# Case Study: Numerical Experiments with Soliton Pulses in Fibers

Key questions:

- how chromatic dispersion can suppress the nonlinear spectral broadening
- the effects of deviations of the input pulses from ideal soliton parameters
- the influence of higher-order dispersion, self-steepening and the delayed nonlinear response
- the limited capability of solitons to adjust their parameters to gradually changing conditions
- the question how soliton energies can be optimized
- some details of higher-order solitons

In optical fibers with anomalous chromatic dispersion, the interesting phenomenon of the formation of soliton pulses can occur. For fundamental solitons, the interplay of chromatic dispersion and the fiber nonlinearity works such that both the temporal and spectral pulse shape remains preserved during propagation, although the mentioned effects separately would lead to substantial changes:

- Chromatic dispersion alone would lead to increasing temporal broadening.
- The Kerr effect alone would lead to self-phase modulation (SPM) and thus to spectral broadening.

In a somewhat simplified picture, the two effects can exactly compensate each other, provided that we have the right shape and energy of the pulses. (More precisely, the total resulting change of spectral phase is not zero, but frequency-independent.) For a fiber with wavelength-independent group velocity dispersion (i.e., second-order dispersion only), that condition is fulfilled if we have sech^{2}-shaped pulses with the soliton pulse energy, which can be calculated as

with the GVD (group velocity dispersion) <$\beta_2$>, the SPM coefficient <$\gamma$>, and the full-width at half-maximum (FWHM) pulse duration <$\tau_{\rm p}$>.

So far, that is all well known, but if we look at soliton phenomena more closely, a lot of additional questions arise, some of which we address in this case study. That way, we can develop a much deeper understanding of solitons. A great tool for such studies is a physical model for numerical simulations of pulse propagation. Compared to analytical calculations, these have the crucial advantage of not requiring many simplifying assumptions. So we are free to study many more details.

## Why is there no spectral broadening?

Common wisdom is that self-phase modulation causes spectral broadening. An obvious question is therefore how dispersion can suppress the nonlinear broadening – shouldn't that be impossible, given that chromatic dispersion is known not to affect the spectral shape, but only the spectral phase?

To get this apparent issue resolved, we need to question the first statement: does self-phase modulation necessarily cause spectral broadening? Maybe only under certain conditions? Indeed, it turns out that the chirp of a pulse, i.e., its variation of instantaneous frequency over time, is the crucial factor influencing the spectral effects of self-phase modulation. It is instructive to try that out in numerical experiments. We initially just assume zero dispersion, just the Kerr nonlinearity, and investigate the evolution of spectral shape and bandwidth within 200 mm of fiber for sech^{2}-shaped 100-fs input pulses with different values of the chirp:

First without initial chirp, where we see the usual spectral broadening:

With an initial up-chirp of 50 THz/ps, we get enhanced spectral broadening:

With an initial down-chirp of −50 THz/ps, we get spectral narrowing:

If we go through a longer fiber, we eventually get spectral broadening again – but there we also have an up-chirp again. Note that self-phase modulation itself affects the chirp, of course.

So we can now understand how chromatic dispersion can suppress the spectral broadening:

- As long as the pulse is unchirped, self-phase modulation does
*not*change its bandwidth. - One can understand that by considering the differential equation for the pulse evolution: the addition to the complex spectral amplitudes is then always 90° out of phase with the existing amplitudes. This changes only the spectral phases, but not the magnitude of the amplitudes.
- However, SPM would normally create an up-chirp, which subsequently allows it to broaden the spectrum.
- Additional anomalous dispersion can prevent the build-up of the chirp and in that way suppress the spectral broadening.
- Of course, that works only if we have the correct pulse shape, which is the sech
^{2}shape under the given simple conditions (no higher-order dispersion, etc.).

## What if the input pulse does not fully satisfy the soliton conditions?

In practice, we can never completely match the soliton conditions concerning pulse shape and pulse energy. For example, if we generate the pulses with a mode-locked laser, they may often significantly deviate from the sech^{2} shape, or the pulse energy launched into the fiber may drift away from the ideal value. What effects will that have?

For investigating this, we use a comprehensive model of a germanosilicate fiber, realized with the software RP Fiber Power. That software offers the Power Form “Passive Fiber for Ultrashort Pulses”, where we can define the details determining the propagating mode in different ways. We begin by defining a step-index refractive index profile of a germanosilicate fiber with parameters close to those of an ordinary single-mode telecom fiber. By directly defining the refractive index profile without a wavelength dependence, we do not get the chromatic dispersion; that we set manually to be −20'000 fs^{2}/m, which is realistic for an assumed wavelength of 1550 nm (in the telecom C-band). (In other cases, we define the profile indirectly via the GeO_{2} doping profile, then getting the full dispersion calculated from that, but in the current case we want to switch higher-order dispersion on or off, for example.) Here are the form settings:

Concerning the fiber nonlinearity, we now only consider the Kerr effect as an instantaneous effect, i.e., without a delayed nonlinear response. Also, we neglect self-steepening. With that, for now we have a quite simplified situation as it is considered in many textbook example cases. We can later check what happens if we subsequently introduce more details.

For the input pulse, we assume a pulse duration of 100 fsm and we initially assume the sech^{2} shape. We let the program automatically calculate the fundamental soliton pulse energy (here: 438 pJ) by entering the following script code:

```
gamma := fiber1.n2_f * (2pi / l_s_c) / A_eff(fiber1.signal)
{ nonlinear coefficient }
E_s := 2 * abs(fiber1.GVD(l_s_c)) / (gamma * (tau_s_in / 1.7627))
{ fundamental soliton pulse energy }
calc cf_set_input("E_s_in", E_s)
```

The code accesses some variables which are related to input fields – for example, `fiber1.n2_f`

is the nonlinear index of the first (and only) fiber, and `tau_s_in`

is the entered initial pulse duration. Finally, it uses the `cf_set_input()`

function to modify the content of an input field.

With that, we obtain the expected result that both the temporal and spectral profile of the pulse remain constant during propagation. We will later see that this actually applies only under our initial simplifying assumptions, but it is nevertheless instructive to first investigate that simplified model in some detail.

Now we can slightly change the input pulse energy, e.g. to 1.1 times the soliton energy. We first look at that case in the time domain for 10 m of fiber, applying a logarithmic color scale:

We see that the pulse now “sheds off” some energy. That leads to a temporal background, which increasingly drifts away from the pulse due to chromatic dispersion. We can better see that in a plot, again with a logarithmic scale:

In the spectral domain, we do not see that much:

Of course, we do not get things drifting away in frequency space.

After some propagation length, we essentially get a real fundamental soliton – but with what parameters? The duration is reduced to 92 fs, so the corresponding energy of the soliton alone (without the background) is (100 fs / 1.1) / 92 fs = 0.988 relative to the full input energy. We see that nearly all of the injected energy ends up in the soliton, and only a small part goes into the temporal background (called the continuum in soliton theory). For a larger energy mismatch, a larger part of the energy can get into that background.

We have also tried with a 10% too low energy; the results (not shown here) look quite similar.

Next, we try what happens if we choose the correct soliton input energy, but a different pulse shape – namely, a Gaussian one. Interestingly, the results look quite different:

Again, nearly all energy gets into the final sech^{2}-shaped soliton pulse. Looking at the diagrams, one may feel that more energy is lost, but note that we have a logarithmic scale spanning a 40-dB range.

One more test is done with an asymmetric pulse shape. We construct this from two Gaussian curves with substantially different widths, one for negative times and one for positive times.

Another possibility is that the input pulses have the right intensity profile and energy, but have some chirp, e.g. resulting from propagation through a dispersion medium. We try that with a chirp of 10 THz/ps, which moderately increases the time–bandwidth product from 0.315 to 0.385:

A general lesson from these tests is that if the input pulses are not very close to the required sech^{2} shape (with flat spectral phase), this will not matter too much; most of the energy will nevertheless propagate as a soliton.

By the way, you see a kind of periodic feature e.g. in the previous diagram and may think that here we see the *soliton period*, but that is not the case. We will later come back to the soliton period.

## What if the fiber has higher-order dispersion?

So far, we have always neglected higher-order dispersion, but now we add a realistic amount of dispersion slope, changing the GVD by −8000 fs^{2}/m per 100 nm wavelength increase. That has a significant, although not dramatic effect on our 100-fs solitons:

The soliton shape changes only slightly due to that substantial TOD. The pulse gets slightly slower – a detail which is somehow related to the spectral asymmetry introduced by the TOD. The impact of TOD would become stronger for still shorter pulses.

## What is the effect of self-steepening?

The self-steepening term is a kind of operator which occurs in the nonlinear term of the propagation equations:

$$\frac{{\partial A}}{{\partial z}} = (\text{linear terms}) + i\gamma \: \left(1 + \frac{i}{\omega_0} \frac{\partial}{\partial t}\right) \: \left(A(z,t)\int\limits_0^\infty {R(\tau )\;{{\left| {A(z,t - \tau )} \right|}^2}{\rm{d}}\tau } \right) $$It is the part with <$\left(1 + \frac{i}{\omega_0} \frac{\partial}{\partial t}\right)$>, occurring before a term involving the response function <$R(\tau)$>. We can interpret it as introducing a frequency dependence of the nonlinear effect; note that <$i \: \partial / \partial t$> corresponds to a factor <$\omega$> in the frequency domain. This term is often neglected because (a) it would often prevent one from finding analytical solutions and (b) it wouldn't change that much as long as the pulse spectrum is not very broad. With numerical simulations we can easily keep that term; it only causes a moderate penalty in terms of computation time.

We try this out with otherwise same settings as before, without higher-order dispersion.

We see essentially two things here. At a very low level on that logarithmic scale, we see some wiggles arising, as again the soliton pulse shape does no more precisely fit the sech^{2} shape, which is often course calculated without considering self-steepening. Second, the pulse now slowly drifts to the right, which indicates a slight reduction in group velocity.

## What is the effect of the delayed nonlinear response on solitons?

So far, we have completely neglected the fact that all optical materials exhibit a nonlinear polarization which does not instantaneously react to optical intensities, but rather with some slight temporal delay. More precisely, we have two different components to the nonlinear response:

- a nearly instantaneous one from the electrons
- a substantially slower one, which also involves microscopic vibrations of the medium

An ultrashort light pulse can excite such ionic vibrations, which then act back on the light for some time. That mechanism is behind stimulated Raman scattering, and it can be described with a response function in the time domain, from which the Raman gain spectrum can be calculated.

The inclusion of the resulting delayed nonlinear response is technically not so easy, but there are efficient numerical methods to do it. Also, there are accurate data available for common optical materials, in particular for fused silica [1], the main components of many glass fibers. We can use all that in our Power Form.

Now we activate the delayed nonlinear response, but no higher-order dispersion, and also let it consider self-steepening. The input pulses are 100-fs sech^{2} pulses at the fundamental soliton energy. We indeed see substantial changes:

What happens here is that the pulse spectrum drifts towards longer wavelengths; that phenomenon is called the soliton self-frequency shift. Further, the increasing wavelength causes a growing reduction in group velocity due to the anomalous dispersion. We investigate this in more detail in a more specialized case study on the soliton self-frequency shift.

You may now wonder whether the textbook examples ignoring the delayed nonlinear response are all unrealistic. That depends, of course, on the circumstances. It turns out that the rate of that wavelength drift depends strongly on the soliton pulse duration. So if we have somewhat longer solitons, say with 500 fs duration, not much drift occurs in a couple of meters of fiber. But for shorter pulses, such effects can soon become rather strong. One should feel somewhat uneasy about using analytical equations without being sure about the limits of their validity, but when working with numerical simulations, one can always test whether switching on another effect would change the results significantly.

Another interesting question is whether we may design a fiber to get more or less of the soliton self-frequency shift:

- What if we double the mode area, while keeping the GVD unchanged (which might be difficult in practice)? This will double the fundamental soliton energy, and result in no change of the nonlinear frequency shift. After all, the given dispersion needs to be balanced by sufficiently strong self-phase modulation, so we cannot simply reduce the strength of nonlinear effects that way.
- For increased or reduced nonlinear effects, we need to modify the magnitude of GVD accordingly. That is a way to go, with some potential for significant changes.
- By changing the material composition of the fiber core, one may change the Raman scattering properties. For strong changes, one would probably need to use non-silica fibers.

## How quickly can solitons adjust to modified conditions?

We have already seen that under many circumstances fundamental solitons do not simply fall apart if their conditions are not well fulfilled; rather, they adjust to the conditions, and only some of the energy may be lost, producing a temporally long background radiation.

We can also consider cases where conditions change continuously during propagation. A simple case is to have a constant amount of propagation losses in the fiber. That leads to an exponential decay of pulse energy and should thus make the solitons gradually longer. But does that also work if we attenuate the pulse energy rapidly?

That can be easily tested with our model. In order to realistically work without the delayed nonlinear response and self-steepening, we use 500-fs pulses. We start with a 1 km long fiber, having a propagation loss of 5 dB/km (which would be rather high for a telecom fiber). If the pulse duration adjusts well, we expect it to arrive at 500 fs · 10^{0}.5 = 1.58 ps.

That works very well: the pulses nearly preserve their sech^{2} shape, just get longer as their energy decays. The final pulse duration is 1.58 ps, exactly as expected.

Now let us try 50 dB/km in a reduced fiber length of 100 m:

Here, we see some deviations, but still at a rather low level (note the logarithmic scale). Let us go further with 500 dB/km over 10 m. The final pulse duration is now only 762 fs, and the time–bandwidth product has increased to 0.463. (For perfect solitons, it is 0.315.) So we see that the solitons cannot adjust to the new conditions that fast.

With substantially shorter pulses, that works much better. For example, with 100-fs solitons (just ignoring that in reality they would show the soliton self-frequency shift), it still works quite well in the highest loss case.

It turns out that one can estimate the required length scale for a close to “adiabatic” adaptation to new conditions using the so-called *soliton period*, which one can calculate with an equation given in the next section. We obtain 6.3 m in our case for 500-fs pulses, but only 0.253 m for 100-fs pulses. (There is a quadratic dependence on the pulse duration.) If we take care not to change conditions substantially within one soliton period, we get close to adiabatic soliton evolution. Long (multi-picosecond) solitons are relatively delicate in that respect, while short solitons can adapt much faster. This is essentially because they are formed by a balance of two strong counter-acting effects, while these effects are rather weak for long solitons.

One can also do the opposite: gradually amplify the solitons in an active fiber and let them get shorter at the same time. We have investigated that in a separate case study.

Another possibility is to have a fiber with continuously changing properties such as the effective mode area and/or the GVD, realized e.g. with tapered fibers.

## What if solitons collide?

Rather interesting phenomena occur when solitons collide with each other. That can happen, for example, if they have different center wavelengths and thus different group velocities. We have also investigated that in a separate case study.

## Couldn't we have much more energetic solitons?

Solitons would appear to be a quite ideal way of transporting light pulses over substantial distances in flexible fiber cables without obtaining any temporal or spectral distortions or broadening. However, soliton pulse energies are often inconveniently low. Even for our example with rather short pulses (100 fs), we have only 438 pJ, while many mode-locked lasers produce many nanojoules, in some cases even multiple microjoules. And for picosecond solitons, we get even lower soliton energies.

The soliton energy equation quite clearly shows our options:

- Stronger chromatic dispersion would help. Unfortunately, it is probably hard to increase the dispersion e.g. by a factor of 5, let alone by several orders of magnitude. Well, one could work on dispersion engineering to explore the limits; see also a case study on that.
- A lower nonlinearity would also help. Only, the nonlinear index of fused silica is already pretty low. There seems to be only one way to reduce it substantially: use air! That can indeed be done even in the context of fibers: there are hollow-core fibers, using special guiding mechanisms, where most of the light field propagates in air, and some fine fused silica structure provides guidance. It is not easy to do, particularly with additional constraints on dispersion, but it is an interesting development.
- Shorter pulses are another possibility. Note, however, that issues with the soliton self-frequency shift or with higher-order dispersion may then get severe.

So far, soliton pulse delivery from mode-locked lasers has not become common, essentially because of the severe limitation in pulse energy.

## Higher-order Solitons

Preserving the temporal and spectral pulse shape over arbitrary propagation distances is possible only with fundamental solitons as discussed so far. However, there are also *higher-order solitons*. We simply get the soliton of order <$k$> by increasing the input pulse energy by the factor <$k^2$>. That leads to a periodic evolution of temporal and spectral shape, with the period being the so-called *soliton period*, calculated with the following equation:

Let us try that out with 500-fs pulses, where the soliton period is 6.32 m, first with the order <$k$> = 2:

We indeed find the expected soliton period of 6.32 m confirmed.

It is nice to see this as an animated spectrogram:

It gets rather fancy if we inspect solitons of 4^{th} order:

But to get realistic, we should turn the delayed nonlinear response and self-steepening on. That changes it completely:

Apparently, the higher-order soliton collapses under the influence of the Raman response. A fundamental soliton is emitted, which then exhibits the soliton self-frequency shift. We cannot have the 4^{th} order soliton stay together and drift as a whole. The second-order soliton seems to be doing better initially, but also soon falls apart:

Note that the Raman response has a strong impact even for a pulse duration where the fundamental soliton is not so strongly affected. Basically, this is because higher-order solitons have broader spectral features and higher peak powers during their evolution, and are thus more sensitive to Raman scattering. Generally, they do not share the great robustness of fundamental solitons.

## Conclusions

The RP Fiber Power software is an invaluable tool for such work – very powerful and at the same time pretty easy to use!

You can learn various things from this study:

- Clean soliton propagation requires input pulses with the correct shape and energy. However, we can still get much of the injected pulse energy propagating as solitons if the energy and/or the pulse shape is not quite correct. That is important in practice.
- Higher-order dispersion and self-steepening modify the behavior. They become more relevant for shorter pulses.
- The delayed nonlinear response, causing stimulated Raman scattering, causes the soliton self-frequency shift. How strong that effect is, depends critically on the pulse duration.
- Fundamental solitons can well adjust their parameters to continuously modified conditions, provided that the changes are not too rapid. The soliton period gives us a good estimate on how slow changes need to be.
- Soliton energies are often lower than desirable. There are limited possibilities to substantially increase them.
- Higher-order solitons are an intriguing phenomenon, but in practice they are easily destabilized by Raman scattering.

## More to Learn

Encyclopedia articles:

### Bibliography

[1] | D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function”, J. Opt. Soc. Am. B 19 (12), 2886 (2002); https://doi.org/10.1364/JOSAB.19.002886 |

[2] | G. P. Agrawal, Nonlinear Fiber Optics, 4th edn., Academic Press, New York (2007) |

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