Page 247 - Phase Space Optics Fundamentals and Applications

228 Chapter Seven

Now following Walther, 10 we identify the expression in the square
−2
−1
−1
brackets as the spectral radiance (W cm sr Hz ) function and de-
note it as follows:

m 1 1
r
r
r
B( s , 0, ˆ n, ) = 2 s + r s12 , s − r s12 , 0,
2 2
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2
r
r
×exp −ik ˆ n · s12 d s12
7 8
m 1 1
r
= s + r s12 , 0, ∗ r s − r s12 , 0,
2 2 2
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2
r
r
×exp −ik ˆ n · s12 d s12 (7.33)
With this definition, the total spectral radiant power is

2
r
r
( ) = B( s , 0, ˆ n, )md d s (7.34)
1/2
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Observe that we were able to do this by defining average and dif-
ference variables. Following this first step, Marchand and Wolf 11 de-
veloped the remaining functions of radiometry, as we shall now do.
The definition of spectral radiance given in Eq. (7.33) almost looks
like an expression of a Wigner distribution 12 W f of a function f (x)
∞

1 1

W f (x, ) = f x + x f ∗ x − x exp(−i2 x ) dx (7.35)
2 2
−∞
In this definition, the variables x and are Fourier conjugate vari-
r
ables. The spectral radiance B( s , 0, ˆ n, ) defined in Eq. (7.33) is similar
to the Wigner distribution, but its arguments s and ˆ n are not Fourier
r
r
conjugate variables; ˆ n is a Fourier conjugate to s12 .
The spectral radiant exitance of a source or the spectral irradiance on a
receiving plane is defined by

r
r
M( s , 0, ) = B( s , 0, ˆ n, )md (7.36)
1/2
In this way, the total spectral radiant power takes the form

2
r
r
( ) = M( s , 0, ) d s (7.37)
A