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

Search WWH ::

Custom Search