Image Processing Reference

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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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