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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.
Contents
- Initial instability and gain
- Mathematical derivation of gain spectrum
- Modulation Instability in Soft Systems
- References
The phenomenon was first discovered − and modelled − 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. It 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 focussing 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, whilst 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
where the oscillatory
where
where the perturbation has been assumed to be small, such that
where
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
This condition describes the requirement for anomalous dispersion (such that
where as noted above,
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 of photoreactive systems, which consequently responds to the time-average intensity of light, in which the femto-second fluctuations cancel out.