A Nd:GdVO4 laser mode locked by self-defocussing cascaded Kerr lens in PPKTP is presented. A strong pulse shortening mechanism is produced by the interplay of group velocity mismatch and the cavity design. The cavity had a repetition rate of 200 MHz and the mode-locked output power was 350 mW. Pulses as short as 2.8 ps were obtained with a bandwidth of 0.6 nm.
©2005 Optical Society of America
1. Introduction and background
Nd:GdVO4 is a relatively new laser crystal with a maximum emission cross section at 1063 nm which is somewhat smaller than that found in Nd:YVO4 but with substantially higher thermal conductivity which facilitates high-power scaling of pulsed lasers . The generation of picosecond pulses by mode-locking Nd:YVO4 lasers has been demonstrated by employing semiconductor saturable absorber mirrors  as well as by using cascaded second order nonlinearity and nonlinear mirror mode-locking [3–5], the technique first pioneered by Stankov et al. . These techniques generated mode-locked pulses with the lengths of around 10 ps. The shortest mode-locked pulses of 2.3 ps in Nd:YVO4 has been generated by using the additive-pulse mode-locking technique . As the gain bandwidth in Nd:GdVO4 is similar to that found in Nd:YVO4 it is possible to generate pulses with the length of about 2 ps, as was demonstrated by Sorokin et al.  with an additive pulse mode-locked laser.
There are three distinct types of modelocking utilising cascaded second order effects, typically second harmonic generation followed by difference frequency generation. With the quadratic polarisation switching technique pulses as short as 2.8 ps has been demonstrated in Nd:YVO4 . The nonlinear mirror mode locking technique has produced pulse lengths down to appproximately 10 ps for Nd-doped crystalline hosts, limited by the group velocity mismatch (GVM) between the fundamental and second harmonic wave. The modulation depth is mainly determined by the conversion efficiency.
If the cascaded nonlinear processes are not perfectly phasematched a nonlinear phase will be imprinted on to the circulating field . The spatial intensity distribution of beam translates as a curved phase front and this cascaded Kerr lens can also be used for cascaded mode-locking, together with an intracavity hard or soft aperture. The modulation depth of this process will depend on the particular cavity design, but as opposed to the case of nonlinear mirror modelocking, it can be large even for small conversion efficiencies. This reduces the need for high intracavity intensities and improves the reliability of the nonlinear crystal. Usually, the nonlinear mirror modelocking and Kerr aperture effects are employed at the same time , the amplitude modulation due to the dichroic mirror being the main mechanism for self-starting and maintaining stable mode locked laser operation.
To the best of our knowledge, we know of no previously reported laser where cascaded Kerr modelocking is the only modelocking mechanism employed. In this work we demonstrate a diode-pumped Nd:GdVO4 laser passively mode locked by employing a defocusing cascaded Kerr lens realized with periodically poled KTP (PPKTP). This technique allowed the generation of a stable continuous pulse train with pulses as short as 2.8 ps and mode-locked bandwidth of 0.6 nm.
2. Experimental results
The cavity design is schematically shown in Fig. 1. The laser has a four-mirror folded z-cavity with two intracavity foci located on the flat input and output couplers M1 and M4, respectively. The cavity length was 740 mm and the distance between M2 (ROC=500 mm) and M3 (ROC=100 mm) was 262 mm. The output coupler was placed 50 mm from M2. The full folding angles at M2 and M3 are kept below 6 degrees in order to minimize astigmatism. All mirrors except the output coupler had high reflectivity at 1063.5 nm, the central wavelength of the laser. The reflectivity of the output coupler was 90%. A Nd3+:GdVO4 laser crystal with 0.3 atm% doping and dimensions 2x2x3 mm is pumped with a fiber-coupled CW diode laser operating at 808 nm. The laser crystal was slightly tilted in order to avoid back reflections. A 2.3 mm long PPKTP crystal with poling period of 9 µm was placed at a distance of 5 mm in front of the output coupler. The exact off-focus position of the PPKTP crystal is important for self-starting of a stable continuous mode locked pulse train generation. The temperature of the PPKTP crystal was kept at 21.8°C, well off the phase-matching peak at 1.8°C (see Fig. 2). At this temperature detuning the cascaded second order interaction emulates the Kerr effect with a negative nonlinear index coefficient.
In order to start mode locking, the laser must be operated close to its stability limit. This was easiest done by decreasing the pump spot radius in the laser crystal from 440 µm at the position of maximum output power to 330 µm, thereby increasing the thermal lens power by a factor of approximately 1.8. At this pump spot size and with a launched pump power of 16.5 W, the laser works in the thermal roll-over regime, as shown in Fig. 3. With increased thermal lens power the cavity mode size in the laser crystal increases and the mode matching is worsened as the stability limit is approached. The diffraction losses increase, since the laser crystal is the smallest aperture in the cavity. In the geometrically unstable regime, the cavity mode roundtrip magnification increases with increased thermal lens power. Our thermal lens measurements indicate that the laser is operating in the unstable mode. This conclusion is also supported by the rather poor efficiency of the laser and by the presence of rings around the central lobe in the far-field beam. The effects of the strong thermal lens are counteracted by the negative cascaded Kerr lens, with decreased diffraction losses, improved mode overlap and increased stability as result. This is the mechanism behind the mode locking. More details and the results of simulations are given in the next section.
When the cavity is aligned and the pump power and spot size are at the right values for mode-locking, a small adjustment of the PPKTP position is needed and the mode-locking self starts. A fast photodiode oscilloscope trace over a 200 µs time scale (longer than the time scale for Q-switching and relaxation oscillations) and with an inset over a few pulses is shown in Fig. 4. The laser average power was about 350 mW and the pulse repetition rate 200 MHz. From the autocorrelation trace we deduced the pulse length of 2.8 ps by assuming a sech2 pulse shape (see Fig. 5). The laser spectra measured in CW and mode-locked operation are displayed in Fig. 6. When operating in the CW mode the Nd3+:GdVO4 laser line of approximately 0.1 nm FWHM could be tuned over 0.6 nm range. This corresponds to the gain bandwidth in the Nd3+:GdVO4. The mode locked spectral width increased more than 6 times to 0.62 nm indicating that the whole available gain spectrum has been mode-locked. The pulses are 1.5 times transform-limited. The modulation in the spectrum comes from the cascaded nonlinear phase shift at phase mismatch close to π. The modulation can be reduced by operating at larger phase mismatch and higher intracavity intensities in order to realize the same cascaded nonlinear phase shift .
3. Mode-locking and pulse shortening mechanisms
In our experiments we utilized only the cascaded nonlinear phase shifts produced in the phase-mismatched second order interaction, which emulated a negative effective Kerr lens. In contrast to ordinary Kerr nonlinearity, the cascaded second order nonlinearity can produce both positive and negative effective third order nonlinearities, with the sign being opposite to that of the phase mismatch ΔkL. The sign and the strength of the resulting Kerr lens can be chosen easily, for instance by temperature tuning. In the small depletion limit of the plane wave approximation, the nonlinear refractive index is proportional to (sinc(ΔkL)-1)/ΔkL. For our PPKTP crystal the maximum effective Kerr coefficient nI2 was estimated to be 2.2·10-13 cm2/W in the plane wave approximation. In our experiment, ΔkL=+5.5 was close to 2π, the phase mismatch value corresponding to full back conversion but not close to π, the value of maximum index modulation in the plane wave approximation. The value was calculated from the crystal length and the temperature offset from phase matching. Solutions of (2+1)D wave equations, outlined below, showed that this operating point corresponds to the maximum of the defocusing cascaded Kerr lens power when the radius of curvature of the spatial phase front is taken into account.
In our laser cavity the negative cascaded Kerr lens counteracts the effects of the strong thermal lens. When the laser cavity operates close to the stability limit the diffraction losses are very sensitive to small changes in the power of the effective Kerr lens. This gives an opportunity for a self-starting mode-locked operation of the laser without employing amplitude modulation with a dichroic mirror as was used in previous schemes [3–6,9]. The effect of the negative cascaded Kerr lens causes the mode size in the laser crystal to decrease thus diminishing the roundtrip diffraction losses in the limiting aperture in the cavity. Our thermal lens measurements were not conclusive as to whether the cavity was geometrically stable or not, so we simulated both cases. In Fig. 7 a simulation of the cavity in the geometrically stable regime shows how the diffraction losses decrease rapidly with increased Kerr lens power. In the geometrically unstable regime, the cavity mode roundtrip magnification increases the diffraction losses. The magnification losses are counteracted by the presence of a Kerr lens, and a weak Kerr lens can even pull the cavity back into the region of geometrically stable operation. This means that the magnification losses vanish, as can be seen in Fig. 8. Note the strong sensitivity on the thermal lens power in both Fig. 7 and Fig. 8.
In order to properly describe phase-mismatched cascaded interaction of the optical waves with large curvature of the phase front one needs to solve a system of (2+1)D wave equations which also take into account different group velocities of the interacting waves. When the pulses are short (<10 ps), the group velocity mismatch (GVM) does play an important role in PPKTP. The coupled wave equations were solved using the SNLO software package developed by A. Smith and co-workers at Sandia National Labs . The calculated phase-front curvature resulting from cascaded interaction is shown Fig. 9 for the case of zero GVM (dotted curve) and the GVM correspondent to that in the PPKTP crystal (solid curve). The temporal profile of the incident pulse is shown also for reference (dashed curve). The negative wave front curvature corresponds to the defocussing Kerr lens. The beam diameter and the phase curvature of the incident beam were 100 µm and 5 mm, respectively. These values correspond to the values obtained in the experimental laser setup. From the graphs in Fig. 9 one can see that the presence of GVM has two important consequencies: first, the negative cascaded phase shift and thus the defocusing lens becomes stronger, and second, the temporal profile of the phase shift becomes bipolar. This means that the initial part of the fundamental pulse will experience a net positive cascaded Kerr lens, while the negative cascaded Kerr lens will affect the trailing part. The physics of this effect is rather simple: the energy converted into second harmonic light travel with a smaller group velocity and is thus transferred into the latter part of the pulse. The net phase change is enhanced for this part of the pulse and the cascaded Kerr lens is stronger than it would be without GVM. The effect of the curvature of the spatial phase front on the cascaded Kerr lens can be seen in Fig. 10. Here the focal length fKerr of the cascaded Kerr lens is shown as a function of phase mismatch for the cases of the phase front curvature of the incident beam being infinity (solid triangles) and 5 mm (solid squares). A sinc2 second harmonic efficency curve (solid line) is also shown for reference. As can be seen from the data when the PPKTP crystal is placed slightly off the beam focus, where the phase curvature is relatively large the maximum power of the cascaded defocusing Kerr lens occurs at the phase mismatch close to ΔkL=+5.5 instead of ΔkL=+π as given by a simple plane-wave theory. This corresponds well to the phase-mismatch observed in our experiment. As was stated before, the mode-locking in our laser is based on the diffraction loss modulation where the self-defocussing lens reduces losses while the focussing lens increases them. The bipolar character of the cascaded nonlinear phase modulation caused by GVM (see Fig. 9) then will serve as the pulse shortening mechanism. Indeed the initial part of the optical pulse will suffer increased losses due to net self-focussing cascaded lens, while the rest of the pulse will see decreased losses. It is a quite remarkable difference from the case of nonlinear-mirror mode-locking where the GVM had a bandwidth limiting effect on the generated pulses. The data in Fig. 11 gives the beam diameter-dependence of the pulse shortening (open triangles), defined here as the integral over positive nonlinear phase modulation normalized to the integral over absolute value of the phase modulation. Also shown in Fig. 11 are the dependences of the optical power of the cascaded self-defocussing lens (solid squares) and the losses due to second harmonic generation (solid circles). For these simulations the optimum phase mismatch of ΔkL=+5.5 has been used. From the dependencies we can see that the laser mode diameter at the crystal should be not less than 100 µm. Smaller beam sizes increases intensity which, in turn, makes the cascaded interaction to take place faster, thus reducing the pulse shortening action effect of GVM and also reducing total negative cascaded Kerr lens. Moreover the second harmonic losses are also increasing for smaller beam diameters. This trend is consistent with our experimental observations. In the laser cavity the beam diameter at the entrance of the PPKTP crystal was larger than 100 µm when the mode-locking self-started. The second harmonic generation losses were rather low, of about 1%.
We have demonstrated a mode-locking technique which employs a negative Kerr lens realized by cascaded interaction in a phase-mismatched PPKTP crystal. The nonlinear phase shift is converted into amplitude modulation partly by the spatial gain aperture of the end pumped laser crystal and partly by compensating for diffraction losses. The group velocity mismatch is found produce a focusing lens in the leading edge of the pulse while it increases the defocusing Kerr lens in the trailing edge, thus forming a very efficient pulse shortening mechanism. The cavity had a repetition rate of 200 MHz and the mode-locked output power was 350 mW. Pulses as short as 2.8 ps have been generated and spectral broadening of 6 times has been observed.
We gratefully acknowledge the Göran Gustafssons foundation and the Carl Tryggers Foundation for financial support.
References and Links
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