Page 208 - Electromagnetics

The implications of causality on the behavior of the constitutive parameters in the
time domain can be easily identified. Writing (2.29) and (2.31) after setting u = t − t

and then u = t , we have

∞

D(r, t) = 0 E(r, t) + 0 χ e (r, t )E(r, t − t ) dt ,
0

∞

J(r, t) = σ(r, t )E(r, t − t ) dt .
0
We see that there is no contribution from values of χ e (r, t) or σ(r, t) for times t < 0.So
we can write
∞

χ e (r, t )E(r, t − t ) dt ,

D(r, t) = 0 E(r, t) + 0
−∞

∞

J(r, t) = σ(r, t )E(r, t − t ) dt ,
−∞
with the additional assumption
χ e (r, t) = 0, t < 0, σ(r, t) = 0, t < 0. (4.30)
By (4.30) we can write the frequency-domain complex permittivity (4.26) as
1 ∞ ∞
c − jωt − jωt
˜ (r,ω) − 0 = σ(r, t )e dt + 0 χ e (r, t )e dt . (4.31)
jω 0 0
In order to derive the Kronig–Kramers relations we must understand the behavior of
c
˜ (r,ω) − 0 in the complex ω-plane. Writing ω = ω r + jω i , we need to establish the
following two properties.
c
Property 1: The function ˜ (r,ω) − 0 is analytic in the lower half-plane (ω i < 0)
except at ω = 0 where it has a simple pole.
We can establish the analyticity of ˜σ(r,ω) by integrating over any closed contour in
the lower half-plane. We have

∞ ∞

˜ σ(r,ω) dω = σ(r, t )e − jωt dt dω = σ(r, t ) e − jωt dω dt . (4.32)
0 0
Note that an exchange in the order of integration in the above expression is only valid
for ω in the lower half-plane where lim t →∞ e − jωt = 0. Since the function f (ω) = e − jωt is

analytic in the lower half-plane, its closed contour integral is zero by the Cauchy–Goursat
theorem. Thus, by (4.32) we have

˜ σ(r,ω) dω = 0.

Then, since ˜σ may be assumed to be continuous in the lower half-plane for a physical
medium, and since its closed path integral is zero for all possible paths , it is by Morera’s
theorem [110] analytic in the lower half-plane. By similar reasoning χ e (r,ω) is analytic
in the lower half-plane. Since the function 1/ω has a simple pole at ω = 0, the composite
c
function ˜ (r,ω) − 0 given by (4.31) is analytic in the lower half-plane excluding ω = 0
where it has a simple pole.

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