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 .