Page 319 - Phase Space Optics Fundamentals and Applications
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300   Chapter Nine


               The change of the self-imaging length, compared to the case of plane
               wave illumination, can be expressed with the help of a magnification
               factor
                                              1
                                      m R =                         (9.37)
                                           1 − z T /R
               For diverging waves R > 0, the self-imaging distance is increased,
               while for converging wavefronts the self-imaging distance is reduced
               compared to plane wave illumination. The Talbot distance can be re-
               covered for R →∞.
                 The modified period of the first self-image can be deduced from the
               new frequency spacing of the   lines; i.e., we can write
                                       1    1    d
                                         =    −                     (9.38)
                                      2d     2d   R
               or

                                        d = m R d                   (9.39)

               Finally, the new radius of curvature at the first self-imaging plane
               becomes

                                    R = R + z p = m R R             (9.40)


               Additional self-imaging planes can be found by substituting R back
               intoEq.(9.36).Foraconvergingwavefrontwefindself-imagingplanes
               with increasing density along the optical axis, the closer they are lo-
               cated to the focal point of the illuminating spherical wave. At the focal
               point we expect to see self-imaging replaced by the Fourier spectrum
               of the grating. In phase space this corresponds to a horizontal shear
               which turns all   lines vertical; i.e., the vertical projections will no
               longer contain any information about the interference terms, but will
               only show the distribution of discrete self-terms.
                 The case of self-imaging under spherical illuminations also serves
               as an example to highlight other important generalizations of Talbot
               self-imaging. For investigating diverging and converging wavefronts,
               we had to drop the requirement of obtaining a perfect replica of the
               complex amplitude. Instead, we accepted a scaled replica as a gener-
               alized self-image. It should be mentioned that the geometric scaling
               under spherical illuminations also applies to the fractional Talbot ef-
               fect and was used to design Talbot array illuminators. 38
                 A further generalization is self-imaging in arbitrary ABCD optical
               systems. Fresnel diffraction from a grating under spherical illumina-
               tion is equivalent to plane wave illumination of the grating followed
               by a system consisting of a parabolic lens and free-space propaga-
               tion. In fact, conditions for obtaining self-images in arbitrary ABCD
               systems have already been investigated. 39
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