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Physical Optics

Lecture 4 : Fourier Optics

2017-04-19

Beate Boehme

Physical Optics: Content

2

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 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)

Object plane aperture image plane

truncated

spherical

wave A(xp,yp)

pp

yyxxR

i

pp

ExP

dydxeyxAyxEpp

ExP

''2

,)','('

where A(xp,yp) describes

transmission and phase (wavefront)

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

DAiry

E(x,y)

3

Resolution – More incoherent points

more independent self luminous points: emission of N spherical waves

summation of intensities

Object plane

aperture image plane

truncated

spherical

wave

A(xp,yp)

DAiry

3. plane

wave

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

4

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

Object plane

aperture

image plane

xp, yp

plane wave

superposition

),(ˆ),( yxEFvvA yx

E(x,y)

dydxeyxEkkA yx ykxki

yx ),(,

vx, yy

y

x

vfy

vfx

'

'

5

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 spatial

frequency 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

6

Plane wave expansion

+1st

-1st

+1st

-1st

+1st

-1st

objectback focal

planeobjective

lens

0th order

Modulation Transfer Function - MTF

Aberration free circular pupil:

Reference frequency

Cut-off frequency:

Analytical representation

'sin' un

f

avo

NAvv

22 0max

2

000 21

22arccos

2)(

v

v

v

v

v

vvHMTF

/ max

00

1

0.5 1

0.5

MTF

Perfect system

Incoherent illumination:

coherent

illumination

The optical system acts as

Low-pass filter with cut-off frequency

7

Calculation of MTF

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

2sina/ sina/

2sina/

𝐴(𝑥′) = 𝐹 𝑥 + 𝑥′∗𝐹 𝑥 − 𝑥′ 𝑑𝑥

8

9

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

10

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%)

1

0,5

Transmission of pupil (pupil function A)

= real function, describes:

ideal optical system

no complex part Autocorrelation real

no modification of phase

no aberrations

MTF

11

Calculation of MTF – complex pupil function

T(xp,yp)

Transmission:

circular pupil

Linear wavefront = phase at pupil

Autocorrelation in x-direction

~ frequency response of the system:

𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp −2𝜋𝑖 𝑊(𝑥𝑃, 𝑦𝑃 )

𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp (−2𝜋𝑖 𝑎𝑥𝑃)

𝑂𝑇𝐹(𝑣𝑥) = 𝐴 𝑥𝑝 +𝜆𝑓2𝑣𝑥 , 𝑦𝑝

∗𝐴 𝑥𝑝 −

𝜆𝑓2𝑣𝑥 , 𝑦𝑝 𝑑𝑥𝑝𝑑𝑦𝑝

MTF = abs (OTF)

0 0.5 1 0

0,5

1

0 0.5 1

2

PTF = angle(OTF) = phase transmission

12

Calculation of MTF – complex pupil function

T(xp,yp)

Transmission:

circular pupil

Linear wavefront = phase at pupil

Spherical phase: Defocus

Autocorrelation in x-direction

~ frequency response of the system:

𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp −2𝜋𝑖 𝑊(𝑥𝑃, 𝑦𝑃 )

𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp (−2𝜋𝑖 𝑎𝑥𝑃)

𝐴 𝑥𝑃𝑦𝑃 = 𝑇 𝑥𝑃𝑦𝑃 ∙ exp (−2𝜋𝑖 𝑎(𝑥𝑃² + 𝑦𝑃²)

𝑂𝑇𝐹(𝑣𝑥) = 𝐴 𝑥𝑝 +𝜆𝑓2𝑣𝑥 , 𝑦𝑝

∗𝐴 𝑥𝑝 −

𝜆𝑓2𝑣𝑥 , 𝑦𝑝 𝑑𝑥𝑝𝑑𝑦𝑝

MTF = abs (OTF)

0 0.5 1

2

PTF = angle(OTF) = phase transmission

13

OTF – Transfer of sine gratings

Transfer of amplitude sine-grating with

Mean intensity and Contrast V

𝐼

𝐼′

x, x’

1/v PTF(v)

𝑉 =𝐼𝑀𝑎𝑥 − 𝐼𝑀𝑖𝑛𝐼𝑀𝑎𝑥 + 𝐼𝑀𝑖𝑛

Object-Contrast

Image-Contrast 𝑉′ = 𝑀𝑇𝐹 𝑣 ∙ 𝑉

𝐼

𝑂𝑇𝐹 = 𝑀𝑇𝐹 𝑣)exp (−𝑖 𝑃𝑇𝐹(𝑣)

For an aberration-free system

PTF = 0

The image has reduced contrast

With aberrations periodic

structures are transferred with

phase shifts.

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 are

blocked

object plane

pupilplane

imageplane

f f f f

u() U (x)1

h()

f f

lightsource

s() U (x)0

T(x)

s

s

Ref: W. Singer

14

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 grating

because higher diffraction orders help “Square wave MTF”

The maximum value of the intensity

is not identical to the contrast value

since the minimal value is finite too

Visibility of rectangular grating

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

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

azimuthal orientation of the object structure „surface MTF“

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

16

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 MTF

poly

MTF

#

122 0max

F

NAvv

17

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

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

18

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

Phase shift in transfer function

19

Resolution Test Chart: Siemens Star

original good system

astigmatism comaspherical

defocusa. b. c.

d. e. f.

20

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

21

Blurred imaging:

- limiting case

- information extractable

Blurred imaging:

- information is lost

- what‘s the time ?

Resolution: Loss of Information

22

Contrast / Resolution of Real Images

resolution,

sharpness

contrast,

saturation

Degradation due to

1. loss of contrast

2. loss of resolution

23

Contrast as a function of spatial frequency

Compromise between

resolution and visibilty

is not trivial and depends

on application

Contrast and Resolution of Real Applications

V

/c

1

010

HMTF

Contrast

sensitivity

HCSF

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

24

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

25

OTF – Calculation from Point spread function

Ideal imaging airy distribution = answer of the optical system to an ideal point source

the system transfer is described in space

frequency space Fourier transform of the airy intensity pattern

Real imaging OTF = Fourier Transform of the point spread function (2D)

in general no radial-symmetric function

aperture image plane

A(xp,yp)

DAiry

DAiry

DAiry

26

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:

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 the

corresponding spatial frequency

PTF = phase transfer function

distinguish: PSF = response answer of a point object

27

),(),(),( yxPTF vvHi

yxMTFyxOTF evvHvvH

Fourier theory of image formation

Coherent and incoherent image formation

28

Fourier Optics – Point Spread Function

Optical system with pupil function P,

Pupil coordinates xp,yp

PSF is Fourier transform

of the pupil function (scaled coordinates)

Intensity of point image

pp

yyyxxxz

ik

pppsf dydxeyxPyxyxgpp ''

,~)',',,(

pppsf yxPFyxg ,ˆ~),(

object

planeimage

plane

source

point

point

image

distribution

29

psfpsfpsfpsf gggyxI *2

),(

Fourier Theory of Incoherent Image Formation

objectintensity image

intensity

single

psf

object

planeimage

plane

Transfer of an extended

object distribution I(x,y)

In the case of shift invariant PSF

(isoplanatism) = convolution of intensities

In frequence domian: Product of

intensity transfer function Hotf(vx,vy)

and object intensity

Absolute value of

the OTF = MTF

Low pass filter of

intensity distribution

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

dydxyxIyyxxgyxI psfinc

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

30

),(),(),( yxobjyxotfyximage vvIvvHvvI

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

31

Fourier Theory of Coherent Image Formation

Transfer of an extended electric field

distribution in object plane E(x,y)

In the case of shift invariant PSF

(isoplanatism) = convolution of fields

Symbol for convolution

object

plane image

plane

object

amplitude

distribution

single point

image

image

amplitude

distribution

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

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

32

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

33

object

plane image

plane

object

amplitude

distribution

single point

image

image

amplitude

distribution

),(,)','( yxEyxgyxE psf

Fourier Theory of Image Formation

object

amplitude

U(x,y)

PSF

amplitude-

response

Hpsf (xp,yp)

image

amplitude

U'(x',y')

convolution

result

object

amplitude

spectrum

u(vx,vy)

coherent

transfer

function

hCTF (vx,vy)

image

amplitude

spectrum

u'(v'x,v'y)

product

result

Fourier

transform

Fourier

transform

Fourier

transform

Coherent Imaging

object

intensity

I(x,y)

squared PSF,

intensity-

response

Ipsf

(xp,y

p)

image

intensity

I'(x',y')

convolution

result

object

intensity

spectrum

I(vx,v

y)

optical

transfer

function

HOTF

(vx,v

y)

image

intensity

spectrum

I'(vx',v

y')

produkt

result

Fourier

transform

Fourier

transform

Fourier

transform

Incoherent Imaging

4.2 Image simulation

34

Comparison Coherent – 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

incoherent coherent

-0.0 5 0 0.0 5

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

-0.0 5 0 0.0 5

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 5

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 5

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

bars resolved bars not resolved bars resolved bars not resolved

35

Incoherent image:

homogeneous areas, good similarity between object

and image, high fidelity

Coherent image:

Granulation of area ranges, diffraction ripple at

edges

incoherent coherent

Coherent – Incoherent Image Formation

incoherent

coherent

36

Image of an edge for

incoherent illumination

No oscillations, smooth distribution

Ideal position of the edge at 50%

Width of the edge transition depends

on PSF of imaging setup

I(x)

x

z

aky

z

aky

z

akySiyI inc

edge 2

12

cos121

2

1)(

Incoherent Image of an Edge

y'0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.1 -0.05 0 0.05 0.1 0.15

derivation of

edge spread

function

edge spread

function

point spread

function

37

Image of an edge for coherent illumination

with integral sine function Si

a: half diameter of aperture

Intensity 25% at edge

Error in detection of edge position for 50%

criterion

Amplitude and intensity distribution:

- oscillations around the edge

- in shadow region hard to resolve

x

E(x)

I(x)

x

2

)( 1

2

1)(

z

akySiyI coh

edge

a

zx

212.0

Coherent Image of an Edge

38

Partial coherent imaging

39

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 additionally

no full constructive interference in the image plane

influence of the illumination onto the image should be considered

object pupil imagelight

source

+ 1

- 1

0

source condenser object lens image

angle shift

40

Heuristic explanation

of the coherence

parameter in a system:

1. coherent:

Psf of illumination

large in relation to the

observation

2. incoherent:

Psf of illumination

small in comparison

to the observation

Coherence parameter s:

describes ratio of

illumination NA to

observation NA

object objective lenscondensersmall stop of

condenser

extended

source

coherent

illumination

large stop of

condenser

incoherent

illumination

Psf of observation

inside psf of

illumination

Psf of observation

contains several

illumination psfs

extended

source

Coherence Parameter

𝜎 =𝑁𝐴(𝑖𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛)

𝑁𝐴(𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛)

41

Partial Coherent Imaging - Example Bar-Pattern

12 mm

object

s = 0.08

pupil intensity

imageobject

spectrum

s = 0.50 s = 1.0

coherence function

Coherence of fields at

two object points

transmission by the

optical system

intensity & amplitude

transmission non-linear

Incoherence:

spreading of edges

Higher coherence:

periodic intensity

modulations (compare

slit with coherent illum.)

Optimization of

coherence settings

Optimization of

illumination

42

Resolution and Contrast for Partial Coherent Imaging

HCTF(s)

1

2

light source

light source

a) partial

coherent

HCTF(s)

light source

light

source

b) incoherent

1

2

x

y

x

y

incoherentpartial

coherent coherentpartial coherent

oblique illumination

coherent oblique

illumination

Transfer of spatial frequencies

depend on illumination settings

and directions

Analytical representation

only for circular Symmetry

possible (Kintner)

Transfer capability depends

on integration overlap of

illumination and detection

pupils

Transmission Cross Correlation

Function (TCC)

43

Partial Coherent Imaging of Siemens Star

coherent partial coherent incoherent

frequency

o = sinu /

frequency

o = 2sinu /

44

Pupil Illumination Pattern

Coherent

Off Axis Annular Annular

Dipole Rotated Dipole

Disk s = 0.5 Disk s = 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

45

Lichtquelle

Kollektor Kondensor Objektiv

Objektebene

Bildebene

Leuchtfeld-

blende

Kondensor-

blende

Apertur-

blende

Application in Microscopy

Koehler illumination

to

image

Microscopic

lens condensor

Object

plane Illumination

aperture

imaging

(microscope)

aperture

collector

Light

source

light field

diaphragm

Light field diaphragm to limit illuminated object area

aperture at the condenser to adjust degree of coherence 𝜎𝑖𝑙𝑙𝑢𝑚 𝜎𝑖𝑚𝑎𝑔𝑖𝑛𝑔

46

Procedure:

1. Aperture stop open, field stop closed: center visible rim of field stop

2. Focussing of condensor: axial shift of condensor, sharpen rim of field stop

3. Open field stop until the visible field is illuminated

4. Close aperture stop to about 2/3

5. Adjust brightness of lamp

Kondensor

Objekt-

ebene

Apertur-

blendeLeuchtfeld-

blende Filter

Kollektor

Lampe

Application in Microscopy

Koehler illumination adjustment

Aperture

stop

condensor

Object

field

stop

collector

lamp

47

Imaging of phase objects

48

Pure phase mask as object constant intensity at image

Variation of focus during observation

Phase structure becomes visilbe

Reason: defocus modifies MTF shift of PFT at pupil zones

Imaging of transparent phase objects 49

Quantitative Phasenmikroskopie

Setup :

anschaulich

Abbildungs-

system

Quelle

Quelle

Messebenen

Objekt

1. System-

messung

2. Abbildung

Verteilung durch

Objekt verändert

Imaging of phase objects 50

Pure phase transmission at object

Approximation: small phase

Modification at pupil: phase mask

For 0. diffraction order:

reduced transmssion a

90°phase shift

approximated intensity at image

)()()( xiexBxP

)(1)( xixE

)'()'(' xiiaxE

)'()'(' xaxI

KondensorBild

Bertrand-

linseObjektiv

Ring-

blendeObjekt

Phasen-

platte

Zernike phase contrast

Phase

object

lens

Pupil with

phase plate

lens

image

condensor

ring

Illumination

51

Klassisches DurchlichtPhasenkontrastbild

Phasenkontrast nach Zernike : Beispiel

Zernike phase contrast 52

Axial resolution and depth of focus

53

Depth of Focus: Geometrical

z

2

object

plane

zgeo

p

entrance

pupil

image

plane

z'geo

p'

exit

pupilsystem

Spot spreading in focus: diameter 2

Detector spatial resolution D

Depth of focus: 2 < D

Axial interval of sharpness. calculated by geometrical optics

54

0

2

12,0 I

v

vJvI

0

2

4/

4/sin0, I

u

uuI

-25 -20 -15 -10 -5 0 5 10 15 20 250,0

0,2

0,4

0,6

0,8

1,0

vertical

lateral

inte

nsity

u / v

Circular homogeneous illuminated

Aperture: intensity distribution

transversal: Airy

scale:

axial: sinc

scale

Resolution transversal better

than axial: x < z

Scaled coordinates according to Wolf :

axial : u = 2 z n / NA2

transversal : v = 2 x / NA

Perfect Point Spread Function

NADAiry

22.1

2NA

nRE

55

Normalized axial intensity

for uniform pupil amplitude

Decrease of intensity onto 80%:

Scaling measure: Rayleigh length

- geometrical optical definition

depth of focus: 1RE

- Gaussian beams: similar formula

22

'

'sin' NA

n

unRu

Depth of Focus: Diffraction Consideration

2

0

sin)(

u

uIuI

2' o

un

R

udiff Run

z

2

1

sin493.0

2

12

focal

plane

beam

caustic

z

depth of focus

0.8

1

I(z)

z-Ru/2 0

r

intensity

at r = 0

+Ru/2

56

Depth of Focus

Depth of focus depends on numerical aperture

1. Large aperture: 2. Small aperture:

small depth of focus large depth of focus

Ref: O. Bimber

57

Ring pupil illumination

Enlarged depth of focus

Lateral resolution constant due to

large angle incidence

Can not be understood geometrically

Depth of focus for Annular Pupil

58

Farfield of a ring pupil:

outer radius aa

innen radius ai

parameter

Ring structure increases with

Depth of focus increases

Application:

Telescope with central obscuration

Intensity at focus

1a

i

a

a

2

121

22

)(2)(2

1

1)(

x

xJ

x

xJxI

r-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I(r)

= 0.01

= 0.25

= 0.35

= 0.50

= 0.70

22 1sin

2

unz

Depth of focus for Annular Pupil

59

Ring shaped masks according to Toraldo :

- discrete rings

- absorbing rings or pure phase shifts

- original setup: only 0 / values of phase

- special case

Fresnel zone plate

Pure phase rings:

amplitude of psf

Extended depth of focus: Toraldo Ring Masks

00

0

0

0

Phasen-

platte

Fokus

n

j j

j

j

j

j

j

i

ukr

ukrJ

ukr

ukrJerE j

1 1

112

1

122

'sin'

'sin'2

'sin'

'sin'2)'(

r

1

2

j

n

3 10

0

60

MTF and bar pattern as a function of defocus

ideal with Mask

Extended depth of focus

61

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)(

3a

Cubic Phase Plate (CCP)

z

y-section

x-section

cubic phase PSF MTF

62

Cubic Phase Plate: PSF and OTF

defocussed focus

Conventional imaging System with cubic phase mask

defocussed focus

63

Conventional microscopic image

Image with phase mask

with / without deconvolution

EdoF: CPP for microscopic Imaging

Ref: E. Dowski

64

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

65

Depth of focus at the human eye –

Correction of presbyopia

66

Human eye and Visual acuity

Distance 5 – 6m

1min for

Visual Acuity

VA 20/20 = 1

1‘ VA = 1

2‘ VA = 0.5

Vitreous

Lens

Cornea

Anterior

chamber

Iris = stop Retina

(Macula)

Intraocular lens:

67

Human eye with monofocal intraocular lens (IOL)

Visus 1 corresponds to

Object @ 1min = 290µm @ 1m (spaces)

paraxial image 4.7µm @ water

period 105 L/mm (rect Bar+space)

corresponds to cone-pitch 2.3µm

@ macula

Pseudophakic eye

With artificial intraocular lens

Correction to refraction = 0

diffraction-limited imaging

Effective focal length 21.7mm @ air

pupil diameter 2 – 6mm

Stop-Diameter 2mm 3mm 4mm 6mm

vg = 2 NA‘ / 260 390 521 780 L/mm

NA‘ = 0.1416

Airy radius 4,8µm 2,4µm 1,6µm

VA = 1

Every day situations a

105L/mm

68

Diffraction-limited PSF @ retina:

Simulation of visual performance

Monofocal imaging

Calculation of imaging to retina:

Airy distribution

Size for 2mm and 4mm pupil:

Axial 291µm 73µm ~1/stop²

radial 4.7µm 2.35µm ~ 1/ stop

distance to zero intensity

Contrast 105L/mm für visual acuity = 1 = 20/20

@ 2 points 99.9% 99.1%

@ sinus 50.8% 75%

3 objects at different distances

F E N at far, near and an intermediate distance

Calculation of intensity at common image plane

Variation of power addition with glasses

69

Monofocal Imaging

Image stack,

Stop 2mm

Inreasing Power of Glasses or addition

70

Monofocal Imaging

Image stack,

Stop 4mm

71

Stop 2mm

Monofocal Imaging

images @ far, intermediate, near with add correction

Stop 4mm

influence on contrast and resolution visible

increase of DOF & less accommodation necessary at small pupils

Simulation of visual performance

72

Far and Near image simultaneously sharp

Power addition 3.75dpt

Separation of light onto

two foci,

greater amount to far

Ideal Bi-focal Imaging

2 Diffraction-limited PSFs in two z-positions

Images for stop 2mm:

Symbol intensities fit to PSF

With a small add power all tree

symbols become visible

simultaneously

Simulation of visual performance

73

Stop 4mm

reduction of contrast barely remarkable (in comparison to monofocal lens)

An ideal bifocal lens would show bad intermediate contrast

An ideal model of real Bifocal lenses is not sufficient

maybe Halos for very high far intensities

Stop 2mm

Ideal Bifocal Imaging

images @ far, intermediate, near with add correction

Simulation of visual performance

74

Stop 4mm

Far Focus Far, symbol for VA 0.5 Add 0.3… 0.4 dpt

Real Bifocal lens

Stop 2mm

Simulation fits

to clinical results

Simulation of visual performance

75

76

MTF: through focus

depends on pupil diameter

With smallest pupil best depth of focus,

No intermediate degradation

two foci, but reduced contrast

MTF-reduction corresponds to

The light propagation to the

other image plane

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400

7

7

3-D imaging

Paraxial monofocal imaging:

Object stack scales to image stack

Influence of the eye by Convolution with incoherent transfer function = PSF

Monofocal IOL: Diffraction-limited PSF

Bifocal IOL: construction of PSF from avaiable data

Object(x,y,z) o PSF(x,y,z) = Image(x,y,z)

Object Image

PSF

Base

PSF

Add Power

@Infinity @4dpt @Infinity @4dpt

monofocal

imaging

O =

77

7

8

3-D imaging

Bifocal Imaging:

Object stack scales to image stack

Influence of the eye by Convolution with incoherent transfer function = PSF

Monofocal IOL: Diffraction-limited PSF

Bifocal IOL: construction of PSF from avaiable data

Object(x,y,z) o PSF(x,y,z) = Image(x,y,z)

Object Image PSF

Base

PSF

Add Power

@Infinity @4dpt @Infinity @4dpt

Bifocal

imaging

O =

2. Image

78

Pure diffractive lens Valle, 2005

symmetric sinus-profile ~cos(r²)

Intensity distribution @ far – near - intermediate

@ 4mm 30% - 30% - 30% sinus-contrast ~ 32%

@ 2mm 40% - 35% - 25% contrast ~ 22%

stop 4mm

stop 2mm

Symmetric intensity distribution

Reduced contrast in comparison to ideal bifocal

lens

Asymmetry for very small pupils

Simulation of visual performance 79

80

Simulation of visual performance

Quartic Axicon Ares, 2005

Wavefont ~ A r4 – B r²

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.020

0.01

0.02

0.03

0.04

0.05

0.06

Punkt - Kontrast K = 0.71382

PSF(r,z) PSF(z) and PSF(r, z = 0)

-1.5 -1 -0.5 0 0.50

0.01

0.02

0.03

0.04

0.05

0.06

Contrast:

2 points 60%

sinus 7%

Visible symbols

for VA = 0.5

80