Optical Waveguides: Numerical Modeling

Nonlinear Properties

Nonlinear effects can have a significant impact on light propagation in optical fibers. While the nonlinear response of optical fibers is relatively weak, tight confinement of light in a micron-size core together with long propagation distances (100 km and more) result in significant nonlinear interactions. Nonlinear effects can be both advantageous and detrimental for fiber devices and transmission systems. On the one hand, nonlinear effects enable new functionalities and devices, such as switching, supercontinuum generation, wavelength conversion, fiber lasers, amplifiers, logic devices, and soliton transmission systems. At the same time, they may be detrimental for optical fiber communication systems.

At high intensities, light-matter interactions become nonlinear as a result of the anharmonic motion of bound electrons in the applied field. As a result, the induced polarization can be written as


where  is the linear susceptibility, and  and  are the second- and third-order nonlinear optical susceptibilities. For simplicity, in Eq. (14) it is assumed that both P and E are scalar, and the medium is dispersionless and lossless. In the most general case,  are tensors of rank j+1 and are frequency-dependent.

The second-order susceptibility, which is responsible for such effects as second-harmonic generation and sum frequency generation, vanishes in optical fibers, owing to the inversion symmetry of silica glass. On the other hand, the third-order susceptibility plays the key role.

Major nonlinear effects in optical fibers include self-phase modulation (SPM), cross-phase modulation (XPM), four-wave mixing (FWM), stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS). Next, we briefly discuss the basics of each of these effects.

Self-Phase Modulation

Self-phase modulation is the change in the phase of an intense optical pulse due to the nonlinear change of the refractive index of the material caused by this intense pulse. Such a self-induced phase shift can be written as (15)


is the nonlinearity coefficient, is the nonlinear coefficient related to , is the wavelength, is the effective core area, P is the peak power, L is the fiber length (assuming lossless case). The SPM results in a spectral broadening of a pulse, which may be disadvantageous for multichannel communication systems, which transmit signals at multiple wavelengths simultaneously.  However, the SPM is extremely useful in many applications, including supercontinuum generation, optical regeneration, and soliton systems.

Cross-Phase Modulation

Cross-phase modulation is the nonlinear phase shift of an optical field at a given frequency (or a given polarization) induced by a co-propagating field at another frequency (or another polarization). The effect of the XPM is twice as strong as that of the SPM as can be seen from the following expression:(16)

where the first term is responsible for the SPM, and the second one is due to the XPM. One of the major effects of the XPM is asymmetric spectral broadening. The XPM leads to many fundamental nonlinear phenomena such as soliton trapping and finds applications in Kerr shutters and intensity discriminators.

Four-Wave Mixing

Four-wave mixing is a third-order parametric process that involves nonlinear interaction of four waves. In this process, two photons at frequencies   and   are annihilated and create two photons at frequencies   and , such that (17)


The FWM process described by Eq. (17) relies on a phase-matching condition given by(18)

Condition (18) is easily satisfied in optical fibers for the partially degenerate case when . The FWM effect is used in many important fiber applications, including wavelength conversion, parametric amplification, and optical regeneration, among others.

Stimulated Raman Scattering

Stimulated Raman scattering is a process in which the incident photon is scattered by a molecule to a lower-frequency photon and is accompanied by a transition of the molecule between two vibrational states. The SRS can strongly affect the performance of multichannel transmission systems by transferring the energy between channels. At the same time, the SRS forms a basis of broadband Raman amplifiers and tunable Raman lasers that are of foremost importance for energy restoration in optical transmission systems and have numerous other applications as well.

Stimulated Brillouin Scattering

Stimulated Brillouin scattering is a nonlinear process that originates from the interaction of light with acoustic phonons and results in generation of a backward-propagating Stokes wave downshifted in frequency from the pump wave. The SBS has a relatively low threshold (lower that that for the SRS) and can be detrimental for fiber-optic transmission systems. However, the SBS finds applications in Brillouin lasers and amplifiers.

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