Image Processing Reference
In-Depth Information
per unit distance, just as a temporal frequency describes some number of tem-
poral cycles or periods of a wave per unit time. If we insert this expression
into the Helmholtz equation, we obtain a differential equation for
E
(
k
x
,
k
y
;
z
,
ω) which is
∂
2
Ek k
(,
;, )
z
ω
xy
(2.79)
+
k
2
Ek k
(,
;, )
z
ω
=
0
z
x
y
∂
2
z
where
k
=+ −+
[
k
(
kk
)]
/
when
kk k
+≤
2
2
212
2
2
2
(2.80)
z
x
y
x
y
k
=+ +−
i
[(
kk k
2
2
)
212
]
/
when
k
2
+>
kk
2
2
(2.81)
z
x
y
x
y
and
k
=
k
0
n
is the wave number in the 0 <
z
<
Z
region. Using the general solu-
tion for this differential equation, the field in this region can be expressed as
∞
∫
ikxkykz
(
++
)
E
(, ,, )
xyz
ω
=
A kk
(
,
;
ω
)
e
dd
kk
x
y
z
xy
xy
−∞
∞
∫
(2.82)
ik xkykz
(
+−
)
dd
+
Bk k
(,
;)
ω
e
k
k
x
y
z
xy
xy
−∞
where
A
(
k
x
,
k
y
; ω) and
B
(
k
x
,
k
y
; ω) are arbitrary functions. This expression is
known as the angular spectrum representation of the
E
field When the refrac-
tive index is real and positive, the
z
-component of the wave vector,
k
z
, is either
real or purely imaginary. Therefore, this expression for the fields represents
that wave field in terms of four types of plane wave solutions:
1.
(
)
12
/
e
ikxky
(
+
)
e
where
k
=+ −+
k
2
kk
2
2
and
kk k
2
+≤
2
2
(2.83)
ik z
x
y
z
z
x
y
x
y
These solutions are homogenous plane waves that propagate from
the boundary plane
z
= 0 toward the boundary plane
z
=
Z
> 0.
2.
12
/
e
ikxky
(
+
)
e
where
k
=+ +−
i
(
kk k
2
2
)
2
and
k
2
+>
kk
2
2
(2.84)
ik z
x
y
z
z
x
y
x
y
These solutions are evanescent waves that decay exponentially from
plane
z
= 0 toward the boundary plane
z
=
Z
> 0.
3.
(
)
12
/
e
ikxky
(
+
)
e
ik z
where
k
=+ −+
k
2
kk
2
2
and
kk k
2
+≤
2
2
(2.85)
x
y
z
z
x
y
x
y
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