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Phase Space in Ultrafast Optics 363

time of arrival of a series of spectral slices (type II, Fig. 11.7b) depend-
ing on the ordering of the filters. There is no difference in principle
between the two possible filter orderings, and thus this type of appa-
ratus should be thought of as one that makes simultaneous measure-
mentsoftheconjugatevariablesrather thansequentialmeasurements.
Fourier’s principle precludes precise simultaneous measurements of
the conjugate variables. Some of the earliest developments in the rep-
resentation and measurement of short optical pulses are based on the
concepts of spectrography in the time-frequency space. 48
The Wigner function of the measurement apparatus for the type I
device, e.g., is

A 2 A t A ∗
W M (t, ; { C , }) = d | ˜ S ( ; C )| dt N t + ; N
2

t

× t − ; exp[i( − )t ] (11.68)
2
In fact, for near-transform-limited input pulses, the apparatus Wigner
function has nearly the same area as the pulse Wigner function it-
self. In principle the Wigner function can be retrieved from the data
by deconvolution, but because of severe signal-to-noise requirements
this approach is impractical. Thus, spectrographic phase-space pulse
characterization techniques supply only qualitative insight into pulse
train statistics. However, in the limit of narrowband filtering, that is,
A
2

| ˜ S ( ; C )| → ( − C ), and if the pulses in the ensemble are as-
sumedtobeidentical,theexperimentaltraceisasimpleconvolutionof
Wigner functions, which can be expressed as a function of the electric
field of the pulse

D ( C , ) = d dt W(t, )W M (t − , − C )
2

A
= dt E in (t)N (t − ) exp(i C t) (11.69)

This set of conditions gives an apparatus function that occupies
the minimum possible area of phase space, and therefore minimally
“smoothes” the signal Wigner function. In this case, the experimental
A
trace is the Gabor spectrogram with a window N .
Type I devices are popular in ultrafast optics. The time-nonsta-
tionaryfilterrequiredforthesedevicescanbeimplementedusingnon-
linear interactions or temporal modulators, and the high-resolution
spectral measurements can be performed by an optical spectrum an-
alyzer (OSA). As an example of the use of a nonlinear interaction to
implement a type I device, let’s consider sum-harmonic generation
frequency resolved optical gating (SHG-FROG), as schematized in