Image Processing Reference
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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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