Theory of Dynamics In-Phase Locking and Quasi-Period Synchronization in Two Mutually Coupled Multi-Quantum-Well Lasers

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1 Commun. Theor. Phys. 55 (2011) Vol. 55, No. 3, March 15, 2011 Theory of Dynamics In-Phase Locking and Quasi-Period Synchronization in Two Mutually Coupled Multi-Quantum-Well Lasers YAN Sen-Lin ( ) Department of Physics and Electronic Engineering, Nanjing Xiaozhuang University, Nanjing , China (Received May 31, 2010) Abstract We study dynamics in two mutually coupling multi-quantum-well lasers. We carry out theoretical and numerical analysis of synchronization, anti-synchronization, in-phase locking in the two identical lasers but detuning, in detain. It is proved that the coupling level determines stability of the lasers by analyzing the eigenvalue equation. Critical case of locking is discussed via the phase difference equation. Quasi-period and stable states in the two lasers are investigated via varying the current, detuning and coupling level. PACS numbers: Jn, Pq, Px Key words: in-phase locking, synchronization, multi-quantum-well laser, coupling 1 Introduction Two coupled oscillators can be locked if they operate with a common frequency, even though their natural solitary frequencies are different. A prominent example is the Adler locking scenario of two instantaneously coupled oscillators (CS). [1] The coupling is of fundamental interest in physics and is still only partially understood. [2 5] Coupled semiconductor lasers are a prototype example of two coupled oscillators [5 9] and, as such, very interesting systems for studying nonlinear dynamics of coupled oscillators. Coupled lasers show a great dynamical variety and be studied between experiment and theory. [4 9] Coupled semiconductor lasers are also of great technological merit because of possible application, such as bistable devices for optical flip-flops, high-frequency generation for optical clocks, or secure communication with a chaotic carrier. [10 15] It is known that even weak coupling between different lasers can lead to complicated dynamics. [3,16 17] Much work has been carried out to study dynamics and synchronization of mutually coupled semiconductor lasers. [7 12,15 21] We are interested in dynamics in two mutually coupled multi-quantum-well (MQW) lasers, which is of fundamental interest in physics and still to be understood. In this paper, we focus on the coupling level influence on the possibility of locking and stability or synchronization in MQW lasers. Firstly, we give the results of the two identical lasers but different detuning, where we find inphase locking and synchronization or anti-synchronization between two lasers. Critical case of locking or stability of the lasers is predicted only by the phase-difference equation or the eigenvalue equations. And quasi-period and stably behaviors in the lasers are illustrated by varying the current, detuning and injection level. A lot of papers reported anti-phase locking in coupling-lasers, [3 13] but we debut the anti-phase locking in theory. Secondly, senlinyan@163.com c 2011 Chinese Physical Society and IOP Publishing Ltd we show how unstability of the two lasers change as the main parameters are changed. We consider the effects of the coupling levels, current, detuning and transportation rate of carriers on the lasers. All these parameters vitally influence dynamics of the two lasers, which is of great importance for understanding more complicate dynamics of compounding laser system. Our study is not only an interesting fundamental physical issue, but can also provide us with new methods for study of CS dynamics in lasers, which is important in any potential application. [16 23] 2 Theoretical Model Figure 1 shows a scheme of two mutually coupled separate confinement hetero-structure (SCH) MQW diode lasers [20 21,24] where the laser 1 injects its lasing into the laser 2 while the laser 2 injects its lasing into the laser 2. The injection quantity controls the mutually coupling level. Then, the model for the lasers 1 and 2 can be described as the following [5 10,19 25] de 1 dφ 1 dn B1 dn 1 de 2 dφ 2 dn B2 = 1 2 (G 1 γ p )E 1 + k τ L E 2 cos(φ 2 φ 1 ), (1a) = 1 2 β c(g 1 γ p ) + k τ L E 2 E 1 sin(φ 2 φ 1 ) ω, (1b) = η i I 1 q γ BQN B1 + γ QB N 1, (1c) = γ BQ N B1 (γ e1 + γ QB )N 1 G 1 V p E 2 1, (1d) = 1 2 (G 2 γ p )E 2 + k τ L E 1 cos(φ 1 φ 2 ), (1e) = 1 2 β c(g 2 γ p ) + k τ L E 1 E 2 sin(φ 1 φ 2 ) + ω, = η i I 2 q γ BQN B2 + γ QB N 2, (1f) (1g)

2 482 Communications in Theoretical Physics Vol. 55 dn 2 = γ BQ N B2 (γ e2 + γ QB )N 2 G 2 V p E 2 2, (1h) where footnotes 1 and 2 represent the lasers 1 and 2, respectively. E and φ are the amplitude and phase of the optical field, and N B, N are carriers in the barrier region and in the active region, respectively. E m is the amplitude of the injection light. The nonlinear mode gain is G = [(Γg 0 v g )/(1 + E 2 /E 2 s )] lg{(n + N s)/(n th + N s )}. Γ = V/V p is the mode confinement coefficient. V is the volume of laser cavity. V p is the mode volume of laser. g 0 is the gain constant. v g is the group velocity of photon in laser cavity, E is normalized in such a way that E = (P/V p ) 1/2 is the amplitude of the optical field with photon number P, and E s is normalized in such a way that E s = (P s /V p ) 1/2 is the amplitude of the optical field at saturation with saturation photon number P s. N s = n s V is the parameter of logarithmic gain with its density n s. N th = n th V is the carrier number at transparency with its density n th. γ p = v g (α m + α int ) is the total photon loss with the group velocity v g. α m is the cavity loss. α int is the internal loss. τ L = 2n g L/c is the round-trip time in the cavity with its length L. c is the light velocity in vacuum. n g = c/v g is the group refractive index. η i is the internal quantum efficiency. I is the drive current and the threshold current is 25 ma. q is the unit charge. β c is the optical linewih enhancement factor. γ BQ is the loss of carriers from the SCH region to the quantum wells and γ QB is the loss of carriers escaping from the active region to the SCH layer. γ e = A nr + B nr (N/V ) + C A (N/V ) 2 is the total carrier loss in the active layer. A nr is the nonradiative recombination rate. B nr is the radiative recombination coefficient. C A is the Auger recombination coefficient. k is the optical injection factor. Fig. 1 A scheme of two mutually coupled MQW lasers. 3 Theoretical Analysis of Stability, In-Phase Locking and Synchronization By forcing the derivatives of Eqs. (1) to zero and presuming the stable points of the two lasers as E 10,20, φ 10,20, N B10,20, and N 10,20. When a symmetric injection is applied to the two identical lasers but detuning, it can be demonstrated that the system possesses two steady state solutions with E 10 = ±E 20. Due to the symmetry of the lasers, under symmetric injection case equal carrier densities and output can be expected on each laser. For the two identical lasers but detuning, we obtain E 2 10(sin Ω 0 β c cosω 0 ) = E 2 20(sin Ω 0 + β c cosω 0 ), (2) where Ω 0 = φ 20 φ 10. Equation (2) points out φ 10 φ 20 = (2n + 1)π/2 (n = 0, ±1, ±2,...), E 10 = E 20, and E 10 = E 20, in-phase locking and synchronization or antisynchronization of the fields in static states, respectively. For dynamical cases, we give the general demonstrations as the following. 3.1 Phase Summation Based on the above, for the two identical lasers in the symmetric injection but detuning case, we can presume: (a) If E 1 (t) = E 2 (t) or E 1 (t) = E 2 (t), synchronization or reverse synchronization of the fields occurs in the two lasers. If the laser emits a sequence of periodic waveform, ( φ 1 + φ 2 ) will vary periodically. From Eq. (1b) and Eq. (1e), we can obtain d(φ 1 + φ 2 ) = β c (G γ p ). (3) The expression of (φ 1 +φ 2 ) may be rewritten as φ 2 +φ 1 = ξ t + H(t) where ξ is one parameter and H(t) is one periodic function or φ 2 = ξφ 1 + H(t). (b) If E 1 (t) = E 2 (t) E 0, or E 1 (t) = E 2 (t) E 0, and the system will become of a stable state. Synchronization or reverse synchronization of the fields occurs also in the two lasers, and we obtain d(φ 1 + φ 2 ) = constant, (4) [φ 1 (t) + φ 2 (t)] is the linear function of time. So we can predict that anti-phase locking in system cannot show. 3.2 Stability, In-Phase Locking, Asymptotic Synchronization or Anti-Synchronization Let Ω(t) = φ 1 (t) φ 2 (t), a = 2 ω, and b = 2k/τ L. From Eq. (1b) and Eq. (1e), a phase difference of the two lasers is obtained by dω = 1 2 β c(g 1 γ p ) 1 2 β c(g 2 γ p ) ( ke2 + + ke ) 1 sin Ω 2 ω. (5) τ L E 1 τ L E 2 Due to the symmetry of the lasers, the system will show two different solutions, defined by the two different values of the phase difference between the two lasers. (i) Static In-Phase Locking Firstly, we discuss static states of the two identical lasers in the symmetric injection but detuning case. (a) If E 10 = E 20 = E 0, let ρ 0 = k/τ L, we obtain a frequency detuning expression as the following ω = ρ 0 sin(ω 0 ). (6)

3 No. 3 Communications in Theoretical Physics 483 Equation (6) is rewritten as φ 20 φ 10 = arcsin ω ρ 0. (7) This is a static in-phase locking expression. It predicates that static in-phase locking is achieved in the two lasers. When a critical condition of k = τ L ω is satisfied, the phase difference becomes of φ 20 φ 10 = (4n+1)π/2 (n = 1, 2,...). (b) If E 10 = E 20 = E 0, for another static state, another frequency detuning expression is given by ω = ρ sin(ω 0 ). (8) And another static in-phase locking expression is given by φ 20 φ 10 = arcsin ω ρ 0. (9) It predicates that another static in-phase locking is achieved. When a critical condition of k = τ L ω is satisfied, it can result in φ 20 φ 10 = (4n + 1)π/2 (n = 0, 1, 2,...). (ii) Dynamical In-Phase Locking Secondly, we discuss dynamical in-phase locking in the two lasers. From Eqs. (1), we obtain d(n B1 N B2 ) d(n 1 N 2 ) d(e 2 1 E2 2 ) = η i I 1 I 2 q γ BQ N B1 + γ QB N 1 + γ BQ N B2 γ QB N 2, (10a) = γ BQ N B1 (γ e1 + γ QB )N 1 G 1 V p E 2 1 γ BQ N B2 +(γ e2 +γ QB )N 2 +G 2 V p E 2 2, (10b) = (G 1 γ p )E k τ L E 1 E 2 cosω (G 2 γ p )E2 2 2k E 1 E 2 cosω. (10c) τ L When I 1 = I 2 = I, namely, the two identical lasers in the symmetric current but detuning case, we can obtain solutions of N 1 = N 2 and E1 2 = E2 2 or E 1 = ±E 2. (a) If E 1 (t) = E 2 (t), we obtain the following phase difference equation dω = b sinω a. (11) Its special solution is Ω = arcsin a b. (12) So dynamical in-phase locking shows while a fixed phase difference exists between the fields of the lasers. When a critical condition of k = τ L ω is satisfied, φ 1 (t) φ 2 (t) = (4n + 1)π/2 (n = 1, 2,...) can show. So the phase difference of the two lasers is a step variable. For a general solution of Eq. (11), we obtain [ b Ω = 2 arctan a + a2 b 2 a2 b tan 2 ] t a 2 + 2nπ + C 1, (a 2 > b 2 ), (13) where C 1 is determined by the starting condition and n may be taken as whole numbers. So the phase difference of the two lasers is periodic or multi-periodic variable while we can predicate the lasers operate in periodic or multiperiodic states. And a tan Ω 2 = et b2 a 2 (b + b 2 a 2 ) b + b 2 a 2 + C 2, e t b 2 a 2 1 (a 2 < b 2 ), (14) where C 2 is determined by the starting condition. When t, we obtain Ω = 2 arctan b + b 2 a 2 + C 2, (a 2 < b 2 ). (15) a So dynamic in-phase locking or asymptotic synchronization of the fields will be achieved and a fixed phase difference is kept between the fields of the lasers. So k = τ L ω is a critical condition for dynamical behavior in the lasers. (b) If E 1 (t) = E 2 (t), we obtain another phase difference equation as the following dω = b sinω a. (16) Its special solution is Ω = arcsin a b. (17) The dynamical in-phase locking will occur. When a critical condition of k = τ L ω is satisfied, the phase difference of φ 1 (t) φ 2 (t) = (4n+1)π/2 (n = 1, 2,...) will show. So the phase difference is a step variable. For general solution of Eq. (11) we obtain [ b Ω = 2 arctan a a2 b 2 a2 b tan 2 ] t a 2 + 2nπ + C 3, (a 2 > b 2 ), (18) where C 3 is determined by the starting condition and n may be taken as whole numbers while the phase difference may perform periodically or multi-periodically. So we can predicate the lasers operate in periodic or multi-periodic states. And a tan Ω 2 = et b2 a 2 ( b + b 2 a 2 ) + b + b 2 + a 2 + C 4, e t b 2 a 2 1 (a 2 < b 2 ), (19) where C 4 are determined by the starting condition. And when t, we obtain Ω = arctan b + b 2 a 2 + C 4, (a 2 < b 2 ). (20) a So the dynamic in-phase locking or asymptotic synchronization of the fields can be obtained, in which a fixed phase difference be kept between the two lasers. So k = τ L ω is a critical condition for dynamic in the lasers. (iii) Coupling Level Affects on Dynamic When I 1 = I 2 = I and k > ω τ L, the two lasers can perform in-phase locking. Using the separating variable

4 484 Communications in Theoretical Physics Vol. 55 method, from Eqs. (10) we obtain the eigenvalue equations as the following For E 1 = ±E 2, we obtain E 2 1 = ±Z 3 /2ρ 0 cosω, (23a) dn B1 + γ BQ N B1 γ QB N 1 = Z 1, or E 2 2 = ±Z 3 /2ρ 0 cosω. (23b) dn 1 de 2 1 dn B2 dn 2 de γ e1 N 1 + G 1 V p E 2 1 = Z 2, (G 1 γ p )E 2 1 = Z 3, + γ BQ N B2 γ QB N 2 = Z 1, + γ e1 N 2 + G 2 V p E 2 2 = Z 2, (G 2 γ p )E 2 2 = Z 3, (21) where Z 1, Z 2, and Z 3 are the eigenvalues un-relational to time. We obtain 2ρ 0 E 1 E 2 cosω = Z 3. (22) It is predicated the two lasers become of a stable state when k > ω τ L. We should point out that the coupling level determines stability of the two lasers while the lasers will perform stably behavior when k > ω τ L. When k < ω τ L, we predicate that the lasers perform unstably, or periodic and multi-periodical behaviors while synchronization or inverse synchronization are achieved but un-locking between the two lasers. So k = τ L ω is a critical condition of the dynamical behavior. The coupling level dominates mainly dynamic in the two lasers. 4 Numerical Results Here, some typical results are introduced by using Eqs. (1) for the two identical lasers but detuning. The laser parameters are listed in Table 1 and the other will be varied, respectively, in following cases. Table 1 The laser parameters. Symbol Value Units Symbol Value Units L 1200 µm A nr s 1 w 1.5 µm B nr cm 3 /s d µm C A cm 6 /s Γ β c 3.5 n g 3.6 η i 0.8 α m 11.5 cm 1 P s α int 20 cm 1 γ BQ s 1 g cm 1 γ QB s 1 n s cm 3 n th cm Coupling Level Influence on Dynamic At first, we focus on effect of the coupling level on dynamical behavior, locking and synchronization in the lasers. We find the coupling level will affect on dynamics, synchronization, locking in the lasers. When we take the parameters I = 50 ma, ω = 1 GHz and k = = ω τ L in the critical case, Fig. 2 shows the synchronization and step in-phases locking, where Fig. 2(a) shows the lasers shifting periodical between stably and multi-periodic states, Fig. 2(b) shows synchronization of the fields, Fig. 2(c) shows the step inphase locking, respectively. It is found that when a multiperiodic state shifts in a stably state, phase difference shows a step variable while in-locking presents again. The injection level controls dynamical behavior, synchronization and locking in the lasers. When the operating parameters are taken as I = 50 ma, ω = 1 GHz and the coupling factor k = 0.18, 0.17, 0.16, 0.15, 0.14, 0.1 and 0.02, respectively, Figs. 3 show phase-trajectories of the laser 1 to illustrate effect of the coupling level on dynamics in the lasers to induce period-5 in (a), period 4 in (b), period-3 in (c), another period-3 in (d), different period-3 in (e), period-2 in (f) and single-period in (g) where anti-synchronization is obtained in the parameter case of Fig. 3 (g) and synchronizations are obtained in the other cases, respectively. When k = 0.22, in-phase locking and synchronization occurs in the two lasers illustrated in Fig. 4. We find that in-phase locking and synchronization or anti-synchronization occur in the lasers when k > = ω τ L and multi-periodic synchronization or multi-periodic anti-synchronization but un-locking occurs when k < = ω τ L, namely, for k > ω τ L, the in-phase locking and asymptotic synchronization of the fields can be achieved, in which a fixed phase difference shows between the fields of the each lasers. And multi-periodic synchronization or multi-periodic antisynchronization but un-locking can be achieved when k < ω τ L. So the coupling level determines stability of the lasers.

5 No. 3 Communications in Theoretical Physics 485 Fig. 2 In-phase locking and synchronization in the critical case (a) The laser output; (b) Synchronization; (c) In-phase locking. Fig. 3 Effect of the coupling level on dynamics in the laser 1. Fig. 4 In-phase locking and synchronization.

6 486 Communications in Theoretical Physics Vol Frequency Detuning Influence on Dynamic Secondly, we discuss effect of the detuning on dynamical behavior in the lasers. When we take the parameters I = 50 ma, k = 0.1, and ω = 0.1 GHz, Fig. 5 shows the anti-synchronization and in-phases locking due to ω < k/τ L where Fig. 5(a) shows the laser 1 output, Fig. 5(b) shows anti-synchronization of the fields and Fig. 5(c) shows the in-phase locking, respectively. Fig. 5 In-phase locking and anti-synchronization. Fig. 6 Effect of the detuning on dynamics in the laser 1. Cases for ω > k/τ L are analyzed via Fig. 6. Figure 6 shows phase-trajectories of the laser 1 to illustrate effect of the detuning on dynamics in the two lasers. When we take the operating parameters as I = 50 ma and k = 0.1 and vary the detuning ω as 0.6 GHz, 0.8 GHz, 1.2 GHz, and 2 GHz, respectively, from Fig. 6(a) to Fig. 6(d) we observe many kinds of dynamical behaviors, such as period-4 in Fig. 6(a), period-3 in Fig. 6(b), period-2 in Fig. 6(c), single-period in Fig. 6(d) and we find multi-period synchronizations in the above cases. When we take the detuning greater than GHz, namely, ω > k/τ L, the lasers will become of periodic or multi-periodic states where the periodic oscillation frequency is close to the detuning when the detuning between 1.4 GHz and 20 GHz. When we take the detuning less than GHz, namely, ω < k/τ L, the lasers become of stably state and can obtain in-phase locking and anti-synchronization or synchronization where anti-synchronization or synchronization is related to the starting condition. 4.3 Current Influence on Dynamic Next, we analyze effect of the current on dynamical behavior in the lasers. When the operating parameters are taken as ω = 0.6 GHz, k = 0.1 and the currents are taken as I = 26 ma, 28 ma, 30 ma, 40 ma, 60 ma, 70 ma, and 80 ma, respectively, Fig. 7 illustrates phasetrajectories of the laser 1 to explain effect of the current on dynamics in the laser where dual-period in (a), period-3 in (b), another period-3 in (c), period-4 in (d) quasi-period- 3 in (e), quasi-period-2 in (f), single-period in (g). We

7 No. 3 Communications in Theoretical Physics 487 observe the synchronization for I = 40 ma, I = 60 ma, I = 70 ma, I = 80 ma, and the anti-synchronization for I = 26 ma, 28 ma, and 30 ma, respectively. We can find the current can affect dynamical behavior of the lasers to induce many quasi-period states when k < ω τ L. Fig. 7 Effect of the current on dynamics in the laser Transportation of Carriers Influence on Dynamic Fig. 8 Effect of the transportation rate of carriers on dynamics in the laser 1.

8 488 Communications in Theoretical Physics Vol. 55 Lastly, we analyze effect of the transportation rate of carriers between the barrier region and in the active region on dynamics in the laser. Due to the transportation rate of carriers being taken as different values, Fig. 8 gives phasetrajectories of the laser 1 to show different dual-period when the operating parameters are taken as ω = 1 GHz, I = 50 ma, k = 0.1, γ BQ is taken as /s for Fig. 8(a) and /s for Fig. 8(b), γ QB is taken as /s for Fig. 8(c), /s for Fig. 8(d), respectively. Where the anti-synchronizations are achieved in cases of Figs. 8(a) and 8(c), the synchronizations are achieved in cases of Figs. 8(b) and 8(d). When ω = 0.9 GHz, I = 40 ma, k = 0.15, γ BQ being taken as /s and /s, γ QB being taken as /s and /s respectively, correspondingly to these cases, Fig. 9 shows different quasi-period where different period-4 in (a), (b), (c), period-3 in (d). Fig. 9 The transportation rate of carriers affects on dynamics. The above numerical results are agreeable with the theory presented in Sec Conclusions This paper studied two mutually coupled MQW lasers and gave the theoretical analysis of the synchronization, anti-synchronization, in-phase locking and stability. Only analysis of the phase difference equation or the eigenvalue equations, we proved that the coupling level control the stability or unstably, or quasi-periodical behaviors and the synchronization or anti-synchronization. We gave also a critical case of k = τ L ω. The theory point out that anti-phase locking cannot show in the lasers. Our theory is simple and practicality for study of complicated system. The theory is also accorded with the numerical results. References [1] R. Adler, Proc. IEEE 61 (1973) [2] H. Mirels, Appl. Opt. 25 (1986) [3] W.C. Weng, Opt. Lett. 10 (1985) 442. [4] Z. Jiang and M. McCall, J. Opt. Soc. Am. B 10 (1993) 155. [5] J. Mulet, C. Masoller, and C.R. Mirasso, Phys. Rev. A 65 (2002) [6] H. Erzgräber, D. Lenstra, and B. Krauskopf, Proc. of SPIE 6184 (2006) [7] G. Carpintero, H. Lamela, M. Leones, C. Simmendinger, and O. Hess, Proc. of SPIE 4283 (2001) 303. [8] H. Erzgräber and S. Wieczorek, Phys. Rev. E 80 (2009) [9] S.P. Hegarty, D. Goulding, B. Kelleher, G. Huyet, M.T. Todaro, A. Salhi, A. Passaseo, and M.D. Vittorio, Opt. Lett. 32 (2007) [10] H.J. Wünsche, S. Bauer, J. Kreissl, O. Ushakov, N. Korneyev, F. Henneberger, E. Wille, H. Erzgräber, M. Peil, W. Elsaor, and I. Fischer, Phys. Rev. Lett. 94 (2005) [11] I. Fischer, Y. Liu, and P. Davis, Phys. Rev. A 62 (2000) [12] J. Mulet, C. Mirasso, T. Heil, and I. Fischer, J. Opt. B: Quantum Semiclass. Opt. 6 (2004) 97. [13] M.T. Hill, H. de Waar, and H.J.S. Dorren, IEEE J. Quantum Electron. 37 (2001) 405. [14] M.T. Hill, H.J.S. Dorren, T. de Vries, X.J.M. Leijtens, J.H. den Besten, B. Smalbrugge, Y. Oei, H. Binsma, G. Khoe, and M.K. Smit, Nature (London) 432 (2004) 206. [15] M. Möhrle, B. Sartorius, C. Bornhol, S. Bauer, O. Brox, A. Sigmund, R. Steingruber, M. Radziunas, and H.J. Wunsche, IEEE J. Select. Topics Quantum Electron. 7 (2001) 217. [16] S. Wieczorek and W.W. Chow, Phys. Rev. Lett. 92 (2004) [17] W.L. Zhang, W. Pan, B. Luo, X.F. Li, X.H. Zou, and M.Y. Wang, J. Opt. Soc. Am. B 24 (2007) [18] G.D. VanWiggeren and R. Roy, Science 279 (1998) [19] J.R. Terry, K. Scott Thornburg, D. DeShazer, G.D. Van- Wiggeren, S. Zhu, P. Ashwin, and Rajarshi Roy, Phys. Rev. E 59 (1999) [20] L. Wu and S.Q. Zhu, Phys. Lett. A 308 (2003) 157. [21] L. Wu and S.Q. Zhu, Commun. Theor. Phys. 41 (2004) 225. [22] T. Matsuura, A. Uchida, and S. Yoshimori, Opt. Lett. 29 (2004) [23] X.M. Liu, X. Yang, F. Lu, et al. Opt. Express 13 (2005) 142. [24] S.L. Yan, Chin. Phys. 16 (2007) [25] L. Wu and S.Q. Zhu, Chin. Phys. 12 (2003) 300.