Nonlinear Fiber Optics - 4 ed. Agrawal

10.5. Polarization Effects 409
Figure 10.20: (a) Average gain spectrum for a dual-pump FOPA for three values of D p ; (b)
average gain spectrum for three initially linear SOPs of the signal when D p = 0.1 ps/ √ km. In
both cases, the isotropic case is also shown. (After Ref. [107]; c○2004 IEEE.)
their frequency difference Ω. The length scale over which such a PMD-induced drift
occurs is governed by the diffusion length L diff = 3(D p Ω) −2 , where D p is the PMD
parameter of the fiber.
Similar to the case of dispersion fluctuations, one can use the stochastic equations
to simulate the FOPA gain spectra numerically for different realizations of the birefringence
distribution along the fiber. The average gain at any signal wavelength can be
calculated by performing an ensemble average over a large number of realizations. In
the case of a single-pump FOPA, the average can be performed analytically [108]. The
analytical as well as numerical results show that random fluctuations of birefringence
reduce the FOPA gain, on average, and distort the gain spectrum such that its flatness
is compromised. A similar behavior is expected for dual-pump FOPAs.
As an example, Figure 10.20(a) shows the impact of PMD on the average gain
spectrum of a dual-pump FOPA for three values of D p with the same parameter values
used for Figure 10.14. More specifically, the 0.5-km-long FOPA is pumped with 0.5 W
of power at two wavelengths of 1520.6 and 1600.6 nm, and both pumps and signal
are copolarized initially. The average gain at any signal wavelength is obtained by
averaging over 50 different realizations of birefringence distribution along the fiber
length. The ideal case of isotropic fiber is also shown for comparison. Similar to the
case of single-pump FOPA, the effect of b 0 is to reduce the nonlinear parameter γ by a
factor of 8/9 [109]. For small values of D p , a lower value of γ reduces the peak gain,
but the spectrum maintains its flat nature in the central region. However, for D p > 0.1
ps/ √ km, a central dip occurs, as seen in Figure 10.20(a). The reason behind the dip
formation is as follows. When the signal frequency is close to one of the pumps, that
pump provides the dominant contribution. However, as signal frequency moves toward
the center, neither of the two pumps remains oriented parallel to the signal, and the
gain is reduced.
An important question is how PMD affects the independence of the FOPA gain
on the signal SOP when the two pumps are orthogonally polarized (either linearly or