Page 250 - Phase Space Optics Fundamentals and Applications

Radiometry, Wave Optics, and Spatial Coherence 231

7.7 Examples
7.7.1 Blackbody Radiation
Mehta and Wolf 13 have calculated the spatial coherence function of
radiation in thermal equilibrium with the walls of a cavity. We can
write it in the form

1 1 sin kr s12
r s + r s12 , r s − r s12 , 0, = 2 S (7.44)
2 2 kr s12

2 2
In this expression r s12 = (x s1 − x s2 ) + (y s1 − y s2 ) is the magnitude
r
of the vector s12 . The function S is defined by
8 h 3 1
S ≡ S( ,T) = 3 (7.45)
h
c exp ( / k B T) − 1
This is in fact the spectral density (properly denoted as du/d = S)
−3
−3
−1
and has units J cm Hz ; u is the energy density (J cm ), and the
frequency is in hertz (Hz) of the blackbody radiation in frequency
space.
Now we follow Palmer and Grant 14 and observe that radiation
in thermal equilibrium with the walls of a cavity escapes (through
a small hole in the wall) with velocity c and spreads in free space
−1
−1
over 4 sr. We multiply Eq. (7.45) by c/4 (cm s sr ) to obtain the
spectral density as accessible to measurement in free space,
c 8 h 3 1
S BB ≡ S BB ( ,T) = 3
h
4 c exp ( / k B T) − 1
2h 3 1
= (7.46)
c 2 exp ( / k B T) − 1
h
−1
−2
−1
It has the units of W cm sr Hz . In this way, the spatial coherence
of blackbody radiation in free space may be written
sin (kr s12 )
R BB (r s12 , ) = 2 S BB ( ,T) (7.47)
kr s12
We use this spatial coherence function in the definition of spectral
radiance in Eq. (7.33). By use of table of integrals, this expression can
be evaluated to give

2
m 1
r
B( s , 0, ˆ n, ) = 2 2 S BB · = S BB ( ,T) (7.48)
2 m
This is the spectral radiance of blackbody radiation in free space
−2
−1
and has the units of W cm sr Hz −1 as pointed out in relation to
Eq. (7.46). It is independent of the average variable s .
r