Nonlinear Fiber Optics - 4 ed. Agrawal

70 Chapter 3. Group-Velocity Dispersion
autocorrelation technique, the pulse is sent through a nonlinear crystal together with
a delayed replica of its own [31]. A second-harmonic signal is generated inside the
crystal only when two pulses overlap in time. Measuring the second-harmonic power
as a function of the delay time produces an autocorrelation trace. The width of this
trace is related to the width of the original pulse. The exact relationship between the
two widths depends on the pulse shape. If pulse shape is known a priori, oritcan
be inferred indirectly, the autocorrelation trace provides an accurate measurement of
the pulse width. This technique can measure widths down to a few femtoseconds but
provides little information on details of the pulse shape. In fact, an autocorrelation
trace is always symmetric even when the pulse shape is known to be asymmetric. The
use of cross correlation, a technique in which an ultrashort pulse of known shape and
width is combined with the original pulse inside a second-harmonic crystal, solves this
problem to some extent. The auto- and cross-correlation techniques can also make
use of other nonlinear effects such as third-harmonic generation [32] and two-photon
absorption [33]. All such methods, however, record intensity correlation and cannot
provide any information on the phase or chirp variations across the pulse.
An interesting technique, called frequency-resolved optical gating (FROG) and developed
during the 1990s, solves this problem quite nicely [34]–[36]. It not only can
measure the pulse shape but can also provide information on how the optical phase and
the frequency chirp vary across the pulse. The technique works by recording a series
of spectrally resolved autocorrelation traces and uses them to deduce the intensity and
phase profiles associated with the pulse. Mathematically, the FROG output is described
by
∫ ∞
2
S(τ,ω)=
∣ A(L,t)A(L,t − τ)exp(iωt)dt
∣ , (3.3.30)
−∞
where τ is an adjustable delay and L is the fiber length. Experimentally, the output pulse
is split into two parts that are combined inside a nonlinear crystal after introducing the
delay τ in one path. A series of the second-harmonic spectra is recorded as τ is varied
from negative to positive values.
The FROG technique has been used to characterize pulse propagation in optical
fibers with considerable success [37]–[42]. As an example, Figure 3.9 shows the measured
FROG traces and the retrieved intensity and phase profiles at the output of a
700-m-long fiber when 2.2-ps pulses with a peak power of 22 W are launched into
it [37]. The parts (c) and (d) show the results of numerical simulations using the NLS
equation with the nonlinear terms included. Such complicated pulse shapes cannot be
deduced from the autocorrelation and spectral measurements alone.
A related technique, known as the cross-correlation FROG, can also be used for
measuring the intensity and phase profiles of ultrashort pulses [40]. It makes use of a
reference pulse and is discussed in Section 12.1.2. Another technique, known as timeresolved
optical gating, can also be employed [43]. In this method, the pulse is passed
through a dispersive medium (e.g., an optical fiber) whose GVD can be varied over a
certain range, and a number of autocorrelation traces are recorded for different GVD
values. Both the intensity and phase profiles can be deduced from such autocorrelation
traces.