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Imaging and Aberration Theory

Lecture 15: Additional topics

2017-02-02

Herbert Gross

Winter term 2016

2

Preliminary time schedule

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

2 26.11. Pupils, Fourier optics, Hamiltonian coordinates

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

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

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

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

6 23.11. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders

7 30.11. Distortion and coma phenomenology, 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 aberrations Dispersion, 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 04.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations

12 11.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, measurement

13 18.01. Point spread function ideal psf, psf with aberrations, Strehl ratio

14 25.01. Transfer function transfer function, resolution and contrast

15 01.02. Additional topics Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability

1. Generalized systems

2. Vectorial aberration theory

3. Field dependence of aberrations

4. Nodal theory

5. Higher order aberrations

6. Aldis theorem

7. Induced aberrations

8. Generalized surface contributions

9. Why aberration theory

3

Contents

Classes according to remaining symmetry

Non-Axisymmetric Systems: Classes and Types

axisymmetric

co-axial

double plane symmetric

anamorphotic

plane symmetric

non-symmetrical

eccentric

off-axis

rot-sym components

3D tilt and decenter

4

Vectorial description

Axis ray as reference

System description by

4-4-matrix

More general : 5x5-calculus

Non-Axisymmetric Systems: Matrix description

image

object

mirror

lens

optical axis

ray

d1

d2

d3

R

DDCC

DDCC

BBAA

BBAA

RR

yyyxyyyx

xyxxxyxx

yyyxyyyx

xyxxxyxx

M'

v

u

y

x

R

5

General Reference and Pupil Re-Scaling

x

y

xP

yp

x'

y'

x'P

y'p

object

plane

entrance

pupil

exit

pupil

image

plane

z

yEnP (rp+Drp)

y'ExP rp

yH

real

ideal

y' (H+DH)

realideal

Wave aberrations: measured as a function of the exit pupil coordinates

Therefore pupil distortion is seen as a change in the entrance

pupil sampling

The changes are in direction (normalized direction vectors)

and in length

6

Wave aberration field

indices

Normalized field vector: H normalized pupil vector: rp

angle between H and rp: q

Expansion according to the invariants for circular symmetric components

Vectorial Aberrations

x

yrp

s

p

s'

p'

xP

yp

x'

y'

x'P

y'p

object

plane

entrance

pupil

exit

pupil

image

plane

z

system

surfaces

P'

P

H

q

nmj

n

pp

m

p

j

klmp rrrHHHWrHW,,

,

mnlmjk 2,2

y

Hrp

field1

1

pupilj

qcos,, 22 ppppp rHrHrrrHHH

7

Transverse Ray Aberrations

Scaled gradient of wavefront:

transverse ray aberrations (L Lagrange invariant)

Expansion

of orders:

with the relations

Two contributions along field vector

and pupil vector

Alternative:

along field and perpendiccular

corresponds to Seidel convention

WL

Wn

RH

pp rr

D1

'

ppprpr rrrHrHpp

2,

p

n

pp

m

p

n

pp

m

p

nmj

j

klm

n

ppr

m

p

nmj

j

klm

m

pr

n

pp

nmj

j

klmpr

rrrrHnHrrrHmHHW

rrrHHHW

rHrrHHWrHW

p

pp

11

,,

,,

,,

2

,

khp rkrhr

0kh

nonorthogonal decomposition orthogonal decomposition

DHDH

Drp

Ah

Ch

Brp

8

Vectorial Aberrations

ord j m n Term scalar Name

0 0 0 0 000W uniform Piston

2

1 0 0 HHW

200 2

200 HW quadratic piston

0 1 0 prHW

111 qcos111 prHW magnification

0 0 1 pp rrW

020 2

020 prW focus

4

0 0 2 2040 pp rrW

4

040 prW spherical aberration

0 1 1 ppp rHrrW

131 qcos131 prHW coma

0 2 0 2222 prHW

q2

22

222 cosprHW astigmatism

1 0 1 pp rrHHW

220 22

220 prHW field curvature

1 1 0 prHHHW

311 qcos3

311 prHW distortion

2 0 0 2400 HHW

4

400 HW quartic piston

6

1 0 2 2240 pp rrHHW

42

240 prHW oblique spherical aberration

1 1 1 ppp rHrrHHW

331 qcos33

331 prHW coma field 3rd

1 2 0 2422 prHHHW

q224

422 cosprHW astigmatism field 4th

2 0 1 pp rrHHW

2

420 24

420 prHW field curvature field 4th

2 1 0 prHHHW

2

511 qcos5

511 prHW distortion field 4th

3 0 0 3600 HHW

6

600 HW piston 6th

0 0 3 3060 pp rrW

6

060 prW spherical aberration 6th

0 1 2 ppp rHrrW

2

151 qcos5

151 prHW coma 6th

0 2 1 2242 ppp rHrrW

q242

242 cosprHW astigmatism 6th

0 3 0 3333 prHW

q333

333 cosprHW trefoil

9

Wave aberration

with shift vector

In 3rd order:

1. spherical

2. coma

3. astigmatism

4. defocus

5. distortion

Systems with Non-Axisymmetric Geometry

q nmj

n

pp

m

pq

j

qqklmp rrrHHHWrHW,,

000,

jjoj HH

p

q

q q

qqqqq

q q

qqqq

q

q

p

q q

qq

q q

qqqq

q q

qq

p

q

qq

q

qq

q

q

ppp

q

qq

q

q

pp

q

qp

rWHW

HWHHWHHW

r

WW

HWWHWW

rWHWHW

rrrWHW

rrWrHW

2

,3110,311

0

2

,31100,3110

2

0,311

2

2

,222,220

0,222,22

2

0,222,220

22

,2220,222

2

0,220

,1310,131

2

,040

2

2

2

1

2

12

2

1

2

1

2

1

,

10

Arbitrary variation of performance over the field of view

- one aberration value is not feasible to describe the distribution over the bundle cross section

- uniformity of aberration variation is important property

- single numbers should be defined to summarize the performance by moments, rms,...

- additional uniformity parameters are necessary to be defined in the merit function

Nodal aberration theory

- vectorial approach on aberration theory describes more general geometries without

symmetry

- nodal points are locations with corrected aberrations in the field

11

Freeform Systems: Performance Assessment

y

x

-8

-6

-4

-2

0

2

4

6

8

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

binodal

points

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

12

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

Field Dependent Zernike Coefficients

Approach to introduce Zernike evaluation into aberrations theory of 6th order:

field dependece of Zernike coefficients

Lengthy analytical formulas

Numerical implementation easy

13

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

Aberration center point

Systems with Non-Axisymmetric Geometry

field

point

y

j

optical

axis ray

image

plane

r

pupil

point

y

H

pupil

plane

e

symmetry

vector

y

aberration

field centreH

o

x

H

jjoj HH

14

Basic concept:

- shifted aberration field centers

- valid for circular symmetric surfaces in general geometry

- every surface has an individual centre of symmetry

- vector summation of individual surface aberration fields

- separation in spherical and aspherical contributions

- multi nodal aberrations in case of decentering

Lack of symmetry:

- individual chief ray height at the surfaces

- field dependence of aberrations fields

Surface in pupil plane: field invariant aberrations

surfaces with distance to pupil: field dependent aberration contribution

Recently extended to freeform surfaces

Nodal Aberration Theory

x x

y y

astigmatisma) centered b) decentered

astigmatism

K. Thompson, Proc. SPIE 7652 (2010)

15

Expanded and rearranged 3rd order expressions:

- aberrations fields

- nodal lines/points for vanishing aberration

Example coma:

abbreviation: nodal point location

one nodal point with

vanishing coma

Nodal Theory

ppp

q

q

q

qq

o

q

qcoma rrrW

W

HWW

,131

,131

,131

)(

131

,131

,131

,131

131 c

q

qq

j

q

q

qq

W

W

W

W

a

pppo

c

coma rrraHWW

131

)(

131

zero

coma

green zero

coma

blue

zero

coma

total

16

Example astigmatism:

abbreviations

General: two nodal points

possible

Special cases

Nodal Theory

q

poq

q

pqoqast rbaHWrHWW22

222

2

222,222

22

,2222

1

2

1

q

q

q

qq

W

W

a,222

,222

222

2

222

,222

,222

2

2

222 aW

W

b

q

q

q

qq

y

x

a222

ib222

-ib222

nodal point 1,

astigmatism corrected

nodal point 2,

astigmatism corrected

constant

astigmatism

image plane

focal

surfaces :

planes

image plane

focal

surfaces :

cones

linear

astigmatism

image plane

focal

surfaces :

parabolas

centered

quadratic

astigmatsim

image plane

focal

surfaces :

complicated

binodal

astigmatism

17

Pseudo-3D-layouts:

eccentric part of axisymmetric system

common axis

Remaining symmetry plane

Schiefspiegler-Telescopes

mirror M1

mirror M3

mirror M2

image

used eccentric subaperture

M1

M3M

2

y

x

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

field points of figure 34-143

18

HMD Projection Lens

eye

pupil

image

total

internal

reflection

free formed

surface

free formed

surface

field angle 14°

y

x

-8

-6

-4

-2

0

2

4

6

8

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

x

-8

-6

-4

-2

0

2

4

6

8

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

binodal

points

-8

-6

-4

-2

0

2

4

6

8

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

-6

-4

-2

0

2

4

6

8

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

-8

-6

-4

-2

0

2

4

6

8

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

astigmatism, 0 ... 1.25 coma, 0 ... 0.34 Wrms

, 0.17 ... 0.58

Refractive 3D-system

Free-formed prism

One coma nodal point

Two astigmatism nodal points

19

Total quality measures, selection depends on application

Circular symmetric systems:

clear definition of optical axis, paraxial range as reference

1. classical primary aberrations, large experience, easy to interpret

2. geometrical representations like transverse or wave aberrations,

fast to calculate, diffraction excluded

3. physical criteria like point spread function (PSF), modulation transfer function (MTF)

General systems with reduced or no symmetry:

1. 3D geometry, freeform surfaces

2. no clear paraxial reference

3. Important: field dependence

Possible general options:

1. geometrical spot diagrams

2. wave aberrations, Zernike representation

3. PSF and MTF

4. Aberration theory of 6th order incorporates field dependence

20

Total Performance Criteria

Sensitivity of surfaces, important for tolerancing,

1. surface contributions, correctability and distribution, induced aberrations

2. structural aberration coefficients (analytic) give relation to system data

Circular symmetric systems:

1. Seidel aberrations in 3rd order (4th order wave aberrations)

2. General aberration theory in 6th order after Shack/Thompson

3. Aldis theorem, but only for one ray

General systems with reduced or no symmetry:

only pure experience on number and location of freeform surfaces as well as complexity and

necessary deviation from circular symmetry

1. 6th order aberration theory for circular symmetric components

2. Wave aberration contribution according to Hopkins / Welford

3. Field dependend Zernike coeffcients

New approaches necessary:

1. Generalization of Aldis theorem

2. Zernike surface contributions

21

Analysis and Sensitivity

Expansion approach for aberrations: cartesian product of invariants of rotational symmetry

Third order aberrations

exponent sum 4

Fifth order aberrations exponent sum 6

Higher Order Aberrations

22

22

222

past

pppcoma

ppsph

yyAW

yxyyCW

yxSW

6223

1

2222

2

2222

1

222

322

pppcomaellcoma

pppskewsphsph

ppskewsphsph

ppplinearcoma

ppzonesphsph

yxyyCW

yxyySW

yxySW

yxyyCW

yxSW

2,,

2

2222pp

pp

yxwyyxxv

yxu

6

5

224

24

33

1

yPW

yyDW

yxyCW

yyAW

yyCW

sphpupspP

pdistdist

pppetzptz

pastast

pcomaellcoma

4

3

222

yPW

yyDW

yxyCW

spP

pdist

ppptz

22

Aldis Theorem

Aldis theorem: surface contribution of transverse aberration of all orders

Calculation by tracing two rays: 1. paraxial marginal ray 2. finite ray

H: Lagrange invariant

A: Paraxial refraction invariant

Transverse aberrations

)( jjjjjjj uchninA

yunH kk

D

D

D

D

D

D

k

j

yjxj

zjzj

jj

yjjj

zkkk

k

j

yjxj

zjzj

jj

xjjj

zkkk

ssss

HyAszA

suny

ssss

xAszA

sunx

1

22

1

22

''

1

''

1

object image

paraxial

marginal ray

arbitrary

finite ray

surfaces 1 2

3 4

5 6

y

y'

u u'

Dx',Dy'

P

P'

23

Properties of Aldis Theorem

Classical Aldis theorem:

1. Advantage of Aldis theorem: contain all orders

2. Larger differences for surfaces/cases with higher order contributions

3. Usually, the reference is the paraxial ray, therefore distortion is taken into account

4. A known formulation is available for aspherical surfaces in centered systems

5. Disadvantage of Aldis theorem: only for one ray

First generalization of Brewer:

Aberrations related on ray data only (real and paraxial)

Therefore applicable for arbitrary surface shapes

Further generalizations:

1. A specialized equation must be used for the case of image in infinity

2. case of non-paraxial image location

3. General 3D geometries and arbitrary surface shapes with the help of parabasal rays

S. Brewer, JOSA 66 (1976) p.8, Surface contribution algorithms for analysis and optimization

24

Example Achromate - Seidel and Aldis contributions at everey surface and in summary

Differences to Seidel terms due to higher

order at cemented surface for larger pupil radii

Aldis Theorem

312

?y’

0.5

-0.5

Transverse

spherical aberrationF/2 Achromat, f’=100

Ref: H. Zügge

- 2

- 1

0

1

2

3

rp

1

Δy'

Surfaces

Sum

1 to 3

- 2

- 1

0

1

2

3

1

Δy'

rp

Surface 1

- 2

- 1

0

1

2

3

Seidel

Aldis

1

Δy'

higher

orders

rp

Surface 2

- 2

- 1

0

1

2

3

1

Δy'

Surface 3

rp

25

Aberration expansion: perturbation theory

Linear independent contributions only in lowest correction order: Surface contributions of Seidel additive

Higher order aberrations (5th order,...): nonlinear superposition - 3rd oder generates different ray heights and angles at next surfaces

- induces aberration of 5th order

- together with intrinsic surface contribution: complete error

Separation of intrinsic and induced aberrations: refraction at every surface in the system

Induced Aberrations

PP'0

initial path

paraxial ray

intrinsic

perturbation at

1st surface

y

1 2 3

y'

intrinsic

perturbation at

2st surface

induced perturbation at 2rd

surface due to changed ray height

change of ray height due to the

aberration of the 1st surface

P'

26

Surface No. j in the system:

intermediate imaging with object, image, entrance and exit pupil

Case a) : object wave perfect, Welford approach

Case b): real wave impinging onto surface:

incidence angles and coordinates changed, induced aberrations taken into account

a) Intrinsic

b) Intrinsic and

induced

Induced Aberrations

entrance

pupil no. j

wave spherical

intermediate

ideal object no. j

surface

index j

exit pupil no. j

wave with intrinsic

aberrations

intermediate

image no. j

entrance

pupil no. j

grid distorted

wave perturbed

intermediate

real object no. j

surface

index j

exit pupil no. j

intrinsic and induced

aberrations intermediate

image no. j

27

Mathematical formulation:

1. incoming aberrations from

previous surface

2. transfer into exit pupil

surface j

3. complete/total aberration

4. subtraction total/intrinsic:

induced aberrations

Interpretation: Induced aberration is generated by pupil distortion together with incoming perturbed

3rd order aberration

Similar effects obtained for higher orders

Usually induced aberrations are larger than intrinsic one

Induced Aberrations

1

1

)5()3(

,

j

i

pipipjentr rWrWrW

pjpj

j

i

pipjpipjexit rWrWrWrrWrW )5()3(

1

1

)5()3()3(

,

1

1

)3()3()5()3(

,,,

j

i

pjipjpj

pjentrpjexitpjcompl

rWrWrW

rWrWrW

1

1

)3()3(

,

j

i

pjipjinduc rWrW

28

Example Gabor telescope - a lens pre-corrects a spherical mirror to obtain vanishing spherical aberration

- due to the strong ray deviation at the plate, the ray heights at the mirror changes

significantly

- as a result, the mirror has induced

chromatical aberration, also the

intrinsic part is zero by definition

Surface contributions and chromatic difference (Aldi, all orders)

Induced Aberrations

1 2 3-1.5

-1

-0.5

0

0.5

1

1.5

= 400 nm

1 2 3-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

= 700 nmmirror

contribution

to color

surfaces surfaces

difference

heigth

difference

with

wavelength= 400 nm

= 700 nm

29

In 3rd order as first perturbation contributions the total system aberration is independent from

the direction

In higher order due to induced aberrations, this reversability is not fulfilled

Important:

1. the light direction can be inverted

2. the coordinate grids and the reference is changing with direction

30

Reversability of Aberrations

paraxrealparaxial start

real back

real start

real back

real start

real back

Example: chromatical aberration for 2x 4f imaging with high/low dispersing lenses

Consequence:

- for systems with large distance of compensating lens groups: system has to be evaluated in

correct order

- for critical systems not the same performance i both directions

31

Reversability of Aberrations

Ref

= 546 nm

=480 nm

=480 nm

-3.00698 -2.40914 -0.67739 -2.8169

Int -2.40914

Ind +0.07955Int -0.67739

Ind +0.26963

SF39

n1 = 20.4

PK50

n2 = 69.74

32

Connection between Wave and Ray Aberrations

'

'

p

p

R Wx

n x

R Wy

n y

D

D

Relation between wave and transverse aberration:

derivative of the wave front according to the pupil coordinates

scaled with radius of reference sphere and index

Approximation for small aberrations and small aperture angles u

A corresponding relation exists between Aldis transverse ray aberration contributions

and Welfords wave aberration contribution

The changing magnification due to the rear system surfaces must be taken into account

yp

z

real ray

wave front W(yp)

R, ideal ray

C

reference

plane

y'D

reference sphere

q

u

Approach of Hopkins / Welford:

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

- 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

34

Options for Surface Contributions

Ord

er

(wav

e a

be

rrat

ion

)

Spat

ial p

up

il re

solu

tio

n

No

n-c

en

tere

d

Fre

efo

rms

Fie

ld d

ep

en

den

ce

Co

mm

en

t

4th order, Seidel 4 N N N Y well known

6th order, Shack/Thompson 6 low Y N Y components circular symmetric

Wavefront, Hopkins/Welford all Y Y Y N one ray only

Aldis all Y N N N one ray only

Aldis generalized all Y Y Y N one ray only

Wavefront all Y Y Y N only numerical, huge information

Zernike high ~ Y Y N problem induced aberrations

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

35

Wave Aberration Additivity

P

arbitrary ray

y

1 2 3

y'

P'

surfaces exit pupil

total Wtot

W1

W2 W3

surface contributions

ray pencil

Surface decomposition of wave aberration

Zernike decomposition of total wave aberration

Zernike decomposition of surface contribution

General relation between coordinates

special case of linear scaling

insertion

case of distortion-free grid projection

For systems with neglectable induced aberrations the bundle diameter scales linear and the

Zernike expansion coefficients are also additive on a normalized bundle radius

If the system suffers from large induced aberrations, the ray grid is distorted and rescaled,

in this case the Zernike coefficients are not exactly additive.

It is also well known, that the Zernikes are changing during propagation

Therefore distance related induced Zernike contributions can be defined in addition

36

Zernike Surface Contributions

G. Dai, Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials,

Appl. Opt. 48 (2009) p.477

s

s

W W

( , ) ( , )p p j p p

j

W x y c Z x y

( , ) ( , )s ps ps js ps ps

j

W x y c Z x y

p p( , y ) , ( , y )ps x p ps y px f x y f x

,p x p p y px m x y m y

( , ) ( , )

( , ), ( , )

p p js ps ps

s j

js x p p y p p

s j

W x y c Z x y

c Z f x y f x y

j js

s

c c

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°

37

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 exactly additive,

difference due to induced aberrations

38

Wavefront Contribution of every Surface

sum of surface contributions

M1 M2 M3

Contributions of the lower

Zernike coefficients per surface

(Fringe convention)

39

Zernike Coefficients per Surface

0

5

10

15

20

25

-3

-2

-1

0

1

2

3

log cj

zernike

index j

astigmatism

surfaces

sum

M1

M2

M3

comaspherical

tilt defocus

Fourier Filtering

Digital optics with pupil phase mask

Primary image blurred

Digital reconstruction with the help of

the system transfer function

Objective tube lens

digital image

Iimage(x') Pupil with

phase mask

transfer function ImageComputer

image digital

restored

Object

image

40

a) object

Image quality with Real Objects

b) good image c) defocussed d) axial chromatic

aberration

e) lateral chromatic

aberration

g) chromatical

astigmatism

f) sphero-

chromatism

41

Real Image with Different Chromatical Aberrations

original object good image color astigmatism 2

6% lateral color axial color 4

42

Time is Over