Nonlinear Fiber Optics - 4 ed. Agrawal

428 Chapter 11. Highly Nonlinear Fibers
fiber dispersion, total optical field at the fiber output is given by
E out (t)=Re{A 1 cos(Δωt)exp(−iω av t)exp[iφ max cos 2 (Δωt)]}, (11.1.4)
where φ max = 2γP av L eff and P av is the average power of the launched signal. It is easy to
see by taking the Fourier transform of Eq. (11.1.4) that the optical spectrum at the fiber
output exhibits peaks at multiples of the beat frequency because of the SPM-induced
phase shift. The ratio of the peak powers depends only on φ max and can be used to
deduce the value of n 2 . In particular, this power ratio for the central and first sideband
is given by [18]
P 0
= J2 0 (φ max/2)+J1 2(φ max/2)
P 1 J1 2(φ max/2)+J2 2(φ max/2) . (11.1.5)
This equation provides φ max from a simple measurement of the power ratio, which can
be used to determine γ and n 2 . For standard telecommunication fibers, the value of n 2
was found to be 2.2 × 10 −20 m 2 /W. This technique was also used to measure n 2 for
several DSFs and DCFs with different amounts of dopants (see Table 11.1). The main
limitation of this technique is that it cannot be used for long fibers for which dispersive
effects become important. In fact, the fiber length and laser powers should be properly
optimized for a given fiber to ensure accurate results [21].
The SPM-induced phase shift can also be measured with an interferometric technique.
In one experiment, the test fiber was placed inside a fiber loop that acted as a
Sagnac interferometer [17]. Mode-locked pulses of ≈5-ps width were launched into
the loop such that pulses acquired a large SPM-induced phase shift in one direction
(say, clockwise). In the other direction, a 99:1 coupler was used to reduce the peak
power so that the pulses acquired mostly a linear phase shift. An autocorrelator at
the loop output was used to deduce the nonlinear phase shift and to obtain n 2 from
it. A CW laser can also be used with a Sagnac interferometer [22]. Its use simplifies
the technique and avoids the uncertainties caused by the dispersive effects. Figure 11.2
shows the experimental setup schematically. The interferometer is unbalanced by using
a fiber coupler that launches unequal amounts of laser power in the counterpropagating
direction, and the transmitted power is measured for low and high values of the input
power. The transmissivity of such a Sagnac interferometer changes at high power levels
because of the SPM-induced phase shift and thus provides a way to measure it. The
measured value of n 2 was 3.1 × 10 −20 m 2 /W at 1064 nm for a fiber whose core was
doped with 20% (by mol) of germania.
A self-aligned Mach–Zehnder interferometer (MZI) has also been used for measuring
n 2 [23]. In this scheme, pulses pass through a MZI twice after being reflected by a
Faraday mirror. The path lengths in the two arms of MZI are different enough that the
two pulses are separated by more than their widths at the MZI output and thus do not
interfere. They pass through the fiber twice and accumulate the SPM-induced nonlinear
phase shift. After a complete round trip. a single pulse has three different delays as
it may pass twice through the long arm, twice through the short arm, or once through
long and short arms. In the last case, the power at the detector depends on the phase
relationship between the interfering signals and can be used to measure the nonlinear
phase shift. Such an interferometer is called self-aligned because the path lengths of
the two interfering signals are automatically matched.