Page 240 - Phase Space Optics Fundamentals and Applications

Radiometry, Wave Optics, and Spatial Coherence 221

7.3 Lambertian Sources
Although sources in general do have position-dependent and/or
angle-dependent properties, it is often advantageous to consider the
limiting case of a source whose properties are independent of posi-
tion and direction. This is an idealization, never realized in practice in
the strict sense but approached in practice to a good approximation.
Consider a source whose spectral radiance is independent of position
r on the source and the direction of observation ˆ n, that is,
ˆ
ˆ
L( r, ˆ n, ) = L 0 ( ) (7.9)
A source described by Eq. (7.9) is called a Lambertian source. For a
planar Lambertian source radiating in the right half space, it follows
from the relationships shown in Table 7.1 that,
ˆ ˆ
= AL 0
ˆ ˆ
M = L 0 (7.10)
ˆ ˆ
I = cos AL 0
In the first two relationships, the factor (sr) is the value of the
hemispherical solid angle with proper account of the cos weighting.
The spectral radiant intensity relationship correctly accounts for the
projection Acos along the viewing direction of the source area A.
Such a source would appear uniformly bright to an observer viewing
it from different directions.

7.4 Mutual Coherence Function
Radiometry and wave optics can be brought together by use of the
mutual coherence function (MCF). It is a statistical quantity, defined in
terms of the wave function. In principle, it can be studied through
optical experiments, and hence it is regarded as an “observable.” We
give a very brief introduction of the mutual coherence function in this
section before proceeding to the radiometry of wave optics. The details
2
of the theory may be found in Born and Wolf, Beran and Parrent, 5
and Marathay. 3
The MCF is defined as a cross-correlation:
T
1
r
∗
r
( 1 , r 2 , ) = ( r 1 ,t) ( r 2 ,t + ) = lim ( 1 ,t) ( 2 ,t + )dt
∗
r
T→∞ T
0
(7.11)
The field is any solution of the time-dependent wave equation. The
MCF contains the field evaluated at two different points at two dif-
ferent times. We have assumed that the field is stationary in time,