Nonlinear Fiber Optics - 4 ed. Agrawal

408 Chapter 10. Four-Wave Mixing
Figure 10.19: Gain spectra for four different pumping schemes for a dual-pump FOPA pumped
with 0.5 W at 1535 and 1628 nm. (After Ref. [104]; c○2004 OSA.)
two pumps are left- and right-circularly polarized. Although the preceding discussion
focuses on signal amplification, the same behavior is expected when the FOPA is used
as a wavelength converter because the idler power P 4 is related to signal power P 3 as
P 4 (L)=P 3 (L) − P 3 (0).
10.5.4 Effect of Residual Fiber Birefringence
As discussed in Section 6.6, most fibers exhibit residual birefringence that varies randomly
along fiber length and may also fluctuate with time. Such birefringence fluctuations
induce PMD and randomize the SOP of any optical field propagating through the
fiber. Random changes in the SOPs of the four waves affect the conservation of angular
momentum during the FWM process and thus degrade the performance of FOPAs
[106]. Such PMD effects have been observed for both single- and dual-pump devices
[67]. The vector theory of FWM can be used to understand the experimental data if it
is extended to include the residual birefringence of fibers.
The extension adds an additional term [b 0 + b 1 (ω j − ω r )σ 1 ]|A j 〉 for j = 1to4
to the right side of Eqs. (10.5.6)–(10.5.9) that converts these equations into a set of
four coupled stochastic equations [107]. Here, ω r is a reference frequency, and it is
practical to choose ω r = ω 1 and adopt a reference pump in which the SOP of one of
the pumps is not affected by birefringence fluctuations. Random variables b 0 and b 1
affect FWM through two different mechanisms. First, b 0 rotates the SOPs of all four
fields on the Poincaré sphere in the same direction, and thus reduces the average gain
roughly by the same amount over the entire gain bandwidth. Second, b 1 causes the
SOPs of the pumps, signal and idler to drift from each other at a rate that depends on