Modulation instability only happens under certain circumstances. The most important condition is
anomalous group velocity dispersion, whereby pulses with shorter
wavelengths travel with higher
group velocity than pulses with longer wavelength. (This condition assumes a
focusing Kerr nonlinearity, whereby refractive index increases with optical intensity.) The instability is strongly dependent on the frequency of the perturbation. At certain frequencies, a perturbation will have little effect, while at other frequencies, a perturbation will
grow exponentially. The overall
gain spectrum can be derived
analytically, as is shown below. Random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum. The tendency of a perturbing signal to grow makes modulation instability a form of
amplification. By tuning an input signal to a peak of the gain spectrum, it is possible to create an
optical amplifier.
Mathematical derivation of gain spectrum The gain spectrum can be derived by starting with a model of modulation instability based upon the
nonlinear Schrödinger equation : \frac{\partial A}{\partial z} + i\beta_2\frac{\partial^2A}{\partial t^2} = i\gamma|A|^2A, which describes the evolution of a
complex-valued slowly varying envelope A with time t and distance of propagation z. The
imaginary unit i satisfies i^2=-1. The model includes
group velocity dispersion described by the parameter \beta_2, and
Kerr nonlinearity with magnitude \gamma. A
periodic waveform of constant power P is assumed. This is given by the solution :A = \sqrt{P} e^{i\gamma Pz}, where the oscillatory 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 :A = \left(\sqrt{P}+\varepsilon(t,z)\right)e^{i\gamma Pz}, where \varepsilon(t,z) is the perturbation term (which, for mathematical convenience, has been multiplied by the same phase factor as A). Substituting this back into the nonlinear Schrödinger equation gives a
perturbation equation of the form :\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 |\varepsilon|^2\ll P. The
complex conjugate of \varepsilon is denoted as \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 :\varepsilon=c_1 e^{i k_m z - i \omega_m t} + c_2 e^{- i k_m^* z + i \omega_m t}, where k_m and \omega_m are the
wavenumber and (real-valued)
angular frequency of a perturbation, and c_1 and 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, \omega_m and 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 c_2=c_1^* and subject to the condition :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 :\beta_2^2\omega_m^2 + 2 \gamma P \beta_2 that is for \omega_m^2 This condition describes the requirement for anomalous dispersion (such that \gamma\beta_2 is negative). The gain spectrum can be described by defining a gain parameter as g \equiv 2|\Im\{k_m\}|, so that the power of a perturbing signal grows with distance as P\, e^{g z}. The gain is therefore given by :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 where as noted above, \omega_m is the difference between the frequency of the perturbation and the frequency of the initial light. The growth rate is maximum for \omega^2=-\gamma P/\beta_2. == Modulation instability in soft systems ==