Further Studies On Modulation Instability In Optical Fibers

Modulation instability is an universal and very important physical phenomenon, which can be observed in many fields, such as fluid dynamics, plasmas, condense-state physics, optics, etc. Modulation instability in optics results from the interplay between the dispersion (or diffraction) and nonlinear effects. It includes temporal, spatial, and spatiotemporal modulation instabilities. The modulation instability occurring in an optical fiber is usually called temporal domain modulation instability. It manifests itself as the occurrence of the spectral side bands and breakup of cw or quasi-cw light radiation into a train of ultrashort pulses in spectral and temporal domain, respectively. This temporal domain modulation instability can be utilized to generate ultrashort optical pulse chains with high repetition rates, construct modulation instability lasers and create temporal domain optical solitons. However, the temporal domain modulation instability is also an important limiting factor for optical fiber communications. The spatial domain modulation instability is closely related to the generation of spatial optical solitons and the filament damages in the laser media of high power lasers. For these reasons, modulation instability has attracted extensive andcontinuous interests. In this paper, the self-phase modulation induced modulation instability and the cross-phase modulation of two optical beams of different wavelengths with the same polarization induced modulation instability are deeply investigated.Adopting the method of the traditional linear-steady analysis, expressing the complex amplitude of optical perturbation as {U₀cos (kz-ΩT) +iV₀sin (kz-ΩT)}or {U₀exp[i(kz-ΩT)] +iV₀exp[i(kz-ΩT)]}, the self-phase modulation induced modulation instability in an optical fiber is studied and the condition and gain spectrum is also obtained. The previous processing of modulation instability can simplify the mathematical process considerably and predict correctly the frequency regions and gain spectrum to some extent when the perturbation wave number k is a pure imaginary number. However, from their mathematical deducing process, it is easy to see that, the traditional processing actually seek a constant wave number k and a constant perturbation gain coefficient. In other words, the traditional processing actually nullifies the first order derivation of the perturbation phase with respect to the propagation distance z and, in turn, implies an extra constraint has been imposed on the evolution equation of perturbation which should be self consistent and have theoretically universal solution. Accordingly, the impacts of phases on the evolutions of the perturbation fields are lost. As a result, some phase related information of the evolutions of the perturbation fields is blacked out. Under these circumstances, traditional studies actually led the discussions to the asymptotic behaviors from the very beginning and then failed to obtain the integrated physical pictures of the initial evolutions of perturbations. In a word, the aim of the predecessors' method of the linear-steady analysis is to describe the initial evolutions of the optical perturbations, but for their defects of mathematical processing,this initial information is lost precisely. Therefore, predecessors' conclusions concluded from the method of the linear-steady analysis are not physically comprehensive. From the view point of theory or practical application, it is necessary to study modulation instability further so as to obtain a thorough and self consistent physical picture.In this work, using a magnitude and phase angle to describe the optical perturbation, starting from the nonlinear Schrodinger equation in an optical fiber, the evolution differential equations for the magnitude and phase of the optical perturbations are deduced and solved in case of small signal approximation. The analytical expressions for the perturbation magnitude, phase and gain coefficient are obtained. Evolutions of the perturbation magnitude, phase, and gain coefficient with the propagating distance for different initial optical perturbations are analyzed and discussed. The asymptotic behaviors of the perturbation phase and gain coefficient are also analyzed and discussed. The results show that, the perturbation evolutions depend on the initial values of perturbations. In the abnormal dispersion region when the perturbation frequency is smaller than the cut-off frequency, where modulation instability is predicted by the predecessors' study to occur, there exist several regions in the phase-distance plane. Perturbation with initial phases falling into these regions will evolve to different asymptotic values eventually. Depending on initial phases, phases may decrease or increase monotonously with the propagation distance before approaching asymptotic values. Similarly, also depending on the initial phases of perturbations, the perturbation magnitudes may increase monotonously or decrease before increasing with the propagation distance. As for the gain coefficient, it is generally a function of the propagation distance and not an invariant parameter as reported in the previous literatures no matter whether in the normalor abnormal dispersion regions of the optical fiber. Moreover, depending on the initial perturbation phases, all the unstable perturbations have the chance to claim a common maximum gain of frequency independent. In the normal dispersion region, as well as in the abnormal region when the perturbation frequency is larger than the cut-off frequency, where the modulation instability is not thought to occur, the perturbation gains oscillate with the propagation distance. In the abnormal dispersion region when the perturbation frequency is smaller than the cut-off frequency, where modulation instability is thought to occur, the gains of perturbations with different initial phases while with the same frequency will evolve to the same positive value (asymptotic value) in different ways. The gains of the unstable perturbations reported in predecessors' researches only represent the asymptotic values of this work. Also depending on the initial phases, the gain coefficients will evolve in different ways before reaching their asymptotic values.According to the traditional analysis, the perturbations with frequencies equal to zero or cut-off frequency have a zero gain coefficient. In this case, one naturally concludes that the magnitudes of the two special perturbations are convergent should not be of unstable types. However, as stated above, the traditional processing of modulation instability has neglected the phase-related information, and then led the discussions to the asymptotic behaviors from the very beginning. Under these circumstances, the initial evolutions of the two special perturbations have been lost. Therefore, it is difficult to judge whether the two perturbations are stable or not. In this work, analytical expressions of the magnitudes, phases and gain coefficients of the two perturbations with zero or cut-off frequency are deduced. The evolutions of the magnitudes, phases and gain coefficients of the two perturbations with the propagation distance are analyzed. Furthermore, the asymptoticbehaviors of the perturbation phases and gain coefficients are analyzed and discussed. The results indicate that, the evolutions of the phases, magnitudes and gain coefficients of the two special perturbations are also closely related to the initial values of the perturbations. The phase of the perturbation with zero (or cut-off) frequency increases (or decreases) with the propagation distance monotonously and tends to its asymptotic value n%+n/2 (or n7c) eventually. Also depending on the initial phases of perturbations, the perturbation magnitudes may increase monotonously or decrease before increase with the propagation distance. The magnitudes of the two perturbations are both divergent when the distance goes to infinity. Depending on the initial phases of the two perturbations, the gain coefficients evolve in different ways before tending to their zero asymptotic values. Accordingly, the two special perturbations can also lead to modulation instability.Mathematically, the perturbations can also be expressed in terms of real and imaginary parts. In this paper, the evolution differential equations for the real and imaginary parts of the optical perturbations are deduced and solved in case of small signal approximation. The analytical expressions for the real and imaginary parts of the optical perturbations are obtained. Their evolutions and coupling relations for different initial values of perturbations are analyzed and discussed. Comparisons have been made with different mathematical forms used to represent perturbation fields in studying the modulation instability happened in optical fibers. The results indicate that, the evolutions of the real and imaginary parts of the perturbations are also closely related to the initial values of the perturbations. Although the two representations, i.e., real plus imaginary parts and magnitude plus phase angle, are mathematically universal and equivalent, the expression using magnitude and phase angle describes the evolutions ofperturbations more clearly and intuitively than that using the real and imaginary part does in a physical sense. The traditional perturbation expressions can, to some extent, correctly predict the frequency regions and asymptotic gain coefficients of modulation instability, and make the mathematical process very simple. But they can not describe the input condition related initial evolutions of perturbations. In conclusion, in case of small signal approximation, to study modulation instability happened in optical fibers both intuitively and comprehensively in a physical sense, the expression based on magnitude and phase angle appears to be a preferable choice.Based on the coupled nonlinear Schrodinger equation in single-mode optical fibers, conditions and gain spectra of modulation instability induced by cross-phase modulation for two optical waves of different frequencies with the same polarizations are further analyzed in this work. The results show that, the conditions and gain spectra of modulation instability are unique only when the two optical waves both propagate in the normal group-velocity dispersion region. When one of the two optical waves propagates in the normal group-velocity dispersion region and the other in the anomalous region, depending on parameters of the two optical waves, perturbation gain spectra may take three possible forms. Under certain circumstances, it is even possible that there are two forms of gain spectra within certain range of perturbation frequency.Starting from the five-order electric polarization, the quintic nonlinear refractive index, overlap integral and quintic nonlinear index induced by cross-phase modulation of two optical waves of different frequencies with the same polarizations in loss single-mode optical fibers are derived. Then, in case of co-existence of cubic and quintic nonlinearity, the coulped nonlinear Schrodinger equations of the slowly varying envelopes for two optical waves and the linearized nonlinearSchrodinger equations for the perturbations are also derived. The synthetic effects of the quintic nonlinear coefficients (size and sign), the dispersion longitudinal varying parameters, the sign of the second order group-velocity dispersion, the size relation of the two perturbation frequencies and the loss on the condition and power gain spectra of modulation instability are analyzed in detail. The results show that, whether in the normal or abnormal group-velocity dispersion region, the positive and the negative quintic nonlinearity will intensify and suppress modulation instability induced by the cross-phase modulation, respectively. Depending on the dispersion longitudinal varying parameters and the size relation of the two perturbation frequencies, the quintic nonlinearity will intensify or suppress modulation instability to different degrees. Also depending on the different parameters, some gain spectra may have no cut-off perturbation frequencies. The loss will suppress the modulation instability.