Page 245 - Phase Space Optics Fundamentals and Applications

226 Chapter Seven

As the angle increases, the far-field condition becomes weaker. For
observation points not satisfying the far-field condition, the higher-
order terms of the stationary-phase approximation cannot be
7
neglected. This fact was realized by Harvey and Shack and they
developed an aberration theory inherent to the diffraction process.
This theory was applied to near-field diffraction whereby Fresnel
diffraction is interpreted as an aberrated form of Fraunhofer diffrac-
tion. In the stationary-phase approximation, there are no restrictions
on the direction cosines p, q and m. Hence, the diffracted field ampli-
tude of Eq. (7.27) is valid over the entire hemisphere, free from any
2
2
2
paraxial restrictions other than p + q + m = 1. For an in-depth
discussion on the application of the stationary-phase approximation
to optical diffraction and imaging, see Mansuripur. 8
9
Later Harvey and coworkers used it to describe diffraction grating
behavior and surface-scattering effects. Next we define radiometric
quantities, starting with the basic relationship given in Eqs. (7.26)
and (7.27).

7.6 Radiometry and Wave Optics
Radiation incident from the left on an aperture in a plane at z = 0
diffracts radiation into the right half-space. The diffracted field re-
sides on a hemisphere of radius r. The origin of the coordinate sys-
tem is in the open aperture. Let d denote a differential element of
2
solid angle. The differential element of area on the hemisphere is r d
2
(cm sr). A radiation detector responds to the ensemble average of the
squared modulus of the optical field; the output is in watts (W). To find
the total power radiated into the right half-space, we integrate over
the hemisphere. The spectral radiant power in the right half-space is
given by

2 2 2 2
( ) = | ( r, )| r sin d d = | ( r, )| r d (7.29)
1/2 1/2
In this expression, the angular brackets denote the ensemble aver-
age. The linear frequency of the radiation is . The integrand contains
the diffracted field ( r, ) as given in Eqs. (7.26) and (7.27). From
Eq. (7.29), it is clear that the ensemble average of the squared modulus of
2
the diffracted field | ( r, )| plays the role of spectral radiance with units
−2
−1
Wcm sr Hz −1 for radiation detection on the surface of the hemi-
sphere.
To reduce the complexity of the resulting equations and for
notational convenience, it is useful to introduce vectors defined
in the following way: The coherence function may involve the