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352   Chapter Eleven





                   Height                       Height


                         L
                             (a)                         (c)




                   Angle                        Angle





                                Height                       Height
                             (b)                         (d)

               FIGURE 11.4 Representations of free-space propagation and propagation in
               a thin lens in geometrical optics and in the associated phase space. The first
               row corresponds to the geometrical optics representations of (a) three
               different rays propagating in free space, where it can be seen that the rays
               before (depicted in gray) and after (depicted in black) conserve their angle
               and acquire an angle-dependent height, and (c) of three different rays
               propagating through a thin lens, where it can be seen that the rays before and
               after the lens conserve their height and acquire a position-dependent angle.
               The second row displays the corresponding phase-space representations
               before (in gray) and after (in black), where the effect of free-space
               propagation is seen in (b) as a shear along the height direction, and the effect
               of the thin lens is seen in (d) as a shear along the angle direction.


                 Because this element is not space-shift-invariant, but rather angle-
               shift-invariant, its form in the conjugate angle space is simpler.


                                         f       if
                                                           2
                             ˜ ˜ K(k x ,k ) =  exp −  (k x − k )   (11.37)
                                  x                      x
                                         k       2k
               The thin lens therefore modifies the wave vector proportionally to
               the lateral position of the ray on the lens, that is, u out = u in − y in /f ,
               but the ray height does not change, that is, y out = y in (Fig. 11.4c).
               This is a shear in the phase space along the wave vector direction
               (Fig. 11.4d). A lens can be combined with free-space propagation
               to rotate the phase-space representation. The transfer matrix di-
               rectly describes modifications of the Wigner function through optical
               systems. 35
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