Dielectric permittivity

Dielectric permittivity Peter Hertel Overview Maxwell’s equations Dielectric permittivity Definition KramersKronig Relation Peter Hertel Onsager Relation University of Osnabrück, Germany Summary Lecture presented at Nankai University, China http://www.home.uni-osnabrueck.de/phertel October/November 2011 Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Make it as simple as possible, but not simpler Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Overview Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Overview • Optics deals with the interaction of light with matter. Dielectric permittivity Overview Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s equations. Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Overview • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s equations. • The Lorentz force on charged particles describes the interaction of light with matter. Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Overview • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s equations. • The Lorentz force on charged particles describes the interaction of light with matter. • We recapitulate Maxwell’s equation in the presence of matter and specialize to a homogeneous non-magnetic linear medium. Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Overview • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s equations. • The Lorentz force on charged particles describes the interaction of light with matter. • We recapitulate Maxwell’s equation in the presence of matter and specialize to a homogeneous non-magnetic linear medium. • We formulate the retarded response of matter to a perturbation by an electric field Dielectric permittivity Overview Peter Hertel Overview Maxwell’s equations • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s Definition KramersKronig Relation Onsager Relation • • Summary • • equations. The Lorentz force on charged particles describes the interaction of light with matter. We recapitulate Maxwell’s equation in the presence of matter and specialize to a homogeneous non-magnetic linear medium. We formulate the retarded response of matter to a perturbation by an electric field It is described by the frequency-dependent susceptibility Dielectric permittivity Overview Peter Hertel Overview Maxwell’s equations • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s Definition KramersKronig Relation Onsager Relation • • Summary • • • equations. The Lorentz force on charged particles describes the interaction of light with matter. We recapitulate Maxwell’s equation in the presence of matter and specialize to a homogeneous non-magnetic linear medium. We formulate the retarded response of matter to a perturbation by an electric field It is described by the frequency-dependent susceptibility The real part and the imaginary part of the susceptibility are intimately related. Dielectric permittivity Overview Peter Hertel Overview Maxwell’s equations • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s Definition KramersKronig Relation Onsager Relation • • Summary • • • • equations. The Lorentz force on charged particles describes the interaction of light with matter. We recapitulate Maxwell’s equation in the presence of matter and specialize to a homogeneous non-magnetic linear medium. We formulate the retarded response of matter to a perturbation by an electric field It is described by the frequency-dependent susceptibility The real part and the imaginary part of the susceptibility are intimately related. If properly generalized to anisotropic media, the susceptibility is a symmetric tensor, provided an exteral magnetic field is reversed. Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Maxwell’s equations Dielectric permittivity Peter Hertel Maxwell’s equations Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary The electromagnetic field E and B accelerates charged particles Dielectric permittivity Peter Hertel Maxwell’s equations Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary The electromagnetic field E and B accelerates charged particles ṗ = q{E(t, x) + v × B(t, x)} Dielectric permittivity Peter Hertel Maxwell’s equations Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary The electromagnetic field E and B accelerates charged particles ṗ = q{E(t, x) + v × B(t, x)} At time t, the particle is at x, has velocity v = ẋ and momentum p. Its electric charge is q. Dielectric permittivity Maxwell’s equations Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary The electromagnetic field E and B accelerates charged particles ṗ = q{E(t, x) + v × B(t, x)} At time t, the particle is at x, has velocity v = ẋ and momentum p. Its electric charge is q. 1 0 ∇ · E = % Dielectric permittivity Maxwell’s equations Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary The electromagnetic field E and B accelerates charged particles ṗ = q{E(t, x) + v × B(t, x)} At time t, the particle is at x, has velocity v = ẋ and momentum p. Its electric charge is q. 1 0 ∇ · E = % 2 ∇·B =0 Dielectric permittivity Maxwell’s equations Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary The electromagnetic field E and B accelerates charged particles ṗ = q{E(t, x) + v × B(t, x)} At time t, the particle is at x, has velocity v = ẋ and momentum p. Its electric charge is q. 1 0 ∇ · E = % 2 ∇·B =0 3 ∇ × E = −∇t B Dielectric permittivity Maxwell’s equations Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary The electromagnetic field E and B accelerates charged particles ṗ = q{E(t, x) + v × B(t, x)} At time t, the particle is at x, has velocity v = ẋ and momentum p. Its electric charge is q. 1 0 ∇ · E = % 2 ∇·B =0 3 ∇ × E = −∇t B 4 (1/µ0 )∇ × B = 0 ∇t E + j Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Maxwell’s equations in matter I Dielectric permittivity Peter Hertel Maxwell’s equations in matter I Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • polarization P is electric dipole moment per unit volume Dielectric permittivity Maxwell’s equations in matter I Peter Hertel Overview Maxwell’s equations Definition • polarization P is electric dipole moment per unit volume KramersKronig Relation • magnetization M is magnetic dipole moment per unit Onsager Relation Summary volume Dielectric permittivity Maxwell’s equations in matter I Peter Hertel Overview Maxwell’s equations Definition • polarization P is electric dipole moment per unit volume KramersKronig Relation • magnetization M is magnetic dipole moment per unit Onsager Relation Summary volume • electric field strength E causes polarization Dielectric permittivity Maxwell’s equations in matter I Peter Hertel Overview Maxwell’s equations Definition • polarization P is electric dipole moment per unit volume KramersKronig Relation • magnetization M is magnetic dipole moment per unit Onsager Relation Summary volume • electric field strength E causes polarization • magnetic induction B causes magnetization Dielectric permittivity Maxwell’s equations in matter I Peter Hertel Overview Maxwell’s equations Definition • polarization P is electric dipole moment per unit volume KramersKronig Relation • magnetization M is magnetic dipole moment per unit Onsager Relation Summary volume • electric field strength E causes polarization • magnetic induction B causes magnetization • % = −∇ · P + %f Dielectric permittivity Maxwell’s equations in matter I Peter Hertel Overview Maxwell’s equations Definition • polarization P is electric dipole moment per unit volume KramersKronig Relation • magnetization M is magnetic dipole moment per unit Onsager Relation Summary volume • electric field strength E causes polarization • magnetic induction B causes magnetization • % = −∇ · P + %f • j = Ṗ + ∇ × M + j f Dielectric permittivity Maxwell’s equations in matter I Peter Hertel Overview Maxwell’s equations Definition • polarization P is electric dipole moment per unit volume KramersKronig Relation • magnetization M is magnetic dipole moment per unit Onsager Relation Summary volume • electric field strength E causes polarization • magnetic induction B causes magnetization • % = −∇ · P + %f • j = Ṗ + ∇ × M + j f • density %f and current density j f of free charges Dielectric permittivity Maxwell’s equations in matter I Peter Hertel Overview Maxwell’s equations Definition • polarization P is electric dipole moment per unit volume KramersKronig Relation • magnetization M is magnetic dipole moment per unit Onsager Relation Summary volume • electric field strength E causes polarization • magnetic induction B causes magnetization • % = −∇ · P + %f • j = Ṗ + ∇ × M + j f • density %f and current density j f of free charges • a vicious circle! Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Maxwell’s equations in matter II Dielectric permittivity Peter Hertel Maxwell’s equations in matter II Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P Dielectric permittivity Peter Hertel Maxwell’s equations in matter II Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement Dielectric permittivity Peter Hertel Maxwell’s equations in matter II Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M Dielectric permittivity Peter Hertel Maxwell’s equations in matter II Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Dielectric permittivity Peter Hertel Maxwell’s equations in matter II Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Dielectric permittivity Peter Hertel Maxwell’s equations in matter II Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Now Maxwell’s equations read Dielectric permittivity Maxwell’s equations in matter II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Now Maxwell’s equations read 1 ∇ · D = %f Dielectric permittivity Maxwell’s equations in matter II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇·B =0 Dielectric permittivity Maxwell’s equations in matter II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇·B =0 3 ∇ × E = −∇t B Dielectric permittivity Maxwell’s equations in matter II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇·B =0 3 ∇ × E = −∇t B 4 ∇ × H = Ḋ + j f Dielectric permittivity Maxwell’s equations in matter II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇·B =0 3 ∇ × E = −∇t B 4 ∇ × H = Ḋ + j f Dielectric permittivity Maxwell’s equations in matter II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇·B =0 3 ∇ × E = −∇t B 4 ∇ × H = Ḋ + j f This is good – only free charges are involved Dielectric permittivity Maxwell’s equations in matter II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • introduce auxiliary field D = 0 E + P • dielectric displacement • introduce auxiliary field H = (1/µ0 )B − M • magnetic field strength Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇·B =0 3 ∇ × E = −∇t B 4 ∇ × H = Ḋ + j f This is good – only free charges are involved and bad – there are more fields than equations Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary James Clerk Maxwell, 1831-1873 Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Dielectric susceptibility I Dielectric permittivity Peter Hertel Dielectric susceptibility I Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Assume a medium which is Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Assume a medium which is • homogeneous Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Assume a medium which is Definition • homogeneous KramersKronig Relation • non-magnetic Onsager Relation Summary Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Assume a medium which is Definition • homogeneous KramersKronig Relation • non-magnetic Onsager Relation • linear Summary Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Assume a medium which is Definition • homogeneous KramersKronig Relation • non-magnetic Onsager Relation • linear Summary • P = 0 χE Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Assume a medium which is Definition • homogeneous KramersKronig Relation • non-magnetic Onsager Relation • linear Summary • P = 0 χE • M =0 Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Assume a medium which is Definition • homogeneous KramersKronig Relation • non-magnetic Onsager Relation • linear Summary • P = 0 χE • M =0 • dielectric susceptibility χ is dimension-less number Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Assume a medium which is Definition • homogeneous KramersKronig Relation • non-magnetic Onsager Relation • linear Summary • P = 0 χE • M =0 • dielectric susceptibility χ is dimension-less number • equivalent D = 0 E Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Assume a medium which is Definition • homogeneous KramersKronig Relation • non-magnetic Onsager Relation • linear Summary • P = 0 χE • M =0 • dielectric susceptibility χ is dimension-less number • equivalent D = 0 E • relative dielectric permittivity = 1 + χ Dielectric permittivity Dielectric susceptibility I Peter Hertel Overview Maxwell’s equations Assume a medium which is Definition • homogeneous KramersKronig Relation • non-magnetic Onsager Relation • linear Summary • P = 0 χE • M =0 • dielectric susceptibility χ is dimension-less number • equivalent D = 0 E • relative dielectric permittivity = 1 + χ • more precisely . . . Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary More precisely Dielectric permittivity More precisely Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Assume linear local relation Dielectric permittivity More precisely Peter Hertel Overview Maxwell’s equations Assume linear local relation Definition KramersKronig Relation Onsager Relation Summary Z P (t, x) = dτ G(τ )E(t − τ, x) Dielectric permittivity More precisely Peter Hertel Overview Maxwell’s equations Assume linear local relation Definition KramersKronig Relation Onsager Relation Summary Z P (t, x) = causality dτ G(τ )E(t − τ, x) Dielectric permittivity More precisely Peter Hertel Overview Maxwell’s equations Assume linear local relation Definition KramersKronig Relation Onsager Relation Summary Z P (t, x) = dτ G(τ )E(t − τ, x) causality G(τ ) = 0 for τ < 0 Dielectric permittivity More precisely Peter Hertel Overview Maxwell’s equations Assume linear local relation Definition KramersKronig Relation Onsager Relation Summary Z P (t, x) = dτ G(τ )E(t − τ, x) causality G(τ ) = 0 for τ < 0 drop x Dielectric permittivity More precisely Peter Hertel Overview Maxwell’s equations Assume linear local relation Definition KramersKronig Relation Onsager Relation Summary Z P (t, x) = dτ G(τ )E(t − τ, x) causality G(τ ) = 0 for τ < 0 drop x Z P (t) = dτ G(τ )E(t − τ ) Dielectric permittivity More precisely Peter Hertel Overview Maxwell’s equations Assume linear local relation Definition KramersKronig Relation Onsager Relation Summary Z P (t, x) = dτ G(τ )E(t − τ, x) causality G(τ ) = 0 for τ < 0 drop x Z P (t) = dτ G(τ )E(t − τ ) G is causal influence, or Green’s functions Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary George Green, 1793-1841 Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Fourier transforms Dielectric permittivity Fourier transforms Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Z f (t) = dω −iωt ˜ e f (ω) 2π Dielectric permittivity Fourier transforms Peter Hertel Overview Maxwell’s equations Definition Z f (t) = dω −iωt ˜ e f (ω) 2π KramersKronig Relation Onsager Relation Summary f˜(ω) = Z dt e +iωt f (t) Dielectric permittivity Fourier transforms Peter Hertel Overview Maxwell’s equations Definition Z f (t) = dω −iωt ˜ e f (ω) 2π KramersKronig Relation Onsager Relation f˜(ω) = Z dt e +iωt f (t) Summary convolution h = g ∗ f , i. e. Z h(t) = dτ g(τ )f (t − τ ) Dielectric permittivity Fourier transforms Peter Hertel Overview Maxwell’s equations Definition Z f (t) = dω −iωt ˜ e f (ω) 2π KramersKronig Relation Onsager Relation f˜(ω) = Z dt e +iωt f (t) Summary convolution h = g ∗ f , i. e. Z h(t) = dτ g(τ )f (t − τ ) then h̃(ω) = g̃(ω)f˜(ω) Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Susceptibility II Dielectric permittivity Susceptibility II Peter Hertel Overview Maxwell’s equations Recall Definition KramersKronig Relation Onsager Relation Summary Z P (t) = dτ G(τ )E(t − τ ) Dielectric permittivity Susceptibility II Peter Hertel Overview Maxwell’s equations Recall Definition Z dτ G(τ )E(t − τ ) KramersKronig Relation P (t) = Onsager Relation Therefore Summary P̃ (ω) = 0 χ(ω)Ẽ(ω) with χ(ω) = 1 G̃(ω) 0 Dielectric permittivity Susceptibility II Peter Hertel Overview Maxwell’s equations Recall Definition Z dτ G(τ )E(t − τ ) KramersKronig Relation P (t) = Onsager Relation Therefore Summary P̃ (ω) = 0 χ(ω)Ẽ(ω) with χ(ω) = 1 G̃(ω) 0 susceptibility χ must depend on frequency ω Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Kramers-Kronig relation I Dielectric permittivity Kramers-Kronig relation I Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Recall Z P (t, x) = dτ G(τ )E(t − τ, x) Dielectric permittivity Kramers-Kronig relation I Peter Hertel Overview Maxwell’s equations Recall Z P (t, x) = dτ G(τ )E(t − τ, x) Definition KramersKronig Relation Onsager Relation Summary G(τ ) = θ(τ )G(τ ) with Heaviside function θ Dielectric permittivity Kramers-Kronig relation I Peter Hertel Overview Maxwell’s equations Recall Z P (t, x) = dτ G(τ )E(t − τ, x) Definition KramersKronig Relation G(τ ) = θ(τ )G(τ ) with Heaviside function θ Onsager Relation Summary Z χ(ω) = du χ(u)θ̃(ω − u) by convolution theorem 2π Dielectric permittivity Kramers-Kronig relation I Peter Hertel Overview Maxwell’s equations Recall Z dτ G(τ )E(t − τ, x) P (t, x) = Definition KramersKronig Relation G(τ ) = θ(τ )G(τ ) with Heaviside function θ Onsager Relation Summary Z χ(ω) = du χ(u)θ̃(ω − u) by convolution theorem 2π 1 0<η→0 η − iω θ̃(ω) = lim Dielectric permittivity Kramers-Kronig relation I Peter Hertel Overview Maxwell’s equations Recall Z dτ G(τ )E(t − τ, x) P (t, x) = Definition KramersKronig Relation G(τ ) = θ(τ )G(τ ) with Heaviside function θ Onsager Relation Summary Z χ(ω) = du χ(u)θ̃(ω − u) by convolution theorem 2π 1 0<η→0 η − iω θ̃(ω) = lim Z χ(ω) = lim 0<η→0 χ(u) du dispersion relation 2π η − i(ω − u) Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Dispersion of white light Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Kramers-Kronig relation II Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Kramers-Kronig relation II Decompose susceptibility in real and imaginary part χ(ω) = χ 0 (ω) + iχ 00 (ω) Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Kramers-Kronig relation II Decompose susceptibility in real and imaginary part χ(ω) = χ 0 (ω) + iχ 00 (ω) Introduce principle value integral Z ω−η Z ∞ Z du du Pr ··· = + ... 2π −∞ ω+η 2π Dielectric permittivity Kramers-Kronig relation II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Decompose susceptibility in real and imaginary part χ(ω) = χ 0 (ω) + iχ 00 (ω) Introduce principle value integral Z ω−η Z ∞ Z du du Pr ··· = + ... 2π −∞ ω+η 2π Employ χ(−ω) = χ(ω)∗ Dielectric permittivity Kramers-Kronig relation II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Decompose susceptibility in real and imaginary part χ(ω) = χ 0 (ω) + iχ 00 (ω) Introduce principle value integral Z ω−η Z ∞ Z du du Pr ··· = + ... 2π −∞ ω+η 2π Employ χ(−ω) = χ(ω)∗ 0 χ (ω) = 2Pr Z du uχ 00 (u) π u2 − ω 2 KKR Dielectric permittivity Kramers-Kronig relation II Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Decompose susceptibility in real and imaginary part χ(ω) = χ 0 (ω) + iχ 00 (ω) Introduce principle value integral Z ω−η Z ∞ Z du du Pr ··· = + ... 2π −∞ ω+η 2π Employ χ(−ω) = χ(ω)∗ 0 Z χ (ω) = 2Pr χ 00 (ω) = 2Pr Z du uχ 00 (u) π u2 − ω 2 KKR du ωχ 0 (u) π ω 2 − u2 inverse KKR Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Hendrik Anthony Kramers (center), Dutch physicist, 1894-1952 Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Ralph Kronig, US American physicist, 1904-1995 Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Time reversal invariance Dielectric permittivity Peter Hertel Time reversal invariance Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) Dielectric permittivity Peter Hertel Time reversal invariance Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) Dielectric permittivity Peter Hertel Time reversal invariance Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} Dielectric permittivity Peter Hertel Time reversal invariance Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} • time reversal invariance requires (E, B) → (E, −B) Dielectric permittivity Peter Hertel Time reversal invariance Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} • time reversal invariance requires (E, B) → (E, −B) • moreover, (%, j) → (%, −j) Dielectric permittivity Time reversal invariance Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} • time reversal invariance requires (E, B) → (E, −B) • moreover, (%, j) → (%, −j) 1 0 ∇ · E = % X Dielectric permittivity Time reversal invariance Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} • time reversal invariance requires (E, B) → (E, −B) • moreover, (%, j) → (%, −j) 1 0 ∇ · E = % X 2 ∇·B =0 X Dielectric permittivity Time reversal invariance Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} • time reversal invariance requires (E, B) → (E, −B) • moreover, (%, j) → (%, −j) 1 0 ∇ · E = % X 2 ∇·B =0 X 3 ∇ × E = −∇t B X Dielectric permittivity Time reversal invariance Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} • time reversal invariance requires (E, B) → (E, −B) • moreover, (%, j) → (%, −j) 1 0 ∇ · E = % X 2 ∇·B =0 X 3 ∇ × E = −∇t B X 4 (1/µ0 )∇ × B = 0 ∇t E + j X Dielectric permittivity Time reversal invariance Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} • time reversal invariance requires (E, B) → (E, −B) • moreover, (%, j) → (%, −j) 1 0 ∇ · E = % X 2 ∇·B =0 X 3 ∇ × E = −∇t B X 4 (1/µ0 )∇ × B = 0 ∇t E + j X Dielectric permittivity Time reversal invariance Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall ṗ = q{E + v × B} • time reversal invariance requires (E, B) → (E, −B) • moreover, (%, j) → (%, −j) 1 0 ∇ · E = % X 2 ∇·B =0 X 3 ∇ × E = −∇t B X 4 (1/µ0 )∇ × B = 0 ∇t E + j X Maxwell’s equations are time reversal invariant Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Onsager symmetry relation Dielectric permittivity Peter Hertel Onsager symmetry relation Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • generalize to a possible anisotropic medium Dielectric permittivity Peter Hertel Onsager symmetry relation Overview Maxwell’s equations • generalize to a possible anisotropic medium Definition KramersKronig Relation Onsager Relation Summary • P̃i (ω) = χij (ω)Ẽj (ω) (sum over j = 1, 2, 3) Dielectric permittivity Peter Hertel Onsager symmetry relation Overview Maxwell’s equations • generalize to a possible anisotropic medium Definition KramersKronig Relation Onsager Relation Summary • P̃i (ω) = χij (ω)Ẽj (ω) (sum over j = 1, 2, 3) • susceptibility is a property of matter in thermal equilibrium Dielectric permittivity Onsager symmetry relation Peter Hertel Overview Maxwell’s equations • generalize to a possible anisotropic medium Definition KramersKronig Relation Onsager Relation Summary • P̃i (ω) = χij (ω)Ẽj (ω) (sum over j = 1, 2, 3) • susceptibility is a property of matter in thermal equilibrium • its value depends on all parameters which affect the equilibrium Dielectric permittivity Onsager symmetry relation Peter Hertel Overview Maxwell’s equations • generalize to a possible anisotropic medium Definition KramersKronig Relation Onsager Relation Summary • P̃i (ω) = χij (ω)Ẽj (ω) (sum over j = 1, 2, 3) • susceptibility is a property of matter in thermal equilibrium • its value depends on all parameters which affect the equilibrium • such as temperature, mechanical strain, external static electric or magnetic fields, . . . Dielectric permittivity Onsager symmetry relation Peter Hertel Overview Maxwell’s equations • generalize to a possible anisotropic medium Definition KramersKronig Relation Onsager Relation Summary • P̃i (ω) = χij (ω)Ẽj (ω) (sum over j = 1, 2, 3) • susceptibility is a property of matter in thermal equilibrium • its value depends on all parameters which affect the equilibrium • such as temperature, mechanical strain, external static electric or magnetic fields, . . . • χij = χij (ω; T, S, E, B, . . . ) Dielectric permittivity Onsager symmetry relation Peter Hertel Overview Maxwell’s equations • generalize to a possible anisotropic medium Definition KramersKronig Relation Onsager Relation Summary • P̃i (ω) = χij (ω)Ẽj (ω) (sum over j = 1, 2, 3) • susceptibility is a property of matter in thermal equilibrium • its value depends on all parameters which affect the equilibrium • such as temperature, mechanical strain, external static electric or magnetic fields, . . . • χij = χij (ω; T, S, E, B, . . . ) • Interchanging indexes and reverting B is a symmetry Dielectric permittivity Onsager symmetry relation Peter Hertel Overview Maxwell’s equations • generalize to a possible anisotropic medium Definition KramersKronig Relation Onsager Relation Summary • P̃i (ω) = χij (ω)Ẽj (ω) (sum over j = 1, 2, 3) • susceptibility is a property of matter in thermal equilibrium • its value depends on all parameters which affect the equilibrium • such as temperature, mechanical strain, external static electric or magnetic fields, . . . • χij = χij (ω; T, S, E, B, . . . ) • Interchanging indexes and reverting B is a symmetry • χij (ω; T, S, E, B, . . . ) = χji (ω; T, S, E, −B, . . . ) Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Lars Onsager, Norwegian/US American physical chemist, 1903-1976 Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Summary Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Summary • Optics deals with the interaction of light with matter. Dielectric permittivity Summary Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s equations. Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Summary • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s equations. • The Lorentz force on charged particles describes the interaction of light with matter. Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Summary • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s equations. • The Lorentz force on charged particles describes the interaction of light with matter. • We recapitulated Maxwell’s equation in the presence of matter and specialized to a homogeneous non-magnetic linear medium. Dielectric permittivity Peter Hertel Overview Maxwell’s equations Definition KramersKronig Relation Onsager Relation Summary Summary • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s equations. • The Lorentz force on charged particles describes the interaction of light with matter. • We recapitulated Maxwell’s equation in the presence of matter and specialized to a homogeneous non-magnetic linear medium. • We described the retarded response of matter to a perturbation by an electric field Dielectric permittivity Summary Peter Hertel Overview Maxwell’s equations • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s Definition KramersKronig Relation Onsager Relation • • Summary • • equations. The Lorentz force on charged particles describes the interaction of light with matter. We recapitulated Maxwell’s equation in the presence of matter and specialized to a homogeneous non-magnetic linear medium. We described the retarded response of matter to a perturbation by an electric field It is described by the frequency-dependent susceptibility Dielectric permittivity Summary Peter Hertel Overview Maxwell’s equations • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s Definition KramersKronig Relation Onsager Relation • • Summary • • • equations. The Lorentz force on charged particles describes the interaction of light with matter. We recapitulated Maxwell’s equation in the presence of matter and specialized to a homogeneous non-magnetic linear medium. We described the retarded response of matter to a perturbation by an electric field It is described by the frequency-dependent susceptibility The real part and the imaginary part of the susceptibility are intimately related (Kramers-Kronig). Dielectric permittivity Summary Peter Hertel Overview Maxwell’s equations • Optics deals with the interaction of light with matter. • Light, as an electromagnetic field, obeys Maxwell’s Definition KramersKronig Relation Onsager Relation • • Summary • • • • equations. The Lorentz force on charged particles describes the interaction of light with matter. We recapitulated Maxwell’s equation in the presence of matter and specialized to a homogeneous non-magnetic linear medium. We described the retarded response of matter to a perturbation by an electric field It is described by the frequency-dependent susceptibility The real part and the imaginary part of the susceptibility are intimately related (Kramers-Kronig). If properly generalized to anisotropic media, the susceptibility is a symmetric tensor, provided an exteral magnetic field is reversed (Onsager).

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