www.iap.uni-jena.de

Lens Design I

Lecture 7: Aberrations II

2019-05-23

Herbert Gross

Summer term 2019

2

Preliminary Schedule - Lens Design I 2019

1 11.04. Basics

Introduction, Zemax interface, menues, file handling, preferences, Editors,

updates, windows, coordinates, System description, 3D geometry, aperture,

field, wavelength

2 18.04. Properties of optical systems IDiameters, stop and pupil, vignetting, layouts, materials, glass catalogs,

raytrace, ray fans and sampling, footprints

3 25.04. Properties of optical systems II

Types of surfaces, cardinal elements, lens properties, Imaging, magnification,

paraxial approximation and modelling, telecentricity, infinity object distance

and afocal image, local/global coordinates

4 02.05. Properties of optical systems IIIComponent reversal, system insertion, scaling of systems, aspheres, gratings

and diffractive surfaces, gradient media, solves

5 09.05. Advanced handling IMiscellaneous, fold mirror, universal plot, slider, multiconfiguration, lens

catalogs

6 16.05. Aberrations IRepresentation of geometrical aberrations, spot diagram, transverse

aberration diagrams, aberration expansions, primary aberrations

7 23.05. Aberrations II Wave aberrations, Zernike polynomials, measurement of quality

8 06.06. Aberrations III Point spread function, optical transfer function

9 13.06. Optimization IPrinciples of nonlinear optimization, optimization in optical design, general

process, optimization in Zemax

10 20.06. Optimization IIInitial systems, special issues, sensitivity of variables in optical systems, global

optimization methods

11 27.06. Advanced handling IISystem merging, ray aiming, moving stop, double pass, IO of data, stock lens

matching

12 04.07. Correction ISymmetry principle, lens bending, correcting spherical aberration, coma,

astigmatism, field curvature, chromatical correction

13 11.07. Correction IIField lenses, stop position influence, retrofocus and telephoto setup, aspheres

and higher orders, freeform systems, miscellaneous

1. Optical path difference

2. Definition of wave aberrations

3. Zernike polynomials

4. Measurement of wave aberrations

3

Contents

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

4

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

5

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

6

Wave Aberration

Definition of the peak valley value

exit

aperture

phase front

reference

sphere

wave

aberration

pv-value

of wave

aberration

image

plane

7

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

8

0),(1

),( dydxyxWF

yxWExP

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

9

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

Tilt of Wavefront

10

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

11

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,

12

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),(

Zernike Polynomials: Different Nomenclatures

13

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

14

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

15

Zernike Polynomials: Explicite Formulas

n mPolar coordinates Interpretation

0 0 1 1 piston

1 1 r sin x

Four sheet 22.5°

1 -1 r cos y2 2 r

22sin 2xy

2 0 2 12

r 2 2 12 2

x y

2 -2 r2

2cos y x2 2

3 3 r3

3sin 32 3

xy x

3 1 3 23

r r sin 3 2 33 2

x x xy

3 -1 3 23

r r cos 3 2 33 2

y y x y

3 -3 r3

3cos y x y3 2

3

4 4 r4

4sin 4 43 3

xy x y

4 2 4 3 24 2

r r sin 8 8 63 3

xy x y xy

4 0 6 6 14 2

r r 6 6 12 6 6 14 4 2 2 2 2

x y x y x y

4 -2 4 3 24 2

r r cos 4 4 3 3 44 4 2 2 2 2

y x x y x y

4 -4 r4

4cos y x x y4 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°

16

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

17

Interferograms of Primary Aberrations

Spherical aberration 1 l

-1 -0.5 0 +0.5 +1

Defocussing in l

Astigmatism 1 l

Coma 1 l

18

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

19

OPD along x- and y-direction for all fields and colors

Wave surface for one wavelength and field point

20

Wave Aberations in Zemax

Change of Wrms value with

1. wavelength

2. field

3. defocus

4. field position

21

Wave Aberations in Zemax

Available:

1. Fringe

2. Standard

3. Tatian-Zernike (ring pupil)

Usual:

Coefficients for one wavelength and one

field point in the image

Also possible:

- changes over the field coordinate

- at every surface in the system

- on a subaperture

Calculation of Strehl ratio in Marechal

approximation

22

Zernike Coefficients in Zemax

New option in Zemax:

- change of selected number of

Zernike coefficients as a function

of the field coordinate

- several settings can be chosen

Shows the uniformity of quality

over the image field

23

Zernike Variation over Field

Rms field map

- change of quality criterion

over field

- several options for criteria:

spot size, wave rms, Strehl

Representation of uniformity

of correction

Example: spot diameter changes

between 6.6 and 8.6 mm

24

Correction Variation over Field