Page 246 - Phase Space Optics Fundamentals and Applications

Radiometry, Wave Optics, and Spatial Coherence 227

variables (x s1 ,y s1 , 0), (x s2 ,y s2 , 0), for which we introduce average and
difference variables:
1 1
x s ≡ (x s1 + x s2 ), y s ≡ (y s1 + y s2 ), x s12 ≡ (x s1 − x s2 ),
2 2
y s12 ≡ (y s1 − y s2 )
Next we introduce vectors s ≡ ˆ ix s + ˆ jy s , r s12 ≡ ˆ ix s12 + ˆ jy s12 , and
r
1
r
we can also have combination vectors s + r s12 = ˆ ix s1 + ˆ jy s1 and r s −
2
1
r
2 s12 = ˆ ix s2 + ˆ jy s2 . On the receiving side use, r ≡ ˆ ix + ˆ jy +
ˆ
kz and the integration over the area elements dx s1 dy s1 dx s2 dy s2 =
2
2
dx s dy s dx s12 dy s12 = d r s d r s12 . The unit normal vector is defined by
ˆ
ˆ
ˆ n = ( ˆ ix + ˆ jy + kz)/r = ˆ ip+ ˆ jq +km, where p, q, and m are the actual
direction cosines. Thus, (xx s12 + yy s12 )/r = ˆ n · r s12 .
Armed with this symbolic notation, we are now ready to proceed
with the formulation of radiometry for wave optics. In the computa-
tion of Eq. (7.29), we need to define the spatial coherence function
∗
(x s1 ,y s1 ,x s2 ,y s2 , 0, ) = (x s1 ,y s1 , 0, ) (x s2 ,y s2 , 0, ) ,
7
1 1 1
r s + r s12 , r s − r s12 , 0, = s + r s12 , 0,
r
2 2 2
8
1
× ∗ r s − r s12 , 0, (7.30)
2
The angular brackets denote the ensemble average. The coherence
function can describe a coherent, partially coherent, or noncoherent
input field in the z = 0 plane. In the computation, we can use the av-
erage and difference variables, as done in Eq. (7.30). The total spectral
radiant power of Eq. (7.29) may be expressed as
⎡

z
1 2 1 1
2
r
r
( ) = r d ⎣ 2 2 s + r s12 , s − r s12 , 0,
1/2 r r 2 2
A
⎤

xx s12 + yy s12 2 2
r
r
×exp −ik d s d s12 ⎦ (7.31)
r
This expression can be regrouped in the following way,
⎡

2 m 1 1
r
r
r
( ) = d s md ⎣ 2 s + r s12 , s − r s12 , 0,
1/2 2 2
A A
⎤
2
r r ⎦ (7.32)
× exp −ik ˆ n · s12 d s12