www.iap.uni-jena.de

Design and Correction of Optical

Systems

Lecture 10: Correction principles I

2019-06-24

Herbert Gross

Summer term 2019

2

Preliminary Schedule - DCS 2019

1 08.04. BasicsLaw of refraction, Fresnel formulas, optical system model, raytrace, calculation

approaches

2 15.04. Materials and ComponentsDispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,

aspheres, diffractive elements

3 29.04. Paraxial OpticsParaxial approximation, basic notations, imaging equation, multi-component

systems, matrix calculation, Lagrange invariant, phase space visualization

4 06.05. Optical SystemsPupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,

photometry

5 13.05. Geometrical AberrationsLongitudinal and transverse aberrations, spot diagram, polynomial expansion,

primary aberrations, chromatical aberrations, Seidels surface contributions

6 20.05. Wave AberrationsFermat principle and Eikonal, wave aberrations, expansion and higher orders,

Zernike polynomials, measurement of system quality

7 27.05. PSF and Transfer functionDiffraction, point spread function, PSF with aberrations, optical transfer function,

Fourier imaging model

8 03.06. Further Performance CriteriaRayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based

criteria, further options

9 17.06. Optimization and CorrectionPrinciples of optimization, initial setups, constraints, sensitivity, optimization of

optical systems, global approaches

10 24.06. Correction Principles ISymmetry, lens bending, lens splitting, special options for spherical aberration,

astigmatism, coma and distortion, aspheres

11 01.07. Correction Principles IIField flattening and Petzval theorem, chromatical correction, achromate,

apochromate, sensitivity analysis, diffractive elements

12 08.07. Optical System ClassificationOverview, photographic lenses, microscopic objectives, lithographic systems,

eyepieces, scan systems, telescopes, endoscopes

1. Introduction

2. Sensitivity and improvement process

3. Special options for spherical aberration

4. Astigmatism

5. Symmetry

6. Coma

7. Distortion

8. Aspheres and higher orders

3

Contents

System Design Phases

1. Paraxial layout:

- specification data, magnification, aperture, pupil position, image location

- distribution of refractive powers

- locations of components

- system size diameter / length

- mechanical constraints

- choice of materials for correcting color and field curvature

2. Correction/consideration of Seidel primary aberrations of 3rd order for ideal thin lenses,

fixation of number of lenses

3. Insertion of finite thickness of components with remaining ray directions

4. Check of higher order aberrations

5. Final correction, fine tuning of compromise

6. Tolerancing, manufactability, cost, sensitivity, adjustment concepts

4

Effectiveness of correction

features on aberration types

Correction Effectiveness

Aberration

Primary Aberration 5th Chromatic

Sp

he

rica

l A

berr

atio

n

Co

ma

Astig

ma

tism

Pe

tzva

l C

urv

atu

re

Dis

tort

ion

5th

Ord

er

Sp

he

rica

l

Axia

l C

olo

r

Late

ral C

olo

r

Se

co

nd

ary

Sp

ectr

um

Sp

he

roch

rom

atism

Lens Bending (a) (c) e (f)

Power Splitting

Power Combination a c f i j (k)

Distances (e) k

Len

s P

ara

mete

rs

Stop Position

Refractive Index (b) (d) (g) (h)

Dispersion (i) (j) (l)

Relative Partial Disp. Mate

ria

l

GRIN

Cemented Surface b d g h i j l

Aplanatic Surface

Aspherical Surface

Mirror

Sp

ecia

l S

urf

ace

s

Diffractive Surface

Symmetry Principle

Acti

on

Str

uc

Field Lens

Makes a good impact.

Makes a smaller impact.

Makes a negligible impact.

Zero influence.

Ref : H. Zügge

5

Strategy of Correction and Optimization

Usefull options for accelerating a stagnated optimization:

split a lens

increase refractive index of positive lenses

lower refractive index of negative lenses

make surface with large spherical surface contribution aspherical

break cemented components

use glasses with anomalous partial dispersion

6

Operationen with zero changes in first approximation:

1. Bending a lens.

2. Flipping a lens into reverse orientation.

3. Flipping a lens group into reverse order.

4. Adding a field lens near the image plane.

5. Inserting a powerless thin or thick meniscus lens.

6. Introducing a thin aspheric plate.

7. Making a surface aspheric with negligible expansion constants.

8. Moving the stop position.

9. Inserting a buried surface for color correction, which does not affect the main

wavelength.

10. Removing a lens without refractive power.

11. Splitting an element into two lenses which are very close together but with the

same total refractive power.

12. Replacing a thick lens by two thin lenses, which have the same power as the two refracting

surfaces.

13. Cementing two lenses a very small distance apart and with nearly equal radii.

Zero-Operations

7

Lens bending

Lens splitting

Power combinations

Distances

Structural Changes for Correction

(a) (b) (c) (d) (e)

(a) (b)

Ref : H. Zügge

8

Removal of a lens by vanishinh of the optical effect

For single lens and

cemented component

Problem of vanishinh index:

Generation of higher orders

of aberrations

9

Lens Removal

1) adapt second radius of curvature 2) shrink thickness to zero

a) Geometrical changes: radius and thickness

b) Physical changes: index

10

Sensitivity by large Incidence

Small incidence angle of a ray:

small impact of centering error

Large incidence angle of a ray:

- strong non-linearity range of sin(i)

- large impact of decenter on

ray angle

Ref: H. Sun

ideal

surface

location

real surface

location:

decentered

ideal ray

real raysurface

normal

ideal ray

real ray

surface

normal

a) small incidence:

small impact of decenter

b) large incidence:

large impact of decenter

i

i

Sensitivity/relaxation:

Average of weighted surface contributions

of all aberrations

Correctability:

Average of all total aberration values

Total refractive power

Important weighting factor:

ratio of marginal ray heights

1 2 3 4 5 6 7 8 91

0

1

1

1

2

1

3

1

4

1

5

1

6

1

7

1

8

1

9

2

0

-50

-40

-30

-20

-10

0

1

0

2

0

3

0

4

0

Sph

1 2 3 4 5 6 7 8 91

0

1

1

1

2

1

3

1

4

1

5

1

6

1

7

1

8

1

9

2

0

-50

-40

-30

-20

-10

0

1

0

2

0

3

0

Ko

m

a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20-2

-1

0

1

2

3

4

Ast

1 2 3 4 5 6 7 8 91

0

1

1

1

2

1

3

1

4

1

5

1

6

1

7

1

8

1

9

2

0

-20

-10

0

1

0

2

0

3

0

4

0

CHL

1 2 3 4 5 6 7 8 91

0

1

1

1

2

1

3

1

4

1

5

1

6

1

7

1

8

1

9

0

1

0

2

0

3

0

4

0

5

0

Inz-

Wi

Sph

Coma

Ast

CHL

incidenceangle

k

j

jj FFF2

1

1h

h jj

Sensitivity of a System

11

Quantitative measure for relaxation

with normalization

Non-relaxed surfaces:

1. Large incidence angles

2. Large ray bending

3. Large surface contributions of aberrations

4. Significant occurence of higher aberration orders

5. Large sensitivity for centering

Internal relaxation can not be easily recognized in the total performance

Large sensitivities can be avoided by incorporating surface contribution of aberrations

into merit function during optimization

Fh

Fh

F

FA

jjj

jj

1

11

k

j

jA

Sensitivity of a System

12

Sensitivity of a System

Ref: H.Züggesurfaces

Representation of wave

Seidel coefficients [l]

Double Gauss 1.4/50

13

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

14

15

Relaxed Design

Photographic lens comaprison

Data:

F# = 2.0

f = 50 mm

Field 20°

Same size and quality

Considerably tigher tolerances in the

first solution

Ref: D. Shafer

Data: Wj, Sj, iMR2 , iCR

2 , Sph, G, A

Relay systems

1. ZeBase M 29 , Takahashi, Olympus

2. Hopkins modified

3. Kidger

Example: Sensitivity of Endoscopic Systems

16

Reality:

- as-designed performance: not reached in reality

- as-built-performance: more relevant

Possible criteria:

1. Incidence angles of refraction

2. Squared incidence angles

3. Surface powers

4. Seidel surface contributions

5. Permissible tolerances

Special aspects:

- relaxed systems does not contain

higher order aberrations

- special issue: thick meniscus lenses

Sensitivity and Relaxation

17

performance

parameter

best

as

built

tolerance

interval

local optimal

design

optimal

design

Possible further criteria for modefied merit function to obtain relaxed systems

1. cos-G-factor of ray bending

2. Squared sum of incidence angles

Target: minimum value for i, i'

3. Optimization of performance and performance change simultaneously

Further Parameter of Sensitivity

18

' 'j j j j j j jn s e n s eG

2 21

1'

2

N

j j

j

i iN

2

1

m

m m

MD p

p

performance

limit

as built

performance

nominal

performance

= 12° = 20°

Relaxed system:

as built performance improved

Typically:

- no or weak correlation to designed

performance

- weak decrease in nominal performance

possible

As Built Performance

19

performance

field size

10 0.5

locally optimized

solution

best as-built

performance

nominal

with

tolerances

average

performance

sensitivity

10 0.5

20

Design Solutions and Sensitivity

Focussing

3 lens with

NA = 0.335

Spherical

correction

with/without

compensation

Red surface:

main correcting

surface

Counterbeding

every lens in

one direction

Microscopic Objective Lens

Incidence angles for chief and

marginal ray

Aperture dominant system

Primary problem is to correct

spherical aberration

marginal ray

chief ray

incidence angle

0 5 10 15 20 2560

40

20

0

20

40

60

microscope objective lens

21

Photographic lens

Incidence angles for chief and

marginal ray

Field dominant system

Primary goal is to control and correct

field related aberrations:

coma, astigmatism, field curvature,

lateral colorincidence angle

chief

ray

Photographic lens

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1560

40

20

0

20

40

60

marginal

ray

22

Variable focal length

f = 15 ...200 mm

Invariant:

object size y = 10 mm

numerical aperture NA = 0.1

Type of system changes:

- dominant spherical for large f

- dominant field for small f

Data:

23

Symmetrical Dublet

f = 200 mm

f = 100 mm

f = 50 mm

f = 20 mm

f = 15 mm

Nofocal

length [mm]

Length [mm]

spherical c9

field curvature c4

astigma-tism c5

1 200 808 3.37 -2.01 -2.27

2 100 408 1.65 1.19 -4.50

3 50 206 1.74 3.45 -7.34

4 20 75 0.98 3.93 2.31

5 15 59 0.20 16.7 -5.33

24

Field vs Aperture Correction

Remote pupil system with decreasing field for one wavelength and 4 lenses

Aperture maximum value for overall diffraction limited correction

Large field angles: lenses bended towards pupil

Old achromate move towards usual achromate

2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

w [°]

D [mm]

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

Spherical Aberration: Lens Bending

20

40

60

X

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

M=0

Spherical

Aberration

(a)

(b) (c)

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

Ref : H. Zügge

25

Aplanatic Surfaces with Vanishing Spherical Aberration

Aplanatic surfaces: zero spherical aberration:

1. Ray through vertex

2. concentric

3. Aplanatic

Condition for aplanatic

surface:

Virtual image location

Applications:

1. Microscopic objective lens

2. Interferometer objective lens

s s und u u' '

s s' 0

ns n s ' '

rns

n n

n s

n n

ss

s s

'

' '

'

'

'

s'

0 50 100 150 200 250 300-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

object

location

S

aplanaticconcentricvertex

linear growing aberrations with

deviation of object location

no shift invariance in z

V C A

26

Aplanatic lenses

Combination of two spherical

corrected surfaces:

one concentric and one aplanatic

surface:

zero contribution of the whole

lens to spherical aberration

Not useful:

1. aplanatic-aplanatic

2. concentric-concentric

bended plane parallel plate,

nearly vanishing effect on rays

Aplanatic Lenses

A-A convergence

enhanced

no effectconvergence

reduced

parallel offset

A-C

C-A C-C

A-VC-V field lens I field lens II

27

Impact of aplanatic lenses in microscopy on magnification

Three cases of typical combinations

28

Aplanatic Lenses

A-C mAC = n

ss‘

n

A-V mAV = n2

ss‘

n

A-C A-V m = n3

nn

' '

'

y nsm

y n s

Interferometer Collimator Lens

Example lens

Aperture NA = 0.5

Spherical correction with

one surface

possible surfaces

under test

Wrms

[l]

0.1

0.08

0.06

0.04

0.02

00 0.03 0.06 0.09 0.150.12

w [°]

diffraction limit

-0.02

0

0.02

sph

-4

-2

0

2

4

coma

surfaces

sumL1 L2 L3 L4lenses

1 2 3 4 5 6 7 8

29

Microscope Objective Lens

Examples of large-working-

distance objective lenses

Aplanatic-concentric shell-lenses

in the front group

Large diameter of the lens L1 M2 M3 M

4

30

Correction of spherical aberration:

Splitting of lenses

Distribution of ray bending on several

surfaces:

- 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)à(b) : 1/4

(c)à(d) : 1/4

(b)à(c) : 1/2

Improvement

Improvement

(e)

0.005 mm

(d)à(e) : 1/75

Improvement

5 mm

Ref : H. Zügge

31

Splitting of lenses and appropriate bending:

1. compensating surface contributions

2. Residual zone errors

3. More relaxed

setups preferred,

although the

nominal error is

larger

Correcting Spherical Aberration : Power Splitting

2.0 mm

0.2 mm

(a)

(b)

(c)

(d)

(e)

0.2 mm

0.2 mm 0.2 mm

Ref : H. Zügge

32

Better correction

for higher index

Shape of lens / best

bending changes

from

1. nearly plane convex

for n= 1.5

2. meniscus shape

for n > 2

Correcting Spherical Aberration: Refractive Index

-8

-6

-4

-2

0

1.4 1.5 1.6 1.7 1.8 1.9

Δs’

4.0

n

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

best shape

best shape

plano-convex

plano-convex

1.686

Ref : H. Zügge

33

Correcting Spherical Aberration: Cementing

0.25 mm0.25 mm

(d)

(a)

(c)

(b)

1.0 mm 0.25 mm

Crown

in front

Filnt

in 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 front

2. bi-convex outer surfaces / meniscus shape

Residual zone error, spherical aberration corrected for outer marginal ray

34

Better correction

for high index also for meltiple

lens systems

Example: 3-lens setup with one

surface for compensation

Residual aberrations is quite better

for 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

35

IR Objective Lens

Aperture f/1.5

Spectral 8 - 12 mm

2 lenses aspherical

ideal

[cyc/mm]

MTF

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

axis

field

solid: tan

dashed: sag

36

Compound Systems

System groups:

1. Afocal zoom telescope

2. Scanning group

3. Reimager

Suitable for cooled matrix detector

37

Bending effects astigmatism

For a single lens 2 bending with

zero astigmatism, but remaining

field curvature

Astigmatism: Lens Bending

Ref : H. Zügge

20

-0. 01 0-0.02-0. 03 0.01-0. 04

Surface 2

Surface 1

Sum

Curvature of surface 1

15

10

5

0

-5

Astigmatism

Seidel coefficients

in [l]

ST

-2.5 0 2.5

ST S T ST ST

-2.5 0 2.5 -2.5 0 2.5 -2.5 0 2.5 -2.5 0 2.5

38

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

l l

l

2

1

3

39

40

Even Aberrations in Symmetrical Systems

Aberrations with even symmetry are doubled

Spherical aberration, Astigmatism, field curvature, axial chromatical aberration

Ref: M. Seesselberg

spherical aberration in an symmetrical system

4 4 9 9W c Z c Z

4 4 9 92 2W c Z c Z

4 4 9 9W c Z c Z

doubled values

41

Odd Aberrations in Symmetrical Systems

Ref: M. Seesselberg

Aberrations with odd symmetry are vanishing

Coma, distortion, transverse chromatical aberration

coma in an symmetrical system

8 8 15 15W c Z c Z 8 8 15 15W c Z c Z

0W

vanishing values

Symmetrical Systems

skew sphericalaberration

Ideal symmetrical systems:

Vanishing coma, distortion, lateral color aberration

Remaining residual aberrations:

1. spherical aberration

2. astigmatism

3. field curvature

4. axial chromatical aberration

5. skew spherical aberration

42

Radii of curvature of wide angle camera lenses - symmetrical setups

Mostly radii 'concentric' towards the stop losition

Locations zj of surfaces normalized for comparison

Nearly linear trend, some exceptions near to the pupil

Stop position centered

43

Wide Angle Lenses - Symmetrical

zjDouble Gauss

Pleogon

Rj

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-250

-200

-150

-100

-50

0

50

100

150

200

250

Biogonzj

stop

Symmetry Principle

Triplet Double Gauss (6 elements)

Double Gauss (7 elements)Biogon

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 nearly

all photographic systems

44

Offner-System

object

image

M

r2

r1

d1

d2

-0.1

0

0.1

-0.1

0

0.1

-0.2

0

0.2

curvature

astigmatism

distortion

M11

M2 M12

sum

Concentric system of Offner:

relation

Due to symmetry:

Perfect correction of field aberrations in third order

21

212

rr

dd

45

Dyson-System

T S

y

-0.10 0-0.20zmirror

object

image

rL

nr

M

Catadioptric system with m = -1 according Dyson

Advantage : flat field

Application: lithography and projection

Relation:

Residual aberration : astigmatism

ML rn

nr

1

46

47

Mono-Centric Systems

Offner

mirror

object

image

rL

nrM

n3

n2

n1

r3

r2

r1

rsph

rm

max

Offner-

Wynne

Dyson

Sutton ball-

lens

Retrofocus I

Retrofocus II

do-nothing lens

(only Petzval)

r2r 1

n n

48

Monocentric System

Nearly perfect system with 4 monocentric surfaces

Data:

f = 25 mm

NA = 0.98

Unfortunately image plane not easily accessible

Only small field possible

No coma due to monocentric behavior

Ref: D. Shafer Proc. SPIE 5865 (2005)

center

Stop

M1

focal plane(curved)

N-BK7N-F2

Mono-Centric Lenses

NewtonSchwarzschild

Stamenov Schmidt

49

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

'y0.05 mm

'y0.05 mm

'y

Pupil section: meridional meridional sagittal

Image height: y’ = 0 mm y’ = 2 mm

Transverse

Aberration:

(a)

(b)

(c)

Wave length:

Achromat

bending

Achromat, splitting

(d)

(e)

Achromat, aplanatic glass choice

Ref : H. Zügge

50

Effect of lens bending on coma

Sign of coma : inner/outer coma

Inner and Outer Coma

outer coma

large

incidence angle

for upper coma ray

inner coma

large

incidence angle

for lower coma ray

0.2 mm'y

0.2 mm'y

0.2 mm'y

0.2 mm'y

From : H. Zügge

51

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

Transverse

Aberration:

Symmetry principle

0.5 mm

'y0.5 mm

'y

(a)

(b)

From : H. Zügge

52

Combined effect, aspherical case prevent correction

Coma Correction: Stop Position and Aspheres

aspheric

aspheric

aspheric

0.5 mm

'y

Sagittal

coma

0.5 mm

'y

Sagittal

comaPlano-convex element

exhibits spherical aberration

Spherical aberration corrected

with aspheric surface

Ref : H. Zügge

53

Distortion and Stop Position

positive negative

Sign of distortion for single lens:

depends on stop position and

sign 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

54

Example: Achromate

Balance :

1. zonal spherical

2. Spot

3. 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)

spherical

aberration

axial

color

Ref : H. Zügge

55

Example: Apochromate

Balance :

1. zonal spherical

2. Spot

3. Secondary spectrum

Coexistence of Aberrations : Balance

SSK2 CaF2 F13

0.120

rP

400 nm

450 nm

500 nm

550 nm

600 nm

650 nm

700 nm

-0.04 0.04 0.08

z

axis

field

0.71°

field

1.0°

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

56