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On the Creation of Solitons in Amplifying Optical Fibers 时间:2018-09-03 22:13   来源:未知   作者:admin   点击: 次   Abstract:We treat the creation of solitons in amplifying fibers. Strictly speaking, solitons are objects in an integrable setting while in real-world systems loss and gain break integrability. That case usually has been treated in the perturbation limit of low loss or gain. In a recent approach fiber-optic solitons were described beyond that limit, so that it became possible to specify how and where solitons are eventually destroyed. Here we treat the opposite case: in the presence of gain, new solitons can arise from an initially weak pulse. We find conditions for that to happen for both localized and distributed gain, with no restriction to small gain. By tracing the energy budget we show that even when another soliton is already present and copropagates, a newly created soliton takes its energy from radiation only. Our results may find applications in amplified transmission lines or in fiber lasers.
1. Introduction
   Solitons are fascinating objects. They arise from a variety of nonlinear wave equations; here we will concentrate on the Nonlinear Schrödinger Equation (NLSE) and fiber-optic solitons as these represent the only type of solitons that has already seen commercial application [1]. Fiber-optic solitons are light pulses which balance the fiber’s dispersion with its nonlinearity such as to stabilize their shape; this makes them eminently suitable as signalling light pulses in optical data transmission. For any other type of soliton a similar interplay of effects produces a similar self-stabilization.
  For the NLSE, Zakharov and Shabat found the soliton solution in their ground-breaking paper [2] (called ZS hereafter). This was followed by an equally important paper by Satsuma and Yajima [3] (hereafter, SY) where the pertaining initial-value problem was solved. Both together established the basics of solitons in fibers as they were suggested in [4]; experimentation commenced a few years later [5].
   When it comes to real-world settings rather than the idealized context of the integrable NLSE, one has to deal with the impact of power loss on solitons. This issue was treated with perturbation methods by several authors [6–11]. However, such approach requires that the loss be weak and can cover neither strong attenuation coefficients nor long distances with weak coefficients. Moreover, it entirely misses the eventual decay of the soliton. For a long time, investigations of lossy fibers beyond the weak-loss limit were confined to numerical simulations.
We could recently demonstrate [12] that SY can be used to cover lossy fiber by interpreting continuous loss as a sequence of infinitely many infinitesimal localized losses, each of which can be treated by SY. It became clear, among other things, what the mechanism for the eventual death of a soliton is. While that paper concentrated on loss, the total accumulated loss factor (called ) can easily be used to describe gain, by letting .
Can solitons be amplified without creating radiation in the process? As shown in [13], that is possible in a very special set of conditions: the soliton needs to have a particular chirp, the fiber parameters must vary along the distance to a certain specification (tapering), and there are constraints on the gain mechanism. Here we consider conventional (unchirped) solitons in conventional (nontapered) fibers, without assumptions about the gain mechanism. Then, radiation-free amplification is possible only in the adiabatic limit, a case of little use in practical terms.