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Design and Correction of Optical

Systems

Lecture 11: Correction principles I

2019-07-01

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. Field flattening and Petzval theorem

2. Field lenses

3. Stop position

4. Chromatical correction

5. Achromate and apochromate

6. Diffractive elements

7. Miscellaneous

3

Contents

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

- no dependence on stop location

Natural behavior: image curved

towards system

Problem: collecting systems

with f > 0:

If only positive lenses:

Rptz always negative

Typical scaling for single lens:

k kkk

kkm

ptz rnn

nnn

R '

''

1

j jjptz fnR

11

optical system

object

plane

ideal

image

plane

real

image

shell

R

4

6.1 nf

Rptz

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 fnR

11

fh

h

f j

j 11

1

5

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 power

Relation between radii

Fn d

nr r d'

( )

( )

1

1 1

r r dn

n2 1

1

21

211

'

'1

rr

d

n

n

fnrnn

nn

R k kkk

kk

ptz

r2

d

r1

2

2)1('

rn

dnF

drr 12

drrn

dn

Rptz

11

)1(1

0

)1(

)1(1

11

2

ndnrrn

dn

Rptz

6

Triplet group with + - +

Effect of distance and

refractive indices

Correcting Petzval Curvature

r 2r 1

collimated

n1

n2

r 3

d/2

7050 10010

-3

10-1

10-2

1/Rpet [1/mm]

r1

[mm]

BK7

SF66 / FK3 / SF66

From : H. Zügge

7

Field Curvature

Correction of Petzval field curvature in lithographic lens

for flat wafer

Positive lenses: Green hj large

Negative lenses : Blue hj small

Correction principle: certain number of bulges

j j

j

n

F

R

1

j

j

jF

h

hF

1

8

Microscope Objective Lens

Possible setups for flattening

the field

Goal:

- reduction of Petzval sum

- keeping astigmatism corrected

d)

achromatized

meniscus lens

a)

single

meniscus

lense

e)

two

meniscus

lenses

achromatized

b)

two

meniscus

lenses

c)

symmetrical

triplet

f)

modIfied

achromatized

triplet solution

9

Field Flatness

Principle of multi-bulges to

reduce Petzval sum

Seidel contributions show principle

-0.2

0

0.2

10 20 30 40 6050

1. bulge 2. bulge 3. bulge1. waist 2. waist

one waist two waists

Petz Petz

1 1

rn

n fp k kk

'

10

Size Reduction by Aspheres

Sensitivity of refractive lenses

for aspheres

5 10 15 20 25 30 35 40 45 50 550

0.1

0.2

0.3

0.40

0.1

0.2

0.3

0.4

0.5

0

2

4

6

8

10

0

0.5

1

1.5

spherical

coma

astigmatism

distortion

sum

1. bulge 2. bulge 3. bulge

pupil

aspherical

11

imageshell

flatimage

fieldlens

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 Lens

12

field lens in

intermediate

image plane

lens L1

lens L2

chief ray

marginal ray

original

pupilshifted pupil

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

13

Field Lens im Endoscope

Ref : H. Zügge

14

Axial chromatical aberration:

- dispersion of marginal ray

- different image locations

Transverse chromatical aberration:

- dispersion of chief ray

- different image sizes

15

Chromatical Aberrations

object

chief ray

marginal

ray

chief rays

marginal rays

l2

l1

ExP

l2

ExP

l1

transverse

chromatical

aberration

axial

chromatical

aberration

ideal

image

l1

ideal

image

l2

Various cases of chromatical aberration correction

16

Chromatical Aberrations

a) axial and lateral color corrected

CRMRFC

FC

b) axial color corrected

F

FC

C

c) lateral color corrected

F C

F C

d) no color corrected

F

C

F C

Compensation of axial colour by

appropriate glass choice

Chromatical variation of the spherical

aberrations:

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-SSK8F2

n = 1.5168 1.6177

= 64.17 49.83

F = 4.47 -3.47

BK7

n= 1.5168 = 64.17

F= 1

(a) (b) (c)

-2.5 0z

r p1

-0.20

1

z

0.200

r p

-100 1000

1

r p

z

486 nm

588 nm

656 nm

Ref : H. Zügge

17

Achromate:

- Axial colour correction by cementing two

different glasses

- Bending: correction of spherical aberration at the

full aperture

- Aplanatic coma correction possible be clever

choice of materials

Four possible solutions:

- Crown in front, two different bendings

- Flint in front, two different bendings

Typical:

- Correction for object in infinity

- spherical correction at center wavelength

with zone

- diffraction limited for NA < 0.1

- only very small field corrected

Achromate

solution 1 solution 2

Crown in

front

Flint in

front

18

Achromate: Realization Versions

Advantage of cementing:

solid state setup is stable at sensitive middle surface with large curvature

Disadvantage:

loss of one degree of freedom

Different possible realization

forms in practice

edge contact cemented

a) flint in front

edge contact cemented contact on axis broken, Gaussian setup

b) crown in front

19

Achromate : Basic Formulas

Idea:

1. Two thin lenses close together with different materials

2. Total power

3. Achromatic correction condition

Individual power values

Properties:

1. One positive and one negative lens necessary

2. Two different sequences of plus (crown) / minus (flint)

3. Large -difference relaxes the bendings

4. Achromatic correction indipendent from bending

5. Bending corrects spherical aberration at the margin

6. Aplanatic coma correction for special glass choices

7. Further optimization of materials reduces the spherical zonal aberration

21 FFF

02

2

1

1

FF

FF

1

21

1

1

FF

2

12

1

1

20

Achromate: Correction

Cemented achromate:6 degrees of freedom:3 radii, 2 indices, ratio 1/2

Correction of spherical aberration:diverging cemented surface with

positive spherical contribution

for nneg > npos

Choice of glass: possible goals1. aplanatic coma correction2. minimization of spherochromatism3. minimization of secondary spectrum

Bending has no impact on chromaticalcorrection:is used to correct spherical aberrationat the edge

Three solution regions for bending1. no spherical correction2. two equivalent solutions3. one aplanatic solution, very stable

case without solution,

only spherical minimum

case with

2 solutions

case with

one solution

and coma

correction

sMR'

R1

aplanatic

case

21

Achomatic solutions in the Glass Diagram

Achromat

flint

negative lenscrown

positive lens

22

Correction axial color

requires a larger

-difference in the glass

map:

if the difference

becomes smaller, the axial

focal powers are increasing

Correction of the spherical

aberration requires a

significant smaller n in

the positive crown lens

23

Achromate

n

25

303540455055601.45

1.5

1.55

1.6

1.65

1.7

1.75

SF15

TIFN5

TIF3

LAK9

N-LAK33

K10

n à 0

n < 0

very

small

smallgood

solution

a)

b)

c)

d)

e)

Achromate

Achromate

Longitudinal aberration

Transverse aberration

Spot diagram

rp

0

486 nm

587 nm

656 nm

0.1 0.2

s'

[mm]

1

axis

1.4°

2°

486 nm 587 nm 656 nm

l = 486 nm

l = 587 nm

l = 656 nm

sinu'

y'

24

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

Cases:

a) simple achromate,

sph corrected, with coma

b) simple achromate,

coma corrected by bending, with sph

c) other glass choice: sph better, coma reversed

d) splitted achromate: all corrected

e) aplanatic glass choice: all corrected

Coma Correction: Achromate

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

25

Fixed flint glass SF12

Variation of index n, optimization of for corrected 3rd order coma:

aplanatic line of possible crown glasses

Real glasses near to the perfect line: spot size different (size of bubbles)

Best choices: SSK50, SK11, L-PHL1 with nearly aplanatic performance

26

Aplanatic Achromate

nd

30354045505560657075801.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

P-PK53

P-SK57

BAF8

BAF9 BAF51

BK10 FK5

PK1

PK2

PK3

PSK2

PSK3

SK2

SK6 SK8

SK11

SK12

SK13

SSK1

SSK3

SSK4A

SSK50

SSK51

SF12

d

BK7

Achromate

Residual aberrations of an achromate

Clearly seen:

1. Distortion

2. Chromatical magnification

3. Astigmatism

27

Spherochromatism

Spherochromatism:

variation of spherical aberration with wavelength,

Alternative notation: Gaussian chromatical error

Individual curve of spherical aberration with color

Conventional achromate:

- coinciding image location for red (C‘) and blue (F‘) on axis (paraxial)

- differences and secondary spectrum for green (e)

- but different intersection lengths

for finite aperture rays

Better balancing with half

spherochromatism on axis

0

aperture

1

0.1 mm

480 nm

546 nm

644 nm

s'sec

s'tot

0.2 mm

s'

rp

1

0.5

0 0.40.2 0.6-0.2

546 nm

480

nm

644 nm

s' in

RU

s'chl

28

Split of cemented surface:

reduced zonal residual aberration

possible

Larger distance of air gap:

reduced spherochromatism

Splitted Achromates

29

a) Classical

achromate

b) Splitted

achromate

c) Splitted

achromate

with large air gap

zone

small

spherochromatism

small

Split of cemented surface:

reduced zonal residual aberration

possible

Larger distance of air gap:

reduced spherochromatism

30

a) Classical

achromate

b) Splitted

achromate

c) Splitted

achromate

with large air gap

zone

small

spherochromatism

small

Spherochromatism: Correction by splitted achromates

red

blue

y

Correction principle:

Different ray heights at second

lens and different dependencies

on ray heights:

Focus

Spherical aberration

2~ 4~

Ref: D. Ochse

30

31

Spherochromatism

Correction of spherochromatism for different axial color corrections

a) no correction

b) achroamte

c) apochromate

z

rp

z

rp

z

rp

z

rp

z

rp

z

rp

a) not corrected b) achromate c) apochromate

spherochromatism

not corrected

spherochromatism

corrected

Ref.: A. Miks

Condition for lateral color correction

32

Perfect symmetrical system

(stop in center of symmetry):

magnification m = -1

front part rear part

l l

l

2

1

3

Lateral color: Correction principles

02 j j

j

j

j

pj F

Axial color contributionRay bundle separation

Use only achromatic elements

or

Use lenses with small contribution to axial color at places with high bundle

separation (e.g. field lens with low power and high )

Ref: D. Ochse

32

33

Lateral color: Stop shift theorem

Spot diagram for maximal field

with Airy diskExample system with four N-BK7 lenses corrected for e-line

Goal: Correct also for C‘ and F‘ lines

BeispielStopshift2.zmxConfiguration 1 of 1

Layout

16.10.2016Total Axial Length: 116.08941 mm

Surface: IMA

100.00

OBJ: 7.0000 (deg)

IMA: 8.535 mm

0.4800

0.5461

0.6438

BeispielStopshift2.zmxConfiguration 1 of 1

Spot Diagram

16.10.2016 Units are µm. Airy Radius: 9.329 µmField : 3RMS radius : 24.956GEO radius : 44.287Scale bar : 100 Reference : Chief Ray

2. Find the stop position for which lateral color vanishes

stop

BeispielStopshift3.ZMXConfiguration 1 of 1

Layout


Surface: IMA

20.00

OBJ: 7.0000 (deg)

IMA: 8.541 mm

0.4800

0.5461

0.6438

BeispielStopshift3.ZMXConfiguration 1 of 1

Spot Diagram


3. Correct longitudinal color there and move stop back

N-SF6stop

BeispielStopshift.ZMXConfiguration 1 of 1

Layout


Surface: IMA

200.00

OBJ: 7.0000 (deg)

IMA: 8.536 mm

0.4800

0.5461

0.6438

BeispielStopshift.ZMXConfiguration 1 of 1

Spot Diagram


1. Ensure the system has longitudinal color

stop

Ref: D. Ochse

33

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 partial

dispertion necessery

Axial Colour: Achromate and Apochromate

l

e

F'

C'

achromateapochromate

residual error

apochromate

644

480

546

residual error

achromate

-100 0 100

singlet

s'

lens

34

Anormal partial dispersion and normal line

Axial Colour: Apochromate

35

Pg,F

0.5000

0.5375

0.5750

0.6125

0.6500

90 80 70 60 50 40 30 20

F2

F5

K7

K10

LAFN7

LAKN13

LAKL12

LASFN9

SF1

SF10

SF11

SF14

SF15

SF2

SF4

SF5

SF56A

SF57

SF66

SF6

SFL57

SK51

BASF51

KZFSN5

KZFSN4

LF5

LLF1

N-BAF3

N-BAF4

N-BAF10

N-BAF51N-BAF52

N-BAK1

N-BAK2

N-BAK4

N-BALF4

N-BALF5

N-BK10

N-F2

N-KF9

N-BASF2

N-BASF64

N-BK7

N-FK5

N-FK51N-PK52

N-PSK57

N-PSK58

N-K5

N-KZFS2

N-KZFS4

N-KZFS11

N-KZFS12

N-LAF28

N-LAF2

N-LAF21N-LAF32

N-LAF34

N-LAF35

N-LAF36

N-LAF7

N-LAF3

N-LAF33

N-LAK10

N-LAK22

N-LAK7

N-LAK8

N-LAK9

N-LAK12

N-LAK14

N-LAK21

N-LAK33

N-LAK34

N-LASF30

N-LASF31

N-LASF35

N-LASF36

N-LASF40

N-LASF41

N-LASF43

N-LASF44

N-LASF45

N-LASF46

N-PK51

N-PSK3

N-SF1

N-SF4

N-SF5

N-SF6

N-SF8

N-SF10

N-SF15

N-SF19

N-SF56

N-SF57

N-SF64

N-SK10

N-SK11

N-SK14

N-SK15

N-SK16

N-SK18N-SK2

N-SK4

N-SK5

N-SSK2

N-SSK5

N-SSK8

N-ZK7

N-PSK53

N-PSK3

N-LF5

N-LLF1

N-LLF6

BK7G18

BK7G25

K5G20

BAK1G12

SK4G13

SK5G06

SK10G10

SSK5G06

LAK9G15

LF5G15

F2G12

SF5G10

SF6G05

SF8G07

KZFS4G20

GG375G34

normal

line

Preferred glass selection for apochromates

36

Relative Partial Dispersion

N-SF1

N-SF6

N-SF57

N-SF66

P-SF68

P-SF67

N-FK51A

N-PK52A

N-PK51

N-KZFS12

N-KZFS4

N-LAF33

N-LASF41

N-LAF37

N-LAF21

N-LAF35

N-LAK10

N-KZFS2

Choice of at least one special glass

Correction of secondary spectrum:

anomalous partial dispersion

At least one glass should deviate

significantly form the normal glass line

Axial Colour : Apochromate

656nm

588nm

486nm

436nm1mm

z0-0.2mm

z-0.2mm

2030405060708090

N-FK51

N-KZFS11

N-FS6

(1)

(2)

(3)

(1)+(2) T

PgF

0,54

0,56

0,58

0,60

0,62

37

Broken-contact achromate:

different ray heights allow fof

correcting spherochromatism

Correction of Spherochromatism: Splitted Achromate

red

blue

y

SK11 SF12

0.4

rp

486 nm

587 nm

656 nm

0.2

z

in [mm]

0

38

Non-compact system

Generalized achromatic condition

with marginal ray hieghts yj

Use of a long distance and

negative F2 for correction

Only possible for virtual imaging

Axial Color Correction with Schupman Lens

first lens

positive

second lens

positive

virtual

imageintermediate

image

wavelength

in mm

0.656

0.486

0.571

50 mm-50 mm 0s

Schupman

lens

achromate

02

2

2

21

1

2

1 Fv

yF

v

y

39

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

40

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

d 1d 2 d 3

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 Surface

41

Principles of Glass Selection in Optimization

field flattening

Petzval curvature

color

correction

index n

dispersion

positive

lens

negative

lens

-

-

+

-

+

+

availability

of glasses

Design Rules for glass selection

Different design goals:

1. Color correction:

large dispersion

difference desired

2. Field flattening:

large index difference

desired

Ref : H. Zügge

42

Lens with diffractive structured

surface: hybrid lens

Refractive lens: dispersion with

Abbe number = 25...90

Diffractive lens: equivalent Abbe

number

Combination of refractive and

diffractive surfaces:

achromatic correction for compensated

dispersion

Usually remains a residual high

secondary spectrum

Broadband color correction is possible

but complicated

refractive

lens

red

blue

green

blue

red

green

red

bluegreen

diffractive

lens

hybrid

lens

R D

453.3

CF

dd

ll

l

Achromatic Hybrid Lens

43

Dispersion by grating diffraction:

Abbe number

Relative partial dispersion

Consequence :

Large secondary spectrum

-P-diagram

Diffractive Optics: Dispersion

330.3''

CF

ee

ll

l

2695.0''

'

',

CF

Fg

FgPll

ll

normal

line

e

0.8

0.6

0.4

0.2-20 0 4020 60 80

DOE

SF59SF10

F4KzFS1

BK7

PgF'

44

Combination of DOE

and aspherical carrier

Diffractive Optics: Singlet Solutions

50 mm

0s'

y'

yp

yp

-0.5 mm50 mm

0s'

y'

yp

yp

-0.5 mm50 mm

0s'

y'

yp

yp

-0.5 mm50 mm

0s'

y'

yp

yp

-0.5 mm

diffractive

surface,

carrier

aspherical

refractive

surface,

aspherical

diffractive

surface, phase

aspherical

all 2. order

d

d

d

d,a

a

45

Data :

- l = 193 nm

- NA = 0.65

- b = 50

- sfree = 7.8 mm

Properties:

- short total track

- extreme large free working distance

- few lenses

Hybrid - Microscope Objective Lens

diffractive

element

46

Relative position of stop inside system

Quantitative measure:

Parameter of excentricity

(h absolut values)

Special cases: c = 1 image plane

c = -1 pupil plane

c = 0 same effective distance from image and pupil

MRCR

MRCR

hh

hh

c

Parameter of Excentricity

hCR(ima)

chief ray

pupil image

marginal ray hMR

surface

no j

hCR

47

Example:

excentricity for all surfaces

Change:

c = +1 ...-1 ...+1

48

Parameter of Excentricity

1 2 3 45 7

9 11 13 15 1719

22 23 26 28 30 3234

stop imageobject

pupil

0 5 10 15 20 25 30 35

-10

0

10

20

30

40

50

0 5 10 15 20 25 30 35

-1

-0.5

0

0.5

1

object

image

chief ray height

marginal ray

height

excentricity

surface

index

surface

index

Influence of Stop Position on Performance

Ray path of chief ray depends on stop position

spotstop positions

49

Example photographic lens

Small axial shift of stop

changes tranverse aberrations

In particular coma is strongly

influenced

stop

Effect of Stop Position

Ref: H.Zügge

50

51

Microscopic Objective Lens Initial System

a) negative

achromate1.

2.b) reversed

positive

achromate

c) symmetric to c)

positive

achromate

d) non-infinite

positive

achromate

e) AC-meniscus

lenses

4.

3.

5.

Design and Correction of Optical Systems - uni-jena.de · Materials and Components Dispersion,...

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