# Case Study: Nonlinear Pulse Compression in a Fiber

## Key questions:

- How well does nonlinear pulse compression work in different pulse duration regimes?
- What influence has the chromatic dispersion? Do we need higher-order dispersion for effective compression?
- To which extent can we compensate a lower input peak power by using a longer fiber?

A common method of temporally compressing ultrashort pulses is to first spectrally broaden the pulses via self-phase modulation in an optical fiber and then more or less correct (flatten) their spectral phase with a dispersive optical element such as a pair of diffraction gratings. That often leads to a reduction of the pulse duration by an order of magnitude or even more, but frequently also to a reduction in pulse quality. For example, the temporal pulse profile may exhibit some substantial pedestal or some side peaks, which for some applications can be problematic.

In this case study, we numerically investigate in some example cases how well that method works in some example cases and what factors influence the results. We look at two cases with rather different pulse energies.

For all the following simulations, we use the RP Fiber Power software, which offers a Power Form titled “Passive fiber for ultrashort pulses”.

## Case 1: 100-nJ 1-ps Pulses at 1050 nm

We first assume that we have unchirped Gaussian input pulses with 100 nJ pulse energy and 1 ps pulse duration at 1050 nm wavelength. That implies a substantial peak power just under 100 nJ / 1 ps = 100 kW.

We calculate the parameters of a germanosilicate single-mode fiber with an assumed core diameter of 8 μm and a constant germanium concentration of 2% within the core. That is found to lead to single-mode behavior (LP_{01} only) at our center wavelength of 1050 nm:

From these data, the program calculates the group velocity dispersion of +20'021 fs^{2}/m at 1050 nm (in the normal dispersion region). (For the pulse propagation, it calculates the dispersion for any required wavelength within the pulse spectrum, so that dispersion with all orders is taken into account.) Further, we assume a nonlinear index of 3 · 10^{−20} m^{2}/W.

We first try the pulse propagation with a fiber length of 500 mm, but find too extreme nonlinear broadening; therefore, we reduce the length to 300 mm. For that, in the time domain we obtain an output pulse with a more than doubled duration of 2.09 ps; this is the effect of chromatic dispersion, which gets substantially increased by the strong spectral broadening. The pulse shape is no more close to Gaussian:

The spectral phase (gray dashed curve) is roughly parabolic in the center, but deviates a lot from that towards the edges. So we are far from getting a linear increase in the instantaneous optical frequency during the pulse.

Note also that the trailing part gets quite steep; here we see some effect of self-steepening, which we have activated for that simulation.

The pulse spectrum shows the typical wiggles resulting from strong self-phase modulation, as long as the influence of chromatic dispersion is not strong:

Another diagram shows nicely how the pulse broadening evolves in the fiber:

Now we try dispersive compression of this pulse, first with second-order dispersion only (i.e., with a wavelength-independent group delay dispersion), and assuming a 20% loss of pulse energy in the pulse compressor:

The resulting compressed pulse:

Note that the software has chosen an asymmetric temporal range for this diagram because it detected that the pulse pedestal extends substantially more towards negative time coordinates. That actually happens only if we turn on the step-steepening effect in the simulation.

Although that pulse has a strongly reduced duration (in terms of full width at half maximum) of 30.6 fs, we see that the pulse quality is not ideal. Unfortunately, even if we reduce the fiber length to 100 mm, which increases the pulse duration to 63.2 fs, the pulse quality is not really better:

That non-perfect pulse quality is actually typical for this technique of pulse compression.

We have so far assumed that the input pulse is unchirped. We can change that:

- If we set an initial up-chirp of 1 THz/ps (which already substantially increases the pulse bandwidth), we obtain somewhat more spectral broadening and pulse lengthening in the fiber, but the compression gives us only a slight further reduction in the compressed pulse duration.
- If we start with a down-chirp of −1 THz/ps, we first get some spectral compression before the spectral broadening sets in:

Another interesting experiment is to start with an unchirped parabolic pulse, although unfortunately that is not so easy to generate. The idea behind this is that for a parabolic pulse, we would get a parabolic phase modulation, as far as that pulse shape is conserved during propagation in the fiber. That parabolic phase profile corresponds to a linear increase in instantaneous frequency. Let us first check the resulting output pulse before compression in the time domain:

So the parabolic profile of power vs. time is indeed approximately conserved. (We get a very steep decay of power at the edges.) Now we inspect the spectral evolution:

Interestingly, we don't get these strong spectral structures anymore. The output pulse profile is relatively smooth, also roughly parabolic:

Why have these spectral wiggles now disappeared? This can be understood as follows:

- For the Gaussian input pulse (and many other pulse shapes), the curvature of the resulting profile of the phase modulation has different signs at different times. The instantaneous frequency first drops, then rises, and then drops again. Therefore, most values of the frequency occur twice during the pulse. Their contributions to the Fourier spectrum interfere with each other (as you can see in the Fourier integral), with the interference conditions varying during the pulse. That causes the spectral wiggles.
- For the parabolic pulse, the instantaneous frequency rises monotonously. Therefore, the mentioned interference effect and the resulting wiggles are eliminated. Only, the pulse shape will not stay perfectly parabolic throughout the fiber.

Now the compressed pulse, when applying an optimized constant GDD:

To improve this further, we now allow optimization also of third-order dispersion:

This is significantly better, although the weak pre-pulse may still be disturbing. Using perfect pulse compression (with arbitrarily high dispersion orders), we also don't get that much better:

So we see that the spectral shape sets the limits for suppressing these side pulses.

By the way, the diagrams show some effects of the finite spectral resolution chosen. We could easily improve that resolution by choosing a longer time trace, and then probably improving the number of points to represent a pulse. The computation time for one case will still be far below one minute.

## Case 2: Weaker Pulses: 10 nJ, 1 ps, 1050 nm

With 10 times less pulse energy than before, we get 10 times less peak power. So we now use a 10 times longer fiber (3 m), leaving the other fiber parameter unchanged. Propagation losses can still be neglected for that length. We again use a 1-ps unchirped Gaussian input pulse, as we initially did in the other case.

One might expect about the same results as before, but this is not the case. First, we look at the temporal pulse profile after the fiber:

This has evolved more towards a flat top and with a substantially longer duration as a result of the stronger effect of chromatic dispersion in the longer fiber.

The pulse spectrum has also changed substantially, no longer having the strong wiggles:

However, that is only true after a sufficiently long propagation length, as the following diagram with the evolution of the pulse spectrum reveals:

Initially, strong wiggles form, but these are then more and more washed out. Also, the magnitude of spectral broadening does not progress much further, as the peak power becomes low.

The compressed pulse (with optimization only of second-order dispersion) has a duration of 81.2 fs:

With a perfect compressor (applying dispersion of any orders), that would look substantially better (pulse duration: 51.2 fs), even though that might be difficult to realize in practice:

We could, of course, try with even longer fibers. However, the following diagram shows that the compressed pulse duration (with perfect compression) already does not full substantially anymore after a few meters (see the dashed green curve):

This is because the pulse duration in the fiber rises, so that the peak power falls and the further nonlinear broadening becomes slow.

## Case 3: Still Weaker Pulses: 1 nJ, 1 ps, 1050 nm

We now consider a case with a still lower pulse energy of 1 nJ only, where the initial peak power is slightly below 1 kW. Again choosing a 10 times longer length (30 m) now really doesn't make sense, as after 10 m already, the nonlinear broadening nearly stops. So we use 10 m. Even in that length, the pulse duration in the fiber rises to 5 ps. The spectrum does not exhibit any wiggles:

Compression with second-order dispersion only leads to pulses with 166 fs duration and significant side lobes:

With perfect compression, the pulse quality would again be much better:

We see that for less energetic pulses we cannot achieve as much compression, basically because a longer fiber is then needed, where the effect of chromatic dispersion becomes more limiting. That, however, could be improved by choosing a fiber with lower dispersion.

## Case 4: 100-nJ 1-ps Pulses at 1550 nm

Now we consider operation at 1550 nm, which is interesting because here the chromatic dispersion of a germanosilicate fiber is in the anomalous regime (with negative GVD). You might expect to see soliton effects there.

For the longer wavelength, we increase the core radius from 4 μm to 6 μm such as to roughly preserve the V number, still staying in the single-mode regime. The effective mode area then rises from 64.5 μm (at 1050 nm) to 142.7 μm. The nonlinear index will also get smaller; we assume 2.7 · 10^{−20} m^{2}/W.

We start with 300 mm fiber length as in case 1. Let us first inspect the spectral evolution:

It turns out that in the first 130 mm length, the spectral broadening works quite similarly as before. We use the interactive pulse display window to conveniently explore pulses at different positions in the fiber, beginning at 130 mm after the entry:

We only see some slight wiggles on the temporal profile. But if we got to 180 mm, these wiggles are already quite strong:

After 220 mm, the spectrum gets very broad, and both temporal and spectral trace become quite irregular:

By the way, in the simulation we now had to enable the delayed nonlinear response, which is associated with Raman scattering, because that has a significant influence on the results (while this was not so in the previous cases). It is also important to add some quantum noise to the input pulse, as the Raman gain will amplify that outside the actual pulse spectrum.

Obviously, this extreme regime is not good for effective pulse compression. We rather stay in the more moderate regime by limiting the fiber length to 120 mm. Here, the dispersive compression with second order only works reasonably well, with a pulse duration of 155 fs:

With perfect compression (not shown here), this does not get substantially better.

So we see that in the anomalous dispersion region we have to be more careful with the choice of fiber length. If we go too far, we get into a regime with complete pulse break-up, which is not suitable for effective pulse compression.

## 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:

- With the highest pulse energy, we get substantial nonlinear broadening before the pulse duration increases a lot. The compressed pulses are then rather short.
- With lower pulse energies, we need to use a longer fiber, and chromatic dispersion then causes more temporal stretching, which reduces the nonlinear effect. Therefore, the compressed pulses get longer.
- Although the spectral phase of the nonlinearly broadened pulses is approximately parabolic, the quality is not ideal when only second-order dispersion is optimized. Even with perfectly shaped dispersion, the achievable pulse quality normally is not very good.

While the basic effects can be understood based on qualitative reasoning, numerical simulations are the only way to reliably predict and optimize the performance. For this, it is vital to have a flexible simulation software.

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