Image Processing Reference

In-Depth Information

14

Chapter 2

when we recognize that the beam in Fig. 2.2 is considered to be infinitely thin; that

is, as it passes through each layer, the layers are homogeneous and without struc-

ture. If we expand the beam size, we find that each part of the beam is affected dif-

ferently as it passes through the atmosphere. This can be illustrated by tracing a

large number of rays distributed over the beam through the atmosphere. In such

cases, it can be much more convenient to identify the points of common phase in

the beam—the wavefront—follow its path through the various layers and note the

phase variations.

Light, on large scales, is governed by the wave equation, which describes how

light propagates:

(
)

2

1

∂

Er

t

(
)

2

∇−

Er

=

0

.

(2.2)

2

2

c

∂

Here
c
is the speed of light,
t
is time,
E
is the energy, and

is the partial derivative

with direction. The solution of the wave equation is basically a sinusoidally varying

wave, which in complex notation is

∇

(

)

(
)

Ee
i

ωϕ
,

t

−−

r

Er

=

(2.3)

0

where
E
0
is the amplitude,

is the angular frequency,
t
is the time,
k
is the wave

number,
x
is the displacement, and

ω

is the phase. Equation 2.3 can be rewritten us-

ing the Euler Relation (Kaplan 1981), reducing the form of the equation to a sinu-

soid. This sinusoidally varying wave and its important elements are detailed in

Fig. 2.3.

ϕ

Figure 2.3
A 3D illustration of a propagating electromagnetic wave.