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# Case Study: Soliton Pulses in a Fiber Amplifier

Key questions:

• Can we adiabatically amplify soliton pulses in a fiber amplifier?
• Which condition needs to be fulfilled to obtain adiabatic soliton evolution?
• What are the limitations?

Soliton pulses can propagate in optical fibers and have some nice properties, such as a clean temporal and spectral pulse shape. Here, we investigate to which extent they would be suitable for amplification in a fiber amplifier. This is inspired by the concept of adiabatic soliton compression, which we try to realize in a modified form. Instead of using a passive fiber with decreasing group velocity dispersion, we use an active fiber where the pulse energy in gradually increased. Couldn't we preserve the soliton shape and even strong pulse compression, given that the soliton pulse duration is inversely proportional to the pulse energy (for given fiber parameters)?

## Setting up the Simulation Model

We can use the Power FormFiber Amplifier for Ultrashort Pulses”. For the fiber, we use some spectroscopic data (transition cross-sections etc.) for erbium in a commercial fiber and freely override other parameters such as erbium density and waveguide parameters. We assume a GVD of −10 000 fs2/m.

### Calculating the Soliton Energy

The input signal pulses should be soliton pulses with 1 ps pulse duration. It is easy to set this and the sech2 pulse shape in the form, but how to get the correct soliton pulse energy? You find the simple equation in the encyclopedia article on solitons, but we don't want to manually calculate that and type it into the form. Instead, we can enter a few lines of script code there:

``````beta2 := rval(stage1.GVD\$, "fs^2/m") * fs^2 { GVD parameter }
gamma := stage1.n2_f * (2pi / signal1.l_s_c) / A_eff(stage1.signal_fw[1])
{ nonlinear coefficient }
E_s := -2 * beta2 / (gamma * (signal1.tau_s_in / 1.7627)) { soliton energy }
show "E_s:   ", E_s:d3:"J"
z_s := -pi * (signal1.tau_s_in / 1.7627)^2 / (2 * beta2) { soliton period }
show "z_s:   ", z_s:d3:"m"
calc cf_set_input('signal1.E_s_in', E_s)``````

This calculates some parameters from some data we already have in the form (center wavelength, nonlinear index) or can be calculated by the program (the effective mode area). We get the soliton energy `E_s` and the soliton period `z_s` and display those with the `show` command in the Output area. The last command puts the calculated value into the input field of the form so that we can see it there.

This may initially seem more complicated than entering a manually calculated value, but if we later want to try with other parameter sets, it is actually quite convenient to have that input value calculated automatically.

Note that the obtained pulse energy is only 18.4 pJ. That's typical for solitons, particularly if the pulse duration is not very short. Could we modify the fiber such as to get e.g. 10 times more energy for the same pulse duration? In principle yes: with 10 times more group velocity dispersion, but that's probably not realistic.

### Amplification of the Pulses

We inject 40 mW pump power at 980 nm in backward direction. The model shows us that with 20 m of fiber (having a relatively low erbium density of 10−24 m2) we get reasonably efficient pump absorption and an output pulse energy of 827 pJ (16.5 dB amplification). But a diagram showing the evolution of pulse parameters disappoints us, if we hoped to preserve the soliton pulse shape:

The pulse duration has been reduced, as we hoped, but not as much as the pulse energy was increased. Also, the green curve shows the time–bandwidth product, but not according to the usual definition; instead of FWHM duration and bandwidth, we use values based on the second moments, which are more appropriate for arbitrary pulse shapes, and a normalization which would deliver 1 for an ideal Gaussian pulse shape. (That curve is not offered by default, but it can be made by inserting one line of script code into the form.) The time–bandwidth product remains constant initially, but strongly rises later on – indicating that we no more have solitons. Why that?

The reason is that we are increasing the pulse energy too rapidly: we should do it closer to adiabatically, giving the soliton more time (propagation distance) to adapt itself. We shouldn't have too much gain within one soliton period, which in our case turns out to be as long as 50.6 m.

Let us first reduce the pulse duration to 500 fs, which also reduces the soliton period to a quarter of the original value and doubles the pulse energy to 36.9 pJ. The result:

Let us inspect the output pulse:

We see that the pulse shape has indeed been strongly distorted. The main pulse is rather short (23.7 fs FWHM), but there is a long pedestal before it and a stronger one after it.

To get better, we should more slowly amplify the pulse. So let us assume a 10 times longer fiber (200 m), but now with 10 times lower Er density. Indeed, it is getting much better:

Note that we neglected the influence of parasitic absorption of pump and signal light in the longer fiber. That could be significant in practice, but could be compensated by using some more power power.

The output pulse also looks far better:

It is just that the center of the spectrum shows some narrowband feature, which is related to a temporally long radiation background, but not carrying much power.

Note that in the time domain the pulses drifted a lot to the left. This is because its center wavelength was slightly reduced, and under conditions of anomalous dispersion that increases the group velocity.

We could in principle use an even longer fiber with very long erbium density, but that fiber may be hard to procure. And surely you don't want to buy that for a few hundred USD per meter...

## Conclusions

You can learn various things from this study:

• In principle, we could adiabatically amplify soliton pulses such that they stay fundamental solitons, with the pulse duration being reduced in inverse proportion to the pulse energy.
• However, soliton pulse energies are rather small and are hard to substantially increase. Also, the required length of the fiber amplifier would be rather long.

## More to Learn

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