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Imaging and Aberration Theory

Lecture 10: Sine condition, aplanatism and isplanatism

2015-01-15

Herbert Gross

Winter term 2014

2

Preliminary time schedule

1 30.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems

2 06.11. Pupils, Fourier optics, Hamiltonian coordinates

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

3 13.11. Eikonal Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media

4 20.11. Aberration expansions single surface, general Taylor expansion, representations, various orders, stop shift formulas

5 27.11. Representation of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations

6 04.12. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders

7 11.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options

8 18.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options

9 08.01. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum

10 15.01. Sine condition, aplanatism and isoplanatism

Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics

11 22.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations

12 29.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, measurement

13 05.02. PSF and transfer function ideal psf, psf with aberrations, Strehl ratio, transfer function, resolution and contrast

14 12.02. Additional topics Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability

1. Pupil aberrations

2. Sine condition

3. Isoplanatism

4. Herschel condition

5. Relation to Fourier optics and phase space

3

Contents

Sine Condition

y y'

n n'

zU U'

y'

y

Lagrange invariante for paraxial angles U, U‘

sin-condition:

extension for finite aperture angle u

Corresponds to energy conservation in the system

Constant magnification for alle aperture zones

Pupil shape for finite aperture is a sphere

Definition of violation of the sine condition:

OSC (offense against sine condition)

OSC = 0 means correction of sagittal coma (aplanatic system)

'sin'

sin

'' un

un

Un

nUm

'sin''sin UynUny

'sin''sin uynuny

4

Optical Sine Condition

Condition for finite angles

Condition for object at infinity

Condition for afocal system

In the formulation

the sagittal magnification

is used

Un

Un

un

num

sin

sin

U

h

u

hf

sin

x'p

x'

y'

x

y

z'

y'p

yp

xp

UU

U’

U’

y

y's

kk h

h

H

H 11

Un

Un

y

ym s

s

sin

sin

5

If for example a small field area and a widespread ray bundle is considered, a perfect

imaging is possible

The eikonal with the expression

can be written for dL=0 as

In the special case of an angle 90°we get with cos(q)=sin(u) the Abbe sine condition

with the lateral magnification

6

Abbe Sine Condition

P

Qs

u dr'

P'

Q's'

u'drq q

rdsnrdsnL

'''d

'cos'cos

'cos''cos

'''

qq

qq

nn

drndrn

rdsnrdsn

rd

rdm

'

'sin'

sin

un

unm

From geometry

Refraction

Division and

substitution

7

Derivation of the Sine Condition

y

Po

P

u

sR

S

Qi

i'

w

w

s'M

s'

u' M'o

M's

P'o

P'

C

y's

ideal

image

plane

sagittal

image

plane

surface

chief ray

marginal

ray

sR

i

R

u

)sin(sin

Rs

i

R

u

M

'

'sinsin

'sin'sin inin

'sin

sin

''sin

sin

sin

'sin'

u

u

n

n

u

u

i

i

sR

Rs M

CP

PP

CM

MM

o

o

o

so '

''

sin ' 'sin 'syn u y n u

sR

y

Rs

y

M

s

'

'

y

y

sR

Rs sM ''

Vectorial Sine Condition

General vectorial sine condition:

spatial frequencies / direction cosines are linear related from entrance to exit pupil

Generalization can be applied for anamorphic systems

object

plane

system

image

plane

yo

zo

sy0

q0

yi

zi

syi

qi

so si

yyoyyi

xxoxxi

csms

csms

8

Transfer of Energy in Optical Systems

Conservation of energy

Invariant local differential flux

Assumption: no absorption

Delivers the sine condition

'22 PdPd

ddudAuuLPd cossin2

T 1

y

dA dA's's

EnP ExP

n n'

F'F

y'

u u'

'sin''sin uynuyn

9

Sine condition fulfilled: linear scaling from entrance to exit pupil

Pupil surface must be sperical

The pupil height scales with the sine of the angle

Pupil Sphere

object

entrance pupil

sphere

image

yo y'

u u'

hEnP=REnP

sin(u)

hExP=RExP

sin(u')

exit pupil

sphere

REnP

RExP

objectyo

equidistant

h =R sin(u)

angle not

equidistant

spherical pupil

surface

10

Sine condition fulfilled: linear scaling from entrance to exit pupil

Offence against the sine condition (OSC):

Exit pupil grid is distorted

Consequences:

1. Photometric effect causes apodization

2. Wave aberration could be calculated wrong

3. Spatial filtering on warped grid

Pupil Distortion

xoxp

sphere

distorted entrance

pupil surface

object

exit pupil

optical

system

sx

u

xp

distorted

exit

pupil

grid

1sin

unf

xD

ap

p

11

Afocal system:

Lagrange invariant

(classical Lagrange invariant for pupil imaging)

Magnification

Sine condition

Fulfillment of the sine condition: linear scaling of entrance to exit pupil

Pupil Distortion

entrance

pupil

xp exit

pupil

x'p

ww'

xp

x'p

optical

system

field

angle

z

''' wxnwxn pp

'w

wm

p

p

xn

xnm

''

12

Typical high-NA system

Virtual pupil located inside

Typical grid distortion

Sine Condition in Microscopic Objective Lens

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1x 10

-3

chief

ray

exit

pupil

rear

stopobject

plane

pupil

13

Photometric effect of pupil distortion:

illumination changes at pupil boundary

Effect induces apodization

Sign of distortion determines the effect:

outer zone of pupil brighter / darker

Additional effect: absolute diameter of

pupil changes

OSC and Apodization

focused +20 mm +50 mm-20 mm-50 mm

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-0.05 - barrel

+0.05 - pincushion

no

distortion

rp

intensity

14

Spherical aberration of the chief ray / pupil imaging

Exit pupil location depends on the field height

Pupil Aberrations

yobject

sP

chief rays

pupil position

pupil

location

15

Pupil Aberration

Interlinked imaging of field and pupil

Distortion of object imaging corresponds to spherical aberration of the pupil

imaging

Corrected spherical pupil aberration: tangent condition

O O’

stop and

entrance pupil

optical system

exit pupil

objectimage

Object imaging Pupil imaging

Blue rays

Red rays

Marginal rays

Marginal raysChief rays

Chief rays

.tan

'tanconst

w

w

16

Eyepiece with pupil aberration

Illumination for decentered pupil :

dark zones due to vignetting

Pupil Aberration

eyepiecelens and

pupil of

the eye

retina

caustic of the pupil

image enlarged

instrument

pupil

17

Wavefront and Spot for Coma

Coma Seidel trans-

verse aberrations

Wavefront for coma

with

Relationship

Here

2

2

3 2 3

3 2

' ' ' (2 cos 2 )

'

' ' ' cos (2 ' ') ' ' cos ' '

' ' ' si ' 'n ' ' ' sinsin 2

p p p Pp

p p pp P

P

P

y S r A P y r D y

x S r P y

C y r

C r ry

q q

q

q

q q

' ',

p p

W y W x

y R x R

y x

W

3 2 3cosp p p p pW r x y yq

2 2

2 2 2

' 2 2 sin cos sin 2

' 3 2 cos 2

p p p p p p p

p

p p p p

p

Wx R R x y Rr Rr

x

Wy R R x y Rr

y

q q q

q

sin , cosp p p p p px r y rq q

Wavefront and Spot for Coma

Schematic geometry:

Notice the doubled revolution in the image plane due to combined effect of azimuthal

rotation and tilt of wavefront

exit

pupiltangential

coma rays

sagittal

coma rays

image

plane

y'

x'

yp

xp

chief ray

coma

spot

90°

0°0°

180°

90°

45°

45° rays

projection

wavefront

with coma

tangential

coma rays

chief ray

wavefront

with coma image

plane

sagittal

coma rays

projection

Tangential and Sagittal Coma

2 terms of tangential transverse aberration:

- Sagittal coma depends on xp, describes the asymmetry

- Tangential coma depends on yp, corresponds to spherical aberration under skew conditions

larger by a factor of 3

Only asymmetry removed with sine condition: sagittal coma vanishes

exit

pupiltangential

coma rays

sagittal

coma rays

image

plane

y'

x'

yp

xp

chief ray

coma

spot0°

90°

45°

wavefront

with coma

ys'

2 2' 3 ' 'p p s ty R x y y y

yt'

Linear coma (all orders)

Transverse aberrations

Sagittal coma yp = 0, xp = a

Sine condition fulfilled:

linear sagittal coma

vanishes

If in addition spherical

aberration is corrected

(aplanatic):

also tangential coma vanishes

Sine Condition and Coma

pupil

n n'

auxiliary axis

lower coma ray

upper coma ray

chief ray

sagittal rays

C

tangential

comasagittal

coma

astigmatic

difference between

coma rays

T

S

center of

curvature

surface R

chief ray

S

T

spot

S

lower

coma ray

upper coma ray

sagittal

rays

auxiliary axis

chief

ray

mppp

m

mppc yxyycyyxW 22')',,(

m

m

ppppm

p

cc yxymxcy

n

R

y

W

n

Ry

12222 )12('''

'

)',,0('

''

'

max,

2

,

yaWan

R

acyn

Ry

c

m

m

msagc

21

Decomposition of coma:

1. part symmetrical around

chief ray: skew spherical

aberration

2. asymmetrical part:

tangential coma

Skew spherical aberration:

- higher order aberration

- caustic symmetric around

chief ray

Skew Spherical aberration

upper

coma ray

chief

ray

lower

coma ray exit

pupil

y'p

ideal image

location

S

sagittal image

point

tangential

image point

T

upper

coma ray

chief

ray

lower

coma ray

exit

pupil

y'p

ideal image

plane

common

intersection

point

2

lowcomupcom

tangcoma

yyy

2

lowcomupcom

skewsph

yyy

22

Perfect imaging on axis due to conic section

- not aplanatic:

linear growth of coma with field size

Aplanatic:

- Perfect stigmatic imaging

on axis, spherical corrected

- linear coma vanishes:

good correction off-axis

but near to axis

- quadratic grows of spot size

due to astigmatism

- aplanatic and perfect

marginal ray quite different

Aplanatic and Perfect Imaging

0

100

50

Dspot

w in °0 1 2

mm]

real rays

ideal

lensideal

rays

real

lens

sin ureal' = 0.894

sin uideal' = 0.707

0

100

50

Dspot

w in °0 1 2

mm]

23

Spherical Corrected Surface

Seidel contribution of spherical aberration

with

Result

Vanishing contribution:

1. first bracket: vertex ray

2. second bracket: concentric

3. bracket: aplanatic surface

Discussion with the Delano formula

2. concentric corresponds to i' = i

3. aplanatic condition corresponds to i' = u

24

jjjj

jjjsnsn

QS1

''

124

j

jh

h

1

jj

jjsR

nQ11

jjjjjj

j

j

jsnsnsR

nh

hS

1

''

1112

2

4

1

0jh

jjjj snsn ''

jj sR

j jjj

j

kkk

SPHSPHuU

uii

iih

n

n

uUn

uUnss

'sin'2

'sin2

2

'sin

''sin''

sin' 111

Aplanatic Surfaces with Vanishing Spherical Aberration

s'

0 50 100 150 200 250 300-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Saplanatic

concentricvertex

oblate ellipsoidoblate ellipsoid prolate ellipsoidhyperboloid+ power series + power series+ power series + power series

sp

he

re

sp

he

re

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

'

' '

'

'

'

25

Aplanatic lenses

Combination of 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 :

parallel offset

A-C :

convergence enhanced

C-C :

no effect

C-A :

convergence reduced

26

General Aplanatic Surface

General approach

of Fermat principle:

aplanatic surface

Cartesian oval,

4th order

Special case OPD = 0:

Solution is spherical aplanatic surface

27

P

S

P'

oval

surface

r

ss'

z

nsnszsrnszrn '')'(')( 2222

2

2

2

2222222

22222222

2222

''

01'/'/'/21'/

'/2'/2'/

0)'/(')(

nn

snr

nn

snz

nnrnnnnzsnnz

nzsnsnnzrzsszrnn

zsnnrnszrn

nsns ''

Isoplanatism

General definition of isoplanatism:

- Invariance of performance for small lateral shifts of the field position

- spherical aberration not necessarily corrected

Usual simple case: near to axis

Consequences:

- vanishing linear growing coma

- caustic symmetrical around chief ray

28

Isoplanatism Condition of Staeble-Lihotzky

Sagittal coma aberration:

from the geometry of the figure and Lagrange invariant

Condition of Staeble-Lihotzky

Problems:

- no quantitative measure

- only tangential rays are considered

- integral criterion

m

un

un

m

sSss

p

p'sin'

sin''''

m

ssS

sS

un

un

m

yy

sphp

p

s'''

''

'sin'

sin''

exit pupil

real

tangential

image plane

ideal

gaussian

image plane

Q' P'

Q'chief ray

optical axis

marginal ray

s

Q't

projection of

sagittal coma ray y'

s'

S'

s'

last

surface

sp'

u'

ys'

ys'

P't

29

Isoplanatism from Wave Aberrations

Lateral shift of object point

Change in image

Change of wave aberration

Isoplanatism: change is equal dW' = dW

30

R

udy

W

P

chief ray dy'' dy'

R'

s'

undydW sin

'

''

'

'''''

R

dsRdym

R

dsRdydy

'sin''

'''sin'''' un

R

dsRdymundydW

01'

''

'sin'

sin1

'sin''

''sin

R

dsR

un

un

m

unR

dsRmun

Isoplanatism

Berek's condition of proportionality

Berek's coincidence condition

Isoplanatism in case of defocussing:

can only be fulfilled in one plane or for telecentricity

31

mun

un

sS

m

sS p

'sin'

sin

''''

.''sin'

sin''' consts

un

un

m

sSs p

p

zs

s

zs

su

zs

s

zs

su

p

p

p

p

p

p

p

p1cos1

''

'1

''

''cos1d

Piecewise Isoplanatism

Invariance of PSF: to be defined

Possible options:

1. relative change of Strehl

2. correlation of PSF's

Examples for microscopic lenses

with and without flattening

correction

In medium field size:

small isoplanatic patches

On axis:

large isoplanatic area

Criteria not useful at the

edge for low performance

Plane MO 100x1.25

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Strehl correlation

Strehl correlation

no plane MO

40x0.85normalized

field position

System

MO plane 100x1.25 isoplanatic patch size in

mm

MO not plane 40x0.85 isoplanatic patch size in

mm

Strehl 1%

Psf correlation

0.5%

Strehl 1%

Psf correlation

0.5%

on axis 70 72 81 100

half field 3.8 3.8 27 3.1

field zone 2.5 2.5 29 39

full field 45 3.8 117 62

32

exit pupil

realtangential

image plane

idealgaussian

image plane

Q' P'

Q'chief ray

optical axis

marginal ray

s

Q'P'

t t

y's

y't

sagittalcoma ray

yo

s'

S's'

Offence Against the Sine Condition

Conradys OSC (offense against sine condition):

- measurement of deviation of sagittal coma

- quantitative validation of the sine condition

Only sagittal coma considered

in case of OSC=0 the Staeble-Lihotzky-

condition is automatically fulfilled

OSC allows for the definition of surface

contribution

''

''

'sin'

sin1

'

''

p

p

t

stOSC

ss

sS

unm

un

y

yy

OSCtppcoma yrryW )0,,(

ExP

yp

z

CR

Q'1

Q's

Q'

P'1

P'

ideal

y's

y'y's

y't

k kkk

CR

kkkkOSC

unh

inQQ

u

w

'''

)'(

sin

sin )(

1

1

'sin'

sin3'

un

unmyyt

33

OSC

Coma and isoplanatism are strongly connected

Vectorial OSC:

linear scaling of spatial frequencies:

perturbation of the linearity

exit

pupiltangential

coma rays

sagittal

coma rays

image

plane

y'

x'

yp

xp

chief ray

coma

spot

apl

p

apl vm

v

1

'

Wvvv yxapl ,

1''

34

Rotation around axis for small angle

calculation of change in wave aberration

Welfords condition

All other conditions can be obtained as special cases:

1. sine condition

2. off axis isoplanatism

3. Herrschel condition

4. Smith cos-invariant

35

General Invariant of Welford

d d dW n p D e n p D e ' ' ' , ' , ' , ,

Surface

n n'

ray

p

axis of

rotation

p'

s

s'

D

d

q

q'

d'

D'

From Eikonal theory: General condition of Smith: Invariance of the scalar product

Special case: P on axis, q = 90°: Abbe sine condition, invariant transverse magnification

Special case: P on axis, q = 0°: Herschel condition, invariant axial magnification

'cos''cos qq drndrn

''' esnesn

dd

Cos-Condition of Smith

P

q

Q

dr

Q'

P'

dr'q'

36

Herschel condition:

Invariance of the depth magnification

In principle not compatible with the sine condition

Therefore a perfect imaging of a volume is impossible

2

'sin''

2sin 22 u

nzu

nz

Herschel Condition

P Qdz Q'P' dz'

u'u

37

Overview on conditions for aberrations and aplanatism-isoplanatism

38

Overview Aplanatism-Isoplanatism

Nr Sine

cond.

Iso-planat cond.

Isoplanatism

condition

Spherical

aberration

Sagittal

coma

Tangential

coma

Imaging system

1 # # # # # general

2a # OSC=0, Conrady # 0 # isoplanatic-I

2b # Staeble-Lihotzky / Berek

# 0 0 isoplanatic-II

3a 0 0 axial aplanatic

3b 0 (skew) 0 0 off-axis aplanatic

Tangential coma

Isoplanatism

Staeble-

Lihotzky

Sagittal coma

Spherical aberration

Isoplanatism

Conrady

OSC

sine condition

off-axis

Aplanatism

0

0 00 0

0

Skew Spherical aberration

sine condition

axial

Aplanatism

0

0 0

Overview on invariants and conditions

39

Overview

only

translation

general invariance

(Welford)

off axis isoplanatismcos-law

(Smith)

translation

along z

change

object

position dy

off axis z-invariance

(Herrschel

axis isoplanatism

(Staeble-Lihotzki /

Berek)

special on

axis y'=0

sine condition

(Abbe)

special for

sph=0

on axis z-invariance

(classical Herrschel)

special on

axis x'=y'=0

only

transverse

translation

Phase Space: 90°-Rotation

Transition pupil-image plane: 90° rotation in phase space

Planes Fourier inverse

Marginal ray: space coordinate x ---> angle q'

Chief ray: angle q ---> space coordinate x'

f

xx'

q

q'

Fourier plane

pupil

image

location

marginal ray

chief

ray

40

Nearfield - Farfield

2f-setup:

Fourier-conjugated

planes

Angle and spatial

frequency are equivalent

Angle- and spatial coor-

dinate are interchanged:

x ---> q'

q ---> x'

Corresponds to nearfield <---> farfield

Relationship:

q v

qq fxf

x','

x'

pupil

Fourier domain

angle/frequency

object space

spatial domain

coordinate x

ff

xuu

41

Product of field size y and numercial aperture is invariant in a paraxial system

The invariant L describes to the phase space volume (area)

The invariance corresponds to

1. Energy conservation

2. Liouville theorem

3. Constant transfer of information

y

y'

u u'

marginal ray

chief ray

object

image

system

and stop

''' uynuynL

Helmholtz-Lagrange Invariant

42

Basic formulation of the Lagrange

invariant:

Uses image heigth,

only valid in field planes

General expression:

1. Triangle SPB

2. Triangle ABO'

3. Triangle SQA

4. Gives

5. Final result for arbitrary z:

Helmholtz-Lagrange Invariant

ExPCR sswy ''''

ExP

CR

s

yw

''

''

s

yu MR

z

y'y yp

pupil imagearbitrary z

chief

ray

marginal

ray

s's'Exp

y'CR

yMR

yCR

B A

O'

Q

S

P

ExPMRExPMR

CR swuwynssws

ynyunL ''''''''

'''''

)(')('' zyuzywnL CRMR

43

Photometry in Phase Space

Radiation transport in optical systems

Phase space area changes its shape

Finite chief ray angle:

parallelogram geometry y

p

2y

2y'

2sinu

2sinu'

sinw'

y

y'

s'

Uw' U'

y

y'

s

lens stop

44

Aberrations in Phase Space

• Angle diviations due to aberrations in the pupil

• Increased spatial extention in the focus region

x

upupil

u

x

ufocus

x

45

Ray Caustic

caustic

• Special case of vanishing determinante of Jacobian matrix: ray caustic

• Singulare solution of the wave equation

• Two ray directions in one point

• Special characterization with Morse- and

Maslov index

x

u

xc

46

1. Slit diffraction

Diffraction angle inverse to slit

width D

2. Gaussian beam

Constant product of waist size wo

and divergence angle qo

q 00w

D

q

q

D D

q

qo

wo

x

z

Uncertainty Relation in Optics

47

Angle u is limited

Typical shapes:

Ray : point (delta function)

Coherent plane wave: horizonthal line

Extended source : area

Isotropic point source: vertical line

Gaussian beam: elliptical area with

minimal size

Range of small etendues: modes, discrete

structure

Range of large etendues: quasi continuum

Light Sources in Phase Space

plane wave

(laser)

x

u

ray

spherical

wave

gaussian

beam

LED

point

source

x

u

discrete

mode

points

quasi continuum

48