The properties of passively mode-locked laser oscillators based on Ytterbium doped gain media are studied theoretically along with experimental data. Based on the chirped-pulse approach limitations due to excessive non-linearities are avoided, opening up new routes for energy scaling of mode-locked solid-state oscillators. Predictions about potential future pulse energies are made and possible experimental problems are discussed.
© 2008 Optical Society of America
In recent years a variety of different concepts for femtosecond laser oscillators with pulse energies up to the mircojoule regime at MHz repetition rates have been developed to overcome the detrimental influence of non-linearities originating from the excessive peak intensities of the intra-cavity laser pulses. With Kerr-lens mode-locked Ti:sapphire oscillators pulse energies of up to 0.5 µJ at 6 MHz with pulse durations around 50 fs have been realized by employing a chirped-pulse scheme, in which the oscillator is operated in a region of net positive intra-cavity dispersion [1, 2, 3]. In this case, the chirp of the intra-cavity pulses significantly lowers the peak intensities. After external dechirping peak powers around 10 MW and intensities beyond 1014 W/cm2 have been demonstrated . After introduction of the similariton laser concept, also operation of passively mode-locked fiber oscillators in the positive dispersion regime opened new perspectives for energy scalling . The pulse dynamics are characterized by large sensitivity to amplitude modulation and stability for a broad range of parameters. Mode-locked operation can be realized in simple, robust cavities and even the dispersion control can be set aside. By use of advanced fiber designs, 400 fs-pulses with an energy of 0.26 µJ at 10 MHz has been demonstrated so far .
Even higher pulse energies have been achieved with laser oscillators based on Yb-doped solid-state gain media which allow for direct pumping with high power laser diodes but suffer from a limited spectral bandwidth. By employing a thin-disk scheme with large laser spot sizes, excessive non-linearities as well as thermal lensing effects can be reduced. With this concept pulse energies in excess of 10 µJ at pulse durations of 790 fs have been realized . At these energies the non-linearity of the air inside the laser cavity requires the laser to be operated under a Helium atmosphere or in vacuum. To avoid this, a new concept based on a so-called active multi-pass cell which allows for 44 passes through the thin-disk was developed, and pulse energies of 13 µJ were reached with pulse durations of 1.3 ps. To achieve stable mode-locked operation group delay dispersion (GDD) values of -191600 fs 2 were required .
Recently, a chirped-pulse setup for a SESAM mode-locked Yb:KYW bulk laser with electro-optical cavity dumping has been demonstrated for the first time . By employing the chirped pulse scheme, the pulse energy from this laser system could be increased by 40 % compared to a similar laser system operated in the solitary (negative dispersion) regime . In this case, the pulse energies were only limited by thermal stress induced by the pump in the crystal but not by non-linearities.
In the case of the Ti:sapphire and Yb-doped fiber oscillators, the positive dispersion regime has been already studied extensively both experimentally and numerically [1, 2, 3, 4, 10, 11, 12], and the pulsing dynamics of these systems is well understood. In this paper we present for the first time a numerical study of Yb-based SESAM mode-locked oscillators operating in the positive dispersion regime. The comparison to the experimental data from  allows for careful adaption of the model parameters. Furthermore, we study the possibility of transferring the chirped-pulse scheme to thin-disk topologies; we can demonstrate that this indeed promises
a large potential for further power scaling.
2. Numerical model and adjustment to the experiment
The numerical model used for these studies is based on a split-step fourier algorithm that solves the well known master equation of mode-locking in combination with the rate equations of the the gain and the saturable absorber. For a more detailed description of the algorithm we refer to  where the same algorithm has already been used to successfully study pulsing dynamics in the solitary mode-locked oscillator.
The key parameters of the numerical model, based on the experimental data from the Yb:KYW bulk laser with cavity dumping described in , are summarized in Table 1 along with their estimated uncertainties. For the SESAM parameters (ΔR, the power modulation depth and Φabs, the saturation fluence) the uncertainties are specified by the manufacturer; the relaxation time τabs has been measured in house with a pump-probe technique. The cavity power losses l are estimated from the reflectivity of the mirrors as well as from the losses in the electro-optical (EOM) cavity-dumper. The beam radii in the gain medium, in the air, in the EOM, and on the SESAM (r gain, r gas, and r BBO) have been calculated using ray-tracing software, and the total intra-cavity dispersion (GDD) β 2 is based on the design values of the laser mirrors. T R is the cavity roundtrip time. The repetition rate (f rep), pulse energy (E pulse), the chirped pulse duration (τ chirped) and the fourier-limited pulse duration (τ FL) were measured in the experiment.
Within the error bars we carefully varied each parameter to make a fit of the simulation results to the experimental measurements. Figure 1 compares an experimentally measured spectrum taken at an intra-cavity GDD of +750 fs2 with the results of the numerical simulations. A good agreement between simulation and experiment can be found for parameters well within the
uncertainties of the experiment (see Table 2, g is the small signal gain). With these parameters, also pulse energies and durations are in agreement: While the experiment yielded pulse energies of 2.2 µJ and a measured chirped pulse duration larger than 5 ps, the simulation resulted in a pulse energy of 2.4 µJ and a chirped pulse duration of 8 ps. The fourier limited pulse duration in the experiment was found to be 400 fs while the simulation, again in good agreement, delivered a fourier-limit of 388 fs.
On top of the typical rectangular shape we obtain both in the experimental and the simulated spectra some asymmetries which are mainly caused by the large third order dispersion imposed by the EOM and to a lesser degree by the SESAM. The structure in the middle of the experimental spectrum could not be directly reproduced in the simulations. We believe that a likely reason for this is that in the experiment a very low energetic second pulse co-propagated in the laser cavity causing some interference fringes in the spectrum.
3. Thin-disk lasers in the positive dispersion regime
After the successful adjustment of the theoretical parameters to the experimental data we studied in more detail the power scaling of chirped-pulse lasers. Since the pulse energy in the Yb:KYW bulk laser is limited by the thermal load of the laser crystal, we focused our work on a thin-disk setup. Since cavity-dumping in a thin-disk setup does not work with high efficiency due to the large laser spot size on the gain medium , we focused our efforts on a regular thin-disk oscillator operated in the positive dispersion regime, which has not been reported so far.
The parameters used in the model were chosen in accordance with a standard Yb-tungstate thin-disk oscillator close to possible experimental parameters and are summarized in Table 3 (τ L and σ L are the upper-state lifetime and the gain cross section of the laser crystal). The transmission of the output coupling mirror was 10% (not included in the losses l). To investigate the energy scalability of this system we varied the small signal gain and the dispersion over a wide range.
The results of this parameter scan are shown in Fig. 2. On the left the evolution of the spectral shape for different values of intra-cavity GDD at pulse energies around 2.1 µJ is shown. For this
plot the GDD was scanned from 855 fs2, which was the smallest value for stable mode-locking, to 2000 fs2 while all other parameters were kept constant.
The right side of Fig. 2 shows how the output spectrum evolves at three different dispersion levels (1050, 1620 and 2000 fs 2) and three different pulse energies (0.5, 2.1 and 3.6 µJ). The broadest spectra can be achieved for low values of intra-cavity GDD (1050 fs 2) and high pulse energies (3.6 µJ). At this optimum point of operation the spectra are slightly M-shaped; with increasing dispersion the spectra become narrower and narrower, rectangular and then almost parabolic and later asymmetric.
The behavior of the spectral shapes are very similar to what has been observed in the experiment  and also to what was calculated for chirped-pulse oscillators in Ti:sapphire lasers . It can be explained by the effects of both higher-order dispersion and non-linearities on the laser pulse inside the cavity.
However, while for Ti:sapphire oscillators third and fourth order dispersion have been found to severely limit the energy scaling abilities of these systems , in our simulations this behavior has not been observed, even for values as high as 10,000 fs 3 or several 100,000 fs4. The reason for which can be seen in the bandwidth of Yb-based gain media which is small compared to Ti-sapphire. This is supported by the fact, that Yb-doped fiber laser can also tolerate for a certain amount of third-order dispersion without loosing pulse quality .
In the next step we explored with our model the regimes of even higher pulse energies that have not been reached so far with oscillators in the solitary regime. Indeed, energy scaling well beyond 10 µJ can be anticipated directly from the oscillator based on the simulations. The first plot in Fig. 3 shows how the required value of positive intra-cavity GDD increases for pulse energies ranging from 0.5 µJ to 20 µJ. This can be explained by the fact that as the pulse energies rise the pulse also needs to be stretched more in order to avoid increasing non-linearities. This is achieved by increasing the intra-cavity GDD. For pulse energies beyond 2 µJ the required GDD needs to be increased almost linearly. Accordingly the second plot of Fig. 3 gives the corresponding pulse duration of the chirped laser pulses which also increases with an almost linear behavior from around 2 ps at 2 µJ to nearly 15 ps at pulse energies around 20 µJ. Most interesting is the Fourier limited pulse length from the last plot of Fig. 3; For pulse energies between 0.5 and 6 µJ the duration of the dechirped pulse decreases from above 700 fs to a minimum around 630 fs which corresponds to a broader spectrum from the increasing pulse energy despite the increase in GDD. For pulse energies above 6 µJ however, the Fourier limit starts to increase again. We believe that this can be explained by the fact that there are two different effects at work here. On the one hand the increase in pulse energy (which is a direct result of an increased small-signal gain) will lead to a broader spectrum as the increased gain allows for laser action of larger parts of the emission spectrum. On the other hand the increasing GDD puts an increasing limit on the spectral width for which the mode-locking condition can be fulfilled. For pulse energies up to 6 µJ the broadening effect governs the evolution of the spectral, leading to a net broadening of the spectrum while for pulse energies above this value the effect of the increasing GDD takes over and leads to a reduction of the spectral width. Nevertheless, even for pulse energies above 15 µJ pulse durations below 650 fs are achievable in the numerical simulations. It should be pointed out that due to the comparatively small bandwidth both the simulation and first experiments  reveal that pulse compression close to the Fourier limit can be easily achieved by a grating pair.
The numerical simulations clearly show that the chirped pulse oscillator scheme seems to be well suited for energy scaling in high power thin-disk lasers based on Yb-doped gain material. Pulse energies up to 20 µJ are achievable in the numerical simulations and no intrinsic limitation of the energy scalability could be found. However, for an actual realization several points need to be considered: To avoid thermal lensing and damage to both the thin-disk and the SESAM at these high powers large spot sizes need to be used which imposes great demands on the surface qualities of the materials used. Furthermore, the amount of GDD required for stable pulsed operation at these high energies is rather high. Simply adding bulk glass as might be feasible in the case of Ti:sapphire is not practical since the high values of GDD would require up to several centimeters of glass to be used in the resonator which would make the resonator design extremely difficult. One might better consider using specially designed positive dispersive chirped mirrors which would allow for an easier resonator design and avoid some problems with extra losses on the bulk material. Finally, the pulse chirp is intrinsic to the positive dispersion regime, and obviously extra-cavity pulse compression via either a prism or a grating compressor is required. However in contrast, current high power solitary lasers require to be operated under a Helium atmosphere [6, 16] or need a rather complex ’active multi-pass cell’ to introduce the vast amount of negative dispersion . Even without pulse compression lasers in the positive dispersion regime can be useful as seed oscillators for amplifier systems, lowering the requirements for the stretching setups.
In conclusion we have reported for the first time on the numerical studies of chirped-pulse oscillators based on Ytterbium doped gain media. In comparison with recent experimental data the results of the numerical simulations clearly point to the fact that the chirped pulse scheme for high power femtosecond lasers holds great promise to reach pulse energies that have so far not been available directly from laser oscillators.
The authors thank Max Lederer for cooperation and support. This work was supported by the Bundesministerium für Bildung und Forschung (BMBF) under contract 13N8723.
References and links
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