Nonlinear Fiber Optics - 4 ed. Agrawal

208 Chapter 6. Polarization Effects
units of Section 5.2, these equations become
( ∂u
i
∂ξ + δ ∂u )
+ 1 ∂ 2 u
∂τ 2 ∂τ 2 +(|u|2 + B|v| 2 )u = 0, (6.5.4)
( ∂v
i
∂ξ − δ ∂v )
+ 1 ∂ 2 v
∂τ 2 ∂τ 2 +(|v|2 + B|u| 2 )v = 0, (6.5.5)
where u and v are the normalized amplitudes of the field components polarized linearly
along the x and y axes, respectively, and
δ =(β 1x − β 1y )T 0 /2|β 2 | (6.5.6)
governs the group-velocity mismatch between the two polarization components. The
normalized time τ =(t − β ¯ 1 z)/T 0 , where β ¯ 1 = 1 2 (β 1x + β 1y ) is inversely related to the
average group velocity. Fiber losses are ignored for simplicity but can be included
easily. The XPM coupling parameter B = 2 3
for linearly birefringent fibers.
When an input pulse is launched with a polarization angle θ (measured from the
slow axis), Eqs. (6.5.4) and (6.5.5) should be solved with the input
u(0,τ)=N cosθ sech(τ), v(0,τ)=N sinθ sech(τ), (6.5.7)
where N is the soliton order. In the absence of XPM-induced coupling, the two polarization
components evolve independently and separate from each other because of
their different group velocities. The central question is how this behavior is affected by
the XPM. This question is answered by solving Eqs. (6.5.4) and (6.5.5) numerically
with B = 2/3 for various values of N,θ, and δ [71]–[73]. The numerical results can be
summarized as follows.
When the two modes are equally excited (θ = 45 ◦ ), the two components remain
bound together if N exceeds a critical value N th that depends on δ; N th ≈ 0.7 for δ =
0.15, but N th ≈ 1 for δ = 0.5. As an example, the top row in Figure 6.13 shows the
amplitudes |u(ξ ,τ)| and |v(ξ ,τ)| as well as the pulse spectra of the two polarization
components at ξ = 5π for N = 0.8 and δ = 0.15. The case in which XPM coupling is
ignored (B = 0) is shown for comparison in the bottom row to reveal how this coupling
affects the pulse evolution in birefringent fibers. As seen clearly in Figure 6.13, when
B = 2/3, the two components are trapped and travel at nearly the same velocity, and
this trapping is a consequence of spectral shifts occurring in opposite directions for
the two components. A similar behavior is observed for larger values of δ but N th is
higher. For values of δ ∼ 1, N th exceeds 1.5. In this case, solitons can form even
when N < N th , but the two components travel at their own group velocities and become
widely separated. When N > N th , the two components remain close to each other, and
the distance between them changes in an oscillatory manner.
When θ ≠ 45 ◦ , the two modes have unequal amplitudes initially. In this case, if
N exceeds N th , a qualitatively different evolution scenario occurs depending on the
value of δ. Figure 6.14 is obtained under conditions identical to those of Figure 6.13
except that θ is reduced to 30 ◦ so that u component dominates. As seen there, for
δ = 0.15 the smaller pulse appears to have been captured by the larger one, and the
two move together. Trapping occurs even for larger values of δ, but only a fraction