Nonlinear Fiber Optics - 4 ed. Agrawal

252 Chapter 7. Cross-Phase Modulation
the XPM-induced phase shift is less than π because of the birefringence-related pulse
walk-off, the technique of cross-splicing can be used to accumulate it over long lengths
[76]. In this technique, the fiber loop consists of multiple sections of polarizationmaintaining
fibers that are spliced together in such a way that the fast and slow axes
are rotated by 90 ◦ in successive sections. As a result, the pump and signal pulses
are forced to pass through each other in each section of the fiber loop, and the XPMinduced
phase shift is enhanced by a factor equal to the number of sections.
7.5.3 XPM-Induced Nonreciprocity
XPM also occurs when two beams having the same (or different) wavelengths are propagated
in opposite directions inside a fiber such that the counterpropagating waves interact
with each other through XPM. Such an interaction can lead to new qualitative
features, manifested through optical bistability and other instabilities when the fiber
is used to construct a nonlinear ring resonator [84]–[95]. Of particular interest is the
XPM-induced nonreciprocity that can affect the performance of fiber gyroscopes [96]–
[101].
The origin of nonreciprocity between two counterpropagating waves can be understood
by following the analysis of Section 7.1. If A 1 and A 2 are the amplitudes of the
forward and backward propagating waves, they satisfy the coupled amplitude equations
similar to Eqs. (7.1.15) and (7.1.16),
± ∂A j
∂z + 1 ∂A j
+ iβ 2 ∂ 2 A j
v g ∂t 2 ∂t 2 + α 2 A j = iγ(|A j | 2 + 2|A 3− j | 2 )A j , (7.5.1)
where the plus or minus sign corresponds to j = 1 or 2, respectively. In the case of
CW beams, this set of two equations is readily solved. If fiber losses are neglected for
simplicity, the solution is given as
A j (z)= √ P j exp(±iφ j ), (7.5.2)
where P j is the peak power and the nonlinear phase shift is given by φ j = γz(P j +2P 3− j )
with j = 1,2. If P 1 ≠ P 2 , the phase shifts φ 1 and φ 2 are not the same for the two
counterpropagating waves. This nonreciprocity is due to the presence of the factor of
two in the XPM term in Eq. (7.5.1).
XPM-induced nonreciprocity can be detrimental for high-precision fiber gyroscopes
used to measure rotation rates as small as 0.01 ◦ per hour [102]. Figure 7.10 shows the
design of a fiber gyroscope schematically. Its operation is based on the Sagnac effect,
known to introduce a rotation-dependent relative phase shift between the counterpropagating
waves [103]. This phase difference is given by
Δφ = φ 1 − φ 2 = γL(P 2 − P 1 )+SΩ, (7.5.3)
where L is the total fiber length, Ω is the rotation rate, and S is a scale factor that
depends on the fiber length L as well as on the radius of the fiber loop [102]. If the
powers P 1 and P 2 were constant, the XPM term in Eq. (7.5.3) would be of little concern.
However, the power levels can fluctuate in practice. Even a power difference of 1 μW