www.iap.uni-jena.de

Lens Design II

Lecture 10: Higher order aberrations

2016-12-21

Herbert Gross

Winter term 2016

2

Preliminary Schedule

1 19.10. Aberrations and optimization Repetition

2 26.10. Structural modifications Zero operands, lens splitting, lens addition, lens removal, material selection

3 02.11. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres

4 09.11. Freeforms Freeform surfaces

5 16.11. Field flattening Astigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses

6 23.11. Chromatical correction I Achromatization, axial versus transversal, glass selection rules, burried surfaces

7 30.11. Chromatical correction II secondary spectrum, apochromatic correction, spherochromatism

8 07.12. Special correction topics I Symmetry, wide field systems,stop position

9 14.12. Special correction topics II Anamorphotic lenses, telecentricity

10 21.12. Higher order aberrations high NA systems, broken achromates, induced aberrations

11 04.01. Further topics Sensitivity, scan systems, eyepieces

12 11.01. Mirror systems special aspects, double passes, catadioptric systems

13 18.01. Zoom systems mechanical compensation, optical compensation

14 25.01. Diffractive elements color correction, ray equivalent model, straylight, third order aberrations, manufacturing

15 01.02. Realization aspects Tolerancing, adjustment

1. Higher order in an achromate

2. Aplanatic-concentric surfaces

3. Merte surface

4. Aspheres

5. High-NA systems

6. Induced aberrations

3

Contents

Taylor expansion of the

sin-function

Definition of allowed error

10-4

Deviation of the various

approximations:

- linear: 5°

- cubic: 24°

- 5th order: 54.2°

4

Paraxial approximation

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

sin(x)

x [°]

exact sin(x)

linear

cubic

5th order

x = 5° x = 24° x = 52° deviation 10-4

Contribution of various orders of the sin-expansion

5

Paraxail Approximation

1st order i

0 5 10 15 20 25 30 35 40 4510

-12

10-10

10-8

10-6

10-4

10-2

100

incidence angle i

3rd order i3/6

5th order i5/120

7th order i7/5040

9th order i9/362880

...36288050401206

sin9753

iiii

ii

Higher Order Aberrations

Relevance of higher order expansion terms

Nearly perfect geometrical imaging possible in the special cases:

1. small aperture

2. small field

(29-6)

field y/w

aperture

DAP/NA2nd order

in NA

4th order

in NA

6th order

in NA....

....

2nd

order

in y

4th

order

in y

6th

order

in y

paraxial

optics

conic mirrors

telescopes

microscopy

high NA

monocentric

systems

photography

wide angle

micro

lithography

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

8

5th Order Aberrations

No

k Field

power

l, Pupil power

m, Azimuthal

power Term Name

1 0 6 0 W rp060

6 Secondary spherical aberration

2 1 5 1 W y rp151

5' cos Secondary coma

3 2 4 2 W y rp242

2 4 2' cos Secondary astigmatis, wing error

4 3 3 3 W y rp333

3 3 3' cos trefoil error, arrow error

5 2 4 0 W y rp240

2 4' Skew spherical aberration

6 3 3 1 W y rp331

3 3' cos Skew coma

7 4 2 2 W y rp422

4 2 2' cos Skew astigmatism

8 4 2 0 W y rp420

4 2' Secondary field curvature

9 5 1 1 W y rp511

5' cos Secondary distortion

Spherical Correction / Higher Orders

Partial correction of residual spherical aberration by 5th order or 5th and 7th order

Different (alternating) sign of coefficients

Residual total error significantly smaller

Large gradients at the edge

3rd and 5th order compensation: residual zonal error at of the pupil radius

9

total

rp

Dz [a.u.]

primary

3rd order

secondary

5th order

1

0.5

0.71

0

total

rp

Dz [a.u.]

primary

3rd order

secondary

5th order

1

0.89

0

a) 2 orders b) 3 orders

0.46

tertiary

7th order

707.02

1

Sine condition fulfilled:

linear coma removed

Remaning aberrations of higher

order:

1. Oblique tangential spherical

2. Oblique sagittal spherical

3. Elliptical coma

10

Higher Order Aberrations

coma 3rd

Dy‘

Dx‘

linear coma 5th

Dy‘

Dx‘

oblique sagittal

spherical 5th

Dy‘

Dx‘

oblique tangential

spherical 5th

Dy‘

Dx‘

elliptical coma 5th

Dy‘

Dx‘

Dy‘

Dx‘

2 2 2

3

2 3 2

3

2 3 2

3

1' sin cos

' '

1' cos 1 cos

' '

p p

p p p

p p p

W c y r y

x c y rn u

y c y rn u

D

D

2 4

2

2 3 2 3

1 2

1 1' 4 sin , ' cos

' ' ' '

p

p p p p

W c y r

x c y r y c y rn u n u

D D

3 2 3 3

4 5

3 2

4

3 2 3 2 2

4 5

1' sin 2

' '

1' 2 cos 2 3 cos

' '

p p p

p p

p p p p

W c y r y c y y

x c y rn u

y c y r c y rn u

D

D

Splitted achromate

Aspherical surface

Higher Order Aberrations: Achromate, Aspheres

achromat broken contact

0.0250-0.025

1

aspheric

a

0.010-0.01

1

0.0020-0.002

1

1

0.0002

1

0.00001

1

0.002

zone

zone zone

air space

ray

Ref : H. Zügge

11

Relaxed System

Example: achromate with cemented/splitted setup

Equivalent performance

Inner surfaces of splitted version more sensitive

Ref: H. Zügge

a ) Cemented achromate f = 100 mm , NA = 0.1

b ) Splitted achromate f = 100 mm , NA = 0.1

-15

-10

-5

0

5

10

1 2 3

-15

-10

-5

0

5

10

1 2 3 4

Seidel coefficient

spherical aberration

spot enlargement for

0.2 ° surface tilt

surface

index

surface

index

Aplanatic Surface

Spherical aberration vanishes for all orders

Aplanatic lens at high NA:

effective real NA is higher than paraxial

Further possible aplanatic lenses

1. less practical importance

2. used in microscopic objective front lens

13

concentric at vertex aplanatic at vertex

real rays

ideal

lensideal

rays

real

lens

sin ureal' = 0.894

sin uideal' = 0.707

Merte Surface

Small difference in refractive

index

Growing higher order

contributions

14

n2n1

Merte

surface

radius

concentric

NA

R

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

-5

10-4

10-3

10-2

10-1

100

Dn

NA = 0.20NA = 0.05 NA = 0.10

c16/c9

c25/c9

c16/c9

c25/c9

c16/c9

c25/c9

Merte surface:

- low index step

- strong bending

- mainly higher aberrations

generated

Higher Order Aberrations: Merte Surface

1

1

O.25

O.25

Merte surface

(a)

(b)

Transverse

spherical aberration

From : H. Zügge

15

Additional degrees of freedom for correction

Exact correction of spherical aberration for a finite

number of aperture rays

Strong asphere: many coefficients with high orders,

large oscillative residual deviations in zones

Location of aspherical surfaces:

1. spherical aberration: near pupil

2. distortion and astigmatism: near image plane

Use of more than 1 asphere: critical, interaction and

correlation of higher oders

SPH

tan u'

corrected

points

residual

spherical

aberration

Aspherical Surfaces

16

Microscope Objective Lens Types

Medium magnification system

40x0.65

High NA system 100x0.9

without field flattening

High NA system 100x0.9

with flat field

Large-working distance

objective lens 40x0.65

Microscope Objective Lens

Examples of large-working-

distance objective lenses

Aplanatic-concentric shell-lenses

in the front group

Large diameter of the lens L1 M

2 M3 M

4

Microscope Lens

Zernike spectrum shows

large higher contributions

Spherical Aberration of Higher Order

(a)

(b) (c)

wavefront surface wavefront section

λ0.1

1 3 5 7 9 11 13 15 17-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

order

sherical aberration

l

Ref: H. Zügge

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

2nd surface

induced perturbation at 2nd

surface due to changed ray height

change of ray height due to the

aberration of the 1st surface

P'

Surface No. j in the system: - intermediate imaging with object, image, entrance and exit pupil

- separate calculations with ideal/real perturbed object point

- pupil distortion must be taken into account

Induced Aberrations

entrance

pupil no. j

grid distorted

wave spherical

intermediate

ideal object no. j

surface

index j

exit pupil no. j

grid uniform

wave with intrinsic

aberrationsintermediate

image no. j

entrance

pupil no. j

grid distorted

wave perturbed

intermediate

real object no. j

surface

index j

exit pupil no. j

grid uniform

wave with intrinsic

aberrationsintermediate

image no. j

Mathematical formulation:

1. incoming aberrations form

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

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

l = 400 nm

1 2 3-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

l = 700 nmmirror

contribution

to color

surfaces surfaces

difference

heigth

difference

with

wavelengthl= 400 nm

l= 700 nm

Dialyt achromate:

Induced axial color at 2nd lens

Correction by distance t

24

Induced Axial Color Aberration

Ref: A. Berner

Several relay systems:

accumulation of residual

aberrations

Growing induced aberrations

Induced Aberrations in Endoscopes

25

W0n0(r)

1 2 3 4 5 6 7 8 9 10 11 125th intrinsic

5th extrinsic

5th total

3rd order

0

0.02

0.04

0.06

0.08

0.1

0.12

surface

index

lens index1 2 3 4 5 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

5th order intrinsic

5th order

extrinsic

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

26

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

27

Reversability of Aberrations

Ref

l = 546 nm

l =480 nm

l =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