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Imaging and Aberration Theory

Lecture 2: Fourier- and Hamiltonian Optics

2014-11-06

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. Definition of aperture and pupil

2. Special ray sets

3. Vignetting

4. Helmholtz-Lagrange invariant

5. Phase space

6. Resolution and uncertainty relation

7. Hamiltonian coordinates

8. Analogy mechanics optics

3

Contents

Classical measure for the opening: numerical aperture

In particular for camera lenses with object at infinity:

F-number

Numerical aperture and F-number are to system properties, they are related to a conjugate object/image location

Paraxial relation

Special case for small angles or sine-condition corrected systems

4

Numerical Aperture and F-number

'sin' unNA DEnP/2

f

image

plane

object in

infinity

u'

EnPD

fF #

'tan'2

1#

unF

'2

1#

NAF

More general definition of the F-number for systems

with finite object location

Effective or working F-number with

and

we get as a relation with the object-in-infinity-case

m is the system magnification, mp is the pupil magnification

5

Generalized F-Number

infinity

s'

f'

u'o u'

finite

objectDExP

mfs 1''

)1('2)1('2'2'sin

mfn

Dm

mf

D

s

Du

EnPpExPExP

p

eff

m

mF

unF

1

'sin'2

1##

Pupil stop defines: 1. chief ray angle w

2. aperture cone angle u

The chief ray gives the center line of the oblique ray cone of an off-axis object point The coma rays limit the off-axis ray cone The marginal rays limit the axial ray cone

Diaphragm in Optical Systems

y

y'

stop

pupil

coma ray

chief ray

marginal ray

aperture

angle

u

field anglew

object

image

6

Imaging on axis: circular / rotational symmetry Only spherical aberration and chromatical aberrations

Finite field size, object point off-axis:

- chief ray as reference

- skew ray bundels:

coma and distortion

- Vignetting, cone of ray bundle

not circular symmetric

- to distinguish:

tangential and sagittal

plane

O

entrance

pupil

y yp

chief ray

exit

pupil

y' y'p

O'

w'

w

R'AP

u

chief ray

object

planeimage

plane

marginal/rim

ray

u'

Definition of Field of View and Aperture

7

Meridional rays: in main cross section plane

Sagittal rays: perpendicular to main cross

section plane

Coma rays: Going through field point

and edge of pupil

Oblique rays: without symmetry

Special rays in 3D

axis

y

x

p

p

pupil plane

object plane

x

y

axissagittal ray

meridional marginal ray

skew raychief ray

sagittal coma ray

upper meridional coma ray

lower meridional coma ray

field point

axis point

Transverse aberrations: Ray deviation form ideal image point in meridional and sagittal plane respectively

The sampling of the pupil is only filled in two perpendicular directions along the axes

No information on the performance of rays in the quadrants of the pupil

Ray-Fan Selection for Transverse Aberration Plots

x

y

pupil

tangential

ray fan

sagittal

ray fanobject

point

9

Pupil sampling for calculation of tranverse aberrations: all rays from one object point to all pupil points on x- and y-axis

Two planes with 1-dimensional ray fans

No complete information: no skew rays

Pupil Sampling

y'p

x'p

yp

xp x'

y'

z

yo

xo

object

plane

entrance

pupil

exit

pupil

image

plane

tangential

sagittal

10

Pupil Sampling

Ray plots

Spot

diagrams

sagittal ray fan

tangential ray fan

yp

Dy

tangential aberration

xp

Dy

xp

Dx

sagittal aberration

whole pupil area

11

Pupil sampling in 3D for spot diagram: all rays from one object point through all pupil points in 2D

Light cone completly filled with rays

Pupil Sampling

y'p

x'p

yp

xp x'

y'

z

yo

xo

object

plane

entrance

pupil

exit

pupil

image

plane

12

Pupil Sampling

13

polar grid cartesian isoenergetic circular

hexagonal statistical pseudo-statistical (Halton)

Fibonacci spirals

Criteria: 1. iso energetic rays 2. good boundary description 3. good spatial resolution

Artefacts due to regular gridding of the pupil of the spot in the image plane

In reality a smooth density of the spot is true

The line structures are discretization effects of the sampling

Pupil Sampling Spot Artefacts

hexagonal statisticalcartesian

14

The physical stop defines the aperture cone angle u

The real system may be complex

The entrance pupil fixes the acceptance cone in the

object space

The exit pupil fixes the acceptance cone in the

image space

Diaphragm in Optical Systems

uobject

image

stop

EnP

ExP

object

image

black box

details complicated

real

system

? ?

Ref: Julie Bentley

15

Relevance of the system pupil :

Brightness of the image Transfer of energy

Resolution of details Information transfer

Image quality Aberrations due to aperture

Image perspective Perception of depth

Compound systems: matching of pupils is necessary, location and size

Properties of the Pupil

16

Entrance and Exit Pupil

exit

pupil

upper

marginal ray

chief

ray

lower coma

raystop

field point

of image

UU'

W

lower marginal

ray

upper coma

ray

on axis

point of

image

outer field

point of

object

object

point

on axis

entrance

pupil

17

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

c

-1 0 +1

18

Example: excentricity for all surfaces

Change: c = +1 ...-1 ...+1

19

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

Pupil Mismatch

Telescopic observation with different f-numbers

Bad match of pupil location: key hole effect

F# = 2.8 F# = 8 F# = 22

a) pupil

adapted

b) pupil

location

mismatch

Ref: H. Schlemmer

20

Artificial vignetting: Truncation of the free area

of the aperture light cone

Natural Vignetting: Decrease of brightness

according to cos w 4 due

to oblique projection of areas

and changed photometric

distances

Vignetting

w

AExp

imaging without vignetting

complete field of view

imaging with

vignetting

imaging with

vignetting

field

angle

D

0.8 Daxis

field

truncation

truncation

stop

21

3D-effects due to vignetting

Truncation of the at different surfaces for the upper and the lower part of the cone

Vignetting

object lens 1 lens 2 imageaperture

stop

lower

truncation

upper

truncation

sagittal

trauncation

chief

ray

coma

rays

22

Truncation of the light cone with asymmetric ray path

for off-axis field points

Intensity decrease towards the edge of the image

Definition of the chief ray: ray through energetic centroid

Vignetting can be used to avoid uncorrectable coma aberrations

in the outer field

Effective free area with extrem aspect ratio:

anamorphic resolution

Vignetting

projection of the

rim of the 2nd lens

projection of the

rim of the 1st lens

projection of

aperture stop

free area of the

aperture

sagittal

coma rays

meridional

coma rayschief

ray

23

Vignetting

Illumination fall off in the image due to vignetting at the field boundary

24

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

Derivation at a single refracting surface: 1. Common height h:

2. Triangles

3. Refraction:

4. Elimination of s, s',w,w'

The invariance corresponds to: 1. Energy conservation

2. Liouville theorem

3. Invariant phase space volume (area)

4. Constant transfer of information

''' uynuynL

Helmholtz-Lagrange Invariant

y

u u'

chief ray

marginal ray

w

h

ss'

w'

n n'

surface

''usush

'

'',s

yw

s

yw

''wnnw

25

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'CRyMR

yCRB A

O'

Q

S

P

ExPMRExPMRCR swuwynssws

ynyunL ''''''''

'''''

)(')('' zyuzywnL CRMR

26

Optical Image formation:

Sequence of pupil and image planes Matching of location and size of image planes necessary (trivial) Matching of location and size of pupils necessary for invariance of energy density In microscopy known as Khler illumination

Nested Ray Path

object

marginal

ray chief ray

1st

intermediate

image

entrance

pupil

stop

exit

pupil

2nd intermediate

image

27

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

28

Laser optics: beam parameter product waist radius times far field divergence angle

Minimum value of L: TEMoo - fundamental mode

Elementary area of phase space: Uncertainty relation in optics

Laser modes: discrete structure of phase space

Geometrical optics: quasi continuum

L is a measure of quality of a beam small L corresponds to a good focussability

ooGB wL q

GBL

12 nwL nnGB

q

Helmholtz-Lagrange Invariant

29

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

30

Direct phase space

representation of raytrace:

spatial coordinate vs angle

Phase Space

z

x

x

u u

x

x1

x'1

x2

x'2

x1 x'1

x2

x'2

u1 u2 u1 u2

11'

2

2'

1

1'

2

2'

space

domain

phase

space

31

Phase Space

z

x

I

x

I

x

u

x

u

x

32

Phase Space

z

x

x

u

1

1

2 2'

2 2'

3 3'4

3

3'

5

4

5

6

6

free

transfer

lens 1

lens 2

grin

lens

lens 1

lens 2

free

transfer

free

transfer

free

transfer

Direct phase space

representation of raytrace:

spatial coordinate vs angle

33

Grin lens with aberrations in phase space:

- continuous bended curves

- aberrations seen as nonlinear

angle or spatial deviations

Phase Space

z

x

x

u u

x

u

x

34

Transfer of Energy in Optical Systems

Conservation of energy

Differential flux (L: radiance)

No absorption

Sine condition

d d2 2 '

ddudAuuLd cossin2

T 1

y

dA dA's's

EnP ExP

n n'

F'F

y'

u u'

'sin''sin uynuyn

35

Geometrical optic: Etendue, light gathering capacity

Paraxial optic: invariant of Lagrange / Helmholtz

General case: 2D

Invariance corresponds to conservation of energy

Interpretation in phase space: constant area, only shape is changed

at the transfer through an optical

system

unD

Lfield

Geo sin2

''' uynuynL

Helmholtz-Lagrange Invariant

space y

angle u

large

aperture

aperture

small

1

23

medium

aperture

y'1

y'2

y'3u1 u2

u3

36

Invariance of Energy:

- constant area in phase space in the geometrical model

- constant integral over density in the wave optical model

Incoherent ensemble, quasi continuum:

Jacobian matrix of transformation relates

the coordinate changes

x

u

imaging in

phase space1

2

3

invariant area

Dx Du = const.

x

v

y

L

y

u

x

LJ

Conservation of Energy

37

Simplified pictures to the changes of the phase space density.

Etendue is enlarged, but no complete filling.

x

u

x

u

x

u1) before array

x

u

x

u

x

u

2) separation of

subapertures

3) lens effect

of subapertures

4) in focal plane 5) in 2f-plane 6) far away

1 2 3 4 5 6

Example Phase Space of an Array

38

Generalization of paraxial picture: Principal surface works as effective location of ray bending

for object points near the optical axis (isoplanatic patch)

Paraxial approximation: plane Can be used for all rays to find

the imaged ray

Real systems with corrected sine-condition (aplanatic):

principal sphere

The pupil sphere can not be used to construct arbitrary ray

paths

If the sine correction is not fulfilled: more complicated

shape of the arteficial surface,

that represents the ray bending

Pupil Sphere

effective surface of

ray bending P'

y

f'

U'

P

39

Pupil sphere: equidistant sine-

sampling

Pupil Sphere

z

object entrance

pupil

image

yo y'

U U'sin(U) sin(U')

exit

pupil

objectyo

equidistant

sin(U)

angle U non-

equidistant

pupil

sphere

40

Aplanatic system:

Sine condition fulfilled Pupil has spherical shape Normalized canonical coordinates for pupil and field

Canonical Coordinates

O'O

y

yphEnP

y'

entrance

pupilexit

pupilobject image

U U'h'ExP

y'p

yp y'p

''sin'

' yun

y

EnP

p

ph

yy

ExP

p

ph

yy

'

''

41

yun

y

sin

Normalized pupil coordinates

Special case: aplanatic imaging

Normalized field coordinates

Special case: paraxial imaging

Reference on chief ray

Reduced image side coordinates

EnP

p

ph

xx

ExP

p

ph

xx

'

''

pp yy '

xx '

CRTCO uununNA sinsinsin

xun

xsag

sin y

uuny CR

tansinsin

Canonical Coordinates

42

EnP

p

ph

yy

ExP

p

ph

yy

'

''

pp xx '

yy '

yun

y

tansin ''sin'

' tan yun

y

xun

xsag

sin'

'sin'' x

unx

sag

Analogy Optics - Mechanics

43

Mechanics Optics

spatial variable x x

Impuls variable p=mv angle u, direction cosine p,

spatial frequency sx, kx

equation of motion spatial

domain

Lagrange Eikonal

equation of motion phase space Hamilton Wigner transport

potential mass m index of refraction n

minimal principle Hamilton principle Fermat principle

Liouville theorem constant number of

particles

constant energy

system function impuls response point spread function

wave equation Helmholtz Schrdinger

propagation variable time t space z

uncertainty h/2p l/2p

continuum approximation classical mechanics geometrical optics

exact description quantum mechanics wave optics