Image Processing Reference

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the magnetization at time
t
depend on the electric and magnetic fields at all

other time instants
t
before
t
(origin of temporal dispersion). As can be seen

in Appendix A, the consequences of causality are far reaching, leading to a

very powerful restriction on the real and imaginary parts of the spectral (i.e.,

Fourier) representation of these susceptibilities. The spectral susceptibility is

an analytic function, and the real and imaginary parts are locked together by

an integral transform relationship or dispersion relationship, also known as

a Hilbert transform.

The medium may also be spatially dispersive in which case the above rela-

tionships would be convolutions over the spatial variable
r
as well. The effects

of spatial dispersion can be most easily understood and observed at inter-

faces between two distinct media where one might expect some reflection

to occur. In media or objects having material fluctuations comparable to the

wavelength of the electromagnetic wave being employed, the “reflections” can

be quite complex and may well be wavelength-dependent. Once a medium

departs from a regular or periodic pattern of material differences, we no lon-

ger talk of “diffraction” but of scattering.

Applying the Fourier transform to the expressions for
D
and
B
, we can write

the constitutive relations in the space-frequency domain as

Dr

(, )

ωε εω ω

=

( ,)(, )

rEr

(2.14)

0

r

Br

(, )

ωµµω ω

=

( ,)(, )

rHr

(2.15)

0

r

with ε
r
= 1 + χ
e
and μ
r
= 1 + χ
m
being the relative dielectric permittivity and

magnetic permeability of the medium, respectively.

Besides polarization and magnetization, the electromagnetic field may

induce currents. The free current density in the material can be divided into

two parts: the conduction current density,
J
c
, induced by an external field and

the source current density
J
s
. The conduction current density is determined

by the electric component of the incident field by

Jr rEr

c
(, )

ωσωω

=

( ,)(, )

(2.16)

Starting from Maxwell's curl equations above and using the expressions for

D
and
B
, one can derive the inhomogeneous wave equations in the space-time

domain for both the electric and magnetic fields Similarly, using Maxwell's

equations and the constitutive relations expressed in the frequency domain,

we obtain the wave equations in the space-frequency domain

∇×∇×

Er rEr

(, )

ω

−

k

2

( ,)(, )

ω ω µµ ωω

=

i

( ,)(, )

r Jr

(2.17)

0

r

s

∇×∇×

Hr rHr

(, )

ω

−

k

( ,)(, )

ω

ω

= ∇×

J r

( ,)

ω

(2.18)

2

s

where
k
=
k
0
n
is the wave number in the medium. The term
k
0
= ω/
c
0
is the

wave number in vacuum in which the velocity of the wave is
c
0
= (ε
0
μ
0
)
−1/2
, and

n
2
= ε
r
μ
r
denotes the square of the refractive index of the material. Making the

substitution ε
r
+
i
σ/ω
0
→ ε
r
, we can incorporate the effect of free conducting

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