Lecture 8: Light propagation in anisotropic media Petr Kužel • Tensors — classification of anisotropic media • Wave equation • Eigenmodes — polarization eigenstates • Normal surface (surface of refractive indices) • Indicatrix (ellipsoid of refractive indices) Anisotropic response Isotropic medium, linear response: = ε χ P 0 E = ε = ε ()+ χ D E 0 1 E Anisotropic medium: the polarization direction does not coincide with the field direction: = ε (χ + χ + χ ) Px 0 11Ex 12Ey 13Ez , = ε ()χ + χ + χ Py 0 21Ex 22Ey 23Ez , ! Tensors = ε ()χ + χ + χ Pz 0 31Ex 32Ey 33Ez . P = ε χ E k⊥D i 0 ij j k is not parallel to S = ε = ε ( + χ ) S⊥E Di ij E j 0 1 ij E j Tensors Transformation of a basis: = f j Aijei Corresponding transformation of a tensor: = ′ ′ = −1 st xk Aki xi xi Aik xk 1 rank (vector) ′ = −1 −1 = ′ nd tkl Aik Ajl tij tij Aki Aljtkl 2 rank p = A A A p′ th i1,i2!in k1i1 k2i2 ! knin k1,k2!kn n rank Tensors: symmetry considerations Intrinsic symmetry: reflects the character of the physical phenomenon represented by the tensor (usually related to the energetic considerations) ε = ε∗ ik ki Extrinsic symmetry: reflects the symmetry of the medium p = A A A p ε = A A ε i1,i2!in k1i1 k2i2 ! knin k1,k2!kn ij ki lj kl Example: two-fold axis along z: −1 0 0 ε = ε = ε ⇒ε=ε = 13 A11 A33 13 – 13 13 31 0 = − A 0 1 0 ε = ε = ε ⇒ε=ε = 23 A22 A33 23 – 23 23 32 0 0 0 1 Tensors: symmetry considerations Second example: higher order axis along z: ε = 2ε + 2ε − ε α α 11 c 11 s 22 2sc 12 cos sin 0 ε = 2ε + 2ε + ε A = − sin α cosα 0 22 s 11 c 22 2sc 12 ε = ε − ε + ()2 − 2 ε 0 0 1 12 sc 11 sc 22 c s 12 The above system of equations leads to: ()ε − ε 2 = − ε 11 22 s 2sc 12 ()ε − ε = 2ε 11 22 sc 2s 12 Solution for a higher (than 2) order axis: sinα ≠ 0 ε ε ε 11 = 22, 12 = 0 ε Dielectric tensor Crystallographic ij Optical symmetry crystallographic system system optical system of axes of axes 2 2 æ n 0 0 ö n 0 0 Axes systems: ç ÷ 2 2 ε = ε isotropic cubic e = e ç 0 n 0 ÷ 0 0 n 0 0 ç ÷ 2 2 ç 0 0 n ÷ 0 0 n • crystallographic è ø 2 2 hexagonal no 0 0 no 0 0 • optical (dielectric) 2 2 uniaxial tetragonal ε = ε0 0 n 0 ε = ε0 0 n 0 o 2 o 2 trigonal 0 0 n 0 0 n • laboratory e e 2 2 nx 0 0 nx 0 0 ε = ε 2 ε = ε 2 orthorhombic 0 0 n y 0 0 0 n y 0 2 2 0 0 nz 0 0 nz 2 ε ε nx 0 0 11 12 0 2 biaxial monoclinic ε = ε0 0 n 0 ε = ε12 ε22 0 y 2 ε 0 0 nz 0 0 33 2 ε ε ε nx 0 0 11 12 13 2 triclinic ε = ε0 0 n 0 ε = ε12 ε22 ε23 y 2 ε ε ε 0 0 nz 13 23 33 Wave equation Maxwell equations for harmonic plane waves: ∧ = ωµ k E 0 H k ∧ H = −ωε⋅ E in the system of principal optical axes 2 nx 0 0 ε = ε 2 0 0 ny 0 2 0 0 nz Wave equation: ∧ ()∧ + ω2µ ε⋅ = ()⋅ − 2 + ω2µ ε⋅ = k k E 0 E k k E k E 0 E 0 or ω2 s()s ⋅ E − E + ε ⋅ E = 0 k 2c2 r Wave equation: continued We define effective refractive index: kc n = ω Wave equation in the matrix form: n2 − n2 ()s2 + s2 n2s s n2s s x y z x y x z Ex n2s s n2 − n2 ()s2 + s2 n2s s E = 0 x y y x z y z y 2 2 2 − 2 ()2 + 2 n sxsz n sy sz nz n sx sy Ez Solutions of this homogeneous system: det = 0. • Effective refractive index n for a given propagation direction • Frequency ω for a given wave vector k • Surface of accepted wave vector moduli for a given ω (normal surface or surface of refractive indices) Wave equation: continued After having developed the determinant one obtains: ω2 2 ω2n2 ω2 2 ω2n2 ω2 2 nx − 2 y − 2 nz − 2 + 2 y − 2 nz − 2 + 2 k 2 k 2 k kx 2 k 2 k c c c c c ω2 2 ω2 2 ω2 2 ω2n2 2 nx − 2 nz − 2 + 2 nx − 2 y − 2 = k y 2 k 2 k kz 2 k 2 k 0 . c c c c or in terms of the effective refractive index: ()()()2 − 2 2 − 2 2 − 2 + nx n ny n nz n 2 []2 ()()()()()()2 − 2 2 − 2 + 2 2 − 2 2 − 2 + 2 2 − 2 2 − 2 = n sx ny n nz n sy nx n nz n sz nx n ny n 0 . This is a quadratic equation in n2, thus it provides two eigenmodes for a given direction of propagation s. Wave equation: continued ≠ In the case when n ni, the wave equation can be further simplified: 2 s2 2 sx + y + sz = 1 2 − 2 2 − 2 2 − 2 2 n nx n ny n nz n Polarization of the electric field for the eigenmodes: n2 − n2 ()s2 + s2 n2s s n2s s x y z x y x z Ex 2 2 − 2 ()2 + 2 2 = n sxsy ny n sx sz n sy sz Ey 0 n2s s n2s s n2 − n2 ()s2 + s2 E x z y z z x y z sx 2 − 2 N nx s For n ≠ n : E = y i 2 − 2 N ny s z 2 − 2 N nz Isotropic case = = ≡ nx ny nz n0 wave equation: ()2 − 2 2 2 = n0 n n 0 Normal surface: degenerated sphere Eigen-polarization: arbitrary (Isotropic materials, cubic crystals) Uniaxial case = ≡ ≡ ≠ nx ny no and nz ne no wave equation: s2 + s2 2 ()2 − 2 1 − x y − sz = no n 2 2 2 0 n ne no Normal surface: sphere + ellipsoid with one common point in the z-direction Sections of the normal surface: s y positive uniaxial crystal sz no no ne ne sx sx Uniaxial case: continued s z S no E D s or k H sx ne Uniaxial case: continued Propagation ordinary ray extraordinary ray (z ≡ optical axis) degenerated case (equivalent to isotropic medium); k // z index n , E ⊥ k o index n , k ⊥ z e index n , E // z o E in the (xy) plane, index n(θ), k: angle θ with z E ⊥ k E in the (kz) plane, E ⋅ k ≠ 0 sin2 θ cos2 θ 1 with + = 2 2 2 ()θ ne no n Biaxial case < < nx ny nz Normal surface has a complicated form Biaxial case: continued Sections of the normal surface by the planes xy, xz, and yz: sy nz n sx = 0: x 2 2 2 1 s + s 1 s2 s − z y − z − y = 0 2 2 2 2 2 ny sx n nx n ny nz sz ny s = 0: y n 2 + 2 2 2 z 1 sx sz 1 sx sz − − − = 0 2 2 2 2 2 sx n ny n nz nx sz ny s = 0: z n 2 2 2 n z 1 s + s 1 s2 s x − x y − x − y = 0 2 2 2 2 2 sy n nz n ny nx Group velocity in anisotropic media Maxwell equations in the Fourier space: ∧ = ωµ ⋅ k E 0 H / H k ∧ H = −ωε⋅ E /⋅ E One can finally obtain: 2k ⋅()E ∧ H S ω = = k ⋅ E ⋅ D + H ⋅ B U This means: S v = ∇ ω = = v g k U e Birefringence Isotropic medium: incident plane wave, reflected plane wave Anisotropic medium: refracted wave is decomposed into the eigenmodes • Tangential components should be conserved on the interface: α = α = α ki sin i k1 sin 1 k2 sin 2 • k1 and k2 are not constant but depend on the propagation direction • Thus we get generalized Snell’s law for an ordinary and an extra- ordinary beam: α = α = ()α α ni sin i no sin 1 n 2 sin 2 • Analytic solution only for special cases • Numerical solution • Graphic solution Birefringence: graphic solution Birefringence: uniaxial crystals > < positive (ne no) negative (ne no) O E optical axis optical axis E O optical axis O optical axis E E O optical axis O optical axis E E O Indicatrix: ellipsoid of indices z 2 2 2 s x + y + z = 2 2 2 1 nx ny nz DII, NII DI, NI.