Page 384 - Phase Space Optics Fundamentals and Applications

Phase Space in Ultrafast Optics 365

of a pulse with second- and third-order spectral phase. Although the
corresponding electric fields and their Wigner functions are asymmet-
ric in time, the corresponding nonlinear spectrograms are not, which
implies that information on the direction of time cannot be recov-
ered directly from the measured spectrogram. Versions of frequency
resolved gating based on other nonlinearities, e.g., third-order non-
A
linearities, are characterized by gates N corresponding to higher-
order products of the unknown electric field E and do not have this
ambiguity. 50 Nonlinear interactions with a known ancillary pulse can
also be used, in which case the experimental trace becomes the spec-
trogram of the electric field of the pulse under test, measured using
the electric field of the known pulse as the gate. 51–53 In these cases,
more intuitive nonlinear spectrograms are usually obtained, allow-
ing, e.g., the visualization of chirp and the interpretation of linear and
nonlinear propagation effects. The previous examples show that the
formalism of type I devices is not limited to amplitude only filters,
as the electric field is in most cases a complex quantity. Scanning of a
phase-only modulation and recording of the associated spectrogram
have also been demonstrated. 54
Figure 11.8c represents a schematic of a type I device based on lin-
ear optics. 55 A temporal modulator is a useful time-nonstationary de-
vice for pulse measurement if its transfer function changes signifi-
cantly during the time scale of the test pulse. While this condition is
difficult to meet for sub-100-fs pulses, the development of fast tem-
poral modulators for optical telecommunications has made possible
the characterization of short optical pulses in the range of hundreds
of femtoseconds to hundreds of picoseconds with completely linear
techniques. Lithium-niobate electrooptic modulators and electroab-
sorption modulators driven by sinusoidal drives at 10 GHz or higher
frequencies provide suitable speeds and magnitudes of phase or am-
plitude modulation. Since nonlinear optics requires high optical in-
tensities or large nonlinearities, linear techniques are advantageous in
terms of sensitivity. The temporal modulator is driven by an electric
signal synchronized to the pulse under test and has a gating function
N A = g which does not depend on the pulse under test. The rela-
tive delay between the pulse and the gate is scanned in the rf domain
using a phase shifter. The OSA after the modulator measures the spec-
trum of the modulated pulse as a function of the optical frequency,
leading to

2

S( , ) = dtE(t)g(t − ) exp(i t) (11.71)

As can be seen in Fig. 11.8d, spectrograms measured with a signal-
independent gate give a better representation of the chirp present on