On the Creation of Solitons in Amplifying Optical Fibers

We treat the creation of solitons in amplifying fibers. Strictly speaking, solitons are objects in an integrable setting while in realworld 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.


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][7][8][9][10][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 Γ < 1) can easily be used to describe gain, by letting Γ > 1.
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 International Journal of Optics 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.

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 ( NL ) [14,15]; in our dimensionless units they both take the form   =  NL =  −2 . 2 appears in the phase term indicating a nonlinear phase rotating at a rate proportional to peak power.The soliton energy is  sol = 2.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  = 0) 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  0 = 1 the soliton is recovered; in the limit  0 → 0 the propagation is linear, that is, purely dispersive.In Section 3.2 we will consider pulses launched with 0 <  < 1/2, motivated by the well-known fact obtained from Inverse Scattering theory [2] that no soliton is formed if  < 1/2.Subsequently, in Section 4.2 we will consider pulses launched with 1/2 <  < 3/2, 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  thr = 1/2 for the first soliton to appear,  thr = 3/2 for the second, and so on.The minimum required gain Γ =  out / in for that is Γ = ( thr /) 2 .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   =   (0) exp() where we introduce a dimensionless gain coefficient  > 0. A characteristic dimensionless gain length is   =  −1 .The gain can be represented in (1) by replacing the RHS with (/2).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.

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 ( = 0),       (, 0)     = International Journal of Optics 3 This is to be compared to an initially sech-shaped pulse which undergoes some shape variations.Pulse wings exhibit some wiggles around  0 = /2 (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  = 0.42 [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.

Evolution of an 𝑁 < 1/2 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  2  because the relevant length scales  NL and   both scale as  −2 .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  = 1 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.Energy conservation ensures that and the peak amplitude evolves as      (, 0)     =     .
Combining ( 10), (11), and (12) we have In Figure 2, ( 13) and ( 14) are plotted for  = 1 and for  = 0.05, . . ., 0.5.In all cases   falls with distance (reflecting the broadening), while   grows.One might suspect that with growing   a soliton might be created once threshold is reached at   = 1/2.However, that is evidently not the case because the number of solitons, here zero, is guaranteed to be preserved.The discussion is only complete with consideration of the chirp, and it is the development of a chirp that counteracts soliton creation.
We concentrate on the central part of the pulse (between the 80% amplitude points) where the phase is very nearly parabolic.From numerical propagation simulation data we obtain   by least-squares fit.We let the chirp coefficient (0, , ) = 0 at launch for all , .As the pulse propagates,  rises linearly, peaks, and then rolls down as () ∝ 1/ when for  → ∞ the pulse duration diverges.By systematic variation of both  and  we determined an expression capturing dependencies on these parameters.The expected inversion symmetry () = −(−) is also built in.This results in with fit parameters  = 1.07 and  = 1.40.Figure 3 shows plots of (15) (solid curves), in comparison to numerical data (dotted).It is clear that ( 15) is a good approximation over the range of interest, that is, 0 ≤  ≤ 0.5 and 0.5 ≤  ≤ 1.5.
One observation to be made is that, during its dispersive propagation, a weak sech pulse with  = 1 will never exceed  ≈ 0.4 which is quite moderate.The other is that chirp values are highest for  → 0: nonlinearity not only slows broadening as asserted in [11], but also reduces chirp.

Soliton Content of Chirped sech-Shaped Pulses.
We are now positioned to determine the soliton content of the pulse under consideration.It is well known [2,3,15] that an unchirped sech pulse with 1/2 <  < 3/2 has pulse energy   = 2 2 and contains exactly one soliton with energy  sol = 2(2 − 1); the balance is radiation with energy  rad = 2( − 1) 2 .We employed the Direct Scattering Transform (DST) technique [22][23][24] to obtain solitonic eigenvalues from pulses as in (8).Results for various chirp values are shown in Figure 4 as data points.We are not concerned here with the real part of the eigenvalue as all velocities are zero; the imaginary part  represents the energy of the pertaining solitons,  sol = 4.This figure is similar to Figure 1 in [18] (with  = /2) and to Figure 1 in [25] (with  0 = /2).It shows that the power contained in a chirped pulse can only partially be converted into a soliton: the more the chirp, the lower the soliton energy.For any  there is a "critical" chirp  crit beyond which soliton formation is not possible.
At  = 0 all traces start at the well-known value and then roll down.Different from [25] we observe that this occurs first quadratically, but beyond the diagonal auxiliary line, linearly.In the quadratic regime, an ansatz fits well, and the same is true for in the linear regime; the crossover occurs at Using these fits, the critical chirp value (at  = 0) is This result coincides with (20) in [25] (their numerical prefactor was 3.28 which agrees within numerical precision); we remark that our two-part function ( 17) and ( 18) fits the data, including those in [25], significantly better than the fit given there.We convinced ourselves that our fit functions hold well for all combinations of 0.5 ≤  ≤ 1.5 and 0.1 ≤  ≤ 10.As an example we specify for the fundamental soliton with standard values  = 1,  = 1 (the reference soliton): it disappears at  crit = 1.67, and at the crossover point its chirp amounts to   =  crit /2.

Amplified Propagation
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 [12]).

4.1.
Starting from Pure Radiation.For weak pulses, that is, pulses with powers below the threshold of soliton generation or just at the threshold at most (0 <  < 1/2), we may approximate that gain acts only on the amplitude scaling factor   , without immediate impact on pulse width, chirp, or other shape characteristics.We start by discussing limiting cases.
High Gain ( → ∞).If amplification is applied more or less stepwise before dispersive spreading has set on, the problem is a straightforward application of [3]: all that is required to generate a soliton is to boost   at least to the threshold value of   =  = 1/2, that is, Γ = 1/(2).Figure 5: Impact of gain: evolution of amplitude scaling parameter   (orange), scaling parameter   (blue), chirp   (green), and soliton content   (red) as calculated here (solid/dash-dotted lines) in comparison to numerical simulation (dots).(a) For the integrable NLSE system with  = 0 and (b) with gain ( = 0.1), soliton appears at   ≈ 5. Initial parameters:  = 0.4,  = 1, and  = 0.
for a second soliton would be at   = 3/2 and in the same example would be reached at   = 2.644  .)Incidentally, a pure soliton is obtained when the gain lifts   to 1.If amplification occurs only after some spreading and chirp has developed, the quoted distance is a lower bound.
Adiabatic Gain ( → 0).The signature of the adiabatic case is that the gain length is very much longer than the characteristic dispersion length, so that   /  → 0. In this limit, no soliton is generated; all of the energy remains in the radiation.We conclude that a soliton count of zero is maintained during adiabatic gain: no soliton will be generated, ever.
Finite Gain.We introduce a gain term into (13), with gain parameter  > 0; it is also possible to let  < 0 in which case  is Beer's coefficient of absorption.The chirp coefficient   and scaling parameter   at position  can be calculated using the initial values of  and  from (15) and The eigenvalue component representing the solitonic energy   can be obtained from the calculated local values given by ( 17) and (18).
In Figure 5(a) the evolution as predicted by this approximation is shown for the integrable NLSE system with  = 0 and at  = 0.4.Parameters   (solid orange line),   (solid blue line),   (solid green line), and   (dash-dotted red line) are shown.In comparison, corresponding simulation results are shown as dots with matching color.Note that formally the "soliton content"   comes out negative when there is no soliton present, while numerics always correctly yields a zero in such case.Theoretical predictions and simulations agree well.
Figure 5(b) shows the same situation as in Figure 5(a) but with a gain of  = 0.1.The soliton content   is shown as solid red curve for meaningful positive values predicting the existence of a soliton.After a certain distance   ≈ 5 the soliton content becomes positive, indicative of the birth of a soliton.This demonstrates that with our formalism the location of the birth of a soliton can be predicted.The slight difference between the predicted positive soliton content and numerical data for  well above the threshold arises from the increasing deviation of the numerical pulse from a pure hyperbolic secant shape.
The position of the soliton's birth,   , is obtained by letting   = 0 in (24) and inserting (15), (22), and (23); we do not spell out the resulting rather unwieldy expression here.  depends on  and : large values of either render it short.Predictions of   (, ) are shown in Figure 6 as solid curves; numerical results from DST are also shown (dots).It is obvious that the general trends are represented well, but there is a tendency that our equations find   systematically high.This is particularly true for long distances, surely because then pulse shape differences can play out more.Nevertheless, the agreement with numerics is still good enough for predictions to be useful.

Starting from One Soliton Plus Radiation.
After discussing how gain can bring a weak ( < 1/2, "subsolitonic") pulse up to the solitonic threshold, we now consider initial conditions 1/2 <  < 3/2 so that we start from a pulse which contains both one soliton and some nonsolitonic radiation.The energy  rad of the latter is given by the difference of the pulse energy   and the soliton energy  sol [3,12]: During amplified propagation, A new (second) soliton will always be created from the radiative part once its energy reaches a threshold, as discussed in the previous section.Note that the radiation has two contributions: (i) from the initial condition as in (25) [3,12], except when  = 1; (ii) from continuous generation during amplified propagation, except in the adiabatic limit [12].This is a consequence of the continuous perturbation of the soliton's characteristic balance between nonlinearity and dispersion.
The main difference to the case of loss as discussed in [12] is that during its propagation the situation becomes more and more diabatic in the lossy fiber, but approaches adiabaticity in the amplifying fiber.This is because in the case of gain the scaling parameter   grows so that the ratio   /  decreases.As shown in [12], the rate of generation of radiative energy is proportional to that ratio.As a result, the radiative fraction of the pulse energy saturates within a few dispersion lengths.For the  = 1 soliton we find that it approaches The situation is compounded by the fact that both contributions to radiation and also the soliton are subject to gain all the time.We will again discuss limiting cases first.
High Gain ( >  2 ).Then   <  D ; in the limit  → ∞ we approach the case of stepwise gain as in SY.No significant change of   and   can occur over   ; only   is raised.Then, the second soliton arises once   = 3/2.Using ( 21) we can predict for the above example of an initially unchirped  = 0.6 sech pulse that for  → ∞ the second soliton appears at   = 1.833   .
Adiabatic Gain ( → 0).In the limit, the soliton does not shed energy to radiation.Thus, both soliton and radiation energy grow exponentially, without redistribution among them.Also, their temporal overlap is minimal because   ≪   , and radiation is rapidly broadened.In consequence, even after a vast propagation distance no further solitons will arise (i.e.,   → ∞).
Small Gain ( <  2 ).The equilibrium between nonlinearity and dispersion is approximately maintained because the soliton can rearrange its width and peak power at a rate that nearly keeps up with the change.Therefore, if there is any sizable amount of radiative energy from the initial condition, it will dominate.
In [12] we derived an expression for the amplitude evolution of a sech pulse in the presence of initial radiation.
Here an extended version in dimensionless form is given including the impact of distributed gain depending on .
The first term on the RHS is the exponentially growing amplitude, under the assumption that the sech shape is basically preserved, and taking the pulse narrowing into account.
Inside the square bracket, the first parenthesis pertains to the initial soliton.The prefactor of the cosine term models the dispersive broadening.The argument of the cosine contains one term for the nonlinear phase rotation and one for the Guoy shift [12]; both together define the beating between soliton and radiation.Incidentally, we find empirically that the nonlinear phase, which follows the peak amplitude, is modelled slightly better with a correction  = 0.03 as indicated, apparently because the sech shape is not strictly preserved.

International Journal of Optics
We put (28) to a numerical test.We considered the propagation of an initially 1/2 <  < 3/2 sech pulse, first without gain by setting  = 0.This is shown in Figure 7(a) for values of  = 0.7, . . ., 1.3.Again solid curves (dots) correspond to theoretical prediction (numerical data), respectively.Pulse parameters oscillation due to beating of the solitonic and dispersive part of the pulse.Incidentally, this figure closely resembles Figure 2 in SY [3], but they only had numerical data.Here the agreement between numerical data and ( 28) is very good.The oscillation frequency is constant because in the absence of gain the nonlinear phase rotates at a constant rate.
In contrast, Figure 7(b) shows similar data but with weak gain ( = 0.05).As the amplitude builds up (on average), the oscillation frequency also increases.Again numerics and theory agree well.
From the amplitude the pulse parameters   and   can be calculated using   and   agree well with numerics.The chirp parameter  is here only obtained from numerics.It is obvious that the chirp oscillates around zero with moderate values resulting in only a slight change of the soliton content.Thus for the calculation of the soliton content the chirp was neglected resulting in the simple expression The soliton eigenvalue is shown in red in Figure 8: predictions of (30) (solid) and numerical results (dotted) are compared and show good agreement.The gain causes the scaling factor   to rise, but not the amplitude scaling factor   which undulates around unity and represents a fundamental soliton.

Generation of a Second
Soliton.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 9 an unchirped sech pulse ( = 1,  = 1.3) was launched into a fiber with  = 0.1 and propagated numerically up to  = 10.0.Undulations indicate beating between constituents, and there may be some amount of energy swapping.That notwithstanding so much is clear: both soliton 1 and radiation grow from their initial value up to the onset of soliton 2 at   = 8 (arrow); beyond that point the radiative energy drops.The second soliton is clearly created from radiation energy, without depletion of the first soliton.
As an example of an actual pulse shape we show the field envelope at  = 9.32 (Figure 9(b)).The pulse shape consists of a narrow, tall peak atop a broad, low-amplitude pedestal.One can fit a sech pulse with   = 0.965 and   = 4.43 with flat phase to the narrow spike, and   = 1.22 and   = 0.185 with parabolic phase ( = 0.076) to the pedestal.Using ( 18) and (30) this suggests eigenvalues of  1 = 2.06 (narrow peak) and  2 = 0.014 (pedestal).DST analysis yields  1 = 2.024 and  2 = 0.029, respectively.In view of approximations made, the agreement is reassuring.One is led to conclude that the high peak almost exclusively represents the higher-energy soliton, while the second soliton and radiation together form the pedestal.Such identification is possible and meaningful because there is a large difference in solitonic energies (almost two orders of magnitude); the resulting different widths lead to minimal temporal overlap.
In Figure 10 we assess the whole range of  values numerically for four values of .Arrows on the right mark the predicted emergence length obtained from (21) for high values of .Lines (III) and (IV) appear in the realm where the initial condition contains one soliton plus some radiation, that is, for 1/2 <  < 3/2, with the exception of  = 1, where the initial condition does not contain radiation.Note that (IV) is a shifted version of (I), and (III) is a mirrored version thereof.These curves refer to the situation where a second soliton is formed from the radiative part.This assumes relatively independent evolution of soliton and radiation, justified for small gain ( ≪ 1) because dispersion broadens the radiative part rapidly so that the mutual overlap is minimal.Therefore, data are closer to (III) and (IV) when the gain is low, while for high gain, data tend towards (II) as expected.There is an interesting transition between both limiting cases, where near  = 1 where the initial condition has no radiation; the radiation generated during propagation becomes dominant.

Conclusion
It is well known from SY [3] that if one considers an unchirped sech pulse of certain width and scales up its energy, above some threshold it contains a soliton, and above certain higher thresholds more solitons appear.The thresholds are given by the amount of radiative energy.
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.
For example, starting from an unchirped sech pulse with  = 0.4, a first soliton arises at   = 0.4463  .

Figure 8 :Figure 9 :
Figure 8: Parameter evolution and soliton content (red) of an initial unchirped  = 1.2 sech pulse in the case without gain (a) and with gain (b).Again solid curves/dots correspond to the model/numerics, respectively.

Figure 11 :
Figure 11: Propagation distance until a new soliton is generated.Distance is normalized to   ; therefore, values also represent the accumulated gain up to that point.Parameter:  as indicated.For reference, auxiliary lines (I-IV) indicate certain limiting cases; see text.