Chapter 10
Dispersive Material
10.1 Introduction
For many problems one can obtain acceptably accurate results by assuming material parameters are
constants. However, constant material parameters are inherently an approximation. For example,
it is impossible to have a lossless dielectric with constant permittivity (except, of course, for free
space). If such a material did exist it would violate causality. (For a material to behave causally, the
Kramers-Kronig relations show that for any deviation from free-space behavior the imaginary part
of the permittivity or permeability, i.e., the loss, cannot vanish for all frequencies. Nevertheless, as
far as causality is concerned, the loss can be arbitrarily small.)
A non-unity, scalar, constant relative permittivity is equivalent to assuming the polarization
of charge within a material is instantaneous and in perfect proportion to the applied electric field.
Furthermore, the reaction is the same at all frequencies, is the same in all directions, is the same for
all times, and the same proportionality constant holds for all field strengths. In reality, essentially
none of the these assumptions are absolutely correct. The relationship between the electric flux
density D and the electric field E can reflect all the complexity of the real world. Instead of simply
having D = ϵE where ϵ is a scalar constant, one can make ϵ a tensor to describe different behaviors
in different directions (off diagonal terms would indicate the amount of coupling from one direction
to another). The permittivity can also be written as a nonlinear function of the applied electric field
to account for nonlinear media. The material parameters can be functions of time (such as might
pertain to a material which is being heated). Finally, one should not forget that the permittivity can
be a function of position to account for spatial inhomogeneities.
When the speed of light in a material is a function of frequency, the material is said to be
dispersive. The fact that the FDTD grid is dispersive has been discussed in Chap. 7. That dispersion
is a numerical artifact and is distinct from the subject of this chapter. We have also considered
lossy materials. Even when the conductivity of a material is assumed to be constant, the material
is dispersive (ref. (5.70) which shows that the phase constant is not linearly proportional to the
frequency which must be the case for non-dispersive propagation).
When the permittivity or permeability of a material are functions of frequency, the material is
dispersive. In time-harmonic form one can account for the frequency dependence of permittivity
by writing
ˆ
D(ω) = ˆϵ(ω)
ˆ
E(ω), where a caret is used to indicate a quantity in the frequency domain.
†
Lecture notes by John Schneider. fdtd-dispersive-material.tex
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