Nonlinear Fiber Optics - 4 ed. Agrawal

Chapter 4
Self-Phase Modulation
An interesting manifestation of the intensity dependence of the refractive index in nonlinear
optical media occurs through self-phase modulation (SPM), a phenomenon that
leads to spectral broadening of optical pulses [1]–[9]. SPM is the temporal analog of
self-focusing that manifests as a narrowing of the spot size of CW beams in a nonlinear
medium with n 2 > 0. SPM was first observed in 1967 in the context of transient selffocusing
of optical pulses propagating in a CS 2 -filled cell [1]. By 1970, SPM had been
observed in solids and glasses by using picosecond pulses. The earliest observation of
SPM in optical fibers was made with a fiber whose core was filled with CS 2 [7]. This
work led by 1978 to a systematic study of SPM in a silica-core fiber [9]. This chapter
considers SPM as a simple example of the nonlinear optical effects that can occur in
optical fibers. Section 4.1 is devoted to the case of pure SPM as it neglects the GVD effects
and focuses on spectral changes induced by SPM. The combined effects of GVD
and SPM are discussed in Section 4.2 with emphasis on the SPM-induced frequency
chirp. Section 4.3 presents two analytic techniques and uses them to solve the NLS
equation approximately. Section 4.4 extends the analysis to include the higher-order
nonlinear effects such as self-steepening.
4.1 SPM-Induced Spectral Changes
A general description of SPM in optical fibers requires numerical solutions of the pulsepropagation
equation (2.3.43) obtained in Section 2.3. The simpler equation (2.3.45)
can be used for pulse widths T 0 > 5 ps. A further simplification occurs if the effect
of GVD on SPM is negligible so that the β 2 term in Eq. (2.3.45) can be set to zero.
The conditions under which GVD can be ignored were discussed in Section 3.1 by
introducing the length scales L D and L NL [see Eq. (3.1.5)]. In general, the pulse width
and the peak power should be such that L D ≫ L > L NL for a fiber of length L. Equation
(3.1.7) shows that the GVD effects are negligible for relatively wide pulses (T 0 > 50 ps)
with a large peak power (P 0 > 1 W).
79