Modulational instability: Difference between revisions

Revision as of 10:11, 3 January 2017

Revision as of 14:00, 3 January 2017

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The gain spectrum can be derived <ref name="agrawal" /> by starting with a model of modulation instability based upon the [[nonlinear Schrödinger equation]]

: <math>\frac{\partial A}{\partial z} + i\beta_2\frac{\partial^2A}{\partial t^2} = i\gamma|A|^2A,</math>

which describes the evolution of a [[complex number|complex-valued]] [[Slowly varying envelope approximation|slowly varying envelope]] <math>A</math> with time <math>t</math> and distance of propagation <math>z</math>. The [[imaginary unit]] <math>i</math> satisfies <math>i^2=-1.</math> The model includes [[group velocity]] dispersion described by the parameter <math>\beta_2</math>, and [[Kerr effect|Kerr nonlinearity]] with magnitude <math>\gamma.</math> A [[periodic function|periodic]] waveform of constant power <math>P</math> is assumed. This is given by the solution

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:<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\ll P.</math> The [[complex conjugate]] of <math>\varepsilon</math> is denoted as <math>\varepsilon^*.</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>\varepsilon=c_1 e^{i k_m z - i \omega_m t} + c_2 e^{- i k_m z + i \omega_m t},</math>

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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 both the requirement for anomalous dispersion (such that <math>\gamma\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 \propto, 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}, &;\,text{for } -\beta_2^2displaystyle \omega_m^2 < - 2 \frac{\gamma P }{\beta_2},
> 0 \\[2ex]
0, &;\,text{for } -\beta_2^2displaystyle \omega_m^2 \ge - 2 \frac{\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. The growth rate is maximum for <math>\omega^2=-\gamma P/\beta_2.</math>

==Breakup==