1. Introduction

The performance of ytterbium-doped chirped-pulse amplification (CPA) fiber systems has developed at a remarkable pace in recent years. Thus, such systems are nowadays able to produce multi-gigawatt peak powers with pulse energies higher than one millijoule [1]. In order to avoid nonlinear pulse distortions and optically induced damage in these systems, the intensity inside the signal core of the amplifying fiber has to be reduced by many orders of magnitude. This is typically achieved by employing fibers with extremely large mode-field diameters (MFDs) and by using long stretched pulse durations. Nevertheless, state-of-the-art fiber CPA systems are starting to face technological limitations in spite of the combination of these two approaches. On the one hand, scaling the MFD is typically restricted by production tolerances of the fiber design. On the other hand, the geometrical dimensions of the laser system restrict the attainable stretched pulse duration. Whereas passive fibers or small-footprint multi-pass grating stretchers can be employed for pulse stretching, compression usually has to be realized using a single-pass grating compressor in order to achieve high efficiency and to be able to handle high peak-powers in the gigawatt range.

Recently, the performance of femtosecond fiber CPA systems has been further improved by using spatially separated amplifiers followed by coherent combination stage [2–4]. In such systems the laser beam is split into N different channels after passing the stretcher, it is amplified in each channel and finally it is coherently combined before the compression stage. This technique can be understood to provide an artificial increase of the MFD and it allows for increasing both the pulse energy and the average power of a system by a factor of N (in the case of negligible losses). However, since this is an interferometric setup, the path length in each channel has to be stabilized, which can be realized by inserting active phase-controlling elements in the channels. Alternatively, in the case of two channels, an elegant solution is the use of the Sagnac geometry, i.e. both counter-propagating beams traverse the same optical paths and, therefore, they can be passively combined [5].

Another possibility to mitigate nonlinear effects even further is to divide and temporally separate the pulses before amplification and to recombine them afterwards. This approach can be seen as an artificial increase of the stretched pulse duration. The initial proposal of this technique involved interferometric free-space delay lines in a double-pass implementation [6]. In this implementation no active phase-control is required, i.e. it allows for a passive coherent combination of the amplified pulses. A similar technique using birefringent crystals for the separation and combination of the pulses was also proposed in [7,8] and it was referred to as divided-pulse amplification (DPA) for the first time. Recently, DPA has been successfully integrated into a Sagnac-amplifier configuration [9]. However, since the delays attainable with birefringent crystal stacks are of only a few picoseconds, this configuration is just applicable to non-stretched pulses. Consequently, the use of DPA in combination with stretched pulses requires free-space delay lines; something that has already been demonstrated using a double-pass fiber-amplifier implementation, again with passive coherent combination of the amplified pulses [10].

In this contribution we analyze the energy-scaling potential of DPA in passive coherent combining setups. The two geometries considered in this work are depicted in Fig. 1. In both cases, a sequence of cascaded Mach-Zehnder-type splitters/combiners is employed for both division and combination. Pulse splitting and combining are achieved by means of polarizing beam splitters (PBSs), i.e. the splitting ratios between each channel can be adjusted by manipulating the state of polarization of the incident beam with a half-wave plate (HWP). The first geometry which is investigated (Fig. 1(a)), is the Sagnac configuration in which the pulses are both temporally and spatially split and combined. The second geometry (Fig. 1(b)) is similar to the setup employed in [10], in which the pulse train is amplified in a double-pass scheme.

 

Fig. 1 Schematic representation of (a) a Sagnac-type DPA setup (the experimental setup) and (b) a double-pass DPA setup.

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In order to understand the limitations of these setups, an analytic model has been developed. This model allows evaluating the dependence of the recombination process on saturation effects, on the total accumulated nonlinear phase and on the contrast ratio of the PBSs. The impact of these effects on the performance of both setups will be discussed and guidelines for a further energy scaling will be given.

2. Experimental investigation of the CPA-DPA technique using a Sagnac geometry

In this section the Sagnac-type DPA setup is investigated and compared to [10] in order to illustrate both the differences between the setups and the behavior in the high energy regime.

A DPA stage with two PBSs (PBS₁ and PBS₂ in Fig. 1(a)) and a Sagnac-type setup including a 1.2 m long ytterbium-doped large-pitch fiber (LPF) [11] with a core size of 90 µm and a MFD of 75 µm is employed for a proof-of-principle experiment. The LPF is pumped from one side with a fiber-coupled pump diode emitting at a wavelength of 976 nm. The stretched pulses from a fiber CPA system, similar to the one described in [1], are used as the seed of the amplification process. These pulses have a FWHM duration of 2 ns, which should in principle allow for higher peak powers and pulse energies than those obtained in [10]. In order to completely separate these pulses, a free-space delay of 7 ns is introduced. After amplification and recombination the output beam leaves the DPA system at the second port of PBS₁ and is finally sent to a dielectric reflection-grating compressor. The compressor consists of two gratings with a groove density of 1740 lines/mm (similar to the stretcher) and possesses a power-independent throughput of 80%.

The repetition rate is set to 20 kHz to allow for high energy extraction. The signal input power (i.e. that incident on PBS₁) is 600 mW. Figure 2(a) depicts the measured output power and the corresponding pulse energy after the compressor as a function of the launched pump power. The results presented in Fig. 2 correspond to the case with homogeneous power division among the pulses. Furthermore, in Fig. 2(a) the four different cases corresponding to each one of the possible beam paths that an undivided pulse can travel in the setup are presented as well. For the selection of the path travelled by the undivided pulse, the orientation angles of the HWPs are set in such a way that no division occurs at the PBSs and that the entire power is directed along one certain path. Each path is identified by a sequence of labels R and T, denoting whether the beam is reflected or transmitted at the different PBSs, when considering propagation towards the Sagnac-type loop. For the backward propagation the opposite path is used. Consequently, there are different polarization-dependent system losses for each one of the possible beam paths. For example, due to the PBSs, the mirrors, and the coupling into the fiber, there is a spread of the average output powers of the four possible single-pulse beam paths that ranges between 16 W and 19 W at a maximum launched pump power of 68 W. For the case of a division into four pulses, at the same pump power, a compressed output power of 10.5 W could be achieved. This corresponds to a pulse energy of 0.52 mJ. The measured autocorrelations of the compressed output pulses at this energy level for both the combined pulse and one of the undivided pulses are depicted in Fig. 2(b). The autocorrelation duration (FWHM) of the combined pulse remained constant at 880 fs for all pump powers, which corresponds to a pulse duration of 600 fs (870 MW peak power assuming a numerically estimated deconvolution factor of approximately 0.682). However, the autocorrelation of the undivided pulse increased steadily with increasing pulse energy up to an duration of 1 ps, which corresponds to a longer pulse duration of 700 fs (950 MW peak power assuming a numerically estimated deconvolution factor of approximately 0.7). As can be seen from Fig. 2(b), the pulse shape of the undivided pulse (red line) is degraded and detrimental pre- and post-pulses appear. The improvement of the pulse quality achieved by DPA can be clearly seen in the autocorrelation trace of the combined pulse (black line). This is because the accumulated total nonlinear phase of the pulse replicas is much lower than that of one single undivided pulse.

 

Fig. 2 (a) Compressed output power as a function of the launched pump power for the combined case (with a homogeneous power division into four pulses) and for each one of the four different amplification paths (R - reflected at PBS, T - transmitted at PBS) that can be traversed by an undivided pulse. (b) Measured autocorrelation traces of the compressed pulses for a maximum pulse energy of 0.5 mJ for both the combination of four pulses and for an undivided pulse case (RR).

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However, it can be seen from Fig. 2(a) that the combined output power after the compressor degrades at high pulse energies. In order to obtain an estimation of the efficiency of the setup, the so-called system efficiency is considered. This parameter is the ratio between the measured average output power of the combined pulses (black curve) and the mean value of the output powers of all the possible single-pulse beams (colored curves). Because of the polarization dependence of the compressor gratings, the measured output power automatically refers to a linear polarization state, i.e. any temporal mismatch at the recombination stages (which would lead to a polarization state different from linear polarization) is already considered in the system efficiency. The calculated system efficiency is shown in Fig. 3(a). For comparison, the measured efficiency for a higher repetition rate of 4 MHz and, therefore, lower pulse energies is also depicted. It can be seen that in the latter case the system efficiency remained approximately constant with the launched pump power at a value of >87%. However, for the high-energy case the system efficiency drops from an initial value of approximately 78% to 60%. Figure 3(b) shows the divided pulses measured with a fast photo-diode before and after the fiber amplifier for one propagation direction through the Sagnac-type loop. Gain saturation, the main reason for the steady degradation of the efficiency, can be clearly seen. The pulse train, which originally contains pulses of equal energy, becomes deformed due to the amplification and saturation processes in the fiber. Unfortunately, since this is a passive setup, such reshaping cannot be compensated for in the combining step. Thus the uneven pulse peak powers result in a difference of the nonlinear phase accumulated by each pulse due to self-phase modulation (SPM) and cross-phase modulation (XPM). This phase difference between the pulses, in turn, spoils the coherent addition.

 

Fig. 3 (a) System efficiencies for both the low-energy case (4 MHz) and the high-energy case (20 kHz) and (b) photo-diode signal of the input pulses and of the saturated amplified pulses at the output of the amplifier for a combined output pulse energy of approximately 0.5 mJ.

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Although at first sight the experiment promised a significant performance improvement with respect to the results obtained in [10] such an improvement could not be achieved. In order to fully understand the causes of the degradation of the system efficiency with the pulse energy, a theoretical model has been developed and will be presented in the following section. Afterwards, the influence of saturation, the acquired nonlinear phase, and the depolarization induced by the components on the combining efficiency, together with the consequences for energy scaling will be discussed.

3. Theoretical description of passive coherent combination DPA setups

In this section the evolution of the pulses through DPA setups, such as those shown in Fig. 1, will be described using an extension of the Jones calculus [12]. In the following, the temporal pulse separation will be assumed to be much larger than the pulse duration. Hence, the stretched pulses only interact at the PBSs.

The theoretical considerations done herein are based on the complex amplitudes of the electric field of the pulses, which can be written as Jones vectors. Furthermore, only the evolution of the pulse peaks is considered, i.e. there is no explicit time dependence of the fields. In order to express the temporal division of one input pulse, different time windows t will be used with t = 1,2,…,2^N for a total number of 2^N pulses after N divisions. Thus, the field in time window t can be written as

Table 1. Jones matrices of the optical elements used in the setup (θ - rotation angle of the optical axis with respect to the p-polarization-axis of the pulse, T_p - transmissivity of the p-component, R_s - reflectivity of the s-component)

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In the following, the calculation formalism will be explained using the double-pass configuration shown in Fig. 1(b). In this particular case the input pulse goes through PBS₀ first, which will also be used to eject the final pulse. Thus after the PBS, using Table 1, the input pulse changes to J_TA_in,t for every t. The description of the following temporal division units, each consisting of one HWP, two PBSs, and a delay line, can be done in two steps. First, the division, which takes place with the help of the HWP and the first PBS of the m^th division unit (m = 1,2,…,N), is calculated resulting in the transmitted (A_T,t) and reflected fields (A_R,t) of each time window:

In the second step, the reflected part is delayed by an amount t_m with t_m being the number of shifted time windows. It should be kept in mind that the DPA principle requires the temporal delays to be different for each delay line (i.e. t_m ≠ t_m’ for m ≠ m’). Finally, both the transmitted and reflected (and subsequently delayed) pulses recombine at the second PBS of the m^th division unit with

In the next step, the generated pulse train is amplified in a fiber amplifier. For simplicity single-pass amplification will be assumed. The input energy E_in,t and the output energy E_out,t are related by the Frantz-Nodvik equation [14]

Besides, the accumulated B-integral of each pulse has to be considered [16]. In order to analyze the impact of the B-integral independently from the fiber used, the ratios between the B-integrals of the pulses are considered. Assuming that each pulse is exponentially amplified, the ratio between the B-integral (B) corresponding to two pulses at time t and t’ is given by

To finally eject the recombined pulse at the output port of the first PBS (PBS₀) and to remove the temporal delays of the pulses, a Faraday rotator is used to rotate the polarization orientation of each pulse by 90° after amplification. The pulse combination is calculated analog to Eqs. (2) to (4).

As a measure of the efficiency, the total system efficiency η_tot can be introduced. It includes the spatial combining efficiency η_spat, which is the total amount ejected at the output port divided by the total amplified output, the temporal combining efficiency η_temp, which accounts for the fraction of temporally uncombined pre- and post-pulses, and the fraction of linear polarization η_lin of the main pulse. Hence, the total system efficiency can be written as

The evolution through a Sagnac-amplifier implementation (Fig. 1(a)) follows a similar procedure. The same amplification model is used except that the amplifier input is determined by the sum of the corresponding counter-propagating pulses entering at the same time (given by the same time window t of the pulses within both branches).

4. Analysis of the impact of saturation, B-integral, and depolarization

In this section the impact of the saturation of the fiber amplifier, the accumulated nonlinear phase difference of the divided pulses and the depolarization (due to the PBSs) on the efficiency of the recombination process will be analyzed for passive DPA implementations using either the double-pass or the Sagnac geometry.

4.1. Double-pass geometry

First the double-pass geometry as depicted in Fig. 1(b) is considered. The calculated total system efficiencies as a function of both the total output pulse energy E_out normalized to the saturation energy E_sat and the maximum B-integral B_max are shown in Fig. 4. A division into two, four and eight pulse replicas is depicted assuming an initial small-signal gain G₁ = 30 dB. The orientation angles of all HWPs are set to |θ| = 22.5° for a homogeneous pulse train and both perfect PBSs with R_s = T_p = 1 and non-perfect PBSs with R_s = 0.99 and T_p = 0.95 are considered.

 

Fig. 4 Simulation of the total system efficiency in a double-pass geometry for a saturated pulse train as a function of the maximum B-integral B_max and of the total output pulse energy E_out normalized to the saturation energy E_sat for a division into two, four and eight pulses (a-c, d-f) with (a-c) R_s = T_p = 1 and (d-f) R_s = 0.99, T_p = 0.95 (HWP orientation angles |θ| = 22.5°, initial small-signal gain of G₁ = 30 dB). For three energies, the saturated shape of the pulse train is shown above the plot (red: p-polarized, blue: s-polarized). Please note that the results displayed in this figure are general and, therefore, independent of the characteristics of the fiber.

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The x-axis of each color-plot shows the total output energy E_out of the amplified pulse train normalized to the saturation energy E_sat ranging from E_out/E_sat = 0.1 to 5, which corresponds to the input E_in/E_sat being varied between 0.0001 and 0.14. For example, assuming E_sat ≈1.4 mJ, which is realistic for a large-pitch fiber with a MFD of 75 µm, this corresponds to input pulse energies ranging from approximately 150 nJ to 190 µJ. The y-axis shows the maximum B-integral accumulated by a single pulse in the pulse train (the B-integrals of the rest of the pulses in the train are calculated using Eq. (8)) ranging from B_max = 0 rad to 7 rad. The calculated total system efficiency η_tot (Eq. (9)) is color-coded and takes into account all losses due to both the spatially and temporally uncombined fractions. Additionally, the corresponding energy distributions of the pulse replicas E_out,t for E_out/E_sat = 0.1, 2, and 4 are depicted above the color maps normalized to the total amount of energy of the pulse train E_out. What can be seen, in general is that the effect of saturation on the total system efficiency becomes more important the higher the input pulse energy is.

First, the case of a division into two pulse replicas using one delay line and perfect PBSs is considered, as depicted in Fig. 4(a). In the limiting case where no nonlinear effects are present (B_max = 0) only a small decrease in efficiency resulting from saturation is observable. This can be explained by the increasing combining losses at the PBS due to the fact that this symmetric setup cannot compensate for the differences in the shape of the pulse train in both propagating directions. In the second limiting case, i.e. at low pulse energies, the shape of the pulse train remains unchanged and high total system efficiencies are obtained even for high values of B_max. This is due to the fact that the difference in the B-integrals between any two pulses is negligible. At higher energy levels the difference in the pulse peaks and, therefore, in the B-integrals between the individual pulses becomes larger. For high B_max values and increasing saturation this leads to a strong degradation of the efficiency. The acquired phase differences lead, in addition to the amplitude difference, to losses at each recombination step, which result in pre- and post-pulses. Unfortunately, only a small amount of this efficiency drop can be compensated by adjusting the HWPs to an intermediate angle between the optimum angles for forward and backward propagation.

The number of pulse divisions is increased in Figs. 4(b)–4(c). However, these figures do not really show the main advantage of higher N-values, i.e. that for a fixed B-integral and for low saturation an N-times higher pulse energy can be extracted from the same amplifiers by splitting the energy into N pulses. It can be seen that when N increases, for small B_max the impact of the saturation of the pulse train leads to a stronger drop in efficiency compared to the two-pulse case due to the increased number of combining processes. Moreover, stronger pre- and post-pulses arise as a consequence of polarization components traveling back the wrong optical path. The efficiency increase in the upper part of Figs. 4(a)–4(c) results from the 2π-periodicity of the coherent addition.

The efficiency is reduced even further if non-perfect PBSs are assumed with e.g. R_s = 0.99 and T_p = 0.95 as depicted in Figs. 4(d) - 4(f). This slightly asymmetric splitting ratio leads to visible changes in the pulse peaks resulting in a degradation of the total system efficiency of 10-20% for the considered divisions becoming worse for a larger number of division units. However, the advantage of such a linear delay-line implementation (Fig. 1(b)) is that the dark ports of each second PBS of a division unit clean the linear polarization state of the pulse replicas by ejecting some of the wrong polarization components, which reduces their impact.

4.2. Sagnac geometry

The situation becomes more complex for a symmetrically pumped Sagnac geometry (Fig. 1(a)), which will be analyzed in the following. In Fig. 5, the calculated total system efficiencies for divisions into two to eight pulse replicas when using perfect PBSs (R_s = T_p = 1) and non-perfect PBSs (R_s = 0.99, T_p = 0.95) are shown. The advantage of this particular implementation is that a lower number of PBSs is needed. In contrast to the double-pass implementation with a splitting in two pulses, the two identical but perpendicularly polarized pulse replicas are amplified while counter-propagating through the loop. Consequently, although there is saturation, the pulse replicas are identically amplified and accumulate exactly the same amount of nonlinear phase leading to a total system efficiency of 100% for all pulse energy and B-integral combinations, as can be seen in Fig. 5(a). Also in the case of four pulses (Fig. 5(b)) a higher efficiency than in the case of the double-pass DPA implementation is obtained. This is due to the fact that the counter-propagating pulse trains ofthe fully symmetric Sagnac loop always recombine perfectly at this recombination step. The drop in efficiency for this particular case results just from the B-integral differences between the first and the second pulse replicas. Again, the increase in efficiency in the upper part of Fig. 5(b) arises from the 2π-periodicity of the coherent addition. However, for a division into eight pulses (Fig. 5(c)) the achievable efficiency decreases further. In this case, after the stepwise recombination, pulse replicas of rotated linear polarization are obtained in each branch before the final recombination step due to the different pulse peak powers, leading to loss fractions propagating back to the input. This has no influence for the division into four pulses due to the advantage of the Sagnac implementation and the use of just one division unit.

 

Fig. 5 Simulation of the total system efficiency in a Sagnac geometry for a saturated pulse train as a function of the maximum B-integral B_max and the total output pulse energy E_out normalized to the saturation energy E_sat for a division into two, four and eight pulses (a-c, d-f) with (a-c) R_s = T_p = 1 and (d-f) R_s = 0.99, T_p = 0.95 (HWP orientation angles |θ| = 22.5°, initial small-signal gain of G₁ = 30 dB). For three energies, the saturated shape of the pulse train is shown above the plot (red: p-polarized, blue: s-polarized). Please note that the results displayed in this figure are general and, therefore, independent of the characteristics of the fiber.

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Additionally, for this implementation non-perfect PBSs are also assumed (R_s = 0.99 and T_p = 0.95). Due to the asymmetric splitting ratio of the PBSs and the fixed HWP orientations (|θ| = 22.5°), the slight change in the pulse peaks lead to a significant degradation of the efficiency even for a division into two pulse replicas. The efficiency drops further for a division into four pulse replicas (Fig. 5(e)) and it becomes too low already for a division into eight pulse replicas (Fig. 5(f)). Thus, any asymmetry between the pulse trains in the Sagnac-type DPA implementation has a stronger impact on the efficiency than in the double-pass DPA implementation.

Finally, these results can be tentatively compared to the experiment described above. A degradation of the system efficiency to 60% for approximately 0.5 mJ (E_out/E_sat < 0.5) of pulse energy was observed, for which the maximum B-integral can be estimated to be at least B_max > 2 rad. Compared to the simulation using non-perfect PBSs (with specifications of R_s and T_p as assumed), which is shown in Fig. 5(e), the total system efficiency is expected to be approximately 70%. The small difference of this theoretical prediction with the experiment can be attributed to the one-sided pump configuration, which leads to B-integral differences even between the counter-propagating pulses.

4.3. Further limiting effects and possible solutions

There are further effects, which may reduce the theoretical efficiency values even further, that have not been considered above. For example, the leading edge of the very first pulse of a pulse train undergoes the highest amplification and saturation is observable already within the pulse. This leads to a deformation of the pulse, which becomes even more considerable the more the fiber amplifier saturates. Since the available gain for the subsequent pulses is exponentially reduced, the pulse shape deformation decreases. This, in turn, has an impact on the combining efficiency. Such an effect becomes also important in the spectral domain when using stretched pulses. The change in the temporal pulse shape also imprints a change in the spectral amplitude which influences the recombination efficiency. Moreover, phase changes within the pulse train due to the Kramers-Kronig relation may also degrade the overall efficiency [10]. Additionally, nonlinear polarization rotation in the fiber may also contribute to further break the symmetry of the passive DPA implementations discussed. Moreover, the occurrence of XPM in the symmetric Sagnac-amplifier may lead to slightly different results when comparing both setups. Hence, all these higher-order effects can lead to a further degradation of the total system efficiency. In order to quantify the impact of these higher-order effects a more detailed analysis is necessary, but this goes beyond the scope of this paper.

With the simplified analysis presented herein the limitations of passive coherent combining implementations using DPA are revealed. Because of the symmetry requirements of such setups any asymmetric change of the pulse train due to amplification dynamics has a strong impact on the overall efficiency that cannot be compensated for. The main reason for this restriction is the lack of degrees of freedom in the division and in the recombination of a pulse train. One possible solution, which has yet to be verified, may be to employ high pulse numbers in these passive setups, reducing the impact of saturation for a given total output pulse energy. However, this would also increase the complexity due to the large number of delay lines required which, additionally, can quickly reach lengths of several meters.

Another solution is to separate the division stage from the recombination stage, which, consequently, makes an active-feedback system necessary. Figure 6 shows a proposed experimental setup. In this case, folded delay lines are used by double-passing the PBSs with the help of quarter-wave plates (QWPs). This allows replacing one mirror of each division unit by a piezo-driven mirror stage. The advantage of such an implementation is the independent manipulation of the input pulse train (with the help of the HWPs of the division unit and/or additional pulse train shaping devices) and the output pulse train, which makes a pre-compensation of saturation effects and of the B-integral differences possible. Such an active DPA scheme would allow for the extraction of unprecedented pulse energies and, therefore, lead to an impressive increase in the peak power emitted from fiber-based systems.

 

Fig. 6 Proposed setup for a DPA implementation with actively stabilized division and combination stages.

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5. Conclusion

In conclusion, an experimental and numerical analysis of the achievable efficiency of passive DPA is presented. Based on a simplified model considering depolarization effects of the PBSs, saturation effects, and differences in the total accumulated nonlinear phases, a degradation of the overall efficiency could be shown. This degradation is unavoidable for high-energy extraction. The reason for this degradation lays in the symmetry requirements of such passive coherent combining setups. Therefore, the passive DPA approach is a concept that works in the low-energy regime but fails in the high-energy regime. Finally, an actively stabilized setup is proposed which is able to circumvent these restrictions. It can be expected that this technique will have a similar impact on fiber based ultrashort-pulse lasers to the recently demonstrated coherent beam combining. Finally, we believe that the combination of this novel technique with the spatial coherent combining approach will bring fiber CPA systems with TW-level peak powers and kW-level average power within reach, i.e. parameters not accessible with any existing laser technology so far.