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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).
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