Page 253 - Phase Space Optics Fundamentals and Applications

234 Chapter Seven

As before, A is the effective area of the open aperture in the z = 0
plane containing the coherent field.

7.7.4 Quasi-Homogeneous Wave-Field
The spatial coherence of a quasi-homogeneous plane wave field has
a factored form

s ( r s1 , r s2 , ) = I s ( r s , ) g s ( r s12 , ) (7.63)

where I s is the ensemble average of the squared modulus of the optical
field and is assumed to be very broad and slowly varying compared
to the coherence function g s ( r s12 , ).
As a simple example, let us use Gaussians to represent the wave
field in the initial plane z = 0,

r 2 s
I s ( r s , ) = I Q exp −
2
2
Q
(7.64)
2
r s12
r
g s ( s12 , ) = exp −
2
2
g
The widths
Q and
g are both frequency -dependent and
Q >>
g .
The first order of business is to obtain the spectral radiance

1 m 2 1 2 2 2
r
B( s , ˆ n, ) = I s ( r s , ) (k
g ) exp − (k
g ) ( p + q ) (7.65)
2 2
It is a Gaussian in angular spectrum space.
The spectral radiant power is given by
1 (k
g ) 2

2
2
2
r
r
( ) = I s ( s , ) d s (k
g ) 2 exp − (1 − ) d
0 2
(7.66)
In doing the angle integral, we substituted = cos . The angle inte-
gral, contained within curly braces, can be evaluated by using Mathe-
2
matica. It can be plotted as a function of (k
g ) /2. For values of
g >
and for values
g >> , the angle integral is rather approximately
unity to a high degree of accuracy. Hence, we may write

2
r
r
( ) = I s ( s , ) d s (7.67)
In like manner, the spectral radiant exitance can be shown to be
M( s , ) = I s ( r s , ) (7.68)
r