ABSTRACT
In linear electrodynamics of a continuous medium the electric current density ~j(~r, t) is proportional to the electric field ~E(~r, t): ~j = σˆ ~E, where σˆ is a linear operator. In the general case the current density takes the following form (see Problem 1.0.5):
jα(~r, t) =
∫ d~r′dt′σαβ(~r, ~r′, t, t′)Eβ(~r′, t′) (2.1)
By introducing the electric induction ~D as
~D(~r, t) = ~E(~r, t) + 4π
dt′~j(~r, t′),
Maxwell’s equations can be written as follows:
~∇× ~E = −1 c
∂ ~B
∂t , (2.2)
~∇× ~B = 1 c
∂ ~E
∂t + 4π
c (~j +~jext) =
= 1
c
∂ ~D
∂t + 4π
c ~jext ≡ 1
c
∂(ǫˆ ~E)
∂t + 4π
c ~jext, (2.3)
where ~jext is the electric current density of external source. The dielectric permeability operator, ǫˆ in equation (2.3), has a form analogous to that in (2.1):
Dα(~r, t) =
∫ d~r′dt′ǫαβ(~r, ~r′, t, t′)Eβ(~r′, t′) (2.4)
In the case of a homogeneous and steady medium the tensor functions, σαβ and ǫαβ in equations (2.1) and (2.4), must have the following form:
σαβ(~r, ~r ′, t, t′) = σαβ(~ρ, τ),
ǫαβ(~r, ~r ′, t, t′) = ǫαβ(~ρ, τ),
~ρ = ~r − ~r′, τ = t− t′
Physics Edition
This makes it convenient to explore Fourier transforms, because in the (~k, ω) representation the integral operators σˆ and ǫˆ in equations (2.1) and (2.4) are
replaced by simple multipliers: the conductivity tensor σαβ(~k, ω), and the
dielectric permeability tensor ǫαβ(~k, ω):
jα(~k, ω) = σαβ(~k, ω)Eβ(~k, ω), (2.5)
Dα(~k, ω) = ǫαβ(~k, ω)Eβ(~k, ω) (2.6)
According to equations (2.1) and (2.4),
σαβ(~k, ω) =
∫ σαβ(~ρ, τ) exp[−i(~k · ~ρ− ωτ)]d~ρdτ
ǫαβ(~k, ω) =
∫ ǫαβ(~ρ, τ) exp[−i(~k · ~ρ− ωτ)]d~ρdτ,
with
ǫαβ(~k, ω) = δαβ + 4πi
ω σαβ(~k, ω) (2.7)
In the (~k, ω) representation Maxwell’s equations (2.2) and (2.3) in the absence of external current are equivalent to the following system of equations for the electric field ~E:
LαβEβ =
( kαkβ − k2δαβ + ω
c2 ǫαβ
) Eβ = 0; (2.8)
therefore, the dispersion equation, which determines a link between the frequency, ω, of the electromagnetic wave in a medium and its wave vector, ~k, takes the form
det||Lαβ(~k, ω)|| = 0 (2.9) In the case of an isotropic and inversion-invariant medium (the latter means that the medium is identical with its stereoisomeric counterpart) the most general form of the dielectric permeability tensor is as follows:
ǫαβ(~k, ω) = ǫ||(k, ω) kαkβ k2
+ ǫ⊥(k, ω) ( δαβ − kαkβ
k2
) (2.10)
The electromagnetic waves in such a medium can be separated into the longitudinal waves (l), with ~E ‖ ~k, and the transverse waves (t), for which ~E ⊥ ~k. The respective dispersion equations, which follow from equations (2.9) and (2.10), read:
ǫ||(k, ωl) = 0, k2 = ω2t c2 ǫ⊥(k, ωt) (2.11)
The damping of the electromagnetic waves is determined by the antihermitian part of the dielectric permeability tensor. Thus, the energy dissipation power per unit volume is equal to
Q = − iω 8π
ǫ (A) αβ E
∗ 0αE0β , (2.12)
where ~E0 is the electric field amplitude of the wave. In the so-called trans-
parency range, where the wave damping is weak, i.e., ǫ (A) αβ ≪ ǫ(H)αβ , the
volumetric energy , W , and the volumetric linear momentum , ~P , of the wave can be defined as
W = 1
16πω
∂
∂ω [ω2ǫ
~k, ω)]E∗0αE0β , ~P = ~k
ω W, (2.13)
where ǫ (H) αβ is the hermitian part of the dielectric permeability tensor.
