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342   Chapter Eleven


               where t = (t 1 + t 2 )/2 and  t = t 1 − t 2 . Then   (t 1 ,t 2 ) is defined as

                         t      t               C(t,  t)
                    t +    ,t −    =                           1/2  (11.7)
                         2      2     [C (t +  t/2, 0) C (t −  t/2, 0)]
                 Using the Schwarz inequality, it is straightforward to show that
               0 ≤| (t +  t/2,t −  t/2)|≤ 1. This leads directly to the inequality

                                              t            t
                                    2
                         0 ≤|C(t,  t)| ≤ C  t +  , 0 C  t −  , 0    (11.8)
                                              2            2
                 Theupperandlowerboundsonthedegreeofcoherencefollowfrom
               Eq. (11.8). However, it is difficult to determine   (t +  t/2,t −  t/2)
               experimentally since it becomes singular for times at which C(t,  t)
               is zero. A more practically useful definition is offered by integrating
               Eq. (11.8) over the entire (t,  t) space and dividing by the quantity on
               the right-hand side, leading to the integral degree of coherence

                                                      2
                                         dt d t|C(t,  t)|
                              0 ≤   =                   ≤ 1         (11.9)
                                                      2
                                             dt C(t, 0)
                 Here and in the remainder of this chapter, all integrals are under-
               stood to be from −∞ to +∞. An integral degree of coherence strictly
               smaller than 1 corresponds to a partially coherent train in which the
               pulse amplitude and/or phase fluctuates, in which case C(t,  t) is the
               fundamental quantity of interest. When   = 1, the ensemble is said
               to be fully coherent (identical pulses) and C(t,  t) factorizes. In the
               latter case the electric field becomes the fundamental quantity of in-
               terest and is readily retrieved from the two-time correlation function
               using

                                     |E(t)|=  C(t, 0)              (11.10)
               and, with t 2 held fixed,

                                     Im [C[(t + t 2 )/2,t − t 2 ]]
                                 −1
                    Arg[E(t)] = tan                        + 	 0    (11.11)
                                     Re [C[(t + t 2 )/2,t − t 2 ]]
               where 	 0 is an undetermined constant. It is important to note that Eqs.
               (11.10) and (11.11) are valid only if the integral degree of coherence
               has been explicitly demonstrated to be equal to unity, which of course
               requires that the two-time correlation function or equivalent repre-
               sentation in frequency or phase space be measured. Thus, in cases
               where an ensemble or train of pulses, rather than an individual pulse,
               is used for application or experimentation, pulse-shape characteriza-
               tion efforts must ultimately be directed toward measurement of the
               ensemble statistics.
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