Page 386 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 367

a phase-only filter, as illustrated in Fig. 11.7 [either a quadratic tempo-
ral phase modulator (type III, Fig. 11.7c) or a quadratic spectral phase
modulator (type IV, Fig. 7d)]. The inclusion of a quadratic phase-only
filter results in a distinctly different interpretation of the measurement,
leading to a fundamentally different inversion algorithm. To see this,
notice that a phase-only filter does not provide any information on
the frequency or the arrival time of a pulse ensemble and hence does
not constitute a measurement of either frequency or time. Therefore, a
tomographic apparatus does not make a simultaneous measurement
of these incompatible variables. Rather, the quadratic-phase modula-
tion acts to rotate the phase space. The square-law detector in com-
bination with the amplitude-only filter records the resulting intensity
distribution.
The measurement function takes the form

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

t

× t − ; t exp[i( − )t ] (11.72)
2
When the phase-nonstationary filter takes the form of Eq. (11.60) and
the amplitude stationary filter that of Eq. (11.62), this reduces to the
form
2
( − C − t)

t
W M (t, ; { C , }) = exp − (11.73)
t
2
This function is different from the filter function of the Gabor trans-
form. Its location in phase space is not determined by the filter
parameters—rather its orientation is. A change in C translates the
entire function along the frequency axis, and a change in alters the

t
orientation of the function about = C . The detected signal is, in
the limit as → 0,

D( C , ) = dt W(t, C + t) (11.74)
t t
This is easily interpreted by transforming the variables to the form

D ( , ) = dt W ( sin + t cos , cos − t sin ) (11.75)
where we have defined tan =− and = C cos and scaled

t
the measured trace by cos( ). The signal D ( , ) is, therefore, a set
of distributions that are marginals of a rotated version of the pulse
Wigner function. This is the essence of tomographic measurements;
indeed, the above formula may be inverted to give the pulse Wigner