Frequency Dispersion and the Analytical Properties of the Permittivity ε(ω). The Propagation of Electromagnetic Waves in Dispersive Media

Recommended textbooks:

Landau L D and Lifshitz E M Electrodynamics of Continuous Media (Oxford: Pergamon)

Lipson S G and Lipson H Optical Physics (Cambridge: Cambridge University Press)

In an isotropic medium without spatial dispersion the relation for the electric induction D(t) and the electric field E(t) takes the form (see (2.4)): D ( t ) = E ( t ) + ∫ 0 ∞ f ( τ ) E ( t - τ ) d τ (4.1) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062882/62037627-852f-4b67-910a-450744f5c7a7/content/math218.tif"/> so the dielectric permeability tensor for such a medium is

ε_αβ = ε(ω)δ_αβ

where the permittivity is ε ( ω ) = 1 + ∫ 0 ∞ f ( τ ) e I ω τ d τ (4.2) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062882/62037627-852f-4b67-910a-450744f5c7a7/content/math219.tif"/> The function ε (ω) is defined such that it is an analytical function in the upper half-plane of the complex variable ω, and ε(ω) → 1 for |ω| j → ∞. The real, (ε'), and the imaginary (ε") parts of this function are linked by Kramers 48 Kronig relations : ε ( ω ) = 1 + 1 π P + ∫ ∞ ε ( ω ) ω - ω d ώ (4.3) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062882/62037627-852f-4b67-910a-450744f5c7a7/content/math220.tif"/> ε " ( ω ) ⁢   = - 1 π P ∫ − ∞ + ∞ ε ⁢     ′ ( ω ⁢ ′ ) - 1 ω ⁢ ′ - ω d ω ⁢ ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062882/62037627-852f-4b67-910a-450744f5c7a7/content/math221.tif"/> These relations are slightly changed for a conducting medium, for which ε(ω) has a pole at ω = 0 (see problem 28).