Modulational instability: Difference between revisions

Revision as of 14:57, 31 July 2009

Revision as of 14:58, 31 July 2009

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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>

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

:<math>g = \begin{cases} 2\sqrt{-\beta_2^2\omega_m^4-2\gamma P \beta_2\omega_m^2} &;\, -\beta_2^2\omega_m^2 - 2 \gamma P \beta_2 > 0 \\ 0 &;\, -\beta_2^2\omega_m^2 - 2 \gamma P \beta_2 \leq 0\end{cases} </math>

where as noted above, <math>\omega_m</math> is the difference between the frequency of the perturbation and the frequency of the initial light.