Page 252 - Phase Space Optics Fundamentals and Applications

Radiometry, Wave Optics, and Spatial Coherence 233

2
2
Owing to the factor m = cos , the noncoherent source will appear
darker when viewed at large angles from the normal.
7.7.3 Coherent Wave Fields
The spatial coherence function for coherent fields assumes a factored
form 15

1 1 1 1
s + r s12 , s − r s12 , 0, = U r s + r s12 U ∗ r s − r s12
r
r

2 2 2 2
× ( − 0 ) (7.57)
1
The first factor U( s + r s12 ) = U( r s1 ) is any solution of the Helmholtz
r
2
equation at frequency 0 . The second factor is the complex conju-
r
gate of the first evaluated at a different point s2 . The delta function
simply emphasizes that the coherent wave field is monochromatic at
frequency 0 .
The spectral radiance function of Eq. (7.33) also assumes a fac-
tored form

m ˆ n 2
r
r
r
B r s , 0, ˆ n, 0 = 2 U ( s1 ) exp −i2 · s1 d s1
0

ˆ n 2
r
r
∗
× U ( r s2 ) exp +i2 · s2 d s2 (7.58)
0
Each integral is a spatial Fourier transform. We can redefine this
expression as
2
m ˆ n ˆ n m ˆ n
r
B( s , 0, ˆ n, 0 ) = 2 ˜ U ˜ U ∗ = ˜ U (7.59)
0 0 2 0
0 0
In this expression the argument ˆ n/ 0 is in fact a two-dimensional
spatial frequency variable.
The spectral radiant power takes the form
2
m ˆ n
( ) = A ˜ U d (7.60)

1 0 0
2
The spectral radiant exitance reads
2
m ˆ n
r
M( s , 0, 0 ) = ˜ U d (7.61)

1 0 0
2
Finally, the spectral radiant intensity takes the form
2
m ˆ n
J ( ˆ n, 0 ) = A ˜ U (7.62)

0 0