Nonlinear Fiber Optics - 4 ed. Agrawal

190 Chapter 6. Polarization Effects
time derivatives can be set to zero in the quasi-CW case. If we also neglect fiber losses,
Eqs. (6.1.15) and (6.1.16) reduce to
dA +
dz
dA −
dz
= iΔβ
2 A − + 2iγ
3 (|A +| 2 + 2|A − | 2 )A + , (6.3.1)
= iΔβ
2 A + + 2iγ
3 (|A −| 2 + 2|A + | 2 )A − . (6.3.2)
Consider first the low-power case and neglect the nonlinear effects (γ = 0). The
resulting linear equations are easily solved. As an example, when the input beam with
power P 0 is σ + -polarized, the solution is given by
A + (z)= √ P 0 cos(πz/L B ), A − (z)=i √ P 0 sin(πz/L B ), (6.3.3)
where the beat length L B = 2π/(Δβ). The state of polarization is generally elliptical
and evolves periodically with a period equal to the beat length. The ellipticity and the
azimuth of the polarization ellipse at any point along the fiber can be obtained using
e p = |A +|−|A − |
|A + | + |A − | , θ = 1 ( )
A+
2 tan−1 . (6.3.4)
A −
Equations (6.3.1) and (6.3.2) can be solved analytically even when nonlinear effects
become important. For this purpose, we use
( ) 3Δβ 1/2
√
A ± =
p± exp(iφ ± ), (6.3.5)
2γ
and obtain the following three equations satisfied by the normalized powers p + and p −
and the phase difference ψ ≡ φ + − φ − :
dp +
dZ = √ 2p + p − sinψ, (6.3.6)
dp −
dZ = −√ 2p + p − sinψ, (6.3.7)
dψ
dZ = p − − p +
√ cosψ + 2(p − − p + ),
p+ p −
(6.3.8)
where Z =(Δβ)z/2. These equations have the following two quantities that remain
constant along the fiber [45]:
p = p + + p − , Γ = √ p + p − cosψ + p + p − . (6.3.9)
Note that p is related to the total power P 0 launched into the fiber through p = P 0 /P cr ,
where P cr is obtained from Eq. (6.3.5) and is given by
P cr = 3|Δβ|/(2γ). (6.3.10)
Because of the two constants of motion, Eqs. (6.3.6)–(6.3.8) can be solved analytically
in terms of the elliptic functions. The solution for p + is [34]
p + (z)= 1 2 p − √ m|q| cn(x), (6.3.11)