Page 359 - Phase Space Optics Fundamentals and Applications

340 Chapter Eleven

its derivative ∂ /∂ is the group delay T( ) at the corresponding
frequency, i.e., the time of arrival of a group of optical frequencies
around .
To reconstruct the temporal electric field, it is necessary and suffi-
cient to measure its Fourier transform for a finite set of frequencies.
9
The Whittaker-Shannon sampling theorem asserts that if the field has
compact support contained in a range t, a sampling of ˜ E( ) at the
Nyquist frequency interval of 2 / t is sufficient for reconstructing
the analytic signal E(t) and consequently the electric field ε(t) exactly.
These equivalent representations of the field in terms of the com-
plementary variables t and suggest that an appropriate phase space
for representing the fields is the two-dimensional chronocyclic phase
space (t, ) in which these variables are arguments of joint time-
frequency distributions describing the pulse field. Such distributions
provide a description that is relevant for measuring pulses using stan-
dard photodetectors. This is because they account properly for the
fluctuations in the set of pulses that contribute to the detected signal.
Further, they also provide an intuitive representation of the temporal
and spectral structure of the pulses, such as a time-dependent fre-
quency or chirp.

11.2.2 Pulse Ensembles and Correlation
Functions
Applications and experiments involving ultrashort optical pulses of-
ten rely on a train or ensemble of pulses rather than a single pulse. In
this case the pulses must be characterized using quantities related
to the ensemble of which they are realizations. The mean electric
field may be defined in some cases for the ensemble as the square
root of the mean intensity times the exponential of the mean phase.
This may be measured directly provided a single-shot measurement
is possible for each pulse of the ensemble; that is, the mean intensity
and phase are obtained respectively by independently averaging the
measured intensities and the measured phases. However, the mean
quantities might not give an insightful picture of the pulse ensemble
(e.g., one could have large variations of the spectral phase from pulse
to pulse and obtain a mean spectral phase that is identically zero).
Furthermore, it is more usual for a multishot measurement to be made
and the detected signal averaged over this sample set of pulses. Even
assuming that the ensemble is ergodic, it is not the case that the field re-
constructed from the averaged signal is the mean field of the ensemble.
In certain cases, though, it is possible to show this directly. Such a mean
electric field is not the most general or useful quantity—often the fluc-
tuations of the pulses are important. When this is the case, the electric
field amplitude and phase of an individual pulse may be meaningless.