
322 OPTICS FOR ENGINEERS
Now, the autocorrelation function can be written by combining Equations 10.51 and 10.52;
(x
1
, y
1
, x
1
, y
1
) =
−jke
−jkz
1
2πz
1
e
−jk
(
x
1
)
2
+
(
y
1
)
2
2z
1
jke
jkz
1
2πz
1
e
jk
x
2
1
+y
2
1
2z
1
×
E
∗
x
, y
,0
E
(
x, y,0
)
e
−jk
(
x
)
2
+
(
y
)
2
2z
1
e
+jk
x
2
+y
2
2z
1
× e
+jk
x
x
1
+y
y
1
z
1
e
−jk
xx
1
+yy
1
z
1
dx
dy
dx dy. (10.53)
Noting that, for example, x
2
−
x
2
=
x + x
x − x
, we can combine terms to arrive at
(x
1
, y
1
, x
1
, y
1
) =
−jke
−jkz
1
2πz
1
e
jk
(
x
1
)
2
+
(
y
1
)
2
2z
1
jke
jkz
1
2πz
1
e
−jk
x
2
1
+y
2
1
2z
1
×
E
∗
x
, y
,0
E
(
x, y,0
)
e
jk
(
x
+x
)(
x
−x
)
+
(
y
+y
)(
y
−y
)
2z
1
× e
jk
(
xx
1
−x
x
1
)
+
(
yy
1
−y
y
1
)
z
1
dx
dy
dx dy (10.54)
Because of the random phases,
E
= 0. If the source points are all incoherent ...