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.
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.
We found that there is a different effect on the output noise, depending if the input noise is caused from the pump or the seed source. Figure 8a shows the simulated output noise as a function of RT for three different combinations of pump and seed noise. Two ranges of operation can be distinguished:
-
(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.
In the following, we focus on the study of pulse amplification at HSPs. As seed noise can be neglected here, all simulations were conducted with pump noise only.
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.
By evaluation of the noise curves as a function of the gain depletion, as shown in Fig. 9b, one finds that the HSPs are located at specific gain depletion values, at higher depletions for lower losses. The output fluence in Fig. 9b was normalized to the total fluence that is stored in the gain material after the pump cycle with
$$\begin{aligned} J_\mathrm{norm} = \frac{J_\mathrm{out}}{J_\mathrm{stor}}. \end{aligned}$$
(5)
The interpretation is straightforward. For the loss-less system, the normalized output fluence increases linearly with the gain depletion and reaches a value of 1 for a gain depletion of 1. In this case, the normalized output fluence is a direct representation of the gain depletion. On introducing losses, this direct representation is not valid anymore. In this specific case, simulating single-pass losses of 10 and 15% leads to a maximum normalized output fluence of 0.51 and 0.39, respectively. This means that although most of the stored energy is extracted (represented by a high-gain depletion), a significant fraction of the stored energy (49 and 61%) is lost and dissipated into the environment.
Figures 10 and 11 further investigate the accessibility and characteristics of HSPs. Figure 10a shows the normalized output noise and output fluence as a function of the pump intensity, for three different RTs. It demonstrates that for a decrease in the RT, the HSP shifts to higher pump intensities. Most importantly, we observe that for an increase in the pump intensity, the RA can produce output pulses with a higher stability (lower output noise at the corresponding HSPs).
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.
Rather counter-intuitively, when operating at RT \(=\) 17 instead of RT \(=\) 15, the output fluence and the normalized output fluence both increase by approximately 1.5% despite more than an order of magnitude less seed fluence (at the corresponding HSP). This is rather unexpected, as one would assume that an increase in RT leads to a decrease in the output pulse fluence due to an increase in accumulated losses. However, we believe that this observation is valid as long as the seed fluence is small compared to the total stored fluence in the gain medium and other loss channels (such as ASE or spontaneous inversion decay during pulse amplification) are negligible. For high-gain RAs with long-lifetime gain materials, such as for Ho:YLF, this precondition is fulfilled in most cases. The above observation can be made traceable as follows:
-
(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.
Measurements
We operated our Ho:YLF RA [6] in regime ① (at a repetition rate of 100 Hz) and measured the output noise and the output pulse energy as a function of various parameters. The RA was seeded with pulses from an Ho:fiber oscillator [20] that produced pulses centered around 2053 nm with a bandwidth of 7 nm. The pulses were stretched with a chirped volume Bragg grating prior seeding. The RA was pumped with a commercial CW Tm:fiber laser that produces up to 120 W of randomly polarized light. Unfortunately, the output power of this laser could not be tuned in fine steps. The measurements presented in Fig. 12a and b, however, required a fine tuning of the pump power. This was realized by a polarization of the pump laser output and by usage of a further polarizer and \(\lambda\)/2-waveplate. The measurements that did not require a variation of the pump power, in Fig. 12c and d, were conducted with 18 W of the unpolarized pump laser output. To avoid water condensation at low crystal temperatures, the complete RA setup was purged with a constant flow of dry air, keeping the humidity level <2%. We observed that purging the RA cavity also significantly improved the stability of the amplified output pulses, related to an improved pump laser stability under dry atmosphere (see Appendix 5, Fig. 14e). If not specified otherwise, the RA was seeded with pulse energies of 690 pJ and the temperature of the thermo-electrically controlled Ho:YLF crystal holder was kept at 18 \(^\circ\)C. Further details about the experimental Ho:YLF RA setup can be found in Appendix 5.
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.
We observed that the minimum noise values measured with linearly polarized light were higher by up to a factor of 3 than the ones measured with randomly polarized light. These noise values, however, cannot be directly compared as the actual noise from the pump laser might be different for the two cases. Unfortunately, no noise values for the unpolarized and polarized pump laser output were available, but we expect two circumstances to contribute to the increased RA output noise observed:
-
(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.
Figure 10e–g shows recorded oscilloscope traces for the intra-cavity pulse build-up typical for operation at a noise minimum and for operation before and after the minimum (with a less or more depleted gain), respectively. These operation points were accessed by controlling the crystal holder temperature while keeping RT fixed. We observe that for T \(=\) T1 (in Fig. 12e), the output pulses exhibit the lowest noise, whereas the pulses exhibit higher noise for T < T1 and T > T1 (in Fig. 12f, g).