Modulational instability: Difference between revisions

Revision as of 22:20, 30 July 2009

Revision as of 14:57, 31 July 2009

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where <math>\omega_m</math> and <math>k_m</math> are the [[frequency]] and [[wavenumber]] of a perturbation, and <math>c_1</math> and <math>c_2</math> are constants. It should be noted that the Nonlinear Schrödinger equation is constructed by removing the [[carrier wave]] of the light being modelled, and so the frequency of the light being perturbed is formally zero. Therefore <math>\omega_m</math> and <math>k_m</math> don't represent absolute frequencies and wavenumbers, but the ''difference'' between these and those of the initial beam of light. It can be shown that the trial function is valid, subject to the condition

:<math>k_m = \pm\sqrt{\beta_2^2\omega_m^4 + 2 \gamma P \beta_2 \omega_m^2}</math>

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 positivenegative, 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 energy 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