Modulational instability: Difference between revisions

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===Mathematical derivation of gain spectrum===

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

:<math>A = \sqrt{P} e^{i\gamma Pz},</math>

where the oscillatory <math>e^{i\gamma Pz}</math> [[wave phase|phase]] factor accounts for the difference between the linear [[refractive index]], and the modified [[refractive index]], as raised by the Kerr effect. The beginning of instability can be investigated by perturbing this solution as

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

where <math>\varepsilon(t,z)</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 Nonlinearnonlinear 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

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

:<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>

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

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>\varepsilon=c_1 e^{i k_m =z - i \pm\sqrt{\beta_2^2\omega_m^4 t} + 2c_2 \gammae^{- Pi \beta_2k_m z + i \omega_m^2 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 Nonlinearnonlinear 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

:<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 \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} &;\, -\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>