Page 205 - Electromagnetics

4.4 Constitutive relations in the frequency domain and the
Kronig–Kramers relations
All materials are to some extent dispersive. If a field applied to a material undergoes
a sufficiently rapid change, there is a time lag in the response of the polarization or
magnetization of the atoms. It has been found that such materials have constitutive
relations involving products in the frequency domain, and that the frequency-domain
constitutive parameters are complex, frequency-dependent quantities. We shall restrict
ourselves to the special case of anisotropic materials and refer the reader to Kong [101]
and Lindell [113] for the more general case. For anisotropic materials we write
˜
˜
˜
P = 0 ¯χ · E, (4.11)
e
˜
˜
M = ˜ ¯χ · H, (4.12)
m
˜
¯
˜
˜
D = ˜ ¯ · E = 0 [I + ˜ ¯χ ] · E, (4.13)
e
˜
˜
˜
¯
B = ˜ ¯µ · H = µ 0 [I + ˜ ¯χ ] · H, (4.14)
m
˜
˜
J = ˜ ¯σ · E. (4.15)
By the convolution theorem and the assumption of causality we immediately obtain the
dyadic versions of (2.29)–(2.31):
t

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

B(r, t) = µ 0 H(r, t) + ¯ χ m (r, t − t ) · H(r, t ) dt ,
−∞
t

J(r, t) = ¯ σ(r, t − t ) · E(r, t ) dt .
−∞
These describe the essential behavior of a dispersive material. The susceptances and
conductivity, describing the response of the atomic structure to an applied field, depend
not only on the present value of the applied field but on all past values as well.
Now since D(r, t), B(r, t), and J(r, t) are all real, so are the entries in the dyadic
matrices ¯ (r, t), ¯µ(r, t), and ¯σ(r, t). Thus, applying (4.3) to each entry we must have
∗
∗
∗
˜ ¯ χ (r, −ω) = ˜ ¯χ (r,ω), ˜ ¯ χ (r, −ω) = ˜ ¯χ (r,ω), ˜ ¯ σ(r, −ω) = ˜ ¯σ (r,ω), (4.16)
e e m m
and hence
∗ ∗
˜ ¯ (r, −ω) = ˜ ¯ (r,ω), ˜ ¯ µ(r, −ω) = ˜ ¯µ (r,ω). (4.17)
If we write the constitutive parameters in terms of real and imaginary parts as

˜ ij = ˜ + j ˜ , ˜ µ ij = ˜µ + j ˜µ , ˜ σ ij = ˜σ + j ˜σ ,

ij ij ij ij ij ij
these conditions become

˜ (r, −ω) = ˜ (r,ω), ˜ (r, −ω) =−˜ (r,ω),

ij ij ij ij
and so on. Therefore the real parts of the constitutive parameters are even functions of
frequency, and the imaginary parts are odd functions of frequency.
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