We explain a pulse compression mechanism reported in picosecond Raman lasers pumped by continuous trains of mode-locked pulses. Our theoretical model is based on transient Raman scattering equations, and shows good agreement with the experimental results. The model reveals that the compression effect is produced by a combination of group velocity walk-off and strong pump pulse depletion. We predict the possibilities and the limitations of this technique for constructing highly efficient, low cost, ultrafast Raman lasers in the visible.
© 2010 OSA
Ultrafast solid-state visible lasers are essential for a range of applications that require tunable short pulses in the yellow-orange range of the spectrum, such as biological imaging by two-photon fluorescence . Ultrashort pulses in this region are generally derived in a variety of ways from Ti:Sapphire lasers; for example optical parametric oscillators (OPOs) are common but can be relatively expensive and hard to use , or propagation in photonic crystal fibers (PCFs) can generate a range of wavelengths but the overall optical efficiency is relatively low .
A simple and reliable solution is to utilize stimulated Raman scattering (SRS) processes, since they allow efficient wavelength conversion from industry standard 532 nm lasers to yellow-orange wavelengths in one step . In particular, SRS in crystalline media has been widely used in different configurations to efficiently generate IR, visible and UV output . Using a cavity around a Raman medium has the advantage of allowing effective control over the conversion and cascading of the SRS process to second and higher Stokes orders, so the desired Stokes order can be selectively output or several wavelengths output simultaneously. A resonator configuration also reduces the lasing threshold, allowing the use of lower power pump lasers.
In the picosecond regime, a short resonator is not effective, but lasing can be easily achieved by using a resonator length that matches the repetition rate of the pump laser so the train of pump pulses and the intracavity Stokes pulses are synchronized. Such ‘synchronously pumped’ Raman lasers have been demonstrated as an efficient route for the generation of picosecond pulses at a range of visible wavelengths . For example, using a 50 mm long KGW crystal pumped by a 532 nm mode-locked picosecond pump laser, we have demonstrated up to 25.6% conversion from 532 nm to 559 nm, with some configurations showing efficient pulse shortening by up to a factor of three when the cavity length is precisely adjusted . Such systems are ideal for applications requiring high peak power at high repetition rates in the visible. However, the dynamics of pulse compression observed in such schemes have not been fully understood. In this paper, we present a numerical model that reproduces the dynamics of synchronously pumped continuous-wave mode-locked Raman lasers and enables us to identify the best regimes for simultaneous pulse compression and maximum efficiency.
2. Description of the model
We considered a standard z-fold cavity setup for the simulations as depicted in Fig. 1 , similar to that used in experimental demonstrations [6–8]. The cavity was composed of 4 mirrors, although all the cavity losses were lumped into the finite reflectance, R, of the output coupler M4. The simulations modeled the Raman interaction occurring between the incident infinite train of pump pulses and the intracavity 1st Stokes pulse.
Since the pulse duration of a typical mode-locked visible pump laser approaches the phonon dephasing time of crystalline Raman laser materials, we used the equations for transient Raman scattering instead of the more common stationary Raman rate equations. For a derivation of the equations describing the transient Raman interaction, see for example Penzkofer et al. . The equations used in our model are as follows:Eqs. (1-3). T2 is the dephasing time of the Raman crystal, and ωS and ωL are the angular frequencies of the Stokes and pump pulses respectively. The parameter gs correspond to the experimentally measured steady-state Raman gain of the crystal. We simulate a quantity Q’, proportional to the phonon excitation Q, so that only gs, rather than the material parameters m, n and (∂α/∂q), appear in our equations. The correspondence between Q’ and the phonon excitation Q is given by:
We averaged over the transverse dimensions appropriately for Gaussian beams, and include changes of transverse beam size along the Raman crystal owing to diffraction. A key feature of the model is that it takes into account dispersion and the group velocity walk-off between the Stokes and pump pulses through the crystal by using different speeds for each of them (vS and vL respectively) and different indices of refraction μS and μL. The effects of the group-velocity walk off in the transient Raman regime have been studied only for single pass Raman generators [10,11]. Previous simulations for synchronously pumped Raman oscillators [12,13] considered experiments using gases or short Raman crystals, often using IR pulses, for which the group velocity walk off between Stokes and pump pulses could be neglected. Consequently, the study of continuous wave picosecond Raman lasers that use high dispersion crystals has been essentially unexplored; it is this regime that enables short-pulse Raman oscillators that also show dramatic pulse compression.
After using the finite difference method to transform the time and space dependent equations into a first order accurate set of time dependent equations on a spatial grid, we solved the equations numerically using a variable time step size Runge-Kutta algorithm. We solve for a sequence of single passes through the crystal for the co-propagating pump and Stokes pulses, using the output Stokes field from one pass as the input Stokes field for the following pass, thus simulating the circulating Stokes field; the simulation is terminated when the Stokes pulse has reached its steady state profile. To avoid numerical dispersion affecting the profile of the resonated Stokes pulse, it is necessary to solve the equations in a frame moving at the Stokes group velocity. The cavity length detuning from perfect matching Δx is simulated by retarding or advancing the Stokes pulse before it is recycled after each round trip. We approximately model the dispersive broadening of the Stokes pulse by applying a discrete broadening after each pass, sufficiently accurate in the picosecond regime where the single pass broadening is small. We used a 48 Intel Xeon core cluster to perform the calculations efficiently, taking typically between 4 and 6 hours to calculate the laser output of a complete set of cavity lengths.
3. Theoretical and experimental results
Simulations were carried out using input parameters to match the experimental conditions presented in . In those experiments, a 50-mm-long KGW crystal anti-reflection coated at 532 nm was used as the SRS gain medium. The pump source was a frequency-doubled CW mode-locked Nd:YVO4 laser. The 2 W of pump radiation was directly focused through a dichroic mirror (M1) into the KGW crystal. The pump pulse duration was 10 ps, at a repetition rate of 80 MHz. The pump light was polarized along the Nm axis matching the 901 cm−1 Raman shift, corresponding to the conversion of 532 nm to 559 nm. The Raman gain gs was set to 11.8 cm/GW taken from , with the passive losses of the resonator estimated to be approximately 3% at 559 nm, while the output coupling was set 10% from the experimental data. The phonon dephasing time used was 1.92 ps according to , although the results were insensitive to small changes.
The cavity length was a crucial experimental parameter determining the behavior and performance of the Raman laser, and so we present the simulation results for a range of cavity length mismatches. Δx is defined as the cavity length difference compared to the one found to have the lowest threshold, with a negative detuning corresponding to a shortened cavity.
The results from the model are plotted with the corresponding experimental results from  in Fig. 2 , showing the modeled output pulse duration (Fig. 2(a), calculated from the output pulse profiles by simulating an autocorrelation measurement to allow direct comparison to the experimental values), and the output power of the laser in Fig. 2(b). The key experimental observations are well reproduced by the model: strong asymmetry in the behaviour for positive and negative detuning, and a narrow region of pulse compression for small positive detunings. This agreement between theoretical and experimental data suggests that the most important physical processes are included in the theoretical model. We see reasonable agreement between theoretical and experimental data, although the model does not fit well for the most positive values of the cavity length detuning where the experimental pulse duration was substantially lengthened. We are still investigating the cause of this discrepancy. However, the efficient pulse compression mechanism is clearly reproduced.
We can use the model to study the behavior of the laser by investigating behavior that is more difficult to observe experimentally. For this laser system, it is the reshaping of the intracavity Stokes pulse during each pass through the Raman material that determines the behavior. In Fig. 3 we present the pulse shapes of the pump and Stokes pulses before and after passing through the crystal for a range of different cavity length mismatches Δx (online these figures are movies showing how the Stokes and pump pulses evolve during a single pass when in the steady-state regime). These pulses are plotted on a spatial axis in a frame of reference that moves with the Stokes velocity through the Raman crystal. The pulses and the moving frame are moving to the right, so that the right-hand edge of the pulses is the leading edge. We label the axis in time as it gives a more intuitive scale.
The requirement in steady state is that the position of the stokes pulse relative to the arriving pump pulse must be the same at the beginning of each round trip, and this must be true for all cavity length mismatches. This is equivalent to saying that the repetition rate of the Stokes laser must match the pump laser exactly. Any advancement or retardation of the Stokes pulse owing to the cavity length mismatch must therefore be exactly compensated on each round trip by reshaping of the Stokes pulse, owing to the gain experienced through the Raman crystal.
If the cavity is too short (so the detuning is negative) the Stokes pulse must be amplified preferentially on its trailing edge so that it is reshaped by the gain to cause an apparent retardation. The Stokes pulse thus appears to move backwards in the frame moving at the Stokes group velocity, exactly compensating the effects of the shortened cavity. This is most apparent for large negative detuning as can be seen in Fig. 3(a) for Δx = −300 μm, where the trailing edge of the Stokes pulses after the crystal appears shifted to the left. As shown in the movies of Fig. 3, the effect of cavity length mismatch as well as the round trip loss transforms the final Stokes pulse back to its initial form. Note that the pump pulse moves slowly to the left because it travels slightly slower than simulation frame that is moving at the Stokes group velocity.
The reshaping of the Stokes pulse to enhance its trailing edge requires that edge to experience the highest gain; it becomes aligned with the peak of the pump pulse in the steady-state regime. Increased gain for the back of the pulse is also naturally enhanced in the transient SRS regime, since accumulated phonons generated by the preceding parts of the pulse remain available to amplify the trailing edge. The pulse has a relatively long leading edge caused by the continual advancement of the Stokes energy when the cavity is shortened.
Figure 3(b) shows the Stokes and pump pulse behavior for the cavity length detuning Δx = −30 μm, which produces the highest output Stokes power. In this case the Stokes and pump pulses are well overlapped through the crystal, and the pump pulse is strongly depleted.
The situation is quite different for positive detunings as it can be seen in Fig. 3(c) for Δx = + 8 μm. In this case, the Stoke pulse arrives a little delayed for each round trip, and to maintain the steady state, the pulse must be reshaped by preferential amplification of the leading edge to cause a compensating advancement. Only very small positive detuning can be tolerated: very limited preferential amplification of the leading edge can occur since in transient SRS there is always the tendency for gain to be lower at the pulse leading edge.
Towards the limit of positive detuning, the leading edge becomes aligned with the peak of the pump pulse to maximize the pulse advancement. This positioning also results in pronounced steepening of the leading edge of the Stokes pulse: it sees the most gain in this regime and becomes intense enough to deplete the energy from the pump pulse as the pump pulse is overtaken due to group velocity walk-off. The leading edge sweeps through the heart of the pump pulse, and the accumulation of energy by the leading edge of the Stokes pulse results in the observed pulse shortening. While the same sweeping action occurs for all cavity length detunings, it is only for the positive detunings that the leading edge is initially positioned near the centre of the pump pulse and so can deplete a significant fraction of the pump pulse energy. At the position for maximum compression to 3 ps, even though the output power is decreased slightly there is still a peak power enhancement of approximately 2.3 times compared to that at the position for maximum output power as can be seen in Fig. 4 .
We now examine the importance of dispersion and the Raman dephasing time for the pulse compression mechanism. Using the Sellmeier equations in , the group delay mismatch between pump and 1st Stokes in KGW is found to be 83 fs/mm, which over the 50 mm confocal length of the cavity waist results in the Stokes pulse overtaking the pump pulse by 4.2 ps on each pass. This is a relatively large fraction of the pump pulse duration and allows efficient compression, since the 3.2 ps compressed pulse can still interact with the majority of the 10 ps pump pulse. In other experiments with a diamond Raman crystal in a similar arrangement, while some pulse shortening was observed, the efficiency in this regime was poor: use of a shorter crystal substantially reduced the walk-off per pass between the pump and Stokes pulses, and this combined with a longer pump pulse duration led to only a small fraction of the pump pulse energy being extracted.
A higher group velocity walk-off enhances the overtaking and pump depletion mechanism (yielding higher compression factors), but at the same time it stretches the Stokes pulses by dispersive broadening. The crystal length optimized for compression is proportional to the pump pulse duration so group delay difference increases proportionally to pump duration. In contrast, the dispersive length (defined as the length needed in the dispersive media to broaden the pulse duration by a factor of ) depends on the pump pulse duration squared. In the experiments, for 10 ps pump pulses the broadening caused by the dispersive media was negligible, and we would benefit from more dispersion. We expect compression to be enhanced for more dispersive Raman materials such as GdVO4. Dispersive broadening will start to play an important role for sub picosecond pump pulses.
The duration of the compressed pulse compared to the dephasing time T2 of the Raman transition is also an important consideration. For the KGW experiment , the 3.2 ps compressed pulse had a similar duration to the 1.92 ps value of T2. As the Stokes pulse duration becomes comparable to T2 or shorter (highly transient SRS regime), the average gain experienced by the pulses is reduced, while the tail of the pulse sees relatively higher gain. These effects tend to hinder further shortening of the Stokes pulse below T2, but the model indicates that it does not represent a limit; substantially higher pump powers are required to further compress the Stokes pulse but we calculate that for twice the maximum pumping power available in that experiment the pulse duration would decrease to 1.4 ps.
In conclusion, we have explained the intracavity pulse dynamics of synchronously- pumped picosecond Raman lasers. Our model confirms that KGW is a suitable material for efficient operation as well as significant pulse compression. More generally, higher dispersion materials (e.g. GdVO4) and materials with shorter dephasing times (LiNbO3) are suggested as promising materials for enhancing this form of pulse compression in future experiments.
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