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Lens Design II

Lecture 4: Freeforms

2018-11-14

Herbert Gross

Winter term 2018

2

Preliminary Schedule Lens Design II 2018

1 17.10. Aberrations and optimization Repetition

2 24.10. Structural modificationsZero operands, lens splitting, lens addition, lens removal, material selection

3 07.11. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres

4 14.11. FreeformsFreeform surfaces, general aspects, surface description, quality assessment, initial systems

5 21.11. Field flatteningAstigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses

6 28.11. Chromatical correction IAchromatization, axial versus transversal, glass selection rules, burried surfaces

7 05.12. Chromatical correction IISecondary spectrum, apochromatic correction, aplanatic achromates, spherochromatism

8 12.12. Special correction topics I Symmetry, wide field systems, stop position, vignetting

9 19.12. Special correction topics IITelecentricity, monocentric systems, anamorphotic lenses, Scheimpflug systems

10 09.01. Higher order aberrations High NA systems, broken achromates, induced aberrations

11 16.01. Further topics Sensitivity, scan systems, eyepieces

12 23.01. Mirror systems special aspects, double passes, catadioptric systems

13 30.01. Zoom systems Mechanical compensation, optical compensation

14 06.01. Diffractive elementsColor correction, ray equivalent model, straylight, third order aberrations, manufacturing

Introduction

Examples and Applications

Surface representations

Modelling of real surfaces

Quality assessment

Strategy of imaging optical design

3

Contents

Asphere

Cylindrical lens

Freeform lens

Axicon

Prisms

4

Optical Components

In the TOP-15 journals

Publications of Freeform Systems (until 2011)

Ref.: W. Ulrich

6

Benefit of Generalized Surfaces

Primary goal from industrial development:

1. reduce number of components (small impact)

2. increase functionality

3. cheaper system (usually a dream of the managers)

4. improved performance

General approach: desired critical properties

1. size/volume

2. Field of view

3. F-number/aperture

Increase in performance with more generalized

surfaces:

1. spherical

2. conics and aspheres

3. freeforms

Ref.: K. Fuerschbach

Criteria for performance: aperture and field

Landscape of solutions

Legend:

RS = rotational symmetric

FF = freeform

NT = non-telecentric

T = telecentric

PN = positive-negative

PP = positive-positive

7

Two-Mirror Systems with Freeforms

Ref.: G. West

8

Structure of the Topic FFS for Imaging

Design methods

- selection of optimal representation

- raytrace

- optimization algorithms

- initial system selection

- best location of FFS

- general rules for usage of FFS in design

Realization aspects

- description of localized deviations

- description of MSF ripple

- real surface analysis in simulation

- tolerancing

- adjustment and alignment of systems

Performance and aberrations

- total quality

- surface contributions

- field dependence

- nodal theory

Overview in issues in optical design with freeform surfaces:

1. Modellierung and plattform

1.1 Mathematical description of the surfaces

1.2 Definition of the tools and the interfaces/formats of excange

(optical design, mechanical design, metrology, manufacturing)

1.3 Development of appropriate performance criteria and presentations for systems free

of symmetry with field dependence

1.4 Adaptation and improvement of optimization algorithms

2. Methods of correction in design

2.1 Gettimng initial systems

2.2 Collection of experience in design strategy

2.3 Development of design rules (?)

2.3 Systematic investigation of system types for freeforms

(in particular mirror systems)

3. Metrology and manufacturing

3.1 Tolerancing of surfaces

3.2 Reference points for positioning

3.3 Data re-import of measured surfaces for performance check

3.4 Support of adjustment and integration

Challenges in Optical Design with Freeforms

9

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

TMAScheimpflug

Yolo telescope anamorphotic camera

traditional

early EUV projectors

Schiefspieglertelecopes

11

Aberration Theory and Symmetry

Seid

el, w

ith

fie

ld

Ara

ki

Vecto

rial I -

wit

h f

ield

Sh

ack / T

ho

mp

so

n / S

asia

n

Vecto

rial II -

wit

h f

ield

Fu

ers

ch

ba

ch

Ald

is t

he

ory

- o

ne

ra

y o

nly

Welf

ord

- o

ne O

PD

po

int

on

ly

Ole

szko

- Z

ern

ike

Para

xia

l -

axis

Para

basal aro

un

d r

eal O

AR

/ C

R

2 x

2

4 ×

4

5 ×

5

Rotational

symmetric

Photographic lens,

microscope, zoom lens

Double plane-

symmetricAnamorphic

Freeform (plane-

symmetric)Scheimpflug

Freeform (non-

symmetric)Cubic phase plate for EDF

Rotational

symmetric

Schiefspiegler telescope,

spherical TMA, HMD

Double plane-

symmetricAnamorphic prism stretcher

Freeform (plane-

symmetric)

Unobscured telescope, TMA

corrected, HMD

Freeform (non-

symmetric)

Alvarez plate system,

panoramic zoom system

Rotational

symmetricYolo telescope, spherical

Freeform (non-

symmetric)Yolo telescope, corrected

Aberration theory

4th order 6th orderexact -

all orders

OAR

Straight

1D bend

2D bend

Sample systemsSurface

symmetry

ray transfer matrix

Reference

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

Imaging reflective:

Mirror system without obscuration, especially TMA (surveying, bird watching, astronomy,

planetarium, fundus camera, panoramic camera, EUV lithography,...)

Head Up Display

Broad-band or short-wavelength systems (x-ray)

Compact folded camera lenses

Grating spectrometer setups

Imaging refractive:

Eyeglasses

Anamorphotic camera systems (object-image format adaptation)

Alvarez and related compensator/adjustment components

Scheimpflug setup for metrology

Prism spectrometer setups

Imaging catadioptric:

Head Mounted Device

Illumination:

Illumination shaping coherent

Illumination shaping incoherent

13

Applications of freeforms

14

Anamorphic refractive systems

Refractive correctors

Head mounted device

Eyeglasses

Refractive and Catadioptric Freeform Systems

eye

pupil

image

total

internal

reflection

free formed

surface

free formed

surface

field angle 14°

15

Telescopes

Spectrometer

Lithographic projection systems

Reflective Freeform Systems

HMD Projection System

Special anatomic requirements

Aspects:

1. Eye movement

2. Pupil size

3. Eye relief

4. Field size

5. See-through / look-around

6. Brightness

7. Weight and size

8. Stereoscopic vision

9. Free-forme surfaces and DOE

spectacles

eye

balleye

axis

earfree space

for HMD

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

3-mirror telescope without

obscuration

Example:

Yolo-telescope

Aberration fields:

1. Spot 2. Coma 3. Astigmatism

Yolo Telescope - system without symmetry

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20y

x

y

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20

x

y

-20 -15 -10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20

18

1. General:

efficient and robust calculation

covering most application needs

simple and robust import and export of data

direktedirect relation to tolerancing

fit of measured data easy possible

simple extension of ROI to larger area for fixing the mounting

2. Raytrace:

fast calculation of intersection points

fast calculation of local slopes

3. Optimization:

significant description of usual surfaces with only a few parameter

fast convergence in optimization

4. Types of surfaces:

smooth surfaces, analytical for aberration correction

multi functional surfaces (multi apertures, segmentation, splitting of bundels)

modeling of local deviations of real surfaces (manufactured)

point cloads from measurements, typically noisy

piecewise continuous surfaces (Fresnel surfaces, arrays, segmented mirrors,...)

Requirement on the Surface Description

19

Extended polynomials

classical non-orthogonal monomial representation

Zernike surface

Only useful for circular pupils and low orders

Splines

Localized description, hard to optimize, good for manufacturing characterization

Generalized Forbes polynomials

Promising new approach, two types, strong relation to tolerancing

Radial basis functions

Non-orthogonal local description approach, good for local effect description

Wavelets

Not preferred for smooth surfaces, only feasible for tolerancing

Fourier representation

Classical description without assumptions, but not adapted to aberrations

Smooth vs segmented, facetted, steps, non-Fermat surfaces

Real world is still more complicated

20

Freeform Systems: Surface Representations

21

Freeform Surfaces

Ref.: D. Ochse

local global

basic shape + deformation terms = freeform

Sphere

Conic

Biconic

…

Along surface normal or

z-direction

orthogonal / non-orthogonal orthogonal / non-orthogonal

sag slope sag slope

Representation classification:

1. Global polar based

2. Global cartesian based

3. Local supporting

4. in particular describing high frequency regular ripple

22

Types of Representations

Global defini-tion

Local defini-tion

Polar based

Carte-sian based

Sag ortho-gonal

Gra-dient ortho-gonal

Boun-dary circ

Boun-dary rect.

Zernike x x x x

Zernike differences x x x x

Generalized Forbes x x x x

Off-axis aspheres x x

Monomials x x

Chebychev x x x x

Legendre x x x x

Fourier x x x x

New polynomials x x x x

Splines x x

Radial basis functions x x x

Wavelets x x x

Generalized scheme

1. basic shape 2. prefactor 3. polynomial expansion

Basic shape:

1. best fit sphere (for extracting the non-spherical contribution)

2. circular conic section (for extracting the non-circular contribution)

3. biconic (to cover the astigmatism and keep it out of the expansion)

Prefactor:

regulates wether the correction is along z or projected normal to the surface

Scaling on a: normalization of the lateral extend to be diameter-independent

Expansion polynomials with coefficients bn:

1. orthogonal in sag (Zernike, Legendre, Chebychev)

2. orthogonal in slope (Forbes)

3. Boundary circular / rectangular

4. different weighting functions (uniform, emphasized boundary, ...)

23

Freeform Surface Representations

2 2

2 22 2

22 2 2 20

1

( , )1 1 1

n n

n

r r

a acr rz x y b P

ac r c r

Extended polynomials

Zernike surfaces

Fourier surface

Cubic spline, locally in patch j,k defined as polynomials of order 3

Expansion into non-orthogonal

local shifted Gaussian

functions (RBF)

Generalized Forbes asphere

24

Freeform Surface Representations

mj

mj

jm

yyxx

yxyxA

ycxc

ycxcyxz

,2222

22

)1()1(11),(

mn

m

nnm

yyxx

yxyxZC

ycxc

ycxcyxz

,2222

22

),()1()1(11

),(

mn

yikxik

nm

yyxx

yx ymxneBycxc

ycxcyxz

,2222

22

Re)1()1(11

),(

mn

w

yy

w

xx

nm

yyxx

yx y

n

x

n

eaycxc

ycxcyxz

,2222

22

22

)1()1(11),(

nk

m

j

m n

jkmnkj yyxxayxz

3

0

3

0

, ),(

2

2

1 0

02

200

22

2

2

2

2

22

2

)sin()cos(

1

1

11),(

a

rQmbma

a

r

a

rQa

rc

a

r

a

r

rc

cryxz

m

n

m n

m

n

m

n

m

n

nn

Generalized approach for

orthogonal surface decomposition

Slope orthogonality is guaranteed

and is related to tolerancing

25

Freeform Systems: Forbes Surfaces

Ref: C.Menke/G.Forbes, AOT 2(2013)p.97

2

2

1 0

02

200

22

2

2

2

2

22

2

)sin()cos(

1

1

11),(

a

rQmbma

a

r

a

rQa

rc

a

r

a

r

rc

cryxz

m

n

m n

m

n

m

n

m

n

nn

Representation of a surface as Fourier decomposition

This kind of description typically contains

three ranges of spatial frequencies:

1. figure

2. mid spatial frequency range (MSF)

3. micro roughness

The phase information of the sine-

components is lost in this chart:

no information on localization of pertur-

bation obtained

In Log-diagram the PSD often is nearly

linear decreasing,

the slope is characteristic for the manu-

facturing process

Oscillations can be identified clearly by

peaks

26

Power Spectral Density

fractal model limiting line

slope m = -1.5...-2.5

log A2

Four

low spatial

frequency

figure error,

loss of

resolution

mid

frequency

rangemicro roughness,

loss of contrast

1/

oscillation of the

polishing machine,

turning ripple

10/D1/D 50/D

larger deviations in K-

correlation approach

growing statistical nature

2

21 1( , ) z( , )

2

x yi x y

PSD x yF x y e dx dyA

Diamond turning or milling creates regular ripple in nearly any case

- reason: point-like tooling and tool vs workpiece oscillations

- in case of final polishing effect is strongly reduced

Depending on the ratio of tool size and surface diameter this structure can not be described

by figure representations

27

Regular Ripple Errors

low

frequency

fit

residual

errors

original

a) b) c) d)

Wave aberration field

indices

Normalized field vector: H

normalized pupil vector: rp

angle between H and rp:

Expansion according to the invariants

for circular symmetric components

Vectorial Aberrations

nmj

n

pp

m

p

j

klmp rrrHHHWrHW,,

,

mnlmjk 2,2

y

Hrp

field1

1

pupil

cos

2

2

pp

ppp

rHrHw

rrrv

HHHu

28

x

yrp

s

p

s'

p'

xP

yp

x'

y'

x'P

y'p

object

plane

entrance

pupil

exit

pupilz

system

surfaces

P'

P

H

z'

image

plane

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 prHW111 cos111 prHW magnification

0 0 1 pp rrW020 2

020 prW focus

4

0 0 2 2040 pp rrW

4

040 prW spherical aberration

0 1 1 ppp rHrrW131 cos131 prHW coma

0 2 0 2222 prHW 2

22

222 cosprHW astigmatism

1 0 1 pp rrHHW220

22

220 prHW field curvature

1 1 0 prHHHW311 cos3

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 rHrrHHW331 cos33

331 prHW coma field 3rd

1 2 0 2422 prHHHW 224

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 cos5

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 cos5

151 prHW coma 6th

0 2 1 2242 ppp rHrrW 242

242 cosprHW astigmatism 6th

0 3 0 3333 prHW 333

333 cosprHW trefoil

30

Vectorial Aberration Contributions

Idea of nodal points:

image points of the tilted component axes

Every component has its individual axis, the aberrations are symmetric around this axis

(circular symmetric sub-system)

The axis are bended towards the image plane

Every circular symmetric component therefore has an individual aberration center sj in

the image plane

The interaction and overlay of the various centers are complicated y

x

lens 2

lens 3

aberration

contribution

lens 3

aberration

contribution

lens 2

s3

s2

lens 1

bended axis

rays

aberration

contribution

lens 1

s1

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

s

)(

131

,131

,131

,131

131 c

q

qq

j

q

q

qq

W

W

W

W

a

ss

pppo

c

coma rrraHWW 131

)(

131

zero

coma

green zero

coma

blue

zero

coma

total

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

- reference on parabasal or finite reference sphere

32

Wave Aberration Additivity

P

arbitrary ray

y

1 2 3

y'

P'

surfaces exit pupil

total Wtot

W1

W2 W3

surface contributions

ray pencil

Optimization of systems with freeform surfaces:

- huge number of degrees of freedom

- large differences in convergence according to surface representation

- local vs global influence functions

- definition of performance and formulation of merit function is complicated and

cumbersome

Classical system matrix for local defined splines is ill conditioned

Starting systems:

- still more important as in conventional optics

- only a few well known systems published

- larger archive for starting systems not available until now

- own experience usually is poor

Best location of FFF surfaces inside the system:

- still more important as in the case of circular symmetric aspheres

- no criteria known until now

33

Freeform Systems: Optimization