www.iap.uni-jena.de

Imaging and Aberration Theory

Lecture 14: Transfer function

2020-01-31

Herbert Gross

Winter term 2019

2

Schedule - Imaging and aberration theory 2019

1 18.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems

2 25.10.Pupils, Fourier optics, Hamiltonian coordinates

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

3 01.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media

4 08.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas

5 15.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations

6 22.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders

7 29.11. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options

8 06.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options

9 13.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spectrum

10 20.12.Sine condition, aplanatism and isoplanatism

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

11 10.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations

12 17.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement

13 24.01. Point spread function ideal psf, psf with aberrations, Strehl ratio

14 31.01. Transfer function transfer function, resolution and contrast

15 07.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsicand induced aberrations, reversability

Fourier method

Optical Transfer function

Contrast and resolution

Quantitative performance assessment

Incoherent image formation

Cascaded systems

Coherent image formation

3D transfer theory and depth resolution

Contents

3

diffracted ray

direction

object

structure

g = 1 /

/ g

k

kT

Definitions of Fourier Optics

Phase space with spatial coordinate x and

1. angle

2. spatial frequency in mm-1

3. transverse wavenumber kx

Fourier spectrum

corresponds to a plane wave expansion

Diffraction at a grating with period g:

deviation angle of first diffraction order varies linear with = 1/g

0k

kv x

xx

),(ˆ),( yxEFvvA yx

A k k z E x y z e dx dyx y

i xk ykx y, , ( , , )

vk 2

vg

1

sin

Arbitrary object expaneded into a

spatial

frequency spectrum by Fourier

transform

Every frequency component is

considered separately

To resolve a spatial detail, at least

two orders must be supported by the

system

NAg

mg

sin

sin

off-axis

illumination

NAg

2

Ref: M. Kempe

Grating Diffraction and Resolution

optical

system

object

diffracted orders

a)

resolved

b) not

resolved

+1.

+1.

+2.

+2.

0.

-2.

-1.

0.

-2.

-1.

incident

light

+1.

0.

+2.

+1.

0.

+2.

-2.

-2. -1.

-1.

+3.+3.

5

Resolution of Fourier Components

Ref: D.Aronstein / J. Bentley

object

pointlow spatial

frequencies

high spatial

frequencies

high spatial

frequencies

numerical aperture

resolved

frequencies

object

object detail

decomposition

of Fourier

components

(sin waves)

image for

low NA

image for

high NA

object

sum

6

pp

vyvxi

pppsfyxOTF dydxeyxINvvH ypxp

2),(),(

),(ˆ),( yxIFvvH PSFyxOTF

p

xp

xpxOTF dx

vfxP

vfxPvH

22)( *

Optical Transfer Function: Definition

Normalized optical transfer function

(OTF) in frequency space

Fourier transform of the Psf-

intensity

OTF: Autocorrelation of shifted pupil function, Duffieux-integral

(general: 2D)

Transfer properties:

PSF: response answer of a point object

OTF: response answer of an extended cosine grating

Absolute value of OTF: modulation transfer function (MTF)

MTF is numerically identical to contrast of the image of a cosine grating at the

corresponding spatial frequency

7

Number of Supported Orders

A structure of the object is resolved, if the first diffraction order is propagated

through the optical imaging system

The fidelity of the image increases with the number of propagated diffracted orders

0. / +1. / -1. order

0. / +1. / -1.

+2. / -2.

order

0. / +1. -1. / +2. /

-2. / +3. / -3.

order

MTF and Contrast

Object

Contrast

Object spectrum

Image spectrum

Image

)2cos()( 0xvacxIobj

c

a

acac

acac

II

IIV

)()(

)()(

minmax

minmax

0022

)0()(ˆ)( vva

vva

vcxIFvI objobj

)()()( vHvIvI MTFobjima

)2cos()(

)(2

)(2

)0(

)(2

)(2

)0()(ˆ

)()(ˆ)(ˆ)'(

00

2

0

2

0

00

1

11

00

xvvHac

evHa

evHa

Hc

vvvHa

vvvHa

vvHcF

vHvIFvIFxI

MTF

xiv

MTF

xiv

MTFMTF

MTFMTFMTF

MTFobjimaima

9

x p

y p

area of

integration

shifted pupil

areas

f x

y f

p

q

x

y

x

y

L

L

x

y

o

o

x'

y'

p

p

light

source

condenser

conjugate to object pupil

object

objective

pupil

direct

light

at object diffracted

light in 1st order

Interpretation of the Duffieux Iintegral

Interpretation of the Duffieux integral:

overlap area of 0th and 1st diffraction order,

interference between the two orders

The area of the overlap corresponds to the

information transfer of the structural details

Frequency limit of resolution:

areas completely separated

10

Duffieux Integral and Contrast

Separation of pupils for

0. and +-1. Order

MTF function

Image contrast for

sin-object I

x

V = 100%

V = 33%V = 50%V = 20%

minI = 0.33 minI = 0.5 minI = 0.66

1

image

contrast

pattern

period

spatialfrequency

100 %

0 NA/2NA/

/NA /2NA8

pupil

diffraction

order

Ref: W. Singer

11

Resolution and Spatial Frequencies

Grating object pupil

Imaging with NA = 0.8

Imaging with NA = 1.3

Ref: L. Wenke

12

Optical Transfer Function of a Perfect System

Aberration free circular pupil:

Reference frequency

Cut-off frequency:

Analytical representation

Separation of the complex OTF function into:

- absolute value: modulation transfer MTF

- phase value: phase transfer function PTF

'sinu

f

avo

'sin222 0

un

f

navvG

2

000 21

22arccos

2)(

v

v

v

v

v

vvHMTF

),(),(),( yxPTF vvHi

yxMTFyxOTF evvHvvH

/ max

00

1

0.5 1

0.5

gMTF

13

14

Calculation of MTF – Some more examples

1-dim case

circular pupil

Ring pupil =

central obscuration

(75%)

Apodization =

reduced transmission

at pupil edge

(Gauss to 50%)

The transfer of frequencies depends on

transmission of pupil

Ring pupil higher contrast near

the diffraction limit

Apodisation increase of contrast at

lower frequencies

1

0,5

MTF

Ref: B. Böhme

I Imax V

0.010 0.990 0.980

0.020 0.980 0.961

0.050 0.950 0.905

0.100 0.900 0.818

0.111 0.889 0.800

0.150 0.850 0.739

0.200 0.800 0.667

0.300 0.700 0.538

Contrast / Visibility

The MTF-value corresponds to the intensity contrast of an imaged sin grating

Visibility

The maximum value of the intensity

is not identical to the contrast value

since the minimal value is finite too

Concrete values:

minmax

minmax

II

IIV

I(x)

-2 -1.5 -1 -0.5 0 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Imax

Imin

object

image

peak

decreased

slope

decreased

minima

increased

15

Modulation Transfer

Convolution of the object intensity distribution I(x) changes:

1. Peaks are reduced

2. Minima are raised

3. Steep slopes are declined

4. Contrast is decreased

I(x)

x

original image

high resolving image

low resolving image

Imax-Imin

Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on the

azimuthal orientation of the object structure

Generally, two MTF curves are considered for sagittal/tangential oriented object structures

Sagittal and Tangential MTF

y

tangential

plane

tangential sagittal

arbitrary

rotated

x sagittal

plane

tangential

sagittal

gMTF

tangential

ideal

sagittal

1

0

0.5

00.5 1

/ max

17

18

Tangential vs Sagittal Resolution

Asymmetry of the PSF

Coma 3rd creates

a 2:3 diameter pattern

Usually coma oriented

towards the axis,

then MTFS > MTFT

(lower row)

Ref: B. Dube, Appl. Opt. 56 (2017) 5661

spatial frequency

spatial frequency

Real MTF of system with residual aberrations:

1. contrast decreases with defocus

2. higher spatial frequencies have stronger decrease

Real MTF

z = 0

z = 0.1 Ru

gMTF

1

0.75

0.25

0.5

0

-0.250 0.2 0.4 0.6 0.8 1

z = 0.2 Ru

z = 0.3 Ru

z = 1.0 Ru

z = 0.5 Ru

-100 -50 0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

gMTF

(z,f)

z in

RU

max

= 0.05

max

= 0.1

max

= 0.2

max

= 0.3

max

= 0.4

max

= 0.5

max

= 0.6

max

= 0.7

max

= 0.8

Zernike

coefficients:

c5 = 0.02

c7 = 0.025

c8 = 0.03

c9 = 0.05

19

Polychromatic MTF

20

Polychromatical MTF:

Cut off frequency depends on

Spectral incoherent weighted

superposition of monochromatic MTF’s

0

)( ),()()( dvgSvg OTF

poly

OTF

0.8

1.0

0.6

0.4

0.2

0

0 max

/2

27.5 Lp/mm

max

55 Lp/mm

gMTF

= 350 nm

= 400 nm

= 450 nm

= 500 nm

= 550 nm

= 600 nm

= 650 nm

= 700 nm

polychromatic

ideal 350 nm

Resolution/contrast criterion:

Ratio of contrasts with/without aberrations for one selected spatial frequency

Real systems:

Choice of several application relevant

frequencies

e.g. photographic lens:

10 Lp/mm, 20 Lp/mm, 40 Lp/mm

Hopkins Factor

)(

)()(

)(

)(

vg

vgvg

ideal

MTF

real

MTFMTF gMTF

ideal

real

gMTF

real

gMTF

ideal

1

0.5

0

21

Optical Transfer Function of a Perfect System

Loss of contrast for higher spatial frequencies

contrast

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ideal

MTF

/max

/max

22

Contrast vs contrast as a function of spatial frequency

Typical: contrast reduced for

increasing frequency

Compromise between

resolution and visibilty

is not trivial and depends

on application

Contrast and Resolution

V

c

1

010

HMTF

Contrast

sensitivity

HCSF

23

Contrast and Resolution

High frequent

structures :

contrast reduced

Low frequent structures:

resolution reduced

contrast

resolution

brillant

sharpblurred

milky

24

Contrast / Resolution of Real Images

resolution,

sharpness

contrast,

saturation

Degradation due to

1. loss of contrast

2. loss of resolution

25

Balance between contrast and resolution: not trivial

Optimum depends on application

Receiver: minimum contrast curve serves as real reference

Most detector needs higher contrast to resolve high frequencies

CSF: contrast sensitivity function

Contrast vs Resolution

gMTF

1 : high contrast

2 :high resolution

threshold contrast a :

2 is better

threshold contrast b :

1 is better

26

Photographic lenses with different performance

27

Modulation Transfer Function

10 c/mm

20 c/mm

40 c/mm

Objektiv 1 f/ 3.5 Objektiv 2

000 0000

10

20

30

40

50

60

70

80

90

100

-25 -20 -15 -10 -5 0 5 10 15 20 25

max. MTF Bildhöhe [mm] max. MTF

MT

F [

%]

b

ei 1

0 , 2

0 , 4

0 L

p/m

m ....... ta

n _

__

sa

g

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Lens 1 f/3.5 Lens 2

Image height

Blurred imaging:

- limiting case

- information extractable

Blurred imaging:

- information is lost

- what‘s the time ?

Resolution: Loss of Information

28

Approximation of the transfer function for large aberrations:

Expansion of the Duffieux-Integral

Approximation fails for

large frequencies

Example photographic lens

Geometrical Approximated Transfer Function

gMTF

0.00 10.00 20.00 30.00 40.00 50.00

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

ideal

axis

full field sag

full field tan

66% field sag

66% field tan

in Lp/mm

ppyxyxyxGTF dydxyvxviyvxvvvg

2

2

11),(

defocus astigmatism

x

coma

xspherical

coma

5. order

x

astigmatism

5. order x

trefoil

x

spherical

5. order

trefoil

5. order

x

astigmatism

y

coma

y

astigmatism

5. order y

trefoil

y

coma

5. order

y

spherical

7. order

coma

7. order

y

coma

7. order

x

astigmatism

7. order xastigmatism

7. order y

trefoil

5. order

y

tilt

x

tilt

y

MTF for Aberrations

OTF is complex function: decomposition into

absolute value (MTF) and phase (PTF)

Phase transfer function:

- corresponds to a lateral offset of the image point

- PTF describes distortion

- can be used to assess coma aberration

Phase Transfer Function

),(),(),( yxPTF vvgi

yxMTFyxOTF evvgvvg

2

)(')(

vgxIxI PTF

PSFPSF

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-6

-4

-2

0

2

4

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

gMTF

() gPTF

()

max

max

spherical aberration : c40

= -0.25

coma

c31

= 0.0

c31

= 0.1

c31

= 0.2

c31

= 0.3

c31

= 0.4

c31

= 0.5

ideal

Phase Mask with cubic

polynomial shape

Effect of mask:

- depth of focus enlarged

- Psf broadened, but nearly constant

- Deconvolution possible

Problems :

- variable psf over field size

- noise increased

- finite chief ray angle

- broadband spectrum in VIS

- Imageartefacts

sonst

xfürexP

xi

0

1)(

3

Cubic Phase Plate (CCP)

z

y-section

x-section

cubic phase PSF MTF

Cubic Phase Mask : PSF and OTF

defocussed focus

Conventional imaging System with cubic phase mask

defocussed focus

defocus negative focussed defocus positive

EDF Example Pictures

Cubic phase plate

- small changes of MTF with defocus

- oscillation of real- and imaginary part:

strong change of PTF

The phase of the Fourier components is

not properly reconstructed:

artefacts in image deconvolution

MTF / OTF for CPP

ohne CCP

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

CCP mit = 1

CCP mit = 2

CCP mit = 4

MTF

Re[OTF]

Im[OTF]

Im[OTF]

Frequenz Re[OTF]

c4 klein

c4 mittel

c4 groß

Real behavior of MTF:

1. invariant MTF for lower spatial frequencies

but re-magnification also increases noise

2. decrease for higher spatial frequencies:

reduction in resolution

MTF / OTF for CCP

0 0.5 1 1.5 2

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

-0.2

0

0.2

0.4

0.6

0.8

1

a = const

c4 wachsend

MTF Re[OTF]

Resolution Estimation with Test Charts

0 1

10

2

3

4

5

6

6

5

4

3

2

1

6

5

4

3

2

2 31

2

3

2

4

5

6

Measurement of resolution with test

charts:

bar pattern of different sizes

two different orientations

calibrated size/spatial frequency

37

Test: Siemens Star

Determination of resolution and contrast

with Siemens star test chart:

Central segments b/w

Growing spatial frequency towards the

center

Gray ring zones: contrast zero

Calibrating spatial feature size by radial

diameter

Nested gray rings with finite contrast

in between:

contrast reversal, pseudo resolution

38

Resolution Test Chart: Siemens Star

original good system

astigmatism comaspherical

defocusa. b. c.

d. e. f.

39

Fourier Optics – Point Spread Function

Point spread function amplitude in an optical system with magnification m

Pupil function P,

Pupil coordinates xp,yp

PSF is Fourier transform

of the pupil function

(scaled coordinates)

pp

myyymxxxz

ik

pppsf dydxeyxPNyxyxgpp ''

,)',',,(

pppsf yxPFNyxg ,ˆ),(

object

planeimage

plane

source

point

point

image

distribution

Fourier Theory of Incoherent Image Formation

objectintensity image

intensity

single

psf

object

planeimage

plane

Transfer of an extended

object distribution Iobj(x,y)

In the case of shift invariance

(isoplanatism):

incoherent convolution

Intensities are additive

In frequency space:

- product of spectra

- linear transfer theory

- spectrum of the psf works as

low pass filter onto the object

spectrum

- Optical transfer function

),(*),()','( yxIyxIyxI objpsfima

dydxyxIyyxxIyxI objpsfima

),()',,',()','(

dydxyxIyyxxIyxI objpsfima

),()','()','(

),(),( yxIFTvvH PSFyxotf

),(),(),( yxobjyxotfyximage vvIvvHvvI

Incoherent Image Formation

object

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.05 0 0.05

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

resolved not resolved Example:

incoherent imaging of bar pattern near the

resolution limit

Incoherent Image Formation

astigmatism comaspherical

aberrationobject ideal

PSF

Example:

incoherent imaging of pattern near the resolution limit with aberrations

Cascaded Optical Systems

Cascaded systems with individual transfer functions:

1. Diffusing screen in intermediate image plane:

Incoherent convolution of individual Psfs

2. Usual case:

Second system works coherent

Wave aberrations can be balanced

Convolution of amplitude Psfs

)()()()( 12 xIxIxIxI objectpsfpsfincoher

2

21 )()()( xAxAxI psfpsfexact

2

2

2

1 )()()( xAxAxI psfpsfapprox

Cascaded Optical Systems

Cascaded systems with individual transfer functions:

Behaviour of the transfer functions in the limiting cases

MTF1

real

MTF2

real

cdef1

= -cdef2

cdef2

= -cdef1

MTFcoh

real

coherent

MTFincoh

real

incoherent

correction

cdef1

= -cdef2

superposed

broadening

cdef1

, cdef2

a) Single

systems

b) Complete

cascaded

system

perfect

Cascaded Optical Systems

Incoherent case:

1. Psf convolution

2. MTF product

system 1 system 2intermediate

image and

diffusing

screen

object

plane

image

planeIpsf,2

Ipsf,1

)()()()( 12 xIxIxIxI objectpsfpsfincoher

)()()()( 12 vIvIvIvI objectpsfpsfincoher

)()()()( 21 xIxIxIxI psfpsfobjectincoher

)()()()( 21 vIvIvIvI psfpsfobjectincoher

Cascaded Optical Systems

Coherent case:

1. Amplitude Psf convolution

2. MTF product

system 1

cdef,2

cdef,1

system 2intermediate

imageobject

plane

image

planeApsf,2

Apsf,1

)()()()(

)()()(

*

2

*

121

2

21

xAxAxAxA

xAxAxI

psfpsfpsfpsf

psfpsfexact

iWiW

iWiWiWiW

exact

evTevT

evTevTevTevTvI

22

2

2

2

1

2

2

2

1

)()(

)()()()()( 2121

Fourier Theory of Coherent Image Formation

Transfer of an extended

object distribution Eobj(x,y)

In the case of shift invariance

(isoplanasie):

coherent convolution of fields

Complex fields additive

In frequency space:

- product of spectra

- linear transfer theory with fields

- spectrum of the psf works as

low pass filter onto the object

spectrum

- Coherent optical transfer

function

object

plane image

plane

object

amplitude

distribution

single point

image

image

amplitude

distribution

dydxyxEyxyxAyxE psfima ),(',',,)','(

dydxyxEyyxxAyxE objpsfima ),(',')','(

),(,)','( yxEyxAyxE objpsfima

),(),( yxEFTvvH PSFyxctf

),(),(),( yxobjyxctfyxima vvEvvHvvE

2

),(),(),( yxobjyxctfyxima vvEvvHvvI

Ideal coherent transfer function:

- corresponds to scaled pupil function

- cut of at pupil edge

- full contrast until edge

Ideal incoherent transfer function:

- convolution of shifted scaled pupils

- smooth decrease to cut-off frequence

Incoherent case:

- higher spatial frequencies resolved

- lower contrast at low frequencies

f

xPvH

p

ctf

)(

2

21

22arccos

2)(

vvvvHotf

Comparison of OTF and CTF

c

H

0 0.5 1

1

coherent

incoherent

50

Ewald Sphere

grating

koutkobj

kin

kin

kobj

kout

Ewald sphere

2objk

L

Assuming an object as grating with period L

Scattering of a wave at the object with

- conservation of energy

- conservation of momentum

The outgoing k-vector must be on a sphere:

Ewald's sphere for possible scattered wave vectors

in outk k

in obj outk k k

51

McCutchen Formula and Axial Resolution

n/

P(z)

zz

z

x

x

P(x)

x

light

cone

Ewald

sphere

transverse

pupil

axial

pupil

cap

nR

2 2 2 2

sin / /x

xv R n NA n NA

Imaging of a plane wave at a volume object

x: minimum value resolution

: maximum interval

Uncertainty relation: x = 1

Radius of the Ewald sphere

generalized 3D pupil: red area

Transverse resolution due to Abbe

Axial resolution:

- height of the cap of the cone

- McCutchen formula

222

2

2

sin11/

1

cos

11

NA

n

NAnn

nRRvz

z

52

3D Transfer Function

z

x

Ewald

sphereforward

2n/

i

i

s

backward

o-max

obj

obj = s - i

Imaging as 3D scattering phenomen

Only special spatial frequencies are allowed due to energy conservation and

momentum preservation

Green circle: supported spatial

frequencies of the transmitted

wave vector

53

3D Transfer Function - Missing Cone

x

z

missing

cone

transfer function

object

x

z

illumination

scattered

wave

transfer function

object

i

s

obs

Realistic case:

finite numerical aperture

Blue cone:

possible incoming wave direction due

to illumination cone

3D coherent transfer function:

limited green area, that fulfills all

conditions

Missing cone:

certain range of spatial axial spatial

frequencies can not be seen in the image,

Example: interfaces of thin coatings are

not seen

3D Fourier transform of pupil with defocussing

M : magnification

pp

yyxxd

ikyxzMz

d

ik

pp

D

psf dydxeeyxPyxApppp

222

2'

23 ),(),(

3D Point Spread Function

x

y

z z'

x'

y'yp

zp

xp

d

d'

object

t(x,y,z)

pupil

P(xp,yp,zp) field

E(x',y',z' )

General coherent 3D transfer function

Fourier transfer of psf field distribuition

Imaging of an transparent object with transmission function T

Offset 1 / : optical path length in thick object

Special case of a single lens with pupil

function P

with

')'()( '2 rderAvH vri

psfcft

zyx

zMvyvxvi

zyxctfzyxobj

ikz

ima dvdvdvevvvHvvvTezyxEzyx

2)1

(2' 1

,,,,)',','(

2,

2)( r

zrzr

rotsym

ctf

vvvPvvH

0

0sin

1

zv

General 3D Transfer Theory

r

z

0.5

1.0

P(r)

0

z0

HCTF