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→Mathematical derivation of gain spectrum: c_2=c_1^* |
→Initial instability and gain: no treshold power for this modulational instability, see Agrawal |
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==Initial instability and gain==
Modulation instability only happens under certain circumstances. The most important condition is ''anomalous group velocity [[dispersion relation|dispersion]]'', whereby pulses with shorter [[wavelength]]s travel with higher [[group velocity]] than pulses with longer wavelength.<ref name="agrawal" /> (This condition assumes a ''focussing'' [[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, whilst at other frequencies, a perturbation will [[exponential growth|grow exponentially]]. The overall [[Gain (electronics)|gain]] spectrum can be derived [[Analytical expression|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.
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