# Case Study: Supercontinuum Generation in a Germanosilicate Single-mode Telecom Fiber

Key questions:

- Can a simple telecom fiber be used to obtain octave-spanning spectra? In which wavelength ranges can this work? Is it essential to work as close as possible to the zero dispersion wavelength?
- How can supercontinuum generation be simulated numerically?
- Does it work similarly for different pulse durations?
- What would be different for photonic crystal fibers?

## Definition of Task

Supercontinuum generation is the strong spectral broadening, for example in a optical fiber, due to intense nonlinear effects. It can involve several physical phenomena:

- self-phase modulation, possibly accompanied by self-steepening
- stimulated Raman scattering due to the delayed nonlinear response
- soliton effects, four-wave mixing and parametric amplification from the interplay of anomalous chromatic dispersion and nonlinearity

Depending on various parameters such as pulse duration and pulse energy, chromatic dispersion properties of the used fiber, etc., the relevance of different physical mechanisms of spectral broadening can be quite different. Therefore, different parameter regimes need to be analyzed separately.

We want to investigate how supercontinuum generation (mostly with 100-fs input pulses) works in germanosilicate single-mode fibers as used for telecom. There are many parameters that could be varied, but here we will try different input wavelengths, pulse energies, and fiber lengths for a given type of fiber: an ordinary single-mode telecom fiber with parameters like those of the common SMF-28.

## Simulations

### The Fiber Parameters

We use the software RP Fiber Power, which offers the Power Form “Passive Fiber for Ultrashort Pulses”. First, we need to define the parameters of the fiber:

- Corning's SMF-28 is a standard single-mode fiber which is widely used in telecommunications. We enter the core radius of 4.1 μm and initially a germania concentration of 4.5%, as this is needed to obtain the known NA of 0.14. This, however, leads to a somewhat small effective mode area of 64.4 μm
^{2}; Corning specified the effective mode field diameter as 10.5 ± 0.5 μm, suggesting a mode area around 87 μm^{2}at 1550 nm. Therefore, let us assume only 3.5% GeO_{2}, leading to an NA of 0.123 and a mode area of 76 μm^{2}– still a bit low, but note that conversion from mode field diameter to mode area also depends on the exact mode shape, which we do not know. - The software calculates the refractive index profile from the germania doping profile, and from that all relevant mode properties. In particular, it can calculate the effective refractive index at any optical frequency, which it then can use to get the full chromatic dispersion profile of the fiber.
- Note that the software will need to calculate the phase constant of the fiber mode for each optical frequency which is relevant for the pulse propagation.
- Anticipating that the pulse spectrum may get so broad that it gets into the absorbing spectral region beyond 2 μm wavelength, we also define an absorption spectrum. For that, we use a simple idealized model containing only propagation losses from Rayleigh scattering and (here more relevant) the infrared absorption. (An additional loss peak due to hydroxyl content of the fiber core could also be included easily, but is largely absent in some optimized fibers, and anyway not very relevant in a few meters of fiber.) Here is the code, which we enter into the form (under “Additional definitions”), so that we can use the function
`alpha(l)`

for the propagation losses:

```
A := 7.81e8 { dB/m }
a_IR := 48.48 { um }
alpha_IR(l) := A * exp(-a_IR / (l / um))
B := 0.95e-3 { dB/(m um^4) }
alpha_sc(l) := B / (l / um)^4
alpha(l) := alpha_IR(l) + alpha_sc(l)
```

- Further, we assume a nonlinear index of 3 · 10
^{−20}m^{2}/W, activate self-steepening and a delayed nonlinear response function for fused silica. That may not be extremely accurate considering that the core contains 3.5% GeO_{2}, but is sufficient for an exploration.

For supercontinuum generation, the chromatic dispersion is particularly important.:

We see that the zero dispersion wavelength is 1313 nm, and we have anomalous dispersion at longer wavelengths. The cut-off wavelength of the next higher mode (LP_{11}) happens to be at a very similar wavelength of 1314 nm, but this is an irrelevant coincidence. It is interesting, however, that by using this mode, e.g. at 1300 nm, we would get strong normal dispersion.

### Settings Related to the Pulse Parameters

We always assume Gaussian pulses with 100 fs pulse duration (full width at half maximum), but with different pulse energies and center wavelengths. It is easy to enter these few parameters in the Power Form.

In addition, we need to define the parameters of the numerical pulse trace used to represent the pulses at any position in the fiber:

- We set the width of the pulse grid to 25 ps. This is much longer than the original pulse duration, but we will see that the violent nonlinear interaction combined with the effect of chromatic dispersion can significantly spread the resulting pulses in the time domain.
- We also need a high temporal resolution to cover the resulting wide spectral range. For this we enter a relatively high number of amplitudes: 2
^{13}. This gives us a wide optical frequency range from 30 THz to 357 THz. - Finally, we should add quantum noise to the input pulse. For short (100-fs) pulses, this is not as important as for picosecond pulses, but it should still be done. The main reason is that there may be a large parametric gain that amplifies quantum fluctuations to macroscopic levels; this effect would be lost by not adding the quantum noise (semiclassically – the software does not use quantum operators!).

### Details of Numerical Simulations

Numerical simulations need to take into account all the above-mentioned ingredients, i.e.,

- the full chromatic dispersion profile,
- nonlinear effects with all relevant details such as the delayed nonlinear response, leading to stimulated Raman scattering, and
- the wavelength-dependent propagation losses.

The article on pulse propagation modeling explains more about the numerical techniques available. In detail, such things are quite sophisticated to set up, especially if you need both reliable and efficient operation for a wide range of parameters. However, if you use a proven software like RP Fiber Power, you do not have to deal with such difficulties. It is also very flexible, for example in defining input data and a wide range of diagrams that can be configured and even modified with completely customized diagrams. With such a tool, one can concentrate on the interesting physics and technological opportunities, rather than on numerical details.

### Input Pulses at 1550 nm

We start our exploration with input pulses at 1550 nm in the telecom C-band. A mode-locked erbium-doped fiber laser, possible in combination with an erbium-doped fiber amplifier for higher pulse energies, can be used as a source. As Figure 2 shows, at 1550 nm we are well beyond the zero dispersion wavelength, i.e. in a region of substantially anomalous dispersion. Here we can expect soliton-like propagation characteristics.

Starting with a fiber length of 1 m, we check how much pulse energy is required for substantial spectral broadening. With 1 nJ, where the initial peak power is 9.39 kW, we already get substantial spectral broadening. This is best seen in a diagram showing the spectral evolution (with a logarithmic color scale, covering a range of 40 dB):

With an increased pulse energy of 3 nJ, the nonlinear interaction gets more extreme:

Now we not only get more broadening towards the longer wavelength side, but also a new spectral component around 943 nm. This strange phenomenon may look like a numerical artifact, but it is real:

- First, the appearance of this component is not sensitive to numerical parameters (longitudinal step size, temporal or spectral range, or resolution), but to changes in chromatic dispersion (e.g., by changes in fiber core diameter). This seems to suggest that some phase matching process is involved.
- It occurs in a similar form even when only the Kerr effect and chromatic dispersion are considered (not the delayed nonlinear response and self-steepening). (A nice feature of numerical simulations is that we can easily turn off certain effects to see if they matter!)
- These circumstances may seem to suggest that we have four-wave mixing: a relatively simple nonlinear phenomenon that requires no more than the Kerr effect and is influenced by chromatic dispersion. However, there does not seem to be a process at work that converts the 1550-nm photons into a higher-energy and a lower-energy photon: the longer-wavelength photons are not seen, and they would also be in the spectral region where the fiber is strongly absorbing. Furthermore, the shorter wavelength peak is not sensitive to infrared absorption: no change when we remove it or make it 100 times stronger.
- Note also the following observations:
- In the simplified simulation without Raman scattering, we have something like higher-order soliton dynamics: strong quasi-periodic spectral “breathing”, i.e. broadening and subsequent spectral contraction (see Figure 5).
- The short-wavelength component appears only after the spectrum has broadened into this region; it just escapes the subsequent spectral contraction. The reason is that the group velocity in this spectral region is much lower than that of the longer wavelength components, allowing that spectral features to drift out of the region where it can interact with the pulse which originally generated it.

It is also interesting to inspect the output of the full simulation in the form of a spectrogram:

One can see that the short-wavelength features has completely drifted away from the other features and can thus no longer interact with them.

We can also check what happens for an even higher input pulse energy of 5 nJ. Let us begin with the spectrogram:

For the first time, we also get significant spectral content at wavelengths below the zero dispersion wavelength, i.e. in the normal dispersion range. Obviously, the entire output is roughly aligned to a parabola-like curve (also drawn here), which is easy to explain: different spectral components have different group velocities; the maximum velocity is at the zero dispersion wavelength (1313 nm), and both shorter and longer wavelength components are slower. The plotted curve shows the group delay versus wavelength over the full fiber length. Note that frequency components that are not generated at the fiber input, but only later, experience a correspondingly smaller group delay as the original light propagated faster. That causes their position in the spectrogram to be shifted slightly to the left relative to the curve.

We can also see some blobs indicating soliton pulses; these are sufficiently separated from the rest to propagate as more or less independent pulses.

We get an octave-spanning spectrum (at least if we consider a logarithmic scale, not e.g. a full width at half maximum of the power spectral density):

We see that a substantially shorter length of fiber would actually be sufficient to obtain strong spectral broadening.

The temporal evolution is also interesting, showing how different parts drift apart due to different group velocities:

A relatively prominent feature in this diagram is a line that – unlike other features – has a significant change in slope. This is a soliton that (as seen in the previous spectral diagram) undergoes a significant soliton self-frequency shift due to stimulated Raman scattering. This also causes the group velocity to gradually decrease, and the time delay at the fiber output to be substantially smaller than it would be if the pulse had that long wavelength to begin with.

## Case Study: Soliton Self-frequency Shift in Glass Fibers

We numerically simulate the soliton self-frequency shift, which is caused by stimulated Raman scattering. Influences like higher-order dispersion are found to be quite relevant.

#pulses#nonlinearities

### Input Pulses at 1350 nm

We now continue our exploration with input pulses at 1350 nm – chosen to be much closer to the zero dispersion wavelength of 1313 nm. We are still in the regime of anomalous dispersion, but it is now much weaker (-2971 fs^{2}/m instead of −21'034 fs^{2}/m). One of the consequences of this is that the pulse energy of a soliton (with a given pulse duration) is much lower. The given input pulse energy thus has the potential to be split into a larger number of solitons.

We start again with a lower pulse energy of 1 nJ. Here we already get a significant broadening of the resulting spectrum into the short-wavelength range with normal dispersion:

The corresponding spectrogram again shows alignment with the dispersion curve:

With 3 nJ input pulse energy, the spectral broadening gets stronger:

The spectrogram also shows more structure than before:

Finally, we inject a 5-nJ pulse, resulting in still more spectral broadening:

The spectrogram also shows more structure than before:

The temporal evolution shows less temporal spreading than for 1550-nm pulses:

Overall, for a 1350-nm pulse we see significant changes compared with 1550 nm, but not really a qualitative difference.

### Input Pulses at 1313 nm

We now try to see what happens when our input pulses are exactly at the zero dispersion wavelength of 1313 nm. We go straight to the highest pulse energy of 5 nJ:

One might have expected to get even more spectral broadening by minimizing the temporal spread. However, we get even less broadening, and the temporal spread is not much reduced because light with a large spectral bandwidth cannot escape the higher-order dispersion:

### Input Pulses at 1000 nm

Finally, we want to see what happens when we inject a pulse at 1000 nm, which is well within the normal dispersion regime. Note that our telecom fiber, designed for the 1.5μm spectral range, is no longer single-mode at 1000 nm: it also supports the LP_{11} mode. In the following, however, we will stick to this design in order not to change the chromatic dispersion for a better comparison with the other results. We simply assume that the input pulse is fully launched in the fundamental mode (LP_{01}).

We try again directly with 5 nJ input pulse energy. The results are completely different from what we had before in the anomalous dispersion region. First, the spectral broadening is much weaker:

The temporal spreading is smooth and quite symmetric:

The spectrogram simply shows a strongly up-chirped output pulse (i.e., with rising optical frequency):

It is interesting that we get so much less spectral broadening (compared to the anomalous dispersion regime) even though the temporal spread within 1 m of the fiber is even slightly less than for a 1550 nm pulse. The reason is that in the anomalous dispersion regime we have higher order soliton effects with strong temporal compression (hence high peak power) of the pulse at some points where the nonlinear interaction becomes quite intense. (Self-phase modulation generates a chirp which subsequently leads to temporal compression by the anomalous dispersion.) This cannot happen in the normal dispersion regime, where the peak power remains much more moderate.

### Using Longer Input Pulses

It is also interesting to explore what happens with longer input pulses, e.g. with a tenfold increased duration of 1 ps. Here we focus on an input wavelength of 1550 nm (in the anomalous dispersion regime).

The increased pulse duration requires several adjustments:

- An input pulse energy of 5 nJ (the highest value used so far) is sufficient for substantial spectral broadening in 1 m of fiber, but the broadening continues well beyond that length. We should therefore use a longer fiber, e.g. 5 m in length.
- We need to consider a much wider temporal range, e.g. 50 ps wide. In order to still have a sufficiently high temporal resolution, we have to increase the number of amplitudes representing a pulse to 2
^{15}. This slows down the computation considerably.

For 5 nJ input energy we get the following:

We do not yet obtain substantial spectral content below 1.3 μm. So we try with a higher energy of 10 nJ:

The spectrogram shows a sequence of solitons at different temporal and spectral positions:

Still, there is not much spectral content at short wavelengths. So we try with 50 nJ, and reduce the fiber length to 1 m, as not much further broadening then happens beyond that:

The spectrogram also shows the improved spectral content at shorter wavelengths:

Interestingly, this also shows a correlation between the long-wavelength blobs, which indicate soliton pulses, and the corresponding more complicated short-wavelength features, which do *not* propagate as solitons as they do in the normal dispersion regime.

We see some significant differences in supercontinuum generation with longer input pulses. Here we start with a rather narrow spectrum, but as we increase the pulse energy to keep the peak power similar to before, we also get a rapid spectral broadening, again eventually getting octave-spanning spectra.

## Conclusions

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

Various conclusions can be drawn from this case study:

- The strong spectral broadening only works with a center wavelength of the input pulses in the anomalous dispersion region. However, the distance from the zero dispersion wavelength is no critical parameter.
- For input pulses from a mode-locked laser in the 1-μm spectral region, an ordinary all-glass fiber is not suitable because it cannot have anomalous dispersion in this region. Therefore, it is necessary to use photonic crystal fibers (PCFs) in this region, which can be tailored for a wide range of dispersion characteristics.
- To understand and optimize supercontinuum generation, numerical simulations with suitable software are indispensable. They easily reveal many features that would be difficult or impossible to access with experiments. The resulting understanding is also essential for predicting and optimizing the performance of devices using supercontinuum generation, such as broadly tunable light sources.

Several other interesting things could also be studied with the same tools:

- We could explore further parameter regions, for example with much longer input pulses. Some aspects may change quite profoundly.
- One should never prematurely assume that the understanding of supercontinuum generation obtained in one parameter regime can be more or less applied to a significantly different parameter range. Note also that we have quite a large parameter space in terms of pulse energy and duration, center wavelength, dispersion properties, and fiber length.
- We could also simulate pulse propagation in photonic crystal fibers with arbitrary chromatic dispersion profiles.
- We could study the effects of small variations in the input pulse parameters on the result. In some parameter regimes (especially for longer input pulses), even quantum fluctuations can cause substantial differences in the details from pulse to pulse, reducing the temporal coherence of the output.
- A quick test shows that omitting the input quantum fluctuations does changes the results somewhat for 1-ps pulses, but not for 100-fs pulses. For longer pulses, that could become quite important.
- Note that statistical evaluations, requiring many pulse propagations, can be quite time-consuming. However, during a night, a computer can generate substantial amounts of interesting data.

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