Modulational instability: Difference between revisions

Revision as of 15:25, 6 April 2020

Revision as of 21:21, 3 May 2020

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{{short description|A phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity,}}

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 [[Frequency spectrum|spectral]]-sidebands and the eventual breakup of the waveform into a train of [[wave packet|pulses]].<ref name="BenjaminFeir">{{cite journal

| doi = 10.1017/S002211206700045X

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

:<math>\beta_2^2\omega_m^2 + 2 \gamma P \beta_2 < 0,</math> {{pad|2em}} that is for {{pad|2em}} <math>\omega_m^2 < -2 \frac{\gamma P}{\beta_2}.</math>

This condition describes the requirement for anomalous dispersion (such that <math>\gamma\beta_2</math> is negative). The gain spectrum can be described by defining a gain parameter as <math>g \equiv 2|\Im\{k_m\}|,</math> so that the power of a perturbing signal grows with distance as <math>P\, e^{g z}.</math> The gain is therefore given by