Crystal Optics

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

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

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

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 ∞ 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 dω −iωt F (t) = f(ω) e 2π • Fourier transform F˜(ω) of F (t) here denoted by f(ω) • the Fourier transform of the dispacement is

di(ω, x) = 0 ij(ω) ej(ω, x)

• ij(ω) = δij + χij(ω) 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 ∞ 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 dω −iωt F (t) = f(ω) e 2π • Fourier transform F˜(ω) of F (t) here denoted by f(ω) • the Fourier transform of the dispacement is

di(ω, x) = 0 ij(ω) ej(ω, x)

• ij(ω) = δij + χij(ω) 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 dω −iωt F (t) = f(ω) e 2π • Fourier transform F˜(ω) of F (t) here denoted by f(ω) • the Fourier transform of the dispacement is

di(ω, x) = 0 ij(ω) ej(ω, x)

• ij(ω) = δij + χij(ω) where χij = G˜ij

Crystal optics Peter Hertel Linear medium

Drude model • Fourier transform Z dω −iωt F (t) = f(ω) e 2π • Fourier transform F˜(ω) of F (t) here denoted by f(ω) • the Fourier transform of the dispacement is

di(ω, x) = 0 ij(ω) ej(ω, x)

• ij(ω) = δij + χij(ω) 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 ∞ 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˜(ω) of F (t) here denoted by f(ω) • the Fourier transform of the dispacement is

di(ω, x) = 0 ij(ω) ej(ω, x)

• ij(ω) = δij + χij(ω) 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 ∞ 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 dω −iωt F (t) = f(ω) e 2π • the Fourier transform of the dispacement is

di(ω, x) = 0 ij(ω) ej(ω, x)

• ij(ω) = δij + χij(ω) 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 ∞ 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 dω −iωt F (t) = f(ω) e 2π • Fourier transform F˜(ω) of F (t) here denoted by f(ω) • ij(ω) = δij + χij(ω) 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 ∞ 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 dω −iωt F (t) = f(ω) e 2π • Fourier transform F˜(ω) of F (t) here denoted by f(ω) • the Fourier transform of the dispacement is

di(ω, x) = 0 ij(ω) ej(ω, 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 ∞ 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 dω −iωt F (t) = f(ω) e 2π • Fourier transform F˜(ω) of F (t) here denoted by f(ω) • the Fourier transform of the dispacement is

di(ω, x) = 0 ij(ω) ej(ω, x)

• ij(ω) = δij + χij(ω) 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 internal energy

Crystal optics Peter Hertel Refraction and absorption

Overview

Permittivity tensor • decompose Maxwell’s equations 0 00 ij = ij + i ij Isotropic medium

Birefringence

Absorption

Drude model • 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 • decompose Maxwell’s equations 0 00 ij = ij + i ij Isotropic medium • refractive part Birefringence ∗ 0 ij + ji Absorption = ij 2 Drude model ∗ • 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

Crystal optics Peter Hertel Refraction and absorption

Overview

Crystal optics Peter Hertel Refraction and absorption

Overview

Permittivity tensor • decompose Maxwell’s equations 0 00 ij = ij + i ij Isotropic medium • refractive part Birefringence ∗ 0 ij + ji Absorption = ij 2 Drude model • absorptive part ∗ ij − 00 = ji ij 2i ∗ • both are Hermitian: Aij = Aji • we shall see later while absorptive part causes absorption Crystal optics Peter Hertel Refraction and absorption

Overview

Permittivity tensor • decompose Maxwell’s equations 0 00 ij = ij + i ij Isotropic medium • refractive part Birefringence ∗ 0 ij + ji Absorption = ij 2 Drude model • 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 00 • assume ij(ω) ≈ 0 for the frequencies under discussion 0 • then ij = ij is hermitian • recall Onsager’s relations

ij(ω; E, B) = ji(ω; E, −B) • where E, B are static external ﬁelds • without external induction ﬁeld ∗ ji = ij = ij • no absorption, no external induction: ij is a real symmetric matrix

• ij can be diagonalized by an orthogonal matrix

Crystal optics Peter Hertel Transparent medium

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model 0 • then ij = ij is hermitian • recall Onsager’s relations

• ij can be diagonalized by an orthogonal matrix

Crystal optics Peter Hertel Transparent medium

Overview

Permittivity tensor 00 Maxwell’s • assume ij(ω) ≈ 0 for the frequencies under discussion equations

Isotropic medium

Birefringence

Absorption

Drude model • recall Onsager’s relations

• ij can be diagonalized by an orthogonal matrix

Crystal optics Peter Hertel Transparent medium

Overview

Permittivity tensor 00 Maxwell’s • assume ij(ω) ≈ 0 for the frequencies under discussion equations 0 • then ij = is hermitian Isotropic ij medium

Birefringence

Absorption

Drude model • where E, B are static external ﬁelds • without external induction ﬁeld ∗ ji = ij = ij • no absorption, no external induction: ij is a real symmetric matrix

• ij can be diagonalized by an orthogonal matrix

Crystal optics Peter Hertel Transparent medium

Overview

Permittivity tensor 00 Maxwell’s • assume ij(ω) ≈ 0 for the frequencies under discussion equations 0 • then ij = is hermitian Isotropic ij medium • recall Onsager’s relations Birefringence ij(ω; E, B) = ji(ω; E, −B) Absorption

Drude model • without external induction ﬁeld ∗ ji = ij = ij • no absorption, no external induction: ij is a real symmetric matrix

• ij can be diagonalized by an orthogonal matrix

Crystal optics Peter Hertel Transparent medium

Overview

Drude model • where E, B are static external ﬁelds • no absorption, no external induction: ij is a real symmetric matrix

• ij can be diagonalized by an orthogonal matrix

Crystal optics Peter Hertel Transparent medium

Overview

Drude model • where E, B are static external ﬁelds • without external induction ﬁeld ∗ ji = ij = ij • ij can be diagonalized by an orthogonal matrix

Crystal optics Peter Hertel Transparent medium

Overview

Drude model • where E, B are static external ﬁelds • without external induction ﬁeld ∗ ji = ij = ij • no absorption, no external induction: ij is a real symmetric matrix Crystal optics Peter Hertel Transparent medium

Overview

Drude model • where E, B are static external ﬁelds • without external induction ﬁeld ∗ ji = ij = ij • no absorption, no external induction: ij is a real symmetric matrix

• ij can be diagonalized by an orthogonal matrix 1 1 = 2 = 3 isotropic medium (glass) 1 2 3 2 = 6= uniaxial (LiNbO3) 1 2 3 3 < < biaxial (KNbO3)

there is a Cartesian coordinate system such that  1 0 0  2 ij =  0 0  0 0 3 There are three cases

Crystal optics Peter Hertel Main axes

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model 1 1 = 2 = 3 isotropic medium (glass) 1 2 3 2 = 6= uniaxial (LiNbO3) 1 2 3 3 < < biaxial (KNbO3)

There are three cases

Crystal optics Peter Hertel Main axes

Overview

Permittivity tensor

Maxwell’s equations there is a Cartesian coordinate system such that Isotropic  1  medium 0 0 2 Birefringence ij =  0 0  3 Absorption 0 0 Drude model 1 2 3 2 = 6= uniaxial (LiNbO3) 1 2 3 3 < < biaxial (KNbO3)

Crystal optics Peter Hertel Main axes

Overview

Permittivity tensor

Crystal optics Peter Hertel Main axes

Overview

Permittivity tensor

Maxwell’s equations there is a Cartesian coordinate system such that Isotropic  1  medium 0 0 2 Birefringence ij =  0 0  3 Absorption 0 0 Drude model There are three cases 1 1 = 2 = 3 isotropic medium (glass) 1 2 3 2 = 6= uniaxial (LiNbO3) Crystal optics Peter Hertel Main axes

Overview

Permittivity tensor

• k × e = ωµ0h

• k × h = −ω0 e √ • With c = 1/ 0µ0 ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Crystal optics Peter Hertel Maxwell’s equations

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • all ﬁelds are of the form −iωt ik · x F (t, x) = f e e

• k × e = ωµ0h

• k × h = −ω0 e √ • With c = 1/ 0µ0 ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Crystal optics Peter Hertel Maxwell’s equations Overview • assume plane waves with ﬁxed angular frequency Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • k × e = ωµ0h

• k × h = −ω0 e √ • With c = 1/ 0µ0 ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Crystal optics Peter Hertel Maxwell’s equations Overview • assume plane waves with ﬁxed angular frequency Permittivity tensor • all ﬁelds are of the form Maxwell’s −iωt ik · x equations F (t, x) = f e e Isotropic medium

Birefringence

Absorption

Drude model • k × h = −ω0 e √ • With c = 1/ 0µ0 ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Birefringence

Absorption

Drude model √ • With c = 1/ 0µ0 ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Birefringence • k × h = −ω0 e Absorption

Drude model • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Birefringence • k × h = −ω0 e Absorption √ • With c = 1/ 0µ0 Drude model ω2 (k × k × e) = − e i c2 ij j • k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Birefringence • k × h = −ω0 e Absorption √ • With c = 1/ 0µ0 Drude model ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Birefringence • k × h = −ω0 e Absorption √ • With c = 1/ 0µ0 Drude model ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj

Birefringence • k × h = −ω0 e Absorption √ • With c = 1/ 0µ0 Drude model ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ Crystal optics Peter Hertel Maxwell’s equations Overview • assume plane waves with ﬁxed angular frequency Permittivity tensor • all ﬁelds are of the form Maxwell’s −iωt ik · x equations F (t, x) = f e e Isotropic medium • k × e = ωµ0h

Birefringence • k × h = −ω0 e Absorption √ • With c = 1/ 0µ0 Drude model ω2 (k × k × e) = − e i c2 ij j • k0 = ω/c vacuum wave number

• k = nk0kˆ refractive index n, propagation direction kˆ • e = eeˆ polarization vector eˆ • to be solved is the mode equation 2 n (kˆ × kˆ × eˆ)i = −ijeˆj • c × b × a = (c · a)b − (c · b)a • kˆ × kˆ × eˆ = (kˆ · eˆ)kˆ − eˆ • mode equation can be written as 2 n (eˆ − (kˆ · eˆ)kˆ)i = ijeˆj • no solution for kˆ k eˆ • therefore kˆ ⊥ eˆ • electromagnetic plane waves in a homogeneous medium are alway transversally polarized

Crystal optics Peter Hertel Maxwell’s equations ctd.

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • kˆ × kˆ × eˆ = (kˆ · eˆ)kˆ − eˆ • mode equation can be written as 2 n (eˆ − (kˆ · eˆ)kˆ)i = ijeˆj • no solution for kˆ k eˆ • therefore kˆ ⊥ eˆ • electromagnetic plane waves in a homogeneous medium are alway transversally polarized

Crystal optics Peter Hertel Maxwell’s equations ctd.

Overview

Permittivity tensor

Maxwell’s equations • c × b × a = (c · a)b − (c · b)a Isotropic medium

Birefringence

Absorption

Drude model • mode equation can be written as 2 n (eˆ − (kˆ · eˆ)kˆ)i = ijeˆj • no solution for kˆ k eˆ • therefore kˆ ⊥ eˆ • electromagnetic plane waves in a homogeneous medium are alway transversally polarized

Crystal optics Peter Hertel Maxwell’s equations ctd.

Overview

Permittivity tensor

Maxwell’s equations • c × b × a = (c · a)b − (c · b)a Isotropic ˆ ˆ ˆ ˆ medium • k × k × eˆ = (k · eˆ)k − eˆ

Birefringence

Absorption

Drude model • no solution for kˆ k eˆ • therefore kˆ ⊥ eˆ • electromagnetic plane waves in a homogeneous medium are alway transversally polarized

Crystal optics Peter Hertel Maxwell’s equations ctd.

Overview

Permittivity tensor

Maxwell’s equations • c × b × a = (c · a)b − (c · b)a Isotropic ˆ ˆ ˆ ˆ medium • k × k × eˆ = (k · eˆ)k − eˆ Birefringence • mode equation can be written as Absorption 2 n (eˆ − (kˆ · eˆ)kˆ)i = ijeˆj Drude model • therefore kˆ ⊥ eˆ • electromagnetic plane waves in a homogeneous medium are alway transversally polarized

Crystal optics Peter Hertel Maxwell’s equations ctd.

Overview

Permittivity tensor

Maxwell’s equations • c × b × a = (c · a)b − (c · b)a Isotropic ˆ ˆ ˆ ˆ medium • k × k × eˆ = (k · eˆ)k − eˆ Birefringence • mode equation can be written as Absorption 2 n (eˆ − (kˆ · eˆ)kˆ)i = ijeˆj Drude model • no solution for kˆ k eˆ • electromagnetic plane waves in a homogeneous medium are alway transversally polarized

Crystal optics Peter Hertel Maxwell’s equations ctd.

Overview

Permittivity tensor

Maxwell’s equations • c × b × a = (c · a)b − (c · b)a Isotropic ˆ ˆ ˆ ˆ medium • k × k × eˆ = (k · eˆ)k − eˆ Birefringence • mode equation can be written as Absorption 2 n (eˆ − (kˆ · eˆ)kˆ)i = ijeˆj Drude model • no solution for kˆ k eˆ • therefore kˆ ⊥ eˆ Crystal optics Peter Hertel Maxwell’s equations ctd.

Overview

Permittivity tensor

Maxwell’s equations • c × b × a = (c · a)b − (c · b)a Isotropic ˆ ˆ ˆ ˆ medium • k × k × eˆ = (k · eˆ)k − eˆ Birefringence • mode equation can be written as Absorption 2 n (eˆ − (kˆ · eˆ)kˆ)i = ijeˆj Drude model • no solution for kˆ k eˆ • therefore kˆ ⊥ eˆ • electromagnetic plane waves in a homogeneous medium are alway transversally polarized • if 1 = 2 = 3 =

• we ﬁnd ij = δij • this is true for an arbitrary Cartesian coordinate system • we say the medium is optically isotropic • the mode equation reads n2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • we ﬁnd ij = δij • this is true for an arbitrary Cartesian coordinate system • we say the medium is optically isotropic • the mode equation reads n2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • this is true for an arbitrary Cartesian coordinate system • we say the medium is optically isotropic • the mode equation reads n2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations

Isotropic medium

Birefringence

Absorption

Drude model • we say the medium is optically isotropic • the mode equation reads n2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium

Birefringence

Absorption

Drude model • the mode equation reads n2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium • we say the medium is optically isotropic Birefringence

Absorption

Drude model • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium • we say the medium is optically isotropic Birefringence

Absorption • the mode equation reads 2 Drude model n (eˆ − (kˆ · eˆ) kˆ) = eˆ √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium • we say the medium is optically isotropic Birefringence

Absorption • the mode equation reads 2 Drude model n (eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium • we say the medium is optically isotropic Birefringence

Absorption • the mode equation reads 2 Drude model n (eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium • we say the medium is optically isotropic Birefringence

Absorption • the mode equation reads 2 Drude model n (eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium • we say the medium is optically isotropic Birefringence

Crystal optics Peter Hertel Optically isotropic medium

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium • we say the medium is optically isotropic Birefringence

Overview 1 2 3 Permittivity • if = = = tensor

Maxwell’s • we ﬁnd ij = δij equations • Isotropic this is true for an arbitrary Cartesian coordinate system medium • we say the medium is optically isotropic Birefringence

Absorption • the mode equation reads 2 Drude model n (eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ k eˆ: no solution √ • kˆ ⊥ eˆ: n = • any polarization eˆ is allowed. • any orthogonal propagation direction kˆ is allowed • hˆ = kˆ × eˆ • kˆ, eˆ, hˆ is right handed set of orthogonal unit vectors • Two eigenvalues of ij are equal, we call them ordinary • the extraordinary eigenvalue is diﬀerent • it belongs to the optical axis (here zˆ)  o 0 0  o ij =  0 0  0 0 e • ordinary beam is polarized ⊥ optical axis √ • eˆ = cos φ xˆ + sin φ yˆ, kˆ = zˆ, no = o • extraordinary beam is polarized k optical axis √ • eˆ = zˆ, kˆ = cos α xˆ + sin α yˆ, ne = e

Crystal optics Peter Hertel Optical axis

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • the extraordinary eigenvalue is diﬀerent • it belongs to the optical axis (here zˆ)  o 0 0  o ij =  0 0  0 0 e • ordinary beam is polarized ⊥ optical axis √ • eˆ = cos φ xˆ + sin φ yˆ, kˆ = zˆ, no = o • extraordinary beam is polarized k optical axis √ • eˆ = zˆ, kˆ = cos α xˆ + sin α yˆ, ne = e

Crystal optics Peter Hertel Optical axis

Overview

Permittivity tensor • Maxwell’s Two eigenvalues of ij are equal, we call them ordinary equations

Isotropic medium

Birefringence

Absorption

Drude model • it belongs to the optical axis (here zˆ)  o 0 0  o ij =  0 0  0 0 e • ordinary beam is polarized ⊥ optical axis √ • eˆ = cos φ xˆ + sin φ yˆ, kˆ = zˆ, no = o • extraordinary beam is polarized k optical axis √ • eˆ = zˆ, kˆ = cos α xˆ + sin α yˆ, ne = e

Crystal optics Peter Hertel Optical axis

Overview

Permittivity tensor • Maxwell’s Two eigenvalues of ij are equal, we call them ordinary equations • the extraordinary eigenvalue is diﬀerent Isotropic medium

Birefringence

Absorption

Drude model • ordinary beam is polarized ⊥ optical axis √ • eˆ = cos φ xˆ + sin φ yˆ, kˆ = zˆ, no = o • extraordinary beam is polarized k optical axis √ • eˆ = zˆ, kˆ = cos α xˆ + sin α yˆ, ne = e

Crystal optics Peter Hertel Optical axis

Overview

Crystal optics Peter Hertel Optical axis

Overview

Crystal optics Peter Hertel Optical axis

Overview

Crystal optics Peter Hertel Optical axis

Overview

Permittivity tensor • Maxwell’s Two eigenvalues of ij are equal, we call them ordinary equations • the extraordinary eigenvalue is different Isotropic medium • it belongs to the optical axis (here zˆ) Birefringence  o 0 0  Absorption = 0 o 0 Drude model ij   0 0 e • ordinary beam is polarized ⊥ optical axis √ • eˆ = cos φ xˆ + sin φ yˆ, kˆ = zˆ, no = o • extraordinary beam is polarized k optical axis √ • eˆ = zˆ, kˆ = cos α xˆ + sin α yˆ, ne = e • what happens if a beam is polarized neither parallel nor perpendicular to optical axis? • e. g. if it is unpolarized • when entering the medium it splits into an ordinary and an extraordinary beam • which propagate with different refractive index • they will leave the the medium at different locations • being polarized • double refraction , or birefringence • calcite (at which birefringence was discovered) • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • e. g. if it is unpolarized • when entering the medium it splits into an ordinary and an extraordinary beam • which propagate with diﬀerent refractive index • they will leave the the medium at diﬀerent locations • being polarized • double refraction , or birefringence • calcite (at which birefringence was discovered) • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium

Birefringence

Absorption

Drude model • when entering the medium it splits into an ordinary and an extraordinary beam • which propagate with diﬀerent refractive index • they will leave the the medium at diﬀerent locations • being polarized • double refraction , or birefringence • calcite (at which birefringence was discovered) • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium • e. g. if it is unpolarized

Birefringence

Absorption

Drude model • which propagate with diﬀerent refractive index • they will leave the the medium at diﬀerent locations • being polarized • double refraction , or birefringence • calcite (at which birefringence was discovered) • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium • e. g. if it is unpolarized Birefringence • when entering the medium it splits into an ordinary and an Absorption extraordinary beam Drude model • they will leave the the medium at diﬀerent locations • being polarized • double refraction , or birefringence • calcite (at which birefringence was discovered) • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium • e. g. if it is unpolarized Birefringence • when entering the medium it splits into an ordinary and an Absorption extraordinary beam Drude model • which propagate with diﬀerent refractive index • being polarized • double refraction , or birefringence • calcite (at which birefringence was discovered) • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium • e. g. if it is unpolarized Birefringence • when entering the medium it splits into an ordinary and an Absorption extraordinary beam Drude model • which propagate with diﬀerent refractive index • they will leave the the medium at diﬀerent locations • double refraction , or birefringence • calcite (at which birefringence was discovered) • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium • e. g. if it is unpolarized Birefringence • when entering the medium it splits into an ordinary and an Absorption extraordinary beam Drude model • which propagate with diﬀerent refractive index • they will leave the the medium at diﬀerent locations • being polarized • calcite (at which birefringence was discovered) • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium • e. g. if it is unpolarized Birefringence • when entering the medium it splits into an ordinary and an Absorption extraordinary beam Drude model • which propagate with diﬀerent refractive index • they will leave the the medium at diﬀerent locations • being polarized • double refraction , or birefringence • elastooptics

Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium • e. g. if it is unpolarized Birefringence • when entering the medium it splits into an ordinary and an Absorption extraordinary beam Drude model • which propagate with diﬀerent refractive index • they will leave the the medium at diﬀerent locations • being polarized • double refraction , or birefringence • calcite (at which birefringence was discovered) Crystal optics Peter Hertel Birefringence

Overview

Permittivity tensor • what happens if a beam is polarized neither parallel nor Maxwell’s equations perpendicular to optical axis? Isotropic medium • e. g. if it is unpolarized Birefringence • when entering the medium it splits into an ordinary and an Absorption extraordinary beam Drude model • which propagate with different refractive index • they will leave the the medium at different locations • being polarized • double refraction , or birefringence • calcite (at which birefringence was discovered) • elastooptics Crystal optics cˆ = zˆ = eê Peter Hertel Overview eˆ(0) Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption Drude model kˆ yˆ = eˆo

eˆ(L)

80mm Crystal optics A 45 degree polarized wave enters the crystal and leaves it at -45 degrees polarization. Peter Hertel

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model Crystal optics

Peter Hertel

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model

Birefringence, or double refraction, by calcite Crystal optics

Peter Hertel

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model

Normally isotropic polymers become birefringent when stressed. Observed with a polarizer. • Only the ordinary beam may be unpolarized • optically biaxial media have three different eigenvalues of permittivity tensor ij • correspondingly three orthogonal directions  1 0 0  2 ij =  0 0  0 0 3 • beams polarized along these√ axes propagate with different refractive indexes na = a • rather difficult to show that there are two optical axes

Crystal optics Peter Hertel Remarks

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • optically biaxial media have three different eigenvalues of permittivity tensor ij • correspondingly three orthogonal directions  1 0 0  2 ij =  0 0  0 0 3 • beams polarized along these√ axes propagate with different refractive indexes na = a • rather difficult to show that there are two optical axes

Crystal optics Peter Hertel Remarks

Overview

Permittivity tensor Maxwell’s • Only the ordinary beam may be unpolarized equations

Isotropic medium

Birefringence

Absorption

Drude model • correspondingly three orthogonal directions  1 0 0  2 ij =  0 0  0 0 3 • beams polarized along these√ axes propagate with diﬀerent refractive indexes na = a • rather diﬃcult to show that there are two optical axes

Crystal optics Peter Hertel Remarks

Overview

Permittivity tensor Maxwell’s • Only the ordinary beam may be unpolarized equations Isotropic • optically biaxial media have three diﬀerent eigenvalues of medium permittivity tensor ij Birefringence

Absorption

Drude model • beams polarized along these√ axes propagate with diﬀerent refractive indexes na = a • rather diﬃcult to show that there are two optical axes

Crystal optics Peter Hertel Remarks

Overview

Absorption • correspondingly three orthogonal directions  1  Drude model 0 0 2 ij =  0 0  0 0 3 • rather diﬃcult to show that there are two optical axes

Crystal optics Peter Hertel Remarks

Overview

Absorption • correspondingly three orthogonal directions  1  Drude model 0 0 2 ij =  0 0  0 0 3 • beams polarized along these√ axes propagate with diﬀerent refractive indexes na = a Crystal optics Peter Hertel Remarks

Overview

Absorption • correspondingly three orthogonal directions  1  Drude model 0 0 2 ij =  0 0  0 0 3 • beams polarized along these√ axes propagate with different refractive indexes na = a • rather difficult to show that there are two optical axes • for simplicity, assume isotropic medium • = 0 + i 00 • nearly transparent medium, 00 0 • recall n¯2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ ⊥ eˆ remains true √ • n¯ = 0 + i 00 is complex √ • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one finds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e • α is absorption constant

Crystal optics Peter Hertel Absorption

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • = 0 + i 00 • nearly transparent medium, 00 0 • recall n¯2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ ⊥ eˆ remains true √ • n¯ = 0 + i 00 is complex √ • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one ﬁnds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e • α is absorption constant

Crystal optics Peter Hertel Absorption Overview • for simplicity, assume isotropic medium Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • nearly transparent medium, 00 0 • recall n¯2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ ⊥ eˆ remains true √ • n¯ = 0 + i 00 is complex √ • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one ﬁnds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e • α is absorption constant

Crystal optics Peter Hertel Absorption Overview • for simplicity, assume isotropic medium Permittivity tensor • = 0 + i 00 Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • recall n¯2(eˆ − (kˆ · eˆ) kˆ) = eˆ • kˆ ⊥ eˆ remains true √ • n¯ = 0 + i 00 is complex √ • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one ﬁnds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e • α is absorption constant

Crystal optics Peter Hertel Absorption Overview • for simplicity, assume isotropic medium Permittivity tensor • = 0 + i 00 Maxwell’s 00 0 equations • nearly transparent medium,

Isotropic medium

Birefringence

Absorption

Drude model • kˆ ⊥ eˆ remains true √ • n¯ = 0 + i 00 is complex √ • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one ﬁnds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e • α is absorption constant

Birefringence

Absorption

Drude model √ • n¯ = 0 + i 00 is complex √ • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one ﬁnds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e • α is absorption constant

Absorption

Drude model √ • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one ﬁnds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e • α is absorption constant

Crystal optics Peter Hertel Absorption Overview • for simplicity, assume isotropic medium Permittivity tensor • = 0 + i 00 Maxwell’s 00 0 equations • nearly transparent medium, Isotropic • recall n¯2(eˆ − (kˆ · eˆ) kˆ) = eˆ medium • kˆ ⊥ eˆ remains true Birefringence √ Absorption • n¯ = 0 + i 00 is complex √ Drude model • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one ﬁnds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e Crystal optics Peter Hertel Absorption Overview • for simplicity, assume isotropic medium Permittivity tensor • = 0 + i 00 Maxwell’s 00 0 equations • nearly transparent medium, Isotropic • recall n¯2(eˆ − (kˆ · eˆ) kˆ) = eˆ medium • kˆ ⊥ eˆ remains true Birefringence √ Absorption • n¯ = 0 + i 00 is complex √ Drude model • With n = 0 one may write 00 n¯ ≈ n + i 2n 00 • with α = k0/n and z = kˆ · x one ﬁnds −iωt ink z −αz/2 E(t, x) = E(0, 0) e e 0 e −αz • S ∝ |E|2, S(z) = S(0) e • α is absorption constant • consider typical electron with mass m and charge q = −e • x is deviation from equilibrium position • damped harmonic oscillatation m(x¨ + Γx˙ + Ω2x) = qE • Fourier transform it m(−ω2 − iΓ + Ω2)x˜ = qE˜ • polarization is P = Nqx with electron density N • susceptibility Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω • static susceptibility Nq2 χ(0) = 2 mΩ 0

Crystal optics Peter Hertel Drude model

Overview

Permittivity tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • x is deviation from equilibrium position • damped harmonic oscillatation m(x¨ + Γx˙ + Ω2x) = qE • Fourier transform it m(−ω2 − iΓ + Ω2)x˜ = qE˜ • polarization is P = Nqx with electron density N • susceptibility Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω • static susceptibility Nq2 χ(0) = 2 mΩ 0

Crystal optics Peter Hertel Drude model

Overview

Permittivity • consider typical electron with mass m and charge q = −e tensor

Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • damped harmonic oscillatation m(x¨ + Γx˙ + Ω2x) = qE • Fourier transform it m(−ω2 − iΓ + Ω2)x˜ = qE˜ • polarization is P = Nqx with electron density N • susceptibility Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω • static susceptibility Nq2 χ(0) = 2 mΩ 0

Crystal optics Peter Hertel Drude model

Overview

Permittivity • consider typical electron with mass m and charge q = −e tensor • x is deviation from equilibrium position Maxwell’s equations

Isotropic medium

Birefringence

Absorption

Drude model • Fourier transform it m(−ω2 − iΓ + Ω2)x˜ = qE˜ • polarization is P = Nqx with electron density N • susceptibility Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω • static susceptibility Nq2 χ(0) = 2 mΩ 0

Crystal optics Peter Hertel Drude model

Overview

Absorption

Drude model • polarization is P = Nqx with electron density N • susceptibility Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω • static susceptibility Nq2 χ(0) = 2 mΩ 0

Crystal optics Peter Hertel Drude model

Overview

Permittivity • consider typical electron with mass m and charge q = −e tensor • x is deviation from equilibrium position Maxwell’s equations • damped harmonic oscillatation Isotropic 2 medium m(x¨ + Γx˙ + Ω x) = qE Birefringence • Fourier transform it Absorption 2 2 ˜ Drude model m(−ω − iΓ + Ω )x˜ = qE • susceptibility Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω • static susceptibility Nq2 χ(0) = 2 mΩ 0

Crystal optics Peter Hertel Drude model

Overview

Permittivity • consider typical electron with mass m and charge q = −e tensor • x is deviation from equilibrium position Maxwell’s equations • damped harmonic oscillatation Isotropic 2 medium m(x¨ + Γx˙ + Ω x) = qE Birefringence • Fourier transform it Absorption 2 2 ˜ Drude model m(−ω − iΓ + Ω )x˜ = qE • polarization is P = Nqx with electron density N • static susceptibility Nq2 χ(0) = 2 mΩ 0

Crystal optics Peter Hertel Drude model

Overview

Permittivity • consider typical electron with mass m and charge q = −e tensor • x is deviation from equilibrium position Maxwell’s equations • damped harmonic oscillatation Isotropic 2 medium m(x¨ + Γx˙ + Ω x) = qE Birefringence • Fourier transform it Absorption 2 2 ˜ Drude model m(−ω − iΓ + Ω )x˜ = qE • polarization is P = Nqx with electron density N • susceptibility Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω Crystal optics Peter Hertel Drude model

Overview

Permittivity • consider typical electron with mass m and charge q = −e tensor • x is deviation from equilibrium position Maxwell’s equations • damped harmonic oscillatation Isotropic 2 medium m(x¨ + Γx˙ + Ω x) = qE Birefringence • Fourier transform it Absorption 2 2 ˜ Drude model m(−ω − iΓ + Ω )x˜ = qE • polarization is P = Nqx with electron density N • susceptibility Ω2 χ(ω) = χ(0) Ω2 − ω2 − iΓω • static susceptibility Nq2 χ(0) = 2 mΩ 0 Crystal optics 10 Peter Hertel

Overview

Permittivity tensor

Maxwell’s equations 5

Isotropic medium

Birefringence

Absorption

Drude model 0

−5 0 0.5 1 1.5 2 2.5 3

Real (blue) and imaginary part (red) of susceptibility χ(ω) relative to χ(0) over ω/Ω. Γ/Ω = 0.1