Physical Optics

Lecture 12: Polarization 2017-05-03 Herbert Gross

www.iap.uni-jena.de

2 Physical Optics: Content

No Date Subject Ref Detailed Content Complex fields, wave equation, k-vectors, interference, light propagation, 1 05.04. Wave optics G interferometry Slit, grating, diffraction integral, diffraction in optical systems, point spread 2 12.04. Diffraction B function, aberrations Plane wave expansion, resolution, image formation, transfer function, 3 19.04. Fourier optics B phase imaging Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point 4 26.04. B resolution resolution, criteria, contrast, axial resolution, CTF Introduction, Jones formalism, Fresnel formulas, birefringence, 5 03.05. Polarization G components Energy, momentum, time-energy uncertainty, photon statistics, 6 10.05. Photon optics D fluorescence, Jablonski diagram, lifetime, quantum yield, FRET Temporal and spatial coherence, Young setup, propagation of coherence, 7 17.05. Coherence G speckle, OCT-principle Atomic transitions, principle, resonators, modes, laser types, Q-switch, 8 24.05. Laser B pulses, power 9 31.05. Gaussian beams D Basic description, propagation through optical systems, aberrations Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy 10 07.06. Generalized beams D beams, applications in superresolution microscopy Apodization, superresolution, extended depth of focus, particle trapping, 11 14.06. PSF engineering G confocal PSF Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects, 12 21.06. Nonlinear optics D CARS microscopy, 2 photon imaging Introduction, surface scattering in systems, volume scattering models, 13 28.06. Scattering G calculation schemes, tissue models, Mie Scattering 14 05.07. Miscellaneous G Coatings, diffractive optics, fibers

D = Dienerowitz B = Böhme G = Gross 3 Contents

. Introduction . Fresnel formulas . Jones calculus . Further descriptions . Birefringence . Components . Applications

4 Scalar / vectorial Optics

. Scalar: Helmholtz equation k

2  ko nE(r)0

. Vectorial: Maxwell equations E

       k  H  D i  j D   E  P     0 r   k  E  B B    H  M k     0 r k  D   i   k  J      k  B 0 J  E H

5 Basic Forms of Polarisation

x 1. Linear

components in phase E z E

y

x

2. circular E E phase difference of 90°

between components z

y

x 3. elliptical E

arbitrary but constant phase E

difference z

y

6 Basic Notations of Polarization

. Description of electromagnetic fields: - Maxwell equations - vectorial nature of field strength    . Decomposition of the field into components EAxcostex Ay cos(t  )ey . Propagation plane wave: - field vector rotates - projection components are oscillating sinusoidal

y

x

z

7 Polarization Ellipse

. Elimination of the time dependence: 2 2 E Ey 2Ex Ey cos 2 Ellipse of the vector E x   sin  A2 A2 A A . Different states of polarization: x y x y - sense of rotation - shape of ellipse

  0°   45°   90°   135°   180°

  225°   270°   315°   360°

8 Polarization Ellipse

. Representation of the state of polarization by an ellipse . Field components

Ex  Ax (cos cos x sin sin x )

Ey  Ay (cos cos y sin sin y ) y . Axes of ellipse: a, b y'

Ax x' . Rotation angle of the field  Ex 2Ax Ay tan2  2 2 cos Ax  Ay Ay E b y  . Angle of eccentricity x

b tan   a a

9 Circular Polarization

. Spiral curve of field vector

. Superposition of left and right handed circular polarized light: resulting linear polarization

y

t2

t3

t1 El  t E x  t

Er

Ref: Manset 10 Circular polarized Light

. Rotation of plane of polarization with t / z . Phase angle 90° . Generation by l/4 plate out of linear polarized light

y

E l t1 SA z E b 1 x E r LA b E 2 l TA Er E

t2 l / 4 - plate y

45°

linear polarizer x

.  Erzeugung : . Polarisator und l / 4 - . Platte 11 Fresnel Formulas

. Schematical illustration of the ray refraction ( reflection at an interface . The cases of s- and p-polarization must be distinguished

a ) s-polarization b) p-polarization

B E r r reflection reflection transmission E r Br transmission

E t B t

i B i' t i i' E t normal to the i normal to the i interface interface

n n' n n' E i B incidence i B incidence i interface E i interface 12 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 ||

  E    E . Boundary condition of Maxwell equations 1 1n 2 2n at a dielectric interface: continuous tangential component of E-field E1t  E2t

E Er r . Amplitude coefficients for r  rTM  reflected field TE E Ee e TM TE n Et transmitted field t   r 1 tTM  rTM 1 TE E TE n' e TE

P 2 Pt ni' cos ' 2 . Reflectivity and transmission Rrr Tt   P ncos i of light power Pe e

13 Fresnel Formulas

. Coefficients of amplitude for reflected rays, s and p

2 2 2 sin(i i') ncosi  n' n sin i ncosi  n'cosi' kez  ktz r      E 2 2 2 sin(i  i') ncosi  n' n sin i ncosi  n'cosi' kez  ktz

tan(i i') n'2cosi  n n'2 n2 sin 2 i n'cosi  ncosi' n'2k  n2 k r     ez tz E|| 2 2 2 2 2 2 tan(i  i') n' cosi  n n' n sin i n'cosi  ncosi' n' kez  n ktz

. Coefficients of amplitude for transmitted rays, s and p

2ncosi 2ncosi 2ncosi 2kez t     E 2 2 2 ncosi  n'cosi' ncosi  n' n sin i ncosi  n'cosi' kez  ktz 2ncosi 2n'ncosi 2ncosi 2n'2k t     ez E|| 2 2 2 2 2 2 n'cosi  ncosi' n' cosi  n n' n sin i n'cosi  ncosi' n' kez  n ktz 14 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 t Reflected s-rays with phase jump of p t (corresponds to r<0) r

. i = 90° No transmission possible Reflected light independent r on polarization

. Brewster angle: completely s-polarized i reflected light Brewster 15 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 R , T R , T 1 1 1 0.9 0.8 0.9

0.8 0.6 0.8 t T 0.4 0.7 0.7 t T 0.2 0.6 0.6 r T(In) 0 0.5 0.5 T(In) -0.2 0.4 0.4

0.3 -0.4 0.3 R r 0.2 0.2 -0.6 R(In) 0.1 0.1 -0.8 R(In) R 0 i 0 i -1 i 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90 16 Jones Vector

. Decomposition of the field strength E   E   A ei x  E   x  x  into two components in x/y or s/p 0    i y  Ey A e    y  . Relative phase angle between

components    y  x

2 2 . Polarization ellipse  E   E  E E  x    y   2 x y cos  sin 2       Ax   Ay  Ax Ay

. Linear polarized light  1  0  cos  E    E    E    0 0 0   0     1  sin 

. Circular polarized light  1 1  1  1  Erz    E    i lz   2   2 i

17 Jones Calculus

. Jones representation of full polarized field:    Ep   J ss J sp   Eop  decomposition into 2 components E  J  E ,       o    J J     Es  ps pp Eos . Cascading of system components:       Product of matrices En J nJ n1 J 2 J 1 E1

E1 System 1 : E2 System 2 : E3 System 3 : E4 En-1 System n : En J1 J2 J3 Jn

. In principle 3 types of components influencing polarization: 1. Change of amplitude polarizer, analyzer 1t 0  J   s  trans    0 1t p    i  e 2 0  2. Change of phase retarder J   ret  i   2   0 e   cos sin    3. Rotation of field components rotator J rot ()  sin cos 

18 Jones Matrices

. Rotated component J() D()J(0)D()

 cos sin  . Rotation matrix D()  sin cos      . Intensity *  I2E1 J JE1

. Three types of components to change the polarization: 1. amplitude: polarizer / analyzer 2. phase: retarder 3. orientation: rotator

. Propagation: 2p i OPL 1 0 1. free space l J PRO  e   0 1

ts 0  rs 0  2. dielectric interface J    J    TRA  0 t  REF  0 r   p   p 

3. mirror 1 0 J SP r   0 1

19 Birefringence: Uniaxial Crystal

. 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 . n  ne  no

ip  n  z e l 0 . Jones matrix J  neo ip  n  z l 0 e

sin22 cos eeii sin  cos   1 J (,)   neo ii22 sin cos ee 1 cos  sin 

. Relative phase angle 2p z  nneo l

20 Descriptions of Polarization

Parameter Properties

1 Ellipticity  , Polarization ellipse only complete polarization orientation 

2 Parameter  Complex parameter only complete polarization

3 Jones vectors Components of E only complete polarization

4 Stokes vectors complete or partial Stokes parameter S ... S o 4 polarization

5 Poincare sphere Points on or inside the only graphical representation Poincare sphere

6 Coherence matrix 2x2 - matrix C complete or partial polarization

 E 21 Stokes Vector

. Description of polarization from the energetic point of view

- So total intensity S0  I(0,0)  I(90,0) 2 2 So  Ex  Ey

- S1 Difference of intensity in x-y linear S1  I(0,0)  I(90,0)

2 2 S1  Ex  Ey

- S2 Difference of intensity linear under 45° / 135° S2  I(45,0)  I(135,0)

S2 2Ex Ey cos

- S Difference of circular components 3 S3  I(45,90)  I(45,90)

S 2E E sin 3 x y

22 Stokes Vector

. Description of a polarization state with Stokes parameterInterpretation: Components of the field on the Poincare sphere . Also partial polarization is taken into account 2 2 2 2 S0  S1  S2  S3 . Relation Unequal sign: partial polarization  S   0    S1  S   S  . Stokes vector 4x1  2     S3  m m m m   00 01 02 03  . Propagation:    m10 m11 m12 m13   S' M  S   S Müller matrix M m m m m   20 21 22 23    m30 m31 m32 m33 

23 Examples of Stokes Vectors

. Linear horizontal / vertical 1  1       1  1 S  S  0  0          0  0 

. LInear 45° 1    0 S  1     0

. Circular clockwise / counter-clockwise

1  1       0   0  S  S  0  0          1 1

24 Partial Polarization

. Partial polarized light: I pol p  degree of polarization I ges p = 0 : un-polarized 2 2 2 p = 1 : fullypolarized S1  S2  S3 p  0 < p < 1 : partial polarized S0

. Determination of Stokes parameter: S0  (1 p)S0  pS0

 Su  S p

. Unpolarized light S1  S2  S3  0

2 2 2 2 . Fully polarized light S0  S1  S2  S3

25 Sphere of Poincare

. Every point on a uni sphere describes one state of polarization . In spherical coordinates:  x  cos 2 sin 2   1     r   ycos 2 cos 2  x2  y2  z 2 left handed z  z   sin 2      circular polarized . Points on z-axis: circular polarized light elliptical . Meridian line: linear polarization polarized

2 . Points inside partial polarization y 2

linear polarized x

right handed circular polarized

26 Partial Polarization on the Poincare Sphere

. Fully polarized: point . Unpolarized: full surface . Partial polarized: probability distribution, points inside

partial fully polarized un-polarized polarized

z z z

P

y y y

x x x

27 Poincare Sphere and Stokes Vector

. Stokes parameter S1 , S2 , S3 : Componenst along axis directions

. Radius of sphere, length of vector: So z . Projection into meridional plane: angle 2 of polarization ellipse . Projection into meridian plane: eccentricity angle 2 S 3 P So

2 S2 S y 1 2

x

28 Polarization

Polarization of a donat mode in the focal region: 1. In focal plane 2. In defocussed plane

Ref: F. Wyrowski

29 Propagation of Polarization

  . Jones matrix changes Jones vector E'  J  E

. Müller matrix changes Stokes vector   S' M S

. Change of coherence matrix  C' J C  J . 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

30 Polarization Raytrace

. 3D calculus in global coordinates according to Chipman incoming outgoing ray ray . Ray trace defines local coordinate system

   sin  s1'      kj-1 kj xL,1    , yL, j  sin  xL,1 , zL,1  sin plane of sin  s1' incidence x        j y x'L,1  xL,1 , y'L,1 sin  xL,1 , z'L,1 s1' j

y' interface j . Transform between local and plane global coordinates  x x x   x y' s'   L,x1 L,y1 L,x1  L,x1 L,x1 1x 1   T 1,in  yL,x1 yL,y1 yL,z1  , T 1,out  xL,y1 y'L,y1 s'1y       sx,in sy,in sz,in   xL,z1 y'L,z1 s'1z 

31 Polarization Raytrace

. Embedded local 2x2 Jones matrix  j j 0  11 12 

J 1,refr  j21 j22 0    0 0 1

t 0 0 r 0 0 . Matrices of refracting surface  p   p  J  0 t 0 , J  0 r 0 and reflection t  s  r  s   0 0 1  0 0 1       . Field propagation E j  P j  E j1

 E   p p p   E   x, j   xx yx zx   x, j1   E    p p p  E  y, j xy yy zx y, j1  E   p p p   E   z, j   xz yz zz   z, j1  . Cascading of operator matrices P  P  P ....P  P total M M 1 2 1 . Transfer properties P T  J T 1 1. Physical changes 1 1,out 1,refr 1,in 2. Geometrical bending effects Q T  J T 1 1 1,out 1,bend 1,in

32 Diattenuation and Retardation

. Change of field strength:  2   calculation with polarization matrix, P E E*  PT  P E T     transmission T  2 * E E  E

T T . Diattenuation D  max min T T max min    . Eigenvalues of Jones matrix i1/ 2 J ret w1/ 2 1/ 2 w1/2 e w1/ 2

. Retardation: phase difference  arg1 arg2  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 RQ1  P total total

Polarization Performance Evaluation

. Müller matrix visualization

Interpretation not trivial

. Retardation and diattenuation map across the pupil

. Polarization Zernike pupil aberration according to M. Totzeck (complicated)

34 Anisotropic Media

. Different types of anisotropic media

Ref: Saleh / Teich 35 Geometry of Raytrace at an Interface

. 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

air crystal crystal axis

a   e wave  normal s' o  ' ' 'eo o  n s incident ray

36 Ray Splitting in Case of Birefringence

. 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

extraordinary ray

crystal axis

ordinary ray

37 Birefringence

. Different directions of E and D

. Ray splitting not identical to wave splitting

E D

ray / phase k / s

H , B energy / wave / P Poynting

38 Anisotropic Media

. Relationship between electric field and   D   E displacement vector 0 r

. Linear relationship of tensor-equation

Dj  jl El   jlmmEl   jlmqmq El  l l,m l,m,q

. 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     2   2   D n s s  E n s s  E E

39 Fresnel or Ray Ellipsoid

. Vanishing solution determinante of the wave equation

k 2 k 2 k 2 x  y  z 0 c2  c2 c2  c2 c2  c2 x y z . Value of speed of ligth depending on the ray direction (phase velocity)

. Alternative: axis 1/n Ey D z displacement vector 1/nz constant energy E density electric field

Ex y 1/ny

1/nx

x

40 Index or Normal Ellipsoid

Dy . Inverse matrix of the dielectric tensor: E electric index or normal ellipsoid field constant energy D  1  density  0 0  displacement  vector  x  Dx   n   1 1 i i  r  0 0    y   1   0 0   z 2 2 2  z  x y z nz

2  2  2 const nx ny nz

. Gives the refractive index as a function of y the orientation ny . Also possible: ellipsoid of k-values nx 2 2 2 2 2 2 kx nx k y ny kz nz 2 2  2 2  2 2 0 n  nx n  ny n  nz x

41 Index Ellipsoid for Uniaxial Crystals

. 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

a) positive birefringence ne > no b) negative birefringence ne < no optical optical axis axis k3 k n0k0

n() k0 e-ray e-ray n0k0



k2 o-ray o-ray n0k0 nek0

42 Wave-Optical Construction of the o/e-direction

. 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

incident ray velocity crystal ellipsoid axis

eo-ray wavefronts o - ray

43 Birefringent Uniaxial Media

. Classical uniaxial media used in polarization components: 1. Quartz, positive birefringent, small difference 2. Calcite, negative birefringent, larger difference e material sign n n o o eo  Calcite negative 1.6584 1.4864 e  o Quarz, SiO2 positive 1.5443 1.5534 quartz calcite

44 Refraction with Birefringence

. 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 parallel rays parallel rays divergent rays perpendicular to wavefront with phase difference in phase

o-ray e-ray

crystal axis crystal crystal axis axis

45 Polarizer

1 c 0  J   s  . Polarizer with attenuation cs/p LIN    0 1 cp 

 cos2  sincos  J ()  . Rotated polarizer P  2  sincos sin  

0 0 . Polarizer in y-direction J (0)  P 0 1  

TA z

y

x 46 Pair Polarizer-Analyzer

z . Polarizer and analyzer with rotation angle  TA E cos . Law of Malus:  Energy transmission TA E

I( ) I cos2  o

y I linear polarizer 

linear x polarizer y  0 90° 180° 270° 360° parallel perpendicular polarizer analyzer

47 Stress Analysis: Polarizer / Analyzer

. 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

TA y linear polarizer y

analyzer

linear x polarizer x

48 Retarder

  i  . Phase difference  between field e 2 0  J   components RET  i   0 e 2  . Retarder plate with rotation angle     cos2   sin 2 ei sincos1 ei  J ( ,)   V  i 2 2 i  sincos1 e  sin   cos e  . Special value: l / 4 - plate generates circular polarized light 1. fast axis y SA z 1 0  J V (0,p / 2)   0 i

2. fast axis 45°

i 11 i J V (p / 4,p / 2)    y 2 i 1   LA

x

49 Rotator

. Rotate the of plane of polarization cos b sin b  . Realization with magnetic field: J ROT     sin b cos b  Farady effect  b  B  LV

b . Verdet constant V

z

y

x

50 Liquid Crystals

. 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 . Parameter1 to describe the order / entropy: S   3cos2  1 2

. Application: - displays - spatial light modulator

51 LCD Cell

. Orientation of the molecules in an external field (voltage)

 molecules glass plate

field : E = 0 field : E > 0

52 Types of Liquid Crystals

nematic smectic A smectic C

53 Types of Liquid Crystals

cholesteric smectic C

54 Liquid Crystal Display

. Transmission changes due to polarization . Black-White switch possible . Reflective or refractive

polarizer 90° - TN-LC cell analyzer

black

polarizer LC cell mirror

white

55

Microscopic Contrast

bright field polarization fluorescence

epi trans illumination illumination

Ref: M. Kempe

56 Differential Interference Contrast

analyzer . Point spread function Wollaston prism i E psfdic(x, y)(1 R)e  Epsf (x  x, y) adjustment phase   Rei  E (x  x, y) psf objective . Parameter:

1. Shear distance shear distance 2. Phase offset x 3. Splitting ratio object

condenser

splitting ratio 1-R R Wollaston prism compensator

polarizer

57 Differential Interference Contrast

. Differential splitting: Large contrast at phase gradients original / shifted differential splitting

58 Differential Interference Contrast

. Lateral shift : preferred direction . Visisbility depends on orientation of details

shift 45° x-shift (0°) y-shift (90°)

59 DIC Phase Imaging

. Orientation of prisms and shift size determines the anisotropic image formation

Ref.: M. Kempe