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Crystal Optics Homogeneous, Anisotropic Media

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Crystal Optics Homogeneous, Anisotropic Media Lecture 3: Crystal Optics Homogeneous, Anisotropic Media Introduction Outline material equations for homogeneous anisotropic media ~ ~ 1 Homogeneous, Anisotropic Media D = E B~ = µH~ 2 Crystals tensors of rank 2, written as 3 by 3 matrices 3 Plane Waves in Anisotropic Media : dielectric tensor 4 Wave Propagation in Uniaxial Media µ: magnetic permeability tensor examples: 5 Reflection and Transmission at Interfaces crystals, liquid crystals external electric, magnetic fields acting on isotropic materials (glass, fluids, gas) anisotropic mechanical forces acting on isotropic materials Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 1 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 2 Properties of Dielectric Tensor Uniaxial Materials Maxwell equations imply symmetric dielectric tensor isotropic materials: n = n = n 0 1 x y z 11 12 13 for any coordinate system T = = @ 12 22 23 A anisotropic materials: 13 23 33 nx 6= ny 6= nz symmetric tensor of rank 2 ) coordinate system exists where uniaxial materials: nx = ny 6= nz tensor is diagonal ordinary index of refraction: orthogonal axes of this coordinate system: principal axes no = nx = ny elements of diagonal tensor: principal dielectric constants extraordinary index of refraction: n = n 3 principal indices of refraction in coordinate system spanned by e z principal axes rotation of coordinate system 0 2 1 around z does not change nx 0 0 ~ 2 ~ anything D = @ 0 ny 0 A E 2 0 0 nz most materials used in polarimetry are (almost) uniaxial x, y, z because principal axes form Cartesian coordinate system Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 3 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 4 Crystals Plane Waves in Anisotropic Media Crystal Axes Terminology Displacement and Electric Field Vectors optic axis is the axis that has a different index of refraction plane-wave ansatz for D~ , E~ , H~ also called c or crystallographic axis ~ ~ i(~k·~x−!t) fast axis: axis with smallest index of refraction E = E0e ray of light going through uniaxial crystal is (generally) split into ~ ~ i(~k·~x−!t) D = D0e two rays ~ H~ = H~ ei(k·~x−!t) ordinary ray (o-ray) passes the crystal without any deviation 0 extraordinary ray (e-ray) is deviated at air-crystal interface no net charges in medium (r · D~ = 0) two emerging rays have orthogonal polarization states D~ · ~k = 0 common to use indices of refraction for ordinary ray (no) and extraordinary ray (ne) instead of indices of refraction in crystal D~ perpendicular to ~k coordinate system D~ and E~ not parallel ) E~ not perpendicular to ~k n < n : negative uniaxial crystal e o wave normal, energy flow not in same direction, same speed ne > no: positive uniaxial crystal Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 5 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 6 Magnetic Field Relation between D~ and E~ combine Maxwell, material equations in principal coordinate constant, scalar µ, vanishing current system density ) H~ k B~ ~ 2 ~ ~ ~ r · H~ = 0 ) H~ ? ~k µDi = µi Ei = n Ei − si E · s i = 1 ··· 3 ~ r × H~ = 1 @D ) H~ ? D~ c @t ~s = ~k=j~kj: unit vector in direction of wave vector ~k ~ µ @H~ ~ ~ r × E = − c @t ) H ? E n: refractive index associated with direction ~s, i.e. n = n(~s) ~ D~ , E~ , and ~k all in one plane 3 equations for 3 unknowns Ei ~ ~ ~ H~ , B~ perpendicular to that plane eliminate E assuming E 6= 0 ) Fresnel equation ~ c ~ ~ 2 2 2 Poynting vector S = 4π E × H s s s 1 x + y + z = ; ~ ~ ~ 2 2 2 2 perpendicular to E and H ) S (in n − µx n − µy n − µz n general) not parallel to ~k 2 = µ energy (in general) not transported in with ni i ~ direction of wave vector k 2 2 “ 2 2”“ 2 2” 2 2 “ 2 2”“ 2 2” 2 2 “ 2 2”“ 2 2” sx nx n − ny n − nz + sy ny n − nx n − nz + sz nz n − nx n − ny = 0 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 7 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 8 Electric Field in Anisotropic Material Non-Absorbing, Non-Active, Anisotropic Materials with arbitrary constant a, electric field vector given by ~k not parallel to a principal axis ) ratio of 2 electric field 0 sx 1 2 2 components k and l n −nx ~ B sy C E = a 2− 2 2 @ n ny A Ek sk n − µl sz = 2 2 2 n −nz El sl n − µk quadratic equation in n ) generally two solutions for given ratio is independent of electric field components direction ~s 2 n and i real ) ratios are real ) electric field is linearly polarized electric field can also be written as in non-absorbing, non-active, anisotropic material, 2 waves 2 ~ propagate that have different linear polarization states and n ~sk E · ~s ~ = different directions of energy flows Ek 2 n − µk direction of vibration of D~ corresponding to 2 solutions are orthogonal to each other (without proof) system of 3 equations can be solved for Ek denominator vanishes if ~k parallel to a principal axis ) treat ~ ~ D1 · D2 = 0 separately Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 9 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 10 Wave Propagation in Uniaxial Media Propagation in General Direction (unit) wave vector direction in spherical coordinates Introduction 0 1 0 1 uniaxial media ) dielectric constants: sx sin θ cos φ ~s = @ sy A = @ sin θ sin φ A 2 s cos θ x = y = no z 2 z = ne θ: angle between wave vector and optic axis second form of Fresnel equation reduces to φ: azimuth angle in plane perpendicular to optic axis 2 2 h 2 2 2 2 2 2 2 2 2i n − no no sx + sy n − ne + sz ne n − no = 0 1 cos2 θ sin2 θ = + 2 2 2 two solutions n1, n2 given by n2 no ne none n2 = n2 n2 (θ) = q 1 o 2 2 2 2 no sin θ + ne cos θ 1 s2 + s2 s2 = x y + z 2 2 2 n2 ne no take positive root, negative value corresponds to waves propagating in opposite direction Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 11 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 12 Ordinary Beam Ordinary and Extraordinary Rays ordinary beam speed independent of wave vector direction ~ 2 ~ ~ from before for µDi = µi Ei = n Ei − si E · ~s ; i = 1 ··· 3 to hold for any ~ 1 cos2 θ sin2 θ direction ~s, Eo · ~s = 0 and Eo;z = 0 = + 2 2 2 n no ne 2 electric field vector of ordinary beam none n2 (θ) = q (with real constant ao 6= 0) 2 2 2 2 no sin θ + ne cos θ 0 sin φ 1 ◦ ~ n2 varies between no for θ = 0 and ne for θ = 90 Eo = ao @ − cos φ A first solution propagates according to ordinary index of refraction, 0 independent of direction ) ordinary beam or ray ordinary beam is linearly polarized second solution corresponds to extraordinary beam or ray ~ Eo perpendicular to plane formed by index of refraction of extraordinary beam is (in general) not the wave vector ~k and c-axis extraordinary index of refraction ~ ~ ~ displacement vector Do = noEo k Eo ~ ~ Poynting vector So k k Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 13 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 14 Dispersion Angle Extraordinary Ray angle between ~k and Poynting vector S~ = angle between E~ and D~ since D~ · ~k = 0 and D~ · D~ = 0 ) unique solution (up to real e e o = dispersion angle constant ae) 0 cos θ cos φ 1 ~ ~ ~ Ee × De (n2 − n2) tan θ sin 2θ n2 − n2 De = ae @ cos θ sin φ A tan α = = e o = e o ~ ~ 2 2 2 2 2 2 2 − sin θ Ee · De ne + no tan θ 2 no sin θ + ne cos θ ~ equivalent expression since Ee · Do = 0, De = Ee 2 0 2 1 no ne cos θ cos φ α = θ − arctan tan θ ~ 2 n2 Ee = a @ ne cos θ sin φ A e −n2 sin θ o for given ~k in principal axis system, α fully determines direction of ~ ~ energy propagation in uniaxial medium uniaxial medium ) Eo · Ee = 0 ~ ~ for θ approaching π=2, α = 0 however, Ee · k 6= 0 for θ = 0, α = 0 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 15 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 16 Propagation Direction of Extraordinary Beam angle θ0 between Poynting vector S~ and optic axis Propagation Along c Axis n2 plane wave propagating along c-axis ) θ = 0 tan θ0 = o tan θ n2 ordinary and extraordinary beams propagate at same speed c e no electric field vectors are perpendicular to c-axis and only depend ordinary and extraordinary wave do (in general) not travel at the on azimuth φ same speed ordinary and extraordinary rays are indistinguishable phase difference in radians between the two waves given by uniaxial medium behaves like an isotropic medium ! (n (θ)d − n d ) example: “c-cut” sapphire windows c 2 e o o do;e: geometrical distances traveled by ordinary and extraordinary rays Christoph U.
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