## Abstract

The dynamical properties of a mode-locked thin disk laser with cavity-dumping in the solitary regime are studied using numerical simulations along with experimental data. Limitations of this system as well as their origin are identified. The results of these investigations agree very well with recently published experimental results. Based on these findings design criteria for future systems are deducted and estimates of possible pulse energies are made.

©2007 Optical Society of America

## 1. Introduction

In recent years a number of different concepts for laser oscillators providing high pulse energies at high repetition rates have been realized. For Ti:sapphire oscillators the use of chirped-pulse oscillators has resulted in pulse energies exceeding 0.5 *µ*J at repetition rates of several MHz with pulse durations of about 50 fs [1, 2, 3]. Using Ytterbium doped gain media in the solitary mode-locking regime two different concepts have provided very promising results: Our group has successfully used a SESAM mode-locked laser with cavity-dumping based on an Yb:KYW bulk crystal to produce pulse energies exceeding 1 *µ*J at repetition rates of 1 MHz with pulse durations of 380 fs [4]. However, in bulk systems the maximum achievable peak power is limited by the B-Integral in the gain medium. By using Yb:YAG thin-disk modules this limitation can be overcome and pulse energies up to 11 *µ*J at a repetition rate of 4 MHz have been demonstrated in SESAM mode-locked oscillators with pulse durations of 790 fs [5]. Recently these two concepts have been merged for the first time and a thin-disk oscillator with cavity-dumping based on Yb:KYW has been built. This resulted in pulse energies up to 3 *µ*J at a repetition rate of 1 MHz with pulse durations of 680 fs [6]. Currently, the limiting factor for the pulse energy in this setup is the dumping ratio which, at 24%, is far lower than in our previous experiments with bulk material where the maximum dumping ratio easily exceeds 50%.

To understand the reasons for this limitation and to find ways of improving the performance of our laser we started to investigate the theoretical properties of such laser systems. In this paper we present a theoretical study based on numerical simulations of cavity dumped systems with special regard to how the implementation of a thin disk scheme affects the dynamic properties of the cavity dumped laser. The comparison between the numerical model and experiments allows for adapting model parameters, determining inherent limits of the current setups and finding possible ways for improvement.

## 2. Numerical Model

It is possible to accurately describe the dynamics of a SESAM mode-locked laser by using three differential equations [7][8][9] which in turn describe the temporal dynamics of the pulse envelope, the laser gain and the absorber loss. The first equation, describing the evolution of the pulse’s envelope, is the master equation of mode-locking:

Here, *T _{R}* is the cavity round-trip time,

*A*(

*t*,

*T*) is the slowly varying field amplitude normalized such that |

*A*(

*T*,

*t*)|

^{2}represents the instantaneous power,

*g*,

*l*are the round-trip gain and loss respectively.

*D*

_{g, f}=

*g*/Ω

^{2}

*+1/Ω*

_{g}^{2}

*the gain and spectral filtering. Ω*

_{f}*is the half-width at half-maximum (HWHM) gain bandwidth and Ω*

_{g}*is the HWHM filter bandwidth.*

_{f}*q*(

*T*,

*t*) denotes the saturable absorber coefficient and

*β*

_{2}the intra cavity group delay dispersion (GDD). The self phase modulation (SPM) coefficient

*γ*is given by the nonlinear refractive index

*n*

_{2}, the center wavelength λ

_{0}, the mode area inside the Kerr-medium

*A*

_{eff,L}, and its double pass length

*ℓ*according to

*γ*=2

*π*

*n*

_{2}

*ℓ*/(λ

_{0}

*A*

_{eff,L}). Non-linearities introduced by the BBO in the Pockels-cell and the ambient air in the resonator are calculated in similar fashion. The gain dynamics of the system is described by

where τ* _{L}* is the upper state lifetime of the laser medium, the pulse energy $E\left(T\right)={\int}_{-{T}_{R}\u20442}^{{T}_{R}\u20442}{\mid A(T,t\prime )\mid}^{2}dt\prime $ and

*E*

_{sat,L}the saturation energy of the laser medium. Finally to describe

the dynamics of the absorber, the third equation is

with the small signal amplitude losses *q*
_{0}, the response time τ* _{abs}*, and the saturation energy

*E*

_{sat,A}. By integrating these equations with a split-step Fourier method along with a periodic loss modulation at an integer multiple of the cavity roundtrip period to model the dumping process it is possible to accurately simulate cavity-dumping in cw mode-locked lasers as has been shown e.g. in [10].

## 3. Comparing Numerics and Experiment

To ensure the validity of our numerical simulations we compared the model to two different laser setups. We started by using the experimental parameters of the thin disk laser system from [6]. The key laser parameters and their estimated uncertainties known from the laser set-up are given in Table 1. For the SESAM parameters the uncertainties are given by the manufacturer. For the modulation depth Δ*R* only the design value is available. The cavity losses *l* are estimated from the reflectivity of the mirrors as well as from the losses on the EOM and the birefringent filter. The beam radii have been calculated using ray-tracing software, and net negative dispersion *β*
_{2} is based on design values of the laser mirrors.

We used these values as a starting point for the simulations. Due to the uncertainties of the experimental values the model parameters have to be varied carefully to make a fit to the measurements. Not surprisingly the most sensitive parameters regarding the pulse energy and

the mode-locking stability are the beam radius on the gain medium *r*
* _{gain}*, the radius on the absorber

*r*, the modulation depth of the absorber Δ

_{abs}*R*and the saturation fluence of the absorber Φ

_{abs}. For the fit of the temporal and spectral properties of the output pulses the radius inside the EOM and the response time of the absorber have been adjusted. The well known experimental dumping parameters (ratio and frequency) are directly included in the simulation. The resulting numerical parameters are summarized in Table 2; they closely resemble the experimental data from Table 1.

In Table 3 we compare the results of the simulation with the parameters from Table 2 and the experiment. The resulting spectra are shown in Fig.1. In the experiment the laser was operated both in air and under a Helium atmosphere. Without the Helium the pulse energy was limited by the non-linearities resulting from the ambient air. Replacing the air in the resonator with Helium severely reduces the non-linearities which has already been reported in [11] and can be understood by looking at the SPM - coefficient (*γ*
_{gas}) for the residual gas. Using the mean beam radius *r _{gas}* inside the cavity the value of

*γ*

_{gas}in air becomes 3·10

^{-3}1/MW while under a helium atmosphere the value decreases to 7·10

^{-6}1/MW – the non-linearities resulting from the residual gas can be reduced by more than two orders of magnitude by using a Helium atmosphere instead of air allowing for higher pump powers. In future experiments one might consider to operate the whole laser system in vacuum. The influence of the ambient gas is an important topic for all high power mode-locked lasers and needs to be studied in further detail in the future.

The numerical parameters for the laser running under a Helium atmosphere are the same as in Table 2 except for the slightly higher small signal gain which reflects the higher pump power and is given in parenthesis.

The spectral position of the Kelly sidebands which is determined primarily by the dumping frequency matches the experimental value very well. The difference in the height can be attributed to the resolution of the spectrometer. For an extensive discussion of the properties of Kelly sidebands in cavity dumped oscillators we refer to [10].

As a second proof for the validity of the numerics we modeled the cavity dumped laser system based on an Yb:KYW bulk crystal as introduced in detail in [4]. Again, the results of the simulation are in excellent agreement with the experimental data both in terms of the pulse spectrum which is shown in Fig. 2 and in terms of pulse duration and energy which is summarized in Table 4 The numerical parameters for the simulation are shown in Table 5. The remaining uncertainties are the same as for the thin disk laser. The radius of the laser mode in the gain crystal (*r _{gain}*) was estimated by the resonator design and by measurements of the beam radius outside the crystal. Further experimental parameters of this laser system can be found in reference [4].

## 4. Limits of cavity-dumping

The excellent agreement between numerical simulations and experimental data forms the basis for the theoretical investigation of the limiting factors for cavity-dumping. This is of particular

interest, because as has already been pointed out, the current pulse energy limitation of our thin disk laser stems from the fact that stable laser operation at dumping ratios beyond 23 % was impossible to achieve – which is remarkably low compared to numbers of up to 80% for the bulk laser system. From this treatment we expect some deeper understanding of the pulse shaping mechanisms and some guidelines how to extract some higher fraction of the intra-cavity pulse energy.

Not surprisingly, the numerical model revealed the same limitation for the maximum dumping ratio for the thin-disk setup between 25% and 30% with the parameter sets close to the experimental values. Higher dumping ratios result in Q-switching instabilities and no stable cw mode-locked laser operation was possible. Typical pulse dynamics from the simulation are shown in Fig.3. On the left hand side the evolution of the intra cavity pulse energy in a stable configuration with the parameters given in Table 2 and a dumping ratio of 10% every 16th roundtrip is shown. Each dot represents the pulse energy after one cavity roundtrip. In contrast, an unstable situation is shown on the right hand side using the same parameters but a dumping ratio of 40%. No stable mode-locking is possible and the laser is Q-switching on much larger timescales. In the experiment this second mode of operation quickly leads to damage in the absorber. Nevertheless the question is, what is the reason for the big difference in the maximum dumping ratio between the bulk and thin-disk setup?

Looking at the two laser setups, the main difference in terms of the gain dynamics is the mode radius in the gain medium. While the mode radius in an Ytterbium bulk laser is typically around 100 *µ*m, the same parameter in a thin disk setup is typically more than 5 times larger. So we chose this parameter as starting point for the numerical study of the differences between the two systems. Figure 4 reveals that the key factor for the stability of a cavity-dumped mode-locked laser is indeed the mode size in the gain medium.

To analyze this situation we used the parameters from Table 2 and varied the mode radius from 540 *µ*m down to 200 *µ*m. Starting from a plateau at around 18% for large mode radii the stability border grows linearly to above 80% as the mode radius is reduced.

It is also important to note that the small signal gain was kept constant while the mode size was scanned. In the experiment this is only possible up to a certain point because the small signal gain is directly linked to the pump power and both saturation effects in the gain medium and the damage threshold of the disk put a limit on the pump power density. With our current disk the maximum pump power density is limited to 4 kW/cm^{2} which would be reached at a mode radius of 350 *µ*m. So for even smaller pump radii the pump power would have to be decreased which would still allow for an increased dumping ratio but at the expense of a reduction in pulse energy.

In the more realistic scenario, starting with the parameters of the thin-disk laser and continuously reducing both pump power and laser mode area on the disk one eventually ends up with a system very similar to the bulk laser. At this end the maximum dumping ratio will easily exceed 50% but the pulse energy will be decreased to about 1 *µ*J.

In a qualitative sense the importance of the mode size can be understood by analyzing its influence on the small signal gain and the saturation energy of the laser [12]

with *R _{p}*, the pump rate of the laser,

*hν*the photon energy and

*m*the number of passes made through the gain medium during one cavity roundtrip (in our case

*m*=4). A larger mode radius on the disk leads to an increase in saturation energy at the expense of a smaller gain. As known from [13] the pulse energy necessary to avoid Q-switching in a SESAM mode-locked laser without cavity-dumping can be calculated from

where *E*
_{sat,A}, the saturation energy of the absorber, is defined as:

Here, an increase in the saturation energy of the laser will result in an increased threshold for stable cw mode-locked operation. For a thin-disk laser like the one described in [5] this is of little consequence since the low output coupling ratio ensures that the intra-cavity pulse energy is usually high enough to avoid Q-switching instabilities. In the case of a cavity-dumped laser however the situation is different, since the small-signal stability analysis behind Eq. 5 is not valid anymore. From our model the numerical value for the threshold was found to be up to an order of magnitude higher in the case of a cavity-dumped laser. This can be understood by the fact that Eq. 5 is only strictly valid for a laser that has reached the steady state regime and has an intra cavity power far beyond the lasing threshold, which is not the case in a cavity dumped laser oscillator. Especially directly after the dumping event, where the intra cavity energy is low, the laser is less stable against Q-switching instabilities.

Still Eq. 5 provides a good starting point for a systematic investigation of the cavity dumped thin-disk oscillator. We started by systematically varying two parameters of the model, the mode area on the gain medium and the modulation depth of the SESAM. The left part of Fig. 5 shows the maximum dumping ratio for different modulation depths Δ*R* and different mode radii *r*
* _{gain}* color coded between blue and red. As a general tendency and not surprising after the discussion of Eq. 5 higher modulation depths lead to lower possible dumping ratios. But nevertheless, even for small modulation depths the mode radius is still the dominant parameter, and for modulation depths exceeding 1.5% the maximum dumping ratio depends almost exclusively on the mode radius.

The right part of Fig.5 shows the maximum stable pulse energy in *µ*J that would be extracted with the dumping ratios shown on the left. The dashed lines indicate some empirical experimental limitations: The vertical line stems from the fact that for modulation depths lower than 0.5% only cw operation of the laser was possible as the loss modulation was not high enough to achieve stable mode-locking. The horizontal line represents the power density limit of the Yb:KYW disk of 4 kW/cm^{2}. So the accessible regime is limited to the upper right quadrant in the two plots. By choosing optimum parameters pulse energies around 6 *µ*J seem to be achievable.

It also needs to be pointed out that even if the limitation of the power density on the disk could be overcome, any substantial increase in pump power would lead to other problems. To fulfill the soliton condition in our cavity dumped oscillator (which is currently done by using home designed chirped mirrors) we need to accumulate a second order dispersion (GDD) of -40000 fs^{2}. As any increase in intra-cavity power results in an increase in the non-linearities in the cavity, those would have to be compensated, which might be hard to achieve by mirrors. One intriguing solution for this issue could be laser operation in the positive dispersion regime, which was already successfully demonstrated in Ti:sapphire oscillators [1][2][3] but to the best of our knowledge never has been shown for thin-disk lasers or Ytterbium based gain media.

The third variable given in equation 5 is the saturation energy of the absorber which forms a third accessible degree of freedom. It can be varied either by choosing a different saturation fluence Φ* _{abs}* or by changing the mode size on the SESAM. However, the saturation fluence is a material parameter and linked to the modulation depth of the absorber. Since we already stated that the modulation depth should be as small as possible the remaining option is to vary the mode size on the SESAM.

In the numerics we again used the set of parameters given in Tab. 2, this time with a fixed mode radius on the disk of *r _{gain}*=450

*µ*m. Varying the mode diameter on the SESAM in the model we found, as anticipated from Eq. 5, that a reduction of the mode size on the absorber leads to a higher stability and thus to an increase in the maximum dumping ratio. Fig. 6 reveals a linear dependence of the stability boundary on

*r*

*. The slope is smaller compared to the effect of the mode size on the disk: To increase the maximum dumping ratio by 10%,*

_{abs}*r*

*needs to be decreased by roughly 25%, while for*

_{abs}*r*

*a reduction of just 10% would be sufficient to get the same effect.*

_{gain}Unfortunately, in the experiment it has been found that the mode size on the SESAM is limited to a very narrow range. Based on the experimental experience with the current thin disk setup stable mode-locked operation is only possible for mode radii in a range roughly between 270 *µ*m and 450 *µ*m. In Fig. 6 this is represented by the two vertical dashed lines. For smaller mode radii the SESAM is prone to damage because of the high intensities, while for larger radii no stable mode-locking was obtained. For different laser setups these values can be expected to vary. However, in general the intensities required for stable mode-locking in a high power SESAM mode-locked lasers have to be close to the damage threshold of the absorber. So this intrinsically does not leave much room for further decreasing the mode radius on the absorber without destroying it. For this reason the mode area on the SESAM does not seem to offer a large potential for optimizing the maximum output pulse energy of the mode-locked thin disk laser.

In addition to the investigations discussed above we varied some other parameters in trying to find out more about the limitations and possible optimizations of this type of laser system. In the experiment the second order dispersion was varied in the range between -20000 fs ^{2} to -60000 fs^{2}, in the simulation even in -10000 fs^{2} to -150000 fs^{2}. Furthermore, in the numerics we varied the cavity round trip time (*T _{R}*) between 50 ns and 80 ns. Even these drastic deviations from the optimum values did not help to improve the output pulse energies nor the overall stability of the laser.

We also varied the dumping frequency. In the experiment rates between 500 kHz and 1 MHz have been tested without significant influence on the pulse energy and the pulsing stability. This is also confirmed by the numerics. Dumping frequencies beyond 1 MHz have been prohibited by limitations of the EOM. For frequencies between 300–400 kHz the laser performance is unstable even at low dumping ratios while at even lower frequencies the maximum possible dumping ratio increases steadily up to values beyond 90 % at 10 kHz and below in accordance with [10]. However, from the application point of view, those low repetition rates are of little interest because amplified laser systems can readily supply pulse energies much higher than that.

## 5. Conclusion

In conclusion we have reported on limitations and possible optimizations of Ytterbium based cavity-dumped mode-locked thin disk laser systems. The detailed comparison between numerical simulation and experimental data verified the model and allowed for the systematic analysis of the stability edges: The maximum dumping ratio in these laser systems has severe limitations; we identified the laser mode size on the disk as the main parameter and showed that a careful optimization of the mode sizes on the disk and on the SESAM should allow for dumping ratios higher than 30% and a doubling of todays pulse energies up to 6 *µ*J. Experimental verification is in progress.

## Acknowledgements

The authors thank Alexander Killi and Max Lederer for cooperation and support. This work was supported by the European Union within Contract No. IST-2005-034562 [Hybrid Integrated BIophotonic Sensors Created by Ultrafast laser Systems (HIBISCUS)] and the Bundesministerium für Bildung und Forschung (BMBF) under contract 13N8723.

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