Nonlinear Fiber Optics - 4 ed. Agrawal

2.3. Pulse-Propagation Equation 33
latter term requires phase matching and is generally negligible in optical fibers. By
making use of Eq. (2.3.4), P NL (r,t) is given by
P NL (r,t) ≈ ε 0 ε NL E(r,t), (2.3.7)
where the nonlinear contribution to the dielectric constant is defined as
ε NL = 3 4 χ(3) xxxx|E(r,t)| 2 . (2.3.8)
To obtain the wave equation for the slowly varying amplitude E(r,t), it is more
convenient to work in the Fourier domain. This is generally not possible as Eq. (2.3.1)
is nonlinear because of the intensity dependence of ε NL . In one approach, ε NL is treated
as a constant during the derivation of the propagation equation [9]. The approach is
justified in view of the slowly varying envelope approximation and the perturbative
nature of P NL . Substituting Eqs. (2.3.2) through (2.3.4) in Eq. (2.3.1), the Fourier
transform Ẽ(r,ω − ω 0 ), defined as
∫ ∞
Ẽ(r,ω − ω 0 )= E(r,t)exp[i(ω − ω 0 )t]dt, (2.3.9)
−∞
is found to satisfy the Helmholtz equation
where k 0 = ω/c and
∇ 2 Ẽ + ε(ω)k0Ẽ 2 = 0, (2.3.10)
ε(ω)=1 + ˜χ (1)
xx (ω)+ε NL (2.3.11)
is the dielectric constant whose nonlinear part ε NL is given by Eq. (2.3.8). Similar to
Eq. (2.1.14), the dielectric constant can be used to define the refractive index ñ and the
absorption coefficient ˜α. However, both ñ and ˜α become intensity dependent because
of ε NL . It is customary to introduce
ñ = n + n 2 |E| 2 , ˜α = α + α 2 |E| 2 . (2.3.12)
Using ε =(ñ + i ˜α/2k 0 ) 2 and Eqs. (2.3.8) and (2.3.11), the nonlinear-index coefficient
n 2 and the two-photon absorption coefficient α 2 are given by
n 2 = 3 8n Re(χ(3) xxxx), α 2 = 3ω 0
4nc Im(χ(3) xxxx). (2.3.13)
The linear index n and the absorption coefficient α are related to the real and imaginary
parts of ˜χ xx
(1) as in Eqs. (2.1.15) and (2.1.16). As α 2 is relatively small for silica fibers,
it is often ignored.
Equation (2.3.10) can be solved by using the method of separation of variables. If
we assume a solution of the form
Ẽ(r,ω − ω 0 )=F(x,y)Ã(z,ω − ω 0 )exp(iβ 0 z), (2.3.14)