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Imaging and Aberration Theory

Lecture 8: Astigmastism and field curvature

2013-12-19

Herbert Gross

Winter term 2013

2

Preliminary time schedule

1 24.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems2 07.11. Pupils, Fourier optics, 

Hamiltonian coordinatespupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates

3 14.11. Eikonal Fermat Principle, stationary phase, Eikonals, relation rays‐waves, geometrical approximation, inhomogeneous media

4 21.11. Aberration expansion single surface, general Taylor expansion, representations, various orders, stop shift formulas

5 28.11. Representations of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations

6 05.12. Spherical aberration phenomenology, sph‐free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders

7 12.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options

8 19.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options9 09.01. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, 

spherochromatism, secondary spoectrum

10 16.01. Further reading on aberrations sensitivity in 3rd order, structure of a system, analysis of optical systems, lens contributions, Sine condition, isoplanatism, sine condition, Herschel condition, relation to coma and shift invariance, pupil aberrations, relation to Fourier optics 

11 23.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations, relation to PSF and OTF

12 30.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, recalculation for offset, ellipticity, measurement

13 06.02. Miscellaneous Intrinsic and induced aberrations, Aldi theorem, vectorial aberrations, partial symmetric systems

1. Geometrical astigmatism

2. Point spread function for astigmatism

3. Field curvature

4. Petzval theorem

5. Correcting field curvature

6. Examples

3

Contents

Reason for astigmatism:

chief ray passes a surface under an oblique angle,

the refractive power in tangential and sagittal section are different

The astigmatism is influenced by the stop position

A tangential and a sagittal focal line is found in different

distances

Tangential rays meets closer to the surface

In the midpoint between both focal lines:

circle of least confusion

Astigmatism

4

Beam cross section in the case of astigmatism:

- Elliptical shape transforms its aspect ratio

- degenerate into focal lines in the focal plane distances

- special case of a circle in the midpoint: smallest spot

y

x

z

tangentialfocus

sagittalfocuscircle of least

confusion

tangentialaperture

sagittalaperture

Astigmatism

5

Primary Aberration Spot Shape for Astigmatism

Seidel formulas for field point only in y‘ considered

Field curvature:circle

Astigmatism:focal line

General spot with defocus:wave aberration

Transverse aberrations

Circle/ring in the pupil delivers an elliptical spoit in the imageSpecial case for c20 = - c22/2:circle of least confusion

PpPppp

PpPppp

ryPryCrSx

yDryPAryCrSy

sin'''2sin'''sin'''

''cos'')''2()2cos2('''cos'''223

3223

0',cos'''2' 2 xryAy Pp

24222

22

'''''

sin'''',cos''''

p

PpPp

ryPyx

ryPxryPy

222

2220),( ycyxcyxW

222020 2,2 ccRyy

WRyxRcx

WRx

r x y2 2 2 1

2)2( 22220

2

220

2

ccRr

yRrc

x

1)()( 2

22

2

222

2

Rrcy

Rrcx

6

At sagittal focus Defocus only in tangential cross section

Astigmatism

Wave aberrationtangential sagittal

22

Primary astigmatism at sagittal focus

Transverse ray aberration y x y

Pupil: y-section x-section x-section

0.01 mm 0.01 mm 0.01 mm

Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 100 cycles per mm

Geometrical spot through focus

0.02

mm

sagittal

tangential

tangential

sagittal

z/mm-0.04 0.040.0-0.02 0.02

1

0

0.5

0 100 200cyc/mm z/mm

1

0

0.5

0.040.0-0.04

Ref: H. Zügge

7

At median focus Anti-symmetrical defocus in T-S-cross

section

Astigmatism

Wave aberrationtangential sagittal

22

Primary astigmatism at medial focus

Transverse ray aberration y x y

Pupil: y-section x-section x-section

0.01 mm 0.01 mm 0.01 mm

Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 100 cycles per mm

sagittaltangentialtangential

sagittal

1

0

0.5

0 100 200cyc/mm z/mm

1

0

0.5

0.040.0-0.04

Geometrical spot through focus

0.02

mm

z/mm-0.04 0.040.0-0.02 0.02

Ref: H. Zügge

8

Imaging of a polar grid in different planes

Tangential focus:

- blur in azumuthal direction

- rings remain sharp

Sagittal focus:

- blur in radial direction

- spokes remain sharp

Astigmatism

9

Astigmatism

Behavior of a local position of the field:1. good resolution of horizonthal structures2. bad resolution of vertical structures

Imaging of polar diagram shows not the classical behaviour of the complete field

10

For an oblique ray, the effective curvatures of the spherical surface depend on azimuth

There are two focal points for sagittal / tangential aperture rays

This splitting occurs already for infinitesimal aperture angles around the chief ray

Intersection lengths along the chief ray: Coddington equations

Right side: oblique power of the spherical interface

The sagittal image must be located on the auxiliary axis by symmetry

Coddington Equations

Rinin

lin

lin cos'cos'cos

''cos'

tan

2

tan

2 R

ininln

ln

sagsag

cos'cos''

'

11

Wavefront for Astigmatiusm with Defocus

Astigmatic wavefront with defocus Purely cylindrical for focal lines in x/y Purely toroidal without defocus: circle of least confusion

12

Psf for Astigmatism

Zernike coefficients c in Pure astigmatism Shape is not circular symmetric due to finite width of focal lines

13

Psf bei Astigmatismus

Ref: Francon, Atlas of optical phenomena

14

Bending effects astigmatism For a single lens 2 bending with

zero astigmatism, but remainingfield curvature

Astigmatism: Lens Bending

Ref : H. Zügge

15

Field Curvature and Image Shells

Imaging with astigmatism:Tangential and sagittal image shell sharp depending on the azimuth Difference between the image shells: astigmatism Astigmatism corrected:

It remains one curved image shell,Bended field: also called Petzval curvature System with astigmatism:

Petzval sphere is not an optimalsurface with good imaging resolution No effect of lens bending on curvature,

important: distribution of lenspowers and indices

16

Astigmatisms and Curvature of Field

ss s

bestsag'' 'tan

2

s s s spet sag pet' ' ' 'tan 3

s s sast sag' ' 'tan

Image surfaces:1. Gaussian image plane2. tangential and sagittal image shells (curved)3. mean image shell of best sharpness4. Petzval shell, arteficial, not a good image

Seidel theory:

Astigmatism is difference

Best image shell

2''3

' tansss sag

pet

17

Correction of Astigmatism and Field Curvature

Different possibilities for the correction of astigmatism and field curvature Two independend aberrations allow 4 scenarious

18

Petzval Shell

The Petzval shell is not a desirable image surface It lies outside the S- and T-shell:

The Petzval curvature is a result of the Seidel aberration theory

19

2''3

' tansss sag

pet

The image splits into two curved shells in the field The two shells belong to tangential / sagittal aperture rays There are two different possibilities for description:

1. sag and tan image shell2. difference (astigmatism) and mean (medial image shell) of sag and tan

Field Curvature

Ref: H. Zügge

20

Basic observation:A plane object gives a curved image

21

Field curvature

Focussing into different planes of a system with field curvature Sharp imaged zone changes from centre to margin of the image field

focused in center(paraxial image plane)

focused in field zone(mean image plane)

focused at field boundary

z

y'

receivingplanes

imagesphere

Field Curvature

22

The image is generated on a curved shell In 3rd order, this is a sphere For a single refracting surface, the Petzval radius is given by

For a system of several lenses, the Petzval curvature is given by

Field for Single Surface

1 1r

nn fp k kk

'

r nrn np '

23

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

24

Petzval Theorem

Elementary derivation by a momocentric system of three surfaces:interface surface with r, object and image surface

Consideration of a skew auxiliary axis

Imaging condition

For the special case of a flat object gives with

rsarsa '',

rnn

sn

sn

'

''

nrnn

Rp

'1

aRa p ,'

25

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

jj

j

fhh

f11

1

26

Flattening Meniscus Lenses

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

27

Group of meniscus lenses

Effect of distance and refractive indices

Correcting Petzval Curvature

r 2r 1

collimated

n n

d

Ref : H. Zügge

28

Triplet group with + - +

Effect of distance and refractive indices

Correcting Petzval Curvature

r 2r 1

collimated

n1

n2

r 3

d/2

Ref: H. Zügge

29

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

30

Conic Sections

222

22

111 yxcyxcz

12

ba

2abc

11

cb

1

1

ca

Explicite surface equation, resolved to zParameters: curvature c = 1 / R

conic parameter Influence of k on the surface shape

Relations with axis lengths a,b of conic sections

Radii of curvature

Parameter Surface shape = - 1 paraboloid < - 1 hyperboloid = 0 sphere > 0 oblate ellipsoid (disc)

0 > > - 1 prolate ellipsoid (cigar )

2/1222/322 11,11 xcc

Rxcc

R ST

31

Astigmatism of Oblique Mirrors

Mirror with finite incidence angle:effective focal lengths

Mirror introduces astigmatism

Parametric behavior of scales astigmatism

2cos

taniRf

i

Rfsag cos2

iRsiRsi

iRss ast

cos22coscos2

sin'22

i0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s / R = 0.2

s / R = 0.4

s / R = 0.6s / R = 1

s / R = 2

s' / R

s'ast

R

focalline L

i

C

s

s'sag

mirror

32

Field Curvature of a Mirror

Mirror: opposite sign of curvature than lens Correction principle: field flattening by mirror

33

Microscope Objective Lens

Possible setups for flattening the field Goal:

- reduction of Petzval sum- keeping astigmatism corrected

Three different classes:1. No effort2. Semi-flat3. Completely flat

d)achromatizedmeniscus lens

a)singlemeniscuslense

e)twomeniscuslensesachromatized

b)twomeniscuslenses

c)symmetricaltriplet

f)modIfiedachromatizedtriplet solution

DS

rel.field

0 0.5 0.707 1

1

0.8

0.6

0.4

0.2

0

diffractionlimit

plane

semiplane

notplane

34

New Achromate

An achromate is typically corrected for axial chromatical aberration The achromatization condition for two thin

lenses close together reads

The Petzval sum usually is negative and the field is curved

A flat field is obtained, if the following condition is fulfilled

This gives the special condition of simultaneous correction of achromatization and flatness of field

perfect image plane

Petzvalshell

y'

f

RP

mean image shell0

2

2

1

1

FF

j jjP fnR

11

02

2

1

1 nF

nF

2

1

2

1

nn

35

New Achromate

This condition correponds to the requirement to find two glasses on onestraight line in the glass map

The solution is well known as simplephotographic lens(landscape lens) K5 F2stop

36

Field Curvature

Correction of Petzval curvature in photographic lens Tessar Positive lenses: green nj small Negative lenses : blue nj large Correction principle: special choice of refractive indices

Cemented component: New Achromate Spherical aberration not correctable in the New Achromate

j j

j

nF

R1

37

Antiplanet

Protar

Dagor

Orthostigmat

Asymmetrical Anastigmatic Doublets

38

Field Curvature

Correction of Petzval field curvature in lithographic lensfor flat wafer

Positive lenses: green hj large, nj large Negative lenses : blue hj small , nj small

Correction principle: certain number of bulges

j j

j

nF

R1

jj

j Fhh

F 1

39

Field Flatness

Principle of multi-bulges toreduce 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 1r

nn fp k kk

'

40

Field Flatness

Effect of mirror on Petzvalsum

Flatness of field for cata-dioptric lenses

-101

concavemirror

1. intermediateimage

2. intermediateimage

41

Field Curvature

Symmetrical system Astigmatism corrected Field curvature remains

fieldcurvature

symmetricalsystem

s'

T S y'

-2 -1 0 +1 +2

42

Astigmatism of Eyeglasses with Rotating Eye

Rotating eye: Astigmatism Coddington equations:

Elliptical line with vanishing astigmatism:Tscherning ellipses

a

object

s'

image

pupil

s

43

Semiconductor laser sources show two types of astigmatism:1. elliptical anamorphotic aperture (not a real astigmatism)2. z-variation of the internal source points in case of index guiding) In laser physics, the quadratic

astigmatic beam shape is not considered to be adegradation, the M2 is constant, it can be corrected by a simple cylindrical optic

Semiconductor Laser Beams

z

z

Q

parallel

Q

perpendicular

semiconductor

junctionplane

44

Anamorphotic or cylindrical/toroidal system are used to get a circular profile form a semiconductor laser Example:

laser beam collimator

Anamorphotic Systems

x

y

x-z-sectionNA = 0.1

y-z-sectionNA = 0.5

z

cylindricalaxi-symmetric

z

45

45° Effects of Anamorphotic Systems

Example of two aspherical cylindricallenses with different focal lengths For high numerical apertures:

no longer additivity and decouplingof x-and y-cross section The wavefront shows the deviations

in the 45° directions

46

Decentered System

Real systems with centering errors:- non-circular symmetric surface on axis- astigmatism on axis- point spread function no longer

circular symmetric

decenterx

surface local astigmatic

47

Astigmatism due to Fabrication Errors

Real surfaces with varying radius of curvature Toroidal surface shape with Rmax, Rmin

Astigmatic effects on axis Irregularity errors in tolerancing

measured by interferometry as ring difference

circular symmetric surface

surface with irregularity error

Rmin

Rmax

48

Astigmatic Correction by Clocking

Two lens surfaces nearby with equal astigmatismdue to manufacturing errors

Azimuthal rotation of one lens against the othercompensates astigmatism in first approximation

This clocking procedure is used in practice foradjusting sensitive systems

49

Example Microscopic lens

Adjusting:1. Axial shifting lens : focus2. Clocking: astigmatism3. Lateral shifting lens: coma

Ideal : Strehl DS = 99.62 %With tolerances : DS = 0.1 %After adjusting : DS = 99.3 %

Adjustment and Compensation

System with tolerancesStrehl: 0,2%

PSF (intensity normalized)

PSF(energy normalized)

Spherical aberrationcompensator

On axis comacompensator

Stop

Obj

ect p

lane

Def

ocus

com

pens

ator

On

axis

ast

igm

atis

mco

mpe

nsat

or N

r. 1

On axis astigmatismcompensator Nr. 2

Ref.: M. Peschka

50

Sucessive steps of improvements

Adjustment and Compensation

PSF(intensitynormalized)

With Tolerances

Step 1 (Z , Z )4 9

Step 2 (Z , Z )7 8

Step 3 (Z , Z )5 6

Step 4 ~ Step 2 (Z , Z )7 8

PSF(energynormalized) Not

adjusted

0.2%21.77%

25.48%

97.2% 99.32%

Stre

hl ra

tio [%

]

0

20

40

60

80

100

Step 1Z , Z4 9

Step 2Z , Z7 8

Step 3Z , Z5 6

Step 4~ Step 2Z , Z7 8

Ref.: M. Peschka

51