Nonlinear Fiber Optics - 4 ed. Agrawal

8.5. Polarization Effects 317
where the effects of linear birefringence are included through the birefringence vector
b l .
The coefficients c 2 , c 3 , and c 4 contain the Fourier transforms, ˜h a (Ω) and ˜h b (Ω)
of the two Raman response functions h a (t) and h b (t) appearing in Eq. (8.5.2). Their
imaginary parts can be used to define the corresponding Raman gains, g a and g b for
the Stokes field as [5]
g j = γ s f j Im[˜h j (Ω)]; ( j = a,b), (8.5.11)
where Ω = ω p − ω s is the Raman shift. For silica fibers, g b /g a is much smaller than 1.
In the case of pump equation (8.5.9), we should replace Ω with −Ω; this change sign
makes g j negative in Eq. (8.5.11). This is expected since the gain for Stokes results in
a power loss for the pump. The real parts of ˜h a (Ω) and ˜h b (Ω) lead to small changes in
the refractive index. Such changes invariably occur and are governed by the Kramers–
Kronig relation. It is useful to introduce two dimensionless parameters δ a and δ b which
are defined as
δ j =(f j /c 1 )Re[˜h j (Ω) − 1]; ( j = a,b). (8.5.12)
Equations (8.5.9) and (8.5.10) appear quite complicated in the Jones-matrix formalism.
They can be simplified considerably if we write them in the Stokes space using
the Pauli spin vector σ. After introducing the Stokes vectors for the pump and Stokes
fields as
P = 〈A p |σ|A p 〉, S = 〈A s |σ|A s 〉, (8.5.13)
and making use of the identities in Eqs. (7.6.12)–(7.6.14), we obtain the following two
vector equations that govern the dynamics of P and S on the Poincaré sphere:
dP
dz + α pP = − ω p
2ω s
[(g a + 3g b )P s P +(g a + g b )P p S − 2g b P p S 3 ]+(ω p b l + W p ) × P,
dS
dz + α sS = 1 2 [(g a + 3g b )P p S +(g a + g b )P s P − 2g b P s P 3 ]+(ω s b l + W s ) × S,
where P p = |P| is the pump power, P s = |S| is the Stokes power, and
(8.5.14)
(8.5.15)
W p = 2γ p
3 [P 3 + 2(1 + δ b )S 3 − (2 + δ a + δ b )S], (8.5.16)
W s = 2γ s
3 [S 3 + 2(1 + δ b )P 3 − (2 + δ a + δ b )P]. (8.5.17)
The vectors W p and W s account for the nonlinear polarization rotation on the Poincaré
sphere through the SPM and XPM effects.
Equations (8.5.14) and (8.5.15) simplify considerably when both the pump and
signal are linearly polarized, as P and S then lie in the equatorial plane of the Poincaré
sphere with P 3 = S 3 = 0. In the absence of birefringence (b l = 0), P and S maintain
their initial SOPs with propagation. When the pump and signal are copolarized, the
two gain terms add in phase, and Raman gain is maximum with a value g ‖ = g a + 2g b .