Image Processing Reference

In-Depth Information

These are plane waves that propagate from
z
=
Z
> 0 toward
z
= 0.

4.

(

)

12

/

e

ikxky

(

+

)

e

where

k

=+ −+

k

kk

and

kk k

+≤

(2.86)

ik z

2

2

2

2

2

2

x

y

z

z

x

y

x

y

These are evanescent waves decaying exponentially from the plane

z
=
Z
toward the plane
z
= 0.

In an absorbing material, the permittivity and the permeability are complex

quantities. Consequently, the wave vector becomes complex and the waves

decay. The propagating and evanescent waves can be considered slow-decaying

and fast-decaying waves.

The angular spectrum representation for a field that propagates into the

half-space
z
≥ 0 and whose sources are located in
z
< 0 is obtained by neglect-

ing the latter term. The spectral amplitudes
A
(
k
x
,
k
y
; ω) of each plane wave

component are given by the Fourier transform of the field in the plane
z
= 0.

Thus, if the field is known in the plane
z
= 0, it is known throughout the half

space
z
> 0 by using this angular spectrum representation.

We also note that the boundary conditions for an electromagnetic field

at the interface between two media are derived from the integral forms of

Maxwell's equations. The boundary conditions are valid for both the time-

dependent fields and their spectral components, and they take the form

ˆ

nEE

×−=

2

(

)

0

(2.87)

1

ˆ

nHHJ

×

(

−

)

=

(2.88)

su

2

1

ˆ

nD D

⋅ −=

2

(

)

ρ
su

(2.89)

1

ˆ

nB B

⋅ −=

2

(

)

0

(2.90)

1

where
n
is the unit vector normal to the interface pointing from the input

medium and into a second medium. The vector
J
su
denotes the surface cur-

rent density, and ρ
su
is the surface charge density. The fields are governed by

Maxwell's equations, which is why the boundary conditions are not indepen-

dent of each other.

2.4 evAneSCent And pRopAgAtIng wAveS

In this chapter, we ignore scattering objects that vary in time and make the

assumption that illumination is by using a plane (quasi-) monochromatic wave.

This greatly simplifies the theoretical model. Assuming linearity, pulsed illu-

mination could be modeled using a set of waves with different frequencies

and an incident nonplane wave can similarly be decomposed into a set of

weighted angular plane waves. The scattering from an object can be assumed

to generate an infinite set of plane waves not all of which are propagating. This

can be expressed mathematically as

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