A Q-switched Nd:YAG laser, an apparatus widely used in industry for cutting, drilling, and marking, has the characteristic of generating various nonlinear dynamical behaviors according to the modulation frequency of Q-switching. Because of this, the operating range of the laser is restricted within a few kHz. For example, when the modulation frequency of Q-switching is higher than 3 kHz, the laser transits from a stable laser output to chaos through period doubling bifurcation as the frequency increases. During the transition, quasi-periodic laser outputs are also generated near 5 kHz, a typical phenomenon in this laser. Of these behaviors, while some chaos control methods to stabilize the chaotic laser outputs have been developed, there was no report of stabilization of quasi-periodicity, which is as much undesirable to the quality of the laser products. In this paper, we aim to stabilize the quasi-periodic behavior of Q-switched Nd:YAG laser outputs. In developing the method, since there is no unstable periodic orbit in a quasi-periodic signal, we choose an arbitrary targeting point on the delayed coordinate (return map) and obtain a control formula from the return map, which perturbs an accessible parameter of the chaotic system. Although our method looks very similar to the return map based chaos control method in the sense of using a return map, it rather belongs to a targeting method because the trajectory is targeted to an arbitrary period orbit. In what follows, first, we illustrate the control method in the Van der Pol oscillator and demonstrate it in a Q-switched Nd:YAG laser experimentally. When the Van der Pol oscillator generates a quasi-periodic signal, we obtain the control formula from the return map of the peaks at each period. In the control formula, we determine a control and a targeting point from the return map of the quasi-periodic signal and obtain such parameters of the control formula from the return map as the slope of the return map around the targeting point and the variation of the return map according to the modulation frequency. When the trajectory is near the control point, the system is perturbed for the trajectory to move to the targeting point according to the stabilization formula. Through this procedure, we can stabilize the quasi-periodic signal to whatever period we want (period 1T, 2T, etc). Very similar is the manner we stabilize the quasi-periodic output of a Q-switched Nd:YAG laser. When the laser operates a quasi-periodic output near 5.5 kHz at 12 Ampere of the pumping current, we obtain a return map from the peak of the Q-switched laser by using a peak-holder circuit. The return map is similar to that of the Van der Pol oscillator. In the experiment, we perturb the modulation frequency by using a voltage-frequency converter. From the return map, we also determine each parametric value of the control formula. When the trajectory of the quasi-periodic output is near the control point we begin to perturb the modulation frequency of the Q-switching to control the laser outputs to a chosen targeting point. By this method, we can easily stabilize the laser output to the desired period-1T, 2T, etc.
