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Modeling of Pulse Amplification

This is part 2 of a tutorial on pulse amplification modeling from Dr. Paschotta. The tutorial has the following parts:

1:  Models for different pulse duration regimes
2:  Gain saturation
3:  Simulating pumping and pulse amplification
4:  Multimode amplifiers
5:  Amplified spontaneous emission
6:  Bulk amplifiers

Part 2: Gain Saturation

Essentially, gain saturation means a reduction of gain caused by the energy extraction by a single pulse or possibly a pulse train. That makes the gain time-dependent during the short time of the pulse, while spontaneous emission or pumping alone would have very little effect during that time.

By the way, there may also be pump saturation effects, meaning saturation of pump absorption at high pump intensities, but that's a different topic and happens over longer time spans.

Does Substantial Gain Saturation Occur?

How to determine whether or not a pulse can cause substantial gain saturation? For that, we can simply calculate the saturation parameter S, defined as the pulse energy divided by the saturation energy of the amplifier. For amplifiers with high gain, that parameter actually increases a lot within the amplifier, and of course we consider it for the amplifier output, not the input.

Cases with Substantial Gain Saturation

In many cases, light pulses in an amplifier have a saturation parameter which is not far below 1. It may even be more than 1; a single light pulse can then extract much of the originally stored energy from the amplifier, causing the gain to nearly completely vanish afterwards. Efficient energy extraction is of course often desired. Obviously, we then need to take into account gain saturation.

Apart from a complication in cases with ultrashort pulses (see below), this is reasonably simple, and is already contained in the earlier mentioned models:

  • The laser gain results from the excitation of laser-active ions in the laser gain medium (e.g. an active fiber). The gain (as well as absorption of pump light, for example), applies to what I call optical channels for pump light, signal light (the pulses) and possibly ASE.
  • The temporal evolution of the excitations is described with rate equations. For example, these contain terms for stimulated emission: light is amplified, and the energy for this comes from laser-active ions going into a lower energy state, i.e., from a reduction of the excitation.

So the model has (a) the excitations and (b) the powers of all optical channels as dynamic variables. We start with some initial values for the excitations (e.g. simply zero excitation in an originally unpumped amplifier), have some given time dependencies for pump and signal inputs (e.g. pumping for some time, then injecting a short signal pulse), and can propagate all dynamical variables over some suitable range of time. As discussed in the next section, it is often appropriate to separately simulate the relatively slow pumping process and the fast pulse amplification process.

As an example for a simulated result (in the nanosecond duration regime, where we apply power propagation), you can see below an amplified pulse where the input signal pulse was triangular. Due to gain saturation, the pulse shape got somewhat deformed.

amplified pulse
Figure 1: Amplification of a triangular input pulse. The blue solid curve shows the output power vs. time, while the dot-dashed blue curve shows the hypothetical result which we would get without gain saturation occurring. The dashed black curve shows the upper-level population (fractional Yb excitation), which somewhat drops during the time of the pulse.

Gain Saturation with Ultrashort Pulses

A substantial technical difficulty arises when we need to simulate strong gain saturation caused by an ultrashort light pulse:

  • Being a time-dependent phenomenon, gain saturation should be treated in the time domain.
  • At the same, we often have a substantial wavelength dependence of the gain, calling for a calculation in the frequency domain.

So for whatever domain we decide, we cannot treat an important aspect!

The author of this article worked on this problem in 2017 and developed a numerical method to treat this situation, which got implemented in the RP Fiber Power software. For the details, read the Photonics Spotlight posting of 2017-08-01 and/or the following article: R. Paschotta, “Modeling of ultrashort pulse amplification with gain saturation”, Opt. Express 25 (16), 19112 (2017), doi:10.1364/OE.25.019112.

Fast Pulse Trains

In case of fast pulse trains, e.g. the output of a mode-locked laser where the pulse repetition rate is usually many megahertz or even multiple gigahertz, we often have the situation that while S is far below 1 – a single pulse cannot cause any significant gain saturation –, the large number of pulses within a millisecond does substantial saturation. Concerning gain saturation, the amplifier then behaves as if it were operated with a continuous-wave signal: there is a slow evolution of gain due to pumping and signal amplification, and no significant variation of gain during one pulse period. For finding the steady state, we may then simply calculate the amplifier gain with a simple continuous-wave model where the signal power is the average power of the pulse train. During the simulation of amplification of a single pulse, we can then neglect gain saturation.

Similarly, we can simulate the evolution of gain after turning on the pump power with a continuous-wave model. For various moments of time, we may then simulate what happens to a single pulse – possibly including chromatic dispersion and nonlinear effects. Requiring only a couple of pulse amplification simulations for getting the full picture, that method is far more efficient than applying a full-blown model to millions of pulses until the steady state is reached.

In relatively rare cases, a continuous-wave simulation does not provide accurate results since the optical spectrum of the signal pulses changes a lot during propagation as a result of nonlinear effects. A solution for that is explained in the next part.

Conclusions

A few conclusions from this part of the tutorial:

  • Gain saturation by a single pulse may be strong or very weak, depending on the situation. Sometimes we have negligible gain saturation by a single pulse in a pulse train, although the pulse train as a whole causes strong saturation.
  • Much simplified models can be applied in certain cases, allowing for very fast computations.

Go to Part 3: Simulating Pumping and Pulse Amplification or back to the start page.


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