Crystal optics Peter Hertel Overview Permittivity tensor Crystal optics Maxwell's equations Isotropic medium Peter Hertel Birefringence Absorption University of Osnabr¨uck,Germany Drude model Lecture presented at APS, Nankai University, China http://www.home.uni-osnabrueck.de/phertel October/November 2011 • Crystal optics investigates the propagation of plane waves in a homogeneous medium • The susceptibility tensor is real and symmetric • three eigenvalues equal: isotropic medium • only two are equal: uniaxial medium • all three are different: biaxial medium • Birefringence • Absorption • Drude model Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations Isotropic medium Birefringence Absorption Drude model • The susceptibility tensor is real and symmetric • three eigenvalues equal: isotropic medium • only two are equal: uniaxial medium • all three are different: biaxial medium • Birefringence • Absorption • Drude model Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations • Crystal optics investigates the propagation of plane waves Isotropic in a homogeneous medium medium Birefringence Absorption Drude model • three eigenvalues equal: isotropic medium • only two are equal: uniaxial medium • all three are different: biaxial medium • Birefringence • Absorption • Drude model Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations • Crystal optics investigates the propagation of plane waves Isotropic in a homogeneous medium medium Birefringence • The susceptibility tensor is real and symmetric Absorption Drude model • only two are equal: uniaxial medium • all three are different: biaxial medium • Birefringence • Absorption • Drude model Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations • Crystal optics investigates the propagation of plane waves Isotropic in a homogeneous medium medium Birefringence • The susceptibility tensor is real and symmetric Absorption • three eigenvalues equal: isotropic medium Drude model • all three are different: biaxial medium • Birefringence • Absorption • Drude model Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations • Crystal optics investigates the propagation of plane waves Isotropic in a homogeneous medium medium Birefringence • The susceptibility tensor is real and symmetric Absorption • three eigenvalues equal: isotropic medium Drude model • only two are equal: uniaxial medium • Birefringence • Absorption • Drude model Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations • Crystal optics investigates the propagation of plane waves Isotropic in a homogeneous medium medium Birefringence • The susceptibility tensor is real and symmetric Absorption • three eigenvalues equal: isotropic medium Drude model • only two are equal: uniaxial medium • all three are different: biaxial medium • Absorption • Drude model Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations • Crystal optics investigates the propagation of plane waves Isotropic in a homogeneous medium medium Birefringence • The susceptibility tensor is real and symmetric Absorption • three eigenvalues equal: isotropic medium Drude model • only two are equal: uniaxial medium • all three are different: biaxial medium • Birefringence • Drude model Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations • Crystal optics investigates the propagation of plane waves Isotropic in a homogeneous medium medium Birefringence • The susceptibility tensor is real and symmetric Absorption • three eigenvalues equal: isotropic medium Drude model • only two are equal: uniaxial medium • all three are different: biaxial medium • Birefringence • Absorption Crystal optics Peter Hertel Overview Overview Permittivity tensor Maxwell's equations • Crystal optics investigates the propagation of plane waves Isotropic in a homogeneous medium medium Birefringence • The susceptibility tensor is real and symmetric Absorption • three eigenvalues equal: isotropic medium Drude model • only two are equal: uniaxial medium • all three are different: biaxial medium • Birefringence • Absorption • Drude model • for a sufficiently weak light wave, the polarization is linear in the electric field strength • however retarded, but local Z 1 Pi(t; x) = 0 dτ Gij(τ) Ej(t − τ; x) 0 • Einstein's summation convention: sum over j from 1 to 3 • Fourier transform Z d! −i!t F (t) = f(!) e 2π • Fourier transform F~(!) of F (t) here denoted by f(!) • the Fourier transform of the dispacement is di(!; x) = 0 ij(!) ej(!; x) • ij(!) = δij + χij(!) where χij = G~ij Crystal optics Peter Hertel Linear medium Overview Permittivity tensor Maxwell's equations Isotropic medium Birefringence Absorption Drude model • however retarded, but local Z 1 Pi(t; x) = 0 dτ Gij(τ) Ej(t − τ; x) 0 • Einstein's summation convention: sum over j from 1 to 3 • Fourier transform Z d! −i!t F (t) = f(!) e 2π • Fourier transform F~(!) of F (t) here denoted by f(!) • the Fourier transform of the dispacement is di(!; x) = 0 ij(!) ej(!; x) • ij(!) = δij + χij(!) where χij = G~ij Crystal optics Peter Hertel Linear medium Overview Permittivity • for a sufficiently weak light wave, the polarization is linear tensor in the electric field strength Maxwell's equations Isotropic medium Birefringence Absorption Drude model • Einstein's summation convention: sum over j from 1 to 3 • Fourier transform Z d! −i!t F (t) = f(!) e 2π • Fourier transform F~(!) of F (t) here denoted by f(!) • the Fourier transform of the dispacement is di(!; x) = 0 ij(!) ej(!; x) • ij(!) = δij + χij(!) where χij = G~ij Crystal optics Peter Hertel Linear medium Overview Permittivity • for a sufficiently weak light wave, the polarization is linear tensor in the electric field strength Maxwell's equations • however retarded, but local Isotropic Z 1 medium Pi(t; x) = 0 dτ Gij(τ) Ej(t − τ; x) Birefringence 0 Absorption Drude model • Fourier transform Z d! −i!t F (t) = f(!) e 2π • Fourier transform F~(!) of F (t) here denoted by f(!) • the Fourier transform of the dispacement is di(!; x) = 0 ij(!) ej(!; x) • ij(!) = δij + χij(!) where χij = G~ij Crystal optics Peter Hertel Linear medium Overview Permittivity • for a sufficiently weak light wave, the polarization is linear tensor in the electric field strength Maxwell's equations • however retarded, but local Isotropic Z 1 medium Pi(t; x) = 0 dτ Gij(τ) Ej(t − τ; x) Birefringence 0 Absorption • Einstein's summation convention: sum over j from 1 to 3 Drude model • Fourier transform F~(!) of F (t) here denoted by f(!) • the Fourier transform of the dispacement is di(!; x) = 0 ij(!) ej(!; x) • ij(!) = δij + χij(!) where χij = G~ij Crystal optics Peter Hertel Linear medium Overview Permittivity • for a sufficiently weak light wave, the polarization is linear tensor in the electric field strength Maxwell's equations • however retarded, but local Isotropic Z 1 medium Pi(t; x) = 0 dτ Gij(τ) Ej(t − τ; x) Birefringence 0 Absorption • Einstein's summation convention: sum over j from 1 to 3 Drude model • Fourier transform Z d! −i!t F (t) = f(!) e 2π • the Fourier transform of the dispacement is di(!; x) = 0 ij(!) ej(!; x) • ij(!) = δij + χij(!) where χij = G~ij Crystal optics Peter Hertel Linear medium Overview Permittivity • for a sufficiently weak light wave, the polarization is linear tensor in the electric field strength Maxwell's equations • however retarded, but local Isotropic Z 1 medium Pi(t; x) = 0 dτ Gij(τ) Ej(t − τ; x) Birefringence 0 Absorption • Einstein's summation convention: sum over j from 1 to 3 Drude model • Fourier transform Z d! −i!t F (t) = f(!) e 2π • Fourier transform F~(!) of F (t) here denoted by f(!) • ij(!) = δij + χij(!) where χij = G~ij Crystal optics Peter Hertel Linear medium Overview Permittivity • for a sufficiently weak light wave, the polarization is linear tensor in the electric field strength Maxwell's equations • however retarded, but local Isotropic Z 1 medium Pi(t; x) = 0 dτ Gij(τ) Ej(t − τ; x) Birefringence 0 Absorption • Einstein's summation convention: sum over j from 1 to 3 Drude model • Fourier transform Z d! −i!t F (t) = f(!) e 2π • Fourier transform F~(!) of F (t) here denoted by f(!) • the Fourier transform of the dispacement is di(!; x) = 0 ij(!) ej(!; x) Crystal optics Peter Hertel Linear medium Overview Permittivity • for a sufficiently weak light wave, the polarization is linear tensor in the electric field strength Maxwell's equations • however retarded, but local Isotropic Z 1 medium Pi(t; x) = 0 dτ Gij(τ) Ej(t − τ; x) Birefringence 0 Absorption • Einstein's summation convention: sum over j from 1 to 3 Drude model • Fourier transform Z d! −i!t F (t) = f(!) e 2π • Fourier transform F~(!) of F (t) here denoted by f(!) • the Fourier transform of the dispacement is di(!; x) = 0 ij(!) ej(!; x) • ij(!) = δij + χij(!) where χij = G~ij • decompose 0 00 ij = ij + i ij • refractive part ∗ ij + 0 = ji ij 2 • absorptive part ∗ ij − 00 = ji ij 2i ∗ • both are Hermitian: Aij = Aji • we shall see later while absorptive part causes absorption • i. e. the conversion of field energy into internal energy Crystal optics Peter Hertel Refraction and absorption Overview Permittivity tensor Maxwell's equations Isotropic medium Birefringence Absorption Drude model • refractive part ∗ ij + 0 = ji ij 2 • absorptive part ∗ ij − 00 = ji ij 2i ∗ • both are Hermitian: Aij = Aji • we shall see later while absorptive part causes absorption • i. e. the conversion of field energy into