Nonlinear Fiber Optics - 4 ed. Agrawal

6.3. Evolution of Polarization State 189
Figure 6.4: Transmittivity T p as a function of input polarization angle θ when the effect of linear
birefringence is included for ΔβL = 2π and φ max = 6.5π. Dashed line shows the behavior when
Δβ = 0. (After Ref. [31]; c○1985 AIP.)
components are excited with equal amplitudes. In practice, this is not the case. The
reason for this discrepancy can be traced back to the neglect of the last term in Eqs.
(6.1.11) and (6.1.12). A more accurate theory should include this term. In the CW
or the quasi-CW case for which the dispersive effects are negligible, Eqs. (6.1.11) and
(6.1.12) can be solved analytically by neglecting the time derivatives and the loss terms.
The analytic solution is given in the next section. Its use shows that Eq. (6.2.18) is quite
accurate in the case of highly birefringent fibers (ΔβL ≫ 1) except near θ = 45 ◦ .In
low-birefringence fibers, the transmittivity can be quite different than that given by
Eq. (6.2.18). Figure 6.4 shows T p as a function of θ for ΔβL = 2π and φ max = 6.5π.
A comparison with the prediction of Eq. (6.2.18) reveals the importance of including
linear birefringence. Physically, the linear and nonlinear birefringence contributions to
the refractive index compete with each other, and both should be included.
6.3 Evolution of Polarization State
An accurate description of the nonlinear polarization effects in birefringent fibers requires
simultaneous consideration of both the modal birefringence and self-induced
nonlinear birefringence [31]–[48]. Evolution of the two polarization components along
such fibers is governed by Eqs. (6.1.11) and (6.1.12) or their variants. However, before
turning to the case of pulse propagation, it is useful to consider how the state of polarization
evolves within the fiber when a CW (or quasi-CW) beam is launched at the
input end.
6.3.1 Analytic Solution
In place of Eqs. (6.1.11) and (6.1.12), it is more convenient to use Eqs. (6.1.15) and
(6.1.16) written in terms of the circularly polarized components. The terms containing