Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 16

(2.7)

∇⋅ B (, ) ω = 0

(2.8)

∇× Er (, ) ωωω = i Br ( ,)

∇× Hr (, ) ωωω =− i Dr ( ,) + Jr (, ) ω (2.9)

In the presence of matter, for example, a scattering object or an (in)homoge-
nous medium, an electromagnetic field induces a polarization P and a magne-
tization M in that object. The vectors D and B take into account this response
of the matter, which is driven by the bound and free electron movement in the
materials involved. D and B are connected to the polarization and magnetiza-
tion by the relationships given here.

Dr (,) t = ε 0 Er Pr ( ,) t + (,) t (2.10)

Br (,) t = µ 0 [ Hr Mr (,) t + ( ,)] t (2.11)

where ε 0 and μ 0 are the electric permittivity and magnetic permeability of in
a vacuum, respectively. The relation between E and D and between H and B
can be very complicated. For example, the permittivity can be a tensor quan-
tity and so the effective permittivity in one coordinate direction can be quite
different in another and we need to understand how an arbitrary incident
E field is affected. Worse than this, all materials are to some extent nonlin-
ear, meaning that the permittivity (or permeability) is a function of the field
amplitude. Even simple models to describe this become rapidly unwieldy
with the permittivity tensor being of rank 3 or 4 depending on whether the
field or field intensity drives the changes. (A rank 3 tensor has 27 terms and
a rank 4 has 81 terms). In this introductory topic, we will ignore nonlinear
phenomena and assume that the incident fields are sufficiently weak and that
they play no significant role. We do expect, however, that the permittivity or
permeability profiles are going to be spatially varying and are probably tenso-
rial in nature.
To begin with, let us consider the electromagnetic field in a simple lin-
ear, isotropic, stationary, spatially nondispersive, but temporally dispersive
medium. In this case, the polarization and the magnetization are connected
to the electric and magnetic field via the following convolution relationships.

t

εχ

Pr (,) t = ( , r t − t ′ ) E r (, ) t ′ d t ′ (2.12)

0 e

−∞

t

∫ χ

Mr (,) t = ( , r r t − t ′ ) (, ) t ′ d t ′ (2.13)

m

−∞

where the response functions χ e and χ m , known as the electric and magnetic
susceptibilities, vanish for t ′ > t . These equations embody one of the most
fundamental principles in physics, namely, causality. The polarization and