Nonlinear Fiber Optics - 4 ed. Agrawal

Chapter 3
Group-Velocity Dispersion
The preceding chapter showed how the combined effects of group-velocity dispersion
(GVD) and self-phase modulation (SPM) on optical pulses propagating inside a fiber
can be studied by solving a pulse-propagation equation. Before considering the general
case, it is instructive to study the effects of GVD alone. This chapter considers the
pulse-propagation problem by treating fibers as a linear optical medium. In Section 3.1
we discuss the conditions under which the GVD effects dominate over the nonlinear
effects by introducing two length scales associated with GVD and SPM. Dispersioninduced
broadening of optical pulses is considered in Section 3.2 for several specific
pulse shapes, including Gaussian and “sech” pulses. The effects of initial frequency
chirping are also discussed in this section. Section 3.3 is devoted to the effects of thirdorder
dispersion on pulse broadening. An analytic theory capable of predicting dispersive
broadening for pulses of arbitrary shapes is also given in this section. Section 3.4
discusses how the GVD limits the performance of optical communication systems and
how the technique of dispersion management can be used to combat them in practice.
3.1 Different Propagation Regimes
In Section 2.3 we focused on the nonlinear Schrödinger (NLS) equation that governs
propagation of optical pulses inside single-mode fibers. For pulse widths >5 ps, we
can use Eq. (2.3.45) in the form
i ∂A
∂z = −iα 2 A + β 2 ∂ 2 A
2 ∂T 2 − γ|A|2 A, (3.1.1)
where A is the slowly varying amplitude of the pulse envelope and T is measured in
a frame of reference moving with the pulse at the group velocity v g (T = t − z/v g ).
The three terms on the right-hand side of Eq. (3.1.1) govern, respectively, the effects
of fiber losses, dispersion, and nonlinearity on pulses propagating inside optical fibers.
Depending on the initial width T 0 and the peak power P 0 of the incident pulse, either
dispersive or nonlinear effects may dominate along the fiber. It is useful to introduce
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