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   E   x y z x y ()x z   x   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   E   x y z x y ()x z   x   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