Numerical analysis of fast saturable absorber mode-locked Yb3+-doped solid state lasers is reported. The analysis includes a special case in which the spectral bandwidth of the short pulse is larger than the fluorescence bandwidth of the gain material. The relationship between the available shortest pulse duration and modulation depth for a standard bulk and thin disk laser geometries with several gain materials are shown. The characteristic phenomena observed in our previous Kerr-lens mode-locked laser experiments were reproduced in the simulation.
© 2015 Optical Society of America
In the last few decades, owing to the appearance of a high power laser diodes (LD) and semiconductor saturable absorber mirrors (SESAM) , investigation of mode-locked Yb laser has significantly advanced. At the same time, new high power laser geometries, such as the thin disk laser, has seen a remarkable development . The combination of these developments with the advantages of Yb3+ ion doping, such as: very small quantum defect; high quantum efficiency and relatively broad gain bandwidth  stable highly efficient high power mode-locked Yb lasers have been realized. Yb:YAG and Yb3+:Lu2O3 are two of the most attractive gain material for thin disk lasers due to their excellent thermal and mechanical properties. 141 W average power with a pulse duration of 748 fs and an optical-to-optical efficiency of more than 40% has been achieved with a Yb3+:Lu2O3 thin disk laser . As high as 80 μJ pulse energy with picosecond pulse duration has been directly obtained from an active multi-pass Yb:YAG thin disk laser . Compared with a fiber laser geometry, the thin disk laser geometry has an advantage in the scope of high pulse energy operation as it can suppress excess nonlinearity inside the cavity because of its thin active material length and large laser mode diameters. The amplification bandwidths of Yb-doped crystalline materials used in thin disk lasers, however, are narrower (e.g. Yb:YAG ~8.5 nm) than that of Yb-doped glass materials and the available shortest pulse durations have been strongly restricted by them. One of the promising ways to obtain short pulse duration from Yb-doped crystalline lasers is utilizing optimized gain materials (broad gain bandwidth. e.g. Yb3+:CaGdAlO4 ) and SESAMs (low two photon absorption ). The first sub-100-fs thin disk laser operation has been reported with a disordered Yb3+:LuScO3 and optimized SESAM by C. J. Saraceno et al. . Another promising way to obtain further short pulse duration is Kerr-lens mode locking (KLM) technique. The KLM Yb3+ lasers have overcome their gain bandwidth limitations (e.g. 71 fs from Yb3+:Lu2O3 , sub-40 fs from Yb:YAG ) where the spectral bandwidths of the pulses could be broader than the emission bandwidths of their gain materials. The first KLM thin disk laser of Yb:YAG crystal (sub 200 fs) has been reported by O. Pronin et al. . Recently, 270 W average power with a pulse duration of 330 fs has been achieved with KLM Yb:YAG thin disk laser . There are many important sophisticated analytical and numerical simulations about fast and slow saturable absorber mode locking, e.g [13–20]. For slow saturable absorber mode locking, the limitation of the short pulse operation was numerically calculated with several parameters by R. Paschotta et al. . The difference between slow and fast saturable absorbers and their analytical and numerical simulation including multi-pulse operation were described by F. X. Kärtner et al. . However, there are few reports addressing mode-locked laser operation where the general gain bandwidth limitation is overcome, in which the spectral bandwidth of the pulses is broader than the gain bandwidth. There are three conditions which should be satisfied during short pulse operation.: Firstly, the short pulse operation should have a lower threshold than CW operation. Secondly, the single short pulse operation should have a lower threshold than multi-pulse operation. Thirdly, the broad spectrum and phase of the short pulse should be sustained and stabilized during operation. The main limiting factor could depend on the method of mode locking, intensity and pulse duration.
This paper aims to deepen understanding and visualize the mechanism and limitation of fast saturable absorber mode-locked Yb3+ lasers with large modulation depth. Numerical simulations of fast saturable absorber soliton-like mode-locked Yb3+ doped solid state lasers based on Split-Step-Fourier (SSF) method are reported. The relationships between the available shortest pulse duration and modulation depth for a standard bulk and thin disk laser geometries, with several gain materials and linear loss, are shown. The simulation includes mode-locked laser operation overcoming the general gain bandwidth limitation and with combined active gain medium. The influence of combined active gain medium and characteristic phenomena observed in our previous Kerr-lens mode-locked laser experiments were successfully reproduced by the simulation.
2. Simulation modelEq. (1) at the right side are a linear loss and linear phase shift, respectively. In the simulation the value of linear phase shift was arranged to shift the pulse position to the center of the calculation time window (20ps). The second term is a dispersed gain. The third term is the cavity dispersion. The fourth term is a saturable absorption and self-phase modulation (SPM). In the numerical simulation, the n + 1th roundtrip pulse can be obtained by applying a transfer function [15, 19] to nth roundtrip pulseEq. (3) are slightly modified from Eq. (1) for our calculation. For example, γ in Eq. (1) equals –γ2 in the Eq. (3) and g (ω) in Eq. (3) is based on a measuredfluorescence data. We adopted three assumptions for the saturable absorber. Firstly, it has an ideal instantaneous response time (fast saturable absorber). Secondly, the saturable absorption becomes 0 for the laser intensity higher than its saturation intensity (Fig. 2(a)). Thirdly, it has no two photon absorption (TPA) loss. TPA would cause another limiting factor of a high peak power short pulse operation, but the main topic of the manuscript is the limitation and mechanism of short pule operation utilizing fast saturable absorbers. This fast saturable absorber is different from a slow saturable absorber, which has a strong absorption only in the front part of the pulse and a long recovery time, leading to typical limitations such as growth of parasitic component within the recovery time . Under the above assumptions the function of the saturable absorption effect γ can be written asFig. 2(b). According to the basic soliton equation, in the double pulse operation we assumed twice pulse duration with half pulse energy. In order to suppress double pulse operation [17, 18] the single pulse operation should have a higher total gain than that in the double pulse operation. As shown in Fig. 2(b), at the S of ~2, the residual saturable loss in the single pulse operation become small and the difference between the single and double pulse operation becomes nearly maximum, so that we keep the S of almost constant value of ~2.0 at the steady state during the simulation by modifying the pump power parameter or Isa. For the cavity dispersion D(ω), second order dispersion D2 and third order dispersion D3 were taken into account. The reabsorption effect was also taken into account in the simulation. The rate equations with reabsorption loss and broad spectral bandwidth are written below17]. In our previous KLM laser experiments, the available shortest pulse duration was always limited by the appearance of CW component instead of Q-switching instability and therefore the assumption would not detract from our simulation results. The effect of Q switching instability was reported in [14, 21] and shows it would limit the small pulse energy instead of the short pulse duration. With these assumptions applied to the rate equations, the saturated gain g(λ) can be expressed by following functionsTable 1.
3. Simulation of standard bulk laser geometry
The gain profile of Yb3+:Lu2O3 used in the simulation is shown in Fig. 3(a). The gain has a peak at the wavelength of 1033 nm with a bandwidth (FWHM) of 13 nm. The emission and absorption cross sections at the peak are 1.2 × 10−20 cm2 and 0.07 × 10−20 cm2, respectively. The nonlinear refractive index is 1.2 × 10−15cm2/W and the repetition rate is 87 MHz. These parameters were decided based on our previous KLM experiment [e.g. 9]. As an initial intra cavity laser state, we used a 500 fs, 40 nJ main pulse with a 20 ps, 1 nJ quasi CW component(Figs. 3(b) and 3(c)). The requirement of the initial state parameters strongly depends on saturation parameter S. If S is too small (typically S<1.5.), we need large pulse energy and/or short pulse duration for initial state to achieve a steady state short pulse operation. If S is larger than 2, much smaller seed pulse energy and longer pulse duration would be fine (e.g. 10 nJ, 2 ps). More detailed investigation of the requirement of a self-starting mode locked operation is an interesting subject, but it is outside the scope of this paper and not easy as we assumed time independent small signal gain. The pump dichroic mirror was assumed to have 99.9% loss below the wavelength of 990 nm. The simulation was done under the variable GVD values with several modulation depths and linear loss. The results of pulse durations in our simulation are in good agreement (< ± 5% difference) with an analytical solution of the pulse durations in soliton mode-locking, e.g. Eq. (17) in ref . for the same parameters (pulse energy, nonlinearity and GVD). The simulated evolution of intracavity pulses with the following parameters, output power of ~1 W, D2 = −1600 fs2, D3 = −9000 fs3, L0 = 7%, and ΔL = 10%, is shown in Figs. 4(a) and 4(b). The pulse grows as it circulates in the cavity and it becomes a very stable pulse at the terminal state. The number of roundtrips is 2000~10000, depending on the parameters. The pulse duration is 58.6 fs and the spectral bandwidth is 19.0 nm, which is 1.46 times broader than the fluorescence bandwidth of the gain material (13 nm). The center wavelength was 1037.7 nm. The simulation results with a spectrum at steady state with a slightly smaller D2 value (D2 = −1550 fs2) is also shown in Fig. 5(a) with the gain profile of Yb3+:Lu2O3. In this case, the spectrum includes a narrow component at its gain peak. This indicates imperfect suppression of background CW-like component. As the time window of the simulation is 20 ps, the CW-like component cannot be fully stabilized and therefore the fine structure of the component in the Fig. 5(a) could include some artificial error, but the total gain and loss for the CW-like component is similar to pure CW. Further decreasing of the D2 value leads to a break of the pulse operation into a CW-like operation and a decrease of the intra cavity power of about40% due to a large increase of the residual saturation loss. In Fig. 5(a), we can see a strong shift of the center wavelength of the pulse from the gain peak position to longer wavelength side. The calculated result of the dependence of the center wavelength on the pulse duration with and without a large reabsorption loss and negative/positive TOD is shown in Fig. 5(b). Under the large reabsorption (in the simulation, we used ~5.3% at 1033-nm peak), the shorter pulse duration leads to a longer center wavelength. The shorter pulses have a broader spectral bandwidth, which suffers reabsorption loss at its shorter wavelength side. Therefore the total gain increases by shifting to the longer wavelength side. At the much shorter pulse duration range (<100 fs), it also shows an influence of TOD. The large positive (negative) TOD leads to a blueshift (redshift). In our previous experiment the cavity has a large negative TOD with large reabsorption loss is in good agreement with the simulated result of ~70 fs pulse duration at the center wavelength of ~1037 nm . It would be noteworthy that all of the characteristic phenomena (The broader spectral bandwidth than the fluorescence bandwidth of the gain material, the large redshift of the center wavelength, the appearance of narrow CW-like component, and the huge decrease of the output power after the transition from short pulse operation to CW operation) observed in our previous KLM laser experiments have been reproduced in the simulation by the influence of the large modulation depth, self-phase modulation, reabsorption and high order dispersion. We define the available shortest pulse duration as the shortest pulse duration without the CW-like component. The shortest pulse duration for each linear loss and modulation depth for fixed gain were estimated by changing the D2 value. The calculated available shortest pulse duration as a function of the modulation depth ΔL under several linear loss values L0 are shown in Fig. 6. For comparison, the simulation results with twice higher intra cavity pulse energies resulting twice higher SPM are also shown in Fig. 6. The results were well approximated by linear lines in log-log scale graphwith the factor of ~ΔL-0.66. In ref , the shortest available pulse duration of slow saturable absorber mode locked lasers was expressed with the different factor of ~ΔL-0.5. This could be caused by a difference of the slow and fast saturable absorbers. Moreover it is worth noting that the results with different modulation depth were also well approximated with a single fitting function. Implying that in this simulation range, the limiting factor of the short pulse operation is common. (Note, the results in right bottom of Fig. 6 have much broader spectral bandwidth than gain bandwidth, while results in upper left of Fig. 6 have narrow spectral bandwidth). Figure 6 indicates that larger linear loss L0 leads to larger modulation depth ΔL for the same pulse duration that is also similar to an analytical solution e.g. Eq. (16) in ref . In addition, the available shortest pulse duration is not strongly affected by the difference of the pulse energy (nonlinearity) in this simulation. It is noteworthy that much larger nonlinearity could affect the available shortest pulse duration, e.g. a laser cavity with additional nonlinear medium to increase SPM, where spectral shape is not Sech2 [22, 23]. The deteils of this, however, are outside the scope of this paper. The available shortest pulse duration also slightly depends on the saturation parameter S. Under the large saturation parameter value, the pulse duration of ~5% shorter values could be obtained, which might be caused by lower residual saturable loss. However, the difference is small and in the real experiment, the excess saturation could lead to some serious problem such as nonlinear absorption, excess Kerr-lens effect, and multi-pulse operation. According to Fig. 6, if we assume 7% total linear loss, a modulation depth of 8% was necessary for our previous Yb3+:Lu2O3 KLM laser operation of the 71 fs pulse duration .
We have also simulated mode-locked lasers based on combined active gain media, where two different kinds of Yb3+ doped gain materials are used in the same cavity to increase its effective gain bandwidth (inhomogeneous broadening) . We used a combined gain profile of Yb3+:Lu2O3 and Yb3+:Sc2O3 (CYb = 3%, 1 mm thickness for each material) shown in Fig. 7(a). The calculated available shortest pulse duration as a function of the modulation depth is shown in Fig. 7(b). For comparison, calculated results of Yb3+:Lu2O3 and Yb:YAG (gain bandwidth of 8.5 nm, CYb = 4%, 2 mm, nonlinear refractive index of 0.65 × 10−15 cm2/W) are also shown in Fig. 7(b). With a 10% modulation depth, the available shortest pulse durations for Yb:YAG, Yb3+:Lu2O3 and Yb3+:Lu2O3/Yb3+:Sc2O3 combined active gain medium are 70 fs, 100 fs, and 45 fs, respectively. The pulse duration of Yb:YAG and Yb3+:Lu2O3 are twice and 1.5 times broader than that of the combined active gain medium. The gain broadening effect in the combined active gain medium seems large on pulse shortening. However, if each peak is not sufficiently saturated, a multi-wavelength laser operation and/or a parasitic lasing component can easily appear due to its inhomogeneous gain property. In addition, if the additional insertion loss of the combined active gain medium is too large, we cannot obtain the advantage of the gain broadening for pulse shortening.
4. Simulation of thin disk laser geometry and multi-pulse operation
In the simulation of the thin disk mode-locked laser, we adopted the following parameters: Yb3+:Y2O3 gain medium with a gain bandwidth of 15 nm, thickness of 300 μm, laser mode area of 785000 μm2, intra-cavity pulse energy of ~6.7 μJ, CYb = 2%, L0 of 7~22%, repetition rate of 45 MHz. The SPM effect of the atmosphere (n2 = 4.2 × 10−19 cm2/W) was also taken into account. The model cavity was also modified for the thin disk laser where the laser beam passes through the gain medium 4 times in a round trip . The total nonlinearity (B integral) of this thin disk laser simulation is ~3 times smaller than aforementioned standard bulk laser geometry. The results of the simulation are shown in Fig. 8. The results imply that the large fast saturable absorption effect enables short pulse operation overcoming the general gain bandwidth limitation even with the thin disk laser geometry that has lower nonlinearity. The fast saturable absorber with a large modulation depth has large pulse shaping and spectral broadening effect that seems key to the pulse shortening. Actually the effect also could be interrupted as other form of nonlinearity (This nonlinearity is not directly depend on B integral value). During the simulation, excess decrease of the nonlinearity (n2 and/or pulse energy) leads to difficulty in converging to the stable soliton mode-locked pulses with Sech2 pulse shape. The increase of pulse duration with low nonlinearity was also reported in slow saturable absorber based soliton mode locked lasers . According to Fig. 8, ~1.5% modulation depth would be enough for a generation of sub-200 fs pulses for 7% linear loss.
The stability against a transition from single pulse mode-locking to multi-pulse mode-locking was also simulated for a large saturation parameter S. In this simulation, we used almost the same parameters as in the standard bulk laser geometry and added additional sub-pulse component to the initial state (Figs. 9(a) and 9(b)). With the sub-pulse component of 4% pulse energy against the main pulse, the modulation depth of 10%, and linear loss of 12%, even if S becomes larger than 20, the sub pulse component disappeared and the single pulse operation (~70 fs) was sustained at steady state. On the other hand, if we decrease the modulation depth under large S parameter (for example, ΔL = 8% with S = ~10), double pulse operation easily developed even with a very small sub-pulse component (30 dB lower pulse energy to the main pulse). Further decrease of the modulation depth easily leads to multi-pulse operation even without any additional sub-pulse components (Fig. 9(c)). This is because the 8% modulation depth is too small to sustain the ~70 fs pulse duration. If S is smaller than ~7, transition to multi-pulse mode locking was not observed, because in soliton mode locking double-pulses have half pulse energy and twice pulse duration, leading a large residual saturation loss in fast saturable absorber mode locking.
In conclusion, we reported numerical analysis on short pulse operation of fast saturable absorber mode-locked Yb3+-doped solid state lasers. The relationship between the available shortest pulse duration and the modulation depth for standard bulk and thin disk laser geometries were shown under several linear loss and gain materials. Large modulation depth seems to be a key to enable a short pulse operation and overcoming the material gain bandwidth limitation. A sufficient value of SPM is required to grow up and sustain the sech2 pulse shape, but large SPM does not seem to be necessary. The characteristic phenomena (broader spectral bandwidth than the fluorescence bandwidth of the gain material, large redshift of the center wavelength, appearance of narrow CW-like component, and huge decrease of the output power after the transition from pulse- to CW operation) observed in our previous Kerr-lens mode-locked laser operations were reproduced in the simulation by the effect of the large modulation depth, self-phase modulation, reabsorption and high order dispersion. The usage of the combined active gain medium was also shown in the simulation. The simulation should be useful to design high power short pulse mode-locked oscillators, with not only Yb3+ doped materials but also other gain materials.
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