Page 248 - Phase Space Optics Fundamentals and Applications

Radiometry, Wave Optics, and Spatial Coherence 229

Alternatively, we can regroup Eq. (7.34) to define the spectral radiant
intensity

2
r
r
J ( ˆ n, ) = m B( s , 0, ˆ n, ) d s (7.38)
A
The total spectral radiant power now reads

( ) = J ( ˆ n, ) d (7.39)
1/2
The definitions given in Eqs. (7.32) to (7.39) form the structure of
wave-theoretic radiometry. We based it on the diffracted field as de-
rived from the stationary-phase expression of Eqs. (7.26) and (7.27).
It is valid over the whole hemisphere of large radius centered on the
open aperture containing the field distribution. Thus, the radiometry
formulated in this way is free from any paraxial restrictions.
Comparison of Eqs. (7.39) and (7.29), suggests the relationship

2 2
J ( ˆ n, ) = r | (r, p, q, )| (7.40)
Alternatively, we can start with the definition of spectral radiant
intensity J ( ˆ n, ) of Eq. (7.38) and use in it the expression of spectral
2
2
r
radiance B( s , 0, ˆ n, ) of Eq. (7.33). The integrals on d r s and d r s12
are identified as the ensemble average of the squared modulus of
the diffracted field of Eq. (7.27), thus reestablishing the relationship
shown in Eq. (7.40). We have just established that for radiation detec-
tion on a hemisphere the spectral radiant intensity is directly related
to the ensemble average of the squared modulus of the diffracted field
multiplied by the square of the radius of the hemisphere. Observe that
2
the squared modulus of the field depends on 1/r and the factor r 2
in Eq. (7.40) indicates that the spectral radiant intensity is a conserved
quantity from one hemisphere to the next concentric hemisphere.
The ensemble average of the squared modulus of the optical field
always plays a role in optical detection. However, which radiometric
quantity it represents depends very much on the geometry and exper-
imental arrangement. In relation to Eq. (7.40), the squared modulus
corresponds to the spectral radiant intensity, since the measurement
was presumably made on the surface of a hemisphere. If the geometry
and the experimental arrangements are changed, the above conclu-
sion will not hold. For example, suppose the measurement is made
on a plane tangent to the hemisphere and perpendicular to the z axis.
When the detector explores the points on the plane, it will subtend a
projected-solid angle md at the origin in the plane of the diffracting
aperture. Let us convert the diffraction pattern on a hemisphere to the