Page 360 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 341

One quantity, perhaps the simplest, that describes some of the sta-
tistical properties of the ensemble is the nonstationary two-time field
correlation function

∗
(t 1 ,t 2 ) = E(t 1 )E (t 2 ) (11.5)

where the angle brackets indicate an average over the ensemble of
pulses and time is referenced to a frame moving at the pulse velocity.
If each pulse in the train is an independent realization of a stochastic
ensemble, the time average is equivalent to an ensemble average, by
definition. This enables the coherence of the train to be defined oper-
ationally in a reasonable way. For a train of identical pulses (t 1 ,t 2 )
factorizes into E(t 1 )E (t 2 ), and the analytic signal E(t) is proportional
∗
to (t, t 2 ), where t 2 is such that E(t 2 ) is nonzero. Thus, any pulse mea-
surement method capable of reconstructing the correlation function
is also capable of returning the electric field when such a description
will suffice.
This definition of the ensemble is suitable for describing a train
of optical pulses for which the pulse-to-pulse temporal phase (and
indeed amplitude) varies more or less randomly. It is always ade-
quate for situations where pulses are measured individually. How-
ever, when averages are taken over trains of pulses, especially those
10
for which the carrier-envelope phase is fixed, the proper description
of the ensemble must involve consideration of the field of the entire
train, which has a nonstationary correlation function. It is formally
quite difficult to formulate rigorously even the simplest of concepts,
such as the spectrum, for a nonstationary field such as this. Proce-
dures along the lines of those developed by Wiener and Khintchine 11
must then be extended to define properly the correlation function of
nonstationary fields. 12
The correlation function (t 1 ,t 2 ) provides a quantitative descrip-
tion of fluctuations from pulse to pulse in the electric field at time
t 1 relative to those at time t 2 . This is a complete description of the
pulse ensemble as long as the fluctuations obey normal (or Gaussian)
statistics. If not, then it is the simplest of a hierarchy of multitime
correlation functions defining the ensemble. The degree to which an
ensemble consists of identical pulses may be obtained from (t 1 ,t 2 )
in terms of an integral degree of temporal coherence , where is
11
readily derived from the time-domain analog of Born and Wolf’s de-
gree of coherence (t 1 ,t 2 ), by first redefining the two-time correlation
function in terms of a center-time coordinate t and a difference-time
coordinate t

C(t, t) = (t 1 ,t 2 ) (11.6)