Page 395 - Phase Space Optics Fundamentals and Applications

376 Chapter Eleven

An entirely analogous argument may be made for temporal shear-
ing interferometers. In this case, the delay line in one arm of the in-
terferometer causes the pulses on recombining at the second beam
splitter to exhibit temporal beats in their intensity that may be re-
solved by a fast time gate. This latter element is the amplitude-only
filter that replaces the spectrometer required in the spectral shearing
interferometer. In this arrangement, a temporal linear phase modu-
lator may be used to provide a “temporal carrier” for the two-time
correlation function in the interference term. This is accomplished by
frequency-shifting one of the pulses with respect to the other by a

shear . The detected signal is then
t

t t t
D t − , t; t = I t − + I t +
2 2 2

+ 2|C(t, t)| cos{arg[C(t, t)] + t}
t
(11.85)
assuming that the time gate is of infinitesimal duration. A similar al-
gorithm, as described for the spectral shearing interferogram, may be
used to extract the temporal phase of the test pulse in this case. In
practice, however, it is very difficult to provide a short enough time
gate to enable this method to work. Nonlinear optical interactions
that cross-correlate the interferogram with the test pulse will not pro-
vide enough temporal resolution to resolve the fringes. Therefore this
method is restricted to pulses whose duration is long enough that an
externally controlled time gate, such as a temporal modulator or a
nonlinear interaction with a short optical pulse, can be used.
As with spectrography, practical implementations of spectral shear-
ing interferometry have been demonstrated with nonlinear optics
and with entirely linear setups. In electrooptic spectral shearing in-
terferometry (EOSI), the spectral shear is obtained by linear tempo-
ral phase modulation, e.g., with lithium-niobate electrooptic phase
modulators. 82,83 A symmetric implementation based on such a mod-
ulator is shown in Fig. 11.12a. The pulse under test is sent into an inter-
ferometer that generates two replicas separated by a delay . One of
the outputs of the interferometer is sent to a phase modulator driven
by a sinusoidal high-frequency modulation. The modulation has a
period 2 and is synchronized so that the two replicas are located at
the two zero crossings of the modulation. In this configuration, the
replicas are sheared by the same amount in opposite directions. The
interferogram measured by an optical spectrum analyzer is

S( ) = ˜ I( + ) + ˜ I( − ) + 2 ˜ I( + ) ˜ I( − )
× cos[ ( + ) − ( − ) + ] (11.86)