Design and Correction of optical Systems

Part 12: Correction of aberrations 1

Summer term 2012Herbert Gross

Overview

1. Basics 2012-04-182. Materials 2012-04-253. Components 2012-05-024. Paraxial optics 2012-05-095. Properties of optical systems 2012-05-166. Photometry 2012-05-237. Geometrical aberrations 2012-05-308. Wave optical aberrations 2012-06-069. Fourier optical image formation 2012-06-1310. Performance criteria 1 2012-06-2011. Performance criteria 2 2012-06-2712. Correction of aberrations 1 2012-07-0413. Correction of aberrations 2 2012-07-1114. Optical system classification 2012-07-18

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12.1 Symmetry principle12.2 Lens bending12.3 Correcting spherical aberration12.4 Coma, stop position12.5 Astigmatism12.6 Field flattening12.7 Chromatical correction12.8 Higher order aberrations

Contents

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Principle of Symmetry

Perfect symmetrical system: magnification m = -1 Stop in centre of symmetry Symmetrical contributions of wave aberrations are doubled (spherical) Asymmetrical contributions of wave aberration vanishes W(-x) = -W(x) Easy correction of:

coma, distortion, chromatical change of magnification

front part rear part

2

1

3

4

Symmetrical Systems

Ideal symmetrical systems: Vanishing coma, distortion, lateral color aberration Remaining residual aberrations:

1. spherical aberration2. astigmatism3. field curvature 4. axial chromatical aberration5. skew spherical aberration

5

Symmetry Principle

Ref : H. Zügge

Application of symmetry principle: photographic lenses Especially field dominant aberrations can be corrected Also approximate fulfillment

of symmetry condition helps significantly:quasi symmetry Realization of quasi-

symmetric setups in nearlyall photographic systems

6

Effect of bending a lens on spherical aberration Optimal bending:

Minimize spherical aberration Dashed: thin lens theory

Solid : think real lenses Vanishing SPH for n=1.5

only for virtual imaging Correction of spherical aberration

possible for:1. Larger values of the

magnification parameter |M|2. Higher refractive indices

Lens Bending

20

40

60

X

M=-6 M=6M=-3 M=3M=0

SphericalAberration

(a)

(b) (c)

(d)-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Ref : H. Zügge

7

Correction of spherical aberration:Splitting of lenses

Distribution of ray bending on severalsurfaces: - smaller incidence angles reduces the

effect of nonlinearity- decreasing of contributions at every

surface, but same sign Last example (e): one surface with

compensating effect

Correcting Spherical Aberration: Lens Splitting

Transverse aberration

5 mm

5 mm

5 mm

(a)

(b)

(c)

(d)

Improvement (a) (b) : 1/4

(c) (d) : 1/4

(b) (c) : 1/2Improvement

Improvement

(e)

0.005 mm

(d) (e) : 1/75Improvement

5 mm

Ref : H. Zügge

8

Splitting of lenses and appropriate bending:1. compensating surface contributions2. Residual zone errors3. More relaxed

setups preferred,although the nominal error islarger

Correcting Spherical Aberration : Power Splitting

Ref : H. Zügge

9

Correcting Spherical Aberration: Cementing

0.25 mm0.25 mm

(d)

(a)

(c)

(b)

1.0 mm 0.25 mm

Crownin front

Filntin front

Ref : H. Zügge

Correcting spherical aberration by cemented doublet: Strong bended inner surface compensates Solid state setups reduces problems of centering sensitivity In total 4 possible configurations:

1. Flint in front / crown in front2. bi-convex outer surfaces / meniscus shape Residual zone error, spherical aberration corrected for outer marginal ray

10

Better correctionfor higher index Shape of lens / best

bending changesfrom1. nearly plane convex

for n= 1.52. meniscus shape

for n > 2

Correcting Spherical Aberration: Refractive Index

-8

-6

-4

-2

01.4 1.5 1.6 1.7 1.8 1.9

∆s’

4.0n

n = 1.5 n = 1.7 n = 1.9 n = 4.0

best shape

best shapeplano-convex

plano-convex

1.686

Ref : H. Zügge

11

Better correctionfor high index also for meltiplelens systems Example: 3-lens setup with one

surface for compensationResidual aberrations is quite betterfor higher index

Correcting Spherical Aberration: Refractive Index

-20

-10

0

10

20

30

40

1 2 3 4 5 6 sum

n = 1.5 n = 1.8

SI ()

Surface

n = 1.5

n = 1.8

0.0005 mm

0.5 mm

Ref : H. Zügge

12

Combination of positiv and negative lens :Change of apertur with factor without shift of image plane

Strong impact on spherical aberration

Principle corresponds the tele photo system

Calculation of the focal lengths

d

s

s'

u'u

Bild-ebene

af'

f'b

sddf

sd

df

b

a

1

1'

1

11'

1

Bravais System

13

Effect of lens bending on coma Sign of coma : inner/outer coma

Inner and Outer Coma

0.2 mm'y

0.2 mm'y

0.2 mm'y

0.2 mm'y

From : H. Zügge

14

Bending of an achromate- optimal choice: small residual spherical

aberration- remaining coma for finite field size Splitting achromate:

additional degree of freedom: - better total correction possible- high sensitivity of thin air space Aplanatic glass choice:

vanishing coma

Coma Correction: Achromat

0.05 mm'y

0.05 mm'y

0.05 mm'y

Pupil section: meridional meridional sagittal

Image height: y’ = 0 mm y’ = 2 mm

TransverseAberration:

(a)

(b)

(c)

Wave length:

Achromat bending

Achromat, splitting

(d)

(e)

Achromat, aplanatic glass choice

Ref : H. Zügge

15

Perfect coma correction in the case of symmetry But magnification m = -1 not useful in most practical cases

Coma Correction: Symmetry Principle

Pupil section: meridional sagittal

Image height: y’ = 19 mm

TransverseAberration:

Symmetry principle

0.5 mm'y

0.5 mm'y

(a)

(b)

From : H. Zügge

16

Combined effect, aspherical case prevent correction

Coma Correction: Stop Position and Aspheres

aspheric

aspheric

aspheric

0.5 mm'y

Sagittalcoma

0.5 mm'y

SagittalcomaPlano-convex element

exhibits spherical aberration Spherical aberration corrected

with aspheric surface

Ref : H. Zügge

17

Bending effects astigmatism For a single lens 2 bending with

zero astigmatism, but remainingfield curvature

Astigmatism: Lens Bending

Ref : H. Zügge

18

Petzval Theorem for Field Curvature

Petzval theorem for field curvature:1. formulation for surfaces

2. formulation for thin lenses (in air)

Important: no dependence on bending

Natural behavior: image curved towards system

Problem: collecting systems with f > 0:If only positive lenses: Rptz always negative

k kkk

kkm

ptz rnnnnn

R '''1

j jjptz fnR11

optical system

objectplane

idealimageplane

realimageshell

R

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Petzval Theorem for Field Curvature

Goal: vanishing Petzval curvature

and positive total refractive power

for multi-component systems

Solution:General principle for correction of curvature of image field:1. Positive lenses with:

- high refractive index- large marginal ray heights- gives large contribution to power and low weighting in Petzval sum

2. Negative lenses with: - low refractive index- samll marginal ray heights- gives small negative contribution to power and high weighting in Petzval sum

j jjptz fnR11

fhh

f j

j 11

1

20

Flattening Meniscus Lenses

Possible lenses / lens groups for correcting field curvature Interesting candidates: thick mensiscus shaped lenses

1. Hoeghs mensicus: identical radii- Petzval sum zero- remaining positive refractive power

2. Concentric meniscus, - Petzval sum negative- weak negative focal length- refractive power for thickness d:

3. Thick meniscus without refractive powerRelation between radii

F n dnr r d

' ( )( )

1

1 1

r r d nn2 1

1

21

211'

'1rrd

nn

fnrnnnn

R k kkk

kk

ptz

r2

d

r1

2

2)1('rn

dnF

drr 12

drrndn

Rptz

11

)1(1

0)1(

)1(1

11

2

ndnrrndn

Rptz

21

Group of meniscus lenses

Effect of distance and refractive indices

Correcting Petzval Curvature

r 2r 1

collimated

n n

d

From : H. Zügge

22

Triplet group with + - +

Effect of distance and refractive indices

Correcting Petzval Curvature

r 2r 1

collimated

n1

n2

r 3

d/2

From : H. Zügge

23

Field Curvature

Correction of Petzval field curvature in lithographic lensfor flat wafer

Positive lenses: Green hj large Negative lenses : Blue hj small

Correction principle: certain number of bulges

j j

j

nF

R1

jj

j Fhh

F 1

24

Effect of a field lens for flattening the image surface

1. Without field lens 2. With field lens

curved image surface image plane

Flattening Field Lens25

Compensation of axial colour byappropriate glass choice

Chromatical variation of the sphericalaberrations:spherochromatism (Gaussian aberration)

Therefore perfect axial color correction(on axis) are often not feasable

Axial Colour: Achromate

BK7 F2

n = 1.5168 1.6200= 64.17 36.37

F = 2.31 -1.31

BK7

n= 1.5168 = 64.17F= 1

(a) (b)

-2.5 0z

r p1

-0.20

1

z0.200

r p

486 nm588 nm656 nm

Ref : H. Zügge

26

Residual spherochromatism of an achromate Representation as function of apeture or wavelength

Spherochromatism: Achromate

longitudinal aberration

0

rp

0.080.04 0.12 z

defocus variation

pupil height :

rp = 1rp = 0.707rp = 0.4rp = 0

0-0.04 0.080.04

587

656

486z

1

486 nm587 nm656 nm

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Broken-contact achromate:different ray heights allow fofcorrecting spherochromatism

Correction of Spherochromatism: Splitted Achromate

red

blue

y

SK11 SF12

0.4

rp

486 nm587 nm656 nm

0.2

zin [mm]

0

28

Non-compact system Generalized achromatic condition

with marginal ray hieghts yj

Use of a long distance andnegative F2 for correction Only possible for virtual imaging

Axial Color Correction with Schupman Lens

first lenspositive

second lenspositive

virtualimage

intermediateimage

wavelengthin m

0.656

0.486

0.571

50 m-50 m 0s

Schupmanlens

achromate

022

22

11

21 F

vyF

vy

29

Effect of different materials Axial chromatical aberration

changes with wavelength Different levels of correction:

1.No correction: lens,one zero crossing point

2.Achromatic correction:- coincidence of outer colors- remaining error for center

wavelength- two zero crossing points

3. Apochromatic correction:- coincidence of at least three

colors- small residual aberrations- at least 3 zero crossing points- special choice of glass types

with anomalous partialdispertion necessery

Axial Colour: Achromate and Apochromate30

Choice of at least one special glass

Correction of secondary spectrum:anomalous partial dispersion

At least one glass should deviatesignificantly form the normal glass line

Axial Colour : Apochromate

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Special glasses and very strong bending allows for apochromatic correction

Large remaining spherical zonal aberration

Zero-crossing points not well distributed over wavelength spectrum

Two-Lens Apochromate

-2mmz

656nm

588nm

486nm

436nmz05m

32

Cemented surface with perfect refrcative index match No impact on monochromatic aberrations Only influence on chromatical aberrations Especially 3-fold cemented components are advantages Can serve as a starting setup for chromatical correction with fulfilled monochromatic

correction Special glass combinations with nearly perfect

parameters

Nr Glas nd nd d d

1 SK16 1.62031 0.00001 60.28 22.32

F9 1.62030 37.96

2 SK5 1.58905 0.00003 61.23 20.26

LF2 1.58908 40.97

3 SSK2 1.62218 0.00004 53.13 17.06

F13 1.62222 36.07

4 SK7 1.60720 0.00002 59.47 10.23

BaF5 1.60718 49.24

Buried Surface33

Field Lenses

Field lens: in or near image planes Influences only the chief ray: pupil shifted Critical: conjugation to image plane, surface errors sharply seen

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Field Lens im Endoscope

without field lenses

with 2 field lenses

with 1 field lens

Ref : H. Zügge

35

Influence of Stop Position on Performance

Ray path of chief ray depends on stop position

spotstop positions

36

Example photographic lens

Small axial shift of stopchanges tranverse aberrations

In particular coma is stronglyinfluenced

stop

Effect of Stop Position

Ref: H.Zügge

37

Distortion and Stop Position

Sign of distortion for single lens:depends on stop position andsign of focal power Ray bending of chief ray defines

distortion Stop position changes chief ray

heigth at the lens

Ref: H.Zügge

Lens Stop location Distortion Examples

positive rear V > 0 tele photo lens negative in front V > 0 loupe positive in front V < 0 retrofocus lens negative rear V < 0 reversed binocular

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Splitted achromate

Aspherical surface

Higher Order Aberrations: Achromate, Aspheres

Ref : H. Zügge

39

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

40

Additional degrees of freedom for correction

Exact correction of spherical aberration for a finitenumber of aperture rays

Strong asphere: many coefficients with high orders,large oscillative residual deviations in zones

Location of aspherical surfaces:1. spherical aberration: near pupil2. distortion and astigmatism: near image plane

Use of more than 1 asphere: critical, interaction andcorrelation of higher oders

Aspherical Surfaces41

Example: Achromate Balance :

1. zonal spherical2. Spot3. Secondary spectrum

Coexistence of Aberrations : Balance

standard achromate

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

compromise :zonal and axial error

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

spot optimizedsecondary

spectrum optimized

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-3

-2

-1

0

1

2

3

sur 1 sur 2 sur 3 sum sur 1 sur 2 sur 3 sur 4 sum

sur 1 sur 2 sur 3 sur 4 sum sur 1 sur 2 sur 3 sur 4 sum

a)

b) c)

sphericalaberration

axialcolor

Ref : H. Zügge

42

Example: Apochromate Balance :

1. zonal spherical2. Spot3. Secondary spectrum

Coexistence of Aberrations : Balance

SSK2 CaF2 F13

0.120

rP

400 nm450 nm500 nm550 nm600 nm650 nm700 nm

-0.04 0.04 0.08z

axis

field0.71°

field1.0°

400 nm 450 nm 500 nm 550 nm 600 nm 650 nm 700 nm

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Summary of Important Topics

Nearly symmetrical system are goord corrected for coma, distortion and lateral color Important influence on correction: bending of a lens Correction of spherical aberration: bending, cementing, higher index Correction of coma: bending, stop position, symmetry Correction of field curvature: thick mensicus, field lens, low index negative lenses with

low ray height Achromate: coincidence of two colors, spherical correction, higher order zone remains Apochromatic correction: three glasses, one with anomalous partial dispersion Remaining chromatic error: spherochromatism Field lenses: adaption of pupil imaging Higher orders of aberrations: occur for large angles Whole system: balancing of aberrations and best trade-off is desired

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Outlook

Next lecture: Part 13 – Correction of Aberrations 21. Date: Wednesday, 2012-07-11Contents: 13.1 Fundamentals of optimization

13.2 Optimization in optical design13.3 Initial setups and correction methods13.4 Sensitivity of systems13.5 Miscellaneous

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