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Physical Optics

Lecture 1: Wave optics

2017-04-05

Herbert Gross

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 D 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

Complex fields, k-vectors and plane waves

Wave equation

Light Propagation

Interference

Interferometry

3

Contents

Quantities: E electrical field t time

H magnetic field e dielectricity

j current m permeability

r charge density c speed of light

Maxwell equations:

current

induction

source free magnetic

charge

continuity of charges

Basic equations for electromagnetism and optics

4

Maxwell Equations

ree Ediv r

0

00 Hdiv r

t

EjHrot r

0ee

t

HErot r

0

tjdiv

r

Alternative formulation of the Maxwell equations in a medium: M magnitization B magnetic induction D electric displacement

The electromagnetic fields form a transverse wave k: wave vector indicates the direction of propagation

Wavelength (scalar)

Maxwell Equations

E

H

k

0

Bk

iDk

BEk

jiDHk

r

EJ

Jk

MHB

PED

r

r

r

ee

0

0

o

o

nkn

ck

2

2 2

o

c c

k

5

Speed of light in medium

in vacuum

constants

Energy density of a field

Local flow of energy:

Poynting vector

Intensity

6

Energy Density and Poynting vector

n

c

nc

rr

0

0000

11

eee

2

0

2

0

2

02

1EHEu

ee

2

0 EecHES k

e

2

002

1EcI e

8

0 0

12.99792458 10o

mc

se

.104

,10...85.8

70

100

Am

Vs

Vm

As

e

Speed of light in medium

Wavelength in medium

Wave vector in medium

Refraction

7

Fields in Dielectric Media

0cc

n

0

n

0k k n

n

Ref: Saleh / Teich

Description of electromagnetic fields:

- vectorial nature of field strength

- decomposition of the field into components

Propagation plane wave:

- field vector rotates

- projection components are oscillating sinusoidal

yyxx etAetAE )cos(cos

z

x

y

Electromagnetic Fields

8

1. Linear components in phase

2. circular phase difference of 90° between components

3. elliptical arbitrary but constant phase difference

x

y

z

E

E

x

y

z

EE

x

y

z

E

E

Basic Forms of Polarisation

9

10

Basic Wave Optics

Scalar wave

phase function

Phase surface:

- fixed phase for one time

- phase surface perpendicular to

unit vektor e

( , )( ) ( ) e i rA r a r

( , )r

A0

2k r r e const

Ref: W. Osten

Wave harmonic in space and time

Representation with complex

exponential function

Coupling of time and space propagation/development

with

Harmonic Wave

Re{F(t)}

t

T

Re{U(z)}

z

t'

Re{F(t)}

Im{F(t)}

Re{F(t)}

Im{F(t)}

t

t'

,E r t U r F t

tetF ti cosRe

zkezUzik

0cosRe 0

20

ck

( )( , ) i k r tE r t e

11

12

Plane Wave

Condition of a plane wave

y

k r = const.

x

z

k

.constrk

znzxU

2cos~,

cossin

2cos~, zxnzxU

13

Plane and Spherical Waves

Plane wave

wave vector k

Spherical wave

)(),( trkiAetrE

)(),( trkier

AtrE

Ref.: B. Dörband

14

Spherical Wave

Field of a spherical wave

radius in spherical coordinates

( )

( , )i kr te

E r tr

222 zyxr

r

Basis: 1. superposition of solutions

2. separability of coordinates

Plane wave

Spectral expansion

A(k): plane wave spectrum

Dispersion relation: Ewald sphere

Transverse expansion

Main idea:

- Field decomposition in plane waves

- Switch into Fourier space of spatial frequencies

- Propagation of plane wave as simple phase factor

- Back transform into spatial domain

- Superposition of plane wave with modified phase

Plane Wave Expansion

kdekArE rki

)(

2

13

trkietrE

,

2

2222

o

zyxc

nkkkk

E x y z A k k z e dk dkx y

i xk yk

x y

x y, , ( , , )

1

22

k k k k k kT z x y z

A k k z E x y z e dxdyx y

i xk ykx y, , ( , , )

15

Propagation of plane waves:

pure phase factor

1. exact sphere

2. Fresnel quadratic approximation

Evanescent waves

components damped in z

important only for near field setups

Propagation algorithm

x-y-sections are coupled

Paraxial approximation

x-y-section decoupled

Plane Wave Expansion

A k k z A k k A k k ex y x y

ik z

x y

ik zk

k

k

kz

x x

, , , , , ,

0 01

2 2

22

22

0,,

0,,,, 2

yx

yx

vvzizik

yx

kkk

iz

zik

yxyx

eevvA

eekkAzkkA

evanescentkkforkkkik yxyxz ,0222

0

22

),(ˆˆ

),(ˆˆ)','(

222/121

1

yxEFeF

yxEFeFyxE

xy

vviz

xy

xy

zik

xy

yx

z

),(ˆˆ)','(22

1 yxEFeFyxE xy

vvzi

xyyx

16

Propagation by Plane / Spherical Waves

Expansion field in simple-to-propagate waves

1. Spherical waves 2. Plane waves

Huygens principle spectral representation

Fourier approach

rdrErr

erE

rrik

2

'

)('

)'(

x

x'

z

E(x)

eikr

r

)(ˆˆ)'( 1 rEFeFrE xy

zik

xyz

x

x'

z

E(x)

eik z z

17

18

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;,;,

19

Angular Spectrum

Plane wave

Wave number

Spatial frequence: re-scaling of k

Fourier transform to get the plane wave spectrum

z

x

n/

x

z

E( , , ) ( , , ) x y zi k x k y k z

x y z A x y z e

2222 nkkkk zyx

k

2

2( , , ) x y zi x y z

E x y z A e

222

yxzn

2, , , , x y zi x y z

x y z x y zE x y z E e d d d

20

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

21

Refraction in k-Space

Plane wave refracted at a plane interface

k-vector should be constant in length

Total internal reflection:

possible if n1 > n2

n2

n1

ky

k1z

k2z

k1y

k1z

k2y

k1y

2

1

1

kz

n2

n1

kz

1. case :

refracted ray

2. case :

beginning

total internal

reflection

3. case :

total internal

reflection

3.

2.

1.

ky

2

1

2

1

2

1

2

1 knkkk zyx

2

2

2

2

2

2

2

2 knkkk zyx

trki

o eEE

22

Evanescent Waves

Usual case in optics:

- refractive index n real

- no damping or loss during light propagation

- spatial frequency real

General case: n and k complex

- damped or evanescent wave

- absorption of the field along the

propagation path

- absorption constant

Damping along z:

- complex refractive index

- absorption constant

- Lambert-Beer law

222

yxzn

, , x yi k x k y zE x y z A e e

222 2

nkk yx

inn 1~

0

0

42

nnk

zeIzI 0)(

1

z

I / I0

1 /

1 / e = 0.368

Metals

Complex refractive index

Alternative formulation:

attenuation

Relation with conductivity

Typical data of some metals

ir ninn ~

inn 1~

e

e

24~ iin

material n k in [m]

gold 0.402 2.54 0.034

silver 0.129 3.25 0.026

aluminum 0.912 6.56 0.013

tungsten 3.50 2.72 0.032

platinum 2.10 3.67 0.023

silicon 4.12 0.048 1.787

23

24

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

25

McCutchen Formula and Axial Resolution

n/

P(z)

zz

Dz

x

x

P(x)

Dx

light

cone

Ewald

sphere

transverse

pupil

axial

pupil

cap

nR

2 2 2 2

sin / /x

xv R n NA n NA

D

Imaging of a plane wave at a volume object

x: minimum value resolution

D: maximum interval

Uncertainty relation: D 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

D

26

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

Maxwell equation for the field E, vectorial

The spatial inhomogeneities couples the field

components

Homogeneous, without charges, non-conductive

separation of vector components, scalar

Time independence:

Wave equation of Helmholtz

In coordinate representation

Wave number in medium refractive index n

Fast z-oscillation separated

Slowly varying envelope approximation

2

22

~~

t

EE

e

tierEtrE )(),(

02 D EkE

o

o

nkn

ck

2

ikzezyxEzyxE ),,(),,(

Wave Equation

0ln2

22 rroo E

t

EE e

ee

0),,(

2

22

2

2

2

2

2

2

Ec

zyxn

z

E

y

E

x

E

2

2

2 22

0

2

22E

zik

E

zE k

n x y

nn Eo

( , )

27

Paraxial approximation

paraxial wave equation

Conditions for scalar approximation:

1. Decoupling of field components,

wavelength small in comparison to free diameter

2. No large angles due to geometry,

Computation of field in large distances z

Scalar Helmholtz equation

a

z

Wave Equation

022

2

2

2

z

Eki

y

E

x

E

2

2

E

zk

E

z

0)(2 D rEnko

k

28

1. Spectral representation or orthogonal expansions:

Plan wave expansion, Fourier method

Expansion into gaussian beams in paraxial systems

Expansion into spherical waves, scattering geometries

Expansion into Eigenmodes for boundary problems,

e.g. in fibers, integrated optics, waveguides

2. Integral representations (field on surfaces)

Kirchhoff diffraction integral, Rayleigh-Sommerfeld I+II

Special approximations: Fresnel-, Collins-, Fraunhofer integral

Debye integral

Richards-Wolf vectorial representation

Boundary edge wave approximation

Method of stationary phase, Saddelpoint method

3. Direct solution of the wave equation (volume solution)

Finite difference method

Finite element method

Radial basis functions

Potential methods

Method of lines

Coherent Numerical Field Propagation

29

Solution Methods of the Maxwell Equations

Maxwell-

equations

diffraction

integrals

asymptotic

approximation

Fresnel

approximation

Fraunhofer

approximation

finite

elements

finite

differences

exact/

numerical1st

approximation

direct

solutions of

the PDE

spectral

methods

plane wave

spectrum

vector

potentials

2nd

approximation

finite element

method

boundary

element

method

hybrid method

BEM + FEM

Debye

approximation

Kirchhoff-

integral

Rayleigh-

Sommerfeld

1st kind

Rayleigh-

Sommerfeld

2nd kind

mode

expansion

boundary

edge wave

30

Method Calculation Properties / Applications

Kirchhoff

diffraction integral E r

iE r

e

r rdF

i k r r

FAP

( ) ( ' )'

'

Small Fresnel numbers,

Numerical computation slow

Fourier method of

plane waves )(ˆˆ)'(21 xEFeFxE zvi

II

Large Fresnel numbers

Fast algorithm

Split step beam

propagation

Wave equation: derivatives approximated

En

yxnkE

z

Eik

z

E

o

1

),(2

2

222

2

2

Near field

Complex boundary geometries

Nonlinear effects

Raytracing Ray line law of refraction

r r sj j j j 1 sin '

'sini

n

ni

System components with a aberrations

Materials with index profile

Coherent mode

expansion

Field expansion into modes

n

nn xcxE )()( dxxxEc nn )()( *

Smooth intensity profiles

Fibers and waveguides

Incoherent mode

expansion

Intensity expansion into coherent modes

n

nn xcxI2

)()(

Partial coherent sources

Wave Optical Coherent Beam Propagation

31

Typical change of the intensity profile

Normalized coordinates

Diffraction integral

32

Fresnel Diffraction

z

geometrical

focus

f

a

z

stop

far zone

geometrical

phase

intensity

a

rr

f

avz

f

au

r

;

2;

22

1

0

20

/

2

0

2 22

)(2

),( rrr

r

devJef

EiavuE

ui

uafi

Split Step Propagators

Calculating the field after every small Dz in many single steps

Algorithms: finite differences of plane waves

The field is known in the complete volume

This time-consuming technique is necessary for non-homogeneous media

z

ys

y

Split-step-propagator: 1. Finite Differences

2. Plane wave expansion

many steps

starting

plane

final plane

inhomogeneous

medium

33

Classical beam propagation: split-step approach

Wave equation

two operators: interfaces (material)

and diffraction

Fourier transform

Plane wave decomposition

)(ˆ, zEFzkE x

0

2

2,,

k

zik

xx

x

ezkEzzkE

D

D

zzkEFzzxE x DD ,ˆ),(' 1

zSezzxEzzxE DDD ˆ

,',

EHED

En

yxn

i

kE

ikz

E

o

ˆˆ

1),(

22

12

22

Eik

ED 2

2

1ˆ

En

yxn

i

kES

o

1

),(

2ˆ

2

2

Split Step Propagators

34

Finite Difference Propagation

Paraxial Helmholtz wave equation

Approximation of derivatives

Implicite scheme, linear system of

equations for one steps Dz

Large Fresnelnumber of one voxel,

Fast evaluation of tridiagonal system

Phase nearly flat, no large tilt

Stability:

1. Explicite schemes, usually unstable

2. Good stable implicite schemes exists

z

lateral x

zn+1

xj x

j+1x

j-1

zn

propagation

x

EE

x

E jj

D

2

11

2

11

2

2 2

x

EEE

x

E jjj

D

Enzxnkx

E

z

Enik oooo 222

2

2

),(2

njnjnj

njnjnj

aEcEaE

aEbEaE

,1,,1

1,11,1,1

az

x

D

D2 2 bz

x

zn n ik nj m o o o

D

D

D2 1

2 2

22,

cz

x

zn n ik nj m o o o

D

D

D2

2 2

22,

35

Focussed gaussian beam with spherical aberration

- asymmety intra- vs. extra focal

- sign of spherical aberration has influence

- broadening of beam waist diameter

- diffraction fringes

Gaussian beam propagation in a parabolic gradient

index medium

Refocussing effects

Beam Propagation Examples

c9 = -0.25

c9 = 0.25

c9 = 0

36

37

Intensity of Superposed Fields

CCD is not able to detect phase due to time averaging

Measuring of intensity with simple detector

Measured intensity is time average

Interferometry and holography:

coding of phase information into measurable intensity variation

Contrast / visibility:

normalized difference of two different intensities

(typically maximum / minimum values)

Value between 0...1

General case of two-wave interference

Ref: W. Osten

22

0

1

2r ot

I P E Ae e

max min

max min

I IC

I I

1 2 121 cosI I I C

38

Interference of Waves

The main property is the phase difference

between two waves

Interference of two waves

special case of equal intensites

Maxima of intensity at even phase differences

Minima of intensity at odd phase differences

Interference of plane waves

Interference of spherical waves:

1. outgoing waves

rotational hyperboloids

2. one outgoing, one incoming wave

rotational ellipsoids Ref: W. Osten

jk j k

1 2 1 2 122 cosI I I I I

2jk N

(2 1)jk N

0 122 1 cosI I

1 2k r k r

1 2k r r

1 2k r r

Two beam interference of two waves:

- propagation in the same direction

- same polarization

- phase difference smaller than axial length of coherence

Coherent superposition of waves

Difference of phase / path difference

Number of fringes

location of same phase

Conrast

122121

2

21

cos2 D

IIII

EEI

122

DDs

sN

D

D

2

12

21

21

minmax

minmax2

II

II

II

IIK

Two Beam Interference

39

40

Interference of Two Waves

Superposition of two plane waves:

1. Intensity

2. Phase difference

Spacing of fringes

Interference of two spherical waves

More complicated geometry

),,(cos2²²),,( 2121 zyxAAAAzyxI D

DD rkkzyxzyxzyx

)(),,(),,(),,( 1212

Ref.: B. Dörband

2sin2 e

ns

41

Two Beam Interference

Interference of two plane waves under different directions

Fringe distance s 1212

2

eenkks

42

Interferometry

Basic idea:

- separation of a wave into two beams (test and reference arm)

-every beam surpasses different paths

- superposition and interference of both beams

- analysis of the pattern

Different setups for:

- the beam splitting

- the superposition

- the referencing

Different path lengths

Difference equivalent of one fringe

Measurement of plates:

Haidinger fringes of equal inclination

Newton fringes of equal thickness

Ref: W. Osten

1 1 2 2 wn t n t N tD

2wt

n

43

Classification of Interferometers

Division of amplitude: - Michelson interferometer

- Mach-Zehnder interferometer

- Sagnac interferometer

- Nomarski interferometer

- Talbot interferometer

- Point diffraction interferometer

Division of wavefront: - Young interferometer

- Rayleigh interferometer

Division of source: - Lloyds mirror

- Fresnel biprism

Ref: R. Kowarschik

44

Localization of Fringes

Interference volume for a plate

Interference volume for a wedge

Ref: R. Kowarschik

volume of

interference

fringes

incident

light back side

reflectedfront side

reflected

volume of

interference

fringes

incident

light

back side

reflected

front side

reflected

test surface

beamsplitter

reference surface

here: flat

illumination

to detector

path difference

mRrm

Test by Newton Fringes

Reference surface and test surface with nearly the same radii

Interference in the air gap

Reference flat or curved possible

Corresponds to Fizeau setup

with contact

Broad application in simple

optical shop test

Radii of fringes

45

Ref: W. Osten

Testing with Twyman-Green Interferometer

Short common path,

sensible setup

Two different operation

modes for reflection or

transmission

Always factor of 2 between

detected wave and

component under test

detector

objective

lens

beam

splitter 1. mode:

lens tested in transmission

auxiliary mirror for auto-

collimation

2. mode:

surface tested in reflection

auxiliary lens to generate

convergent beam

reference mirror

collimated

laser beam

stop

46

Testing with Fizeau Interferometer

Long common path, quite insensitive setup

Autocollimating Fizeau surface quite near to test surface, short cavity length

Imaging of test surface on detector

Straylight stop to bloc unwanted light

Curved test surface: auxiliary objective lens (aplanatic, double path)

Highest accuracy

detector

beam

splitter

collimatorconvex

surface

under test

light

source

Fizeau

surface

auxiliary lens

stop

47

Interferograms of Primary Aberrations

Spherical aberration 1

-1 -0.5 0 +0.5 +1

Defocussing in

Astigmatism 1

Coma 1

48

Intensity of fringes

I(x,y,t) intensity of fringes

V(x,y) contrast of pattern

W(x,y) phase function to be found

(x,y,t) reference phase

Rs(x,y) multiplicative speckle noise

IR(x,y,t) additive noise

Tracing of fringes:

- time consuming method, interpolation, indexing of fringes, missing lines

Fourier method:

-wavelet method

- FFT Method

- gradient method

- fit of modal functions

Evaluation of Fringes

),,(),(),,(),(cos),(1),(),,( 0 tyxIyxRtyxyxWyxVyxItyxI RS

49

Real Measured Interferogram

Problems in real world measurement:

Edge effects

Definition of boundary

Perturbation by coherent

stray light

Local surface error are not

well described by Zernike

expansion

Convolution with motion blur

Ref: B. Dörband

50

Interferogram - Definition of Boundary

Critical definition of the interferogram boundary and the Zernike normalization

radius in reality

51

52

Interferometry

General description of the measurement quantity:

superpostion of spatially modulated signal and noise

Io: basic intensity, source

T: transmission of the system, including speckle

: phase, to be found

IN: noise, sensor, electronics, digitization

Signal processing, SNR improvement:

- filtering

- background subtraction

Ref: W. Osten

0( , ) ( , ) ( , ) cos ( , ) ( , )NI x y I x y T x y x y I x y

original signal

filtered signal

background

processed signal

53

Interferometry

perfect interferogram

reduced contrast due

to background intensity

with speckle

with noise

Ref: W. Osten