9.1. FIBER SOLITONS 405
Chappen to have opposite signs so thatβ 2 Cis negative. The nonlinear phenomenon
of SPM imposes a chirp on the optical pulse such thatC>0. Sinceβ 2 <0 in the 1.55-
μm wavelength region, the conditionβ 2 C<0 is readily satisfied. Moreover, as the
SPM-induced chirp is power dependent, it is not difficult to imagine that under certain
conditions, SPM and GVD may cooperate in such a way that the SPM-induced chirp is
just right to cancel the GVD-induced broadening of the pulse. The optical pulse would
then propagate undistorted in the form of a soliton.
9.1.1 Nonlinear Schrodinger Equation ̈
The mathematical description of solitons employs the nonlinear Schr ̈odinger (NLS)
equation, introduced in Section 5.3 [Eq. (5.3.1)] and satisfied by the pulse envelope
A(z,t)in the presence of GVD and SPM. This equation can be written as [10]
∂A
∂z+
iβ 2
2∂^2 A
∂t^2−
β 3
6∂^3 A
∂t^3=iγ|A|^2 A−α
2A, (9.1.1)
where fiber losses are included through theαparameter whileβ 2 andβ 3 account
for the second- and third-order dispersion (TOD) effects. The nonlinear parameter
γ= 2 πn 2 /(λAeff)is defined in terms of the nonlinear-index coefficientn 2 , the optical
wavelengthλ, and the effective core areaAeffintroduced in Section 2.6.
To discuss the soliton solutions of Eq. (9.1.1) as simply as possible, we first set
α=0 andβ 3 =0 (these parameters are included in later sections). It is useful to write
this equation in a normalized form by introducing
τ=
t
T 0, ξ=
z
LD, U=
A
√
P 0
, (9.1.2)
whereT 0 is a measure of the pulse width,P 0 is the peak power of the pulse, andLD=
T 02 /|β 2 |is the dispersion length. Equation (9.1.1) then takes the form
i∂U
∂ξ−
s
2∂^2 U
∂τ^2+N^2 |U|^2 U= 0 , (9.1.3)
wheres=sgn(β 2 )=+1or−1, depending on whetherβ 2 is positive (normal GVD) or
negative (anomalous GVD). The parameterNis defined as
N^2 =γP 0 LD=γP 0 T 02 /|β 2 |. (9.1.4)It represents a dimensionless combination of the pulse and fiber parameters. The phys-
ical significance ofNwill become clear later.
The NLS equation is well known in the soliton literature because it belongs to a
special class of nonlinear partial differential equations that can be solved exactly with
a mathematical technique known as theinverse scattering method[11]–[13]. Although
the NLS equation supports solitons for both normal and anomalous GVD, pulse-like
solitons are found only in the case of anomalous dispersion [14]. In the case of normal
dispersion (β 2 >0), the solutions exhibit a dip in a constant-intensity background.
Such solutions, referred to as dark solitons, are discussed in Section 9.1.3. This chapter
focuses mostly on pulse-like solitons, also calledbrightsolitons.