Page 239 - Phase Space Optics Fundamentals and Applications

220 Chapter Seven

Total radiant power in terms of L is given by

(z) = L( r, ˆ n) cos d dA (7.7)
A 1/2
In this expression, the z value indicates the source location on the z
axis. The symbol A under the integral stands for the integration over
the area of the source, and the 1/2 under the integral implies the
angle integration is limited to the right half space, 0 ≤ ≤ /2 and
0 ≤ ≤ 2 . The product cos d is called the projected solid angle
differential element. Following Eq. (7.7), the spectral radiant power
ˆ
ˆ
(z, ) is found by using the spectral radiance L( r, ˆ n, ) with units of
−1
−2
−1
Wcm sr Hz .
Far enough away from the source, the details of the source struc-
ture become less important but the characteristic angular distribu-
tion of radiation assumes importance. To describe this situation, the
term radiant intensity I (z, ˆ n) with units W sr −1 is used. Its solid angle
integral over the right half-space yields the total radiant power

(z) = I (z, ˆ n) d (7.8)
1/2
ˆ
The solid angle integral over the spectral radiant intensity I(z, ˆ n, ) with
−1
units of W sr Hz −1 yields the total spectral radiant power.
Among the various functions defined above, the basic one is the
ˆ
spectral radiance L( r, ˆ n, ). In terms of it, the other functions may
be derived by regrouping and performing the appropriate integral.
The interrelationships among the spectral functions are displayed in
Table 7.1.

ˆ
L( r, ˆ n, ), [W cm −2 sr −1 Hz −1 ]

ˆ ˆ
(z, ) = L( r, ˆ n, ) cos d dA
A 1/2

ˆ ˆ ˆ ˆ
M( r, ) = L( r, ˆ n, ) cos d I(z, ˆ n, ) = cos L( r, ˆ n, ) dA
1/2 A

ˆ ˆ ˆ ˆ
(z, ) = M( r, ) dA (z, ) = I(z, ˆ n, ) d
A 1/2

TABLE 7.1 Interrelationships among the Spectral Functions of
Conventional Radiometry