Nonlinear Fiber Optics - 4 ed. Agrawal

8.5. Polarization Effects 315
maximum for an optimum fiber length, a feature similar to that of compressors based
on higher-order solitons. This behavior is easily understood by noting that GVD reduces
the XPM-induced chirp to nearly zero at the point of maximum compression
(see Section 3.2). The main point to note from Figure 8.24 is that weak input pulses
can be amplified by 50–60 dB while getting compressed simultaneously by a factor of
10 or more. The quality of compressed pulses is also quite good with no pedestal and
little ringing. The qualitative features of pulse compression remain nearly the same
when pulse widths or group velocities of the pump and signal pulses are not the same,
making this technique quite attractive from a practical standpoint. Simultaneous amplification
and compression of picosecond optical pulses were indeed observed in a 1996
experiment [186].
8.5 Polarization Effects
In the scalar approach used so far, it has been implicitly assumed that both pump and
signal are copolarized and maintain their state of polarization (SOP) inside the fiber.
However, unless a special kind of polarization-maintaining fiber is used for making
Raman amplifiers, residual fluctuating birefringence of most fibers changes the SOP of
any optical filed in a random fashion and leads to polarization-mode dispersion (PMD),
a phenomenon discussed in Section 7.7.1. In this section we develop a vector theory
of SRS. It turns out that the amplified signal fluctuates over a wide range if PMD
changes with time, and the average gain is significantly lower than that expected in the
absence of PMD [187]–[190]. Such features have also been observed experimentally
[191]–[193].
8.5.1 Vector Theory of Raman Amplification
The starting point is Eq. (2.1.10) for the third-order nonlinear polarization induced
inside the fiber material (silica glass) in response to an optical field E(r,t). In the
scalar case, this equation leads to Eq. (2.3.32) as only the component χ xxxx (3) is relevant
for the nonlinear response. In the vector case, the situation is much more complicated.
Using Eq. (2.3.31) together with E(t)= 1 2 [E(t)exp(−iω 0 t)+c.c.], the slowly varying
part of polarization can be written in the form
P NL
i (t)= ε 0
4 ∑ j
∑
k
∫ t
∑ χ (3)
ijkl E j(t) R(t −t 1 )Ek ∗ (t 1)E l (t 1 )dt 1 , (8.5.1)
l
−∞
where E j is the jth component of the slowly varying field E and the spatial coordinate
r has been suppressed to simplify the notation.
The functional form of the nonlinear response function R(t) is similar to that given
in Eq. (2.3.38) and is given by [5]
R(t)=(1 − f R )δ(t)+ f a h a (t)+ f b h b (t), (8.5.2)
where t e is assumed to be nearly zero, f R ≡ f a + f b is the fractional contribution added
by silica nuclei through the Raman response functions h a (t) and h b (t), normalized