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Phase Space in Ultrafast Optics    355


                                                        Phase
                                                       modulator



                                                   φ



                                                               t
                              (a)                         (c)

                    ω                         ω








                                        t                         t
                               (b)                       (d)

               FIGURE 11.5 Representations of the effect of (a) dispersive propagation in a
               two-grating compressor and (c) propagation in a quadratic temporal phase
               modulator. (b) Dispersive propagation leads to a shear of the chronocyclic
               representation along the time axis. (d) The quadratic temporal phase
               modulator leads to a shear of the chronocyclic representation along the
               frequency axis.



               belonging to the transfer matrix



                                               1    0
                                  T modulator =                    (11.51)
                                              −	     1
                                                 t
               This adds a time-dependent instantaneous frequency −	 t to the

                                                                 t
               pulse, as is observed in Fig. 11.5d. Analogy to the combined action
               of free-space propagation followed by a lens suggests that prop-
               agation in a time-stationary dispersive element (such as a grating
               pair compressor) and passage through a time-nonstationary phase
               modulator (such as an electrooptic crystal driven by a temporal
               parabolic voltage) will cause a rotation of the phase-space distri-
               bution of the temporal ray bundle in the chronocyclic phase space.
               The combination of dispersive propagation 	     ,1  followed by a tem-
               poral lens 	 , again followed by dispersive propagation 	     ,2 , leads

                          t
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