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Lens Design II

Lecture 3: Aspheres

2018-11-07

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

1. Aspheres

2. Conic sections

3. Forbes aspheres

4. System improvement by aspheres

5. Aspheres in Zemax

3

Contents

4

Aspherical Correction

Correction of spherical aberration by

an asphere

Ref: A. Herkommer

a) spherical

lens

b) aspherical

lens

refraction too

strong

asphere reduces

power

2

2 21 1 1

c rz

c r

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 )

5

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

r

2

2 21 1 1

crz

c r

6

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

7

Aspherical Surface Types

Polynomial Aspherical Surface

Standard rotational-symmetric description

22 4

2 20

(r)1 1 1

Mm

m

m

crz a r

c r

Ref: K. Uhlendorf

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

r ... radial distance to optical axis

c curvature

conic constant

am aApherical coefficients

2 2r x y

8

r

z(r)

r 4

r 6

r 8

r 10

r 12

r 14

r 16

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)

9Ref: K. Uhlendorf

9

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

10

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

11

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

12

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

13

Perfect stigmatic imaging on axis:

elliptical front surface

Asphere: Perfect Imaging on Axis

concentric

elliptical

14

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]

15

Xray telescopeWolter type I

Nested shells with gracing incidence

Increase of numerical aperture by several shells

Gracing Incidence-Xray Telescope

detector

hyperboloids Wolter type I

rays

paraboloids

nested cylindrical

shells

towards paraboloid

focus point

Woltertyp

1. Paraboloid

2. Hyperboloid

Gracing Incidence-Xray Telescope

Aspheres - Geometry

z

r

aspherical

contour

spherical

surface

z(r)

height

r

deviation

z

Reference: deviation from sphere

Deviation z along axis

Better conditions: normal deviation rs

18

sphere

z

r

perpendicular

deviation rs

deviation z

along axis

height

r

tangente

z(r)

aspherical

shape

Improvement by higher orders

Generation of high gradients

Aspherical Expansion Order

r

z(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]

19

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

r r

residual spherical

transverse aberrations

Corrected

points

with

r' = 0

paraxial

range

r' = c dzA/dr

zA

perfect

correcting

surface

corrected points

residual angle

deviation

real asphere with

oscillations

points with

maximal angle

error

20

Polynomial Aspherical Surface

Forbes Aspheres - Qcon

New orthogonalization and normalization using Jacobi-polynomials Qm

requires normalization

radius rmax

(1:1 conversion to standard

aspheres possible)

• Mean square slope

24 2

max max2 2

0

(r) / r / r1 1 1

M

m m

m

crz r a Q r

c r

M

m

m ma0

5/

Ref: K. Uhlendorf

21

r

z(r)

r 4Q0(r)

r 4Q1(r)

r 4Q2(r)

r 4Q3(r)

r 4Q4(r)

r 4Q5(r)

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 2

22

max max0

2 2 2 20 max0 0

1

(r) 1 1 1

M

m m

m

r r

r rc r rz a B

rc r c r

Ref: K. Uhlendorf

22

r

z(r)

u = (r/rmax)2

u(u-1)B0(u)

u(u-1)B1(u)

u(u-1)B2(u)

u(u-1)B3(u)

u(u-1)B4(u)

u(u-1)B5(u)

23

Forbes Aspheres

Strong asphere Qcon Mild asphere Qbfs

sag along z-axis difference to best fit sphere

sag along local surface normal

slope orthogonal not slope orthogonal

true polynom not a polynomial due to projection

type Q 1 in Zemax type Q 0 in Zemax

direct tolerancing of coefficients no direct relation of coefficients to slope

2 2

22

max max0

2 2 2 20 max0 0

1

(r) 1 1 1

M

m m

m

r r

r rc r rz a B

rc r c r

24 2

max max2 2

0

(r) / r / r1 1 1

M

m m

m

crz r a Q r

c r

r

z(r)

u = (r/rmax)2

u(u-1)B0(u)

u(u-1)B1(u)

u(u-1)B2(u)

u(u-1)B3(u)

u(u-1)B4(u)

u(u-1)B5(u)r

z(r)

r 4Q0(r)

r 4Q1(r)

r 4Q2(r)

r 4Q3(r)

r 4Q4(r)

r 4Q5(r)

24

Selection of Asphere Types

Correction of Retro focus type camera lens

F# = 2.8 , d=21 , 2w = 94°

Considerably better resukt and faster optimization

by the use of Q aspheres

Ref: C. Menke

a) standard asphere b) Qbfs asphere

Asphere far from pupil:

- ray bundels of field points

separated

- field dependend correction

- also impact on distortion

Asphere near pupil:

- all ray bundels equally affected

- problem field angles: coma

25

Impact of Asphere

surface 2

surface 15

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

26

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

27

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

28

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

29

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

30

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

31

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

32

Selection of one aspherical

surface in a photographic lens

33

Aspherization of a Camera Lens

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.05

0.1

0.15

0.2

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.05

0.1

0.15

0.2

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.1

0.2

0.3

0.4

1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.5

1

1.5

spherical

aberration

coma

astigmatism

distortion

surface

index

surface

index

surface

index

surface

index

S14S 9 S 11S 5S 2

spherical system:

197 nm

surface 2:

196 nmsurface 5:

185 nm

surface 9:

187 nm

surface 11:

278 nm

surface 14:

178 nm

Handy Phone Objective lenses

Examples

Ref: T. Steinich

US 7643225L = 4.2 mm , F'=2.8 , f = 3.67 mm , 2w=2x34°

US 6844989L = 6.0 mm , F'=2.8 , f = 4.0 mm , 2w=2x31°

EP 1357414L = 5.37 mm , F'=2.88 , f = 3.32 mm , 2w=2x33.9°

Olympus 2L = 7.5 mm , F'=2.8 , f = 4.57 mm , 2w=2x33°

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

35

Setting of surface properties

Surface properties and settings

36

Standard spherical and conic sections

Even asphere classical asphere

Paraxial ideal lens

Paraxial XY ideal toric lens

Coordinate break change of coordinate system

Diffraction grating line grating

Gradient 1 gradient medium

Toroidal cylindrical lens

Zernike Fringe sag surface as superposition of Zernike functions

Extended polynomial generalized asphere

Black Box Lens hidden system, from vendors

ABCD paraxial segment

37

Important Surface Types

37