Image Processing Reference
InDepth Information
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 quasimonochromatic 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 quasimonochromatic 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 timeaver
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 timeharmonic fields, the timedepen
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 timedependent 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 socalled
homogenous Helmholtz equations.
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