Page 238 - Phase Space Optics Fundamentals and Applications

Radiometry, Wave Optics, and Spatial Coherence 219

For a planar source of area Aand in the plane z = constant, the total
radiant power is defined by the integral over the area of the source

(z) = M(x, y, z) dx dy (7.3)
A
On the other hand, the radiation from the source is received on a
surface. It is described by the term radiant incidence or irradiance E with
−2
units of W cm . It too is a function of position r. The spectral irradiance
−2
−1
ˆ E( r, ) has units of W cm Hz .
Let us reconsider the radiant exitance M. It is a function of position
ˆ
r = ˆ ix + ˆ jy + kz (7.4)
From every point (x, y) on the source, it may channel different
amounts of radiation energy in different directions ˆ n with compo-
nents
ˆ
ˆ n = ˆ ip + ˆ jq + km
(7.5)
ˆ
= ˆ i(sin cos ) + ˆ j(sin sin ) + k cos
where and are the polar angles, is measured from the z axis, and
the azimuthal angle is measured from the x axis in the xy plane.
The differential element of solid angle d with units of steradians (sr)
about the direction ˆ n is

dp dq
d = = sin d d (7.6)
m
This equality is established by using the Jacobian of the change of
variables from ( p, q)to ( , ).
To account for the variation of the source properly as a function
of position and direction, a term called radiance L( r, ˆ n) is used, with
−1
−2
units of W cm sr . An observer looking at the source in the direction
of ˆ n sees the projected area dA proj = dA cos as shown in Fig. 7.1.

dA proj = dA cosq
ˆ n

dA q dW

FIGURE 7.1 Projected area.