Electromagnetic Waves - Introduction to Imaging from Scattered Fields - page 27

[ ∇+

2

k ωω

2

( )] Er (, ) = 0 (2.73)

[ ∇+

2

k ω

2

( )] Hr (, ) ω = 0 (2.74)

The most useful solution for these Helmholtz equations is the quasi-mono-
chromatic plane wave and the spatial part of the solutions is

Er (, ) ω = E ( ω ) e i

kr ⋅

(2.75)

0

Hr (, ) ω = H ( ω ) e i

kr ⋅

(2.76)

0

where E 0 (ω) and H 0 (ω) are the complex electric and magnetic field amplitudes,
that is, they are complex, having a magnitude and phase. The vector k is the
wave vector which must satisfy

k ⋅ = k

2 εωµω

() () (2.77)

0 r r

This expression is very important since it effectively describes all the prop-
agation characteristics of the medium and is known as the dispersion rela-
tionship, indicating the necessary relationship that must exist between the
frequency and the wave number. Just to confuse matters, the relation between
the real and imaginary parts of the fields, as a function of frequency, also
satisfy a dispersion relation (the Hilbert transform mentioned earlier), and if
the medium or scatterer is discrete, that is, is embedded in a homogeneous
background, then one can show that the real and imaginary parts of the field
as a function of k will also satisfy a similar Hilbert transform relation. We
will revisit this kind of dispersion relationship later in the topic, since it pro-
vides a key to not only recovering unmeasured phase information from mea-
sured intensity data, but also toa mechanism for solving the inverse scattering
problem.
Finally, we note the concept of the angular plane wave representation of a
scattered or propagating field By decomposing an arbitrary field into a super-
position of waves in space, just as one might in frequency, we gain a pow-
erful tool and additional insight. The angular spectrum representation is a
particularly useful technique for understanding the propagation of optical
fields in homogenous media. It is the expansion of an electromagnetic field in
a weighted summation of plane waves with variable amplitudes and propaga-
tion directions.
If we consider a quasi-monochromatic field in a region 0 ≤ z ≤ Z with sources
located outside of this region, then the spatial part of the field satisfies the
Helmholtz equations. We can express the field by a Fourier integral

∞
∫

(2.78)

; ,) (

ikxky + )

E (, ,, ) xyz ω = E kk ( , z ω e dd k k

x y

xy xy

−∞

Here k x and k y are spatial frequencies corresponding to the real space coor-
dinates x and y . Spatial frequencies describe some number of spatial periods