Modulational instability: Difference between revisions

Revision as of 03:06, 18 December 2016

Revision as of 03:07, 18 December 2016

Line 93:

:<math>A = \left(\sqrt{P}+\varepsilon(t,z)\right)e^{i\gamma Pz}</math>

where <math>\varepsilon\left(t,z\right)</math> is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as <math>A</math>). Substituting this back into the Nonlinear Schrödinger equation gives a [[perturbation theory|perturbation equation]] of the form

:<math>\frac{\partial \varepsilon}{\partial z}+i\beta_2\frac{\partial^2\varepsilon}{\partial t^2}=i\gamma P \left(\varepsilon+\varepsilon^*\right)</math>

where the perturbation has been assumed to be small, such that <math>\varepsilon^2\approx 0</math>. 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

Line 107:

:<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 power be exceeded. The gain spectrum can be described by defining a gain parameter as <math>g \equiv \Im [ 2|k_m| ]</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>