Image Processing Reference

In-Depth Information

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).

15

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