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Physical Optics

Lecture 12: Polarization

2017-05-03

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 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

Introduction

Fresnel formulas

Jones calculus

Further descriptions

Birefringence

Components

Applications

3

Contents

Scalar:

Helmholtz equation

Vectorial:

Maxwell equations

Scalar / vectorial Optics

0)(2 rEnko

k

E

H

k

0

Bk

iDk

BEk

jiDHk

EJ

Jk

MHB

PED

r

r

0

0

4

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

5

Description of electromagnetic fields:

- Maxwell equations

- 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

Basic Notations of Polarization

6

Elimination of the time dependence:

Ellipse of the vector E

Different states of polarization:

- sense of rotation

- shape of ellipse

0° 45° 90° 135° 180°

225° 270° 315° 360°

2

2

2

2

2

sincos2

yx

yx

y

y

x

x

AA

EE

A

E

A

E

Polarization Ellipse

7

Representation of the state of polarization by an ellipse

Field components

Axes of ellipse: a, b

Rotation angle of the field

Angle of eccentricity

a

btan

Polarization Ellipse

)sinsincos(cos xxxx AE

)sinsincos(cos yyyy AE

cos2

2tan22

yx

yx

AA

AA

x

y

x'

y'

a

b

Ax

Ay

Ex

Ey

8

Circular Polarization

Spiral curve of field vector

Superposition of left and right handed circular

polarized light:

resulting linear polarization

Ref: Manset

tx

y

E

Er

El

t

t1

t2

t3

9

Rotation of plane of polarization with t / z

Phase angle 90°

Generation by l/4 plate out of linear polarized light

Erzeugung :

Polarisator und l / 4 -

Platte

b1

x

y

E

Er

El

t1

El

Er E

t2

b2

SA z

y

x

TA

45°

l / 4 - plate

linear

polarizer

LA

Circular polarized Light

10

Fresnel Formulas

n n'

incidence

reflection transmission

interface

E

B

i

i

E

B

r

r

E

Bt

t

normal to the interface

i

i'i

a) s-polarization

n n'

incidence

reflection

transmission

interface

B

E

i

i

B

E

r

r

B

E t

t

normal to the interface

i'i

i

b) p-polarization

Schematical illustration of the ray refraction ( reflection at an interface

The cases of s- and p-polarization must be distinguished

11

Fresnel Formulas

Electrical transverse polarization

TE, s- or -polarization, E perpendicular to incidence plane

Magnetical transverse polarization

TM, p- or p-polarization, E in incidence plane

Boundary condition of Maxwell equations

at a dielectric interface:

continuous tangential component of E-field

Amplitude coefficients for

reflected field

transmitted field

Reflectivity and transmission

of light power

TEe

rTE

E

Er

1 TETEe

tTE r

E

Et 1

' TMTM r

n

nt

nn EE 2211

tt EE 21

TMe

rTM

E

Er

2r

e

PR r

P

2' cos '

cos

t

e

P n iT t

P n i

||

12

Fresnel Formulas

Coefficients of amplitude for reflected rays, s and p

Coefficients of amplitude for transmitted rays, s and p

tzez

tzezE

kk

kk

inin

inin

innin

innin

ii

iir

'cos'cos

'cos'cos

sin'cos

sin'cos

)'sin(

)'sin(

222

222

tzez

tzezE

knkn

knkn

inin

inin

innnin

innnin

ii

iir

22

22

2222

2222

||'

'

'coscos'

'coscos'

sin'cos'

sin'cos'

)'tan(

)'tan(

tzez

ezE

kk

k

inin

in

innin

in

inin

int

2

'cos'cos

cos2

sin'cos

cos2

'cos'cos

cos2

222

tzez

ezE

knkn

kn

inin

in

innnin

inn

inin

int

22

2

2222||

'

'2

'coscos'

cos2

sin'cos'

cos'2

'coscos'

cos2

13

Fresnel Formulas

Typical behavior of the Fresnel amplitude coefficients as a function of the incidence angle

for a fixed combination of refractive indices

i = 0

Transmission independent

on polarization

Reflected p-rays without

phase jump

Reflected s-rays with

phase jump of p

(corresponds to r

Fresnel Formulas: Energy vs. Intensity

Fresnel formulas, different representations:

1. Amplitude coefficients, with sign

2. Intensity cefficients: no additivity due to area projection

3. Power coefficients: additivity due to energy preservation

r , t

0 10 20 30 40 50 60 70 80 90-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

i

t

r

r

t

R , T

i0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T(In)

T(In)

R(In)

R(In)

R , T

i0 10 20 30 40 50 60 70 80 90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

T

R

R

15

Decomposition of the field strength E

into two components in x/y or s/p

Relative phase angle between

components

Polarization ellipse

Linear polarized light

Circular polarized light

y

x

i

y

i

x

y

x

eA

eA

E

EE

0

xy

0

10E

1

00E

sin

cos0E

iErz

1

2

1

iElz

1

2

1

Jones Vector

222

sincos2

yx

yx

y

y

x

x

AA

EE

A

E

A

E

16

Jones representation of full polarized field:

decomposition into 2 components

Cascading of system components:

Product of matrices

In principle 3 types of components influencing polarization:

1. Change of amplitude polarizer, analyzer

2. Change of phase retarder

3. Rotation of field components rotator

Jones Calculus

1121 EJJJJE nnn

p

s

trans t

tJ

10

01

cossin

sincos)(rotJ

2

2

0

0

i

i

ret

e

eJ

os

op

ppps

spss

s

p

oE

E

JJ

JJ

E

EEJE ,

System 1 :

J1

E1 System 2 :

J2

E2 System 3 :

J3

E3 System n :

Jn

En-1 EnE4

17

Rotated component

Rotation matrix

Intensity

Three types of components to change the polarization:

1. amplitude: polarizer / analyzer

2. phase: retarder

3. orientation: rotator

Propagation:

1. free space

2. dielectric interface

3. mirror

)()0()()( DJDJ

cossin

sincos)(D

1

*

12 EJJEI

Jones Matrices

10

012 OPLiPRO eJ

l

p

p

s

TRA t

tJ

0

0

p

s

REF r

rJ

0

0

10

01rJ SP

18

Birefringence: index of refraction depends on field orientation

Uniaxial crystal:

ordinary index no perpendicular to crystal axis

extra-ordinary index ne along crystal axis

Difference of indices

Jones matrix

Relative phase angle

0

0

i n z

neo i n z

eJ

e

p

l

p

l

2 2

2 2

sin cos sin cos 1( , )

sin cos 1 cos sin

i i

neoi i

e eJ

e e

2

e o

zn n

p

l

oe nnn

Birefringence: Uniaxial Crystal

19

Descriptions of Polarization

E

Parameter Properties

1

Polarization ellipse

Ellipticity ,

orientation only complete polarization

2

Complex parameter

Parameter

only complete polarization

3

Jones vectors

Components of E

only complete polarization

4

Stokes vectors

Stokes parameter So ... S4

complete or partial

polarization

5

Poincare sphere

Points on or inside the

Poincare sphere only graphical representation

6

Coherence matrix

2x2 - matrix C

complete or partial

polarization

20

Description of polarization from the energetic point of view

- So total intensity

- S1 Difference of intensity in x-y linear

- S2 Difference of intensity linear under 45° / 135°

- S3 Difference of circular components

)0,90()0,0(0 IIS

)0,90()0,0(1 IIS

)0,135()0,45(2 IIS

)90,45()90,45(3 IIS

Stokes Vector

22

yxo EES

22

1 yx EES

cos22 yxEES

sin23 yxEES

21

Description of a polarization state with Stokes parameterInterpretation:

Components of the field on the Poincare sphere

Also partial polarization is taken into account

Relation

Unequal sign: partial polarization

Stokes vector 4x1

Propagation:

Müller matrix M

2

3

2

2

2

1

2

0 SSSS

3

2

1

0

S

S

S

S

S

Stokes Vector

S

mmmm

mmmm

mmmm

mmmm

SMS

33323130

23222120

13121110

03020100

'

22

Linear horizontal / vertical

LInear 45°

Circular clockwise / counter-clockwise

0

0

1

1

S

0

0

1

1

S

0

1

0

1

S

1

0

0

1

S

1

0

0

1

S

Examples of Stokes Vectors

23

Partial polarized light:

degree of polarization

p = 0 : un-polarized

p = 1 : fullypolarized

0 < p < 1 : partial polarized

Determination of Stokes parameter:

Unpolarized light

Fully polarized light

Partial Polarization

ges

pol

I

Ip

pS S S

S

12

2

2

3

2

0

pu SS

pSSpS

000 )1(

0321 SSS

2

3

2

2

2

1

2

0 SSSS

24

Every point on a uni sphere describes one state of polarization

In spherical coordinates:

Points on z-axis: circular polarized light

Meridian line: linear polarization

Points inside

partial polarization

Sphere of Poincare

2sin

2cos2cos

2sin2cos1

222

z

y

x

zyxr

y

x

z

right handed

circular polarized

left handed

circular polarized

elliptical

polarized

linear

polarized

2

2

25

Fully polarized: point

Unpolarized: full surface

Partial polarized: probability distribution, points inside

y

x

z

P

y

x

z

y

x

z

fully polarized un-polarizedpartial

polarized

Partial Polarization on the Poincare Sphere

26

Stokes parameter S1 , S2 , S3 :

Componenst along axis directions

Radius of sphere, length of vector: So

Projection into meridional plane:

angle 2 of polarization ellipse

Projection into meridian plane:

eccentricity angle 2

Poincare Sphere and Stokes Vector

So

S1

S2

S3

y

x

z

P

2

2

27

Polarization

Polarization of a donat mode in the focal region:

1. In focal plane 2. In defocussed plane

Ref: F. Wyrowski

28

Jones matrix changes Jones vector

Müller matrix changes Stokes vector

Change of coherence matrix

Procedure for real systems:

1. Raytracing

2. Definition of initial polarization

3. Jones vector or coherence matrix local on each ray

4. transport of ray and vector changes at all surfaces

5. 3D-effects of Fresnel equations on the field components

6. Coatings need a special treatment

7. Problems: ray splitting in case of birefringence

JCJC'

SMS

'

EJE

'

Propagation of Polarization

29

3D calculus in global coordinates according to Chipman

Ray trace defines local coordinate system

Transform between local and

global coordinates

Polarization Raytrace

inLLinjL

in

inL szxsy

ss

ssx

1,1,,

1

11, ,,

'

'

'',',' 11,1,1,1,1, szxsyxx LLinLLL

zzLzL

yyLyL

xxLxL

out

inzinyinx

zLyLxL

xLyLxL

in

syx

syx

syx

T

sss

yyy

xxx

T

11,1,

11,1,

11,1,

,1

,,,

1,1,1,

1,1,1,

,11

''

''

''

,

plane of

incidence

interface

plane

incoming

ray

outgoing

ray

kj-1 kj

xj

yj

y'j

30

Embedded local 2x2 Jones matrix

Matrices of refracting surface

and reflection

Field propagation

Cascading of operator matrices

Transfer properties

1. Physical changes

2. Geometrical bending effects

Polarization Raytrace

1,

1,

1,

,

,

,

1

jz

jy

jx

zzyzxz

zxyyxy

zxyxxx

jz

jy

jx

jjj

E

E

E

ppp

ppp

ppp

E

E

E

EPE

121 .... PPPPP MMtotal

100

00

00

,

100

00

00

s

p

rs

p

t r

r

Jt

t

J

100

0

0

2221

1211

,1 jj

jj

J refr

1

,1,1,11

inrefrout TJTP

1

,1,1,11

inbendout TJTQ

31

Change of field strength:

calculation with polarization matrix,

transmission T

Diattenuation

Eigenvalues of Jones matrix

Retardation: phase difference

of complex eigenvalues

To be taken into account:

1. physical retardance due to refractive index: P

2. geometrical retardance due to geometrical ray bending: Q

Retardation matrix

Diattenuation and Retardation

EE

EPPE

E

EPT

T

*

*

2

2

minmax

minmax

TT

TTD

2/12/12/12/12/1 wewwJ

i

ret

21 argarg

totaltotalPQR

1

32

Müller matrix visualization

Interpretation not trivial

Retardation and diattenuation map

across the pupil

Polarization Zernike pupil aberration

according to M. Totzeck (complicated)

Polarization Performance Evaluation

34

Anisotropic Media

Different types of anisotropic media

Ref: Saleh / Teich

Classical geometry of birefringent refraction refers on interface plane

Grandfathers method:

Calculation iterative due to non-linear equations in prism coordinates

History: formulas according to Muchel / Schöppe

Only plane setup considered, crystal axis in plane of incidence

Geometry of Raytrace at an Interface

incidentray

e

o

crystalaxis

wavenormal

'

o

''eo

air crystal

a

s'

s

n

35

Incident plane wave

Different refractive indices for polarization direction

Important: relative orientation against crystal axis

Arbitrary incidence: splitting of rays

- ordinary ray: perpendicular to optical axis

- extra ordinary ray: in plane of incidence

Ray Splitting in Case of Birefringence

crystal axis

extraordinary ray

ordinary ray

36

Birefringence

Different directions of E and D

Ray splitting not identical to wave splitting

37

k / s

P

DE

H , B energy / wave /

Poynting

ray / phase

Relationship between electric field and

displacement vector

Linear relationship of tensor-equation

First term, coefficient :

local direction, birefringence

Second term, coefficient :

gradient of field, third order, optical activity, polarization rotated

Third term, :

forth order, intrinsic birefringence, spatial dispersion of birefringence

General:

Due to the tensor properties of the coefficients, all these effects are anisotropic

The field E and the displacement D are no longer aligned

Anisotropic Media

ED r

0

qml

lqmjlmq

ml

lmjlm

l

ljlj EEED,,,

EEssnEssnD

22

38

Vanishing solution determinante of the

wave equation

Value of speed of ligth depending on the ray

direction (phase velocity)

Alternative: axis 1/n

Fresnel or Ray Ellipsoid

022

2

22

2

22

2

z

z

y

y

x

x

cc

k

cc

k

cc

k

1/nx

x

y

z

1/ny

1/nz

Ey

electric

field

constant

energy

density

Ex

E

D

displacement

vector

39

Inverse matrix of the dielectric tensor:

index or normal ellipsoid

Gives the refractive index as a function of

the orientation

Also possible: ellipsoid of k-values

Index or Normal Ellipsoid

constn

z

n

y

n

x

zyx

2

2

2

2

2

2

z

y

x

r

100

01

0

001

1iin

x

y

z

ny

nz

nx

022

22

22

22

22

22

z

zz

y

yy

x

xx

nn

nk

nn

nk

nn

nk

Dy

electric

fieldconstant

energy

density

Dx

D

E

displacement

vector

40

Special case uniaxial crystal:

ellipsoid rotational symmetry

no ordinary direction valid for two directions

ne extra ordinary valid for only one direction

Two cases:

ne > no : positive ( prolate, cigar )

ne < no : negativ ( oblate, disc )

Arbitrary orientation : intersection points

Index Ellipsoid for Uniaxial Crystals

optical

axis

a) positive birefringence ne > no

o-ray

e-ray

optical

axis

b) negative birefringence ne < no

o-ray

e-ray

k2

k3

kn0k0

n0k0

n() k0

n0k0 nek0

41

Incident plane wave

Osculating tangential plane at the ordinary index-sphere:

defines normal to o-ray direction

Osculating tangential plane at the index o/e-index ellipsoid:

normal to e-direction

Wave-Optical Construction of the o/e-direction

crystal

axis

incident ray

o - rayeo-ray

velocity

ellipsoid

wavefronts

42

Classical uniaxial media used in polarization components:

1. Quartz, positive birefringent, small difference

2. Calcite, negative birefringent, larger difference

Birefringent Uniaxial Media

oe

e

o

quartz calcite

material sign no neo

Calcite negative 1.6584 1.4864

Quarz, SiO2 positive 1.5443 1.5534

43

Optical symmetry axis of crystal

material breaks symmetry

Split of rays depends on the

axis orientation

Split of field into two orthogonal

polarisation components

Energy propagation (ray,

Poynting) in general not

perpendicular to wavefront

Refraction with Birefringence

divergent rays

crystal

axis

parallel rays

with phase difference

crystal

axiscrystal

axis

parallel rays

in phase

o-ray e-ray

44

Polarizer with attenuation cs/p

Rotated polarizer

Polarizer in y-direction

p

s

LIN c

cJ

10

01

z

y

x

TA

2

2

sincossin

cossincos)(PJ

10

00)0(PJ

Polarizer

45

Polarizer and analyzer with rotation

angle

Law of Malus:

Energy transmission

TA

z

y

x

TA

linear

polarizer y

linear

polarizer

E

E cos

2cos)( oII

I

0 90° 180° 270° 360°

Pair Polarizer-Analyzer

parallel

polarizer

analyzer

perpendicular

46

Crossed pair of polarizer - analyzer plates

No sample: transmission zero

Polarized sample between the plates:

- spatial resolved rotation of polarization

- analysis of stress and strain

TA z

y

x

TA

linear

polarizer x

linear

polarizer y

analyzer

Stress Analysis: Polarizer / Analyzer

47

Phase difference between field

components

Retarder plate with rotation angle

Special value:

l / 4 - plate generates circular polarized light

1. fast axis y

2. fast axis 45°

2

2

0

0

i

i

RET

e

eJ

z

y

x

SA

LA

ii

ii

Vee

eeJ

22

22

cossin1cossin

1cossinsincos),(

iJ V

0

01)2/,0( p

1

1

2

1)2/,4/(

i

iiJ V pp

Retarder

48

Rotate the of plane of polarization

Realization with magnetic field:

Farady effect

Verdet constant V

bb

bb

cossin

sincosROTJ

z

y

x

b

VLB

b

Rotator

49

Types of LC media:

- nematic long molecules, oriented in one direction

- smectic long molecules, oriented in one direction in several layers

different types A, B, C

- discotic plate-shapes molecules, oriented in a plane

- cholesteric skrew-shaped molecules

Functional principle:

- interaction creates domaines

- orientation of the domaines by voltage

- anisotropic behavior, birefringence

- strong influence of temperature

Parameter to describe the order / entropy:

Application:

- displays

- spatial light modulator

Liquid Crystals

1cos32

1 2 S

50

Orientation of the molecules in an external field (voltage)

LCD Cell

moleculesglass plate

field : E = 0 field : E > 0

51

Types of Liquid Crystals

nematic smectic A smectic C

52

Types of Liquid Crystals

smectic C

cholesteric

53

Transmission changes due to polarization

Black-White switch possible

Reflective or refractive

Liquid Crystal Display

polarizer analyzer90° - TN-LC cell

black

white

polarizer LC cell mirror

54

epi

illumination trans

illumination

Microscopic Contrast

Ref: M. Kempe

polarization bright field fluorescence

55

Differential Interference Contrast

),(

),()1(),(

yxxEeR

yxxEeRyxE

psf

i

psf

i

psfdic

condenser

object

Wollaston

prism

Wollaston

prism

compensator

objective

polarizer

analyzer

shear

distance

x

adjustment

phase

splitting

ratio

R1-R

Point spread function

Parameter:

1. Shear distance

2. Phase offset

3. Splitting ratio

56

original / shifted differential splitting

Differential Interference Contrast

Differential splitting:

Large contrast at phase

gradients

57

Differential Interference Contrast

Lateral shift : preferred direction

Visisbility depends on orientation of details

shift 45° x-shift (0°) y-shift (90°)

58

59

DIC Phase Imaging

Orientation of prisms and shift size determines the anisotropic image formation

Ref.: M. Kempe