# Study on laser characteristics of Ho:YLF regenerative amplifiers: Operation regimes, gain dynamics, and highly stable operation points

## Abstract

We present a comprehensive study of laser pulse amplification of Ho:YLF regenerative amplifiers (RAs) with respect to operation regimes, gain dynamics, and output pulse stability. The findings are expected to be more generic than for this specific gain material. Operation regimes are distinguished with respect to pulse energy and the appearance of pulse instability, and are studied as a function of the repetition rate, seed energy, and pump intensity. The corresponding gain dynamics are presented, identifying highly stable operation points related to high-gain build-up during pumping and high-gain depletion during pulse amplification. Such operation points are studied numerically and experimentally as a function of several parameters, thereby achieving, for our Ho:YLF RA, highly stable output pulses with measured fluctuations of only 0.19% (standard deviation).

## 1 Introduction

The first experimental and numerical work dealing with the onset of bifurcation and chaotic output pulses was published by Dörring et al. [3] for a Yb:glass RA. The susceptibility of RAs to show unstable output pulses was connected to the repetition rate of the RAs and the excited state lifetime of the gain medium (\(\tau _\mathrm{gain}\)). For repetition rates on the order of the inverse lifetime 1/\(\tau _\mathrm{gain}\), RAs can show unstable output pulses.

It was later shown, but not quantified, that the seed energy also affects the onset of bifurcation, and that higher seed energies allow operation at higher repetition rates without the presence of bifurcation [4]. The first experimental hint that there is an inherently stable RT beyond the bifurcation instability, as illustrated in Fig. 1 at high RTs, was demonstrated in [3]. This was more precisely studied for an Nd:YVO\(_4\) RA, showing bifurcation for a large range of RTs, prior to the start of further inherently stable RTs [5]. More recently, this was demonstrated, for the first time, for an Ho:YLF RA [6], achieving high pulse energies by overcoming bifurcation instability at a repetition rate of 1 kHz. Furthermore, the measurements and simulations showed that the output pulse stability is highest at a specific RT close to the maximum average output pulse energy, independent of the presence of bifurcation in the system at an earlier RT [6].

RAs lasing at a wavelength of 2 \(\upmu\)m and based, for example, on Ho:YLF, Ho:YAG [7], and Tm:YAP [8] have recently garnered attention as a viable driving source for mid-IR optical parametric amplifiers (OPAs) [9]. They can be used in conjunction with highly efficient non-oxide non-linear crystals (such as ZnGeP2 [9, 10]), as opposed to the more conventional 1-\(\upmu\)m pump sources. Unfortunately, Ho:YLF RAs, with an upper state lifetime of about 15 ms (or equivalently, an inverse lifetime of \(\sim\)67 Hz), greatly suffer from the onset of bifurcation at the demonstrated repetition rates. Typically, Ho:YLF RAs were either operated at relatively low repetition rates [9], thus completely suppressing the onset of bifurcation, at high repetition rates until the onset of bifurcation [11], or directly in the bifurcation at a stable double-pulsing state [12]. In the latter case, pulse-picking only the higher energy pulse at half the repetition rate allows the extraction of stable and high-energy pulses [12, 13]. By employing high pump intensities, we demonstrated operation of our Ho:YLF RA up to repetition rates of 750 Hz without any sign of bifurcation, which is more than an order of magnitude higher than the inverse lifetime [6]. This indicates that the pump intensity also plays a crucial role in the onset of bifurcation.

The numerical models that have been reported and used for the simulation of RAs either focused on a rather qualitative understanding of pulse instability and the underlying gain dynamics [3, 4], or on the reproduction of concrete laboratory measurements [11, 14, 15]. In the first case, rather simplistic mathematical expressions were used, neglecting for example an explicit dependence of the pump intensity on the gain build-up. In the second case, more complex and computationally intense numerical models were employed. Typically, they consist of a split-approach, simulating the pulse amplification with Frantz–NodvikFN equations and the gain build-up with rate-equations. RA operation was then considered as a function of few parameters in a rather limited parameter space. Hence, the reported findings provide, in both cases, only a partial understanding of the onset of pulse instability and the corresponding gain dynamics. As shown in Fig. 1, it is essential that any numerical model used to study pulse stability of RAs includes the effect of pump noise. Notably, there are no numerical studies for pulse amplification in RAs that include this noise source, to the best of our knowledge.

In this paper, we present a comprehensive, mostly numerical, study of pulse amplification with continuous wave (CW)-pumped Ho:YLF RAs in regards of operation regimes, gain dynamics, and output pulse stability. The simulation model is completely based on modified and computationally fast FN equations, allowing for an efficient variation of RA parameters in a large parameter space.

The simulations distinguish four different RA operation regimes exhibiting different output pulse characteristics as a function of the pump intensity, repetition rate, seed fluence, and pulse gain. These operation regimes represent inherent limitations in RA amplification in terms of stable output pulse energy and average output power. We analyze the onset of bifurcation and we empirically find that for high-gain RAs, the repetition rate at which bifurcation appears scales linearly with the pump intensity and follows a power law with the seed energy. We furthermore present the RA gain dynamics characterized by the gain build-up during the pump phase, and gain depletion during the pulse amplification. Upon adding pump noise to the simulations, we identify an operation point that offers the highest output pulse stability, in the following called high stability point (HSP), located at a high-gain build-up and high-gain depletion.

We analyze such HSPs numerically and experimentally as a function of various RA parameters. We thus achieve with our laboratory Ho:YLF RA highly stable output pulses with measured energy fluctuations of only 0.19% (standard deviation). Although this study was conducted with an Ho:YLF RA, the results are considered more generic and provide a more complete and general understanding of pulse amplification in RAs, independent of the gain material.

The paper has been structured in the following manner: Sect. 2 briefly describes the simulation model that is used for this study. In Sect. 3, the different RA operation regimes are distinguished and discussed, and the onset of bifurcation is studied as a function of the pump intensity and seed fluence. Section 4 describes the gain dynamics for the operation regimes and identifies HSPs. Analogous HSPs are studied numerically and experimentally as a function of various RA parameters in Sect. 5.

## 2 Numerical simulation model

This chapter summarizes the general simulation procedure used for the simulations in this paper. The model simulates consecutive pump and pulse amplification cycles, which allows a statistical analysis of a large ensemble of amplified output pulses. The model is sketched out in Fig. 2. At the beginning of the simulations, a random value for the initial inverted fraction \(\beta\) in the laser crystal can be used as a starting value. During a pumping cycle, the inverted fraction increases to the inverted fraction \(\beta _\mathrm{p}\), which is then fed into the consecutive amplification cycle as the starting value. Then, during the amplification cycle, the inverted fraction decreases to the inverted fraction \(\beta _\mathrm{a}\), which is again fed into the consecutive pumping cycle, and so on.

Noise originating from the pump and seed source is included by a Gaussian distributed variation of the pump and seed fluences for consecutive pump and amplification cycles. In this paper, we refer to the noise of the pump and seed source as the standard deviation of consecutive pump and seed fluences in percent of the corresponding mean fluence. Equivalently, the standard deviation of the output pulse fluence in percent of the mean output fluence is used as a measure for the output noise. The simulation framework was already presented in detail in [16], where the model was spectrally generalized and used to study spectral shaping effects in Ho:YLF RAs. Here, we used the monochromatic version as it is computationally less expensive. The equations used for this work can be found in Appendix 1, along with further simulation details.

Although the simulation model is quite simplistic, it captures the essential physics necessary for this study. The model does not consider, for example, temporal or spatial effects, nor amplified spontaneous emission (ASE), up-conversion, and thermal and non-linear effects. ASE could, for example, limit RA operation as it could cause a cavity-dumped background when low seed pulse fluences and/or a high number of RTs are used.

### 2.1 Analysis of the gain dynamics

To analyze the gain dynamics of RAs, the simulation results are analyzed in terms of the normalized gain *g*\(_\mathrm{p}\) and the gain depletion *g*\(_\mathrm{depl}\). The normalized gain *g*\(_\mathrm{p}\) represents the amount of stored energy in the gain medium after the pump process. It can have values between 0 and 1, where 0 represents no and 1 represents the maximum possible energy stored in the gain medium. The gain depletion *g*\(_\mathrm{depl}\) represents the amount of extracted energy during the pulse amplification. It can have values between 0 and 1, where 0 represents no and 1 represents complete energy extraction. The equations used to calculate *g*\(_\mathrm{p}\) and *g*\(_\mathrm{depl}\) can be found in the Appendix 3.

## 3 Operation regimes of regenerative amplifiers

This chapter numerically studies pulse amplification in four characteristic operation regimes as a function of the repetition rate, pump intensity, seed pulse fluence, and pulse gain. The regimes were generally observed in experiments, for example, in [5, 19], but there has not been a study of these regimes as a function of the pump intensity, repetition rate, and seed fluence.

*I*\(_\mathrm{pump}\) as the pump intensity,

*J*\(_\mathrm{seed}\) as the seed pulse fluence and b as a fitting parameter. The relation suggests a linear dependence of BT with the pump intensity and a dependence on the seed fluence that follows a power law. To verify this assumption, we fitted the simulation results from Fig. 4a with

*f*\(_1\) and

*f*\(_2\) represent the dependency on the pump intensity and the seed fluence with

*m*,

*n*,

*c*,

*d*, and

*b*as fitting parameters. The fitted BT curves are also shown in Fig. 4a, with the solid lines. The fitted parameter values for

*m*,

*n*,

*c*,

*d*, and

*b*are listed in Table 1. We find good agreement for the shown seed fluences that are much smaller than the total stored fluence in the gain medium, in this particular case, for seed fluences <1000 nJ/cm\(^2\). Figure 4b presents the operation regimes ①–④ for a pump intensity of 15 \(\times\) PT, now for a larger range of seed fluences and repetition rates. The repetition rate was normalized with the inverse excited state life time of the gain medium 1/\(\tau _\mathrm{gain}\) and the pump intensity (in multiples of PT). We observe that for seed fluences >1000 nJ/cm\(^2\), the simulated results for BT and the fitted curve (orange dashed line) start to deviate. For seed fluences exceeding a certain cut-off value (here 10\(^6\) nJ/cm\(^2\)), no pulse instability appears anymore (regime ②), independent of the repetition rate.

*cut-off gain*below which no pulse instability exists anymore (at any RT), observed for gains \(\lesssim\)100. Similar to BT as a function of the seed fluence, we found that BT as a function of the gain also follows a power law.

### 3.1 Typical output, normalized gain, and gain depletion values

*g*\(_\mathrm{p}\) and the gain depletion

*g*\(_\mathrm{depl}\), demonstrating that there are gain dynamics at play that are characteristic for the regimes ① to ④. The regimes are separated in the figure with the thin vertical dotted lines.

Regime ① is characterized by a saturated normalized gain (blue solid line) and high-gain depletion (green dotted line), as shown in Fig. 4b. This means that a high fluence is stored in the gain medium during the pump process and most of it is depleted in the following amplification process. Consequently, high output fluences (black dashed-dotted line) can be extracted (Fig. 4a). Throughout regime ①, the output fluence is saturated and remains nearly constant with an increase in the repetition rate, while the average output intensity (red dashed line) increases almost linearly.

In regime ②, the pump intensity is no longer sufficient to saturate the normalized gain during the pump processes and, consequently, the output fluence decreases (black dashed-dotted line). Here, we observe the onset of bifurcation. The peak in the average output intensity (red dashed line) may represent the ’best’ compromise between high output fluence extraction and high average output intensity. The transition between regimes ② and ③ is marked by a sudden step in the normalized gain (blue solid line) and gain depletion (green dotted line). This corresponds to a jump of the yellow star from an RT *after-pulse instability* in Fig. 3c to an RT *before-pulse instability* in Fig. 3d. Consequently, less gain is depleted and the normalized gain after the successive pump processes increases.

The operation regimes ③ and ④ are characterized by low gain depletion and, consequently, low output fluence, as only a small amount of the stored fluence is depleted. We observe that the output fluence (black dashed-dotted line) decreases almost logarithmically with the repetition rate. At the beginning of regime ③, the average output intensity (red dashed line) is low but increases with an increase in the repetition rate. This is caused by the shift of the yellow star in Fig. 3d towards the peak of the average pulse energy (red line in Fig. 3e) as the repetition rate increases. In regime ④, the normalized gain (blue solid line) stays nearly constant, while the average output intensity (red dashed line) reaches its saturated maximum.

## 4 Gain dynamics and high stability points HSPs

The focus of this chapter lies on, first, exposing the actual gain dynamics during RA operation and, second, the identification of operation points with highly stable output pulse energies. The procedure to unravel the gain dynamics from the direct simulation results is explained in Figs. 6a and b. RA amplification was simulated as a function of the pump intensity and RT, and the corresponding output noise of consecutive output pulses was calculated, represented in Fig. 6a by the color code. There are areas that are inherently stable (in deep blue) and areas that show pulse instability (all other colors). The operation regimes ①–④ are marked accordingly in the figure.

For each value pair of RT and pump intensity (meaning for each pixel in Fig. 6a), the corresponding average normalized gain *g*\(_{p}\) and gain depletion *g*\(_\mathrm{{depl}}\) was calculated (in accordance to the equations in Appendix 2) based on the inversion build-up and depletion recorded during the simulations. Figure 6b presents the noise values from Fig. 6a, now redrawn as a function of the corresponding gain values *g*\(_{p}\), and *g*\(_\mathrm{{depl}}\), and hence, highlights the underlying gain dynamics. The spiky appearance of the border between the stable and the unstable area is an effect of the RT that can only be varied in the simulations in multiple integers of a single RT.

Figure 6b–d demonstrates the effect of the seed fluence on the gain dynamics and the borders of the operation regimes. For an increase of the seed fluence from 1000 nJ/cm\(^2\) (in Fig. 6b) to 10\(^5\) nJ/cm\(^2\) (in Fig. 6c), the inherently stable areas (in deep blue) grow. Contrary, for a decrease of the seed fluence to 10\(^{-1}\) nJ/cm\(^2\) (in Fig. 6d), the inherently stable areas shrink. For the white areas in Fig. 6b–d, the simulation did not produce values.

In general, the ranges producing inherently stable output pulses are characterized either by low gain depletion (little of the stored fluence extracted from the gain medium) or by high-gain depletion (most of the stored fluence extracted).

The simulations presented in Fig. 6 were conducted without pump noise and, hence, demonstrate the inherent susceptibility of an RA to show pulse instability. In the next step, the simulations of Fig. 6a and b were repeated, but now including noise from the pump source (simulated with 3% standard deviation). The general RA gain dynamics are not affected by this rather small-scale pump intensity fluctuation, however, only by the inclusion of pump noise, the emergence of a specific highly stable operation point can be observed.

Figure 7a presents the results of the repeated simulations from Fig. 6a, including pump noise. The areas that were inherently stable in Fig. 6a (in deep blue) now also exhibit a certain amount of output noise. Notably, there is a specific area in Fig. 7a, labeled with high stability point (HSP), where the output noise is minimum (in deep blue). Figure 7b demonstrates the corresponding gain dynamics. It demonstrates that the observed HSP is localized at a specific gain depletion value (in this case, *g*\(_\mathrm{{depl}}\) \(\sim\)0.85). The exact position mainly depends on the cavity losses and pump intensity, which will be discussed in Sect. 5. It is important to note, that the HSP can only be accessed if the pump intensities are sufficiently high to drive the RA into the operation regimes ① or ②.

## 5 Noise optimization at high stability points (HSPs)

The previous Sect. 4 demonstrated that the HSP corresponds to operation with high values of normalized gain and gain depletion (Fig. 7a, b). It seems plausible provided that sufficient pump intensity is employed to drive the system to high normalized gains *g*\(_\mathrm{p}\), any parameter that allows for an adjustment of the gain depletion *g*\(_\mathrm{{depl}}\) should in principle allow to access the HSP.

This section presents both numerical and experimental results for an Ho:YLF RA that was optimized for operation at an HSP. In Sect. 5.1, the noise at the HSP is studied numerically as a function of the RA parameters RT, single-pass losses, pump intensity, and seed fluence. In Sect. 5.2, laboratory measurements for operation at an analogous HSP are presented as a function of the parameters RT, pump intensity, seed fluence, and crystal holder temperature.

As a measure for the output noise, we use as before the standard deviation of the output pulse fluence in percent of the mean output fluence (for the simulations), or an equivalent expression in terms of energy (for the measurements). The numerical and the experimental Ho:YLF laser were both operated in regime ①; however, they differ in their operational parameters. Consequently, the measurements focus on qualitative agreement with the simulations.

### 5.1 Simulations

The simulation approach is analogous to the approach in Sect. 4 to study the gain dynamics. RA operation was simulated as a function of various parameters and post-analysis of the simulation results yielded in the underlying gain dynamics.

- (1)
For up to RT = 13, the gain is not yet depleted and the pulse energy increases with an increase in RT. Here, both the pump and the seed noise contributes to the output noise.

- (2)
Once the RA approaches the maximum pulse energy, the effect of the seed noise (green curve in Fig. 8a) becomes small compared to the effect of the pump noise (black curve in Fig. 8a). In this range, the pump noise completely dominates the output noise, i.e., it becomes almost identical to that for the RA exhibiting both pump and seed noise (black and red line in Fig. 8a).

For the local maxima (RT \(=\) 15) and minima (RT \(=\) 18) of the output noise in Fig. 8a, the output noise was found to be linearly dependent on the input pump noise, as shown in Fig. 8b. Because of the observed linear dependence, all output noise values are normalized in the following with the input pump noise. The resulting ’*normalized output noise*’ can be considered as the noise-suppression factor by which the input noise is suppressed in the output noise. A normalized output noise of 1 represents equal input pump and output noise. Figure 8a, b emphasizes that to realize an RA that produces a highly stable output pulse energy, there is, besides operation at an HSP, also a need for a low-noise pump source. Furthermore, for the extraction of pulse energies close to gain depletion, a low-noise seed source is of less significance than a low-noise pump source.

Figure 9a demonstrates the physical origin of HSPs. Figure 9a shows simulation results as a function of the pump intensity, for the three different single-pass losses 0, 10, and 15%, for operation at a fixed RT (simulated at RT = 19). Notably, the simulations were conducted as a function of the pump intensity and not as a function of RT. In terms of accessing an HSP, both parameters can be used, but only the pump intensity can be varied in sufficiently small steps to resolve the noise fine structure, as will be further emphasized in Fig. 10a and b. Figure 9a demonstrates that for a system without losses, no intermediate HSP is present. For increasing losses, the HSP becomes more and more pronounced. Consequently, the maximum output pulse fluence decreases with an increase in the losses.

These curves are redrawn in Fig. 10b as a function of the gain depletion, with the output fluence normalized according to Eq. (5). When comparing Fig. 10a and b, we find that for higher pump intensities, the HSPs are located at higher gain depletions and provide an increased output fluence and normalized output fluence. We assume that the main reasons for the increase in the output fluences are twofold. They are, first, due to an increase in the stored fluence in the gain medium (due to the higher pump intensity) and, second, due to a decrease in the accumulated losses (due to the lower RT).

In the following, we try to quantify these contributions by comparing to output fluence in Fig. 10a and the normalized output fluence in Fig. 10b. While the output fluence is affected by the pump intensity and the accumulated losses, the normalized output fluence is mainly affected by the accumulated losses alone (due to its normalization to the stored fluence in the gain medium). For operation at an HSP, we observe that the maximum normalized output fluence increases in Fig. 10b by \(\sim\)8% (comparing RT \(=\) 17 with RT \(=\) 19). In comparison, the output fluence in Fig. 10a increases by \(\sim\)18%, slightly more than twice as much. Hence, we conclude for this specific RA that the 8% increase is contributed by the decreased accumulated losses and the 10% (the difference between the 18 and the 8%) is contributed by the increased stored fluence.

Analogous to Fig. 10, Fig. 11 shows results for RA operation as a function of the seed fluence for three RTs. Figure 11a presents the normalized output noise and output fluence, and Fig. 11b evaluates and presents the same data as a function of the gain depletion. Analog to the results in Fig. 10, increasing RT shifts the HSP to a decreased seed pulse fluence.

- (1)
Considering a loss-less system, the stored fluence in the gain medium represents the maximum output pulse fluence that can be extracted. Unlike higher pump intensities, higher seed fluences do not change the stored fluence in the gain medium.

- (2)
When losses are included, the extractable output fluence is lowered by the accumulated losses. The simulations suggest, however, that a lower or higher seed fluence does not change the absolute amount of losses. This can be seen in Fig. 11d, where the lines for the normalized output fluence for the different RTs overlap each other.

- (3)
From Fig. 11d, it can be seen for higher RTs that the output fluence at the corresponding HSP shifts closer to the maximum of the output fluence curve. This leads to an increase in the output fluence for lower seed fluences if the RA is operated at the corresponding HSP. This behavior agrees with measurements as well, which are discussed in Fig. 12c.

### 5.2 Measurements

The presented pulse energies correspond to the uncompressed output, measured with a commercial calibrated energy meter. For each measurement, 3000 consecutive output pulses were recorded (30 s measuring time at a repetition rate of 100 Hz) and analyzed with respect to pulse energy and output noise. Figure 12a presents the measured output noise and output pulse energy as a function of the pump power for RT \(=\) 12 and RT \(=\) 14. The minimum noise values are 0.76 and 0.95% for 12 and 14 RTs, respectively. As predicted by the simulations in Fig. 10a, for an increase in the pump power, the amplified output pulses can be coupled out at an earlier RT. At the corresponding HSPs, the output noise decreases and the output pulse energy increases for an increase in the pump power. In this particular case, the stability was improved by \(\sim\)20% and the pulse energy increased by \(\sim\)50%. The range of pump powers for operation at an HSP can be quite narrow. This is demonstrated in Fig. 12a with the two points P1 and P2, representing, for RT = 12, the output noise for the pump powers 15.7 W and 15.5 W, respectively. On a reduction of the pump power by only 1.3% (from P1 with 15.7 W to P2 with 15.5 W), the output noise increased by \(\sim\)100 %, from 0.76 to 1.49%.

Figure 12b demonstrates the output noise and output energy as a function of RT, for the pump powers 15.7 W, 15.1 W and 14.1 W. The two points P1 and P2 represent the same points than in Fig. 12a. The figure emphasizes that, while for operation at 15.7 W, the lowest noise HSP can be accessed for RT \(=\) 12, for a pump power of 15.5 W, there is no RT with a similar low output noise. The reason is that due to the function principle of RAs, the RT can only operated with multiple integers of a single RT, and hence, cannot be fine-tuned. Consequently, the lowest HSP cannot be accessed anymore if the pump power is slightly off its optimum value and RT is the only other parameter used to control the RA.

Figure 12c studies the effect of the seed pulse energy on the output noise and the output energy for different RTs. As suggested by the simulations in Fig. 11a, we observe a slightly lower noise at the HSP for an increase in the seed energy, at a lower RT. Furthermore, the output energy at the corresponding HSPs shifts closer to the maximum of the corresponding output pulse energy curve. This leads to the counter-intuitive effect that at the corresponding HSPs, the output pulse energy increases despite lower seed energy. In particular, for operation at the HSPs, we measured noise values of 0.24 and 0.27% for the seed pulse energies of 280 and 19 pJ, with corresponding output pulse energies of 2.45 and 2.5 mJ, respectively. In this case, operation at the higher RT resulted in a stability improvement of 11%. At the same time, despite more than an order of magnitude more seed energy, the amplified output energy decreased by 2%.

During operation with the Ho:YLF RA, we experimentally discovered that a variation of the Ho:YLF crystal holder temperature allows a noise optimization similar to the optimization described in dependence of the pump power and seed energy. Figure 12d shows the output noise and the pulse energy as a function of the crystal holder temperature for RT \(=\) 11 to RT \(=\) 14. Analog to the previous noise curves in Fig. 10a–c, there is a distinct noise minimum, in this case at a specific crystal holder temperature. For a decrease in the crystal holder temperature, we also observe a significant increase in the output pulse energy. With a decrease in the crystal holder temperature from 18 to \(-16~^\circ\)C, the output pulse energy increases almost linearly by a factor of 2.6, from 2.5 to 6.43 mJ.

Currently, the temperature dependence of the emission and absorption cross sections is not implemented in the simulation model. However, based on the observations and findings described in this paper, we propose the following physical interpretation. We observe that with a decrease in the Ho:YLF crystal temperature, the seed pulses experience a stronger amplification, caused by higher absorption cross sections at the pump wavelength and higher emission cross sections at the seed wavelength at lower temperatures [9, 21]. Consequently, the HSPs (optimum gain depletion) is reached with fewer RTs and the pulses can be coupled out earlier. Hence, the output energy increases at lower temperatures due to fewer accumulated losses.

- (1)
The pump laser was operated for Fig. 12a and 12b at 32 W, almost twice the value used for the measurements with random polarization at 18 W in Fig. 12c and d. Therefore, the higher noise level of the RA output could be explained with different noise levels of the pump.

- (2)
The fraction of the polarized light that is used to pump the RA could fluctuate, thus introducing an additional source of pump noise.

## 6 Conclusions

We presented a comprehensive analysis of pulse amplification in Ho:YLF RAs. Although this study was conducted with an Ho:YLF RA, the findings are expected to be more generic and lead to a more complete understanding of RA operation regimes, gain dynamics, and low-noise operation.

The employed numerical simulation model, based on the iterative Frantz–Nodvik(FN) formalism, simulates consecutive pump and amplification cycles. The model also accounts for noise originating from the pump and seed sources. The model is computationally fast, which is important for simulations involving a large number of consecutive pump and amplification cycles. This is crucial for a statistical analysis of results when RA parameters are varied finely or over a large range.

In Sect. 3, we numerically identified RA operation regimes with respect to the repetition rate, pump intensity, seed fluence, and pulse gain. We recognized four different operation regimes that show different output pulse characteristics. In regime ①, for low repetition rates, no bifurcation instability exists. In regime ②, bifurcation appears, but the highest average pulse energy is at an inherently stable RT beyond the bifurcation instability. In regime ③, the highest output energy at an inherently stable RT is limited due to the onset of pulse instability. In regime ④, the highest average pulse energy can be extracted at an RT before-pulse instability appears, and consequently, the instability can be practically ignored under such circumstances. Our simulations showed that the bifurcation threshold (BT), separating regimes ① and ②, shifts to higher repetition rates for increased pump intensities and higher seed fluences. The results indicate that the simplistic assumption that the onset of bifurcation is decided by the inverse lifetime of the gain medium (1/\(\tau _\mathrm{gain}\)) is a limiting case for an RA with low pump intensity and high-gain (low seed fluence). We empirically find for high-gain RA with gains >10\(^6\), a linear proportionality between BT and the pump intensity, and a proportionality following a power law for the seed fluence.

In Sect. 4, we presented a numerical analysis of the gain dynamics in RAs operating in regimes ①–④ with respect to gain build-up and gain depletion in the gain medium. On including noise in the simulations, we observed the emergence of HSPs that allow the extraction of highly stable output pulses with a high pulse energy. This operation point corresponds to a high-gain build-up during RA pump and a high-gain depletion during pulse amplification.

In Sect. 5, we numerically and experimentally studied the output noise and output pulse energy at HSPs. The fundamental cause for the observed HSPs was numerically identified to be the RA losses. This means for every real laboratory RA that is operated in the regimes ① or ② that there always is an HSPs located in close proximity to the highest output pulse energy. Our findings demonstrate that any RA parameter that allows a fine-adjustment of the gain depletion can be used to access and optimize these HSPs. To experimentally study analogous HSPs with an Ho:YLF RA, we varied the pump intensity, seed fluence, number of round trips, and crystal holder temperature. Optimization of these parameters led to highly stable output pulses from our Ho:YLF RA with measured pulse energy fluctuations of only 0.19% (standard deviation). The noise curves show very characteristic noise minima that can be quite narrow (noise increases by 100% for a change in the pump power of only \(\sim\)1.3%). Therefore, the lowest noise operation points can be easily missed if the RA operational parameters are chosen as slightly different from their optimum values, or if they drift with time and cause the RA output noise to move up and down the corresponding noise curves.

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. The authors thank Haider Zia for helpful discussions.

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