Small Signal Analysis Of Modulation Instability For Kerr Medium

The research on modulation instability in the time domain can be very helpful inimproving the performance of optical communication system. It can also facilitatethe researches in the field of ultra short pulse. In the space domain, modulationinstability will induce small-scale self-focusing which usually causes the filamentdestruction of optical elements and thus limits the output power of the system. Theresearch on modulation instability is necessary. Starting from nonlinear Schr?dinger equation, we have got the analyticalsolutions representing the real and imaginary parts of a perturbation field travelingin optical fibers under small signal approximation and discussed the variationproperty of the real part, imaginary part, the mod, the argument and gain of theperturbation field. Our research shows that: at the initial stage, the waver number kvaries as z changes. Only when z is sufficiently long does z approximate to aconstant; the initial phase of the perturbation field has an important influence on theevolution property and the initial vale of the perturbation gain. For a specific initialphase, the gain of a perturbation at any frequency will reach a common maximumvalue. In the negative dispersion region, when the perturbation frequency is less thanthe cut-off frequency, the gain expression obtained by previous researches is only III四川大学硕士学位论文the maximum value in our expression when the transmission distance is very long.For different initial phase, the gain of the dispersion of the same frequency willreach the same asymptotic value after various evolution process and differenttransmission distance; When the perturbation frequencies are different, theasymptotic values of the gain are different. The analytical solutions for the evolutionof perturbation field obtained by previous researchers are thus incomplete.We have also investigated the problem of small-scale self-focusing under smallsignal approximation and conducted relevant numerical calculation. The analyticalexpressions for the gain coefficient of the self-focusing ripple amplitude have beenobtained. The influence of the initial phase and frequency mod the perturbation gainduring the transmission process has been calculated. Our research shows: In the timedomain, the initial phase of the perturbation has important influences on both theinitial value and the evolution property of the perturbation gain. The gain may beeither greater or less than zero, or equals zero, and have similar evolution propertywith the time domain. Our calculation also indicates that the perturbation with afrequency different from the frequency of the maximum gain may also reach themaximum gain coefficient. This is a result different from the prediction of the BTtheory, and consequently the selection of the length of the gain medium inhigh-power laser system should be treated with caution.