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Advanced Lens Design

Lecture 7: Aspheres and freeforms

2013-11-26

Herbert Gross

Winter term 2013

1. Aspheres

2. Conic sections

3. Forbes aspheres

4. System improvement by aspheres

5. Free form surfaces

2

Contents

22 yxz

222

22

111 yxc

yxcz

22

22 yxRRRRz xxyy

Conic section

Special case spherical

Cone

Toroidal surface with

radii Rx and Ry in the two

section planes

Generalized onic section without

circular symmetry

Roof surface

2222

22

1111 ycxc

ycxcz

yyxx

yx

z y tan

3

Aspherical Surface Types

222

22

111 yxc

yxcz

1

2

b

a

2a

bc

1

1

cb

1

1

ca

Explicite surface equation, resolved to z

Parameters: curvature c = 1 / R

conic parameter

Influence of on the surface shape

Relations with axis lengths a,b of conic sections

Parameter Surface shape

= - 1 paraboloid

< - 1 hyperboloid

= 0 sphere

> 0 oblate ellipsoid (disc)

0 > > - 1 prolate ellipsoid (cigar )

4

Conic Sections

Conic aspherical surface

Variation of the conical parameter

Aspherical Shape of Conic Sections

z

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y

22

2

111 yc

cyz

Equation

c : curvature 1/Rs

: eccentricity ( = -1 )

radii of curvature :

22

2

)1(11 cy

ycz

2

tan 1

s

sR

yRR

2

32

tan 1

s

sR

yRR

vertex circle

parabolic

mirror

F

f

z

y

R s

C

Rsvertex circle

parabolic

mirror

F

y

z

y

ray

Rtan

x

Rsag

tangential circle

of curvature

sagittal circle of

curvature

Parabolic Mirror

Simple Asphere – Parabolic Mirror

sR

yz

2

2

axis w = 0° field w = 2° field w = 4°

Equation

Radius of curvature in vertex: Rs

Perfect imaging on axis for object at infinity

Strong coma aberration for finite field angles

Applications:

1. Astronomical telescopes

2. Collector in illumination systems

Simple Asphere – Elliptical Mirror

22

2

)1(11 cy

ycz

F

s

s'

F'

Equation

Radius of curvature r in vertex, curvature c

eccentricity

Two different shapes: oblate / prolate

Perfect imaging on axis for finite object and image loaction

Different magnifications depending on

used part of the mirror

Applications:

Illumination systems

Equation

c: curvature 1/R

: Eccentricity

22

2

)1(11 cy

ycz

ellipsoid

F'

F

e

a

b

oblate

vertex

radius Rso

prolate

vertex

radius Rsp

Ellipsoid Mirror

Aspheres - Geometry

z

y

aspherical

contour

spherical

surface

z(y)

height

y

deviation

z

sphere

z

y

perpendicular

deviation rs

deviation z

along axis

height

y

tangente

z(y)

aspherical

shape

Reference: deviation from sphere

Deviation z along axis

Better conditions: normal deviation rs

Perfect stigmatic imaging on axis:

Hyperoloid rear surface

Strong decrease of performance

for finite field size :

dominant coma

Alternative: ellipsoidal surface on front surface and concentric rear surface

Asphere: Perfect Imaging on Axis

1

1

1

1

1

2

2

2

2

n

ns

r

n

s

n

sz

ns

z

r

F

0

100

50

Dspot

w in °0 1 2

m]

Perfect stigmatic imaging on axis:

elliptical front surface

Asphere: Perfect Imaging on Axis

concentric

elliptical

Improvement by higher orders

Generation of high gradients

Aspherical Expansion Order

r

y(r)

0 0.2 0.4 0.6 0.8 1-100

-50

0

50

100

12. order

6. order

10. order8. order

14. order

2 4 6 8 10 12 1410

-1

100

101

102

103

order

kmax

Drms

[m]

Aspheres: Correction of Higher Order

Correction at discrete sampling

Large deviations between

sampling points

Larger oscillations for

higher orders

Better description:

slope,

defines ray bending

y y

residual spherical

transverse aberrations

Corrected

points

with

�y' = 0

paraxial

range

�y' = c dzA/dy

zA

perfect

correcting

surface

corrected points

residual angle

deviation

real asphere with

oscillations

points with

maximal angle

error

Polynomial Aspherical Surface

Standard rotational-symmetric description

0

0,5

1

1,5

2

0 0,2 0,4 0,6 0,8 1 1,2

h

h^4

h^6

h^8

h^10

h^12

h^14

h^16

M

m

m

mhahc

hhz

0

42

22

2

111)(

Ref: K. Uhlendorf

Basic form of a conic section superimposed by a Taylor expansion of z

h ... Radial distance to optical axis

... Curvature

c ... Conic constant

am ... Apherical coefficients

15

Polynomial Aspherical Surface

Forbes Aspheres

From the standard description no info about maximum deviation and how strong the

the asphere is

Departure over 64.1mm CA ??

* Fig. 2.5 of Advanced Optics Using Aspherical Elements (SPIE Press, 2008)

Editors R. Hentschel, B. Braunecker, H. Tiziani.

Some recently patented aspheres. All lengths are in mm, and Ak in units of mm1–k.

Accuracy / sig. digits ??

16 Ref: K. Uhlendorf

Polynomial Aspherical Surface

Forbes Aspheres - Qcon

New orthogonalization and normalization using Jacobi-polynomials Qm

requires normalization radius hmax

(1:1 conversion to standard aspheres possible)

• Mean square slope

M

m

mm hhQahhhc

hhz

0

2

max

4

max22

2

//111

)(

-1

-0,5

0

0,5

1

1,5

2

0 0,2 0,4 0,6 0,8 1 1,2

h

h^4*Q0

h^4*Q1

h^4*Q2

h^4*Q3

h^4*Q4

h^4*Q5

M

m

m ma0

5/

17 Ref: K. Uhlendorf

Polynomial Aspherical Surface

Forbes Aspheres - Qbfs

Limit gradients by special choice of the scalar product

(1:1 conversion to standard aspheres not possible)

• Mean square slope

M

m

mah0

22

max/1

2

max

022

0

22

0

2

0 /:mit 1

1

11)( hhuuBa

h

uu

h

hhz

M

m

mm

-0,5

0

0,5

0 0,2 0,4 0,6 0,8 1 1,2

h

u(1-u)B0

u(1-u)B1

u(1-u)B2

u(1-u)B3

u(1-u)B4

u(1-u)B5

18 Ref: K. Uhlendorf

Polynomial Aspherical Surface

Forbes Aspheres - Qbfs

Same aspheres in terms of Qbfs. Each am is given in units of nm.

Easier to interpret

usually more efficient in optimization

19 Ref: K. Uhlendorf

Polynomial Aspherical Surface

Other descriptions

2222

6

6

4

4

2

4

4

2

2

2

1

1

02

zyxs

scscsC

sbsbB

kA

CBzAz

M

m

N

n

nm

ij

M

m

m

m zhahahk

hz

0 1

2

0

2

22

2

1110

)(

)(

tgz

tfh

Superconic (Greynolds 2002)

• Implicit z-polynomial asphere (Lerner/Sasian 2000)

• Truncated parametric Taylor (Lerner/Sasian 2000)

20 Ref: K. Uhlendorf

Correction on axis and field point

Field correction: two aspheres

Aspherical Single Lens

spherical

one aspherical

double aspherical

axis field, tangential field, sagittal

250 m 250 m 250 m

250 m 250 m 250 m

250 m 250 m 250 m

a

a a

Reducing the Number of Lenses with Aspheres

Example photographic zoom lens

Equivalent performance

9 lenses reduced to 6 lenses

Overall length reduced

Ref: H. Zügge

a) all spherical

9 lenses

Vario Sonnar 3.5 - 6.5 / f = 28 - 56

b) with 3 aspheres

6 lenses

length reduced

aspherical

surfaces

Reducing the Number of Lenses with Aspheres

Example photographic zoom lens

Equivalent performance

9 lenses reduced to 6 lenses

Overall length reduced

Ref: H. Zügge

436 nm

588 nm

656 nm

xpyp

xy

axis field 22°

xpyp

xy

xpyp

xy

axis field 22°

xpyp

xy

A1A3

A2

a) all spherical, 9 lenses

b) 3 aspheres, 6 lenses,

shorter, better performance

Photographic lens f = 53 mm , F# = 6.5

Reducing the Number of Lenses with Aspheres

Binocular Lenses 12.5x

Nearly equivalent performance

Distortion, Field curvature and pupil aberration slightly improved

1 lens removed

Better eye relief distance

a) Binocular 12.5x, all spherical

b) Binocular 12.5x, 1 aspherical surface

field curvature in dptr distortion in %

-2 0 +2 -5 0 +5

yytan sag

-2 0 +2 -5 0 +5

yytan sag

Lithographic Projection: Improvement by Aspheres

Considerable reduction

of length and diameter

by aspherical surfaces

Performance equivalent

2 lenses removable

a) NA = 0.8 spherical

b) NA = 0.8 , 8 aspherical surfaces

-13%-9%

31 lenses

29 lenses

Ref: W. Ulrich

Location depending on correction target:

spherical : pupil plane

coma and astigmatism: field plane

No effect on Petzval curvature

Best Position of Aspheres

-0.5 0 0.5 1-0.2

-0.1

0

0.1

0.2

0.3

0.4

spherical

coma

astigmatism

distortion

d/p'

aspherical

effect

Example:

Lithographic lens

Sensitivities for aspherical correction

Aspherical Sensitivity

S1 S5 S12S16

S23 S28S4stop

5 10 15 20 25 30 350

1

2

3

5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

spherical

aberration

coma

astigmatism

distortion

surface

index

surface

index

surface

index

surface

index

Strong asphere : Turning points z''=0

Deviation from sphere z

Realization Aspects for Aspheres

asphere

0 2 4 6 8 10 12 14 16

0 2 4 6 8 10 12 14 16

0 2 4 6 8 10 12 14 16

-2

-1

0

-0.2

0

0.2

-0.05

0

0.05

r

r

r

profile z(r)

1. derivative z'(r)

2. derivative z''(r)

r

r

0 2 4 6 8 10 12 14 16

0 2 4 6 8 10 12 14 16

profile deviation z(r)

1. derivative z'(r)

0

1

2

3

0

0.5

1

Aplanatic Telescope with two aspheres

Point-by-point determination of aplanatic imaging conditions

Aplanatic Aspherical Systems

secondary

mirror

primary

mirror

F

fP

DP

d

uP u'

fS

image

asphere 1asphere 2

ray 1

ray 2

Special correcting free shaped aspheres:

Inversion of incoming wave front

Application: final correction of lithographic systems

Aspheres – Correcting Residual Wave Aberrations

conventional lenslens with correcting

surface

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

31

Freeform Systems: Surface Representations

Extended polynomials in x,y:

Zernike expansion

Extended Forbes asphere

Expansion in other orthogonal

polynomial systems:

Legendre, Chebychev, ...

Fourier expansion

Expansion into non-orthogonal

local shifted Gaussian

functions (RBF)

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

32

Freeform Systems: Equations of Description

2,

,2222

22

)1()1(11),(

mn

mn

mn

xxxx

yxyxa

xcxc

ycxcyxz

02222

22

),()1()1(11

),(j

jj

xxxx

yxyxZc

xcxc

ycxcyxz

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

n

k

m

j

m n

jkmnkj yyxxayxz

3

0

3

0

, ),(

mn

w

yy

w

xx

nm

yyxx

yx y

n

x

n

eaycxc

ycxcyxz

,2222

22

22

)1()1(11),(

mn

yikxik

nm

yyxx

yx ymxneBycxc

ycxcyxz

,2222

22

Re)1()1(11

),(

Generalized approach for

orthogonal surface decomposition

Slope orthogonality is guaranteed

and is related to tolerancing

33

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

History:

- exact solutions of Fermat-priciple for one wavelength and only a few field points

corresponding to the number of surfaces

- development of algorithms for illumination tailoring

- mostly methods are applicable for illumination and imaging

Dimension:

- 2D is much easier / 3D is complicated and often not unique

SMS-method of Minano

- construction of the surfaces ray by ray with simple procedure

- approved method in illumination and imaging

Ries tailoring

- method used since longer time

- exact algorithm not known

Oliker-Method for illumination

approximation of smooth surface by sequence of parabolic arcs

Reality:

- due to finite size of surcse sna dbroadband applications tailored methods are only useful

for finding a good starting system for optimization

34

Freeform Systems: Exact Tailoring

Arbitrary variation of performance over the transverse cross section

- 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

New aberration types are hard to interpret

Nodal aberration theory

- vectorial approach on aberration theory describes

more general geometries without

symmetry

- nodal points are locations with corrected

aberrations in the field

35

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

Basic idea of tailoring:

- point-wise mapping of surfaces

- fulfillment of Fermat principle

- fulfillment of correct photometry

36

Freeform Systems: Exact Tailoring

Ref: Winston/Minano/Benitez, Nonimaging optics

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

37

Freeform Systems: Optimization

38

Freeform Optical Systems

Optical system

concept

Optical design

model

Automatic

optimization

Merit function

System

constraints

Technological

conditions

Freeform

surfaces

Surface

metrology

Scattered

point cloud

Fitting model

CAD modelPerformance

measurement

System

integration

CMM

manufacturing

Quality

requirements

Performance

calculationVerification

optical design

surface

manufacturing

and metrology

system

integration and

metrology

M1

M2(stop)

M3

General purpose:

- freeform surfaces are useful for compact systems with small size

- due to high performance requirements in imaging systems and limited technological

accuracy most of the applications are in illumination systems

- mirror systems are developed first in astronomical systems with complicated symmetry-free

geometry to avoid central obscuration

HMD

Head mounted device with extreme size constraints

HUD

Head up display, only few surfaces allowed

Schiefspiegler

- astronomical systems without central obscuration

- EUV mirror systems for next generation lithography systems

Illumination systems

Various applications, smooth and segmented

39

Freeform Systems: Applications