1. Introduction

Numerical models are widely used for predicting and analyzing the performance of laser amplifiers of various kinds. In simple simulations of pulse amplification, where spectral properties of the amplified light are not of interest, it is easy to take into account gain saturation, which becomes substantial if the pulse fluence in the gain medium exceeds the gain saturation fluence. For that purpose, one can simply include the degree of excitation of laser-active ions at various positions in the active medium as an additional set of dynamical variables, the temporal evolution of which is governed by rate equations. However, problems arise in the context of ultrashort laser pulses: one usually needs to treat effects like chromatic dispersion and frequency-dependent gain in the frequency domain, while gain saturation can be treated only in the time domain. Notably, that problem cannot simply be solved by frequent switching between time and frequency domain (e.g., with an FFT algorithm), i.e., by using a split-step Fourier algorithm with small step size, since the problem occurs already for the interaction of the tiniest volume of gain material with the optical pulse.

We first review some previous work in that context. G. P. Agrawal [1] introduced a method based on a propagation equation in the time domain (Eq. (11)), which can naturally contain a time-dependent gain g(t). The time-dependent gain is calculated (Eq. (10)) as

Yahel et al. [3] used a completely different approach. Here, the pulses have been described with time-frequency distributions, i.e. on two-dimensional grids involving both the time and frequency dimension, so that simultaneously time- and frequency-dependent gain can be included. That kind of description has been taken over from radiative transfer theory, which does not take into account coherent effects. Given that ultrashort pulses are in most cases highly coherent, and that they must of course obey the time-frequency uncertainty (which however is not guaranteed by the equations of motion used for time-frequency distributions), it is not clear in which cases the validity of results can be trusted.

In [4], an again totally different approach was taken for simulating chirped-pulse amplification in fiber amplifiers. Essentially, short temporal sections of the amplified pulse are considered as quasi-monochromatic, so that simple rate equations can be applied. It is not completely accurate to consider only the instantaneous frequency of the pulse, effectively ignoring e.g. the finite spectral width even of an unchirped pulse, but the accuracy may well be sufficient for typical chirped-pulse amplifiers. However, the method would not be usable for a general-purpose numerical calculator, as it requires quite specific assumptions on the amplified pulse.

Another possibility would be to use Maxwell–Bloch equations. Here, the full microscopic dynamics of the gain medium can in principle be included. However, that approach is hardly practical for typical cases of solid-state bulk or fiber lasers, where a huge number of laser-active ions are interacting with the laser pulse in complicated ways, which one would certainly prefer not to model in detail on a microscopic level.

2. New simulation method

It appears that none of the previously described simulation methods would have been suitable for an all-purpose simulation software [5], which is expected to work with virtually arbitrary light pulses and shapes of gain spectra. A rate equation model for the gain dynamics (rather than Maxwell–Bloch equations, for example) must be used, because that is not overly complex and suitable for the usually available spectroscopic data (transition cross-sections and upper-state lifetime(s)), and ultrashort pulses must be numerically represented by one-dimensional complex arrays in the time or frequency domain rather than by time–frequency distributions, for example, in order to limit memory consumption and computation time, and also to safely avoid inconsistences (e.g. violations of time-frequency uncertainty). Properly taking into account the wavelength dependence is indispensable – not only concerning the light amplification, but also the calculation of gain saturation, as the saturation fluence can be vary substantially within the optical spectrum of a pulse. Therefore, a new simulation method has been developed, which is described in the following.

As usual, the used pulse propagation algorithm involves the repeated propagation of pulses through short segments of an optical fiber, for example, where each segment is automatically (depending on essential pulse parameters, local gain etc.) ensured to be short enough such that the amplifier gain, dispersive effects and nonlinear phase shifts per propagation step are reasonably weak. The split-step-Fourier method is used, i.e., effects like self-phase modulation are treated in the time domain, while effects like chromatic dispersion are treated in the frequency domain; fast Fourier transforms (FFTs) are used to regularly switch between those domains. With conventional methods, one would have to decide between either applying the laser amplification in the time domain – properly describing gain saturation but not the gain spectrum – or in the frequency domain – obtaining the proper frequency dependence but no time-dependent gain due to saturation. The new method for simulating the gain effects, however, is integrated into the split-step algorithm as one of the time-domain effects, although it also uses the frequency domain internally. Note that we do not need to integrate e.g. dispersive and nonlinear effects into our algorithm for laser gain, since such effects can be applied separately before or after treating the gain.

We first consider the basic principle of treating the laser gain before addressing important technical details. First, the pulse in the time domain is divided into multiple slices, with their number determined such that the energy in each slice is small enough to saturate the gain only weakly (e.g., by up to 5%). Each temporal slice is then subject to an FFT to get to the frequency domain, where the frequency-dependent gain – including gain saturation – by that temporal slice is applied, and finally the slice is transformed back to the time domain. Finally, the resulting processed slices are combined to obtain the complete trace of temporal amplitudes of the pulse after one numerical step.

An important detail is that instead of using a simple “cutting” in the time domain, the slices of width T are generated by multiplication of the full temporal pulse trace with a “soft” window function of the form

Due to the “soft” windowing, the effectively obtained time dependence of the gain is smooth rather than occurring in discrete steps, which would constitute disturbing artifacts.

The gain saturation has been implemented in two different versions. The simpler one is used when the overall model is based on a simplified gain model, containing only a single metastable level (the upper laser level) of the laser-active ions. In that case, the change of excitation level caused by each spectral amplitude can be calculated easily, using pre-computed wavelength-dependent saturation energies. (In extreme spectral regions, where transition cross sections are small, large saturation energies lead to correspondingly weak saturation.) In case of a more sophisticated gain model, allowing for arbitrary (user-defined) metastable levels and transitions between those, more sophisticated calculations are required, possibly involving multiple transitions for one wavelength, each one having different saturation characteristics. Although that is not simple to do, it is noteworthy that the chosen modeling approach is not limited to a simplified rate equation model.

For any kind of gain model, the gain saturation caused by one temporal slice (i.e., the local reduction in excitation density) may be applied either after applying the whole gain spectrum to the slice, or each time after a single spectral component has been amplified. (Note that in any case this does not lead to substantially different results because the optical energy per slice is limited, as mentioned before.) The latter method has been selected; here, the order in which spectral components are processed (i.e., from lower to higher frequencies or vice versa) has some influence on the result. For minimizing numerical artifacts, that order can be reversed e.g. after each spatial propagation step. With that approach, one probably obtains overall smaller numerical errors for a certain numerical step size, but the computation time per step is a bit longer.

Another detail to be observed is that the chosen window function attenuates the energy in one slice by the factor 3/8 (comparing with a rectangular window of width T). On the other hand, as each slice effectively represents only a time interval of width T / 2, the attenuation factor with respect to that spacing is 3/4. Therefore, the effective optical powers for gain saturation have to be increased by the factor 4/3 in order to obtain the correct strength of saturation.

3. Example case

As an example case, we consider the amplification of a Gaussian 1.5-ns pulse with 15 µJ energy, a chirp of + 4 GHz/ps and a center wavelength of 1030 nm in a fiber amplifier. Without the strong chirp, the pulse duration would be ≈80 fs. The amplifier consists of a 0.8-m long double-clad fiber with 10 µm core radius and 10 µm mode field radius. It is initially brought to an Yb excitation state of roughly 50% with a 975-nm pump source, such that a gain of 15.5 dB at 1030 nm is reached. The model, implemented with the RP Fiber Power software [5] which uses the described algorithm, used time and frequency traces with 2¹⁷ complex amplitudes, divided into 2 ∙ 8 + 1 = 17 overlapping temporal slices, each being 0.5 ns long. (The total length of the time trace is 8 ∙ 0.5 ns = 4 ns.) It shows amplification of the pulse to 172 µJ, which is ≈1.7 times the saturation energy of 103 µJ at 1030 nm. Figure 1 shows the output power vs. time in comparison to the hypothetical case without gain saturation. Figure 2 shows the spectrum of the output pulse. From both figures one can see that the long-wavelength components, passing the fiber first due to the up-chirp of the pulse, experienced a gain similar to the initial gain, while the short-wavelength components saw an already saturated gain. These results fully agree with what would be qualitatively expected. Also note that due to the “soft” windowing used in the algorithm, the reduction of gain is continuous rather than step-wise.

 

Fig. 1 Temporal evolution of output power and wavelength in an example case. Then shown wavelength is calculated from the instantaneous frequency, showing the up-chirp of the pulse.

Download Full Size | PDF

 

Fig. 2 Optical spectrum of the output pulse.

Download Full Size | PDF

The slight wiggles on the spectrum of the output pulse (Fig. 2) are not caused by the used algorithm; they resulted from strong self-phase modulation in conjunction with the not perfectly smooth temporal shape of the pulse, which itself is related to the shape of the gain spectrum. When artificially omitting self-phase modulation in the simulation of propagation in the fiber, the wiggles disappear; they reappear in similar form when the pulse is thereafter sent through an optical element with self-phase modulation only. These tests confirm that the wiggles are not a numerical artifact, but a true feature of the system.

In principle, the chosen method would reach a limitation if an extremely high peak power of the pulse would require extremely short temporal slices in order to avoid strong gain saturation within one slice. That would become problematic if the slices were so short that the corresponding frequency resolution becomes too coarse. The phase relaxation time of the laser-active ions in the model might then effectively be assumed to be shorter than it actually is. However, that would happen only for a slice length well below 1 ps, as the amplifier gain usually does not vary much within 1 / 1 ps = 1 THz of optical frequency. Indeed, the “temporal memory” (the phase relaxation time) of typical solid-state gain media is below 1 ps. On the other hand, even an extremely high peak power of several megawatts, close to the threshold for catastrophic self-focusing in a typical optical fiber, would imply only a few microjoules within 1 ps, and that is still well below typical values of the gain saturation energy. Normally, the peak power even has to be kept much lower than a megawatt in order to avoid excessive pulse distortion by self-phase modulation. Therefore, the discussed limitation is hardly of practical relevance for solid-state gain media. There might be problems, however, for laser dyes, having far higher emission cross sections; such gain media exhibit gain saturation already at much lower pulse fluences.

Note that other limitations for such models (not related to the new method) may arise from the typically used rate equation models, which neglect the finite (but often not accurately known) relaxation time within Stark level manifolds of the gain system. However, at least the power limitations for optical fibers in combination with typical gain saturation energies again mean that this type of limitation will rarely be relevant.

4. Conclusions

In conclusion, a new kind of numerical model has been described, which can simulate the amplification of intense ultrashort pulses (possibly stretched to nanosecond durations), where both the full spectral dependencies and the time-dependence of the amplifier gain due to substantial gain saturation can be properly described. It has been shown that a fundamental limitation of the used algorithm, related to the limited spectral resolution associated with short temporal slices, is not relevant for practical cases at least in the context of optical fibers. The algorithm is well suited particularly for simulations in the context of chirped-pulse amplifier systems.