Page 387 - Phase Space Optics Fundamentals and Applications

368 Chapter Eleven

distribution using the inverse Radon transform. 62 The quadratic tem-
poral phase modulator of a type III device rotates the input pulses so
as to map time into frequency, and a spectrometer is used to resolve
the spectrum of the output pulses. Similarly, the quadratic spectral
phase modulator of a type IV device rotates the input pulses so as to
map frequency into time, and a time gate is used to resolve the time-
dependent intensity of the output pulses. A combination of quadratic
temporal and spectral phase modulations in series can also be used
to rotate the Wigner function, which avoids the requirement for large
63
amplitude of a single quadratic modulation. The Wigner function of
an optical source can, in principle, be completely reconstructed from a
set of its projections determined by using a type III or type IV device.
If the optical source is a train of identical optical pulses, the electric
field can be determined from the Wigner function reconstructed using
a large number of its projections.
Simplified versions of chronocyclic tomography have been experi-
mentally demonstrated. The first one rotates the Wigner function by
◦
90 and measures its frequency marginal using a spectrometer, which
is a scaled representation of the temporal intensity of the pulse under
test. This is known as the time-to-frequency converter because it is ef-
fectively a procedure to map the temporal intensity of the input pulse
onto the spectral intensity of another signal. 39 The second one rotates
the Wigner function by a small angle, 64,65 so that Eq. (11.75) becomes

D( C , ) = dt W(t + C , C − t ) (11.76)
Development of this equation leads to the result

∂ D ∂

( C , 0) =− ˜ I( C ) ( C ) (11.77)

∂ ∂ C
Equation (11.77) indicates that the spectrally resolved changes of the
optical spectrum of the pulse after small amounts of quadratic tempo-
ral phase modulation are algebraically linked to the optical spectrum
and spectral phase of the pulse. Since ˜ I( C ) can be directly measured
with a spectrometer [e.g., the spectrometer used to measure the signal
of Eq. (11.76) when the modulation is turned off], the spectral phase
can be obtained by solving the second-order differential equation,
Eq. (11.77). Figure 11.9 is a schematic of an implementation of sim-
plified chronocyclic tomography. The left-hand side of Eq. (11.77) is
obtained as a finite difference of the optical spectra obtained after
two small quadratic temporal phase modulations of opposite signs.
Quadratic modulation is obtained by synchronization with the max-
imum and minimum of a sinusoidal phase modulation obtained in a
lithium niobate electrooptic phase modulator. In this symmetric con-
figuration, the optical spectrum is simply obtained as the average of