Page 397 - Phase Space Optics Fundamentals and Applications

378 Chapter Eleven

2 is the second-order dispersion of the chirped pulse. For pulses in
the hundreds of femtoseconds regime and shorter, spectral shearing
interferometryhasprovedtobearobustandeffectivemethodofdeter-
mining the amplitude and phase of optical, infrared, UX, and indeed
EUV pulses.
Electric field reconstruction with spectral shearing interferometry
relies primarily on the steps described in Fig. 11.11. Once the spectral
phase of the two-time correlation function ( + ) − ( − ) has
been extracted, it can be integrated into the spectral phase of the pulse
under test ( ). The reconstruction algorithm is therefore direct and
algebraic, even when the time-nonstationary filters are synthesized
using nonlinear optics. Spectral shearing interferometry can also be
used without a delay between the two interfering pulses, in which
case the phase of the interferometric component can be retrieved by
scanning the relative phase of the interfering pulses. 89

11.4 Conclusions
A phase-space representation of ultrashort optical pulses is useful for
three reasons. First, it provides a useful tool for visualizing pulsed
fields and enables an intuitive way to understand central concepts
such as chirp, group delay, and instantaneous frequency. Second, it
enables representation of pulse ensembles in terms of the lowest-
order correlation function of the ensemble. Third, it provides a simple
framework for understanding measurement methods that are based
on square-law detectors, which are universal in optics. In this chapter,
we set out the basic definitions required to define the chronocyclic
phase space, and its space-time extension, as well as developed a cat-
alog of modern measurement techniques in terms of phase space (or
correlation space) distributions. This catalog can be shown to be com-
plete and can be understood in terms of manipulation and sectioning
of the phase-space distributions by means of linear filters and photo-
detectors. This clarifies an important misconception about ultrafast
measurements—that they require a nonlinear response somewhere in
the apparatus. In fact, a linear filter with a nonstationary response is
sufficient. Indeed the minimum necessary conditions for obtaining a
signal that encodes sufficient information to invert the electric field
of the pulse must contain at least one time-stationary and one time-
nonstationary filter. Further, we have shown how all the currently
most popular methods can be incorporated into this framework. Be-
cause it is necessary to synthesize a nonstationary filter by means of
nonlinear optics when one is working with ultrashort pulses (i.e., with
durations below 100 fs), inverting the data from the measured signal
sometimes requires iterative algorithms, though in some cases it is