

600 nm

y' [m]

field height y/ymax

-15 -12 -9 -6 -3 0 3 6 9 12 15

Airy1

31

Spot Diagram

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

32

Spot Diagram

Variation of field and color

Scaling of size:

1. Airy diameter (small circle)

2. 2nd moment circle (larger circle, scales with wavelength)

3. surrounding rectangle 486 nm 546 nm 656 nm

axis

fieldzone

fullfield

33

Gaussian Moment of Spot

Spot pattern with transverse aberrations xj and yj

1. centroid

2. 2nd order moment

3. diameter

Generalized:

Rays with weighting factor gj:

corresponds to apodization

Worst case estimation:

size of surrounding rectangle Dx=2xmax, Dy = 2ymax

xN

xS jj

1

yN

yS jj

1

M rN

x x y yG j S j Sj

22 21

GMD 2

M rN

g x x y yG

G

j j S j S

j

22 21

34

Practical problem in analysis of classical

spot diagrams:

relation between deviations and pupil location

is lost

Idea of Kingslake:

transverse aberrations of spot points drawn

in pupil intersection points

x and y at every surface in the pupil

sampling grid

r at all surfaces in the pupil sampling grid

Problems:

1. proper representation of quite different

scales

2. distorted grid in case of induced aberrations

35

Kingslakes Diagram

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

xp

yp

J. Palmer, Lens aberration data, Monographs on Apllied Optics, A. Hilger, 1971

Extension of Kingslakes

representation for surface

contributions

Problem:

compaction of high

complexity,

limited clearness

36

Kingslakes Diagram Extended

x

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

M3

M1

M2

Aberrations of a Single Lens

Single plane-convex lens,

BK7, f = 100 mm, l = 500 nm

Spot as a function of field position

Coma shape rotates according to circular symmetry

Decrease of performance with the distance to the axis

Example HMD without symmetry

x

y

y

x

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

eye

pupil

image

total

internal

reflection

free formed

surface

free formed

surface

field angle 14°

37

negativ spherical aberration

intrafocal: compact broadened spot with bright edge

extrafocal: ring structure

positiv spherical aberration

intrafocal: ring structurewith bright center

extrafocal: ring structure with bright outer ring

coma

bending of caustic

shifted center of gravity

Caustic of Spherical Aberration and Coma

Ref: W. Singer

spherical negativ

spherical positiv

coma

z

38

Wave Aberration

Definition of the peak valley value

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

39

Wave Aberrations

Classification of wave aberrations for one image point:

Zernike polynomials

Mean root square of wave front error

Normalization: size of pupil area

Worst case / peak-valley wave front error

Generalized for apodized pupils (non-uniform illumination)

dydxAExP

ppppmeanpp

ExP

rms dydxyxWyxWA

WWW222 ,,

1

pppppv yxWyxWW ,,max minmax

pppp

w

meanppppExPw

ExP

rms dydxyxWyxWyxIA

W2)(

)(,,,

1

40

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

41

Typical Variation of Wave Aberrations

Microscopic objective lens:

Changes of rms value of wave aberration with

wavelength

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

Representation of the wave aberration

with field position

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

Wrms [l]

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

diffraction limit

field

w [°]0 2.8 5.6 8.4 11.2 14

l= 480 nm

l= 546 nm

l= 644 nm

42

Typical Variation of Wave Aberrations

Representation of the wave aberration

for defocussing at several field points

- decrease of performance with field

height

- field curvature

Wavefront over the pupil as surface

Wprms [l]

-0.4 -0.2 0 0.2 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

axis

zone

full field

axis field zone full field+1 l

-1 l

0

43

Typical Variation of Wave Aberrations

Representation of the wave aberration

as a function of field and wavelength

for a microscopic lens

Analysis:

1. diffraction limited correction near to

axis for medium wavelength range

2. no flattening

3. blue edge more critical than red edge

0.06

0.09

0.12

0.1

5

0.15

0.1

8

0.180.21

0.24

0.0

714

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64

y'

l

[m]

diffraction

limit

44

Contributions of the lower Zernike coefficients per surface,

In logarithmic scale not additive

(Fringe convention)

Error in additivity due to

numerical reasons for

astigmatism

Effect of induced aberrations

and grid distortion in the

range of l / 20 in this case

45

Zernike Coefficients per Surface

3

Log cjs

index j

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

cj [l]

index j

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25-3

-2

-1

0

1

2M1 M1 M3

tilt def ast com sph

Example system: plane symmetric TMA system

nearly diffraction limited correction for a small field

of view

M1: off axis asphere, M2, M3: freeforms

F-number 1.8, field -1°...+1°

46

TMA System

x = -1° x = +1°

y = +1°

x = 0°

y = -1°

y = 0°

field

angles x/y

M1

circular

symmetric

asphere

M2

pupil

freeform M3

freeform

image

M2

freeform

M3

freeform

M1

asphere

Surface contributions of every mirror with parabasal reference

pupil rescaling neglected

Dominating astigmatism

Sum of wave aberration not additive,

difference due to induced aberrations

47

Wavefront Contribution of every Surface

sum of surface contributions

M1 M2 M3

Low order Zernikes as a function of the field position

Completly different distributions,

Complete characterization gives a huge amount of detailed information.

Also analytical solution for lower orders provided in the literature

48

Zernikes as Function of the Field

R. Gray, C. Dunn, K. Thompson, J. Rolland, Opt. Expres 20(2012) p. 16436, An analytic expression for the field

dependence of Zernike polynomilas in rotational symmetric optical systems

astigmatismcomaspherical aberration

49

PSD Ranges

Typical impact of spatial frequency

ranges on PSF

Low frequencies:

loss of resolution

classical Zernike range

High frequencies:

Loss of contrast

statistical

Large angle scattering

Mif spatial frequencies:

complicated, often structured

fals light distributions

log A2

Four

low spatial

frequency

figure errormid

frequency

range micro roughness

1/l

oscillation of the

polishing machine,

turning ripple

10/D1/D 50/D

larger deviations in K-

correlation approach

ideal

PSF

loss of

resolution

loss of

contrast

large

angle

scattering

special

effects

often

regular

Spatial Frequency of Wavefront Aberrations

The spatial frequency determines the effect of the wave front aberration

Characteristic ranges, scaled on the diameter of the pupil:

- figure error : Zernike

causes resolution loss

- midfrequency range

- high frequency : roughness

causes contrast loss

log I(r)

0 5 10 15 20 2510

-6

10-5

10-4

10-3

10-2

10-1

100

r/rairy

g/D = 0.5

g/D = 4

g/D = 2

g/D = 1

g/D = 0.25

g/D = 0.07g/D = 0.1

50

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

Spot Pattern of a HS - WFS

a) spherical aberration b) coma c) trefoil aberration

Aberrations produce a distorted spot pattern

Calibration of the setup for intrinsic residual errors

Problem: correspondence of the spots to the subapertures

21

1121 )''(''

yy

yssss y

Hartmann Method

Similar to Hastmann Shack Method with simple hole mask and two measuring

planes

Measurement of spot center position as geometrical transverse aberrations

Problems: broadening by diffraction

Hartmann

screen with

hole mask

test optic

y

focal

plane

first measuring

plane

second measuring

plane

reference

plane

s'1

y

2

s'

s'

y1

y2

u'

Hartmann Method

Real pinhole pattern with signal

Problems with cross talk and threshold

L

threshold

peak

minimum

c)

D

d a

b

a) b)

Testing with Twyman-Green Interferometer

Short common path,

sensible setup

Two different operation

modes for reflection or

transmission

Always factor of 2 between

detected wave and

component under test

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

Interferograms of Primary Aberrations

Spherical aberration 1 l

-1 -0.5 0 +0.5 +1

Defocussing in l

Astigmatism 1 l

Coma 1 l

56

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

57