Dielectric Permittivity

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Dielectric permittivity

Peter Hertel

Overview Maxwell’s Dielectric permittivity equations

Deﬁnition

Kramers- Kronig Peter Hertel Relation

Onsager Relation University of Osnabr¨uck,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

Kramers- Kronig Relation

Onsager Relation

Summary

Make it as simple as possible, but not simpler • 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 • 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

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • 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 • 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

Overview • Optics deals with the interaction of light with matter.

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • 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

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • 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

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager Relation

Summary • 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

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulate Maxwell’s equation in the presence of Relation matter and specialize to a homogeneous non-magnetic Summary linear medium. • 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

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulate Maxwell’s equation in the presence of Relation matter and specialize to a homogeneous non-magnetic Summary linear medium. • We formulate the retarded response of matter to a perturbation by an electric ﬁeld • 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

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulate Maxwell’s equation in the presence of Relation matter and specialize to a homogeneous non-magnetic Summary linear medium. • We formulate the retarded response of matter to a perturbation by an electric ﬁeld • It is described by the frequency-dependent susceptibility • If properly generalized to anisotropic media, the susceptibility is a symmetric tensor, provided an exteral magnetic ﬁeld is reversed.

Dielectric permittivity

Peter Hertel Overview

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulate Maxwell’s equation in the presence of Relation matter and specialize to a homogeneous non-magnetic Summary 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

Peter Hertel Overview

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulate Maxwell’s equation in the presence of Relation matter and specialize to a homogeneous non-magnetic Summary 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. 1 0∇ · E = % 2 ∇ · B = 0

3 ∇ × E = −∇t B

4 (1/µ0)∇ × B = 0 ∇t E + j

The electromagnetic ﬁeld E and B accelerates charged particles

p˙ = q{E(t, x) + v × B(t, x)}

At time t, the particle is at x, has velocity v = x˙ and momentum p. Its electric charge is q.

Dielectric permittivity

Peter Hertel Maxwell’s equations

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary 1 0∇ · E = % 2 ∇ · B = 0

3 ∇ × E = −∇t B

4 (1/µ0)∇ × B = 0 ∇t E + j

p˙ = q{E(t, x) + v × B(t, x)}

At time t, the particle is at x, has velocity v = x˙ and momentum p. Its electric charge is q.

Dielectric permittivity

Peter Hertel Maxwell’s equations

Overview

Maxwell’s equations The electromagnetic ﬁeld E and B accelerates charged Deﬁnition particles Kramers- Kronig Relation

Onsager Relation

Summary 1 0∇ · E = % 2 ∇ · B = 0

3 ∇ × E = −∇t B

4 (1/µ0)∇ × B = 0 ∇t E + j

At time t, the particle is at x, has velocity v = x˙ and momentum p. Its electric charge is q.

Dielectric permittivity

Peter Hertel Maxwell’s equations

Overview

Maxwell’s equations The electromagnetic ﬁeld E and B accelerates charged Deﬁnition particles Kramers- Kronig Relation p˙ = q{E(t, x) + v × B(t, x)} Onsager Relation

Summary 1 0∇ · E = % 2 ∇ · B = 0

3 ∇ × E = −∇t B

4 (1/µ0)∇ × B = 0 ∇t E + j

Dielectric permittivity

Peter Hertel Maxwell’s equations

Overview

Maxwell’s equations The electromagnetic ﬁeld E and B accelerates charged Deﬁnition particles Kramers- Kronig Relation p˙ = q{E(t, x) + v × B(t, x)} Onsager Relation At time t, the particle is at x, has velocity v = x˙ and Summary momentum p. Its electric charge is q. 2 ∇ · B = 0

3 ∇ × E = −∇t B

4 (1/µ0)∇ × B = 0 ∇t E + j

Dielectric permittivity

Peter Hertel Maxwell’s equations

Overview

1 0∇ · E = % 3 ∇ × E = −∇t B

4 (1/µ0)∇ × B = 0 ∇t E + j

Dielectric permittivity

Peter Hertel Maxwell’s equations

Overview

1 0∇ · E = % 2 ∇ · B = 0 4 (1/µ0)∇ × B = 0 ∇t E + j

Dielectric permittivity

Peter Hertel Maxwell’s equations

Overview

1 0∇ · E = % 2 ∇ · B = 0

3 ∇ × E = −∇t B Dielectric permittivity

Peter Hertel Maxwell’s equations

Overview

1 0∇ · E = % 2 ∇ · B = 0

3 ∇ × E = −∇t B

4 (1/µ0)∇ × B = 0 ∇t E + j • polarization P is electric dipole moment per unit volume • magnetization M is magnetic dipole moment per unit volume • electric ﬁeld strength E causes polarization • magnetic induction B causes magnetization • % = −∇ · P + %f • j = P˙ + ∇ × M + jf • density %f and current density jf of free charges • a vicious circle!

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • magnetization M is magnetic dipole moment per unit volume • electric ﬁeld strength E causes polarization • magnetic induction B causes magnetization • % = −∇ · P + %f • j = P˙ + ∇ × M + jf • density %f and current density jf of free charges • a vicious circle!

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition • polarization P is electric dipole moment per unit volume

Kramers- Kronig Relation

Onsager Relation

Summary • electric ﬁeld strength E causes polarization • magnetic induction B causes magnetization • % = −∇ · P + %f • j = P˙ + ∇ × M + jf • density %f and current density jf of free charges • a vicious circle!

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition • polarization P is electric dipole moment per unit volume

Kramers- Kronig • magnetization M is magnetic dipole moment per unit Relation volume Onsager Relation

Summary • magnetic induction B causes magnetization • % = −∇ · P + %f • j = P˙ + ∇ × M + jf • density %f and current density jf of free charges • a vicious circle!

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition • polarization P is electric dipole moment per unit volume

Kramers- Kronig • magnetization M is magnetic dipole moment per unit Relation volume Onsager Relation • electric ﬁeld strength E causes polarization Summary • % = −∇ · P + %f • j = P˙ + ∇ × M + jf • density %f and current density jf of free charges • a vicious circle!

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition • polarization P is electric dipole moment per unit volume

Kramers- Kronig • magnetization M is magnetic dipole moment per unit Relation volume Onsager Relation • electric ﬁeld strength E causes polarization Summary • magnetic induction B causes magnetization • j = P˙ + ∇ × M + jf • density %f and current density jf of free charges • a vicious circle!

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition • polarization P is electric dipole moment per unit volume

Kramers- Kronig • magnetization M is magnetic dipole moment per unit Relation volume Onsager Relation • electric ﬁeld strength E causes polarization Summary • magnetic induction B causes magnetization • % = −∇ · P + %f • density %f and current density jf of free charges • a vicious circle!

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition • polarization P is electric dipole moment per unit volume

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition • polarization P is electric dipole moment per unit volume

Kramers- Kronig • magnetization M is magnetic dipole moment per unit Relation volume Onsager Relation • electric ﬁeld strength E causes polarization Summary • magnetic induction B causes magnetization • % = −∇ · P + %f • j = P˙ + ∇ × M + jf • density %f and current density jf of free charges Dielectric permittivity

Peter Hertel Maxwell’s equations in matter I

Overview

Maxwell’s equations

Deﬁnition • polarization P is electric dipole moment per unit volume

Kramers- Kronig • magnetization M is magnetic dipole moment per unit Relation volume Onsager Relation • electric ﬁeld strength E causes polarization Summary • magnetic induction B causes magnetization • % = −∇ · P + %f • j = P˙ + ∇ × M + jf • density %f and current density jf of free charges • a vicious circle! • introduce auxiliary ﬁeld D = 0E + P • dielectric displacement

• introduce auxiliary ﬁeld H = (1/µ0)B − M • magnetic ﬁeld strength

1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j

Now Maxwell’s equations read

This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j

• dielectric displacement

• introduce auxiliary ﬁeld H = (1/µ0)B − M • magnetic ﬁeld strength Now Maxwell’s equations read

This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary ﬁeld D = 0E + P Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j

• introduce auxiliary ﬁeld H = (1/µ0)B − M • magnetic ﬁeld strength Now Maxwell’s equations read

This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary ﬁeld D = 0E + P Deﬁnition • dielectric displacement Kramers- Kronig Relation

Onsager Relation

Summary 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j

• magnetic ﬁeld strength Now Maxwell’s equations read

This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager Relation

Summary 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j

Now Maxwell’s equations read

This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j

Now Maxwell’s equations read

This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j

This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary Now Maxwell’s equations read 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary Now Maxwell’s equations read 1 ∇ · D = %f 3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇ · B = 0 f 4 ∇ × H = D˙ + j This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j This is good – only free charges are involved and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j and bad – there are more ﬁelds than equations

Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j This is good – only free charges are involved Dielectric permittivity

Peter Hertel Maxwell’s equations in matter II

Overview

Maxwell’s equations • introduce auxiliary field D = 0E + P Definition • dielectric displacement Kramers- Kronig • introduce auxiliary field H = (1/µ0)B − M Relation

Onsager • magnetic ﬁeld strength Relation

Summary Now Maxwell’s equations read 1 ∇ · D = %f 2 ∇ · B = 0

3 ∇ × E = −∇t B f 4 ∇ × H = D˙ + j 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

Kramers- Kronig Relation

Onsager Relation

Summary

James Clerk Maxwell, 1831-1873 • homogeneous • non-magnetic • linear

• P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . .

Assume a medium which is

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • homogeneous • non-magnetic • linear

• P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • non-magnetic • linear

• P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation

Onsager Relation

Summary • linear

• P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation • non-magnetic

Onsager Relation

Summary • P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation • non-magnetic

Onsager Relation • linear

Summary • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation • non-magnetic

Onsager Relation • linear Summary • P = 0χE • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation • non-magnetic

Onsager Relation • linear Summary • P = 0χE • M = 0 • equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation • non-magnetic

Onsager Relation • linear Summary • P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number • relative dielectric permittivity = 1 + χ • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation • non-magnetic

Onsager Relation • linear Summary • P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • more precisely . . .

Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation • non-magnetic

Onsager Relation • linear Summary • P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ Dielectric permittivity

Peter Hertel Dielectric susceptibility I

Overview

Maxwell’s equations Assume a medium which is Deﬁnition • homogeneous Kramers- Kronig Relation • non-magnetic

Onsager Relation • linear Summary • P = 0χE • M = 0 • dielectric susceptibility χ is dimension-less number

• equivalent D = 0E • relative dielectric permittivity = 1 + χ • more precisely . . . Assume linear local relation

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 More precisely

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig 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 More precisely

Overview

Maxwell’s Assume linear local relation equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary 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 More precisely

Overview

Maxwell’s Assume linear local relation equations

Deﬁnition

Kramers- Z Kronig P (t, x) = dτ G(τ)E(t − τ, x) Relation

Onsager Relation

Summary 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 More precisely

Overview

Maxwell’s Assume linear local relation equations

Deﬁnition

Kramers- Z Kronig P (t, x) = dτ G(τ)E(t − τ, x) Relation

Onsager Relation causality Summary drop x Z P (t) = dτ G(τ)E(t − τ)

G is causal inﬂuence, or Green’s functions

Dielectric permittivity

Peter Hertel More precisely

Overview

Maxwell’s Assume linear local relation equations

Deﬁnition

Kramers- Z Kronig P (t, x) = dτ G(τ)E(t − τ, x) Relation

Onsager Relation causality Summary G(τ) = 0 for τ < 0 Z P (t) = dτ G(τ)E(t − τ)

G is causal inﬂuence, or Green’s functions

Dielectric permittivity

Peter Hertel More precisely

Overview

Maxwell’s Assume linear local relation equations

Deﬁnition

Kramers- Z Kronig P (t, x) = dτ G(τ)E(t − τ, x) Relation

Onsager Relation causality Summary G(τ) = 0 for τ < 0

drop x G is causal inﬂuence, or Green’s functions

Dielectric permittivity

Peter Hertel More precisely

Overview

Maxwell’s Assume linear local relation equations

Deﬁnition

Kramers- Z Kronig P (t, x) = dτ G(τ)E(t − τ, x) Relation

Onsager Relation causality Summary G(τ) = 0 for τ < 0

drop x Z P (t) = dτ G(τ)E(t − τ) Dielectric permittivity

Peter Hertel More precisely

Overview

Maxwell’s Assume linear local relation equations

Deﬁnition

Kramers- Z Kronig P (t, x) = dτ G(τ)E(t − τ, x) Relation

Onsager Relation causality Summary 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

Kramers- Kronig Relation

Onsager Relation

Summary

George Green, 1793-1841 Z dω −iωt f(t) = e f˜(ω) 2π

Z +iωt f˜(ω) = dt e f(t)

convolution h = g ∗ f, i. e. Z h(t) = dτ g(τ)f(t − τ)

then

h˜(ω) =g ˜(ω)f˜(ω)

Dielectric permittivity

Peter Hertel Fourier transforms

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary Z +iωt f˜(ω) = dt e f(t)

convolution h = g ∗ f, i. e. Z h(t) = dτ g(τ)f(t − τ)

then

h˜(ω) =g ˜(ω)f˜(ω)

Dielectric permittivity

Peter Hertel Fourier transforms

Overview

Maxwell’s equations Z dω −iωt f(t) = e f˜(ω) Deﬁnition 2π Kramers- Kronig Relation

Onsager Relation

Summary convolution h = g ∗ f, i. e. Z h(t) = dτ g(τ)f(t − τ)

then

h˜(ω) =g ˜(ω)f˜(ω)

Dielectric permittivity

Peter Hertel Fourier transforms

Overview

Maxwell’s equations Z dω −iωt f(t) = e f˜(ω) Deﬁnition 2π Kramers- Kronig Relation Z Onsager +iωt Relation f˜(ω) = dt e f(t) Summary then

h˜(ω) =g ˜(ω)f˜(ω)

Dielectric permittivity

Peter Hertel Fourier transforms

Overview

Maxwell’s equations Z dω −iωt f(t) = e f˜(ω) Deﬁnition 2π Kramers- Kronig Relation Z Onsager +iωt Relation f˜(ω) = dt e f(t) Summary convolution h = g ∗ f, i. e. Z h(t) = dτ g(τ)f(t − τ) Dielectric permittivity

Peter Hertel Fourier transforms

Overview

Maxwell’s equations Z dω −iωt f(t) = e f˜(ω) Deﬁnition 2π Kramers- Kronig Relation Z Onsager +iωt Relation f˜(ω) = dt e f(t) Summary convolution h = g ∗ f, i. e. Z h(t) = dτ g(τ)f(t − τ)

then

h˜(ω) =g ˜(ω)f˜(ω) Recall Z P (t) = dτ G(τ)E(t − τ)

Therefore

P˜ (ω) = 0χ(ω)E˜ (ω) with 1 χ(ω) = G˜(ω) 0 susceptibility χ must depend on frequency ω

Dielectric permittivity

Peter Hertel Susceptibility II

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary Therefore

P˜ (ω) = 0χ(ω)E˜ (ω) with 1 χ(ω) = G˜(ω) 0 susceptibility χ must depend on frequency ω

Dielectric permittivity

Peter Hertel Susceptibility II

Overview

Maxwell’s equations Recall Deﬁnition Z Kramers- P (t) = dτ G(τ)E(t − τ) Kronig Relation

Onsager Relation

Summary susceptibility χ must depend on frequency ω

Dielectric permittivity

Peter Hertel Susceptibility II

Overview

Maxwell’s equations Recall Deﬁnition Z Kramers- P (t) = dτ G(τ)E(t − τ) Kronig Relation Onsager Therefore Relation

Summary P˜ (ω) = 0χ(ω)E˜ (ω) with 1 χ(ω) = G˜(ω) 0 Dielectric permittivity

Peter Hertel Susceptibility II

Overview

Maxwell’s equations Recall Deﬁnition Z Kramers- P (t) = dτ G(τ)E(t − τ) Kronig Relation Onsager Therefore Relation

Summary P˜ (ω) = 0χ(ω)E˜ (ω) with 1 χ(ω) = G˜(ω) 0 susceptibility χ must depend on frequency ω Recall Z P (t, x) = dτ G(τ)E(t − τ, x)

G(τ) = θ(τ)G(τ) with Heaviside function θ

Z du χ(ω) = χ(u)θ˜(ω − u) by convolution theorem 2π

1 θ˜(ω) = lim 0<η→0 η − iω

Z du χ(u) χ(ω) = lim dispersion relation 0<η→0 2π η − i(ω − u)

Dielectric permittivity

Peter Hertel Kramers-Kronig relation I

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary G(τ) = θ(τ)G(τ) with Heaviside function θ

Z du χ(ω) = χ(u)θ˜(ω − u) by convolution theorem 2π

1 θ˜(ω) = lim 0<η→0 η − iω

Z du χ(u) χ(ω) = lim dispersion relation 0<η→0 2π η − i(ω − u)

Dielectric permittivity

Peter Hertel Kramers-Kronig relation I Recall Overview Z Maxwell’s equations P (t, x) = dτ G(τ)E(t − τ, x)

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary Z du χ(ω) = χ(u)θ˜(ω − u) by convolution theorem 2π

1 θ˜(ω) = lim 0<η→0 η − iω

Z du χ(u) χ(ω) = lim dispersion relation 0<η→0 2π η − i(ω − u)

Dielectric permittivity

Peter Hertel Kramers-Kronig relation I Recall Overview Z Maxwell’s equations P (t, x) = dτ G(τ)E(t − τ, x)

Deﬁnition

Kramers- Kronig Relation G(τ) = θ(τ)G(τ) with Heaviside function θ Onsager Relation

Summary 1 θ˜(ω) = lim 0<η→0 η − iω

Z du χ(u) χ(ω) = lim dispersion relation 0<η→0 2π η − i(ω − u)

Dielectric permittivity

Peter Hertel Kramers-Kronig relation I Recall Overview Z Maxwell’s equations P (t, x) = dτ G(τ)E(t − τ, x)

Deﬁnition

Kramers- Kronig Relation G(τ) = θ(τ)G(τ) with Heaviside function θ Onsager Relation

Summary Z du χ(ω) = χ(u)θ˜(ω − u) by convolution theorem 2π Z du χ(u) χ(ω) = lim dispersion relation 0<η→0 2π η − i(ω − u)

Dielectric permittivity

Peter Hertel Kramers-Kronig relation I Recall Overview Z Maxwell’s equations P (t, x) = dτ G(τ)E(t − τ, x)

Deﬁnition

Kramers- Kronig Relation G(τ) = θ(τ)G(τ) with Heaviside function θ Onsager Relation

Summary Z du χ(ω) = χ(u)θ˜(ω − u) by convolution theorem 2π

1 θ˜(ω) = lim 0<η→0 η − iω Dielectric permittivity

Peter Hertel Kramers-Kronig relation I Recall Overview Z Maxwell’s equations P (t, x) = dτ G(τ)E(t − τ, x)

Deﬁnition

Kramers- Kronig Relation G(τ) = θ(τ)G(τ) with Heaviside function θ Onsager Relation

Summary Z du χ(ω) = χ(u)θ˜(ω − u) by convolution theorem 2π

1 θ˜(ω) = lim 0<η→0 η − iω

Z du χ(u) χ(ω) = lim dispersion relation 0<η→0 2π η − i(ω − u) Dielectric permittivity

Peter Hertel

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary

Dispersion of white light Decompose susceptibility in real and imaginary part χ(ω) = χ 0(ω) + iχ 00(ω) Introduce principle value integral Z du Z ω−η Z ∞ du Pr ··· = + ... 2π −∞ ω+η 2π Employ χ(−ω) = χ(ω)∗

Z du uχ 00(u) χ 0(ω) = 2Pr KKR π u2 − ω2

Z du ωχ 0(u) χ 00(ω) = 2Pr inverse KKR π ω2 − u2

Dielectric permittivity

Peter Hertel Kramers-Kronig relation II

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary Introduce principle value integral Z du Z ω−η Z ∞ du Pr ··· = + ... 2π −∞ ω+η 2π Employ χ(−ω) = χ(ω)∗

Z du uχ 00(u) χ 0(ω) = 2Pr KKR π u2 − ω2

Z du ωχ 0(u) χ 00(ω) = 2Pr inverse KKR π ω2 − u2

Dielectric permittivity

Peter Hertel Kramers-Kronig relation II Decompose susceptibility in real and imaginary part Overview Maxwell’s χ(ω) = χ 0(ω) + iχ 00(ω) equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary Employ χ(−ω) = χ(ω)∗

Z du uχ 00(u) χ 0(ω) = 2Pr KKR π u2 − ω2

Z du ωχ 0(u) χ 00(ω) = 2Pr inverse KKR π ω2 − u2

Dielectric permittivity

Peter Hertel Kramers-Kronig relation II Decompose susceptibility in real and imaginary part Overview Maxwell’s χ(ω) = χ 0(ω) + iχ 00(ω) equations Deﬁnition Introduce principle value integral Kramers- Kronig Z du Z ω−η Z ∞ du Relation Pr ··· = + ... Onsager 2π −∞ ω+η 2π Relation

Summary Z du uχ 00(u) χ 0(ω) = 2Pr KKR π u2 − ω2

Z du ωχ 0(u) χ 00(ω) = 2Pr inverse KKR π ω2 − u2

Dielectric permittivity

Summary Employ χ(−ω) = χ(ω)∗ Z du ωχ 0(u) χ 00(ω) = 2Pr inverse KKR π ω2 − u2

Dielectric permittivity

Summary Employ χ(−ω) = χ(ω)∗

Z du uχ 00(u) χ 0(ω) = 2Pr KKR π u2 − ω2 Dielectric permittivity

Summary Employ χ(−ω) = χ(ω)∗

Z du uχ 00(u) χ 0(ω) = 2Pr KKR π u2 − ω2

Z du ωχ 0(u) χ 00(ω) = 2Pr inverse KKR π ω2 − u2 Dielectric permittivity

Peter Hertel

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary

Hendrik Anthony Kramers (center), Dutch physicist, 1894-1952 Dielectric permittivity

Peter Hertel

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary

Ralph Kronig, US American physicist, 1904-1995 • time reversal (t, x) → (−t, x) • consequently (v, p) → (−v, −p) • recall p˙ = 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 Time reversal invariance

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • consequently (v, p) → (−v, −p) • recall p˙ = 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 Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • recall p˙ = 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 Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig Relation

Onsager Relation

Summary • 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 Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager Relation

Summary • 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 Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager • time reversal invariance requires (E, B) → (E, −B) Relation

Summary 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 Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager • time reversal invariance requires (E, B) → (E, −B) Relation • moreover, (%, j) → (%, −j) Summary 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 Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager • time reversal invariance requires (E, B) → (E, −B) Relation • moreover, (%, j) → (%, −j) Summary

1 0∇ · E = % 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 Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager • time reversal invariance requires (E, B) → (E, −B) Relation • moreover, (%, j) → (%, −j) Summary

1 0∇ · E = % X 2 ∇ · B = 0 X 4 (1/µ0)∇ × B = 0 ∇t E + j X Maxwell’s equations are time reversal invariant

Dielectric permittivity

Peter Hertel Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager • time reversal invariance requires (E, B) → (E, −B) Relation • moreover, (%, j) → (%, −j) Summary

1 0∇ · E = % X 2 ∇ · B = 0 X 3 ∇ × E = −∇t B X Maxwell’s equations are time reversal invariant

Dielectric permittivity

Peter Hertel Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager • time reversal invariance requires (E, B) → (E, −B) Relation • moreover, (%, j) → (%, −j) Summary

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 Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager • time reversal invariance requires (E, B) → (E, −B) Relation • moreover, (%, j) → (%, −j) Summary

1 0∇ · E = % X 2 ∇ · B = 0 X 3 ∇ × E = −∇t B X 4 (1/µ0)∇ × B = 0 ∇t E + j X Dielectric permittivity

Peter Hertel Time reversal invariance

Overview

Maxwell’s equations • time reversal (t, x) → (−t, x) Deﬁnition • consequently (v, p) → (−v, −p) Kramers- Kronig • recall p˙ = q{E + v × B} Relation

Onsager • time reversal invariance requires (E, B) → (E, −B) Relation • moreover, (%, j) → (%, −j) Summary

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 • generalize to a possible anisotropic medium

• P˜i(ω) = χij(ω)E˜j(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 Onsager symmetry relation

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • P˜i(ω) = χij(ω)E˜j(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 Onsager symmetry relation

Overview

Maxwell’s equations • generalize to a possible anisotropic medium Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • 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 Onsager symmetry relation

Overview

Maxwell’s equations • generalize to a possible anisotropic medium Deﬁnition ˜ ˜ Kramers- • Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3) Kronig Relation

Onsager Relation

Summary • 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 Onsager symmetry relation

Overview

Maxwell’s equations • generalize to a possible anisotropic medium Deﬁnition ˜ ˜ Kramers- • Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3) Kronig Relation • susceptibility is a property of matter in thermal equilibrium

Onsager Relation

Summary • 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 Onsager symmetry relation

Overview

Summary equilibrium • 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 Onsager symmetry relation

Overview

Summary equilibrium • such as temperature, mechanical strain, external static electric or magnetic ﬁelds, . . . • Interchanging indexes and reverting B is a symmetry

• χij(ω; T, S, E, B,... ) = χji(ω; T, S, E, −B,... )

Dielectric permittivity

Peter Hertel Onsager symmetry relation

Overview

Summary equilibrium • such as temperature, mechanical strain, external static electric or magnetic ﬁelds, . . .

• χij = χij(ω; T, S, E, B,... ) • χij(ω; T, S, E, B,... ) = χji(ω; T, S, E, −B,... )

Dielectric permittivity

Peter Hertel Onsager symmetry relation

Overview

Summary 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

Peter Hertel Onsager symmetry relation

Overview

Summary 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

Kramers- Kronig Relation

Onsager Relation

Summary

Lars Onsager, Norwegian/US American physical chemist, 1903-1976 • 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 • 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).

Dielectric permittivity

Peter Hertel Summary

Overview

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • 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 • 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).

Dielectric permittivity

Peter Hertel Summary

Overview • Optics deals with the interaction of light with matter.

Maxwell’s equations

Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • 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).

Dielectric permittivity

Peter Hertel Summary

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- Kronig Relation

Onsager Relation

Summary • 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).

Dielectric permittivity

Peter Hertel Summary

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager Relation

Summary • 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).

Dielectric permittivity

Peter Hertel Summary

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulated Maxwell’s equation in the presence of Relation matter and specialized to a homogeneous non-magnetic Summary linear medium. • 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).

Dielectric permittivity

Peter Hertel Summary

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulated Maxwell’s equation in the presence of Relation matter and specialized to a homogeneous non-magnetic Summary linear medium. • We described the retarded response of matter to a perturbation by an electric ﬁeld • 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).

Dielectric permittivity

Peter Hertel Summary

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulated Maxwell’s equation in the presence of Relation matter and specialized to a homogeneous non-magnetic Summary linear medium. • We described the retarded response of matter to a perturbation by an electric ﬁeld • It is described by the frequency-dependent susceptibility • If properly generalized to anisotropic media, the susceptibility is a symmetric tensor, provided an exteral magnetic ﬁeld is reversed (Onsager).

Dielectric permittivity

Peter Hertel Summary

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulated Maxwell’s equation in the presence of Relation matter and specialized to a homogeneous non-magnetic Summary 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

Peter Hertel Summary

Overview • Optics deals with the interaction of light with matter. Maxwell’s • Light, as an electromagnetic ﬁeld, obeys Maxwell’s equations equations. Deﬁnition

Kramers- • The Lorentz force on charged particles describes the Kronig Relation interaction of light with matter.

Onsager • We recapitulated Maxwell’s equation in the presence of Relation matter and specialized to a homogeneous non-magnetic Summary 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).