In the fields of nonlinear optics and fluid dynamics, modulational instability or sideband instability is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.

It is widely believed that the phenomenon was first discovered − and modeled − for periodic surface gravity waves (Stokes waves) on deep water by T. Brooke Benjamin and Jim E. Feir, in 1967. Therefore, it is also known as the Benjamin−Feir instability. However, spatial modulation instability of high-power lasers in organic solvents was observed by Russian scientists N. F. Piliptetskii and A. R. Rustamov in 1965, and the mathematical derivation of modulation instability was published by V. I. Bespalov and V. I. Talanov in 1966. Modulation instability is a possible mechanism for the generation of rogue waves.

Initial instability and gain

Modulation instability only happens under certain circumstances. The most important condition is anomalous group velocity dispersion, whereby pulses with shorter wavelengths travel with higher group velocity than pulses with longer wavelength. (This condition assumes a focusing Kerr nonlinearity, whereby refractive index increases with optical intensity.)

The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, while at other frequencies, a perturbation will grow exponentially. The overall gain spectrum can be derived analytically, as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum.

The tendency of a perturbing signal to grow makes modulation instability a form of amplification. By tuning an input signal to a peak of the gain spectrum, it is possible to create an optical amplifier.

Mathematical derivation of gain spectrum

The gain spectrum can be derived by starting with a model of modulation instability based upon the nonlinear Schrödinger equation

which describes the evolution of a complex-valued slowly varying envelope. The imaginary unit The model includes group velocity, and Kerr nonlinearity A periodic is assumed. This is given by the solution

phase factor accounts for the difference between the linear refractive index, and the modified refractive index, as raised by the Kerr effect. The beginning of instability can be investigated by perturbing this solution as

). Substituting this back into the nonlinear Schrödinger equation gives a perturbation equation of the form

The complex conjugate Instability can now be discovered by searching for solutions of the perturbation equation which grow exponentially. This can be done using a trial function of the general form

are the wavenumber and (real-valued) angular frequency are constants. The nonlinear Schrödinger equation is constructed by removing the carrier wave don't represent absolute frequencies and wavenumbers, but the difference and subject to the condition

This dispersion relation is vitally dependent on the sign of the term within the square root, as if positive, the wavenumber will be real, corresponding to mere oscillations around the unperturbed solution, whilst if negative, the wavenumber will become imaginary, corresponding to exponential growth and thus instability. Therefore, instability will occur when

1. that is for

The gain is therefore given by

Modulation instability in soft systems

Modulation instability of optical fields has been observed in photo-chemical systems, namely, photopolymerizable medium. Modulation instability occurs owing to inherent optical nonlinearity of the systems due to photoreaction-induced changes in the refractive index. Modulation instability of spatially and temporally incoherent light is possible owing to the non-instantaneous response of photoreactive systems, which consequently responds to the time-average intensity of light, in which the femto-second fluctuations cancel out.

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Further reading