This work demonstrates the suppression of chaos in a Nd:YVO4 laser by biharmonical pump modulation (the first for chaos-inducing and the second for chaos-suppressing). The laser exhibits chaotic behavior when only the first signal is applied for pump modulation and its frequency is adjusted close to the relaxation-oscillation frequency. Adding the second signal with subhamonic and a specific phase difference to the first modulation signal will reshape the modulated waveform of the pump beam to suppress the aforementioned chaotic behavior. The initial phase of the second harmonic perturbation plays an important role in the suppression of chaos. This result is confirmed by numerical simulation.
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Since Ott, Grebogi and Yorke demonstrated the first report to control chaos, the control or suppression of chaos has attracted great research interest. These authors showed that a periodic orbit in the chaotic attractor can be stabilized by adding small time-dependent perturbations in an accessible system parameter . Various schemes of control chaos, such as time-delayed feedback, occasional proportional feedback, and non-feedback, have been proposed in the subsequent research . In lasers, a chaotic Nd:YAG laser system that contains an intracavity KTP crystal has been controlled using occasional proportional feedback . A non-feedback scheme that employs a weak periodic perturbation has been adopted to stabilize a chaotic CO2 laser .
The non-feedback control is mainly based on periodic excitation of the system. The suppression of multimode laser chaos by applying near-resonant perturbation to a modulated microchip LiNdP4O12 laser with Doppler-shifted light feedback has been observed . The second subharmonic modulation suppresses chaotic laser behavior that is associated with a modulation mechanism. Suppression of chaos in a Nd:YVO4 microchip laser is accomplished by injection of one of the periodic orbits into the bifurcation region of another chaotic system driven by pump modulation . Theoretically, a study of control of chaos in a model of a multimode neodymium-doped yttrium aluminum garnet laser using periodic perturbations of accessible control parameters had been proposed . The modulation of losses is a parametric modulation, while modulating the pump is an example of an external modulation. A small modulation of either the losses or the pump can eliminate chaos or make the system even more chaotic.
Moreover, Chacón presented that the geometrical resonance provides the mechanism underlying the non-feedback control of chaos . When a system is modulated by two external harmonic forces, one induces chaos and the other can suppress chaos if the geometrical resonance is satisfied. Further, the initial phase of second harmonic perturbation can play a switching role in the suppression and enhancement of chaos in nonautonomous system . The phase differences in bihamonic modulation and amplitude modulation had been reported in CO2 laser [10,11], loss modulation, i.e. parametric modulation, was concentrated. However, this important role was not discussed in the aforementioned studies of solid-state lasers with pump modulation in detail.
This work examines the suppression of chaos in a Nd:YVO4 laser with a biharmonic pump modulation in which a system with an external modulation is focused. A laser typically behaves chaotic when the modulation frequency is adjusted close to the relaxation-oscillation frequency [12,13]. However, adding the second sinusoidal modulation to reshape the waveform of pump modulation will suppress the chaotic dynamics to be periodic. The initial phase difference of these two modulated signals and modulation depths, defined as the modulation amplitude divided by the average value of the pump intensity, determine the region of the suppressing chaos. Numerical simulation supports the suppression of chaos.
Figure 1 schematically depicts the experimental setup for the Nd:YVO4 laser with pump modulation. A diode laser with a wavelength of 808 nm and a maximum output power of 1 W was collimated by an objective lens with numerical aperture of 0.47, beam shaped by an anamorphic prism pair, and focused onto the Nd:YVO4 crystal by another objective lens with a focal length of 8 mm. The Nd:YVO4 crystal had dimensions and 2 at.% Nd3+ doping. One side of the Nd:YVO4 crystal had an antireflection coating at 808 nm and a high reflection coating at 1064 nm. This side was also acted as an end mirror of the laser cavity. The other side had an anti-reflection coating at 1064 nm to reduce the effect of intracavity etalon. A concave mirror with a radius of curvature of Rc = 80 mm and a reflectivity of 90% was used as the other end mirror and the output coupler. The signal of the intensity and spectrum was measured by a high-speed photodetector together with an oscilloscope and a RF spectrum analyzer. A cavity length of 61.27 mm was determined by the beating frequency of longitudinal modes, in which the g1g2 parameters of the cavity configurations are about 0.255 as considering the thermal effect.
The diode laser was connected to two function generators to produce modulated output for pumping. Since modulating the pump is an example of an external modulation , a biharmonically driving system is formed. One function generator, labeled as FG1, generated a modulation to induce the chaos, and the other function generator, labeled as FG2, generated a modulation to suppress the chaos excited by FG1. Sinusoidal signals were chosen to be the external driving forces in this work. The modulation frequency, fmi, and the modulation depth, pmi (defined as the modulation amplitude of intensity divided by the average intensity for the pump laser) were used to characterize the modulated signal. Here, the index i = 1, 2 refers to the qualities of the signals generated from FG1 and FG2, respectively. Two function generators had a synchronizing connection, the initial phase of the signal from FG1 was set to zero and used as reference. The initial phase of the signal from FG2, ϕ, was adjustable and employed to examine the phase effect on the suppression of chaos. The resolution of the phase is 1° in the function generator. The initial phase ϕ is significant to determine the mixed waveform to be periodic only if fm1/fm2 is rational. In this work the frequencies were chosen as fm2 = fm1/2. Since the pump power is a scalar quantity, it is expected that the dynamical behavior of a laser by using one pump modulation of pm1sin(2πfm1t) + pm2sin(2πfm2t + ϕ) is equivalent to that of a laser simultaneously pumped by two modulated sources, one is pm1sin(2πfm1t) and the other is pm2sin(2πfm2t + ϕ). Moreover, such a laser system can also be viewed as the pump source of the laser modulated by a reshaping waveform with the frequency of fm2. It is discussed that varying pm2 and ϕ will reshape the sinusoidal waveform generated by FG1 to suppress the chaotic behavior induced by the fm1 resonating to the relaxation oscillation frequency.
In a single modulation, i.e. pm2 = 0, a period-doubling route to chaos was observed in the system that was typical in a laser with pump modulation, when fm1 approached the relaxation oscillation frequency of the laser and pm1 exceeds a bifurcating threshold [12,13]. The threshold of modulation depth for chaotic behavior is pm1 = 14.3% at the relaxation oscillation frequency of 550 KHz. This threshold increases as modulation frequency is detuned away from the relaxation oscillation frequency. Thus, fm1 is fixed to 550 KHz in the following experiments for the optimal resonance condition.
When pm1 = 15.7%, the output intensity of the laser is chaotic under a single modulated pump. Next, the FG2 was opened, and the waveform was set to fm2 = fm1/2 = 275 KHz and pm2 = 32.3%. Figure 2 displays the output and pump intensities of the laser with various initial phases. The output intensity is suppressed to periodic oscillations for ϕ = 220° and ϕ = 255°. However, the oscillation is not a typical period 2 as ϕ = 255°, since the low peak is not located at the middle of two high peaks and the period of two high peaks is around the period of the waveform of FG2. This train may be suitable to be called as period 1 associated with a double-peak waveform. Under this viewpoint, the system transfer to the stable orbit with the period of 1/fm2. These two types of output intensities are the major time evolutions as suppressing the chaos and are different to the evolution in the period-doubling route to chaos under a single pump modulation, presented elsewhere , having a modulated sinusoidal waveform. The stable orbit with the period of 1/fm1 bifurcates to chaos under the single resonant modulation with fm1, and then the system suppresses to another stable orbit with the period of 1/fm2 if we add the second modulation with fm2 = fm1/2. Moreover, the suppressed behavior presents in a specific region of ϕ for a fixed pm2, and away this region chaos keeps, such as ϕ = 260° shown in Fig. 2(c). Comparing with the pump waveforms corresponding to ϕ = 255° and ϕ = 260°, as shown in Fig. 2(e) and 2(f), respectively, there are very similar but the dynamical behavior are very different. Apparently, the suppressing behavior strongly depends on the initial phase ϕ. One of the boundary of the transition from the period to the chaos is ϕ = 258°. Around this boundary, the dynamics behaves an intermittently switch between the period and the chaos, which is different to the period-doubling route to chaos in single modulation,
Figure 3 , with various pm1, plots the boundary of chaos-suppressed region obtained by varying simultaneously the pm2 and ϕ. The chaos-suppressed region decreased as the pm1 increased. The chaos-suppressed region decreased as the pm1 increased and disappears if pm1 exceeds 30.4%. For each pm1, there is a minimal requirement of pm2 to suppress the chaos. The suppressed range of phase diminishes as pm2 increases. The maximum modulation depth of pm2 is 44.4% occurring around the initial phase ϕ = 216° and this value is independent on pm1. Because a modulation frequency near half of the relaxation oscillation frequency can also induce chaos in single pump-modulation system, a large pm2 will exchange the role from chaos-suppressing to chaos-inducing to make the suppressed region vanish. Experimental results show that the initial phase ϕ plays an important role in the suppression of chaos, and the suppression falls away if this phase cannot satisfy a specific condition. This phase-dependent characteristic agrees with the suppression of the chaos provided by the geometrical resonance, proposed by Chacón . However, a closed region and high modulation depth in this work is different to a tongue shape and a low modulation perturbation in Ref . Additionally, the similar suppressed region also exists at the center of ϕ = 36°. When the initial phase ϕ shifts π, the pump waveform is the same as the one without shift beside an initial time delay of 1/2fm2 in time evolution. It is expected that the same dynamical behaviors will be found. We focus on showing the result relating to the center at ϕ = 216° because the suppressed region contains the negative ϕ if we choose the results relating to the center at ϕ = 36°.
3. Numerical simulations
Via numerical simulation, the laser model was constructed based on the rate equation and the generalized Huygens diffraction integral [13,14]. According to other studies , the optical field, En, and the total atomic population in the excited state ΔNn,i are governed by
Considering a bi-sinusoidal pump modulation, the pumping rate should be represent to
Here, tr is the round-trip time and the spatial distribution of the pumping rate Rpm is set to the Gaussian profile. This study considers a spot size of the pump beam of 300μm at P = 1W. The relation between the pump power, P, and pumping rate was also shown in Ref. 14. The parameters of the laser are σ = 2.5 × 10−18 cm2 and 1/γ = 50μs, and the reflectivity of the output coupler is 90%. The cavity length L = 59.6 mm equal to g1g2 = 0.255 to match the experimental value. Based on these laser parameters, the resonant frequency is fm1 = 758 KHz, and the second modulation frequency is fm2 = fm1/2 = 379 KHz. Figure 4 demonstrates the boundary of chaos-suppressed region with various pm1. This region decreases as pm1 increases and diminishes toward the maximal pm2 = 30% with the initial phase of 216°. This phase and the suppressed region agree with those of experiment well. The modulation frequency and the modulation depth depend on the pump rate, but these values seemly minimize the influence of the suppressed characteristics relating to the initial phase.
This work investigates the suppression of chaos in a Nd:YVO4 laser with the pump modulated by bi-sinusoidal waveforms. The modulation frequency of the first waveform is adjusted close to the relaxation-oscillation frequency, and then the laser typically behaves chaotic when modulation depth thus increases to a given threshold. However, the chaos can be suppressed by adding the second sinusoidal modulation to reshape the waveform of the pump laser under subharmonic frequency. The initial phase and the modulation depth of the second harmonic perturbation must satisfy specific conditions to achieve the suppression of chaos. The numerical simulation for the suppressed region agrees closely with that obtained by experimental result.
The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC 97-2112-M-006-017-MY3 and NSC 97-2112-M-029-001-MY3.
References and links
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