Nonlinear Fiber Optics - 4 ed. Agrawal

134 Chapter 5. Optical Solitons
This nonlinear equation can be solved by multiplying it by 2(dV /dτ) and integrating
over τ. The result is
(dV/dτ) 2 = 2KV 2 −V 4 +C, (5.2.19)
where C is a constant of integration. Using the boundary condition that both V and
dV /dτ vanish as |τ|→∞, C is found to be 0. The constant K is determined from the
condition that V = 1 and dV/dτ = 0 at the soliton peak, assumed to occur at τ = 0.
Its use provides K = 1 2
, and hence φ = ξ /2. Equation (5.2.19) is easily integrated to
obtain V (τ) =sech(τ). We have thus recovered the solution in Eq. (5.2.16) using a
simple technique.
In the context of optical fibers, the solution (5.2.16) indicates that if a hyperbolicsecant
pulse, whose width T 0 and the peak power P 0 are chosen such that N = 1inEq.
(5.2.3), is launched inside an ideal lossless fiber, the pulse will propagate undistorted
without change in shape for arbitrarily long distances. It is this feature of the fundamental
solitons that makes them attractive for optical communication systems [76].
The peak power P 0 required to support the fundamental soliton is obtained from Eq.
(5.2.3) by setting N = 1 and is given by
P 0 = |β 2|
γT 2
0
≈ 3.11 |β 2|
γTFWHM
2 , (5.2.20)
where the FWHM of the soliton is defined using T FWHM ≈ 1.76T 0 from Eq. (3.2.22).
Using typical parameter values, β 2 = −1 ps 2 /km and γ = 3W −1 /km for dispersionshifted
fibers near the 1.55-μm wavelength, P 0 is ∼1 W for T 0 = 1 ps but reduces to
only 10 mW when T 0 = 10 ps because of its T0 −2 dependence. Thus, fundamental
solitons can form in optical fibers at power levels available from semiconductor lasers
even at a relatively high bit rate of 20 Gb/s.
5.2.3 Higher-Order Solitons
Higher-order solitons are also described by the general solution in Eq. (5.2.8). Various
combinations of the eigenvalues η j and the residues c j generally lead to an infinite
variety of soliton forms. If the soliton is assumed to be symmetric about τ = 0, the
residues are related to the eigenvalues by the relation [79]
c j = ∏N k=1 (η j + η k )
∏ N k≠ j |η j − η k | . (5.2.21)
This condition selects a subset of all possible solitons. Among this subset, a special
role is played by solitons whose initial shape at ξ = 0isgivenby
u(0,τ)=N sech(τ), (5.2.22)
where the soliton order N is an integer. The peak power necessary to launch the Nthorder
soliton is obtained from Eq. (5.2.3) and is N 2 times that required for the fundamental
soliton. For the second-order soliton (N = 2), the field distribution is obtained