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Case Study: Parabolic Pulses in a Fiber Amplifier

intro picture

Key questions:

  • What pulse energies are realistic for a femtosecond fiber amplifier?
  • What dispersion characteristics do we need for effective temporal compression after the fiber?
  • What influence does the fiber's doping density have?

When ultrashort pulses propagate in a fiber amplifier with normal chromatic dispersion, this can happen in a regime where parabolic pulses are formed. Here, the pulses asymptotically approach a state where both the pulse duration and pulse bandwidth grow in proportion to the cubic root of the pulse energy, and acquire a nicely linear up-chirp. Such pulses are immune against pulse break-up and can relatively easily be subject to dispersive pulse compression. Therefore, this regime is interesting for obtaining relatively short pulses with high energy – although not as much as with chirped-pulse amplification – and this with a relatively simple amplifier setup. Another nice feature is that the details of the seed laser pulses are not critical.

In this case study, we explore how well this works for a typical active fiber and answer a couple of relevant questions – for example, what is the impact of substantial spectral broadening (implying a large variation of amplifier gain within the spectrum), how far we can get into a strongly nonlinear regime while preserving pulse quality, how the performance will depend on the available seed pulses and on the doping density of the used active fiber.

Setting up the Simulation Model

For the modeling, we use the software RP Fiber Power, which offers a Power Form titled “Fiber amplifier for ultrashort pulses”. We can just fill in all the parameters, including those for configuring some diagrams, and get it calculated.

For an initial simulation, we assume unchirped seed pulses with 200 fs duration and an energy of 0.1 nJ at 1060 nm. We want to amplify these in an ytterbium-doped single-mode fiber. Instead of using data for a particular commercial fiber, for our exploration we work with a generic data set where we adjust various parameters for a typical Yb-doped germanosilicate single-mode fiber. We assume a core radius of 4 μm and a moderate Yb doping density of 5 · 1024 m−3, which implies sufficient pump absorption within a couple of meters of fiber. Further, we assume a group velocity dispersion of +10 000 fs2/m and a nonlinear index of 3 · 10−20 m2/W. For now, we apply backward pumping with 500 mW at 940 nm, a reasonable pump power for core pumping.

Estimating the Possible Pulse Energy

A first question is what kind of output pulse energies are realistic with this approach. Could they be so high that a single pulse extracts a substantial part of the energy stored in the fiber? For that, the output pulse energy would need to exceed the saturation energy of the fiber, which the simulation model reports to be 34.6 μJ. Dividing that by a roughly estimated output pulse duration of a few picoseconds (after some temporal stretching in the fiber), we get a peak power of the order of 10 MW – which will never work; it would even destroy the fiber via catastrophic nonlinear self-focusing!

We will thus need to limit the peak power to a regime which is several orders of magnitude lower. That implies much lower pulse energies in the nanojoule region, and substantial average output powers will be possible only in conjunction with a high pulse repetition rate. A first very rough guess can be that it may work around 10 MHz · 30 nJ = 300 mW, where the peak power is of the order of 30 nJ / 3 ps = 10 kW. A gain of 10 lg (30 nJ / 0.1 nJ) = 24.8 dB is realistic for a single amplifier stage.

Tests with Repetitive Operation

So we set up the simulation for repetitive operation of the amplifier at 10 MHz. Being interested mostly in the steady state, we let the model calculate the initial state of the fiber based on the average signal input power of 10 MHz · 0.1 nJ = 1 mW: It does a continuous-wave amplifier calculation with that, not considering any pulse-specific effects, and takes the resulting Yb excitation profile for the following simulation of pulse amplification.

Here is the Power Form with the made inputs and various calculated outputs:

Power Form with input and output values
Figure 1: The Power Form with various input and output values.

The model shows that for a fiber length of 5 m we can absorb most pump power. However, we obtain only 7.3 nJ output pulse energy; the amplifier gain is obviously quite low, but why? The diagram showing the initial state (before arrival of the pulse) reveals the reason:

initial state of the amplifier
Figure 2: Initial state of the amplifier.

We see that we get strong amplified spontaneous emission (ASE), particularly in forward direction. How can this happen despite the quite low gain of our signal pulses? By inspection of the ASE output spectrum (not shown here) we find that it is around the 975-nm peak. To solve that, we can use a pump source at 975 nm, with which it is impossible to generate 975-nm ASE. But the simulation shows that the fiber then needs to be substantially longer – another surprise, considering that the absorption cross-section is much higher at 975 nm than at 940 nm! The explanation for that is strong absorption of the pump saturation at the applied power level.

With 8 m fiber length, we now reach 33 nJ output pulse energy, corresponding to 330 mW average output power. Let us inspect the pulses we get in that regime, using the interactive pulse display window (although the Power Form also offers some diagrams):

output pulses for operation with 10 MHz
Figure 3: Output pulses for amplifier operation with 10 MHz.

We can also inspect the temporal and spectral evolution with color diagrams:

temporal evolution
Figure 4: Temporal evolution of the pulses along the fiber.
spectral evolution
Figure 5: Spectral evolution of the pulses along the fiber.

We see that the pulses get increasing time delays. This is because the spectrum drifts towards shorter wavelengths, where the amplifier gain is higher, and normal chromatic dispersion then causes the pulses to get somewhat slower.

Dispersive Pulse Compression

The temporal pulse shape (seen in the top area) is not quite parabolic, as expected. The also shown instantaneous wavelength deviation (from the center wavelength of 1060 nm) shows that the pulse initially has the long-wavelength components, and the short-wavelength components follow later (up-chirp of the instantaneous optical frequency). It turns out that the longer-wavelength components have substantially more amplifier gain, and this leads to the quite asymmetric temporal pulse shape.

The pulse spectrum (lower area) is rather broad, already going well beyond the region where we have substantial amplifier gain. Still, the spectral phase (shown in green) looks like parabolic, suggesting that dispersive compression may work well. However, as the spectral phase varies by hundreds of radians within the spectrum, we cannot tell from that whether it is sufficiently close to parabolic for effective pulse compression. So we enter the following code into the form for execution after the simulation:

calc (pp_compress(2); store_pulse(2); tau_c2 := tau_p())
calc (pp_compress(3); store_pulse(3); tau_c3 := tau_p())
show "tau_c2:  ", tau_c2:d3:"s"
show "tau_c3:  ", tau_c3:d3:"s"

This will try dispersive pulse compression, first using only second-order dispersion, then also third-order dispersion, in both cases numerically optimized. (Possible power losses at the compressor are ignored.) It stores the pulses as numbers 2 and 3 and displays the resulting FWHM pulse durations.

For second-order dispersion only, we get 69.7 fs output pulse duration. Although that doesn't sound bad, the resulting pulse is not so great:

compressed pulse
Figure 6: Pulse after compression with second-order dispersion only.

Even with third-order dispersion (not shown here), this cannot be remedied. Apparently, we are in a too strongly nonlinear regime here, where we cannot get nice parabolic pulses. So let's try with a higher pulse repetition rate of 30 MHz, which leads to a more moderate peak power and nonlinear spectral broadening.

pulses for 30 MHz
Figure 7: Die output pulses with 30 MHz before compression.

The pulses are now a bit closer to parabolic, although still substantially “tilted” due to stronger amplification of the shorter-wavelength part, coming at later times.

With second-order dispersion, we still do not get good pulse compression:

pulse with 30 MHz, second-order compression
Figure 8: The output pulse with 30 MHz, compressed with second-order dispersion only.

With third-order compression, however, it works quite well, only with some weak pulse pedestals:

pulse with 30 MHz, third-order compression
Figure 9: The output pulse with 30 MHz, compressed with second- and third-order dispersion.

We get a compressed pulse duration of 61.9 fs and a reasonably low time–bandwidth product of 0.66, consistent with the fairly flat spectral phase. Note that the pulse's center wavelength is shifted substantially away from that of our input pulses at 1060 nm.

We ignored the self-steepening effect (related to the frequency dependence of the nonlinear interaction) in those simulations, but a quick test shows that this has only a minor effect in this regime of operation.

Influence of the Fiber's Doping Density

Normally, we could e.g. double the doping density of the fiber and then just use half of the fiber length. But in our case, that would modify both the strength of nonlinear and dispersive effects. Without a simulation, it would be hard to reliably check what effect that would have on the performance. So how will that work out in the end?

We can simply try it. The result is that spectral broadening gets even stronger, and the compressed pulse duration (with second- and third-order dispersion) leads to 44.1 fs pulse duration. So we have more spectral broadening despite using a shorter fiber – because the weaker chromatic dispersion causes less temporal pulse stretching, which implies a higher peak power in the fiber.

Sensitivity to Input Pulse Parameters

We can now also check how important parameters of our input pulses are. For example, if we double their duration to 400 fs, we just get a little less energy and even slightly shorter pulses. Alternatively, if we use 50 pJ instead of 100 pJ, the pulse energy only drops to 10.3 nJ, and the compressed pulse duration is slightly increased to 65 fs. So we see that indeed the pulse parameters of the used seed laser are not that important in the parabolic pulse regime.

Further Exploration

One may further explore various aspects; some examples for questions which could be addressed:

  • How much higher pulse energies become possible by using a fiber with larger mode area?
  • How does it work for shorter or longer signal wavelengths?
  • How far you can get by optimizing the Yb doping concentration (which we found to have an impact on the pulse broadening and compression)
  • How will the pulse parameters evolve after turning on the pump power of the amplifier?

Such questions are highly relevant for the development of such devices, and for many of them there is no other convenient way of clarification than doing such numerical simulations.

You may also be interested in a case study where the amplification of soliton pulses is investigated.


RP Fiber Power

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:

  • We quite easily find the regime where approximately parabolic pulses are formed by the interplay of Kerr nonlinearity and normal chromatic dispersion.
  • In detail, however, various effects play a role; for example, the wavelength-dependent gain can lead to an asymmetric temporal pulse shape.
  • In a regime with too strong nonlinear effects, high quality dispersion compression becomes difficult.

Obviously, numerical simulations, to be performed with a flexible simulation software, are vital for analyzing the performance and developing optimized fiber amplifier designs.

More to Learn

Encyclopedia articles:


[1]D. Anderson et al., “Wave-breaking-free pulses in nonlinear-optical fibers”, J. Opt. Soc. Am. B 10 (7), 1185 (1993); https://doi.org/10.1364/JOSAB.10.001185
[2]K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers”, Opt. Lett. 21 (1), 68 (1996); https://doi.org/10.1364/OL.21.000068
[3]M. E. Fermann et al., “Self-similar propagation and amplification of parabolic pulses in optical fibers”, Phys. Rev. Lett. 84 (26), 6010 (2000); https://doi.org/10.1103/PhysRevLett.84.6010

(Suggest additional literature!)

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