Page 321 - Phase Space Optics Fundamentals and Applications
P. 321

302   Chapter Nine

                                               n
                                        d 1,2
                      n 2
                      n 1,2
                      n 1
                      n 0,2
                      n 0,1

                      n
                       0
                                                                  x
                      n 0,–1

                      n –1

                                    S

               FIGURE 9.10 Self-imaging based on the Montgomery condition.




               that the condition in Eq. (9.43) automatically fulfills the self-imaging
               condition for all other cross-terms with m  = 0.



          9.10 Self-Imaging and Incoherent
                 Illumination
               Phase-space optics allows one to describe both coherent and partially
               coherent signals with one consistent formalism. While it is not the
               purpose of this discussion to explore partially coherent optics in phase
               space, the study of self-imaging conditions allows us to take a sneak
               peak at how problems involving incoherent signals can be addressed.
                 To this end we consider the setup in Fig. 9.11, which is a double-
               grating configuration. The first grating G 1 is illuminated with inco-
               herent quasi-monochromatic light. At distance z L a second grating is
               located in the front focal plane of a Fourier lens f . This is one of vari-
               ous systems for studying the Lau effect. 24  The first grating G 1 serves
               as a periodically modulated incoherent light source, which is used to
               illuminate a the second grating G 2 located at some distance z L along
               the optical axis. For a discrete set of grating separations it is possible to
               observe a pattern of high-contrast fringes in the Fraunhofer diffraction
               plane of the 2- f system.
                 The Lau effect easily rivals the Talbot effect in terms of its mathe-
               matical beauty and its potential for applications. Perhaps due to its
               slightly higher complexity, however, the Lau effect has unquestion-
               ably received much less attention than coherent self-imaging.
   316   317   318   319   320   321   322   323   324   325   326