Nonlinear Fiber Optics - 4 ed. Agrawal

10.5. Polarization Effects 407
to study how the pump SOPs change with propagation inside the fiber [104]. The
results show that the two pumps launched with orthogonal SOPs maintain their initial
orthogonality only when they are linearly or circularly polarized at z = 0. This can
also be inferred from Eqs. (10.5.19) and (10.5.20) directly. When pumps are linearly
polarized, S 13 = S 23 = 0; when they are circularly polarized, ê 3 × S j = 0. In both
cases, the derivative dS j /dz vanishes for j = 1 and 2, ensuring that the pumps remain
orthogonality polarized.
We use the same technique to see how the Stokes vector of the signal and idler are
affected by the XPM induced by the two pumps. It turns out that, if the two pumps
remain orthogonally polarized along the fiber, the SOPs of the signal and idler are not
affected by them. In physical terms, even though each pump rotates the signal’s Stokes
vector on the Poincaré sphere, rotations induced by two equal-power pumps cancel
each other when pumps are orthogonally polarized. Thus, only the linear and circular
SOPs for the orthogonally polarized pumps can provide polarization-independent gain.
In these two specific pumping configurations, the vector problem reduces to a scalar
problem and can be solved analytically.
The analytic solution of the signal and idler equations, obtained by solving Eqs.
(10.5.8) and (10.5.9), provides the following expressions for the signal gain:
G s ≡ 〈A ( )
3(L)|A 3 (L)〉
〈A 3 (0)|A 3 (0)〉 = 1 + 1 + κ2
4g 2 sinh 2 (gL), (10.5.21)
where κ and g are defined as
κ = Δk + r κ γ(P 1 + P 2 ), g =[(r g γ) 2 P 1 P 2 − (κ/2) 2 ] 1/2 . (10.5.22)
The constants r κ and r g depend on whether the two pumps are linearly or circularly
polarized. For linearly polarized pumps, r κ = 1 and r g = 2/3. When the pumps are
circularly polarized, r κ = 2/3 and r g = 4/3.
Equation (10.5.21) should be compared with Eq. (10.4.5), obtained in the scalar
case in which the two pumps are linearly copolarized and generate the signal and idler
that are polarized linearly in the same direction. Effectively, r κ = 1 and r g = 2 in this
case. When the signal is orthogonally polarized with respect to linearly copolarized
pumps, one finds that r κ = 5/3 and r g = 2/3. Such changes in r κ and r g for different
pumping configurations indicate that FWM efficiency can change considerably
depending on the SOPs of the pump at the input end.
Figure 10.19 shows the gain spectra for several pumping schemes by focusing on a
500-m-long dual-pump FOPA and using parameter values γ = 10 W −1 /km, λ 0 = 1580
nm, β 30 = 0.04 ps 3 /km, and β 40 = 1.0 × 10 −4 ps 4 /km. The pump wavelengths (1535
and 1628 nm) and powers (P 1 = P 2 = 0.5 W) are chosen such that the FOPA provides
a fairly flat gain of 37 dB over a wide wavelength range when the two pumps
as well as the signal are linearly copolarized (dotted curve). Of course, this gain is
highly polarization-dependent. When the signal is polarized orthogonal to the copolarized
pumps, the gain is reduced to almost zero in the central part (thin solid curve).
When orthogonal linearly polarized pumps are used, the gain spectrum is still wide
and flat (dashed curve), but the gain is relatively small (around 8.5 dB). However, as
shown by the solid curve, the FOPA gain can be increased from 8.5 to 23 dB if the