Modeling of Pulse Amplification
2: Gain saturation
3: Simulating pumping and pulse amplification
4: Multimode amplifiers
5: Amplified spontaneous emission
6: Bulk amplifiers
Part 1: Models for Different Pulse Duration Regimes
For successful physics modeling, it is always advisable to start with a careful consideration on the type of model used, i.e., essentially on which physical effects need to be taken into account. The best suited model is often not the most comprehensive one, but the one which contains everything relevant while avoiding any unnecessary complications – thus not only saving computation time but also helping the user to focus the attention on what matters.
In the context of pulse amplification, we need to consider whether chromatic dispersion and nonlinear effects need to be taken into account, and how to treat gain saturation. Another question is whether propagation times need to be taken into account. Some of those questions are not trivial to answer.
A newcomer may easily decide for a technical approach which is either over-simplified, i.e., not delivering correct results, or is unnecessarily complicated and time-consuming to work with. Therefore, the benefits of using some professional high quality software for such things are not only that the appropriate types of models are available and correctly implemented, but also that the technical support can be very helpful for finding the best path forward.
Chromatic dispersion effects are based on the fact that various details of propagation are dependent on the optical frequency, and that a light pulse generally has an optical spectrum spanning some frequency range.
For the amplification of ultrashort pulses, i.e., light pulses with picosecond or femtosecond durations, chromatic dispersion effects are usually quite relevant. The group velocity of light varies sufficiently much within the frequency range of the optical spectrum of the pulses to lead to substantial changes of the temporal pulse shape. In the normal dispersion regime, for example, the shorter-wavelength components need more time to get through an amplifier than the longer-wavelength components, and that typically leads to a “smearing out” of the power vs. time, i.e., to pulse broadening. This simple picture is quite intuitive but neglects the interplay with nonlinear effects (see below), which can make things rather more complicated. For example, you may even get pulse compression effects.
In regimes with longer pulse durations – for example, one nanosecond or longer –, chromatic dispersion is usually not having significant effects. Even if the pulse are not transform-limited, i.e., having a broader optical bandwidth than the minimum value dictated by Fourier transform principles, the resulting variations of group delay within the spectrum are not large enough to become relevant for the resulting output pulse shape.
Light pulses can have some optical energy concentrated to a short time span, which implies a high peak power – often many orders of magnitude higher than the average power of a pulse train. In addition, laser pulses usually propagate with rather small effective mode area through an optical amplifier – particularly if it is a fiber amplifier. As a result, the pulses can have very high optical intensities, which easily lead to substantial nonlinear effects – particularly when the gain medium is long, as it tends to be for a fiber amplifier. Even in the nanosecond pulse duration regime, and certainly in the picosecond and femtosecond regime, nonlinear effects then become quite relevant. For ultrashort pulses, that is often a substantial factor limiting the possible amplifier performance.
One might think that we thus basically always need to taken into account optical nonlinearities when modeling the amplification of short light pulses. That is not true, however. Even if the Kerr effect causes strong self-phase modulation on the pulses, in the nanosecond regime that usually does not have an impact on the temporal pulse shape or duration. Also, the resulting spectral broadening is typically too weak to affect the amplifier gain through its wavelength dependence. Therefore, even if nonlinear effects are strong in the sense of strong self-phase modulation, we may fully neglect this in a simulation model – making it such that it only calculates the evolution of optical power while not considering the optical phase. The following diagram shows an example for that; here we did a full simulations including nonlinear and dispersive effects, but would have obtained essentially the same output pulse shape with simple power propagation.
If the resulting self-phase modulation is of interest, we can still calculate that separately based on the evolution of peak power as obtained from the model. More precisely, we need the B integral (calculated for the pulse peak), and the model can deliver that based on the calculated power evolution. Based on the B integral, we can just add the time-dependent nonlinear phase shift to the obtained power vs. time, and from that we can also calculate the optical spectrum in a fraction of a second.
In the picosecond or femtosecond pulse duration regime, that approach would not work, since the nonlinear phase changes have a stronger impact on the optical spectrum and thus on the effects of chromatic dispersion. (The nonlinear phase modulation is not only tentatively stronger, but also much faster in this regime!) Even the effect of spectral broadening on the effective amplifier gain may be significant. In total, we often get a complicated interaction of nonlinear and dispersive effects, which may not only lead to temporal broadening, but even to complete break-up of pulses. Further, that pulse break-up may even be sensitive to quantum noise influences.
Another sometimes important nonlinearity is stimulated Raman scattering (SRS). Fully simulating that requires a sophisticated model with time-consuming computations. However, in most cases there is only one simple question to answer: is SRS expected to cause trouble by substantially converting signal light to longer wavelengths? To answer that, we only need to calculate the Raman gain resulting from the obtained signal peak power. As a simple rule of thumb, once that remains below 70 dB, not much will happen. For pulses, we can actually often even tolerate some more Raman gain. A simple model with power propagation may then be sufficient; otherwise, we need a full ultrashort pulse propagation model. Both are described in the following.
Simple Models with Power Propagation
If dispersive and nonlinear effects can be neglected, we can use a dynamic simulation model based on simple power propagation. (“Dynamic” simply means that we deal with time-dependent quantities.) Here, a light pulse is characterized simply by its optical power vs. time – at least if it is a single-mode model. (Section 4 discusses multimode cases.) One may also use a beam propagation model for multimode cases, where for each <$z$> position we have a two-dimensional array of optical intensities instead of simple one power value; that leads to more time-consuming, but not fundamentally more difficult simulations as long as we have quasi-monochromatic light.
In the single-mode case, we have simple differential equations for the powers of all involved optical channels, as I call it – for example, just one pump and one signal channel. We might also have two pumps (e.g. counterpropagating) and multiple signals at different wavelengths, and also possible many more channels for ASE – just more differential equations, but nothing fundamentally different. These differential equations also contain the local excitation of the gain medium, which itself may be influenced by gain saturation (see the next part), so that effectively we have coupled differential equations.
The solution of those equations as an initial value problem is not particularly difficult. Basically, for every moment of time we can calculate the first temporal and spatial derivatives of all powers and the temporal derivatives of the excitations, and use those values to propagate all these quantities forward in time and space. That works even if counterpropagating waves are involved.
It is quite simple to select a fiber, enter various operation parameters and start the simulation, producing some numerical outputs in the form itself (output fields with gray background), and as diagrams. One of them is shown here; more details on this are explained in the next part.
Are Propagation Times Relevant?
It takes some time for light pulses to propagate through an amplifier, and in principle that may be relevant for correctly simulating its function. Consider, for example, a case with a counterpropagating pulsed pump, where a signal pulse arriving at the output end with some time delay. At that end, the pump light enters the fiber (without a time delay). So the propagation time modifies the temporal relation of signal and pump. However, that turns out to be a fully negligible aspect for basically all fiber amplifiers. This is simply because propagation times in a few meters of fiber are only of the order of 10 ns, while upper-state lifetimes and typical pumping times are hundreds of microseconds or more. So we can safely ignore propagation times in our models – simulating it as if pulses could instantly travel through the whole fiber.
The situation is rather different in Q-switched fiber lasers, for example; there, the round-trip time is very relevant for the resulting pulse duration. In the RP Fiber Power software, it is thus possible to make dynamic models where propagation times are taken into account. Only, that enforces smaller temporal steps and may thus slow down the calculations substantially. So it is good to know that we usually don't need that for amplifier models.
How to Propagate Ultrashort Pulses
For ultrashort pulses, where the optical phase as a function of time or optical frequency is also very important, we need a substantially more sophisticated type of simulation model. Such a model is implemented in our software, and its operation principle is briefly described in the following:
- The pulse can be represented in the time domain by an array of complex amplitudes. That array must span a sufficiently wide time range (typically several times the pulse duration), such that the pulse stays sufficiently far away from the edges during the whole propagation. Further, the array must have sufficiently many amplitudes for having a good enough temporal resolution.
- For the amplitudes, we do not directly use the rapidly oscillating electric field, but rather a “slowly varying amplitude” (well, actually it is not that slow…) where we take out the fast optical oscillation at the center frequency.
- Some effects can be applied in the time domain – for example, self-phase modulation from the Kerr effect.
- Using a Fast Fourier Transform (FFT) algorithm, the software can get these amplitudes into the frequency domain. There, one can conveniently apply frequency-dependent effects like amplifier gain and chromatic dispersion. A split-step Fourier method is used to regularly switch between time and frequency domain and appropriately apply all interactions for each spatial propagation step.
- Even stimulated Raman scattering and self-steepening can be taken into account. It is only that SRS simulations require a quite fine temporal resolution (in order to cover the relevant spectral range) and thus tend to be relatively slow.
To control all that, our software provides various functions
- for subsequently sending a pulse through various elements (e.g. an amplifying fiber, a passive fiber or some optical filter),
- for retrieving calculated pulse properties, and
- for controlling various other things, such as saving or retrieving whole pulses, switching between pulses and setting a start pulse.
Such functions can be used in scripts. Again, our software comes with many demo scripts, e.g. one for a fiber amplifier for (not ultrashort) pulses, also containing a custom form:
An an example, the following diagram shows how the pulse parameters evolve along the active fiber of a fiber amplifier.
A few conclusions from this part of the tutorial:
- In the nanosecond pulse duration regime, we can normally neglect chromatic dispersion and optical nonlinearities when simulating the power evolution. This means that we can do simple power propagation. Based on the calculated powers, we can still calculate some nonlinear effects if needed.
- For picosecond and femtosecond pulses, we usually need to fully treat nonlinear and dispersive effects in the amplifier model. That is possible with a more sophisticated type of simulation model.
By the way, in most cases we also need a model which can cope with quasi-three-level laser gain media, as most fiber amplifiers have such transitions. However, this issue not specific for pulse amplification.
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