## Abstract

Continuously pumped regenerative amplifiers are subject to energy instability at high pulse repetition rates due to period doubling bifurcation. Theoretical and experimental data are presented, in order to differentiate and understand instability effects in Nd:YVO_{4} regenerative amplifier, and possible techniques for performance optimization are analyzed. An increase in the seed pulse energy is demonstrated to improve amplification dynamics. Addition of a preamplifier is shown as an efficient way to achieve seed energy high enough to provide stable operation at repetition rates up to 200 kHz with average output power near the theoretical limit.

©2009 Optical Society of America

## 1. Introduction

Regenerative amplifiers (RA) are extensively used for amplification of ultrashort pulses generated by mode-locked oscillators. Presently this technique enables production of lasers operating at repetition rates up to hundreds of kilohertz and generating high energy, up to millijoule level, pulses. Regenerative amplifiers are an important part of most picosecond and femtosecond industrial solid state laser systems. Both high system efficiency and stable output parameters over a wide range of pulse repetition rates are essential for this actively developing field. For creation of power-efficient systems, laser gain media which may be directly pumped by laser diodes is advantageous. Ytterbium doped materials represent one such family and are widely used for amplification of femtosecond pulses [1,2]. Furthermore, systems based on Yb:YAG in thin disk configuration are scalable to high average power [3]. Laser materials doped with neodymium are well suited for picosecond pulse durations and are competitive for moderate power. Systems based on Nd:YVO_{4} and Nd:GdVO_{4} crystals operate at repetition rates up to 200 kHz and produce more than 10 W of output power [4,5]. Long lifetime of the upper laser level typical of both Nd and Yb ions is an important advantage because it supports accumulation of substantial population inversion under continuous laser diode pumping. However, this long inversion lifetime may also cause stability problems at high repetition rates. Continuously pumped regenerative amplifiers demonstrate peculiar pulse amplification dynamics when the pulse repetition period becomes comparable or shorter than the gain relaxation time. Period doubling bifurcations develop generating alternating energy pulses or even sequences of pulses having chaotic energy distribution.

To date, only a few articles have been dedicated to this phenomenon despite its critical influence on the performance of regenerative amplifiers. Complicated dynamic behavior has been observed for a system based on Ytterbium doped glass, and the role of period doubling has been investigated both theoretically and experimentally [6]. However, one of the important parameters, the seed pulse energy, was left beyond the scope. The experiments were confined to studying cavity dumping of the Q-switched laser, an approximately equivalent to the RA seeded by extremely low pulse energy. Our recent theoretical analysis of RA operation revealed the importance of the seed energy and demonstrated that increase in the seed energy helps in eliminating the instabilities [7]. In the present paper we experimentally verify effectiveness of this approach of stability improvement for the Nd:YVO_{4} regenerative amplifier.

This paper begins by describing the experimental setup and presenting results demonstrating peculiar behavior of the RA seeded by a conventional mode-locked laser of moderate power. Then, using numerical modeling, we consider stability problems of regenerative amplification and possible solutions. System performance improvement achieved by utilization of a preamplifier is presented and discussed in the next section. We conclude the paper by summarizing experimental and theoretical results.

## 2. Regenerative amplifier seeded with moderate energy pulses. Stability problems

The schematic diagram of the experimental setup used for investigation of regenerative amplification is shown in Fig. 1 .

A diode pumped passively mode locked Nd:YVO_{4} laser was used as a master oscillator. It generated a continuous pulse train with repetition frequency of 82 MHz and average power of 300 mW. The laser was able to produce optical pulses with duration as short as 6 ps. The short pulses were used in experiments where dynamics peculiar to high peak intensities were of interest. However, most of the amplification investigations were focused on the “pure” dynamics not disturbed by optical nonlinearities. These experiments were carried out with 58 ps duration pulses obtained by installing an etalon in the oscillator cavity (the etalon narrows the bandwidth, thus widening the pulse duration).

A pulse picker was used to select pulses for further amplification and in this way to control the effective repetition frequency of the seed source. The pulse picker was an electro-optical switch based on the RTP Pockels cell. In the initial experiments optical pulses selected by the pulse picker were directed to the regenerative amplifier. The seed pulse energy was 3.2 nJ at the input of RA.

The regenerative amplifier was comprised of an optical resonator containing the gain medium (Nd:YVO_{4} crystals) and an electro-optical switch for control of the resonator quality. The laser crystal was continuously pumped by the fiber coupled laser diode module with fiber core diameter of 400 µm and numerical aperture of 0.22. Optimal pump power, providing maximum output in TEM_{00} mode, was set to be 44 W. The electro-optical switch consisted of a BBO Pockels cell, a quarter-wave plate and a thin-film polarizer.

Operation of a regenerative amplifier in general may be regarded as a succession of operation cycles consisting of a pump stage and an amplification stage. During the pump stage, before voltage is applied to the Pockels cell, laser action is suppressed by high resonator losses and the gain medium accumulates population inversion. The amplification stage starts when the seed pulse is injected into the resonator and the quarter-wave voltage is applied to the Pockels cell. The intracavity losses become minimal and are kept low for some preset time while the optical pulse circulating in the resonator is amplifying. As soon as the intracavity energy reaches a desired level the Pockels cell voltage is switched off. This dumps the amplified pulse out of the cavity counter-propagating to the seed pulse input. The output radiation is diverted from the input signal path by a circuit containing the Faraday rotator, half-wave plate and polarizer.

The total multi-pass gain of RA depends on the number of cavity round trips (NRT) which is determined by the amplification-stage duration. This important parameter is easily controlled by setting the time interval during which the high voltage is applied to the Pockels cell. Selection of proper NRT is a routine procedure for optimization of the RA. During the amplification stage, the intracavity pulse energy grows until the gain becomes equal to the resonator losses and then the pulse energy decays. The optimal NRT corresponds to the optical pulse dumping at the moment when its energy has reached the peak value. However, such a simple approach works well for low repetition rates only. At higher repetition rates, when the pump stage is comparable or shorter than the gain relaxation time, the operation cycles become interdependent. The equilibrium between population inversion depletion caused by amplification and inversion restoring caused by pumping may become unstable. This often leads to violation of the single-energy regime and to generation of periodically alternating high/low energy pulses, or more complicated instability patterns [7]. As an illustration, Fig. 2 shows oscilloscope screen shots of the RA output pulse train in typical single-energy and period doubling regimes. Since energy bifurcation is unacceptable for most laser applications, optimal operation implies not only the highest output power but also a single-energy regime. It is natural to define the corresponding number of round trips as optimal. At high repetition rates the dependence of output parameters on NRT is more complicated, and therefore system optimization becomes more complicated. Experimentally obtained diagrams of the RA average output power and pulse energy versus NRT demonstrating these peculiarities are presented in Fig. 3 . The specific dumping frequencies were chosen to describe the most relevant cases of RA behavior. The single-peaked dependence inherent to low repetition rates appears at 10 kHz [Fig. 3(a)]. The average power and the pulse energy reach the maximum values simultaneously, when NRT is equal to ten. At 20 kHz the situation is different [Fig. 3(b)]. The shape of the energy curve shows that the system undergoes bifurcation in the 9–13 NRT range. However, in this case the period doubling does not affect the system performance because the output power reaches its maximum value in a single-energy regime.

Instability effects become more pronounced at higher pulse repetition rates. The period doubling not only breaks the energy stability but also distorts the curve of the average power. This curve now has two explicit peaks [Figs. 3(c) and 3(d)]. The first peak, corresponding to the maximum power, is located in a period doubling zone, whereas the second one is just over the instability edge. The optimal regime is obtained in the vicinity of the bifurcation point. At 75 kHz the optimal NRT is equal to 48. This point is close to the second power peak, on the right side of the period doubling zone [Fig. 3(c)]. For 90 kHz repetition rate the optimal NRT is equal to 13 and is situated right before the first bifurcation point [Fig. 3(d)]. The experimentally observed problems at high repetition rates are the following: (i) the output energy exhibits unacceptable fluctuations when the RA generates the highest average power, (ii) the highest stable pulse energy is reached close to the instability edge.

## 3. Theoretical analysis of system operation and improvement possibilities

One may expect that it is possible to improve RA behavior by selecting more appropriate parameters governing the system. We leave out of the scope optimization the pump characteristics and the geometry of the optical resonator allowing more power in TEM_{00} mode, as this does not relate directly to the RA dynamics. The parasitic intracavity losses, although formally a governing parameter, are also not analyzed. They should be simply reduced as much as technically possible. Our goal is to maximize extraction of the given stored population inversion as a stable train of output pulses. The experiments described in the previous section demonstrate that variation of only NRT does not solve the stability problem. So, we proceeded to investigate what advantages increasing the seed energy can give. For this purpose we numerically evaluated the influence of the pulse repetition rate and NRT on the system operation for three seed pulse energies differing by about two orders of magnitude. These particular energies and the remaining fixed parameters were chosen so that they corresponded to the conditions of the experiment. They are summarized in Table 1
.

The steady state small signal gain, determined by the pump intensity, was measured directly. The parasitic intracavity losses were derived from the specifications of the optical components. The Nd:YVO_{4} crystal properties necessary for modeling (emission cross section and gain relaxation time) were taken from [8].

The modeling approach was described in detail in our previous paper [7]. The optical pulse evolution within a single RA operation cycle was analyzed in the approximation of rate equations. The discrete-time dynamical system approach was applied for consideration of the interference of successive cycles and for evaluating stability of operation points. The corresponding data are mapped in the space of control parameters – NRT and the repetition rate [Fig. 4(a) ]. This space is split into two parts; a zone of stable operation and an instability zone of period doubling. The curve separating those zones (further referred as separatrix) represents a manifold of bifurcation points in the parameter space. The instability zone occupies the right upper part of the plot. This region shrinks as the seed energy increases.

It is important to evaluate not only the stability at some given control parameters (operation points) but also the corresponding output power. The dynamics of regenerative amplification and output characteristics of the system are determined by the set of control parameters alone [7] in contrast to e.g. bi-stability effects where initial conditions may also influence the operation. Therefore, we can theoretically determine the highest obtainable average power assuming the absence of period doubling. The calculations show that this power is invariant under the seed energy provided that other governing parameters are fixed. Achieving the calculated maximum average power implies the best possible utilization of the stored pump energy, i.e. it assures the highest power efficiency. The diagram of theoretically achievable power versus repetition rate will be given in the next section for comparison with the experimental data. Now we can consider the manifold of points in the parameter space where this power is achieved. The corresponding NRT providing the highest output power (NRT^{MAX}) was calculated under the same assumption of maintaining stability. However, in contrast to the maximum power itself, the NRT^{MAX} is dependent on the seed pulse energy. The parameter separatrix together with the curve of NRT^{MAX} represent stability diagram of the RA operation [Fig. 4(a)]. The approach of stability diagrams forms a more systematic concept of the system optimization and specifically allows estimation of the seed level which may enable one to avoid instability effects, and so reach the theoretically possible average power.

The system capabilities are completely exploited when the operation is stable at NRT providing the highest output power. This condition is realized when the curve of NRT^{MAX} is outside the instability region. At low seed energy of 11 pJ the appropriate repetition rates should be less than 20 kHz. For the medium seed level of 1.1 nJ this range increases to 25 kHz. At higher repetition rates the NRT^{MAX} curve enters the instability zone. Consequently, the optimal NRT can be obtained at the stability edge, either on the upper or lower branch of the separatrix.

Now we can understand the relevant cases of system behavior, presented in previous section and corresponding to the medium seed value, by using stability diagrams. The trajectories of the operating points corresponding to NRT variation at a constant repetition rate are presented in Fig. 4(b). This trajectory at 10 kHz does not pass the instability zone. At 20 kHz the optimal operating point is above the instability zone. At both 75 kHz and 90 kHz the optimal NRT is on the stability edge and is rather far from the point of the highest attainable power. These results are in good agreement with the experimental data.

An important consequence of the numerical simulation is that for 240 nJ or higher seed energies the NRT^{MAX} curve does not enter the instability zone in the whole range of repetition rates. So, for our laser system we have found a seed value sufficient to eliminate negative features of amplification dynamics, and thus to completely exploit the system capabilities.

## 4. Regenerative amplifier operation with pre-amplified seed

Application of a more powerful master oscillator and/or decrease of the pulse repetition frequency by making the laser cavity longer are the straightforward ways to achieve higher seed energy. We omit challenging approaches such as mode-locked lasers with an extra-long optical resonator or with cavity dumping. Simple estimations show that in order to provide 240 nJ pulses the master oscillator operating in a CW mode locking regime with a reasonable repetition frequency of 50 MHz should generate average power of 12 W. On the other hand the useful part of this power is much lower, e.g. only 48 mW even operating at 200 kHz.

As an alternative of such a prodigal approach a double pass Nd:YVO_{4} preamplifier installed behind the pulse picker was used. High emission cross section of the Nd:YVO_{4} crystal and operation at relatively low input average power make this system efficient. Only 2 W of pumping was sufficient to achieve a gain coefficient of more than two orders of magnitude. The pulse energy at the input of the RA was 3.2 nJ when the preamplifier was disabled. The energy of the pre-amplified pulse reached 1.1 µJ at 10 kHz and steadily decreased with the repetition rate to 370 nJ at 200 kHz. In the most critical point at 68 kHz [see Fig. 4(a)] the measured seed energy was 700 nJ, well above the calculated value that ensures stable operation. Such an excess seed level was helpful, because the seed energy was not completely exploited. In general, it is difficult to avoid mode mismatching between a seed laser and optical cavity of the RA in both spatial and spectral domains. Spectral mismatching can exist even with identical gain media because of e.g. different temperatures of laser crystals of the master oscillator and RA. Mode mismatching reduces the effective seed energy. For our system we estimated the effective seed values to be about three times less than the measured ones. This was taken into account so that the seed values chosen for calculations (listed in Tab. 1) corresponded to those used in the experiments: pre-amplified seed, unamplified seed and attenuated seed. This set of RA input signals covers the functionally important range. The value of the unamplified seed, 3.2 nJ, is of the same order of magnitude as the pulse energy of commonly used moderate power solid-state picosecond lasers. Operation with the seed energy intentionally attenuated to 32 pJ provides opportunity to evaluate typical behavior of the RA seeded with potentially attractive low power sources, e.g. with ultrafast laser diodes, which would substantially reduce system size and complexity.

Experimental dependences of the RA average output power versus repetition rate for these values of seed energy are presented in Fig. 5(a) . The measurements were performed at the optimal round trip number. The NRT was set for maximum average power while maintaining stable operating regime for every repetition rate. The operation was considered stable when the standard deviation of the pulse energy did not exceed 1%. The output is virtually independent of the seed level for low repetition rates. However, at higher rates there is a drop in power for low and medium seed levels. The most significant power decrease appears in the 80–95 kHz range, then the output power steadily grows as the repetition rate increases.

This non-monotonic behavior of the power curve can be explained by considering the evolution of the operating point in the parameter space. As an example, we examined the medium seed case [Fig. 4(b)]. The operating points coincide with the theoretical curve of NRT^{MAX} until the latter enters into the instability zone (at 25 kHz). Then there are two possible trajectories of the operating point – the upper and lower branches of the separatrix. Up to 80 kHz the upper branch has an advantage by providing a higher output energy [optimal NRT position is similar to Fig. 2(c)]. However as the repetition rate increases the stable operating point moves further away from the NRT^{MAX} position, resulting in lower output energy. Consequently, starting from 85 kHz the optimal NRT switches to the lower border of the instability region [typical example is Fig. 2(d)]. In this regime the operating point gradually comes closer to the NRT^{MAX} position, and consequently the output power steadily increases with increasing repetition rate.

The power curve obtained with the preamplifier has no signs of downward excursion. This seed energy is sufficient to maintain stable operation at maximum power. Generally these experimental data are in good agreement with the theoretical curve of the available power. Some deviations observed at low repetition rates, we suppose, are caused by the Kerr effect. This nonlinearity can act as additional intensity-dependent intracavity losses. The Kerr effect requires higher pulse energies, and accordingly its influence is more pronounced at lower repetition rates. Spectrum broadening of the output pulse is evidence of the Kerr effect. We measured optical spectrum transforming from 0.05 nm at repetition rates above 40 kHz up to 0.15 nm at 10 kHz, thus implicating the Kerr effect for deviations at low repetition rates.

We also compared performances of the RA seeded by high and medium pulse energy for the functionally important case of shorter optical pulses. Typically about 9 ps pulse duration was obtained at the output of the RA seeded by the 6 ps pulse. This duration is close to the minimum value supported by the gain bandwidth of the Nd:YVO_{4} crystal in high total-gain applications such as regenerative amplification. The measurements were constrained to dumping rates above 50 kHz. Nevertheless, the intensities were substantial, and the Kerr effect influence was so strong that it eventually resulted in decrease of the output power [Fig. 5(b)]. The average power obtained with the pre-amplified seed was slightly lower than that theoretically predicted below 80 kHz and the difference reached 6.7% at 50 kHz. However, the comparison of these characteristics with those obtained at a medium seed level shows that the benefit of the preamplifier is even more pronounced in case of shorter pulses. The difference is related to a large decrease of the average power below 85 kHz for the case of the medium seed. This phenomenon can be explained by returning again to stability diagrams. The optimal operating points for the short pulse experiments were always settled along the lower branch of the parameter separatrix [Fig. 4(b)]. This regime gave a significantly lower output power than that obtained with the 58 ps pulse duration (obtained at the upper branch). However, attempts to operate at the upper branch (optimum for long pulses) resulted in an even larger decrease of the output. In order to understand this difference we estimated the *B* integral, the conventional quantitative gauge of the Kerr effect. The *B* integral calculated at 50 kHz repetition rate gave values of 1.3 (acceptable) and 7.6 (problematic) at transition from low to upper separatrix branch respectively. The reason for so large *B* integral value in the latter case is that operation behind the peak of power at larger NRT implies multiple round trips for the high intensity optical pulse. Consequently, the Kerr effect influence increases which eventually makes this regime highly inefficient. The high seed energy gives additional advantage at shorter pulses due to a significantly lower value of the optimal NRT.

## 5. Conclusion

Performance of diode pumped solid state regenerative amplifiers is essentially limited by instabilities inherent at high repetition rates. These limitations strongly depend on the seed pulse energy. A substantial shortfall of the average power in respect of theoretical limits has been observed for the Nd:YVO_{4} amplifier seeded by nanojoule-level pulse energy at repetition rates exceeding 40 kHz. This was caused by unacceptable energy fluctuations in regimes, which potentially could provide the highest average power. Furthermore, the maximum of stable energy was reached at the border between stable and unstable regimes. In general, operation at the margin of stability incurs challenges for robust operation in real systems. Even slight changes to control parameters may result in system instability. Therefore reliably stable operation generally requires setting the operating parameters well away from the instability border, but this in turn leads to a further reduction of the laser output power.

An increase in the seed pulse energy improves the amplifier operation by avoiding bifurcation of the output pulse energy. Moreover, when the seed energy is high enough the system performance can reach levels determined theoretically on an assumption of absence of instabilities. Utilization of a preamplifier is a simple and efficient way to obtain the required seed level. The performance of Nd:YVO_{4} based RA seeded by pre-amplified pulses has demonstrated that the system capabilities are thoroughly exploited. The average power reached the theoretically possible value and good energy stability was maintained. This approach can be useful for creating regenerative amplifiers with improved robustness, stability, and wall-plug efficiency, thoroughly utilizing the potential performance of the systems.

## Acknowledgments

The authors wish to acknowledge the technical assistance of Juozas Verseckas from EKSPLA UAB in preparation of the experimental setup and Lucian Hand from Altos Photonics Inc. for fruitful discussions of the manuscript. This work was partially financed by the Eurostars Project E!4335-UPLIT.

## References and links

**1. **T. Miura and S. Ito, “High-energy and high-power Yb:KGW femtosecond regenerative amplifier,” Proc. SPIE **7203**, 72030U (2009). [CrossRef]

**2. **D. Nickel, C. Stolzenburg, A. Giesen, and F. Butze, “Ultrafast thin-disk Yb:KY(WO4)2 regenerative amplifier with a 200-kHz repetition rate,” Opt. Lett. **29**(23), 2764–2766 (2004). [CrossRef] [PubMed]

**3. **J. Kleinbauer, D. Eckert, S. Weiler, and D. Sutter, “80 W ultrafast CPA-free disk laser,” Proc. SPIE **6871**, 68711B (2008). [CrossRef]

**4. **J. Kleinbauer, R. Knappe, and R. Wallenstein, “A powerful diode-pumped laser source for micro-machining with ps pulses in the infrared, the visible and the ultraviolet,” Appl. Phys. B **80**(3), 315–320 (2005). [CrossRef]

**5. **J. Kleinbauer, R. Knappe, and R. Wallenstein, “13-W picoseconds Nd:GdVO4 regenerative amplifier with 200-kHz repetition rate,” Appl. Phys. B **81**(2-3), 163–166 (2005). [CrossRef]

**6. **J. Dörring, A. Killi, U. Morgner, A. Lang, M. Lederer, and D. Kopf, “Period doubling and deterministic chaos in continuously pumped regenerative amplifiers,” Opt. Express **12**(8), 1759–1768 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1759. [CrossRef] [PubMed]

**7. **M. Grishin, V. Gulbinas, and A. Michailovas, “Dynamics of high repetition rate regenerative amplifiers,” Opt. Express **15**(15), 9434–9443 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-15-9434. [CrossRef] [PubMed]

**8. **R. D. Peterson, H. P. Jenssen, and A. Cassanho, “Investigation of the spectroscopic properties of Nd:YVO4,” Proc. OSA TOPS, Advanced Solid-State Lasers, M.E. Fermann and L.R. Marshall, eds., 68, 294 (2002).