# Modulation instability

Original Entry: Ian Burgess, fall 2009

## Definition

Modulation instability occurs in a variety of nonlinear systems where fluctuations due to noise experience gain through the nonlinearity. The term is most commonly used in the context of nonlinear optics whereby the spatial profile of a broad beam, or the temporal profile of a long pulse can spontaneously break up into an array of smaller (shorter) beams (pulses), each having a characteristic size where the effects of the nonlinear interaction reach equilibrium with diffraction. The simplest type of such a nonlinear material is one which displays an optical Kerr effect, where the index increases linearly with intensity. Temporal envelope fluctuations in a long pulse propagating in a fiber under the Kerr effect grow at a rate that depends on their frequency, with a maximum gain occurring at a specific frequency <math>\omega_{m}</math>, which depends on the strengths of dispersion and the nonlinearity (see ref. 1 for a detailed mathematical treatment of this problem). The spectrum of the pulse after propagation shows the appearance of two sidebands at <math>\omega_{0}\plusminus\omega_{m}</math>.

## Reference

G.P. Agrawal, *Nonlinear Fiber Optics*, 3rd Ed. Academic Press, 2007.