Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 15

Tw o

Electromagnetic Waves

2.1 MAxwell'S equAtIonS
The behavior of the electromagnetic field in any medium is governed by
Maxwell's equations. The macroscopic Maxwell's equations in SI units are
given by

∇⋅ D (,) rt = ρ ( ,) rt (2.1)

∇⋅ B (,) rt = 0 (2.2)

δ B (,) rt

t

(2.3)

∇× E (,) rt =−

δ

∂ D (,) rt

t

(2.4)

∇× H (,) rt = + J (,) rt

∂

where D denotes the electric displacement, B the magnetic induction, E the
electric field, and H the magnetic field The sources of the field are the free
charge density, ρ, and the free current density, J .
These are continuous functions in macroscopic electrodynamics. The above
equations are written in the space-time domain, but the electromagnetic field
and, in particular, its interaction with matter are more easily analyzed in
the space-frequency domain. To do this, we use t he spectral representation
of time-dependent fields, that is, the spectrum F ( r ,ω) of an arbitrary time-
dependent field F ( r , t ) that is given by its Fourier transform (see Appendix A * ).
Conversely, if F ( r ,ω) is known, the time-dependent field can be calculated by
the inverse Fourier transform as shown here:

∞

∫

F (,) rt = F r ( ,) ω e

− it ω d

ω (2.5)

−∞

Applying the Fourier transform to Maxwell's equations, one obtains them
in the space-frequency domain, and these obviously hold for the spectral
components of the electromagnetic field

∇⋅ Dr (, ) ωρω = ( ,) r (2.6)

* Note that in 1-D F ( r,t ) → F ( x,t ), the Fourier transform of which is F ( k, ω).

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