Nonlinear Fiber Optics - 4 ed. Agrawal

406 Chapter 10. Four-Wave Mixing
Figure 10.18: Parametric gain as a function of ellipticity angle for κ = 0 (solid curve) and κ ≠ 0
(dashed curve) when the two pumps are elliptically polarized with orthogonal SOPs.
they can make the pump SOPs nonorthogonal even if they were orthogonal initially at
z = 0.
To study such polarization changes, it is useful to write Eqs. (10.5.6) and (10.5.7)
in the Stokes space after introducing the Stokes vectors of the two pumps as
S 1 = 〈A 1 |σ|A 1 〉, S 2 = 〈A 2 |σ|A 2 〉, (10.5.18)
where σ = σ 1 ê 1 + σ 2 ê 2 + σ 3 ê 3 is the Pauli spin vector in the Stoke space. Recall that
a Stokes vector moves on the surface of the Poincaré sphere; it lies in the equatorial
plane for linearly polarized light and points toward a pole for circularly polarized light.
In the convention adopted here, the north pole corresponds to left-circular polarization.
Recall also that orthogonally polarized pumps are represented by two Stokes vectors
that point in opposite directions on the Poincaré sphere.
By differentiating S 1 and S 2 and using Eqs. (10.5.6) and (10.5.7), one can show
that the Stokes vectors for the two pumps satisfy
dS 1
dz = 2γ
3 [(S 13 + 2S 23 )ê 3 − 2S 2 ] × S 1 , (10.5.19)
dS 2
dz = 2γ
3 [(S 23 + 2S 13 )ê 3 − 2S 1 ] × S 2 , (10.5.20)
where S j3 is the third component of S j (along the ê 3 direction). This equation shows
that the SPM rotates the Stokes vector along the vertical direction. In contrast, the two
XPM terms combine such that the XPM rotates the Stokes vector along a vector that lies
in the equatorial plane. Equations (10.5.19) and (10.5.20) can be solved numerically