Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 17

the magnetization at time t depend on the electric and magnetic fields at all
other time instants t before t (origin of temporal dispersion). As can be seen
in Appendix A, the consequences of causality are far reaching, leading to a
very powerful restriction on the real and imaginary parts of the spectral (i.e.,
Fourier) representation of these susceptibilities. The spectral susceptibility is
an analytic function, and the real and imaginary parts are locked together by
an integral transform relationship or dispersion relationship, also known as
a Hilbert transform.
The medium may also be spatially dispersive in which case the above rela-
tionships would be convolutions over the spatial variable r as well. The effects
of spatial dispersion can be most easily understood and observed at inter-
faces between two distinct media where one might expect some reflection
to occur. In media or objects having material fluctuations comparable to the
wavelength of the electromagnetic wave being employed, the “reflections” can
be quite complex and may well be wavelength-dependent. Once a medium
departs from a regular or periodic pattern of material differences, we no lon-
ger talk of “diffraction” but of scattering.
Applying the Fourier transform to the expressions for D and B , we can write
the constitutive relations in the space-frequency domain as

Dr (, ) ωε εω ω = ( ,)(, ) rEr (2.14)

0 r

Br (, ) ωµµω ω = ( ,)(, ) rHr (2.15)

0 r

with ε r = 1 + χ e and μ r = 1 + χ m being the relative dielectric permittivity and
magnetic permeability of the medium, respectively.
Besides polarization and magnetization, the electromagnetic field may
induce currents. The free current density in the material can be divided into
two parts: the conduction current density, J c , induced by an external field and
the source current density J s . The conduction current density is determined
by the electric component of the incident field by

Jr rEr

c (, )

ωσωω = ( ,)(, ) (2.16)

Starting from Maxwell's curl equations above and using the expressions for
D and B , one can derive the inhomogeneous wave equations in the space-time
domain for both the electric and magnetic fields Similarly, using Maxwell's
equations and the constitutive relations expressed in the frequency domain,
we obtain the wave equations in the space-frequency domain

∇×∇× Er rEr (, ) ω − k

2

( ,)(, ) ω ω µµ ωω = i ( ,)(, ) r Jr (2.17)

0 r s

∇×∇× Hr rHr (, ) ω − k ( ,)(, ) ω ω = ∇× J r ( ,) ω (2.18)

2

s

where k = k 0 n is the wave number in the medium. The term k 0 = ω/ c 0 is the
wave number in vacuum in which the velocity of the wave is c 0 = (ε 0 μ 0 ) −1/2 , and
n 2 = ε r μ r denotes the square of the refractive index of the material. Making the
substitution ε r + i σ/ω 0 → ε r , we can incorporate the effect of free conducting