Nonlinear Fiber Optics - 4 ed. Agrawal

26 Chapter 2. Pulse Propagation in Fibers
relations given by [1]
D = ε 0 E + P, (2.1.5)
B = μ 0 H + M, (2.1.6)
where ε 0 is the vacuum permittivity, μ 0 is the vacuum permeability, and P and M are
the induced electric and magnetic polarizations. For a nonmagnetic medium such as
optical fibers, M = 0.
Maxwell’s equations can be used to obtain the wave equation that describes light
propagation in optical fibers. By taking the curl of Eq. (2.1.1) and using Eqs. (2.1.2),
(2.1.5), and (2.1.6), one can eliminate B and D in favor of E and P and obtain
∇ × ∇ × E = − 1 ∂ 2 E
c 2 ∂t 2 − μ ∂ 2 P
0
∂t 2 , (2.1.7)
where c is the speed of light in vacuum and the relation μ 0 ε 0 = 1/c 2 was used. To
complete the description, a relation between the induced polarization P and the electric
field E is needed. In general, the evaluation of P requires a quantum-mechanical
approach. Although such an approach is often necessary when the optical frequency is
near a medium resonance, a phenomenological relation of the form (1.3.1) can be used
to relate P and E far from medium resonances. This is the case for optical fibers in the
wavelength range 0.5–2 μm that is of interest for the study of nonlinear effects. If we
include only the third-order nonlinear effects governed by χ (3) , the induced polarization
consists of two parts such that
P(r,t)=P L (r,t)+P NL (r,t), (2.1.8)
where the linear part P L and the nonlinear part P NL are related to the electric field by
the general relations [2]–[4]
P L (r,t) =ε 0
∫ t
−∞
P NL (r,t) =ε 0
∫ t
−∞
χ (1) (t −t ′ ) · E(r,t ′ )dt ′ , (2.1.9)
dt 1
∫ t
−∞
dt 2
∫ t
−∞
dt 3
× χ (3) (t −t 1 ,t −t 2 ,t −t 3 ) . E(r,t 1 )E(r,t 2 )E(r,t 3 ). (2.1.10)
These relations are valid in the electric-dipole approximation and assume that the
medium response is local.
Equations (2.1.7)–(2.1.10) provide a general formalism for studying the third-order
nonlinear effects in optical fibers. Because of their complexity, it is necessary to make
several simplifying approximations. In a major simplification, the nonlinear polarization
P NL in Eq. (2.1.8) is treated as a small perturbation to the total induced polarization.
This is justified because the nonlinear effects are relatively weak in silica fibers.
The first step therefore consists of solving Eq. (2.1.7) with P NL = 0. Because Eq.
(2.1.7) is then linear in E, it is useful to write in the frequency domain as
∇ × ∇ × Ẽ(r,ω)=ε(ω) ω2
Ẽ(r,ω), (2.1.11)
c2