Nonlinear Fiber Optics - 4 ed. Agrawal

7.6. Polarization Effects 257
Figure 7.11: Evolution of pump SOP (solid curve) and probe SOP (dashed curve) on the
Poincaré sphere as a function of τ at a distance of ξ = 20. Parts (a) and (b) show the front
and back faces of the Poincaré sphere, respectively. Both pulses are assumed to be Gaussian of
the same width but are launched with different SOPs.
for both pulses, the Jones vectors for the two input pulses have the form
( )
)
( ) ( )
cosφ
|A 1 (0,τ)〉 = P 1/2
isinφ 0
exp
(− τ2
cosθ
, |A 2 (0,τ)〉 = P 1/2
2
sinθ 20 exp − τ2
2r 2 ,
(7.6.16)
where φ is the ellipticity angle for the pump, θ is the angle at which probe is linearly
polarized from the x axis, and r = T 2 /T 0 is the relative width of the probe compared
to the pump. Figure 7.11 shows how the pump SOP (solid curve) and the probe SOP
(dashed curve) change on the Poincaré sphere with τ at a distance of ξ = 20, assuming
φ = 20 ◦ , θ = 45 ◦ , r = 1, and μ = 0.1 (L W = L/2). The pump SOP varies in a simple
manner as it rotates around ê 3 and traces a circle on the Poincaré sphere. In contrast,
the probe SOP traces a complicated pattern, indicating that the SOP of the probe is
quite different across different parts of the probe pulse. Such polarization changes
impact the XPM-induced chirp. As a result, the spectral profile of the probe develops
a much more complicated structure compared with the scalar case. We focus next on
such spectral effects.
7.6.3 Polarization-Dependent Spectral Broadening
In general, Eq. (7.6.8) should be be solved numerically except when the pump maintains
its SOP. As discussed earlier, this happens if the pump pulse is linearly or circularly
polarized when it is launched into the fiber. To gain some physical insight, we
first consider the case in which both the pump and probe fields are linearly polarized
at the input end, but the probe is oriented at an angle θ with respect to the pump. The
Jones vectors for the two input fields are then obtained from Eq. (7.6.16) after setting
φ = 0.