Nonlinear Fiber Optics - 4 ed. Agrawal

1.3. Fiber Nonlinearities 15
Section 2.3 for its derivation)
n 2 = 3 8n Re(χ(3) xxxx), (1.3.3)
where Re stands for the real part and the optical field is assumed to be linearly polarized
so that only one component χ xxxx (3) of the fourth-rank tensor contributes to the
refractive index. The tensorial nature of χ (3) can affect the polarization properties of
optical beams through nonlinear birefringence. Such nonlinear effects are covered in
Chapter 6.
The intensity dependence of the refractive index leads to a large number of interesting
nonlinear effects; the two most widely studied are known as self-phase modulation
(SPM) and cross-phase modulation (XPM). Self-phase modulation refers to the
self-induced phase shift experienced by an optical field during its propagation in optical
fibers. Its magnitude can be obtained by noting that the phase of an optical field
changes by
φ = ñk 0 L =(n + n 2 |E| 2 )k 0 L, (1.3.4)
where k 0 = 2π/λ and L is the fiber length. The intensity-dependent nonlinear phase
shift, φ NL = n 2 k 0 L|E| 2 , is due to SPM. Among other things, SPM is responsible for
spectral broadening of ultrashort pulses [25] and formation of optical solitons in the
anomalous-dispersion regime of fibers [26].
Cross-phase modulation refers to the nonlinear phase shift of an optical field induced
by another field having a different wavelength, direction, or state of polarization.
Its origin can be understood by noting that the total electric field E in Eq. (1.3.1)
is given by
E = 1 2 ˆx[E 1 exp(−iω 1 t)+E 2 exp(−iω 2 t)+c.c.], (1.3.5)
when two optical fields at frequencies ω 1 and ω 2 , polarized along the x axis, propagate
simultaneously inside the fiber. (The abbreviation c.c. stands for complex conjugate.)
The nonlinear phase shift for the field at ω 1 is then given by
φ NL = n 2 k 0 L(|E 1 | 2 + 2|E 2 | 2 ), (1.3.6)
where we have neglected all terms that generate polarization at frequencies other than
ω 1 and ω 2 because of their non-phase-matched character. The two terms on the righthand
side of Eq. (1.3.6) are due to SPM and XPM, respectively. An important feature of
XPM is that, for equally intense optical fields of different wavelengths, the contribution
of XPM to the nonlinear phase shift is twice that of SPM. Among other things, XPM is
responsible for asymmetric spectral broadening of copropagating optical pulses. Chapters
6 and 7 discuss the XPM-related nonlinear effects.
1.3.2 Stimulated Inelastic Scattering
The nonlinear effects governed by the third-order susceptibility χ (3) are elastic in the
sense that no energy is exchanged between the electromagnetic field and the dielectric
medium. A second class of nonlinear effects results from stimulated inelastic scattering
in which the optical field transfers part of its energy to the nonlinear medium. Two