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Physical Optics

Lecture 3: Fourier optics

2018-04-25

Herbert Gross

Physical Optics: Content

2

No Date Subject Ref Detailed Content

1 11.04. Wave optics GComplex fields, wave equation, k-vectors, interference, light propagation,

interferometry

2 18.04. Diffraction GSlit, grating, diffraction integral, diffraction in optical systems, point spread

function, aberrations

3 25.04. Fourier optics GPlane wave expansion, resolution, image formation, transfer function,

phase imaging

4 02.05.Quality criteria and

resolutionG

Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point

resolution, criteria, contrast, axial resolution, CTF

5 09.05. Photon optics KEnergy, momentum, time-energy uncertainty, photon statistics,

fluorescence, Jablonski diagram, lifetime, quantum yield, FRET

6 16.05. Coherence KTemporal and spatial coherence, Young setup, propagation of coherence,

speckle, OCT-principle

7 23.05. Polarization GIntroduction, Jones formalism, Fresnel formulas, birefringence,

components

8 30.05. Laser KAtomic transitions, principle, resonators, modes, laser types, Q-switch,

pulses, power

9 06.06. Nonlinear optics KBasics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,

CARS microscopy, 2 photon imaging

10 13.06. PSF engineering GApodization, superresolution, extended depth of focus, particle trapping,

confocal PSF

11 20.06. Scattering LIntroduction, surface scattering in systems, volume scattering models,

calculation schemes, tissue models, Mie Scattering

12 27.06. Gaussian beams G Basic description, propagation through optical systems, aberrations

13 04.07. Generalized beams GLaguerre-Gaussian beams, phase singularities, Bessel beams, Airy

beams, applications in superresolution microscopy

14 11.07. Miscellaneous G Coatings, diffractive optics, fibers

K = Kempe G = Gross L = Lu

Introduction

Optical transfer function

Resolution

Image formation

Phase imaging

3

Contents

diffracted ray

direction

object

structure

g = 1 /

/ g

k

kx

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

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

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

8

Propagation of Plane Waves

Phase of a plane wave

The spectral component is simply multiplied by a phase

factor in during propagation

the function h is the phase function

Back-transforming this into the spatial domain:

Propagation corresponds to a convolution

with the impulse response function

Fresnel approximation for propagation:

zz z0 1

x

z

zheeee yx

nzi

zinzi

iyx

z

;,

222

22

cos2

2

1 0 0, , , , , ; , ,zi z

x y x y x y x yE z E z e h z E z

...2

11 222

222

yxyx

00

2

002

1200

20

20

22

0;,1

;, dydxeeyxUezi

zyxUyyxxi

zyxi

zyx

zzi

P

1 0, , , ; , ,E x y z H x y z E x y z

yx

yxi

yx ddezhzyxH yx2;,;,

),(ˆ),( yxIFvvH PSFyxOTF

*

2

( ) ( )2 2

( )

( )

x xp p p

OTF x

p p

f v f vP x P x dx

H v

P x dx

Optical Transfer Function: Definition

Normalized optical transfer function (OTF) in frequency space:

Fourier transform of the Psf- intensity

Absolute value of OTF: modulation transfer function MTF

Gives the contrast at a special spatial frequency of a

sine grating

OTF: Autocorrelation of shifted pupil function, Duffieux-integral

Interpretation: interference of 0th and 1st diffraction of

the light in the pupil

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

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

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

spatial

frequency

100 %

0 NA/2NA/

/NA /2NA8

pupil

diffraction

order

Ref: W. Singer

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

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

n sin ' 1o

Airy

na uv

f D

max 0

2 2 sin '2

na n uv v

f

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

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

16

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

Contrast and Resolution

High frequent

structures :

contrast reduced

Low frequent structures:

resolution reduced

contrast

resolution

brillant

sharpblurred

milky

17

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

18

Contrast / Resolution of Real Images

resolution,

sharpness

contrast,

saturation

Degradation due to

1. loss of contrast

2. loss of resolution

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

20

Phase Space: 90°-Rotation

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

Planes Fourier inverse

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

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

f

xx'

'

Fourier plane

pupil

image

location

marginal ray

chief

ray

Resolution – More incoherent points

more independent self luminous points: emission of N spherical waves

summation of intensities

First point (green)

Object planeaperture image plane

truncated

spherical

wave

xp, yp

DAiry

plane

wave

Ref: B. Böhme

Resolution – More incoherent points

more independent self luminous points: emission of N spherical waves

summation of intensities

First point (green)

Second point (blue)

Object planeaperture image plane

truncated

spherical

wave

xp, yp

DAiry

2. plane

wave

DAiry

Ref: B. Böhme

Resolution – More incoherent points

more independent self luminous points: emission of N spherical waves

summation of intensities

First point (green)

Second point (blue)

Third point (red)

Object plane

aperture image plane

truncated

spherical

wave

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 Ref: B. Böhme

Fourier Optics – Point Spread Function

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 I(x,y)

In the case of shift invariance

(isoplanasy):

incoherent convolution

Intensities are additive

dydxyxIyyxxgyxI psfinc

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

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

dydxyxIyyxxgyxI psfinc

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

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

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

29

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

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

-0.05 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

-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

Partial Coherent Imaging

Complete description of an optical system:

1. Light source

2. Illumination system, amplitude response hill

3. Transmission object

4. Observation / imaging system with amplitude response hobs

sourceillumination

system

observation

system

pupil Psensor

object

plane

image

planeillumination

field

xs , ys

Is hill Iihobs

Io

xp , yp xi , yi

obs

ill

u

u

sin

sin

Coherence Parameter

Finite size of source : aperture cone with angle uill

Observation system: aperture angle uobs

Definition of coherence parameter :

Ratio of numerical apertures

Limiting cases:

coherent = 0

uill << uobs

incoherent

= 1

uill >> uobs

object imagesource

xi , yixo , yo

lens

uobs

uill

illumination observation

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

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

Simple setup with symmetrical system with

accessible Fourier plane

Different focal lengths of the subsystems allows

a get a magnification m > 1

4 - f - Setup

y

x

y'

x'yp

xp

object

plane

Fourier

plane

image

plane

f

f

f

f

front

system

rear

system

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

pupil complete open

pupil truncated by slit

image

scanned image with stripes

spectrum of the

scanned image

filtered spectrum

false orders suppressed

filtered image without artefacts

Fourier Filtering

Imaging of part of the moon by

scanning

Artefacts by scanning stripes visible

Stripes corresponds to a pointed line

in the Fourier spectrum

If the diffraction orders are blocked /

filtered out, the image quality is

improved

grating

object objective

lens

diffraction

spectrumimage

Abbes Diffraction Theory of Image Formation

Wave optical interpretation of the optical image formation

The objekt 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

There are special assumptions for

the validity of Abbe's formula

There are three options for improving

the resolution:

1. lowering the wavelength

2. increasing the index of refraction

3. using larger aperture cone angles

objunx

sin

61.0

Location of diffraction orders in the

back focal plane

Increasing grating angle with spatial

frequency of the grating

39

Abbe Theory of Microscopic Resolution

+1st

-1st

+1st

-1st

+1st

-1st

objectback focal

planeobjective

lens

0th order

Fouriertheorie of 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

Microscopic Imaging

41

image

contrast

pattern

period

spatial

frequency

100 %

0 NA/2NA/

/NA /2NA8

pupil

diffraction

order

-1

1

object planesource

pupil plane

image plane

0

imaginglens

y

x

y'

x'yp

xp

object

plane

Fourier

plane

image

plane

f

f

f

f

front

system

rear

system

Resolution and Spatial Frequencies

Grating object pupil

Imaging with NA = 0.8

Imaging with NA = 1.3

Ref: L. Wenke

42

Imaging of Phase Structures

Example

Pure phase object

Intensity in the image

1. image stack for 5

defocus positions

2. Image after phase

retrieval

3.difference

44

Zernike Phase Contrast

Principle

Illumination and ring

1. decentered

2. centered

Ref.: M. Kempe

Conventional bright fieldPhase contrast

Zernike Phase Contrast

Quelle: http://www.mikrofoto.de/ordner3/phase.html

Phase Contrast Microscopy

Adjustment of phase ring and illumination

Comparison: brightfield – phase contrast

Ref: M. Kempe

Differential Interference Contrast

• Contrast of phase objects

can also be obtained by

interference of sheared

beams

• In DIC the sheared beams

can be created and the

beams can be overlapped

by Wollaston prisms

(orthogonal polarization)

• Interference of two beams

with displacement 𝛿𝑥 by

analyzer

phase gradient imaging

2

,,, yxxryxryxI yxirr ,exp

x

rxyxryxxr

,,

2

22,

xxryxI

Ref: M. Kempe

Differential Interference Contrast

Lateral shift : preferred direction

Visisbility depends on orientation of details

shift 45° x-shift (0°) y-shift (90°)

49

DIC Phase Imaging

Orientation of prisms and shift size determines the anisotropic image formation

Ref.: M. Kempe