Nonlinear Fiber Optics - 4 ed. Agrawal

Chapter 7
Cross-Phase Modulation
So far in this book, only a single electromagnetic wave is assumed to propagate inside
optical fibers. When two or more optical fields having different wavelengths propagate
simultaneously inside a fiber, they interact with each other through the fiber nonlinearity.
In general, such an interaction can generate new waves under appropriate conditions
through a variety of nonlinear phenomena such as stimulated Raman or Brillouin
scattering, harmonic generation, and four-wave mixing; these topics are covered in
Chapters 8 to 12. The Kerr-type nonlinearity can also couple two optical fields through
cross-phase modulation (XPM) without inducing any energy transfer between them. In
practice, XPM always accompanies self-phase modulation (SPM) when two or more
optical fields are launched simultaneously into an optical fiber. From a physical perspective,
XPM occurs because the effective refractive index seen by an optical beam
in a nonlinear medium depends not only on the intensity of that beam but also on the
intensity of other copropagating beams [1].
The XPM-induced coupling among multiple optical fields gives rise to a number
of interesting nonlinear effects in optical fibers. The coupling between two fields of
different wavelengths is considered in Section 7.1, where a set of two coupled nonlinear
Schrödinger (NLS) equations is derived, assuming that each wave maintains its state of
polarization. These equations are used in Section 7.2 to discuss how the XPM affects
the phenomenon of modulation instability. Similar to the analysis in Section 6.4, this
instability can occur even in the normal-dispersion regime of an optical fiber. Section
7.3 focuses on soliton pairs whose members support each other through XPM. The
effects of XPM on the shape and the spectrum of copropagating ultrashort pulses are
described in Section 7.4. Several applications of XPM-induced coupling in optical
fibers are discussed in Section 7.5. A vector theory of XPM is developed in Section
7.6 and is used to discuss several new effects such as polarization-dependent spectral
broadening, pulse trapping, and optical wave breaking. Section 7.7 extends this theory
to the case of birefringent fibers. A pump–probe configuration is used to reveal that
random changes in the birefringence of a fiber along its length lead to the phenomenon
of intrapulse depolarization.
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