Physical Optics
Lecture 7: Polarization 2018-05-23 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 11.04. Wave optics G interferometry Slit, grating, diffraction integral, diffraction in optical systems, point spread 2 18.04. Diffraction G function, aberrations Plane wave expansion, resolution, image formation, transfer function, 3 25.04. Fourier optics G phase imaging Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point 4 02.05. G resolution resolution, criteria, contrast, axial resolution, CTF Energy, momentum, time-energy uncertainty, photon statistics, 5 09.05. Photon optics K fluorescence, Jablonski diagram, lifetime, quantum yield, FRET Temporal and spatial coherence, Young setup, propagation of coherence, 6 16.05. Coherence K speckle, OCT-principle Introduction, Jones formalism, Fresnel formulas, birefringence, 7 23.05. Polarization G components Atomic transitions, principle, resonators, modes, laser types, Q-switch, 8 30.05. Laser K pulses, power Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects, 9 06.06. Nonlinear optics K CARS microscopy, 2 photon imaging Apodization, superresolution, extended depth of focus, particle trapping, 10 13.06. PSF engineering G confocal PSF Introduction, surface scattering in systems, volume scattering models, 11 20.06. Scattering L calculation schemes, tissue models, Mie Scattering 12 27.06. Gaussian beams G Basic description, propagation through optical systems, aberrations Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy 13 04.07. Generalized beams G beams, applications in superresolution microscopy 14 11.07. Miscellaneous G Coatings, diffractive optics, fibers
K = Kempe G = Gross L = Lu 3 Contents
. Introduction . Fresnel formulas . Jones calculus . Further descriptions . Birefringence . Components . Applications 4 Scalar / vectorial Optics
. Scalar: Helmholtz equation k
2 ko nE(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 EAx costex 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 tan 2 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
El
t1 SA z E b1 x E r LA b2 El
TA Er E
t2 l / 4 - plate y
45°
linear polarizer x 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 . Amplitude coefficients for r r rTE TM reflected field E Ee e TE TM 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 Rrr Tt P ncos i of light power Pe e 13 Fresnel Formulas
. Coefficients of amplitude for reflected rays, s and p
sin(i i') ncosi n'2 n2 sin 2 i ncosi n'cosi' k k r ez tz E 2 2 2 sin(i i') ncosi n' n sin i ncosi n'cosi' kez ktz
tan(i i') n'2 cosi n n'2 n2 sin 2 i n'cosi ncosi' 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 ncosi' n' kez n ktz
. Coefficients of amplitude for transmitted rays, s and p
2ncosi 2ncosi 2ncosi 2k t ez E 2 2 2 ncosi n'cosi' ncosi n' n sin i ncosi n'cosi' kez ktz 2ncosi 2n'ncosi 2ncosi 2n'2 k t ez E|| 2 2 2 2 2 2 n'cosi ncosi' n' cosi n n' n sin i n'cosi ncosi' 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 ei x E x x into two components in x/y or s/p 0 E i y y Aye . 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 0 E0 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: E p 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 nJ n1 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 t 0 r 0 J s J s 2. dielectric interface TRA REF 0 t p 0 rp
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 ii22 sin cos ee 1 cos sin
2p z . Relative phase angle nn l eo 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 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)
S3 2Ex Ey sin 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 cos2 sin 2 1 r y cos2 cos2 x2 y 2 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
xL,x1 xL, y1 xL,x1 xL,x1 y'L,x1 s'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 j1 E p p p E x, j xx yx zx x, j1 Ey, j pxy pyy pzx Ey, j1 Ez, j pxz pyz pzz Ez, j1 . Cascading of operator matrices Ptotal P M P M 1 ....P2 P1
. 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 Tmax Tmin . Eigenvalues of Jones matrix i1/ 2 J ret w1/ 2 1/ 2 w1/ 2 e w1/ 2
. Retardation: phase difference arg1 arg2 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 Q1 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
D j jl El g jlmm El jlmqmq El l l,m l,m,q . First term, coefficient : local direction, birefringence . Second term, coefficient g: 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 D n2s s E n2 s s E E 39 Fresnel or Ray Ellipsoid
. Vanishing solution determinante of the wave equation
2 2 2 kx k y kz 2 2 2 2 2 2 0 c cx c cy c cz . 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 displacement 0 0 vector
x Dx 1 1 ni 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 c p
cos2 sincos J () . Rotated polarizer P 2 sincos sin
. Polarizer in y-direction 0 0 J P (0) 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
2 I( ) Io cos
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 sincos1 ei J ( ,) V i 2 2 i sincos1 e sin cos e . Special value: l / 4 - plate generates circular polarized light 1. fast axis y
1 0 SA z J V (0,p / 2) 0 i
2. fast axis 45°
i 11 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 LV
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 3cos2 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 E psf (x x, y) adjustment phase i R e E psf (x x, y)
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