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Phase Space in Ultrafast Optics 361

It is easy to show that if all the filters are time-stationary, then
:
2
W M (t, ; {p k }) = | ˜ S k ( ; {p k })| (11.66)
k
and for nonstationary filters
:
2
W M (t, ; {p k }) = |N k (t; {p k })| (11.67)
k
In both cases, it is clear that Dconsists of an overlap of a marginal of the
pulse Wigner function with the measurement Wigner function, and
therefore it returns no information on the phase of the field. An appa-
ratus consisting of only one class of filters will not work. What is sur-
prising is that an apparatus consisting of at least one time-stationary
and one time-nonstationary filter yields a signal from which the field
can be reconstructed.
Because of the need to explore the entire region of the time-
frequency phase space occupied by the pulse, a measurement scheme
in which the smallest possible number of elements is present must
therefore be an apparatus containing at least two filters of the classes
described previously, each characterized by one parameter. From
Fig. 11.6 it is clear that there are two general two-filter strategies.
The first consists of two filters in sequence, say, in the upper arm
of the interferometer, with the lower arm not used at all. This class
of devices may be called phase-space methods, since it turns out that
they make measurements directly on a phase-space representation of
the test pulse. The second category may be labeled interferometric or
in-parallel methods, since these devices use one filter in each of the
upper and lower arms of the interferometer of Fig. 11.6.

11.3.3 Phase-Space Methods
The analysis of phase-space techniques is found in Ref. 47. Our dis-
cussion follows this framework. There are two subclasses of phase-
space techniques—those that make simultaneous measurements of
the complementary variables and t, recording thereby one of the
phase-space distributions, and those that record marginals of the
Wigner function, following a rotation in the phase space, leading to
a set of spectral or temporal intensities parameterized by the rota-
tion angle. The former method is known as spectrographic while the
latter is referred to as tomographic. For each of these subclasses there
are two possible filter orderings, resulting in a total of four types of
phase-space measurement.
Taking into consideration the amplitude- and phase-only filter sub-
classes, there are a number of possible ways to arrange the filters to
make up a minimalist scheme. But it is completely ineffective to allow