Nonlinear Fiber Optics - 4 ed. Agrawal

34 Chapter 2. Pulse Propagation in Fibers
where Ã(z,ω) is a slowly varying function of z and β 0 is the wave number to be determined
later, Eq. (2.3.10) leads to the following two equations for F(x,y) and Ã(z,ω):
∂ 2 F
∂x 2 + ∂ 2 F
∂y 2 +[ε(ω)k2 0 − ˜β 2 ]F = 0, (2.3.15)
2iβ 0
∂Ã
∂z +(˜β 2 − β 2 0 )Ã = 0. (2.3.16)
In obtaining Eq. (2.3.16), the second derivative ∂ 2 Ã/∂z 2 was neglected since Ã(z,ω)
is assumed to be a slowly varying function of z. The wave number ˜β is determined
by solving the eigenvalue equation (2.3.15) for the fiber modes using a procedure similar
to that used in Section 2.2. The dielectric constant ε(ω) in Eq. (2.3.15) can be
approximated by
ε =(n + Δn) 2 ≈ n 2 + 2nΔn, (2.3.17)
where Δn is a small perturbation given by
Δn = n 2 |E| 2 + i ˜α
2k 0
. (2.3.18)
Equation (2.3.15) can be solved using first-order perturbation theory [10]. We first
replace ε with n 2 and obtain the modal distribution F(x,y), and the corresponding wave
number β(ω). For a single-mode fiber, F(x,y) corresponds to the modal distribution
of the fundamental fiber mode HE 11 given by Eqs. (2.2.12) and (2.2.13), or by the
Gaussian approximation (2.2.14). We then include the effect of Δn in Eq. (2.3.15). In
the first-order perturbation theory, Δn does not affect the modal distribution F(x,y).
However, the eigenvalue ˜β becomes
where
˜β(ω)=β(ω)+Δβ(ω), (2.3.19)
∫∫ ∞
Δβ(ω)= ω2 n(ω) −∞ Δn(ω)|F(x,y)|2 dxdy
c 2 ∫∫
β(ω) ∞ . (2.3.20)
−∞ |F(x,y)|2 dxdy
This step completes the formal solution of Eq. (2.3.1) to the first order in perturbation
P NL . Using Eqs. (2.3.2) and (2.3.14), the electric field E(r,t) can be written
as
E(r,t)= 1 2 ˆx{F(x,y)A(z,t)exp[i(β 0z − ω 0 t)] + c.c.}, (2.3.21)
where A(z,t) is the slowly varying pulse envelope. The Fourier transform Ã(z,ω −ω 0 )
of A(z,t) satisfies Eq. (2.3.16), which can be written as
∂Ã
∂z = i[β(ω)+Δβ(ω) − β 0]Ã, (2.3.22)
where we used Eq. (2.3.19) and approximated ˜β 2 − β 2 0 by 2β 0( ˜β − β 0 ). The physical
meaning of this equation is clear. Each spectral component within the pulse envelope
acquires, as it propagates down the fiber, a phase shift whose magnitude is both frequency
and intensity dependent.