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Radiometry, Wave Optics, and Spatial Coherence 235
The spectral radiant intensity takes the form
1 2 1 2 2 2 2
r
r
J ( ˆ n, ) = (k
g ) exp − (k
g ) p + q I s ( s , ) d s
2 2
(7.69)
2
2
2
The factor ( p + q ) = (sin ) shows the dependence on angle
which varies from 0 to /2.
The set of Eqs. (7.65) to (7.69) describes the radiometry with respect
to the initial plane z = 0. After propagation the Gaussians retain their
form but the scale changes. For example, I s after propagation takes
the form
2
1 r 1
I ( r, ) = I Q 2 exp − 2 2 (7.70)
1 + z 2
1 + z
Q
The parameter z is defined by z ≡ z/(k
Q
g ) in which z is the
location of the plane parallel to the initial plane z = 0. The peak value
2
2
is reduced by the factor 1/(1 + z ), and the variance
is increased by
g
2
the factor 1 + z . The spatial coherence function scales in an analogous
manner but acquires a linear phase factor. For details, see Ref. 3 where
more references to the literature are included.
Acknowledgments
Randal Johnson, MacAulay-Brown, Inc., for help in the design of the
figures. Juliet Hughes, University of Arizona, for help in the layout of
the manuscript
References
1. R. C, Jones, “Terminology in photometry and radiometry,” J. Opt. Soc. Am.
53(11): 1314–1315 (1963).
2. Max Born and Emil Wolf, Principles of Optics, Pergamon Press, New York, 1959.
3. Arvind S. Marathay, Elements of Optical Coherence Theory, Wiley, New York, 1982,
Chapter 3, pp. 29–32.
4. I. S. Sokolnikoff and E. M. Redheffer, Mathematics of Physics and Modern Engi-
neering, McGraw-Hill, New York, 1958, p. 322.
5. M. J. Beran and G. B. Parrent, Jr., Theory of Partial Coherence, Prentice-Hall,
Englewood Cliffs, N.J., 1964; see also “incoherent source,” ref. 3, Marathay,
p. 78.
6. The details of stationary-phase calculation are described in Chapter 6, “Diffrac-
tion,” by A. S. Marathay, J. F. McCalmont, and J. Shiefman, Optical Engineer’s
Desk Reference, Edited by William L. Wolfe, Pub. Optical Society of America and
The International Society for Optical Engineering, 2003.
7. James Harvey and Roland Shack, “Aberrations of diffracted wave fields,” Appl.
Opt. 17: 3003 (1978).