^{1}

^{1}

^{1}

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.

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 [

For the NLSE, Zakharov and Shabat found the soliton solution in their ground-breaking paper [

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 [

We could recently demonstrate [

Can solitons be amplified without creating radiation in the process? As shown in [

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.

The Nonlinear Schrödinger Equation (NLSE) in normalized form 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

In Section

If the gain is localized, it follows from SY that the gain enhances the power at least up to the threshold

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 [

This is to be compared to an initially sech-shaped pulse which undergoes some shape variations. Pulse wings exhibit some wiggles around

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 [

The effect of chirp on the soliton content of a pulse has been studied before. In 1979 Hmurcik and Kaup [

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.

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

The propagation of a “weak” pulse in this framework is fully described by just three parameters: amplitude scaling factor

In the linear limit, by way of our approximation, the amplitude evolution is well described [

Amplitude evolution of weak sech pulses (

Energy conservation ensures that

Evolution of amplitude scaling factor

Chirp evolution for

We concentrate on the central part of the pulse (between the

We are now positioned to determine the soliton content of the pulse under consideration. It is well known [

Soliton energy eigenvalues when the chirp is varied. Data shown as dots for

This figure is similar to Figure

At

We convinced ourselves that our fit functions hold well for all combinations of

We will consider gain in its simplest form, that is, as multiplication of amplitude with a certain gain factor, but without complications like gain saturation, frequency-dependent (selective) gain, and so on. In a purely linear system, it is immaterial whether gain acts at the beginning or at the end of the propagation or is distributed over the distance: both dispersion and gain are linear processes, and they commute. As soon as there are nonlinear phase shifts (or other consequences of nonlinearity), it makes a great difference whether gain is localized or distributed (for the analogous case of loss this was demonstrated in [

For weak pulses, that is, pulses with powers below the threshold of soliton generation or just at the threshold at most (

The eigenvalue component representing the solitonic energy

In Figure

Impact of gain: evolution of amplitude scaling parameter

Figure

The position of the soliton’s birth,

The distance until a first soliton appears (

After discussing how gain can bring a weak (

The main difference to the case of loss as discussed in [

In [

We put (

(a) Evolution of pulse amplitude from model (solid curve) and numerics (dots) for initial values of

In contrast, Figure

From the amplitude the pulse parameters

Parameter evolution and soliton content (red) of an initial unchirped

So far we discussed the evolution of the first soliton; now we turn to a discussion of the second soliton. We traced the energy partition between solitons and radiation by evaluating the nonlinear spectrum. For Figure

(a) Evolution of the energies of both solitons and radiation at

As an example of an actual pulse shape we show the field envelope at

In Figure

Position of creation of the second soliton (dots) as a function of gain parameter

Solitons arise from the radiative energy once there is “enough of it.” This implies that, beginning at the point of soliton creation, radiative energy is depleted; usually the depletion rate is initially faster than the restoration rate by gain. This is in fact visible in all calculations. Indeed, the partition of energy between radiation and soliton has some oscillation with the beat frequency as seen in Figure

The normalized position of first and second soliton generation is shown in an overview in Figure

Propagation distance until a new soliton is generated. Distance is normalized to

Lines (III) and (IV) appear in the realm where the initial condition contains one soliton plus some radiation, that is, for

It is well known from SY [

We were curious to see how this translates into the case of an optical fiber with gain. That situation is more complex because (i) chirp develops during propagation, and (ii) radiation can come from two different sources: the initial condition and the generation during amplified propagation. These circumstances render it less than obvious whether gain leads to the creation of solitons.

By writing an expression for the chirp and by tracking the energy budget, we determined the distance for a new soliton to appear. We verified that the soliton is formed from radiative energy; if another soliton is already present, its energy is not depleted. As long as the fiber length exceeds the creation distance, a soliton will be created under almost all circumstances, the sole exception being the limiting case of adiabatic amplification where the creation distance diverges. As prior investigations treated the problem within the adiabatic approximation, predictions as given here have not been possible before.

In any realistic setting, solitonic data transmission will incorporate both loss and gain in the fiber. In the design of such systems it is mandatory to understand the impact of those factors well, and not just in the somewhat unrealistic adiabatic limit. Our results may also be of interest for investigations of amplification in a fiber laser when the formation of solitonic pulses is desired.

The authors declare that there are no conflicts of interest regarding the publication of this paper.