Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 18

electrons that might exist in the medium, as described by a very classical
model for conduction known as the Drude model.
It is important to stress that these wave equations, which describe the prop-
agation of an electromagnetic field, are valid in an inhomogeneous or scatter-
ing media. If we know how the permittivity and permeability (or equivalently
the refractive index) vary as a function of space and frequency, we can predict
how a wave that is incident in that medium will propagate and scatter. The
converse, which is the focus of this topic, is a much harder problem.

2.2 gReen'S FunCtIon
Green's function is very important and gives the electromagnetic field pro-
duced by a point source for a given medium or object. In free space or a vac-
uum, Green's function from a point source is an outgoing spherical wave.
Here I is the unit dyad; a dyadic Green's function

is made of three vector Green's functions. Thus, in a homogeneous (infinite)
medium, Green's function,

G is the Green dyadic and

G , satisfies the equation.

(2.19)

∇×∇× Grr (, ,) ′ ω − k 2 () ω ωδ G rr (, ,) ′ = Ir r ( − ′ )

where I is the unit tensor and δ denotes the delta function describing the point
source located at some position r ′. A solution of this equation can be formally
written as

1
2

�

Grr (, ,)

′ =+∇∇

ω I Grr (, ,) ω

′ n i (2.20)

k

where n i is a unit vector and where

e ik
4π|

|| rr − ′

Grr (, ,) ′ω =

(2.21)

rr − ′ |

is the outgoing scalar Green's function that satisfies the wave equation. In
terms of Green's function the solution to

∇×∇× Er rEr (, ) ω − k ( ,)(, ) ω ω µµ ωω = i ( ,)(, ) r Jr

s

(2.22)

2

0 r

for the current distribution J s , located in some volume V , becomes

∫

Er Er (, ) ω = ( ,) ω + i ωµ µω () (, ,) () Grr ω J s r ′ d r

′

3

′ (2.23)

0 0 r

V

where r is a point located outside V . The field E 0 is the incident wave, which
is typically assumed to be a plane wave, and which one could imagine as
being created by a point source at infinite distance away from the medium in
volume V .