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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.