Page 211 - Electromagnetics

These are the Kronig–Kramers relations, named after R. de L. Kronig and H.A. Kramers
who derived them independently. The expressions show that causality requires the real
and imaginary parts of the permittivity to depend upon each other through the Hilbert
transform pair [142].
It is often more convenient to write the Kronig–Kramers relations in a form that
employs only positive frequencies. This can be accomplished using the even–odd behavior
c
of the real and imaginary parts of ˜ . Breaking the integrals in (4.35)–(4.36) into the
ranges (−∞, 0) and (0, ∞), and substituting from (4.27), we can show that
2 ∞ ˜ (r, )
c
c
˜ (r,ω) − 0 =− P.V. d , (4.37)
2
π 0 − ω 2
2ω ∞ ˜ (r, ) σ 0 (r)
c
c
˜ (r,ω) = P.V. d − . (4.38)
2
π 0 − ω 2 ω
The symbol P.V. in this case indicates that values of the integrand around both = 0
and = ω must be excluded from the integration. The details of the derivation of
(4.37)–(4.38) are left as an exercise. We shall use (4.37) in § 4.6 to demonstrate the
Kronig–Kramers relationship for a model of complex permittivity of an actual material.
c
We cannot specify ˜ arbitrarily; for a passive medium ˜ c must be zero or negative at
all values of ω, and (4.36) will not necessarily return these required values. However, if
c
we have a good measurement or physical model for ˜ , as might come from studies of the
absorbing properties of the material, we can approximate the real part of the permittivity
using (4.35). We shall demonstrate this using simple models for permittivity in § 4.6.
The Kronig–Kramers properties hold for µ as well. We must for practical reasons
consider the fact that magnetization becomes unimportant at a much lower frequency
than does polarization, so that the infinite integrals in the Kronig–Kramers relations
should be truncated at some upper frequency ωmax . If we use a model or measured
values of ˜µ to determine ˜µ , the form of the relation (4.37) should be [107]

2 ω max ˜µ (r, )

˜ µ (r,ω) − µ 0 =− P.V. d ,
2
π 0 − ω 2
where ωmax is the frequency at which magnetization ceases to be important, and above
which ˜µ = µ 0 .

4.5 Dissipated and stored energy in a dispersive medium
Let us write down Poynting’s power balance theorem for a dispersive medium. Writing
i
c
J = J + J we have (§ 2.9.5)

∂D ∂B
i c
− J · E = J · E +∇ · [E × H] + E · + H · . (4.39)
∂t ∂t
We cannot express this in terms of the time rate of change of a stored energy density
because of the difficulty in interpreting the term
∂D ∂B
E · + H · (4.40)
∂t ∂t

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