# Case Study: Soliton Self-frequency Shift

Key questions:

- How does the soliton self-frequency shift work?
- What parameters determine how strong it is?
- How does it work closer to the zero dispersion wavelength?

We consider an interesting phenomenon with soliton pulses in optical fibers (with anomalous chromatic dispersion), which is particularly relevant for short pulse durations.

In a simple picture, considering only the Kerr nonlinearity of a fiber as an instantaneous nonlinear effect, both the temporal and spectral envelope of a fundamental soliton is preserved during propagation. However, the fiber also has a delayed nonlinear response, related to microscopic vibrations, which gives rise to stimulated Raman scattering, and that has an effect on the solitons: there is Raman gain for its lower-frequency components, so that those get amplified at the expense of the higher-frequency components. In effect, the whole optical spectrum of the solitons drifts towards longer wavelengths. In a sufficiently long fiber, that drift can be substantial: easily several nanometers or even more than 100 nm.

Textbooks often investigate soliton effects based on simplified models – for example, with no higher-order chromatic dispersion and with a simplified form of the Raman response. Here, however, we use numerical simulations where we not require such simplifications. That allows us to investigate various additional details, which may be relevant in practice.

## Setting up the Simulation Model

We use the software RP Fiber Power, which offers the Power Form “Passive Fiber for Ultrashort Pulses”. We assume that we have a step-index germanosilicate fiber, where we get the refractive index profile calculated from a given germania doping profile. From that, we get the group velocity dispersion of the LP_{01} mode calculated by the mode solver. Here are the form settings:

For the input pulses, we initially set a duration of 500 fs and a sech^{2} shape. Normally, one would need to calculate the correct pulse energy for a fundamental soliton and enter that in the form – which would be quite inconvenient. Note that the form was not specifically made for soliton simulations, but we can enter some script code at various places in the form to obtain additional functionality which is very convenient for our specific case. The following code does exactly what we need:

```
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. For the used pulse energy equation, see the encyclopedia article on solitons.

For the nonlinear response, in the form we set the nonlinear index, activate the self-steepening effect (essentially, considering the frequency dependence of the nonlinear effect), activate the delayed nonlinear response, and for its shape select “full data for fused silica”. These data, which are far more detailed and realistic than frequently used models, have been taken from a paper [1]. (We could also define any other Raman response function.) Here is how that looks in the form:

## Case 1: Pulses at 1550 nm

With that, we are already set up for simulating the pulse propagation. We now consider 500-fs pulses with an initial center wavelength of 1550 nm. This is in the telecom C band, where the chromatic dispersion of our step-index fiber is in the strongly anomalous region with −20'456 fs^{2}/m.

Let us inspect some diagrams showing what happens. First, we get a perhaps surprising temporal evolution of the pulse in the fiber:

The pulse more and more drifts away; this is because as its center wavelength increases, its group velocity decreases – after all, we have anomalous chromatic dispersion. Eventually, the pulse reaches the end of the simulated time trace and then comes back from the other end of it; that is a numerical artifact which we could avoid by using a longer numerical time trace. Well, although the drift is understandable, we want to suppress it in the following simulations, which we can simply do in the form: We can enter a setting which re-centers the time-domain pulse shape after every 10 m of fiber, for example. (We could do it more often; it just then takes a little more computation time.)

Now we inspect the spectral evolution:

As expected, the pulse spectrum continuously drifts towards longer wavelengths. Within a kilometer of fiber, we get a shift by 5.6 nm or −0.7 THz.

What if we use half the initial pulse duration, i.e., 250 fs? That has various consequences:

- The soliton pulse energy will be doubled to 179 pJ, so the peak power will be even quadrupled. That means substantially stronger nonlinear effects.
- The pulse bandwidth is also doubled.
- The Raman gain grows with the frequency difference – at least up to the optimum frequency around 13 THz, and our initial pulse bandwidth is still only 1.26 THz. So that is another reason why the frequency shift should get larger.

An equation from a textbook [2] predicts that the frequency shift scales with the inverse fourth power of the pulse duration, but our simulation shows that it is actually less rapidly growing. For 250 fs, our simulation shows a shift of 58.2 nm or −7 THz. So the frequency shift got 10 times larger, while we may have expected a factor 2^{4} = 16. Why that?

Let us look at the new spectral evolution:

We see that the spectral shift slows down substantially. Also, the spectral bandwidth is significantly reduced – from the initial 1.26 THz to 0.91 THz. We can explain this as follows:

- Our simulation considers the full chromatic dispersion profile, i.e., not a constant GVD as in simple textbook examples. The magnitude of GVD (in units of ps / (nm km)) rises with increasing wavelength.
- What does that mean for solitons? For the given pulse energy, the fundamental soliton must get longer as the GVD increases, which also implies a reduced bandwidth.
- That also reduces the peak power, and that together with the reduced bandwidth decelerates the further frequency shift.

So far, we have neglected any propagation losses in the fiber, which is not fully realistic. For a good telecom fiber, that loss will be of the order of only 0.2 dB in 1 km. (Of course, it would be easy to include any wavelength dependence of that loss in the simulation.) It turns out that this would not have a severe effect, just slightly decreasing the frequency shift. But we leave that turned on for the following.

Let us also try what happens for an initial pulse duration of only 100 fs:

This is getting quite extreme. We had to use a doubled temporal resolution for covering such a large wavelength range. Note that the software automatically takes finer spatial step in the fiber, as it recognizes that to be necessary, and thus gets noticably slower. At these long final wavelength, the propagation losses of the fiber actually rise substantially; taking that into account, the wavelength drift would slow down further.

We could try this for more pulse durations. The most natural way to display the results is a diagram where we plot the frequency shift versus the initial pulse duration. This rather special type of diagram is not directly offered by our Power Form, but we can create it by inserting a couple of lines of script code:

```
diagram 25:
x: 100, 1000, log
"pulse duration (fs)", @x
y: 0.03, 30, log
"abs. value of frequency shift within 1 km of fiber (THz)", @y
frame
hx
hy
! for r := lg(CS_x2) to lg(CS_x1) step -(lg(CS_x2) - lg(CS_x1)) / 10 do
begin
tau0 := expd(r) * fs; { do equal steps on the logarithmic axis }
E0 := 2 * abs(fiber1.GVD(l_s_c)) / (gamma * (tau0 / 1.7627));
CreateInputPulse(E0, tau0, l_s_c);
{ startpulse_s(E0, tau0, 0); {}
SimulatePulse(0, 0);
df := f_m() - c / l_s_c;
setcolor(blue);
point(ReIm(tau / fs, -df / THz), "O");
end;
```

The result (obtained in a few minutes on an ordinary office PC):

In the right part, we have a nearly linear variation on the logarithmic scale, corresponding to the mentioned inverse fourth power dependence, but in the left part it levels down. By the way, the used realistic shape of the Raman gain also causes a slight deviation from that simple power law.

## Case 2: Pulses at 1350 nm

Now we start with a substantially shorter initial wavelength of 1350 nm. We find that this works very differently, resulting in a rather small wavelength shift of only 0.8 nm for 500-fs pulses:

We can understand that as follows:

- At 1350 nm, which is not far from the zero dispersion wavelength, the GVD is nearly an order of magnitude weaker: only −2594 fs
^{2}/m. - As a result, the fundamental soliton energy gets much smaller than before, and the same holds for the peak power.
- This results in a much weaker frequency shift.

We also try that out for 100-fs pulses:

The wavelength shift again gets larger, although by far not as large as before with a 1550-nm start wavelength. In addition, we now observe some wiggle structure which we did not see before. What's that?

We try out some things to understand that:

- First, we turn off the delayed nonlinear response and the self-steepening effect. You can't do that in an experiment, but in a simulation this is easy. The result is that although the wavelength drift disappears, the wiggles remain in a similar form. So it is not the Raman scattering.
- Next, we can eliminate higher-order dispersion. For that, we can directly define the refractive index profile (without a wavelength dependence), using the refractive index of core and cladding, as previously calculated for 1350 nm. In addition, we manually enter the constant GVD of −2594 fs
^{2}/m. Now, indeed, the wiggles have gone! - If we turn on the delayed nonlinear response and self-steepening again, but still work without higher-order dispersion, we get much more wavelength shift than before, and some weak wiggles:

So we can conclude the following:

- Higher-order dispersion (mostly of third order) is largely responsible for the observed wiggles. Compared with the weak GVD, it plays a substantial role for short and correspondingly broadband pulses. We see that neglecting this leads to quite unrealistic results.
- Basically, the oscillationen arise because the input soliton pulse shape has been calculated based on the assumption of constant GVD (i.e., no higher-order dispersion). Actual solitons of the fiber with higher-order dispersion would have a slightly different shape.

## 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:

- Solitons are often calculated for constant GVD and other simplifying assumptions. In practice, one has various additional effects, such as higher-order dispersion, which can play a substantial role.
- Numerical simulations can be more realistic, taking into account the full dispersion profile, and spectral loss profile, etc.
- We have seen that the soliton self-frequency shift slows down substantially if the magnitude of GVD increases with wavelength. That is actually easy to understand.
- When starting with a wavelength closer to the zero dispersion wavelength, higher-order dispersion directly becomes quite relevant, significantly changing the soliton profile and complicating the situation.

## 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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