Content deleted Content added
m →Initial instability and gain: Fixing links to disambiguation pages, replaced: gain → gain using AWB |
m →Mathematical derivation of gain spectrum: Typo fixing, typo(s) fixed: Therefore → Therefore, using AWB |
||
Line 91:
where the perturbation has been assumed to be small, such that <math>\epsilon^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
:<math>\epsilon=c_1 e^{i k_m z - i \omega_m t} + c_2 e^{- i k_m z + i \omega_m t}</math>
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. 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 negative, the wavenumber will become [[imaginary number|imaginary]], corresponding to exponential growth and thus instability. Therefore, instability will occur when
| |||