Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 29

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