International Journal of Optics

Volume 2018, Article ID 9452540, 11 pages

https://doi.org/10.1155/2018/9452540

## On the Creation of Solitons in Amplifying Optical Fibers

Institut für Physik, Universität Rostock, 18059 Rostock, Germany

Correspondence should be addressed to Fedor Mitschke; ed.kcotsor-inu@ekhcstim.rodef

Received 20 December 2017; Accepted 4 February 2018; Published 7 March 2018

Academic Editor: Murugan Senthil Mani Rajan

Copyright © 2018 Christoph Mahnke et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

In the general case of more-than-vanishing gain, the question naturally and unavoidably arises: if one starts with a weak (nonsolitonic) pulse and boosts its power, can a soliton be created? Or, alternatively, if one starts with a soliton and amplifies its power, can a second soliton arise?

The answer is straightforward only if one has localized (stepwise) gain acting on an unchirped pulse as this special case can be treated with SY. For chirped pulses and distributed gain the answer is not obvious at all. We point out that clarification of this issue may be relevant to several contexts. In today’s fiber links, optical gain is often provided by Raman amplification which is distributed over a long fiber length. Can a soliton, after suffering from some severe perturbation, be restored to obtain solitonic properties again? Or if one considers a fiber laser, if the gain fiber segment is fed with a weak pulse, can a soliton arise? (In the case of a laser, resonator boundary conditions apply; our treatment covers processes in the gain fiber only.)

In the present paper we address the problem of soliton creation from gain, both localized and distributed. Where closed solutions do not exist, we resort to fits of observed behavior, to arrive at statements with some predictive power.

#### 2. Basic Definitions

The Nonlinear Schrödinger Equation (NLSE) in normalized form is [2, 14, 15] where is distance, is time, and is the amplitude envelope in a comoving frame of reference. Equation (1) describes the propagation of light in optical fibers in the absence of loss and gain, at anomalous dispersion. Its fundamental soliton solution has the form when it rests in the center of the frame of reference. is the scaling parameter that sets both peak amplitude and inverse duration. That appears in both reflects the soliton’s hallmark balance between dispersive and nonlinear effects, a fact that has also been expressed as an “area theorem” [16] that the product of peak amplitude and width—or, alternatively, peak power times width squared—takes a constant value [14, 15, 17]. Frequently the same is expressed using characteristic length scales for dispersion () and nonlinearity () [14, 15]; in our dimensionless units they both take the form . appears in the phase term indicating a nonlinear phase rotating at a rate proportional to peak power. The soliton energy is .

We will investigate the evolution of a pulse that is initially not a soliton. In its fullest generality the problem would be intractable, and so we make one simplifying assumption: we consider pulses that have sech amplitude envelope and are initially unchirped. To describe nonsolitonic pulses, we break the link between peak amplitude and inverse duration by introducing an amplitude scaling parameter . As pulse parameters may evolve during propagation, we abbreviate for . Hence, the initial condition (the launch pulse at ) is written as On the RHS we use the soliton order which compares the relative strength of dispersive and nonlinear effects [14, 15]. The integer number closest to is the soliton count [3]. if and only if the pulse is chirp-free. As we always consider chirp-free initial conditions, we do not need to make the distinction at the launch point, but during propagation, may and will differ from . For the soliton is recovered; in the limit the propagation is linear, that is, purely dispersive.

In Section 3.2 we will consider pulses launched with , motivated by the well-known fact obtained from Inverse Scattering theory [2] that no soliton is formed if . Subsequently, in Section 4.2 we will consider pulses launched with , that is, from the regime, where there is exactly one soliton, to determine whether a second soliton can arise from gain.

If the gain is localized, it follows from SY that the gain enhances the power at least up to the threshold for the first soliton to appear, for the second, and so on. The minimum required gain for that is . If the gain is not localized but distributed, more of it is required to reach threshold because there are two opposing trends: the pulse energy grows exponentially with distance as where we introduce a dimensionless gain coefficient . A characteristic dimensionless gain length is . The gain can be represented in (1) by replacing the RHS with . Energy growth brings the pulse closer to the threshold of soliton generation. At the same time, the pulse undergoes dispersive broadening which renders it temporally broader, lets the peak power droop, and—most importantly—creates a chirp. Power boost helps chirp hinder soliton creation. Therefore we must first discuss the formation of chirp in Section 3.1.

#### 3. The Situation without Gain or Loss

##### 3.1. A Note on Chirp

We start with a comment on the linear, purely dispersive limit. The sech shape typical for solitons is mathematically less convenient than a Gaussian: an initially Gaussian pulse, upon dispersion, remains Gaussian everywhere, except for rescaling, but underneath the Gaussian envelope a linear chirp (quadratic phase profile) develops. According to [14] but written in dimensionless units, its evolution is given by with arbitrary amplitude . At center (),

This is to be compared to an initially sech-shaped pulse which undergoes some shape variations. Pulse wings exhibit some wiggles around (after one soliton period), but asymptotically the shape is not much different from a chirped sech again at least in its central part. Unfortunately, no closed analytic expression is available to write this evolution. We therefore make the simplifying assumption for the linear regime that the pulse envelope never strays far from the sech shape; indeed assume that the shape is constant except for scaling. This forces us to take into account the different spreading rates of the two shapes by introducing a scaling factor for the amplitude and phase evolution in the form By varying for the best match of both evolutions we find [12].

We now proceed by considering a pulse amplitude that is not vanishingly small. Then there is a nonlinear contribution to the pulse evolution. It was observed early on in numerical work [11] that in the anomalous-dispersion regime considered here this contribution always serves to reduce the broadening. Here, however, we need to evaluate the chirp.

The effect of chirp on the soliton content of a pulse has been studied before. In 1979 Hmurcik and Kaup [16] started from the observation that the McCall-Hahn area theorem in self-induced transparency only applies to unchirped pulses and found that chirped pulses need to be stronger in order to generate the same effect. Desem and Chu [18] 1986 specifically talked about fiber-optic solitons, considered ZS soliton eigenvalues, and found how for (in current usage of terminology) different soliton orders the power going into the soliton is diminished when chirp is introduced, to the point that a soliton vanishes at some particular value of the chirp, beyond which all power goes into radiation (the soliton order is closely related to the area theorem). Also in 1986, Blow and Wood [19] pursued the relation between the solitonic energy eigenvalue and the pulse’s bandwidth. In 1987, Mamistov and Sklyarov [20] determined threshold values of phase modulation beyond which the soliton count in a pulse would be reduced. Kaup et al. [21] 1994 introduced a simplified pulse shape (rectangular amplitude profile, piecewise-linear antisymmetric phase profile) which allowed getting further insight. They found that, for different soliton orders, at the point where the soliton vanishes the phase slope was always close to a characteristic value.

In all cases the results indicate that comparing an unchirped with a chirped pulse of the same energy, a lesser fraction of energy is available for soliton formation in the chirped case. This will become apparent in detail below.

##### 3.2. Evolution of an sech Pulse

Chirp is detrimental to soliton formation; energy gain is conducive to it. With energy growing exponentially with distance and the chirp growing at a much lesser rate (this will be detailed in Figure 3), there will always be a crossover point where the advantage from gain outweighs the disadvantage from chirp. The smaller the gain factor, the longer the distance over which the chirp can develop. Our discussion will thus concentrate on the run-up distance to soliton formation, that is, the position in the amplifying fiber where a soliton is created.

The propagation of a “weak” pulse in this framework is fully described by just three parameters: amplitude scaling factor , soliton scaling parameter , and chirp coefficient . The latter is defined as where is the optical phase. A chirped sech pulse at position is then written as We now follow the evolution of such pulse. It will broaden with distance due to dispersion, but depending on the nonlinear Kerr effect will counteract and slow down the broadening [11]. Due to energy conservation, the peak amplitude will droop accordingly.

In the linear limit, by way of our approximation, the amplitude evolution is well described [12] by with a small amplitude factor in the numerator. In the denominator, the relevant variable is because the relevant length scales and both scale as . To find a useful expression for the evolution for amplitudes larger than in the linear limit we numerically checked how scales with and find that makes a very good fit. Data in Figure 1 test this assertion for and several values of ; we have also checked for different values (not shown). A comparison is shown between plots of (10) (solid curves) and a full numerical simulation which provides a quite accurate reference (dashed curves). The agreement is convincing, confirming that our approximations are quite reasonable.