Nonlinear Fiber Optics - 4 ed. Agrawal

5.5. Higher-Order Effects 157
where a p , T p , C p , and φ p represent the amplitude, width, chirp, and phase of the pulse.
We also allow for the temporal shift q p of the pulse envelope and the frequency shift Ω p
of the pulse spectrum. All six parameters may change with z as the pulse propagates
through the fiber.
Using the definitions of moments in Sections 4.3.1 and 4.4.2 and the moment
method, we obtain the following set of equations for the evolution of pulse parameters
[167]:
dT p
dz =(β 2 + β 3 Ω p ) C p
, (5.5.3)
T p
( )
dC p 4
dz = β2
π 2 +C2 p + 4 ¯γP 0 T 0
π 2 + 6Ω2 p
T p π 2 (2β 2 + β 3 Ω p )
+ β 3
( 4
π 2 + 3C2 p
Tp
2 )
Ωp
2T 2 p
dq p
dz = β 2Ω p + β 3
2 Ω2 p + β 3
6T 2 p
+ 48 ¯γP 0 T 0
π 2 , (5.5.4)
ω 0 T p
)
(1 + π2
4 C2 p + ¯γP 0 T 0
, (5.5.5)
ω 0 T p
dΩ p
dz
= − 8T R ¯γP 0
15
T 0
Tp
3
+ 2 ¯γP 0 T 0 C p
, (5.5.6)
3ω 0
T 3 p
where ¯γ = γ exp(−αz). As in Section 4.3.1, we have ignored the phase equation. The
amplitude a p can be determined from the relation E 0 = 2P 0 T 0 = 2a 2 p(z)T p (z), where E 0
is the input pulse energy.
It is evident from Eqs. (5.5.3) through (5.5.6) that the pulse parameters are affected
considerably by the three higher-order terms in Eq. (5.5.1). Before considering their
impact, we use these equations to determine the conditions under which a fundamental
soliton can form. Equation (5.5.3) shows that pulse width will not change if the chirp
C p remains zero for all z. The chirp equation (5.5.4) is quite complicated. However, if
we neglect the higher-order terms and fiber losses (α = 0), this equation reduces to
dC p
dz = ( 4
π 2 +C2 p
)
β2
T 2 p
+ 4γP 0
π 2 T 0
T p
. (5.5.7)
It is clear that if β 2 > 0, both terms on the right side are positive, and C p cannot remain
zero even if C p = 0 initially. However, in the case of anomalous dispersion (β 2 < 0),
the two terms cancel precisely when pulse parameters initially satisfy the condition
γP 0 T0 2 = |β 2|. It follows from Eq. (5.2.3) that this condition is equivalent to setting
N = 1.
It is useful in the following discussion to normalize Eq. (5.5.1) using dimensionless
variables ξ and τ defined in Eq. (5.2.1). The normalized NLS equation takes the form
i ∂u
∂ξ + 1 ∂ 2 u
2 ∂τ 2 + ∂ 3 u
|u|2 u = iδ 3
∂τ 3 − is ∂
∂τ (|u|2 u)+τ R u ∂|u|2
∂τ , (5.5.8)
where the pulse is assumed to propagate in the region of anomalous GVD (β 2 < 0) and
fiber losses are neglected (α = 0). The parameters δ 3 , s, and τ R govern, respectively,