physical optics 3 diffraction, resolution and image

$: Physical optics 3 Diffraction, Resolution and image$
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Physical Optics

Lecture 3 : Diffraction, resolution and Fourier Optics

2017-04-19

Beate Boehme

Physical Optics: Content2

No Date Subject Ref Detailed Content

1 05.04. Wave optics G Complex fields, wave equation, k-vectors, interference, light propagation, interferometry

2 12.04. Diffraction B Slit, grating, diffraction integral, diffraction in optical systems, point spread function, aberrations

3 19.04. Fourier optics B Plane wave expansion, resolution, image formation, transfer function, phase imaging

4 26.04. Quality criteria and resolution B Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-

point resolution, criteria, contrast, axial resolution, CTF

5 03.05. Polarization G Introduction, Jones formalism, Fresnel formulas, birefringence, components

6 10.05. Photon optics D Energy, momentum, time-energy uncertainty, photon statistics, fluorescence, Jablonski diagram, lifetime, quantum yield, FRET

7 17.05. Coherence G Temporal and spatial coherence, Young setup, propagation of coherence, speckle, OCT-principle

8 24.05. Laser B Atomic transitions, principle, resonators, modes, laser types, Q-switch, pulses, power

9 31.05. Gaussian beams D Basic description, propagation through optical systems, aberrations

10 07.06. Generalized beams D Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy beams, applications in superresolution microscopy

11 14.06. PSF engineering G Apodization, superresolution, extended depth of focus, particle trapping, confocal PSF

12 21.06. Nonlinear optics D Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects, CARS microscopy, 2 photon imaging

13 28.06. Scattering G Introduction, surface scattering in systems, volume scattering models, calculation schemes, tissue models, Mie Scattering

14 05.07. Miscellaneous G Coatings, diffractive optics, fibers

D = Dienerowitz B = Böhme G = Gross

Diffraction at slit

Intensity after a slit with a screen far behind a

Destructive interference

dr

am d )sin( min

1

4

2sin~

xxI

axwhere

sinc(x) ]²

Intensity distribution:

Sinc-function = sinus cardinalis= (german) Spaltfunktion= „slitfunction“

For instance a = 4 = 4*500nm = 2µm

First zero crossing@ 25cm

With a = 20µm First zero crossing@ 2,5cm

d = 100cm

rdcmcm

a 10025

4

cmcm

a 1005,2

40

4

-1

sin(4x)

|1/(4x)|

sin(4x)4x

-1 10

1

sin(4x)4x[ ]²

2sin~

xxI

axwhere

sinc(x) ]²

Sinc-function= sinus cardinalis= (german) Spaltfunktion= „slitfunction“

Sinc-function

sinc(x)

[sinc(x)]²

Slit 4

Slit 1

5

Diffraction at slit

-2 -1 0 1

0

1

-2 -1 0 1 20

1

Transmission

Intensity at screen

)sin()cos( xdxx The sinc-function is the

fourier-transformof the slit function

222/

2/

)2/sin(~)cos(~)( xdxxI

a

a

xx edxe

Variable x@ Object

Variable @ Screen

dr

ad

min1sin

6

Diffraction at circular aperture

-2 -1 0 1

0

1

-2 -1 0 1 20

1

Transmission

Intensity at screen

The Airy-distribution follows from the2D-Fourier-transform of the circular aperture

Variable r@ Object

Variable @ Screen

DA

DNA

22.1sin min

1

'sin' unNA

2

1

2

22)(

NAr

NArJrI

Similar, but about 20% morespreaded distributionnon-equidistant zeros

Fourier Transformation 7

Corresponds to the description of a function as sum of sine or cosine-functions:

Real numbers: sinus and cosine transformation

Description of a (periodic) function as sum of cosine- or sine-functions

Complex formulation: Fourier transformation (FFT)

Corresponds to principle of superposition of electric fields

~ cos

~ sin

~ cos

~

8

Fourier Transformation

Some functions and their Fourier transformations:

Sine

Two sine-functions, 10% difference in frequencyequal amplitudes

Fourier transformation is additive

time or spatial

coordinatefrequency

v0

frequencyv0

9


Some functions (signals) and their Fourier transformations (spectra):

Uniform, constant function

pulse vica versa

Gaussian function exp(- a t ²)

Exponential exp( - | t |)

x frequency

vv0x

delta-function@ v = 0

Uniform spectrum

Gauss

Lorentz very slow . 2 . decrease

[ 1+ (2pv)²]

10


Some functions (signals) and their Fourier transforms (spectra):

rectangular

Single pulse

Sum of M = 2s+1 equidistant pulsesM = 5

M = 15

x

vv0x

Sinc(v)

Uniform spectrum

Sin(Mv)/sin(v)

v

x

11

Fourier Transformation - Properties

linearity: The FFT of the sum of 2 functions is the sum of their FFTsFFT( a +b ) = FFT(a) + FFT (b)

Scaling FFT( f(sx) ) = F(v/s) 1/s if f is scaled by s, then F is scaled by 1/sFFT (f(-x)) = F(-v)

Convolution: The convolution in time / space is equivalent to a multiplication in Frequency domainif FFT (a(x)) = A(v) and C (v) = A(v) B(v)

FFT (b(x)) = B(v) c (x) = a(x) b(x)

Symmetry for real functions: if f(x) is real F(-v) = F*(v) if f(x) is real and f(x)=f(-x) F(-v) = F(v) = F*(v)

F is also real and symmetric Correlation Theorem

if c is the correlation between a and b, then for FFTs is validC(v) = B(v)* A(v)

thus with real functions a,b,cc = a b C(v) = B(v) A(v)

= Δ Δ

c() = Δ · ∗ Δ

Fourier-Theory - Diffraction at sinus-grating 12

x vv0

x v

Sinusoidal transmission

Plus offset

= transmission function

Due to symmetry for real, symmetric function the orders 0, +1 and -1 occur

Limited extension of grating corresponds to multiplication with rectangle function no ideal delta-functions,

but broadening corresponding to Airy

v

13

Fourier Transformation – Consequences


Uniform, constant function

Impulse Vica versa

Gaussian function exp(- a t ²)

Exponential exp( - | t |)

A plane wave with infinite extend Could be focused to an infinitesimal small focus

An ideal point source produces a plane wave

A Gaussian intensity distributionremains Gaussian during propagation

A faster decay of intensity or signal causes a broader spectrum

14

Fourier Transformation - Consequences


rectangular

Single puls


M = 15

Sin(Mv)/sin(v)

The diffraction patter of a slit is the square of the sinc-function. A smaller slit causes a more extended pattern (Scaling theorem)

the resolution of a grating increases with the number of periods, because the lines in the spectrum become narrower

15

Fourier Transformation - Consequences

rectangular


Convolution of rectangle with pulses Product of sinc & grating function (delta-functions) as diffraction pattern

Sin(Mv)/sin(v)

Width of slits defines occurrence of ordersNumber of slits defines fidelity

Sinc(v)

Diffraction at grating – Complex Field 16

1. Width of slits defines occurrence of orders

2. Number of slits defines fidelity

3. Period of grating defines distance of orders

1-2-3-4 1-2-3-45 5

Between the maxima are 5 periods –The number increases with number of periods M

Diffraction at grating – Intensity 17

1. Width of slits defines occurrence of orders

2. Number of slits defines fidelity

3. Period of grating defines distance of orders

1-2-3-4 1-2-3-45 5

Between the maxima are 5 periods –The number increases with number of periods M

18

-2 -1 0 1 20

0.5

1

-2 -1 0 1 2-1

0

1

-2 -1 0 1 2-0.5

0

0.5

-2 -1 0 1 2-1

0

1

-2 -1 0 1 2-0.5

0

0.5

-2 -1 0 1 2-1

0

1

-2 -1 0 1 2-0.5

0

0.5

-2 -1 0 1 2-1

0

1

-2 -1 0 1 2-0.5

0

0.5

-2 -1 0 1 2-1

0

1

Description of (periodic) functions as sum of cos-functions

The object structures can be described As sum of cosine-functions

with their frequency spectrum

as Fourier transforms

Cos_trafo.m

=

=

=

=

+

+

+

+

cos(x) where x = x/2

- sin(3x)/3

+ sin(5x)/5

- sin(7x)/7

+ sin(9x)/9

Fourier Components

Diffraction in optical systems

self luminous point: emission of spherical wave optical system: only a limited solid angle is propagated truncation of the spherical wave results in a finite angle light cone in the image space: uncomplete constructive interference of partial waves

spreaded image point the optical systems works as a low pass filter limited resolution Field in the image plane ~ Fourier transformation of the complex pupil function A(xp,yp)

19

Object plane aperture image plane

truncatedspherical

wavexp, yp

pp

yyxxR

i

ppExP

dydxeyxAyxEpp

ExP''2

,)','(

where A(xp,yp) describes

transmission and phase (wavefront)

2* )','()','( yxEEEyxI

DAiry

19

Large astronomical telescopes consist of two collecting mirrors

Primary and secondary mirror Secondary mirror defines central

obscuration for incoming light secondary mirror

M2 / f2

primarymirrorM1, f1

image

bd

11

Resolution – example Telescope

20

Central obscuration

Spider obscuration

Circular apertureno obscuration

Circular aperture28% central obscuration


three spider vanes


four spider vanes

Resolution - Example Telescope

21

a) f/28 b) f/56 c) f/80

Resolution - Example camera lens

22

F-Number as measure for camera aperture

#1

2

F# high small aperture low resolution high depth of focus

F# low large aperture high resolution low depth of focus

Remark: in general no diffraction-limited images with cameras

a

f

a b

Resolution – Two points

Two independent self luminous points: emission of two spherical waves Incoherent superposition summation of intensities

23

Object plane aperture image plane

truncatedspherical

wavexp, yp

DAiry

Δ 0,5 0,61Δ 0,385

0,47Contrast V = 0,15Imin = 0,735 I0 Contrast V = 0

Sparrow-CriterionRayleigh-Criterion

planewave

24

Resolution – Two points

25

Resolution – Two points with incoherent illumination

Resolution and Spatial Frequencies Grating object pupil

Imaging with NA = 0.8

Imaging with NA = 1.3

Ref: L. Wenke

26

Resolution – More incoherent points

more independent self luminous points: emission of N spherical waves summation of intensities First point (green)

27

Object planeaperture image plane

truncatedspherical

wavexp, yp

DAiry

planewave


more independent self luminous points: emission of N spherical waves summation of intensities First point (green) Second point (blue)

28

Object planeaperture image plane

truncatedspherical

wavexp, yp

DAiry

2. planewave

DAiry


more independent self luminous points: emission of N spherical waves summation of intensities First point (green) Second point (blue) Third point (red)

29

Object plane

aperture image plane

truncatedspherical

wave

xp, yp

DAiry

3. planewave

DAiry

DAiry

In the aperture (pupil plane) we observe a plane wave for each object point For N points N independent plane waves wit different directions Diffraction for all waves Superposition of the point images

Incoherent points in image plane

Superposition of image intensities

30

image plane

truncatedspherical

wave

DAiry

DAiry

DAiry

-10

1

Overlay of point images31

1

-1 0 1

0

-1 0 1

0

1

-1 0 1

0

1

-1 0 1

Visibility = Contrast:

Imax=1

Imin=0

V = 1

Imax=1

Imin=0.735

V = (1-0.735)/(1+0.735)V = 0.15

Image Contrast32

-1 0 1

0

1

-1 0 1

Contrast

Imax=1

Imin=0.735

The definition of contrast refers to dark intensity = 0

e.g. without any light Intensity = 1, which occurs with

an extended object

In metrology these values are often unknown: CCD-cameras do Subtraction of dark current contrast enhancement

The definition is valid for extended periodical structures, where the edges of the structure have no influence onto the values at the center

Resolution – Vice Versa discusssion

Plane waves with different directions in the object plane Focused, convergent waves in the pupil plane Coordinate of focus depends on direction of plane wave Limitation of directions by the aperture Superposition of the transmitted plane waves in the image

Plane waves can be thought of generated by a grating, illuminated with a plane wave Far field diffraction pattern in the pupil

33

Object plane

apertureimage plane

xp, yp

plane wavesuperposition

Resolution – Vice Versa discusssion

Plane waves with different directions in the object plane Focused, convergent waves in the pupil plane Coordinate of focus depends on direction of plane wave Limitation of directions by the aperture Superposition of the transmitted plane waves in the image

34

Object plane

apertureimage plane

xp, yp

plane wavesuperposition

Obscuration of waves with high angle:

Plane waves can be thought of generated by a grating, illuminated with a plane wave Far field diffraction pattern in the pupil

Sine-grating in the object plane Two diffraction orders:

0. order = transmitted light +1. order - 1. order

Increasing diffraction angle with smaller period g / increasing spatialfrequency v = 1/g Location of diffraction orders in the

back focal plane depends on grating period

The sine-grating can only be reproduced in the image, if orders 0, +1 and -1 are transmitted There is a minimum period

which can be transmitted

35

Plane wave expansion

+1st

-1st

+1st

-1st

+1st

-1st

objectback focal

planeobjective

lens

0th order

diffracted ray direction

objectstructure

g = 1 /

/ g

k

k x

Definitions of Fourier Optics

Phase space with spatial coordinate x and1. angle 2. spatial frequency in mm-1

3. transverse wavenumber kx

are related by grating equation

Fourier spectrum

corresponds to a plane wave expansion

For focussing of diffraction orderswith lens to final image distance

0kkv x

xx

),(ˆ),( yxEFvvA yx

A k k z E x y z e dx dyx yi xk ykx y, , ( , , )

vk 2

vg

1sin

36

yx vfyvfx ',' focal length f

y'

planewave

spherical wave fronts

z

yp

Abbes Diffraction Theory of Image Formation

Wave optical interpretation of the optical image formation The object is considered as a superposition of several gratings (Fourier picture) Every grating diffracts the light in the various orders Only those orders are contributing to the image, which lie inside the cone of the

numerical aperture of the optical system There is a limiting spatial frequency and a minimum feature size, that is resolved by the

system

+1.order

-1.order

37

Imaging of a crossed grating object Spatial frequency filtering by a slit:

Case 1: - pupil open- Cross grating imaged

Case 2: - truncation of the pupil by a split- only one direction of the grating is

resolved

Fourier Filtering

38

Fourier Decomposition

Ref: D.Aronstein / J. Bentley 39

40

-2 -1 0 1 20

0.5

1

-2 -1 0 1 2-1

0

1

-2 -1 0 1 2-0.5

0

0.5

-2 -1 0 1 2-1

0

1

-2 -1 0 1 2-0.5

0

0.5

-2 -1 0 1 2-1

0

1

-2 -1 0 1 2-0.5

0

0.5

-2 -1 0 1 2-1

0

1

-2 -1 0 1 2-0.5

0

0.5

-2 -1 0 1 2-1

0

1

Description of slit transmission as sumof cosine-functions

All object structures can be described as sum of cosine-functions

Cos_trafo.m

=

=

=

=

+

+

+

+

cos(x) where x = x/2

- sin(3x)/3

+ sin(5x)/5

- sin(7x)/7

+ sin(9x)/9

Fourier Components

Modulation transfer function – coherent case41

Frequencyv = 0

Transmission of Frequencies for sine-grating

with collimated / coherent illumination

+1st

-1st

+1st

-1st

+1st

-1st

objectback focal

planeobjective

lens

0th order

v = 4 Lines/mm

v = 7L/mm

v = 12L/mm

FrequencyLines/mm

Image Contrast

1

coherent:

apertureplane sin

1

vg

Number of Supported Orders

A structure of the object is resolved, if the first diffraction order is propagatedthrough the optical imaging system

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

42

Off-Axis illumination With an off-axis illumination the transmitted

and diffracted orders shift at the pupil The 2. order walks into the pupil, if Ill = NA With a set of illumination directions with

NAillumination = Naimaging e.g. approx. incoherent Incoherent cut off frequency

= two times the coherent cut off

NAg

2

Modulation transfer – off axis illumination

opticalsystem

object

diffracted ordersa) resolved

b) not resolved

+1.

+1.+2.

+2.

0.

-2.

-1.

0.

-2.

-1.

incidentlight

43

NAg

Coherent – collimated illumination

-1.

-1.

0.+2.+1.

0.

+1.

-2.

+2.+1.

NA

g NA

g

2

Modulation Transfer Function - MTF

Aberration free circular pupil:Reference frequency

Cut-off frequency:

Analytical representation

'sin' un

favo

NAvv 22 0max

2

000 21

22arccos2)(

vv

vv

vvvHMTF

/ max0

0

1

0.5 1

0.5

MTF

44

Perfect system

Incoherent illumination:

coherentillumination

Incoherent Image Formation

One illumination point generates a plane wave in the object space Diffraction of the wave at the object structure Diffraction orders occur in the pupil Constructive interference of all supported diffraction orders in the image plane Too high spatial

frequencies areblocked

object plane

pupilplane

imageplanef f f f

u() U (x)1

h()

f f

lightsource

s() U (x)0

T(x)

s

s

Ref: W. Singer

45

Calculation of MTF

46

MTF describes transmission of sine gratings by the optical system Description in frequency space Calculation and explanation as description of point image in frequency space

= spectrum of PSF

Alternative calculation: Autocorrelation of pupil function = overlap integral as function of shift

For 1-dim pupil: autocorrelation of two Top-hat functions = triangle function

For 2-dim circular pupil: autocorrelation of two circles:

proportional to the overlapping surface triangle-similar at center slow decrease to zero

2sin/sin/

2sin/

∗

pp

vyvxipppsfyxOTF dydxeyxINvvH ypxp

2),(),(

),(ˆ),( yxIFvvH PSFyxOTF

p

xp

xpxOTF dxvfxPvfxPvH

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:OTF: in general complex function, describes transfer of amplitude and phase

response answer of an extended cosine grating MTF = modulation transfer function (MTF) = Absolute value of OTF

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

PTF = phase transfer function

distinguish: PSF = response answer of a point object

47

),(),(),( yxPTF vvHiyxMTFyxOTF evvHvvH

MTF of a Perfect System Loss of contrast for higher spatial frequencies

48

Contrast and Resolution

High frequent structures :contrast reduced

Low frequent structures:resolution reduced

contrast

resolution

brillant

sharpblurred

milky

49

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 sine grating

Contrast of an corresponding rectangular grating is higher than for the sine gratingbecause higher diffraction orders help “Square wave MTF”

The maximum value of the intensityis not identical to the contrast valuesince the minimal value is finite too

Visibility of rectangular grating

minmax

minmax

IIIIV

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

peakdecreased

slopedecreased

minimaincreased

50

Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on theazimuthal orientation of the object structure „surface MTF“

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

Sagittal and Tangential MTF

y

tangentialplane

tangential sagittal

arbitraryrotated

x sagittalplane

tangential

sagittal

gMTF

tangential

ideal

sagittal

1

0

0.5

0 0.5 1 / max

51

Polychromatic MTF

Cut off frequency depends on

Polychromatic MTF: Spectral incoherent weighted superposition of monochromatic MTF’s

Example: uncorrected axial color F (486), D(587), C(656nm) with SF6 instead SF5

0

)( ),()()( dvHSvH MTFpoly

MTF

52

#122 0max F

NAvv

contrast decreases with defocus higher spatial frequencies have

stronger decrease Zero values in MTF indicate

phase shift of OTF contrast reversal

Real MTF

z = 0

z = 0.1 Ru

gMTF1

0.75

0.25

0.5

0

-0.250 0.2 0.4 0.6 0.8 1

z = 0.2 Ruz = 0.3 Ru

z = 1.0 Ruz = 0.5 Ru

53

Siemens star – experiment

54

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 resolutionPhase shift in transfer function

55

Resolution Test Chart: Siemens Star

original good system

astigmatism comaspherical

defocusa. b. c.

d. e. f.

56

Resolution Estimation with Test Charts

0 1

10

2

3

4

5

6

65

4

3

2

1

6

5

4

3

22 3

1

2

3

2

456

Measurement of resolution with test charts: bar pattern of different sizes two different orientations calibrated size/spatial frequency

57

Contrast / Resolution of Real Images

Degradation due to1. loss of contrast2. loss of resolution

58

Contrast as a function of spatial frequency

Compromise betweenresolution and visibiltyis not trivial and dependson application

Contrast and Resolution of Real Applications

59

Real systems:

Limited contrast sensitivity of detectors

for instance: 8Bit = 256ct limit 1/256 for contrast

contrast sensitivity may depend on direction and spatial frequency

Image processing with contrast enhancement

Human eye: about 0,25% contrast sensitivity v/vreal

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 frequenciesCSF: 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

60

Fourier theory of image formation

Coherent and incoherent image formation

61

Fraunhofer Point Spread Function

Mathematical formulation of the Huygens-principle Rayleigh-Sommerfeld diffraction integral

for small Fresnel number at far field Fraunhofer approximation linear part of expansion

Optical systems: Numerical aperture NA in image space Pupil amplitude/transmission/illumination T(xp,yp) Wave aberration W(xp,yp)

Transition from exit pupilto image plane

Point spread function (PSF) Fourier transform of the complex pupil function A

12

z

rN p

F

),(2),(),( pp yxWipppp eyxTyxA

pp

yyxxR

iyxiW

ppExP

dydxeeyxTyxEpp

ExPpp''2

,2,)','(

dydxrr

erEirE d

rrki

O

cos'

)()'('

62

Fourier Optics – Point Spread Function

Optical system with pupil function P,Pupil coordinates xp,yp

PSF is Fourier transformof the pupil function (scaled coordinates)

Intensity of point image

pp

yyyxxxzik

pppsf dydxeyxPyxyxg pp '',~)',',,(

pppsf yxPFyxg ,ˆ~),(

objectplane

imageplane

sourcepoint

pointimagedistribution

63

psfpsfpsfpsf gggyxI *2),(

Fourier Theory of Coherent Image Formation

Transfer of an extended electric fielddistribution in object plane E(x,y)

In the case of shift invariant PSF (isoplanatism) = convolution of fields

Symbol for convolution

dydxyxEyxyxgyxE psf ),(',',,)','(

dydxyxEyyxxgyxE psf ),(',')','(

64

E(x,y)

),(,)','( yxEyxgyxE psf

Fourier Theory of Coherent Image Formation

Convolution in spatial domain description of field as sum of frequency (grating) components transition to frequency domain by Fourier Transformation

Convolution in space corresponds to product of spectra Coherent optical transfer function

spectrum of the PSF works as low pass filter onto the object spectrum

),(),( yxgFTvvH PSFyxctf ),(),(),( yxobjyxctfyxima vvEvvHvvE 2

),(),(),( yxobjyxctfyxima vvEvvHvvI

65

objectplane image

plane

objectamplitude

distribution

single pointimage

imageamplitudedistribution

),(,)','( yxEyxgyxE psf

Fourier Theory of Incoherent Image Formation

objectintensity image

intensity

singlepsf

objectplane

imageplane

Transfer of an extendedobject distribution I(x,y) In the case of shift invariant PSF

(isoplanatism) = convolution of intensities

In frequence domian: Product ofintensity transfer function Hotf(vx,vy)and object intensity

Absolute value ofthe OTF = MTF

Low pass filter ofintensity distribution

),(*),()','( yxIyxIyxI objpsfimage

dydxyxIyyxxgyxI psfinc

),(),',,'()','(2

66

),(),(),( yxobjyxotfyximage vvIvvHvvI

Modulation Transfer

Convolution of the object intensity distribution I(x) changes:1. Peaks are reduced2. Minima are raised3. Steep slopes are declined4. Contrast is decreased

67

Comparison Coherent – Incoherent Image Formationobject

‐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

incoherent coherent

‐0.0 5 0 0.0 50

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

‐0.0 5 0 0.0 50

0.1

0.2

0.3

0.4

0.5

0.6

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bars resolved bars not resolved bars resolved bars not resolved

68

Incoherent image: homogeneous areas, good similarity between objectand image, high fidelity

Coherent image:Granulation of area ranges, diffraction ripple atedges

Coherent – Incoherent Image Formation69

Fourier Theory of Image Formation

Coherent Imaging

objectintensity

I(x,y)

squared PSF,intensity-responseIpsf (xp,yp)

imageintensityI'(x',y')

convolution

result

objectintensityspectrum

I(vx,vy)

opticaltransferfunction

HOTF (vx,vy)

imageintensityspectrumI'(vx',vy')

produkt

result

Fouriertransform

Fouriertransform

Fouriertransform

Incoherent Imaging

4.2 Image simulation70

Partial Coherent Imaging

Every object point is illuminated by an angle spectrum due to the finite extend of the source In the pupil the diffraction orders are broadened no full constructive interference in the image plane

source condenser object lens image

angle shift

71

Heuristic explanationof the coherenceparameter in a system:

1. coherent:Psf of illuminationlarge in relation to theobservation

2. incoherent:Psf of illuminationsmall in comparison to the observation

Coherence parameter : describes ratio of illumination NA to observation NA

object objective lenscondensersmall stop of condenser

extended source

coherentillumination

large stop of condenser

incoherentillumination

Psf of observation inside psf of illumination

Psf of observation contains several illumination psfs

extended source

Coherence Parameter

72

Resolution of two points for partial coherent illumination in distance r' Circular source with parameter Resolution pre-factor Distance of objects for identical contrast (intensity drop to 73%)

1. coherent = 0 :

2. Incoherent >> 1 :

3. Optimum = 1.46 :

Resolution for Partial Coherent Imaging

'sin81.0'

ur

'sin61.0'

ur

'sin56.0'

ur

73

Resolution and Contrast for Partial Coherent Imaging

Transfer of spatial frequenciesdepend on illumination settingsand directions

Analytical representationonly for circular Symmetrypossible (Kintner)

Transfer capability dependson integration overlap ofillumination and detectionpupils

incoherentpartial

coherent coherentpartial coherent

oblique illuminationcoherent oblique

illumination

74

Transmission Cross Correlation Function (TCC)

Typical change oftransfer capability

Partial coherent:

75

Partial Coherent Imaging of Siemens Star

76

Pupil Illumination Pattern

Coherent

Off Axis Annular Annular

Dipole Rotated Dipole

Disk = 0.5 Disk = 0.8

6 -Channel

Variation of the pupil illumination Enhancement of resolution Improvement of contrast Object specific optimization

Often the best compromise with partial coherent illumination where slightly increased intensity at the

edges

In microscopy the adjustment of Koehler illumination corresponds to choosing this setup

Ref: W. Singer

77

Dark field illumination in microscopy

Foucault knife edge method for aberration measurement

Schlieren method for measurement of striae and inhomogeneities in materials

Zernike contrast method in microscopy

Use of apodization for suppression of diffraction rings, resolution enhancing masks

Beam clean up of laser radiation with kepler system and mode stop

Edge enhancement techniques in lithography

Oblique illumination for resolution enhancement in microscopy

Schmidt corrector plate in astronomical telescopes

Pupil filter masks to generate extended depth of focus

Resolution enhancement by structured illumination

Applications of Fourier Filtering Techniques

78

physical optics 3 diffraction, resolution and image

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