Intrinsic instability and locking of pulsation frequencies in free-running two-mode class-B lasersMathematics and Statistics
Intrinsic instability and locking of pulsation frequencies in free-running two-mode class-B lasers
Free-running multimode class-B lasers utilizing Fabry-Perot cavities have been most successfully described by Tang, Statz, and deMars (TSD) rate equations . Both a linear [2,3] and nonlinear  analysis of the TSD equations for an arbitrary number of lasing modes give rise to self-organized dynamics such as antiphase but not to any kind of unstable steady states.
Among many kinds of solid-state laser materials belong-ing to class-B , widely used Nd:YVO4 (yttrium orthovana-date) laser crystals, which are of significant current techno-logical interest as diode-pumped microchip lasers, possess unique spectroscopic properties such as reduced Stark split-ting and substantial transfer of lattice energy to Nd ions . These features result from an unusual crystal field which is strongly modified by lattice vibrations (e.g., phonons) [5,6]. Consequently, the YVO4 host causes substantially large in-homogeneous broadening . In addition, Raman spectro-scopic studies showed that microinhomogeneities depend on the method of crystal growth .
Recently, quasiperiodic and chaotic evolutions as well as a p /q -type locking of modal pulsation frequencies (i.e., in-termode parametric resonance [8,9]) have been observed in a laser-diode (LD) -pumped free-running Nd:YVO4 laser op-erating in a two-mode regime without any additional external influence factors . The p /q -type locking implies that the first lasing mode exhibits period-q pulsations and the second lasing mode does period-p pulsations, in which the repetition periods of the two modes are the same. Self-induced spiking oscillations and associated instabilities have also been dem-onstrated using the same laser free running in a three-mode regime . These experimental observations shed light on the fundamental understanding of laser stability since it is commonly believed that spontaneous instabilities are impos-sible in free-running class-B lasers without any additional degree of freedom such as external modulation, light injec- tion, intracavity second-harmonic generation, delayed feed- back, etc. Various physical mechanisms have been thus in-corporated to the TSD rate equations with the hope of predicting unstable steady solutions. Those additional mechanisms explored up to date are phase-sensitive coupling between the complex modal amplitudes [12-14] via the long-wavelength population grating [12,15,16], population diffusion , and longitudinal inhomogeneous pumping [15,16]. However, none of the above listed mechanisms suits the conditions of our experiments [10,11]. Since in the Nd:YVO4 laser the phonons of the host lattice strongly modify the crystal field, this results in a third-order nonlin-earity. Such an intensity-dependent nonlinearity is thought to induce a cross-gain mechanism of direct mode-mode cou-pling which could play a key role in triggering spontaneous instabilities.
In this paper we present detailed analytical and numerical studies of the dynamics of a two-mode free-running laser-diode-pumped microchip Nd:YVO4 laser together with ex-perimental results. The experimental setup and observations for the two-mode regime are presented in Sec. II. Section III is devoted to an asymptotic analysis of model rate equations with a higher-order cross-gain mode-coupling term which was proposed to describe a class of inhomogeneously broad-ened lasers like that under consideration. The analysis clari-fies the origin of the instability occurrence. Finally, we show the simulation results in Sec. IV. The instability threshold and the observed phenomenon of p /q -type locking of pulsa-tion frequencies are wellreproduced by our simulations.
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