Image Processing Reference

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[

∇+

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

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