Wave Propagation in Uniaxial Media. Reflection and Transmission at Interfaces
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1 Lecture 5: Crystal Optics Outline 1 Homogeneous, Anisotropic Media 2 Crystals 3 Plane Waves in Anisotropic Media 4 Wave Propagation in Uniaxial Media 5 Reflection and Transmission at Interfaces Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 1
2 Homogeneous, Anisotropic Media Introduction material equations for homogeneous, anisotropic media D = ɛ E B = µ H tensors of rank 2, written as 3 by 3 matrices ɛ: dielectric tensor µ: magnetic permeability tensor for the following, assume µ = 1 examples: crystals, liquid crystals external electric, magnetic fields acting on isotropic materials (glass, fluids, gas) anisotropic mechanical forces acting on isotropic materials Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 2
3 Properties of Dielectric Tensor Maxwell equations imply symmetric dielectric tensor ɛ 11 ɛ 12 ɛ 13 ɛ = ɛ T = ɛ 12 ɛ 22 ɛ 23 ɛ 13 ɛ 23 ɛ 33 symmetric tensor of rank 2 coordinate system exists where tensor is diagonal orthogonal axes of this coordinate system: principal axes elements of diagonal tensor: principal dielectric constants 3 principal indices of refraction in coordinate system spanned by principal axes n 2 x 0 0 D = 0 ny 2 0 E 0 0 nz 2 x, y, z because principal axes form Cartesian coordinate system Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 3
4 Uniaxial Materials isotropic materials: n x = n y = n z anisotropic materials: n x n y n z uniaxial materials: n x = n y n z ordinary index of refraction: n o = n x = n y extraordinary index of refraction: n e = n z rotation of coordinate system around z has no effect most materials used in polarimetry are (almost) uniaxial Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 4
5 Crystals Crystal Axes Terminology optic axis is the axis that has a different index of refraction also called c or crystallographic axis fast axis: axis with smallest index of refraction ray of light going through uniaxial crystal is (generally) split into two rays ordinary ray (o-ray) passes the crystal without any deviation extraordinary ray (e-ray) is deviated at air-crystal interface two emerging rays have orthogonal polarization states common to use indices of refraction for ordinary ray (n o ) and extraordinary ray (n e ) instead of indices of refraction in crystal coordinate system n e < n o : negative uniaxial crystal n e > n o : positive uniaxial crystal Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 5
6 Plane Waves in Anisotropic Media Displacement and Electric Field Vectors plane-wave ansatz for D, E, H E = E 0 e i( k x ωt) D = D 0 e i( k x ωt) H = H 0 e i( k x ωt) no net charges in medium ( D = 0) D k = 0 D perpendicular to k D and E not parallel E not perpendicular to k wave normal s = k/ k, energy flow in different directions, at different speeds Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 6
7 Magnetic Field constant, scalar µ, vanishing current density H B H = 0 H k H = 1 c D t E = µ c H t H D H E D, E, and k all in one plane H, B perpendicular to that plane Poynting vector S = c 4π E H perpendicular to E and H S (in general) not parallel to k energy (in general) not transported in direction of wave vector k Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 7
8 Relation between D and E combine Maxwell, material equations in principal coordinate system ( ( )) D i = ɛ i E i = n 2 E i s i E s i = 1 3 s = k/ k : unit vector in direction of wave vector k n: refractive index associated with direction s, i.e. n = n( s) 3 equations for 3 unknowns E i eliminate E assuming E 0 Fresnel equation s 2 x n 2 ɛ x + s2 y n 2 ɛ y + s2 z n 2 ɛ z = 1 n 2 with n 2 i = ɛ i s 2 x n2 x n 2 n 2 y n 2 n 2 z + s 2 y n2 y n 2 n 2 x n 2 n 2 z + s 2 z n2 z n 2 n 2 x n 2 n 2 y = 0 Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 8
9 Electric Field in Anisotropic Material electric field can also be written as ( ) n 2 s k E s E k = n 2 ɛ k equivalent to (a a constant) E = a s x n 2 n 2 x s y n 2 n 2 y s z n 2 n 2 z quadratic equation in n generally two solutions for given direction s system of 3 equations can be solved for E k denominator vanishes if k parallel to a principal axis treat separately Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 9
10 Non-Absorbing, Non-Active, Anisotropic Materials k not parallel to a principal axis ratio of 2 electric field components k and l E k = s ( k n 2 ) ɛ ( l ) E l s l n 2 ɛ k ratio is independent of electric field components n 2 and ɛ i real ratios are real electric field is linearly polarized in non-absorbing, non-active, anisotropic material, 2 waves propagate that have different linear polarization states and different directions of energy flows direction of vibration of D corresponding to 2 solutions are orthogonal to each other (without proof) D 1 D 2 = 0 Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 10
11 Wave Propagation in Uniaxial Media Introduction uniaxial media dielectric constants: ɛ x = ɛ y = n 2 o ɛ z = n 2 e second form of Fresnel equation reduces to ( ) [ ( ) ( n 2 no 2 no 2 sx 2 + sy 2 n 2 ne 2 two solutions n 1, n 2 given by ) + s 2 zn 2 e ( )] n 2 no 2 = 0 n 2 1 = n 2 o 1 n 2 2 = s2 x + s 2 y n 2 e + s2 z n 2 o Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 11
12 Propagation in General Direction (unit) wave vector direction in spherical coordinates s x sin θ cos φ s = s y s z = sin θ sin φ cos θ θ: angle between wave vector and optic axis φ: azimuth angle in plane perpendicular to optic axis 1 n2 2 = cos2 θ no 2 + sin2 θ ne 2 n o n e n 2 (θ) = no 2 sin 2 θ + ne 2 cos 2 θ take positive root, negative value corresponds to waves propagating in opposite direction Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 12
13 Ordinary and Extraordinary Rays from before 1 n2 2 = cos2 θ no 2 + sin2 θ ne 2 n o n e n 2 (θ) = no 2 sin 2 θ + ne 2 cos 2 θ n 2 varies between n o for θ = 0 and n e for θ = 90 first solution propagates according to ordinary index of refraction, independent of direction ordinary beam or ray second solution corresponds to extraordinary beam or ray index of refraction of extraordinary beam is (in general) not the extraordinary index of refraction Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 13
14 Ordinary Beam ordinary beam speed independent of wave vector direction ( ( )) for D i = ɛ iei = n 2 Ei s i E s, i = 1 3 to hold for any direction s, E o s = 0 and E o,z = 0 electric field vector of ordinary beam (with real constant a o 0) E o = a o sin φ cos φ 0 ordinary beam is linearly polarized E o perpendicular to plane formed by wave vector k and c-axis displacement vector D o = n o Eo E o Poynting vector S o k Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 14
15 Extraordinary Ray since D e k = 0 and D e D o = 0 unique solution (up to real constant a e ) cos θ cos φ D e = a e cos θ sin φ sin θ since E e D o = 0, D e = ɛe e ne 2 cos θ cos φ E e = a ne 2 cos θ sin φ no 2 sin θ uniaxial medium E o E e = 0 however, E e k 0 Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 15
16 Dispersion Angle angle between k and Poynting vector S = angle between E and D = dispersion angle E e D ( e n 2 e no) 2 tan α = E e D e equivalent expression = (n2 e n 2 o) tan θ n 2 e + n 2 o tan 2 θ = sin 2θ 2 ( ) n 2 α = θ arctan o ne 2 tan θ n 2 o sin 2 θ + n 2 e cos 2 θ for given k in principal axis system, α fully determines direction of energy propagation in uniaxial medium for θ approaching π/2, α = 0 for θ = 0, α = 0 Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 16
17 Propagation Direction of Extraordinary Beam angle θ between Poynting vector S and optic axis tan θ = n2 o ne 2 tan θ ordinary and extraordinary wave do (in general) not travel at the same speed phase difference in radians between the two waves given by ω c (n 2(θ)d e n o d o ) d o,e : geometrical distances traveled by ordinary and extraordinary rays Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 17
18 Propagation Along c Axis plane wave propagating along c-axis θ = 0 ordinary and extraordinary beams propagate at same speed c n o electric field vectors are perpendicular to c-axis and only depend on azimuth φ ordinary and extraordinary rays are indistinguishable uniaxial medium behaves like an isotropic medium example: c-cut sapphire windows Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 18
19 Propagation Perpendicular to c Axis plane wave propagating perpendicular to c-axis θ = π/2 E o = sin φ cos φ 0 E o perpendicular to plane formed by k and c-axis electric field vector of extraordinary wave 0 E e = 0 1 E e parallel to c-axis direction of energy propagation of extraordinary wave parallel to k since E e D e Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 19
20 Phase Delay between Ordinary and Extraordinary Rays ordinary and extraordinary wave propagate in same direction ordinary ray propagates with speed c n o extraordinary beam propagates at different speed c n e E o, E e perpendicular to each other plane wave with arbitrary polarization can be (coherently) decomposed into components parallel to E o and E e 2 components will travel at different speeds (coherently) superposing 2 components after distance d phase difference between 2 components ω c (n e n o )d radians phase difference change in polarization state basis for constructing linear retarders Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 20
21 Summary: Wave Propagation in Uniaxial Media ordinary ray propagates like in an isotropic medium with index n o extraordinary ray sees direction-dependent index of refraction n 2 (θ) = n o n e no 2 sin 2 θ + ne 2 cos 2 θ n 2 n o n e direction-dependent index of refraction of the extraordinary ray ordinary index of refraction extraordinary index of refraction θ angle between extraordinary wave vector and optic axis extraordinary ray is not parallel to its wave vector angle between the two is dispersion angle tan α = (n2 e n 2 o) tan θ n 2 e + n 2 o tan 2 θ Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 21
22 Reflection and Transmission at Uniaxial Interfaces General case from isotropic medium (n I ) into uniaxial medium (n o, n e ) θ I : angle between surface normal and k I for incoming beam θ 1,2 : angles between surface normal and wave vectors of (refracted) ordinary wave k 1 and extraordinary wave k 2 phase matching at interface requires ki x = k 1 x = k 2 x x: position vector of a point on interface surface n I sin θ I = n 1 sin θ 1 = n 2 sin θ 2 n 1 = n o : index of refraction of ordinary wave n 2 : index of refraction of extraordinary wave Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 22
23 Ordinary and Extraordinary Rays ordinary wave Snell s law sin θ 1 = n I n 1 sin θ I law for extraordinary ray not trivial n I sin θ I = n 2 (θ(θ 2 )) sin θ 2 (in general) θ 2 and therefore k 2 will not determine direction of extraordinary beam since Poynting vector (in general) not parallel to wave vector solve for θ 2 determine direction of Poynting vector special cases reduce complexity of equations Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 23
24 Extraordinary Ray Refraction for General Case c x c y n o 2 n2 e cot θ 2 = s n 2 o ± n n2 e +n2 e c2 x n 2 e n2 o o propagation vector of extraordinary ray sin 2 n θ o n 2 e 2 n2 o I no 2 + c2 x ne 2 n2 o S x = cos α cos θ 2 + sin α sin θ 2 `cx sin θ 2 c y cos θ 2 q cz 2 + `c x sin θ 2 c y cos θ 2 2 S y = cos α sin θ 2 sin α cos θ 2 `cx sin θ 2 c y cos θ 2 q cz 2 + `c x sin θ 2 c y cos θ 2 2 sin α S z = c z q cz 2 + `c x sin θ 2 c y cos θ 2 2 cx 2 + c2 y c optic axis vector c = (c x, c y, c z ) T S propagation direction of extraordinary ray S = (S x, S y, S z ) T θ I angle between k I and interface normal θ 2 angle between k e and interface normal α dispersion angle Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 24
25 Normal Incidence normal incidence θ I = 0, θ 1 = θ 2 = 0 choose plane formed by surface normal and crystal axis both wave vectors and ordinary ray not refracted extraordinary ray refracted by dispersion angle α ( ) n 2 α = θ arctan o ne 2 tan θ Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 25
26 Optic Axis in Plane of Incidence and Plane of Interface θ + θ 2 = π/2 cot θ 2 = ne n o cot θ 1 θ 1 : angle between surface normal and ordinary ray or wave vector (sin θ I = n o sin θ 1 ) extraordinary wave sees equivalent refractive index ( ) n y = ne 2 + sin 2 θ I 1 n2 e no 2 direction of Poynting vector S x = cos(θ 2 + α) S y = sin(θ 2 + α) S z = 0 determine dispersion angle α and add to θ 2 to obtain direction of extraordinary ray Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 26
27 Optic Axis Perpendicular to Plane of Incidence c-axis perpendicular to plane of incidence θ = π 2, n 2 ( π 2 ) = ne n I sin θ I = n e sin θ 2 extraordinary wave vector obeys Snell s law with index n e θ = π 2 dispersion angle α = 0 Poynting vector wave vector, extraordinary beam itself obeys Snell s law with n e double refraction only for non-normal incidence Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 27
28 Interface from Uniaxial Medium to Isotropic Medium ordinary ray follows Snell s law transmitted extraordinary wave vector and ray coincide exit of extraordinary wave on interface defined by extraordinary ray extraordinary wave vector follows Snell s law with index n 2 (θ) n I sin θ E = n 2 sin θ U n I index of isotropic medium θ E angle of wave/ray vector with surface normal in isotropic medium n 2, θ U corresponding values for extraordinary wave vector in uniaxial medium n 2 is function of θ normally already known from beam propagation in uniaxial medium θ U is function of geometry of interface, plane-parallel slab of uniaxial medium, θ E = θ I, (in general) extraordinary beam displaced on exit Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics 28
29 Total Internal Reflection (TIR) TIR also in anisotropic media n o n e one beam may be totally reflected while other is transmitted principal of most crystal polarizers example: calcite prism, normal incidence, optic axis parallel to first interface, exit face inclined by 40 extraordinary ray not refracted, two rays propagate according to indices n o,n e at second interface rays (and wave vectors) at 40 to surface nm: n o = , n e = requirement for total reflection n U ni sin θ U > 1 with n I = 1 extraordinary ray transmitted, ordinary ray undergoes TIR Christoph U. Keller, C.U.Keller@astro-uu.nl Lecture 5: Crystal Optics n o, n e c 40 o e
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