Mathematical derivation of gain spectrum

The gain spectrum can be derived ^[3] by starting with a model of modulation instability based upon the nonlinear Schrödinger equation

${\displaystyle {\frac {\partial A}{\partial z}}+i\beta _{2}{\frac {\partial ^{2}A}{\partial t^{2}}}=i\gamma |A|^{2}A,}$

which describes the evolution of a complex-valued slowly varying envelope ${\displaystyle A}$ with time ${\displaystyle t}$ and distance of propagation ${\displaystyle z}$. The imaginary unit ${\displaystyle i}$ satisfies ${\displaystyle i^{2}=-1.}$ The model includes group velocity dispersion described by the parameter ${\displaystyle \beta _{2}}$, and Kerr nonlinearity with magnitude ${\displaystyle \gamma .}$ A periodic waveform of constant power ${\displaystyle P}$ is assumed. This is given by the solution

${\displaystyle A={\sqrt {P}}e^{i\gamma Pz},}$

where the oscillatory ${\displaystyle e^{i\gamma Pz}}$ 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

${\displaystyle A=\left({\sqrt {P}}+\varepsilon (t,z)\right)e^{i\gamma Pz},}$

where ${\displaystyle \varepsilon (t,z)}$ is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as ${\displaystyle A}$). Substituting this back into the nonlinear Schrödinger equation gives a perturbation equation of the form

${\displaystyle {\frac {\partial \varepsilon }{\partial z}}+i\beta _{2}{\frac {\partial ^{2}\varepsilon }{\partial t^{2}}}=i\gamma P\left(\varepsilon +\varepsilon ^{*}\right),}$

where the perturbation has been assumed to be small, such that ${\displaystyle |\varepsilon |^{2}\ll P.}$ The complex conjugate of ${\displaystyle \varepsilon }$ is denoted as ${\displaystyle \varepsilon ^{*}.}$ 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

${\displaystyle \varepsilon =c_{1}e^{ik_{m}z-i\omega _{m}t}+c_{2}e^{-ik_{m}^{*}z+i\omega _{m}t},}$

where ${\displaystyle k_{m}}$ and ${\displaystyle \omega _{m}}$ are the wavenumber and (real-valued) angular frequency of a perturbation, and ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$ 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, ${\displaystyle \omega _{m}}$ and ${\displaystyle k_{m}}$ 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, provided ${\displaystyle c_{2}=c_{1}^{*}}$ and subject to the condition

${\displaystyle k_{m}=\pm {\sqrt {\beta _{2}^{2}\omega _{m}^{4}+2\gamma P\beta _{2}\omega _{m}^{2}}}.}$

This dispersion relation is vitally dependent on the sign of the term within the square root, as if positive, the wavenumber will be real, corresponding to mere oscillations around the unperturbed solution, whilst if negative, the wavenumber will become imaginary, corresponding to exponential growth and thus instability. Therefore, instability will occur when

${\displaystyle \beta _{2}^{2}\omega _{m}^{2}+2\gamma P\beta _{2}<0,}$   that is for   ${\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}}.}$

This condition describes the requirement for anomalous dispersion (such that ${\displaystyle \gamma \beta _{2}}$ is negative). The gain spectrum can be described by defining a gain parameter as ${\displaystyle g\equiv 2|\Im \{k_{m}\}|,}$ so that the power of a perturbing signal grows with distance as ${\displaystyle P\,e^{gz}.}$ The gain is therefore given by

${\displaystyle g={\begin{cases}2{\sqrt {-\beta _{2}^{2}\omega _{m}^{4}-2\gamma P\beta _{2}\omega _{m}^{2}}},&{\text{for }}\displaystyle \omega _{m}^{2}<-2{\frac {\gamma P}{\beta _{2}}},\\[2ex]0,&{\text{for }}\displaystyle \omega _{m}^{2}\geq -2{\frac {\gamma P}{\beta _{2}}},\end{cases}}}$

where as noted above, ${\displaystyle \omega _{m}}$ is the difference between the frequency of the perturbation and the frequency of the initial light. The growth rate is maximum for ${\displaystyle \omega ^{2}=-\gamma P/\beta _{2}.}$