Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric permittivity

Peter Hertel

University of Osnabruck, Germany

Lecture presented at Nankai University, China

http://www.home.uni-osnabrueck.de/phertel

October/November 2011

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Make it as simple as possible, but not simpler

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Overview

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulate Maxwell’s equation in the presence ofmatter and specialize to a homogeneous non-magneticlinear medium.

• We formulate the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related.

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed.

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations

The electromagnetic field E and B accelerates chargedparticles

p = q{E(t,x) + v ×B(t,x)}

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

1 ε0∇ ·E = %

2 ∇ ·B = 0

3 ∇×E = −∇tB4 (1/µ0)∇×B = ε0∇tE + j

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations

The electromagnetic field E and B accelerates chargedparticles

p = q{E(t,x) + v ×B(t,x)}

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

1 ε0∇ ·E = %

2 ∇ ·B = 0

3 ∇×E = −∇tB4 (1/µ0)∇×B = ε0∇tE + j

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations

The electromagnetic field E and B accelerates chargedparticles

p = q{E(t,x) + v ×B(t,x)}

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

1 ε0∇ ·E = %

2 ∇ ·B = 0

3 ∇×E = −∇tB4 (1/µ0)∇×B = ε0∇tE + j

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations

The electromagnetic field E and B accelerates chargedparticles

p = q{E(t,x) + v ×B(t,x)}

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

1 ε0∇ ·E = %

2 ∇ ·B = 0

3 ∇×E = −∇tB4 (1/µ0)∇×B = ε0∇tE + j

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations

The electromagnetic field E and B accelerates chargedparticles

p = q{E(t,x) + v ×B(t,x)}

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

1 ε0∇ ·E = %

2 ∇ ·B = 0

3 ∇×E = −∇tB4 (1/µ0)∇×B = ε0∇tE + j

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations

The electromagnetic field E and B accelerates chargedparticles

p = q{E(t,x) + v ×B(t,x)}

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

1 ε0∇ ·E = %

2 ∇ ·B = 0

3 ∇×E = −∇tB4 (1/µ0)∇×B = ε0∇tE + j

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations

The electromagnetic field E and B accelerates chargedparticles

p = q{E(t,x) + v ×B(t,x)}

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

1 ε0∇ ·E = %

2 ∇ ·B = 0

3 ∇×E = −∇tB

4 (1/µ0)∇×B = ε0∇tE + j

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations

The electromagnetic field E and B accelerates chargedparticles

p = q{E(t,x) + v ×B(t,x)}

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

1 ε0∇ ·E = %

2 ∇ ·B = 0

3 ∇×E = −∇tB4 (1/µ0)∇×B = ε0∇tE + j

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter I

• polarization P is electric dipole moment per unit volume

• magnetization M is magnetic dipole moment per unitvolume

• electric field strength E causes polarization

• magnetic induction B causes magnetization

• % = −∇ · P + %f

• j = P +∇×M + j f

• density %f and current density j f of free charges

• a vicious circle!

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB

4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

This is good – only free charges are involved

and bad – there are more fields than equations

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Maxwell’s equations in matter II

• introduce auxiliary field D = ε0E + 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 = −∇tB4 ∇×H = D + j f

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

James Clerk Maxwell, 1831-1873

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dielectric susceptibility I

Assume a medium which is

• homogeneous

• non-magnetic

• linear

• P = ε0χE

• M = 0

• dielectric susceptibility χ is dimension-less number

• equivalent D = εε0E

• relative dielectric permittivity ε = 1 + χ

• more precisely . . .

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

More precisely

Assume linear local relation

P (t,x) =

∫dτ G(τ)E(t− τ,x)

causality

G(τ) = 0 for τ < 0

drop x

P (t) =

∫dτ G(τ)E(t− τ)

G is causal influence, or Green’s functions

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

More precisely

Assume linear local relation

P (t,x) =

∫dτ G(τ)E(t− τ,x)

causality

G(τ) = 0 for τ < 0

drop x

P (t) =

∫dτ G(τ)E(t− τ)

G is causal influence, or Green’s functions

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

More precisely

Assume linear local relation

P (t,x) =

∫dτ G(τ)E(t− τ,x)

causality

G(τ) = 0 for τ < 0

drop x

P (t) =

∫dτ G(τ)E(t− τ)

G is causal influence, or Green’s functions

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

More precisely

Assume linear local relation

P (t,x) =

∫dτ G(τ)E(t− τ,x)

causality

G(τ) = 0 for τ < 0

drop x

P (t) =

∫dτ G(τ)E(t− τ)

G is causal influence, or Green’s functions

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

More precisely

Assume linear local relation

P (t,x) =

∫dτ G(τ)E(t− τ,x)

causality

G(τ) = 0 for τ < 0

drop x

P (t) =

∫dτ G(τ)E(t− τ)

G is causal influence, or Green’s functions

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

More precisely

Assume linear local relation

P (t,x) =

∫dτ G(τ)E(t− τ,x)

causality

G(τ) = 0 for τ < 0

drop x

P (t) =

∫dτ G(τ)E(t− τ)

G is causal influence, or Green’s functions

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

More precisely

Assume linear local relation

P (t,x) =

∫dτ G(τ)E(t− τ,x)

causality

G(τ) = 0 for τ < 0

drop x

P (t) =

∫dτ G(τ)E(t− τ)

G is causal influence, or Green’s functions

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

More precisely

Assume linear local relation

P (t,x) =

∫dτ G(τ)E(t− τ,x)

causality

G(τ) = 0 for τ < 0

drop x

P (t) =

∫dτ G(τ)E(t− τ)

G is causal influence, or Green’s functions

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

George Green, 1793-1841

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Fourier transforms

f(t) =

∫dω

2πe−iωt

f(ω)

f(ω) =

∫dt e

+iωtf(t)

convolution h = g ∗ f , i. e.

h(t) =

∫dτ g(τ)f(t− τ)

then

h(ω) = g(ω)f(ω)

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Fourier transforms

f(t) =

∫dω

2πe−iωt

f(ω)

f(ω) =

∫dt e

+iωtf(t)

convolution h = g ∗ f , i. e.

h(t) =

∫dτ g(τ)f(t− τ)

then

h(ω) = g(ω)f(ω)

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Fourier transforms

f(t) =

∫dω

2πe−iωt

f(ω)

f(ω) =

∫dt e

+iωtf(t)

convolution h = g ∗ f , i. e.

h(t) =

∫dτ g(τ)f(t− τ)

then

h(ω) = g(ω)f(ω)

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Fourier transforms

f(t) =

∫dω

2πe−iωt

f(ω)

f(ω) =

∫dt e

+iωtf(t)

convolution h = g ∗ f , i. e.

h(t) =

∫dτ g(τ)f(t− τ)

then

h(ω) = g(ω)f(ω)

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Fourier transforms

f(t) =

∫dω

2πe−iωt

f(ω)

f(ω) =

∫dt e

+iωtf(t)

convolution h = g ∗ f , i. e.

h(t) =

∫dτ g(τ)f(t− τ)

then

h(ω) = g(ω)f(ω)

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Susceptibility II

Recall

P (t) =

∫dτ G(τ)E(t− τ)

Therefore

P (ω) = ε0χ(ω)E(ω)

with

χ(ω) =1

ε0G(ω)

susceptibility χ must depend on frequency ω

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Susceptibility II

Recall

P (t) =

∫dτ G(τ)E(t− τ)

Therefore

P (ω) = ε0χ(ω)E(ω)

with

χ(ω) =1

ε0G(ω)

susceptibility χ must depend on frequency ω

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Susceptibility II

Recall

P (t) =

∫dτ G(τ)E(t− τ)

Therefore

P (ω) = ε0χ(ω)E(ω)

with

χ(ω) =1

ε0G(ω)

susceptibility χ must depend on frequency ω

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Susceptibility II

Recall

P (t) =

∫dτ G(τ)E(t− τ)

Therefore

P (ω) = ε0χ(ω)E(ω)

with

χ(ω) =1

ε0G(ω)

susceptibility χ must depend on frequency ω

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation I

Recall

P (t,x) =

∫dτ G(τ)E(t− τ,x)

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

χ(ω) =

∫du

2πχ(u)θ(ω − u) by convolution theorem

θ(ω) = lim0<η→0

1

η − iω

χ(ω) = lim0<η→0

∫du

2π

χ(u)

η − i(ω − u)dispersion relation

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IRecall

P (t,x) =

∫dτ G(τ)E(t− τ,x)

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

χ(ω) =

∫du

2πχ(u)θ(ω − u) by convolution theorem

θ(ω) = lim0<η→0

1

η − iω

χ(ω) = lim0<η→0

∫du

2π

χ(u)

η − i(ω − u)dispersion relation

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IRecall

P (t,x) =

∫dτ G(τ)E(t− τ,x)

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

χ(ω) =

∫du

2πχ(u)θ(ω − u) by convolution theorem

θ(ω) = lim0<η→0

1

η − iω

χ(ω) = lim0<η→0

∫du

2π

χ(u)

η − i(ω − u)dispersion relation

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IRecall

P (t,x) =

∫dτ G(τ)E(t− τ,x)

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

χ(ω) =

∫du

2πχ(u)θ(ω − u) by convolution theorem

θ(ω) = lim0<η→0

1

η − iω

χ(ω) = lim0<η→0

∫du

2π

χ(u)

η − i(ω − u)dispersion relation

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IRecall

P (t,x) =

∫dτ G(τ)E(t− τ,x)

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

χ(ω) =

∫du

2πχ(u)θ(ω − u) by convolution theorem

θ(ω) = lim0<η→0

1

η − iω

χ(ω) = lim0<η→0

∫du

2π

χ(u)

η − i(ω − u)dispersion relation

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IRecall

P (t,x) =

∫dτ G(τ)E(t− τ,x)

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

χ(ω) =

∫du

2πχ(u)θ(ω − u) by convolution theorem

θ(ω) = lim0<η→0

1

η − iω

χ(ω) = lim0<η→0

∫du

2π

χ(u)

η − i(ω − u)dispersion relation

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Dispersion of white light

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation II

Decompose susceptibility in real and imaginary part

χ(ω) = χ ′(ω) + iχ ′′(ω)

Introduce principle value integral

Pr

∫du

2π· · · =

(∫ ω−η

−∞+

∫ ∞ω+η

)du

2π. . .

Employ

χ(−ω) = χ(ω)∗

χ ′(ω) = 2Pr

∫du

π

uχ ′′(u)

u2 − ω2KKR

χ ′′(ω) = 2Pr

∫du

π

ωχ ′(u)

ω2 − u2inverse KKR

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IIDecompose susceptibility in real and imaginary part

χ(ω) = χ ′(ω) + iχ ′′(ω)

Introduce principle value integral

Pr

∫du

2π· · · =

(∫ ω−η

−∞+

∫ ∞ω+η

)du

2π. . .

Employ

χ(−ω) = χ(ω)∗

χ ′(ω) = 2Pr

∫du

π

uχ ′′(u)

u2 − ω2KKR

χ ′′(ω) = 2Pr

∫du

π

ωχ ′(u)

ω2 − u2inverse KKR

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IIDecompose susceptibility in real and imaginary part

χ(ω) = χ ′(ω) + iχ ′′(ω)

Introduce principle value integral

Pr

∫du

2π· · · =

(∫ ω−η

−∞+

∫ ∞ω+η

)du

2π. . .

Employ

χ(−ω) = χ(ω)∗

χ ′(ω) = 2Pr

∫du

π

uχ ′′(u)

u2 − ω2KKR

χ ′′(ω) = 2Pr

∫du

π

ωχ ′(u)

ω2 − u2inverse KKR

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IIDecompose susceptibility in real and imaginary part

χ(ω) = χ ′(ω) + iχ ′′(ω)

Introduce principle value integral

Pr

∫du

2π· · · =

(∫ ω−η

−∞+

∫ ∞ω+η

)du

2π. . .

Employ

χ(−ω) = χ(ω)∗

χ ′(ω) = 2Pr

∫du

π

uχ ′′(u)

u2 − ω2KKR

χ ′′(ω) = 2Pr

∫du

π

ωχ ′(u)

ω2 − u2inverse KKR

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IIDecompose susceptibility in real and imaginary part

χ(ω) = χ ′(ω) + iχ ′′(ω)

Introduce principle value integral

Pr

∫du

2π· · · =

(∫ ω−η

−∞+

∫ ∞ω+η

)du

2π. . .

Employ

χ(−ω) = χ(ω)∗

χ ′(ω) = 2Pr

∫du

π

uχ ′′(u)

u2 − ω2KKR

χ ′′(ω) = 2Pr

∫du

π

ωχ ′(u)

ω2 − u2inverse KKR

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Kramers-Kronig relation IIDecompose susceptibility in real and imaginary part

χ(ω) = χ ′(ω) + iχ ′′(ω)

Introduce principle value integral

Pr

∫du

2π· · · =

(∫ ω−η

−∞+

∫ ∞ω+η

)du

2π. . .

Employ

χ(−ω) = χ(ω)∗

χ ′(ω) = 2Pr

∫du

π

uχ ′′(u)

u2 − ω2KKR

χ ′′(ω) = 2Pr

∫du

π

ωχ ′(u)

ω2 − u2inverse KKR

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Ralph Kronig, US American physicist, 1904-1995

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Time reversal invariance

• 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 ε0∇ ·E = % X

2 ∇ ·B = 0 X

3 ∇×E = −∇tB X

4 (1/µ0)∇×B = ε0∇tE + j X

Maxwell’s equations are time reversal invariant

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Onsager symmetry relation

• generalize to a possible anisotropic medium

• Pi(ω) = χij(ω)Ej(ω) (sum over j = 1, 2, 3)

• susceptibility is a property of matter in thermal equilibrium

• its value depends on all parameters which affect theequilibrium

• such as temperature, mechanical strain, external staticelectric 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, . . . )

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

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

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).

Dielectricpermittivity

Peter Hertel

Overview

Maxwell’sequations

Definition

Kramers-KronigRelation

OnsagerRelation

Summary

Summary

• Optics deals with the interaction of light with matter.

• Light, as an electromagnetic field, obeys Maxwell’sequations.

• The Lorentz force on charged particles describes theinteraction of light with matter.

• We recapitulated Maxwell’s equation in the presence ofmatter and specialized to a homogeneous non-magneticlinear medium.

• We described the retarded response of matter to aperturbation by an electric field

• It is described by the frequency-dependent susceptibility

• The real part and the imaginary part of the susceptibilityare intimately related (Kramers-Kronig).

• If properly generalized to anisotropic media, thesusceptibility is a symmetric tensor, provided an exteralmagnetic field is reversed (Onsager).