Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 26

that is the cause of the scattering and that is the function we wish to recover,
whereas our measurements are of the product U ψ and we do not know ψ. If ψ
is known, for example, whether it is a constant or a plane wave, then we can
invert this for the scattering function.

2.3 plAne wAveS
Let us now consider a quasi-monochromatic wave. A truly monochromatic
wave is not physical since it would be noncausal. However, it is very con-
venient to consider an extremely narrowband source that approximately
oscillates at a single constant frequency. Our knowledge of Fourier analy-
sis tells us that a finite bandwidth source can always be represented by a
weighted sum of single frequency waves. This greatly simplifies our analysis
and facilitates our intuitive understanding of scattering mechanisms and
inverse scattering procedures. We have already neglected nonlinear pro-
cesses in our media and so we have a linear system, but we do still allow for
temporal dispersion, that is, that different frequencies might see different
refractive index values.
A quasi-monochromatic field component at a frequency ω will have the
form

Fr Fr (,) t = ω ω ( ,) e it (2.68)

where the spatial part F ( r , ω) is complex, that is, it has a real and imaginary
part or, equivalently, a magnitude and a phase. At high frequencies, for exam-
ple, greater than 1 THz, we tend to measure the averaged energy or time-aver-
aged Poynting vector associated with the electromagnetic field, E × H , which
in homogeneous measurement space is proportional to | E | 2 . Consequently,
when we calculate this value, we lose the very important information about
the complex field, namely, its phase. For time-harmonic fields, the time-depen-
dent Maxwell's equations take the form

∇⋅ Dr (, ) ωρω = ( ,) r

(2.69)

∇⋅ B (, ) ω = 0 (2.70)

∇× Er (, ) ωωω = i Br ( ,) (2.71)

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

(2.72)

These equations are equivalent to those of the spectral components of an
arbitrary time-dependent field shown above. Conse q uently, th e solutions for
E ( r , ω) and H ( r , ω) are identical to the solutions for E (, ω and H (, ).

ω
For dielectric materials the free charge density ρ and the source current
density J s are zero. This is valid also for materials having appreciable conduc-
tivity, that is, for metals. If the medium is homogenous, that is, if the material
parameters do not depend on r , the wave equations reduce to the so-called
homogenous Helmholtz equations.