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Spontaneous self-organisation in chaotic laser mode-mode interaction
15October 1996 OPTICS COMMUNICATIONS ELSEVIER
OpticsCommunications131 (1996) 89-94
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Spontaneous self-organisation in chaotic laser mode-mode interaction D.Y. Tang, N.R. Heckenberg Physics Department, The Universityof Queensland, Brisbane, OLD 4072, Austr~dia
Received 18 December1995;revisedversionreceived3 April 1996;accepted3 April 1996
Abstract We have experimentallystudied the dynamics of the interaction between two globally coupled chaotically emitting laser modes. We show that even in the chaotic case, the laser can show spontaneous self-organisation.In the case of our laser, this is expressed as the chaotic mode intensity pulsations of the two individual laser modes being either in step or alternatiitg.
Recently dynamics of globally coupled multimode lasers has been intensively studied, as in these lasers several modes coexist and under the nonlinear interaction between them, self-organised states can spontaneously be formed, which manifest different kinds of collective effects. Studies of collective effects of globally coupled multi-mode lasers have made significant progress in the past years. Collective effects such as cooperative laser frequency locking [1,2], antiphase dynamics [3-6], gain circulation [7] and energy sharing among coexisting laser modes [8] have been experimentally revealed in different kinds of globally coupled multi-mode lasers and been well explained theoretically. These studies have not only greatly improved our understanding of multi-mode laser dynamics, but, because of the direct relation between the dynamics of the globally coupled multi-mode laser to that of globally coupled nonlinear oscilla~*ors, have also made significant contributions to the study of nonlinear dynamical systems.
So far, studies of collective effects in globally coupled multi-mode lasers mainly concentrated on the laser mode interaction, in situations where the interacting modes are intrinsically stable. By "intrinsically stable" we mean that without mode interaction each laser mode involved would lase in its stable steady state. Although in previous studies of collective effects of multi-mode lasers, chaotic states of laser modes were observed occasionally, these chaotic states were caused by the nonlinear mode interaction, not due to the individual laser mode dynamics itself. Studies of chaotic laser dynamics have shown that under suitable conditions, lasers can also operate in intrinsically chaotic states, where eve~a without the existence of mode interactions, each mode of the laser behaves chaotically. Obviously, due to the intrinsic instability of the individual interacting laser modes, the interaction between laser modes could become even more complicated. However it would be interesting to ask: do collective effects still exist in the chaotic laser mode interac-
0030-4018/96/$12.00 Copyright© 1996ElsevierScienceB.V. All fightsreserved. Pll S0030-4018(96)00246-5
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D.Y. Tang, N.R, Heckenberg / Optics Communications 131 (1996) 89-94
tion? Since the dynamics of globally coupled multimode lasers operating in their intrinsically chaotic state is similar to that of globally coupled chaotic oscillators, and dynamics of such nonlinear systems is attracting more and more attention in nonlinear dynamics studies [9-1 l l, an understanding of the dynamics of chaotically interacting laser modes would have also practical importance in many disciplines of nonlinear science. In order to explore the dynamics of chaotic laser mode interactions, we have experimentally studied the dynamics of an optically pumped NH 3 bidirectional (i.e. two mode) ring laser in the parameter range where chaotic single mode dynamics is found. In this paper we show the dynamics of this laser, and in particular we demonstrate that even with the chaotic laser mode interaction, collective effects still exist in the globally coupled multi-mode laser. In the case of our laser, this is manifested as the chaotic mode intensity variations of each individual mode being either inphase or antiphase. Our experimental setup used has the same configuration as described in Ref. [12]. In brief it consists of an optically pumped bidirectional ISNl-13 ring laser operating on the wavelength of 153 p.m. The pump laser is an isotropic carbon dioxide (13CO2) laser, whose cavity has a Littrow structure. The pump laser frequency is monitored through a COs Lamb-dip cell, and the pump frequency can be finely tuned through a piezo ceramic mirror' mount. The far infrared (FIR) emissions of the bidirectional NH~ ring laser in both directions were simultaneously detected with two Schottky barrier diodes. The electrical signal from each diode was low-noise preamplified and then stored with a DASP c',,rd in a computer. In a previous paper [13], we have reported experimental studies on the nonlinear mode-mode interaction between the forward and backward modes of the le.ser. In that experiment, in order to get a clear picture of nonlinear mode interaction of the laser, we intentionally operated the laser in the parameter range where stable single mode dynamics was ensured. Our results suggested that the interaction between the two laser modes is mainly the mode competition resulting from sharing the same population inversion and mode coupling caused by the spatial population grating formed in the laser. Through these nonlinear
mode interactions the two counterpropagating modes of the laser become globally mutually coupled. Under the global coupling the intensity evolution of each mode can show strong antiphase dynamics. We have observed both periodic and chaotic antiphase dynamics in the laser. Under special conditions even an antiphase period-doubling route to chaos at the mode alternation frequency was revealed in the laser. It is to be noted, however, that in that experiment the observed chaotic mode dynamics is the result of strong nonlinear mode interaction between the counter-propagating laser modes, which has no relation to the single mode laser chaos which can be shown by the laser under different conditions. This was clearly proved by the time scale of the observed mode intensity pulsation. We have shown in that experiment that the observed mode intensity pulsation frequency changes with mode interaction strength, and the maximum value of the frequency observed is about 0.7 MHz, which is still lower than the normal single mode instability frequency of the laser at that gas pressure. In the present experiment, in contrast, in order to experimentally study the chaotic laser mode interaction, we have operated the laser in the single mode chaos parameter regime. In this parameter regime, when the laser operates in a single mode (either forward or backward mode) state, the mode evolution shows chaotic dynamics. Further, under suitable conditions, the backward mode of the laser can show 'spiral chaos' dynamics well described by the Lorenz-Haken equations [ ! 4,15]. When the laser is operating in the single mode chaos regime, experimentally we found that, depending on the pump laser frequency detuning relative to the pump line centre and the FIR laser cavity detuning, the laser can show different chaotic dynamics. With a relatively large pump frequency detuning and the FIR laser cavity tuned close to or in resonance with one of the gain peaks, the laser will show essentially single mode 'spiral' chaos in the strong mode, and the weak mode will normally be suppressed. However, if the frequency separation between the two modes is not big enough, at the points where the strong chaotic mode intensity goes close to zero, the weak laser mode will no longer be suppressed, and will build up transiently. Even a weak, transient existence of the counterpropagating mode can cause the dynamics of the chaotic laser
D.¥. Tang. N.R. Heckenberg / Optics Communications 131 (1996) 89-94
mode to greatly deviate from the single mode Lorenz-like chaos, as we have shown in Ref. [15]. We are here more interested in the interaction of two coexisting chaotic modes. To this end we tune the pump laser frequency closer to the pump line centre. It was found experimentally that with,'.n a certain pump frequency detuning range, both laser modes can lase simultaneously, with both mode emissions being chaotic. With fixed pump laser intensity, we have studied the interaction of the two coexisting chaotic modes for different pump frequency detuning, FIR laser cavity detuning and gas pressure. Under different laser operation conditions, two kinds of relationship between the mode intensity variations of the chaotic modes were observed. Fig. 1 shows as an example one of these mode intensity variation relations observed. Fig. 1 corresponds to a laser state where the intensity evolutions of the modes are not very chaotic. The mode intensity variation in Fig. 1 has the following characteristics: The fundamental intensity pulsations of both modes have the same frequency. Studying the behaviour of this fundamental mode intensity pulsation shows that its frequency clearly depends on the gas pressure, but remains relatively independent of the pump frequency and laser cavity detuning. This behaviour together with the frequency value suggests strongly that this fundamental mode intensity pulsation is caused by single mode laser chaos of each individual mode. This experimental result shows that in this parameter regime, the intrinsic
(a)
(b)
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Time (/zS) Fig. I. Mode intensity evolutions of the laser as it is in an in-step collective state. Gas pressure is 6 Pa, pump laser intensity is 4.3 W / c m 2. (a) Mode intensity evolution of the forward mode. (b) Mode intensity evolution of the backward mode.
91
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Frequency(MHz) Fig. 2. Fourier spectra calculated from the mode intensity evolution shown in Fig. I. (a) and (b) correspond to the trace of (a) and (b) in Fig. I respectively.
mode instability plays the predominant role in determining the mode intensity variation. Although the dynamics of each individual mode is unstable, the intensity pulsations between them keep exactly in time, indicating that the variation of each mode is not independent of the other. Another interesting feature of the mode intensity variation is the antiphase relationship between the intensity pulse height of the two modes. It is easy to see that when the intensity pulse height in one mode is higher, then the corresponding intensity pulse in the other mode is lower. Because of this an:iphase pulse height relation between them, the mode intensity variations of the two modes have totally different forms, and hence return maps. In order to find out the frequency composition of the mode intensity evolution, so as to understand the
92
D.Y. Tang, N.R. Heckenberg / Optics Communications 131 (1996) 89-.04
possible mechanism of the mode interaction, we have calculated the Fourier spectrum of each mode intensi~ evolution, as shown in Fig. 2. Although the de~led mode intensity variation with time of each mode is different, their Fourier spectra show in contrast the same characteristics. In each spectrum there exist three independent frequency components, and the value of each frequency component is the same for the two modes. It is easy to recognise that f~ is the frequency which corresponds to the fundamental intensity pulsation. As we have pointed out above, it is introduced by the single mode instability. f2 is the frequency responsible for the observed slow antiphase intensity pulse height variation, f2 exists even when the interacting modes are in their steady state. As we have understood from previous experiments [13], f2 is caused by the mode competition and coupling between the two modes, f3 is a frequency which has also been observed in the steady state mode interaction of the laser [13]. Its existence can be interpreted as the result of another laser instability particular to the optically pumped bidirectional ring laser. As reported in Ref. [13], f3 results in a periodic mode intensity modulation with unsymmetrical modulation strength in each mode. However, the modulation caused by f3 is not obvious in the mode intensity evolution. All other significant spectral components shown in Fig. 2 are nonlinear combinations of these three independent frequency components; their existence can be attributed to the result of nonlinear interaction between the modes. From the frequency composition of the mode intensity evolution, we see that the observed mode dynamics is the result of mutual interaction among the above described three mode interaction mechanisms. Since all the three spectral components show mainly line structure, it is plausible that dynamics caused by each individual mechanism i~ not chaotic. But we emphasize that under the coexistence of them the dynamics of each mode is chaotic. We have calculated the autocorrelation and some other metric behaviours of each laser mode, and all results show that this state is a weakly ch~o~ic state. The formation of the observed chaos is probably due to the well.known quasi-periodic route to chaos, since now there exist three independent frequencies in the system. Reducing the cavity detuning or pump frequency
(a)
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(b)
0
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Time (,uS) Fig. 3. Mode intensity evolution of the laser as it is in a more chaotic in-step collective state. Gas pressure and pump laser intensity is the same as in Fig. 1, but the cavity detuning is slighdy reduced. (a) Mode intensity evolution of the forward mode. (b) Mode intensity evolution of the backward mode.
detuning slightly further, the state shown in Fig. 1 becomes more chaotic. A typical time series is shown in Fig. 3. Comparing the intensity evolution between Fig. 1 and Fig. 3 it can be seen that the mode competition in Fig. 3 becomes more significant: while in Fig. I intensity pulse heights merely show antiphase behaviour, now this competition occasionally results in almost total suppression of the intensity pulsation in the other mode. We have analysed the Fourier spectrum and the autocorrelation function of Fig. 3. The Fourier spectrum has now a very broadened continuous structure, showing that it has become completely chaotic. Despite the generally continuous structure of the spectrum, three local spectral maxima located respectively in the positions of fl, f2 and f3 are still clearly visible in the spectrum. The calculated autocorrelation function confirms further the chaoticity of the state. Although the state is very chaotic now, the individual intensity pulses between the two modes still remain exactly in step, indicating that it is an intrinsic behaviour of mode interaction. Depending on the laser conditions, another case of relative mode intensity evolution, as shown in Fig. 4 and Fig. 5, was observed. As in Fig. 1 we have shown in Fig. 4 the case in which the mode intensity is not very chaotic. In contrast to the mode intensity evolution shown in Figs. 1 and 3, where the mode intensity pulsations of the two modes are
D.Y. Tang. N.R. Heckenberg / Optics Communications 131 (1996) 89-94
Ca)
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Time ( ~ ) Fig. 4. Mode intensity evolutions of the laser as it is in an alternating collective state. Gas pressure is 6.5 Pa, pump laser intensity is 4 W / c m 2. (a) Mode intensity evolution of the forward mode. (b) Mode intensity evolution of the backward mode.
exactly in step, the mode intensity pulsations of the two modes are now exactly out of step. Another significant difference to that of Fig. 1 and Fig. 3 is that the intensity pulse heights of the modes do not show antiphase behaviour, instead sometimes, with one pulse time delay, the intensity variations of these two modes show a strong similarity. Analysis of the Fourier spectra show that there still exist three independent frequencies, which can be identified again as having the same origins as described in the case of Fig. 1, although now the effect caused by the existence of f2 is no longer obvious in the mode inten(~)
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Time (/aS) Fig. 5. Mode intensity evolutions of the laser as it is in a more chaotic alternating collective state. Gas pressure is 6 Pa, pump laser intensity is 4 W/cm 2. (a) Mode intensity evolution of the forward mode. (b) Mode intensity evolution of the backward mode.
93
sity evolution. The coexistence of these three independent frequencies suggests that the mode interaction resulting in the mode iatensity evolution of Fig. 4 has the same mechanisms as those of Fig. 1. The state shown in Fig. 4 is again a three frequency weak quasi-periodic chaos state. Fig. 5 is a completely chaotic state of the out-of-step mode intensity pulsation. It can again be seen that even though the mode intensity variation is very chaotic, the out-ofstep relationship between the two mode intensity variations remains unchanged. We have checked these experimental findings under other accessible experimental conditions, and as long as the modulation effect caused by the existence of f3 can be neglected, the observed mode intensity variation will show one of the above two situations. It might be though that mode variations caused by the intrinsic mode instability could occur independently of other modes, however, this experimental result shows clearly that even when the interacting laser modes are intrinsically unstable, under a nonlinear mode interaction, any variation of an individual mode will no longer be independent but rather correlated, and depending on the particular mode interaction, the total system will evolve into different collective states. Under the nonlinear interaction of our laser, two kinds of collective states were observed. In one the individual mode intensity pulsations are in step, in the other the mode intensity pulsations are alternate or out of step. We note that mathematically the intensity variations of each mode in the above two collective states can be generally described as i a ( t ) - Aa(t)~t) and lb(t)----Ab(t)f(t + 8), where f(t)=f(t + T) is a periodic function of time with period T = 1/ft, Aa(t) and A b(t) are the functions which describe the chaotic modulation of the mode intensity, and 8 is a constant. In the in-step collective state 8 = 0, and in the out-of-step state ,~-- T/2. From the mathematical expression of the mode variations, one can see that the two collective states can alternatively be described as two special cases of phase synchronisation between the two mode intensity variations. From the mode competition point of view, the behaviour associated with each of the observed collective laser states also seems understandable. In the in-step collective state, when the intensity of one mode has its maximum, the intensity of the other
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D.Y. Tang. N.R. lleckenberg / Optics Communications 131 (1996) 89-94
mode will have its maximum as well. Although the exact value of mode intensity varies with time, at any time two modes coexist, so this is in a way equivalent to the case of two modes in their steady state, so in the in-step collective state strong mode competition as observed in the steady state case exists, and this gives rise to the strong antiphase pulse height relation. In contrast, in the out-of-step cclleetive state, because mode intensity variations of the two modes are out of phase, when the pulse intensity of one mode reaches its maximum, the pulse intensity of the other mode will be at its minimum, so that at any time there seems just one mode existing in the system, and consequently there is no strong type of mode competition like in the steady state case. We can also understand the observed mode behaviours in each of the collective states from the point of view, that both modes share the population inversion of the laser. Due to the intrinsic mode instability, the population inversion varies now with time. If the two modes pulse together, it means that the two modes coexist all the time, and they will share the same population inversion all the time, so consequently there exists strong competition. Whereas in the case of the alternating state, the modes exist alternately, and at each particular time just one mode exists, so there exists no strong mode competition. This is why in this case sometimes the mode intensity evolution shows strong similarities. It can be seen that in both collective states, the intensity time evolution of each mode is not the same, despite the fact that they are dynamically coupled. We have compared some metric behaviours such as Fourier spectrum, autocorrelation and mutual information calculated from each of the individual mode intensity evolutions. In contrast, these statistical behaviours of the mode intens'.'ty evolution do show the same characteristics. It can be seen that the dynamics of coupled chaotic systems is normally more complicated than that of each sub chaotic system before coupling. Recently, Pecora and Caroll have proposed a method to synchronise two chaotic attractors [16]. By using a special electric circuit they have also successfully demonstrated such chaotic synchronisation [17]. A crucial requirement of their chaos synchmnisation is that one can separate a stable subsystem from a chaotic system, and link the stable subsystem with ~ common drive
signal to another near identical chaotic system to be synchronised. Obviously our laser system does not fulfil this condition. However, under certain conditions, we have also realised synchronisation between the two mode chaotic dynamics in our laser. Details of this experimental result will be reported elsewhere. It is, however, worth emphasis that chaotic synchronisation is also a special case of collective laser effects, which can spontaneously form under strong mutual laser mode interaction. In conclusion, we have experimentally studied the dynamics of interactions between two chaotically varying laser modes. We have shown experimentally that under the mutual chaotic mode interaction, the dynamics of each mode becomes more complicated than its original chaotic dynamics. However, we observed that even under the chaotic laser mode interaction, the laser shows spontaneous self-organisation, which appears in the case of our laser as either in-step or alternating intensity pulsations in the two modes.
References [I] L.A. Lugiato, C. Oldano and L.M. Narducci, J. Opt. Soc. Am. B 5 (1988) 879. [2] Chr. Tamm, Phys. Rev. A 38 (1988) 5960. [3] K. Wiesenfeld, C. Bracikowski, G. James and R. Roy, Phys. Rev. Lett. 65 (1990) 1749. [4] K. Otsuka, P. Mandel, M. Oeorgiou and C. Etrich, Jpn. J. Appl. Phys. 32 (I 993) L318. [5] S. Bielawski, D. Derozier and P. Glorieux, Phys. Rev. A 4~~, (1992) 281 I. [6] N.B. An and P. Mandel, Optics Comm. 112 (1994) 225. [7] K. Otsuka and Y. Aizawa, Phys. Rev. Lett. 72 (1994) 2701. [8] C. Bracikowski and R. Roy, Phys. Rev. A 43 (1991) 6455. [9] lgnacio Pastor-Diaz and Antonio L6pez-Fmguas, Phys. Rev. E 52 (1995) 1480. [ I0] K. Kaneko, Physica D 54 (I 99 I) I. [I I] H.G. Winful and L. Rahman, Phys. Rev. Lea. 65 (1990) 1575. [12] Tin Win, M.Y. Li, J.T. Males, N.R. Heckenberg and C.O. Weiss, Optics Comm. 103 (1993) 479. [13] D.Y. Tang, R. Dysktra and N.R. Heckenberg, Optics Comm., accepted. [14] C.O. Weiss and J. Brock, Phys. Rev. Lett. 57 (1986) 2804. [15] D.Y. Tang, C.O. Weiss, E. Rold~m and G.J. de Valctircel, Optics Comm. 89 (1992) 47. [16] L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. [17] L.M. Pecora and T.L. Carroll, IEEE. Trans. Circuits Systems 38 (1992) 453.
OpticsCommunications131 (1996) 89-94
,,
Spontaneous self-organisation in chaotic laser mode-mode interaction D.Y. Tang, N.R. Heckenberg Physics Department, The Universityof Queensland, Brisbane, OLD 4072, Austr~dia
Received 18 December1995;revisedversionreceived3 April 1996;accepted3 April 1996
Abstract We have experimentallystudied the dynamics of the interaction between two globally coupled chaotically emitting laser modes. We show that even in the chaotic case, the laser can show spontaneous self-organisation.In the case of our laser, this is expressed as the chaotic mode intensity pulsations of the two individual laser modes being either in step or alternatiitg.
Recently dynamics of globally coupled multimode lasers has been intensively studied, as in these lasers several modes coexist and under the nonlinear interaction between them, self-organised states can spontaneously be formed, which manifest different kinds of collective effects. Studies of collective effects of globally coupled multi-mode lasers have made significant progress in the past years. Collective effects such as cooperative laser frequency locking [1,2], antiphase dynamics [3-6], gain circulation [7] and energy sharing among coexisting laser modes [8] have been experimentally revealed in different kinds of globally coupled multi-mode lasers and been well explained theoretically. These studies have not only greatly improved our understanding of multi-mode laser dynamics, but, because of the direct relation between the dynamics of the globally coupled multi-mode laser to that of globally coupled nonlinear oscilla~*ors, have also made significant contributions to the study of nonlinear dynamical systems.
So far, studies of collective effects in globally coupled multi-mode lasers mainly concentrated on the laser mode interaction, in situations where the interacting modes are intrinsically stable. By "intrinsically stable" we mean that without mode interaction each laser mode involved would lase in its stable steady state. Although in previous studies of collective effects of multi-mode lasers, chaotic states of laser modes were observed occasionally, these chaotic states were caused by the nonlinear mode interaction, not due to the individual laser mode dynamics itself. Studies of chaotic laser dynamics have shown that under suitable conditions, lasers can also operate in intrinsically chaotic states, where eve~a without the existence of mode interactions, each mode of the laser behaves chaotically. Obviously, due to the intrinsic instability of the individual interacting laser modes, the interaction between laser modes could become even more complicated. However it would be interesting to ask: do collective effects still exist in the chaotic laser mode interac-
0030-4018/96/$12.00 Copyright© 1996ElsevierScienceB.V. All fightsreserved. Pll S0030-4018(96)00246-5
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D.Y. Tang, N.R, Heckenberg / Optics Communications 131 (1996) 89-94
tion? Since the dynamics of globally coupled multimode lasers operating in their intrinsically chaotic state is similar to that of globally coupled chaotic oscillators, and dynamics of such nonlinear systems is attracting more and more attention in nonlinear dynamics studies [9-1 l l, an understanding of the dynamics of chaotically interacting laser modes would have also practical importance in many disciplines of nonlinear science. In order to explore the dynamics of chaotic laser mode interactions, we have experimentally studied the dynamics of an optically pumped NH 3 bidirectional (i.e. two mode) ring laser in the parameter range where chaotic single mode dynamics is found. In this paper we show the dynamics of this laser, and in particular we demonstrate that even with the chaotic laser mode interaction, collective effects still exist in the globally coupled multi-mode laser. In the case of our laser, this is manifested as the chaotic mode intensity variations of each individual mode being either inphase or antiphase. Our experimental setup used has the same configuration as described in Ref. [12]. In brief it consists of an optically pumped bidirectional ISNl-13 ring laser operating on the wavelength of 153 p.m. The pump laser is an isotropic carbon dioxide (13CO2) laser, whose cavity has a Littrow structure. The pump laser frequency is monitored through a COs Lamb-dip cell, and the pump frequency can be finely tuned through a piezo ceramic mirror' mount. The far infrared (FIR) emissions of the bidirectional NH~ ring laser in both directions were simultaneously detected with two Schottky barrier diodes. The electrical signal from each diode was low-noise preamplified and then stored with a DASP c',,rd in a computer. In a previous paper [13], we have reported experimental studies on the nonlinear mode-mode interaction between the forward and backward modes of the le.ser. In that experiment, in order to get a clear picture of nonlinear mode interaction of the laser, we intentionally operated the laser in the parameter range where stable single mode dynamics was ensured. Our results suggested that the interaction between the two laser modes is mainly the mode competition resulting from sharing the same population inversion and mode coupling caused by the spatial population grating formed in the laser. Through these nonlinear
mode interactions the two counterpropagating modes of the laser become globally mutually coupled. Under the global coupling the intensity evolution of each mode can show strong antiphase dynamics. We have observed both periodic and chaotic antiphase dynamics in the laser. Under special conditions even an antiphase period-doubling route to chaos at the mode alternation frequency was revealed in the laser. It is to be noted, however, that in that experiment the observed chaotic mode dynamics is the result of strong nonlinear mode interaction between the counter-propagating laser modes, which has no relation to the single mode laser chaos which can be shown by the laser under different conditions. This was clearly proved by the time scale of the observed mode intensity pulsation. We have shown in that experiment that the observed mode intensity pulsation frequency changes with mode interaction strength, and the maximum value of the frequency observed is about 0.7 MHz, which is still lower than the normal single mode instability frequency of the laser at that gas pressure. In the present experiment, in contrast, in order to experimentally study the chaotic laser mode interaction, we have operated the laser in the single mode chaos parameter regime. In this parameter regime, when the laser operates in a single mode (either forward or backward mode) state, the mode evolution shows chaotic dynamics. Further, under suitable conditions, the backward mode of the laser can show 'spiral chaos' dynamics well described by the Lorenz-Haken equations [ ! 4,15]. When the laser is operating in the single mode chaos regime, experimentally we found that, depending on the pump laser frequency detuning relative to the pump line centre and the FIR laser cavity detuning, the laser can show different chaotic dynamics. With a relatively large pump frequency detuning and the FIR laser cavity tuned close to or in resonance with one of the gain peaks, the laser will show essentially single mode 'spiral' chaos in the strong mode, and the weak mode will normally be suppressed. However, if the frequency separation between the two modes is not big enough, at the points where the strong chaotic mode intensity goes close to zero, the weak laser mode will no longer be suppressed, and will build up transiently. Even a weak, transient existence of the counterpropagating mode can cause the dynamics of the chaotic laser
D.¥. Tang. N.R. Heckenberg / Optics Communications 131 (1996) 89-94
mode to greatly deviate from the single mode Lorenz-like chaos, as we have shown in Ref. [15]. We are here more interested in the interaction of two coexisting chaotic modes. To this end we tune the pump laser frequency closer to the pump line centre. It was found experimentally that with,'.n a certain pump frequency detuning range, both laser modes can lase simultaneously, with both mode emissions being chaotic. With fixed pump laser intensity, we have studied the interaction of the two coexisting chaotic modes for different pump frequency detuning, FIR laser cavity detuning and gas pressure. Under different laser operation conditions, two kinds of relationship between the mode intensity variations of the chaotic modes were observed. Fig. 1 shows as an example one of these mode intensity variation relations observed. Fig. 1 corresponds to a laser state where the intensity evolutions of the modes are not very chaotic. The mode intensity variation in Fig. 1 has the following characteristics: The fundamental intensity pulsations of both modes have the same frequency. Studying the behaviour of this fundamental mode intensity pulsation shows that its frequency clearly depends on the gas pressure, but remains relatively independent of the pump frequency and laser cavity detuning. This behaviour together with the frequency value suggests strongly that this fundamental mode intensity pulsation is caused by single mode laser chaos of each individual mode. This experimental result shows that in this parameter regime, the intrinsic
(a)
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Time (/zS) Fig. I. Mode intensity evolutions of the laser as it is in an in-step collective state. Gas pressure is 6 Pa, pump laser intensity is 4.3 W / c m 2. (a) Mode intensity evolution of the forward mode. (b) Mode intensity evolution of the backward mode.
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Frequency(MHz) Fig. 2. Fourier spectra calculated from the mode intensity evolution shown in Fig. I. (a) and (b) correspond to the trace of (a) and (b) in Fig. I respectively.
mode instability plays the predominant role in determining the mode intensity variation. Although the dynamics of each individual mode is unstable, the intensity pulsations between them keep exactly in time, indicating that the variation of each mode is not independent of the other. Another interesting feature of the mode intensity variation is the antiphase relationship between the intensity pulse height of the two modes. It is easy to see that when the intensity pulse height in one mode is higher, then the corresponding intensity pulse in the other mode is lower. Because of this an:iphase pulse height relation between them, the mode intensity variations of the two modes have totally different forms, and hence return maps. In order to find out the frequency composition of the mode intensity evolution, so as to understand the
92
D.Y. Tang, N.R. Heckenberg / Optics Communications 131 (1996) 89-.04
possible mechanism of the mode interaction, we have calculated the Fourier spectrum of each mode intensi~ evolution, as shown in Fig. 2. Although the de~led mode intensity variation with time of each mode is different, their Fourier spectra show in contrast the same characteristics. In each spectrum there exist three independent frequency components, and the value of each frequency component is the same for the two modes. It is easy to recognise that f~ is the frequency which corresponds to the fundamental intensity pulsation. As we have pointed out above, it is introduced by the single mode instability. f2 is the frequency responsible for the observed slow antiphase intensity pulse height variation, f2 exists even when the interacting modes are in their steady state. As we have understood from previous experiments [13], f2 is caused by the mode competition and coupling between the two modes, f3 is a frequency which has also been observed in the steady state mode interaction of the laser [13]. Its existence can be interpreted as the result of another laser instability particular to the optically pumped bidirectional ring laser. As reported in Ref. [13], f3 results in a periodic mode intensity modulation with unsymmetrical modulation strength in each mode. However, the modulation caused by f3 is not obvious in the mode intensity evolution. All other significant spectral components shown in Fig. 2 are nonlinear combinations of these three independent frequency components; their existence can be attributed to the result of nonlinear interaction between the modes. From the frequency composition of the mode intensity evolution, we see that the observed mode dynamics is the result of mutual interaction among the above described three mode interaction mechanisms. Since all the three spectral components show mainly line structure, it is plausible that dynamics caused by each individual mechanism i~ not chaotic. But we emphasize that under the coexistence of them the dynamics of each mode is chaotic. We have calculated the autocorrelation and some other metric behaviours of each laser mode, and all results show that this state is a weakly ch~o~ic state. The formation of the observed chaos is probably due to the well.known quasi-periodic route to chaos, since now there exist three independent frequencies in the system. Reducing the cavity detuning or pump frequency
(a)
"~
(b)
0
5
l0
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20
Time (,uS) Fig. 3. Mode intensity evolution of the laser as it is in a more chaotic in-step collective state. Gas pressure and pump laser intensity is the same as in Fig. 1, but the cavity detuning is slighdy reduced. (a) Mode intensity evolution of the forward mode. (b) Mode intensity evolution of the backward mode.
detuning slightly further, the state shown in Fig. 1 becomes more chaotic. A typical time series is shown in Fig. 3. Comparing the intensity evolution between Fig. 1 and Fig. 3 it can be seen that the mode competition in Fig. 3 becomes more significant: while in Fig. I intensity pulse heights merely show antiphase behaviour, now this competition occasionally results in almost total suppression of the intensity pulsation in the other mode. We have analysed the Fourier spectrum and the autocorrelation function of Fig. 3. The Fourier spectrum has now a very broadened continuous structure, showing that it has become completely chaotic. Despite the generally continuous structure of the spectrum, three local spectral maxima located respectively in the positions of fl, f2 and f3 are still clearly visible in the spectrum. The calculated autocorrelation function confirms further the chaoticity of the state. Although the state is very chaotic now, the individual intensity pulses between the two modes still remain exactly in step, indicating that it is an intrinsic behaviour of mode interaction. Depending on the laser conditions, another case of relative mode intensity evolution, as shown in Fig. 4 and Fig. 5, was observed. As in Fig. 1 we have shown in Fig. 4 the case in which the mode intensity is not very chaotic. In contrast to the mode intensity evolution shown in Figs. 1 and 3, where the mode intensity pulsations of the two modes are
D.Y. Tang. N.R. Heckenberg / Optics Communications 131 (1996) 89-94
Ca)
,
0
m
i
s
s
5
s
s
10
15
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Time ( ~ ) Fig. 4. Mode intensity evolutions of the laser as it is in an alternating collective state. Gas pressure is 6.5 Pa, pump laser intensity is 4 W / c m 2. (a) Mode intensity evolution of the forward mode. (b) Mode intensity evolution of the backward mode.
exactly in step, the mode intensity pulsations of the two modes are now exactly out of step. Another significant difference to that of Fig. 1 and Fig. 3 is that the intensity pulse heights of the modes do not show antiphase behaviour, instead sometimes, with one pulse time delay, the intensity variations of these two modes show a strong similarity. Analysis of the Fourier spectra show that there still exist three independent frequencies, which can be identified again as having the same origins as described in the case of Fig. 1, although now the effect caused by the existence of f2 is no longer obvious in the mode inten(~)
0
10
2O
30
Time (/aS) Fig. 5. Mode intensity evolutions of the laser as it is in a more chaotic alternating collective state. Gas pressure is 6 Pa, pump laser intensity is 4 W/cm 2. (a) Mode intensity evolution of the forward mode. (b) Mode intensity evolution of the backward mode.
93
sity evolution. The coexistence of these three independent frequencies suggests that the mode interaction resulting in the mode iatensity evolution of Fig. 4 has the same mechanisms as those of Fig. 1. The state shown in Fig. 4 is again a three frequency weak quasi-periodic chaos state. Fig. 5 is a completely chaotic state of the out-of-step mode intensity pulsation. It can again be seen that even though the mode intensity variation is very chaotic, the out-ofstep relationship between the two mode intensity variations remains unchanged. We have checked these experimental findings under other accessible experimental conditions, and as long as the modulation effect caused by the existence of f3 can be neglected, the observed mode intensity variation will show one of the above two situations. It might be though that mode variations caused by the intrinsic mode instability could occur independently of other modes, however, this experimental result shows clearly that even when the interacting laser modes are intrinsically unstable, under a nonlinear mode interaction, any variation of an individual mode will no longer be independent but rather correlated, and depending on the particular mode interaction, the total system will evolve into different collective states. Under the nonlinear interaction of our laser, two kinds of collective states were observed. In one the individual mode intensity pulsations are in step, in the other the mode intensity pulsations are alternate or out of step. We note that mathematically the intensity variations of each mode in the above two collective states can be generally described as i a ( t ) - Aa(t)~t) and lb(t)----Ab(t)f(t + 8), where f(t)=f(t + T) is a periodic function of time with period T = 1/ft, Aa(t) and A b(t) are the functions which describe the chaotic modulation of the mode intensity, and 8 is a constant. In the in-step collective state 8 = 0, and in the out-of-step state ,~-- T/2. From the mathematical expression of the mode variations, one can see that the two collective states can alternatively be described as two special cases of phase synchronisation between the two mode intensity variations. From the mode competition point of view, the behaviour associated with each of the observed collective laser states also seems understandable. In the in-step collective state, when the intensity of one mode has its maximum, the intensity of the other
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D.Y. Tang. N.R. lleckenberg / Optics Communications 131 (1996) 89-94
mode will have its maximum as well. Although the exact value of mode intensity varies with time, at any time two modes coexist, so this is in a way equivalent to the case of two modes in their steady state, so in the in-step collective state strong mode competition as observed in the steady state case exists, and this gives rise to the strong antiphase pulse height relation. In contrast, in the out-of-step cclleetive state, because mode intensity variations of the two modes are out of phase, when the pulse intensity of one mode reaches its maximum, the pulse intensity of the other mode will be at its minimum, so that at any time there seems just one mode existing in the system, and consequently there is no strong type of mode competition like in the steady state case. We can also understand the observed mode behaviours in each of the collective states from the point of view, that both modes share the population inversion of the laser. Due to the intrinsic mode instability, the population inversion varies now with time. If the two modes pulse together, it means that the two modes coexist all the time, and they will share the same population inversion all the time, so consequently there exists strong competition. Whereas in the case of the alternating state, the modes exist alternately, and at each particular time just one mode exists, so there exists no strong mode competition. This is why in this case sometimes the mode intensity evolution shows strong similarities. It can be seen that in both collective states, the intensity time evolution of each mode is not the same, despite the fact that they are dynamically coupled. We have compared some metric behaviours such as Fourier spectrum, autocorrelation and mutual information calculated from each of the individual mode intensity evolutions. In contrast, these statistical behaviours of the mode intens'.'ty evolution do show the same characteristics. It can be seen that the dynamics of coupled chaotic systems is normally more complicated than that of each sub chaotic system before coupling. Recently, Pecora and Caroll have proposed a method to synchronise two chaotic attractors [16]. By using a special electric circuit they have also successfully demonstrated such chaotic synchronisation [17]. A crucial requirement of their chaos synchmnisation is that one can separate a stable subsystem from a chaotic system, and link the stable subsystem with ~ common drive
signal to another near identical chaotic system to be synchronised. Obviously our laser system does not fulfil this condition. However, under certain conditions, we have also realised synchronisation between the two mode chaotic dynamics in our laser. Details of this experimental result will be reported elsewhere. It is, however, worth emphasis that chaotic synchronisation is also a special case of collective laser effects, which can spontaneously form under strong mutual laser mode interaction. In conclusion, we have experimentally studied the dynamics of interactions between two chaotically varying laser modes. We have shown experimentally that under the mutual chaotic mode interaction, the dynamics of each mode becomes more complicated than its original chaotic dynamics. However, we observed that even under the chaotic laser mode interaction, the laser shows spontaneous self-organisation, which appears in the case of our laser as either in-step or alternating intensity pulsations in the two modes.
References [I] L.A. Lugiato, C. Oldano and L.M. Narducci, J. Opt. Soc. Am. B 5 (1988) 879. [2] Chr. Tamm, Phys. Rev. A 38 (1988) 5960. [3] K. Wiesenfeld, C. Bracikowski, G. James and R. Roy, Phys. Rev. Lett. 65 (1990) 1749. [4] K. Otsuka, P. Mandel, M. Oeorgiou and C. Etrich, Jpn. J. Appl. Phys. 32 (I 993) L318. [5] S. Bielawski, D. Derozier and P. Glorieux, Phys. Rev. A 4~~, (1992) 281 I. [6] N.B. An and P. Mandel, Optics Comm. 112 (1994) 225. [7] K. Otsuka and Y. Aizawa, Phys. Rev. Lett. 72 (1994) 2701. [8] C. Bracikowski and R. Roy, Phys. Rev. A 43 (1991) 6455. [9] lgnacio Pastor-Diaz and Antonio L6pez-Fmguas, Phys. Rev. E 52 (1995) 1480. [ I0] K. Kaneko, Physica D 54 (I 99 I) I. [I I] H.G. Winful and L. Rahman, Phys. Rev. Lea. 65 (1990) 1575. [12] Tin Win, M.Y. Li, J.T. Males, N.R. Heckenberg and C.O. Weiss, Optics Comm. 103 (1993) 479. [13] D.Y. Tang, R. Dysktra and N.R. Heckenberg, Optics Comm., accepted. [14] C.O. Weiss and J. Brock, Phys. Rev. Lett. 57 (1986) 2804. [15] D.Y. Tang, C.O. Weiss, E. Rold~m and G.J. de Valctircel, Optics Comm. 89 (1992) 47. [16] L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. [17] L.M. Pecora and T.L. Carroll, IEEE. Trans. Circuits Systems 38 (1992) 453.