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

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Properties of Dielectric Tensor Uniaxial Materials Maxwell equations imply symmetric dielectric tensor isotropic materials: n = n = n   x y z 11 12 13 for any coordinate system T = =  12 22 23  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  2  around z does not change nx 0 0 ~ 2 ~ anything D =  0 ny 0  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−ωt) fast axis: axis with smallest index of refraction E = E0e ray of light going through uniaxial crystal is (generally) split into ~ ~ i(~k·~x−ωt) D = D0e two rays ~ H~ = H~ ei(k·~x−ωt) 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 (∇ · 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

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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 ~ ~ ~ ∇ · H~ = 0 ⇒ H~ ⊥ ~k µDi = µi Ei = n Ei − si E · s i = 1 ··· 3 ~ ∇ × H~ = 1 ∂D ⇒ H~ ⊥ D~ c ∂t ~s = ~k/|~k|: unit vector in direction of wave vector ~k ~ µ ∂H~ ~ ~ ∇ × 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  sx  2 2 components k and l n −nx ~  sy  E = a 2− 2 2  n ny  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

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Wave Propagation in Uniaxial Media Propagation in General Direction (unit) wave vector direction in spherical coordinates Introduction     uniaxial media ⇒ dielectric constants: sx sin θ cos φ ~s =  sy  =  sin θ sin φ  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  sin φ  ◦ ~ n2 varies between no for θ = 0 and ne for θ = 90 Eo = ao  − cos φ  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

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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)  cos θ cos φ  ~ ~ ~ Ee × De (n2 − n2) tan θ sin 2θ n2 − n2 De = ae  cos θ sin φ  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  2  no ne cos θ cos φ α = θ − arctan tan θ ~ 2 n2 Ee = a  ne cos θ sin φ  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 ω (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

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Propagation Perpendicular to c Axis plane wave propagating perpendicular to c-axis ⇒ θ = π/2 Phase Delay between Ordinary and Extraordinary Rays ordinary and extraordinary wave propagate in same direction  sin φ  ~ ordinary ray propagates with speed c Eo =  − cos φ  no extraordinary beam propagates at different speed c 0 ne ~ ~ ~ ~ Eo, Ee perpendicular to each other ⇒ plane wave with arbitrary Eo perpendicular to plane formed by k and c-axis polarization can be (coherently) decomposed into components ~ ~ electric ﬁeld vector of extraordinary wave parallel to Eo and Ee  0  2 components will travel at different speeds ~ Ee =  0  (coherently) superposing 2 components after distance d ⇒ phase ω 1 difference between 2 components c (ne − no)d radians phase difference ⇒ change in polarization state ~ Ee parallel to c-axis basis for constructing linear retarders direction of energy propagation of extraordinary wave parallel to ~k ~ ~ since Ee k De

Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 19 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 20 Reﬂection and Transmission at Uniaxial Interfaces Summary: Wave Propagation in Uniaxial Media ordinary ray propagates like in an isotropic medium with index no General case extraordinary ray sees direction-dependent index of refraction from isotropic medium (nI) into uniaxial medium (no, ne) ~ none θI: angle between surface normal and kI for incoming beam n2 (θ) = q n2 sin2 θ + n2 cos2 θ θ1,2: angles between surface normal and wave vectors of o e ~ ~ (refracted) ordinary wave k1 and extraordinary wave k2 phase matching at interface requires n2 direction-dependent index of refraction of the extraordinary ray no ordinary index of refraction ~k · ~x = ~k · ~x = ~k · ~x ne extraordinary index of refraction I 1 2 θ angle between extraordinary wave vector and optic axis ~x: position vector of a point on interface surface extraordinary ray is not parallel to its wave vector

angle between the two is dispersion angle nI sin θI = n1 sin θ1 = n2 sin θ2

(n2 − n2) tan θ n = n : index of refraction of ordinary wave tan α = e o 1 o 2 2 2 ne + no tan θ n2: index of refraction of extraordinary wave

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Extraordinary Ray Refraction for General Case

s “ ” n2n2+n2c2 n2−n2 Ordinary and Extraordinary Rays “ 2 2” o e e x e o 2 “ 2 2”“ 2 2” cx cy no − ne ± no 2 − no − ne − no cx + cy sin θI cot θ2 = ordinary wave ⇒ Snell’s law 2 2 “ 2 2” no + cx ne − no

nI propagation vector of extraordinary ray sin θ1 = sin θI n1 ` ´ sin α sin θ2 cx sin θ2 − cy cos θ2 Sx = cos α cos θ2 + q 2 ` ´2 law for extraordinary ray not trivial cz + cx sin θ2 − cy cos θ2 ` ´ sin α cos θ2 cx sin θ2 − cy cos θ2 Sy = cos α sin θ2 − q c2 + `c sin θ − c cos θ ´2 nI sin θI = n2 (θ(θ2)) sin θ2 z x 2 y 2 sin α Sz = cz ∗ q 2 ` ´2 ~ cz + cx sin θ2 − cy cos θ2 (in general) θ2 and therefore k2 will not determine direction of extraordinary beam since Poynting vector (in general) not parallel T to wave vector ~c optic axis vector ~c = (cx , cy , cz ) ~ ~ T solve for θ2 ⇒ determine direction of Poynting vector S propagation direction of extraordinary ray S = (Sx , Sy , Sz ) ~ special cases reduce complexity of equations θI angle between kI and interface normal ~ θ2 angle between ke and interface normal α dispersion angle

Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 23 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 24 Normal Incidence Optic Axis in Plane of Incidence and Plane of Interface θ + θ = π/2 ⇒ cot θ = ne cot θ 2 2 no 1 θ1: angle between surface normal and ordinary ray or wave vector (sin θI = no sin θ1) extraordinary wave sees equivalent refractive index s n2 = 2 + 2 θ − e ny ne sin I 1 2 no

direction of Poynting vector normal incidence ⇒ θI = 0, θ1 = θ2 = 0 choose plane formed by surface normal and crystal axis Sx = cos(θ2 + α)

both wave vectors and ordinary ray not refracted Sy = sin(θ2 + α)

extraordinary ray refracted by dispersion angle α Sz = 0

n2 determine dispersion angle α and add to θ to obtain direction of α = θ − arctan o tan θ 2 2 extraordinary ray ne

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Interface from Uniaxial Medium to Isotropic Medium ordinary ray follows Snell’s law transmitted extraordinary wave vector and ray coincide Optic Axis Perpendicular to Plane of Incidence exit of extraordinary wave on interface deﬁned by extraordinary ray π π c-axis perpendicular to plane of incidence ⇒ θ = 2 , n2 2 = ne extraordinary wave vector follows Snell’s law with index n2 (θ)

nI sin θI = ne sin θ2 nI sin θE = n2 sin θU

extraordinary wave vector obeys Snell’s law with index ne π θ = 2 ⇒ dispersion angle α = 0 nI index of isotropic medium Poynting vector k wave vector, extraordinary beam itself obeys θE angle of wave/ray vector with surface normal in isotropic medium n2, θU corresponding values for extraordinary wave vector in Snell’s law with ne uniaxial medium double refraction only for non-normal incidence n2 is function of θ normally already known from beam propagation in uniaxial medium

θU is function of geometry of interface,

plane-parallel slab of uniaxial medium, θE = θI, (in general) extraordinary beam displaced on exit Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 27 Christoph U. Keller, Utrecht University, [email protected] Lecture 3: Crystal Optics 28 Total Internal Reﬂection (TIR)

TIR also in anisotropic media

no 6= ne ⇒ one beam may be totally reﬂected while other is transmitted principal of most crystal polarizers

example: calcite prism, normal incidence, 40° optic axis parallel to ﬁrst interface, exit face ◦ inclined by 40 n , n o e ⇒ extraordinary ray not refracted, two rays propagate according to indices no,ne 40° e at second interface rays (and wave vectors) at 40◦ to surface c o 632.8 nm: no = 1.6558, ne = 1.4852 requirement for total reﬂection nU sin θ > 1 nI U with nI = 1 ⇒ extraordinary ray transmitted, ordinary ray undergoes TIR

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