Elsevier

Physics Letters A

Volume 378, Issue 44, 12 September 2014, Pages 3269-3273
Physics Letters A

Phase synchronization in a two-mode solid state laser: Periodic modulations with the second relaxation oscillation frequency of the laser output

https://doi.org/10.1016/j.physleta.2014.09.035Get rights and content

Highlights

  • We study the intrinsic phase synchronization in a periodically pump-modulated two-mode solid state laser.

  • The empirical mode decomposition method is utilized to define the intrinsic phase synchronization.

  • The degree of phase synchronization is quantified by a proposed synchronization coefficient.

Abstract

Phase synchronization (PS) in a periodically pump-modulated two-mode solid state laser is investigated. Although PS in the laser system has been demonstrated in response to a periodic modulation with the main relaxation oscillation (RO) frequency of the free-running laser, little is known about the case of modulation with minor RO frequencies. In this Letter, the empirical mode decomposition (EMD) method is utilized to decompose the laser time series into a set of orthogonal modes and to examine the intrinsic PS near the frequency of the second RO. The degree of PS is quantified by means of a histogram of phase differences and the analysis of Shannon entropy.

Introduction

Synchronization in nonlinear systems has been extensively studied because of its critical importance in a wide variety of disciplines, including physics [1], [2], mathematics [3], biology [4], physiology [5], ecology [6], chemistry [7], and others [8]. The phenomenon is caused by interactions between systems and can be classified into several categories, depending on the emerged correlations among the systems. The obvious one is complete synchronization (CS), in which interacting systems adjust their states and finally converge to a single trajectory [9]. Although the definition of CS is extremely simple, the requirements for achieving CS are relatively strict and therefore other synchronous phenomena, especially phase synchronization (PS) [10], are more common in both artificial and natural systems.

PS has been one of the most intriguing subjects in nonlinear science since the pioneering work done on it in 1996 [11]. PS is a type of synchronization that reflects rhythms identification of interacting systems, but the amplitudes of the systems remain uncorrelated. The mathematical definition of PS is |nϕ1mϕ2|<c, where ϕ1 and ϕ2 are instantaneous phases of interacting systems; n and m are integers, and c is a constant. Because of the universality of PS, it has been studied experimentally in various systems, such as plasma systems [12] and fluid systems [13]. PS has also been realized in a variety of lasers owing to its potential practical applications, examples of which include optically coupled Nd:YAG lasers [14], electronically coupled Nd:YAG lasers [15], the cw CO2 laser that is driven by an intracavitary electro-optic modulator (EOM) [16], the Nd:YAG laser that is driven by an intracavitary acousto-optic modulator (AOM) with two frequencies [17], and others [18].

Recently, interest has been extended to PS in a periodically pump-modulated frequency-doubled Nd:YAG laser [19]. Ahlborn and Parlitz applied recurrence analysis and pseudo ensemble averaging to define and quantify PS in the system, with a focus on the emergent synchronous region, which is called the Arnold tongue. The Arnold tongue occurs close to the modulating frequency of f1 MHz, which is exactly the main relaxation oscillation (RO) frequency of the laser output in the absence of modulation. Subsequently, Lin et al. used a similar technique and realized PS in a two-mode microchip Nd:YVO4 laser experimentally and theoretically [20]. These results are crucial to increasing the superimposed common output of several lasers that are driven by periodic pump modulation.

An interesting dynamic feature of a multi-mode laser is the presence of more than one frequency peak in the power spectrum of the laser output. Accordingly, the multi-mode laser essentially has several RO frequencies. The way in which the system responds to external periodic driving becomes a very interesting question in the study of PS. The power spectrum of the free-running two-mode microchip Nd:YVO4 laser presents two distinct RO frequencies. Although PS has been demonstrated when the periodic pump is modulated to the first RO frequency of the free-running laser [20], according to standard analysis, this phenomenon is invisible close to the second RO frequency. In this Letter, the empirical mode decomposition (EMD) method [21] is used to elucidate the intrinsic PS in the vicinity of the second relaxation frequency. Hilbert transform is used to define the phases of, and the phase differences between, the decomposed mode of the laser output and the modulating signal. Further analysis reveals the existence of a robust Arnold tongue in the frequency region of interest.

This Letter is organized as follows. Section 2 introduces the two-mode solid state laser model and presents its power spectrum. In Section 3, the EMD process is used to decompose the laser output and the Hilbert transform is used to calculate its instantaneous phase. In Section 4, the appearance of PS is confirmed and PS is quantified by an analysis of phase difference. Finally, a brief conclusion is drawn.

Section snippets

Two-mode solid-state laser model

The scaled Tang–Statz–deMars (TSD) two-mode laser model has been demonstrated to be able to reproduce the dynamics of the periodically modulated microchip Nd:YVO4 laser [20], [22]. The TSD model with an externally driving term isdn0dt=w0[1+F(t)]n0k=1Nγk(n012nk)sk,dnmdt=γmn0smnm(1+k=1Nγksk),dsmdt=K[γm(n012nm)1]sm, where N=2 is the total number of lasing modes, and the subscript m=1,2 is the index of the corresponding mode. n0 is the spatially averaged population inversion density with

Empirical mode decomposition

Empirical mode decomposition (EMD) is adopted here to define explicitly the instantaneous phase and calculate the phase difference between the driver F(t) and the laser output. Huang et al. proposed this method for analyzing non-stationary and nonlinear time series [21]. The method has since been applied to study the phase correlation and dynamic properties of financial data [24], [25], [26], cardiorespiratory synchronization [27], and human ventricular fibrillation [28]. The EMD method is

Synchronization analysis

Since the purpose of this study is to explore the dynamics of a laser system in response to a periodic driver of vfr2, IMF c2(t) is used as the relevant mode for further analysis. The instantaneous phase of c2(t) is constructed from an analytical signal z(t), which is defined asz(t)=c2(t)+iy(t)=A(t)eiϕc2(t), where the function y(t) is the Hilbert transform of c2(t); A(t) is the amplitude, and ϕc2(t) is the phase of c2(t) [8]. To determine how ϕc2(t) is related to the phase of the periodic

Conclusion

Phase synchronization in a solid-state laser system that is driven by a sinusoidal signal with a frequency close to the second RO frequency of the laser system, was numerically studied. The EMD method and the synchronization coefficient, determined by using Shannon entropy are applied to quantify the degree of PS. Close to the second RO frequency of the synchronization diagram, an Arnold tongue is clearly observed, demonstrating PS. This phenomenon has been shown to be robust against variations

Acknowledgements

This work was supported by the National Science Council of Republic of China (Taiwan) under the Grant No. 102-2112-M-035-002, and NCTS of Taiwan. We thank Dr. Ming-Chya Wu for helpful discussions on the EMD method. Ted Knoy is appreciated for his editorial assistance.

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