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Antiphase dynamics in an optically pumped bidirectional ring laser
a __ __ ~
ELSEVIER
15 May 1996
OPTICS COMMUNICATIONS Optics Communications 126 (1996) 318-325
Full length article
Antiphase dynamics in an optically pumped bidirectional ring laser D.Y. Tang, R. Dykstra, N.R. ~~ckenberg Physics Depurtment,
The Uniuersity
of Queenslund, Brisbutzr,
Qkl4072,
Australia
Received 19 October 1995;revised version received 12 January 1996;accepted 14 January 1996
Abstract
We present experimental evidence of antiphase dynamics in an optically pumped NH, bidirectional ring laser. We show that under strong mode-mode coupling between the forward and backward modes of the laser, the steady output in both directions of the laser becomes dynamically unstable. Depending on the mode interaction strength, the intensity evolution of each mode can be periodic or chaotic. Under certain conditions a period-doubling route to chaos was observed. However, no matter what state the taser is, even under the influence of another ins~bility, the intensity variations of the two modes caused by the mode interaction are always antiphase, showing that they are in a collective self-organized state.
Recently, there has been increased interest in nonlinear dynamics of multi-mode lasers, because in these lasers many modes coexist and under the nonlinear interaction between them a series of new dynamical effects, which can be described as results of collective self-organization of modes, appear [l41. Among these collective effects of multi-mode Iasers, the effect of antiphase dynamics has attracted special attention. Initially an antiphase effect was found by Hadley and Beasley in numerical studies of circuits containing coupled Josephson junctions [5]. It was described as a state that is periodic in time with each oscillator having the same wave form but being shifted by I/N of a period from its neighbor, where N is the total number of oscillators. The first experimental. observation of this effect in an optical system was reported by Wiesenfeld et al. [6] in a multi-mode Nd:YAG laser with a KTP frequency doubling crystal in its cavity. They have shown that the intensity variations of the laser modes on the two orthogonal polarizations are antiphase. Following this
experiment, antiphase dynamics of multi-mode lasers were further observed in a modulated microchip solid-state laser [7,8] and Nd-doped optical fiber laser [9]. Depending on the behavior of a multi-mode laser, there can be different manifestations of antiphase dynamics. It was shown that when the output of a multi-mode laser is time dependent, in the case of periodic laser intensity evolution, antiphase dynamics has the form as initially described by Hadley et al. [6]; while in the case of the chaotic intensity domain, antiphase dynamics means that each mode displays as many frequencies with broadened peaks as there are modes, whereas the total intensity displays only one broad peak at the largest frequency [7]. Antiphase dynamics was also observed in multimode lasers with steady intensity output [8,10]. The nature of antiphase dynamics in multi-mode lasers was intensively studied theoretically by Mande1 and Otsuka et al, [I I-141. It was shown that the intensity antiphase evolution of individual laser
0030-4018/96/$12.00 0 1996Elsevier Science B.V. All rights reserved PI1 SOO30-4018(96)00063-6
D.Y. Tang et ul./ Optics Communicurions 126 (19961318-325
modes is due ta the strong mode-mode coupling caused by strong cross mode saturation. Although antiphase dynamics has been observed experimentally in several lasers 16-91, all the cases so far reported have the same characteristic, namely the decay rate of the atomic pol~ization of these lasers is far bigger than the cavity decay rate, so it plays no role in the laser dynamics. This means that in modeling the dynamics of these lasers the polarization can be adiabatically eliminated from the laser equations - the so called class-B situation. As a further experimental confirmation of antiphase dynamics of multi-mode lasers, we show in this paper antiphase dynamics in the more complex class-C situation where the dynamics of the polarization is important, namely, the optically pumped NH, bidirectional ring laser. We demonstrate experimentally that under strong mode-mode interaction between the forward and backward emission of the laser, the stationary bidirectional emission of the laser becomes dynamically unstable and a mode alternation effect similar to that observed in the bidirectional class-B CO, laser [ 151appears, which in principle is the antiphase dynamics in this laser. Depending on the experimental conditions, the alternation between modes can be periodic or chaotic. Period-doubling route to chaos has also been observed in the mode alternation frequency which shows furthermore that this effect is a dete~inistic effect of the laser under strong mode interaction. The behavior of optically pumped ring lasers has been studied previously by a number of authors. Because of the gain anisotropy caused by longitudinal optical pumping, the optically pumped ring laser has the property of unidirectional emission [16,17]. Recently it was also theoretically pointed out by Vilaseca et al. [18] and experimentally confirmed by Wazen et al. [ 191 that optically pumped ring lasers can also exhibit stationary bidirectional operation. The reason for the stations bidirectional emission of the laser is that the nonIine~ com~tition and coupling between the forward and backward emission of the laser could compensate for the directional gain anisotropy. In fact as shown by Vilaseca et al., bidirectional emission of optically pumped ring lasers is not abnormal and can happen in a wide range of laser conditions. The dynamical aspects of optically pumped bidirectional ring lasers were first studied
_?I?
experimentally by Klische and Weiss [20] and then by Abraham and Weiss [21]. They observed in an optically pumped FIR bidirectional ring laser. intensity self pulsing, period-doubling to chaos, frequency locking and periodic windows in the chaotic regimes. However, in their ex~~ments, because measurements were made of only one direction of emission, the relation between the intensity evolution of the two modes was not clear. Also from their experiments the mechanism which caused the observed dynamics is not clear. In the past we have also made an intensive experimental study of the chaotic dynamics of an optically pumped single mode ring laser, and analyzed the Lorenz like spiral chaos [22], period-doubling chaos [23] and intermittency [24] observed in this laser. Here we note, however, that in those ex~riments the unidirectional emission of the laser is achieved through de~ning the pump laser frequency far away from the pump line center. Because of the Doppler effect on the gas molecules and the velocity selectiveness of optical pumping, the effective gain frequencies for the forward and backward emissions of the laser are widely separated and at any one time only one of them can be located in the laser cavity mode frequency linewidth range and excited. This unidirectional operation of the laser is different from that described in Ref. [17] where the unidirectional operation of the laser is due to mode com~tition, where the strong mode suppresses the weak one. As will be shown below the unidirectional operation mechanism described in Ref. [ 171 is no longer valid when the mode interaction becomes strong, while the one used in our experiments is valid even when the pump is very strong and the laser emission becomes unstable as shown in Ref.
La. The setup used in this experiment was the same as reported in Ref. [25]. Briefly it consists of a “bEI3 laser optically pumped by an isotopic 13C02 laser, whose frequency is con~olled by using Doppler-free spectroscopy (CO, Lamb-dip cell). The frequency of the CO, laser is finely tunable using a piezo ceramic mirror mount. The cavity of the NH, laser is a 2 meter long triangular ring and has a confocal cavity structure. The FIR laser emission has a wavelength of 153 p,m. Through tuning the pump laser frequency to or close to the pump line center, FIR lasing in both directions of the ring can be achieved.
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The intensity output o the FIR laser in both directions is detected simt ltaneously by two Schottky barrier diodes. The sig ials from the Schottky diodes are preamplified and sampled with two 25 MS/s DASP A/D cards wh ch were simultaneously triggered. A RF-spectrum analyzer and an oscilloscope are used in the experim :nt to monitor in real time the spectrum of laser emission in one direction and the intensities of both mod :s. From our previous experimental studies on the single mode chaotic dy ramics of this laser, we know that in the lower gas jressure range, the laser can fulfill the single mode 1iser instability conditions and displays intrinsic laser :haotic dynamics. In the present experiment we are more interested in the collective effects of multi-inode operation, namely we want to study the laser behavior under the forward and backward mode in eraction. As the first step of our studies we hope to get a clear picture of this mode-mode interactior and avoid the involvement of any intrinsic laser intability in the interaction. To this end we have concucted our experiment in the extremely high gas pn ssure range (S-10 Pa). Because the homogeneously broadened line width of the laser is determined by the gas pressure broadening, which becomes lari.er for higher gas pressure, in this gas pressure it is expected that the laser cavity will no longer be a “ Iad cavity”, so no intrinsic single mode laser instability can happen. This is confirmed by the exper mental result that at this gas pressure we have never observed the single mode laser instability which i/e observed in the lower gas pressure range (2-6 Pa:. Depending on the la,;er operation conditions, different operation modes Jredicted theoretically for an optically pumped ring laser can be achieved. When the interaction betweeii the FIR beams is weak, unidirectional operation of the laser as reported in Ref. [17] is observed. ‘With strong pump intensity, depending on the pump frequency and FIR laser cavity detuning, we have realized stationary bidirectional emission as reported in Ref. [ 191. However, we found that the statior ary bidirectional emission of the laser becomes unstable when the interaction between the forward and backward emissions is further increased. Fig. 1 shows a mode instability caused by this strong mode-mod{: interaction. Under strong mode-mode coupling cf the laser, the intensity of
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Time (@) of the laser observed under strong Fig. 1. Mode alternation forward and backward mode interaction. Pump intensity is 5 W/cm’, gas pressure is IO Pa.
each mode switches alternatively. We note that a similar kind of antiphase mode switching has also been observed in other kinds of bidirectional ring lasers, such as the CO, bidirectional ring laser [15], which was described as deterministic mode altemation. We point out that this effect is in fact the antiphase dynamics in ring lasers which is a collective self-organized state formed under strong modemode coupling. The mode alternation shown in Fig. 1 is regular. But with increased mode interaction, the regular mode switching does not remain, instead a chaotic mode intensity alternation as shown in Fig. 2 happens. Figs. 2 A and 2B are two laser states which were obtained with fixed pump laser frequency and different cavity detuning. Obviously changing cavity detuning changes mode switching frequency significantly. Another obvious difference between the mode evolution shown in Fig. 2 and Fig. 1 is that, while in Fig. 1 when a mode is switched on, just a damped relaxation oscillation exists in it, in Fig. 2 this damped relaxation oscillation becomes undamped periodic mode intensity modulation. Comparing the modulation frequency in Figs. 2A and 2B, there is no clear change indicating that this mode modulation frequency does not depend on the cavity frequency change. To understand the mechanism of this mode modulation, we have experimentally studied its behavior. Fig. 3 shows for example a pure undamped mode
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Time (@) Fig. 2. Chaotic mode alternation of the laser. Pump intensity is 6.7 W/cm’, gas pressure is 8 Pa. From A to B the pump frequency detuning is fixed, just the laser cavity detuning is tuned.
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Time (&) Fig. 3. Periodic mode intensity self-modulation observed under strong pump strength and weak mode interaction. Pump intensity is 6.7 W/cm*, gas pressureis 8 Pa.
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modulation of the modes obtained with the same pump intensity but relatively larger pump laser frequency detuning relative to the pump frequency center than the state in Fig. 2. Because with larger pump frequency detuning, the frequency separation between the effective gain of forward and backward emission is bigger, under interaction with laser cavity mode both emissions cannot be simultaneously strong, and consequently the interaction between them becomes weaker. The experimental result that there is no mode intensity switching in Fig. 3 and that when the pump frequency detuning is reduced, the mode alternation shown in Fig. 2 sets in indicates that this mode self modulation effect is not directly related to the mode interaction strength, it is an intrinsic mode instability of the laser. Several typical characteristics of this mode modulation are worthy of note. As shown in Fig. 3, the modulation frequency of each mode is the same, but there is a phase difference between them. Further experimental researches show that the phase difference varies with the laser operation conditions, and the modulation frequency depends significantly on the pump frequency detuning relative to the pump line center. For the present gas pressure and pump intensity, changing pump frequency can change the modulation frequency continuously from close to zero to maximum of - 6 MHz. The modulation strength on each mode intensity is normally different, modulation just on one mode was also experimentally observed showing evidence of asymmetry between the two modes. This kind of mode intensity modulation disappears, when the effective pump intensity is below a certain value as shown in Fig. 1. Because the mode interaction strength is closely related to the pump strength, experimentally in order to get strong mode interaction, a strong pump strength value is also needed. This has the consequence that most generally both effects described above coexist in the laser, and what we observed in the laser intensity evolution is in fact this result. Fig. 2A shows that when both of these two effects are weak, the interaction between them is also weak, and the final result of their coexistence is just a linear superposition of their effects. However, when one of them becomes strong, the influence of one on the other becomes also visible as shown in Fig. 2B. The experimental result that intensity of one mode is
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modulated while intensity of the other mode is suppressed indicates again that this mode modulation is not caused by mode frequency beating as could happen in ring laser due to backscatterin~ of external mirror or detector, but a kind of intrinsic mode instability of the laser. Fig. 2 shows that de:;pite the existence of another mode instability in the laser, the intensity evolution caused by the mode interaction remains antiphase and this antiphase can be chaotic. Periodic antiphase mode intensity changes under the coexistence of these two effects have also been observed. Fig. 4 shows for example some of these periodic antiphase states. These periodic ar tiphase mode intensity variations were obtained when the mode intensity modulation frequency is tuned to be a harmonic of the mode alternation frequency and it eventually locked to this harmonic frequency component. Due to asymmetry between the two modes and the existence of mode modulation instability, the ex~rimentally observed periodic antiphase intensity pulses could have different, sometimes even complicated, forms, which reflects the complexity of the antiphase dynamics in the laser. The states shown in Fig. 4 show antiphase behavior between each intensity pulse, i.e. when intensity variation in on: mode increased, the intensity in other mode decreases. These results were formed because in all these states the antiphase effect caused by mode interaction is stronger than the mode modulation effect caused by the intrinsic instability of the laser. PIhen the relation is reversed, namely the m~ulation effect is stronger than the antiphase one, the result shown in Fig. 5 is then observed, which shows not the mode intensity change is antiphase, but the intensity pulse height variation between the two modes is antiphase. Fig. 5A shows a quasi-periodic mode intensity evolution, which is the result that the mote moduIation frequency is irrational to the mode aI,:emation frequency. Fig. 5B is the case that the rnolzie modulation frequency is locked to the second harmonics of the antiphase mode change frequency. Under the coexistence of both mode instability effects, usually complicated quasi-periodic mode intensity variation was observed. When pump frequency detuning or FIR cavity frequency detuning is changed, the laser show:; a very complicated global bifurcation diagram. However, under the condition
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Time &Us) Fig. 4. Examples of periodic antiphase states of the laser observed under coexistence of mode alternation and self-modutation effects.
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Fig. 5. Antiphase states of the laser in which the self-modulation effect is stronger than the mode alternation effect. Pump intensity is 5.5 W/cm*, gas pressure is 10 Pa. (A) The modulation frequency is irrational to the mode alternation frequency. (B) The modulation frequency is locked to the second harmonics of the mode alternation frequency.
that the mode modulation frequency is frequency locked to a harmonic of the mode alternation frequency, a period-doubling route to chaos in the mode alternation frequency is usually observed in a very narrow parameter range. Fig. 6 shows one of this period-doubling route to chaos. Fig. 6 shows that with increased interaction strength, period-doubling to chaos appears simultaneously in both modes, and especially in all steps of the scenario the intensity changes between the two modes keep strictly antiphase, even in the chaotic state. The clear perioddoubling route to chaos suggests strongly that the observed antiphase dynamics between the two modes
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effect of the laser, and not a stochastic switching induced by noise as observed in other kinds of bidirectional ring lasers [26-2X]. The always antiphase relationship between the mode intensities shows also that the dynamics of each mode is not independent of the other, but strongly correlated. We have analyzed the Fourier spectrum, autocorrelation and mutual information of the chaotic state of each mode, we found that they all have exactly the same behaviors. This result demonstrates that for a coupled multi-component dynamical system, to get insight into the dynamics of the total system it is enough to measure just one dynamical variable. At first sight the chaotic states shown in Fig. 6D have chaotic variation both in intensity pulse height and duration. this is clearly different to the chaotic state of single mode period-doubling chaos. We believe that it would be interesting to compare the dynamics of these two forms of laser period-doubling chaos and find out their dynamical difference and relations. The characterization of this perioddoubling chaos and its comparison with the single mode laser period-doubling chaos is in progress. Before we end this paper, we note that Zeghlache et al. [29] and Hoffer et al. [30] have theoretically studied in detail the dynamics of bidirectional class-B and class-C ring lasers respectively, they have observed in their models periodic and chaotic mode alternation, coexistence of damped or undamped mode intensity modulation with mode alternation etc. To some extent our experimental results show a strong similarity to those results they found in their models, despite the fact that our laser is an optically pumped class-C laser, which in principle is more complicated. The similarity between the observed dynamical phenomena of our laser and those observed in their theoretical models indicates that the mode interaction in our laser is probably mainly due to the cross saturation of the coexisting modes and strong coupling caused by spatial population inversion grating in the laser, and under the present experimental conditions that effects specifically related to the polarization and optical pumping in our laser are not responsible for the observed antiphase dynamics. In conclusion we have experimentally studied the dynamics of an optically pumped bidirectional NH, ring laser under strong mode-mode interaction. We
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Time (pS) Fig. 6. Antiphase period-doubling route to chaos caused by the mode-mode interaction of the laser. Pump intensity is 5 W/cm*, gas pressure is 10 Pa; (A) antiphase period one state, (B) antiphase period-two state, (C) antiphase period-four state, (D) antiphase chaotic state.
show that under strong interaction between the forward and backward emissions, the stationary bidirectional emission of the laser becomes unstable and instead an antiphase dynamics of the laser appears, which is exhibited as a low frequency mode intensity alternation between the modes. Depending on the mode interaction strength, the antiphase dynamics of the laser can be periodic or chaotic. We have also shown ex~~mentally that under strong pumping, there exists another mode instability in the laser exhibited as a periodic mode intensity self-modulation. The mode modulation frequency depends on the pump frequency detuning. Because the mode interaction strength and pump strength are closely related, generally both effects coexist in the laser. An an-
tiphase period-doubling route to chaos has been experimentally observed, which suggests that the observed antiphase effect is a deterministic behavior of the laser.
References [I]
K. Otsuka and Y. Aizawa, Phys. Rev. Lett. 72 (1994) 270. K. Otsuka, Phys. Rev. Lett. 67 (1991) 1090. C. Bracikowski and R. Roy, Phys. Rev. A 43 (1991) 6455. P. Mandel and J. Wang, Optics Lett. 19 (1994) 533. P. Hadley and M.R. Beasley, Appl. Phys. Lett. 50 (1987) 621. [6] K. Wiesenfeld. C. Bracikowski, G. James and R. Roy, Phys. Rev. Lett. 6.5 (1990) 1749. [2] [3] [4] [5]
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Otsuka, P. Mandel, M. Georgiou and C. Etrich, Jpn. J. Appt. Phys. 32 ( 1993) 13 18. [S] K. Otsuka, M. Georgiou and P. Mandel, Jpn. J. Appl. Phys. 31 (1992) L12.50. 191 S. Bielawski, D. Derozier and P. Giorieux, Phys. Rev. A 46 (1992) 2811. [IO] K. Otsuka. P. Mandel, S. Bielawski, D. Derozier and P. Glorieux. Phys. Rev. A 46 (1992) 1692. [I I] J.Y. Wang and P. Mandel, Phys. Rev. A 48 (1993) 671. [Ill] D. Pieroux, P. Mandel and K. Otsuka, Optics Comm. 108 ( 1998 273. [ 131 N. Ba. Andrid P. Mandel, Optics Comm. I 12 t 1994) 235. (141 P. Mandel, C. Etrich and K. O&t&a, IEEE. J. Quantum Electron. 29 ( 1993) 836. [IS] G.L. Lippi, J.R. Tredicce, N.B. Abraham and F.T. Arecchi. Optics Comm. 53 f 1985) 129. [ 161 J. Heppner and C.O. Weiss, Appi. Phys. Lett. 33 (1978J590. [I71 K. Matsushima, N. Higashida, N. Sokabe and T. Ariyasu. Optics Comm. I17 (1995) 462. (181 R. Vilaseca, J. Martorell. R. Corbahkr and P. Laguarta, Optics Comm. 70 (1989) 131. (71 K.
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[ 191 P. Wazen and G.L. Bourdet, IEEE. 3. Qu~tum Ekctron. 27 11991) 152. [20] W. Klische and CO. Weiss, Phys. Rev. A 31 (1985) 4049. [21] N.B. Abraham and C.O. Weiss, Optics Comm. 68 (1988) 437. 1221 D.Y. Tang. CO. Weiss, E. RoldtIn and G.J. de Valckcel, Optics Comm. 89 ( 1992) 47. [23] D.Y. Tang, M.Y. Li, J.T. Males. N.R. Heckenberg and C.O. Weiss, Phys Rev. A 52 (1995) 717. [24] D.Y. Tang, J. Pujol and CO. Weiss, Phys. Rev. A 44 f I99 I) RX. 1251Tin Win, M.Y. LI, J.T. Males, N.R. Heckenberg and C.O. Weiss, Optics Comm. 103 (199.1) 479. [26] S. Singh and L. Mandei, Phys. Rev. A 20 ( 1979) 2459. 1271R. Roy and L. Mandel, Optics Comm. 34 it980) 133. [ZS] P. Lett and L. Mandel, J. Opt. Sot. Am. B 2 (1985) 1615. (291 H. Zeghlache, P. Mandel, N.B. Abraham, L.M. Hoffer, G.L. Lippi and T. Mello. Phys. Rev. A 37 (1988) 470. [301 L.M. Hoffer and N.B. Abraham, Optics Comm. 74 (1989) 261.
ELSEVIER
15 May 1996
OPTICS COMMUNICATIONS Optics Communications 126 (1996) 318-325
Full length article
Antiphase dynamics in an optically pumped bidirectional ring laser D.Y. Tang, R. Dykstra, N.R. ~~ckenberg Physics Depurtment,
The Uniuersity
of Queenslund, Brisbutzr,
Qkl4072,
Australia
Received 19 October 1995;revised version received 12 January 1996;accepted 14 January 1996
Abstract
We present experimental evidence of antiphase dynamics in an optically pumped NH, bidirectional ring laser. We show that under strong mode-mode coupling between the forward and backward modes of the laser, the steady output in both directions of the laser becomes dynamically unstable. Depending on the mode interaction strength, the intensity evolution of each mode can be periodic or chaotic. Under certain conditions a period-doubling route to chaos was observed. However, no matter what state the taser is, even under the influence of another ins~bility, the intensity variations of the two modes caused by the mode interaction are always antiphase, showing that they are in a collective self-organized state.
Recently, there has been increased interest in nonlinear dynamics of multi-mode lasers, because in these lasers many modes coexist and under the nonlinear interaction between them a series of new dynamical effects, which can be described as results of collective self-organization of modes, appear [l41. Among these collective effects of multi-mode Iasers, the effect of antiphase dynamics has attracted special attention. Initially an antiphase effect was found by Hadley and Beasley in numerical studies of circuits containing coupled Josephson junctions [5]. It was described as a state that is periodic in time with each oscillator having the same wave form but being shifted by I/N of a period from its neighbor, where N is the total number of oscillators. The first experimental. observation of this effect in an optical system was reported by Wiesenfeld et al. [6] in a multi-mode Nd:YAG laser with a KTP frequency doubling crystal in its cavity. They have shown that the intensity variations of the laser modes on the two orthogonal polarizations are antiphase. Following this
experiment, antiphase dynamics of multi-mode lasers were further observed in a modulated microchip solid-state laser [7,8] and Nd-doped optical fiber laser [9]. Depending on the behavior of a multi-mode laser, there can be different manifestations of antiphase dynamics. It was shown that when the output of a multi-mode laser is time dependent, in the case of periodic laser intensity evolution, antiphase dynamics has the form as initially described by Hadley et al. [6]; while in the case of the chaotic intensity domain, antiphase dynamics means that each mode displays as many frequencies with broadened peaks as there are modes, whereas the total intensity displays only one broad peak at the largest frequency [7]. Antiphase dynamics was also observed in multimode lasers with steady intensity output [8,10]. The nature of antiphase dynamics in multi-mode lasers was intensively studied theoretically by Mande1 and Otsuka et al, [I I-141. It was shown that the intensity antiphase evolution of individual laser
0030-4018/96/$12.00 0 1996Elsevier Science B.V. All rights reserved PI1 SOO30-4018(96)00063-6
D.Y. Tang et ul./ Optics Communicurions 126 (19961318-325
modes is due ta the strong mode-mode coupling caused by strong cross mode saturation. Although antiphase dynamics has been observed experimentally in several lasers 16-91, all the cases so far reported have the same characteristic, namely the decay rate of the atomic pol~ization of these lasers is far bigger than the cavity decay rate, so it plays no role in the laser dynamics. This means that in modeling the dynamics of these lasers the polarization can be adiabatically eliminated from the laser equations - the so called class-B situation. As a further experimental confirmation of antiphase dynamics of multi-mode lasers, we show in this paper antiphase dynamics in the more complex class-C situation where the dynamics of the polarization is important, namely, the optically pumped NH, bidirectional ring laser. We demonstrate experimentally that under strong mode-mode interaction between the forward and backward emission of the laser, the stationary bidirectional emission of the laser becomes dynamically unstable and a mode alternation effect similar to that observed in the bidirectional class-B CO, laser [ 151appears, which in principle is the antiphase dynamics in this laser. Depending on the experimental conditions, the alternation between modes can be periodic or chaotic. Period-doubling route to chaos has also been observed in the mode alternation frequency which shows furthermore that this effect is a dete~inistic effect of the laser under strong mode interaction. The behavior of optically pumped ring lasers has been studied previously by a number of authors. Because of the gain anisotropy caused by longitudinal optical pumping, the optically pumped ring laser has the property of unidirectional emission [16,17]. Recently it was also theoretically pointed out by Vilaseca et al. [18] and experimentally confirmed by Wazen et al. [ 191 that optically pumped ring lasers can also exhibit stationary bidirectional operation. The reason for the stations bidirectional emission of the laser is that the nonIine~ com~tition and coupling between the forward and backward emission of the laser could compensate for the directional gain anisotropy. In fact as shown by Vilaseca et al., bidirectional emission of optically pumped ring lasers is not abnormal and can happen in a wide range of laser conditions. The dynamical aspects of optically pumped bidirectional ring lasers were first studied
_?I?
experimentally by Klische and Weiss [20] and then by Abraham and Weiss [21]. They observed in an optically pumped FIR bidirectional ring laser. intensity self pulsing, period-doubling to chaos, frequency locking and periodic windows in the chaotic regimes. However, in their ex~~ments, because measurements were made of only one direction of emission, the relation between the intensity evolution of the two modes was not clear. Also from their experiments the mechanism which caused the observed dynamics is not clear. In the past we have also made an intensive experimental study of the chaotic dynamics of an optically pumped single mode ring laser, and analyzed the Lorenz like spiral chaos [22], period-doubling chaos [23] and intermittency [24] observed in this laser. Here we note, however, that in those ex~riments the unidirectional emission of the laser is achieved through de~ning the pump laser frequency far away from the pump line center. Because of the Doppler effect on the gas molecules and the velocity selectiveness of optical pumping, the effective gain frequencies for the forward and backward emissions of the laser are widely separated and at any one time only one of them can be located in the laser cavity mode frequency linewidth range and excited. This unidirectional operation of the laser is different from that described in Ref. [17] where the unidirectional operation of the laser is due to mode com~tition, where the strong mode suppresses the weak one. As will be shown below the unidirectional operation mechanism described in Ref. [ 171 is no longer valid when the mode interaction becomes strong, while the one used in our experiments is valid even when the pump is very strong and the laser emission becomes unstable as shown in Ref.
La. The setup used in this experiment was the same as reported in Ref. [25]. Briefly it consists of a “bEI3 laser optically pumped by an isotopic 13C02 laser, whose frequency is con~olled by using Doppler-free spectroscopy (CO, Lamb-dip cell). The frequency of the CO, laser is finely tunable using a piezo ceramic mirror mount. The cavity of the NH, laser is a 2 meter long triangular ring and has a confocal cavity structure. The FIR laser emission has a wavelength of 153 p,m. Through tuning the pump laser frequency to or close to the pump line center, FIR lasing in both directions of the ring can be achieved.
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The intensity output o the FIR laser in both directions is detected simt ltaneously by two Schottky barrier diodes. The sig ials from the Schottky diodes are preamplified and sampled with two 25 MS/s DASP A/D cards wh ch were simultaneously triggered. A RF-spectrum analyzer and an oscilloscope are used in the experim :nt to monitor in real time the spectrum of laser emission in one direction and the intensities of both mod :s. From our previous experimental studies on the single mode chaotic dy ramics of this laser, we know that in the lower gas jressure range, the laser can fulfill the single mode 1iser instability conditions and displays intrinsic laser :haotic dynamics. In the present experiment we are more interested in the collective effects of multi-inode operation, namely we want to study the laser behavior under the forward and backward mode in eraction. As the first step of our studies we hope to get a clear picture of this mode-mode interactior and avoid the involvement of any intrinsic laser intability in the interaction. To this end we have concucted our experiment in the extremely high gas pn ssure range (S-10 Pa). Because the homogeneously broadened line width of the laser is determined by the gas pressure broadening, which becomes lari.er for higher gas pressure, in this gas pressure it is expected that the laser cavity will no longer be a “ Iad cavity”, so no intrinsic single mode laser instability can happen. This is confirmed by the exper mental result that at this gas pressure we have never observed the single mode laser instability which i/e observed in the lower gas pressure range (2-6 Pa:. Depending on the la,;er operation conditions, different operation modes Jredicted theoretically for an optically pumped ring laser can be achieved. When the interaction betweeii the FIR beams is weak, unidirectional operation of the laser as reported in Ref. [17] is observed. ‘With strong pump intensity, depending on the pump frequency and FIR laser cavity detuning, we have realized stationary bidirectional emission as reported in Ref. [ 191. However, we found that the statior ary bidirectional emission of the laser becomes unstable when the interaction between the forward and backward emissions is further increased. Fig. 1 shows a mode instability caused by this strong mode-mod{: interaction. Under strong mode-mode coupling cf the laser, the intensity of
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each mode switches alternatively. We note that a similar kind of antiphase mode switching has also been observed in other kinds of bidirectional ring lasers, such as the CO, bidirectional ring laser [15], which was described as deterministic mode altemation. We point out that this effect is in fact the antiphase dynamics in ring lasers which is a collective self-organized state formed under strong modemode coupling. The mode alternation shown in Fig. 1 is regular. But with increased mode interaction, the regular mode switching does not remain, instead a chaotic mode intensity alternation as shown in Fig. 2 happens. Figs. 2 A and 2B are two laser states which were obtained with fixed pump laser frequency and different cavity detuning. Obviously changing cavity detuning changes mode switching frequency significantly. Another obvious difference between the mode evolution shown in Fig. 2 and Fig. 1 is that, while in Fig. 1 when a mode is switched on, just a damped relaxation oscillation exists in it, in Fig. 2 this damped relaxation oscillation becomes undamped periodic mode intensity modulation. Comparing the modulation frequency in Figs. 2A and 2B, there is no clear change indicating that this mode modulation frequency does not depend on the cavity frequency change. To understand the mechanism of this mode modulation, we have experimentally studied its behavior. Fig. 3 shows for example a pure undamped mode
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modulation of the modes obtained with the same pump intensity but relatively larger pump laser frequency detuning relative to the pump frequency center than the state in Fig. 2. Because with larger pump frequency detuning, the frequency separation between the effective gain of forward and backward emission is bigger, under interaction with laser cavity mode both emissions cannot be simultaneously strong, and consequently the interaction between them becomes weaker. The experimental result that there is no mode intensity switching in Fig. 3 and that when the pump frequency detuning is reduced, the mode alternation shown in Fig. 2 sets in indicates that this mode self modulation effect is not directly related to the mode interaction strength, it is an intrinsic mode instability of the laser. Several typical characteristics of this mode modulation are worthy of note. As shown in Fig. 3, the modulation frequency of each mode is the same, but there is a phase difference between them. Further experimental researches show that the phase difference varies with the laser operation conditions, and the modulation frequency depends significantly on the pump frequency detuning relative to the pump line center. For the present gas pressure and pump intensity, changing pump frequency can change the modulation frequency continuously from close to zero to maximum of - 6 MHz. The modulation strength on each mode intensity is normally different, modulation just on one mode was also experimentally observed showing evidence of asymmetry between the two modes. This kind of mode intensity modulation disappears, when the effective pump intensity is below a certain value as shown in Fig. 1. Because the mode interaction strength is closely related to the pump strength, experimentally in order to get strong mode interaction, a strong pump strength value is also needed. This has the consequence that most generally both effects described above coexist in the laser, and what we observed in the laser intensity evolution is in fact this result. Fig. 2A shows that when both of these two effects are weak, the interaction between them is also weak, and the final result of their coexistence is just a linear superposition of their effects. However, when one of them becomes strong, the influence of one on the other becomes also visible as shown in Fig. 2B. The experimental result that intensity of one mode is
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modulated while intensity of the other mode is suppressed indicates again that this mode modulation is not caused by mode frequency beating as could happen in ring laser due to backscatterin~ of external mirror or detector, but a kind of intrinsic mode instability of the laser. Fig. 2 shows that de:;pite the existence of another mode instability in the laser, the intensity evolution caused by the mode interaction remains antiphase and this antiphase can be chaotic. Periodic antiphase mode intensity changes under the coexistence of these two effects have also been observed. Fig. 4 shows for example some of these periodic antiphase states. These periodic ar tiphase mode intensity variations were obtained when the mode intensity modulation frequency is tuned to be a harmonic of the mode alternation frequency and it eventually locked to this harmonic frequency component. Due to asymmetry between the two modes and the existence of mode modulation instability, the ex~rimentally observed periodic antiphase intensity pulses could have different, sometimes even complicated, forms, which reflects the complexity of the antiphase dynamics in the laser. The states shown in Fig. 4 show antiphase behavior between each intensity pulse, i.e. when intensity variation in on: mode increased, the intensity in other mode decreases. These results were formed because in all these states the antiphase effect caused by mode interaction is stronger than the mode modulation effect caused by the intrinsic instability of the laser. PIhen the relation is reversed, namely the m~ulation effect is stronger than the antiphase one, the result shown in Fig. 5 is then observed, which shows not the mode intensity change is antiphase, but the intensity pulse height variation between the two modes is antiphase. Fig. 5A shows a quasi-periodic mode intensity evolution, which is the result that the mote moduIation frequency is irrational to the mode aI,:emation frequency. Fig. 5B is the case that the rnolzie modulation frequency is locked to the second harmonics of the antiphase mode change frequency. Under the coexistence of both mode instability effects, usually complicated quasi-periodic mode intensity variation was observed. When pump frequency detuning or FIR cavity frequency detuning is changed, the laser show:; a very complicated global bifurcation diagram. However, under the condition
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Time &Us) Fig. 4. Examples of periodic antiphase states of the laser observed under coexistence of mode alternation and self-modutation effects.
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Fig. 5. Antiphase states of the laser in which the self-modulation effect is stronger than the mode alternation effect. Pump intensity is 5.5 W/cm*, gas pressure is 10 Pa. (A) The modulation frequency is irrational to the mode alternation frequency. (B) The modulation frequency is locked to the second harmonics of the mode alternation frequency.
that the mode modulation frequency is frequency locked to a harmonic of the mode alternation frequency, a period-doubling route to chaos in the mode alternation frequency is usually observed in a very narrow parameter range. Fig. 6 shows one of this period-doubling route to chaos. Fig. 6 shows that with increased interaction strength, period-doubling to chaos appears simultaneously in both modes, and especially in all steps of the scenario the intensity changes between the two modes keep strictly antiphase, even in the chaotic state. The clear perioddoubling route to chaos suggests strongly that the observed antiphase dynamics between the two modes
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effect of the laser, and not a stochastic switching induced by noise as observed in other kinds of bidirectional ring lasers [26-2X]. The always antiphase relationship between the mode intensities shows also that the dynamics of each mode is not independent of the other, but strongly correlated. We have analyzed the Fourier spectrum, autocorrelation and mutual information of the chaotic state of each mode, we found that they all have exactly the same behaviors. This result demonstrates that for a coupled multi-component dynamical system, to get insight into the dynamics of the total system it is enough to measure just one dynamical variable. At first sight the chaotic states shown in Fig. 6D have chaotic variation both in intensity pulse height and duration. this is clearly different to the chaotic state of single mode period-doubling chaos. We believe that it would be interesting to compare the dynamics of these two forms of laser period-doubling chaos and find out their dynamical difference and relations. The characterization of this perioddoubling chaos and its comparison with the single mode laser period-doubling chaos is in progress. Before we end this paper, we note that Zeghlache et al. [29] and Hoffer et al. [30] have theoretically studied in detail the dynamics of bidirectional class-B and class-C ring lasers respectively, they have observed in their models periodic and chaotic mode alternation, coexistence of damped or undamped mode intensity modulation with mode alternation etc. To some extent our experimental results show a strong similarity to those results they found in their models, despite the fact that our laser is an optically pumped class-C laser, which in principle is more complicated. The similarity between the observed dynamical phenomena of our laser and those observed in their theoretical models indicates that the mode interaction in our laser is probably mainly due to the cross saturation of the coexisting modes and strong coupling caused by spatial population inversion grating in the laser, and under the present experimental conditions that effects specifically related to the polarization and optical pumping in our laser are not responsible for the observed antiphase dynamics. In conclusion we have experimentally studied the dynamics of an optically pumped bidirectional NH, ring laser under strong mode-mode interaction. We
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Time (pS) Fig. 6. Antiphase period-doubling route to chaos caused by the mode-mode interaction of the laser. Pump intensity is 5 W/cm*, gas pressure is 10 Pa; (A) antiphase period one state, (B) antiphase period-two state, (C) antiphase period-four state, (D) antiphase chaotic state.
show that under strong interaction between the forward and backward emissions, the stationary bidirectional emission of the laser becomes unstable and instead an antiphase dynamics of the laser appears, which is exhibited as a low frequency mode intensity alternation between the modes. Depending on the mode interaction strength, the antiphase dynamics of the laser can be periodic or chaotic. We have also shown ex~~mentally that under strong pumping, there exists another mode instability in the laser exhibited as a periodic mode intensity self-modulation. The mode modulation frequency depends on the pump frequency detuning. Because the mode interaction strength and pump strength are closely related, generally both effects coexist in the laser. An an-
tiphase period-doubling route to chaos has been experimentally observed, which suggests that the observed antiphase effect is a deterministic behavior of the laser.
References [I]
K. Otsuka and Y. Aizawa, Phys. Rev. Lett. 72 (1994) 270. K. Otsuka, Phys. Rev. Lett. 67 (1991) 1090. C. Bracikowski and R. Roy, Phys. Rev. A 43 (1991) 6455. P. Mandel and J. Wang, Optics Lett. 19 (1994) 533. P. Hadley and M.R. Beasley, Appl. Phys. Lett. 50 (1987) 621. [6] K. Wiesenfeld. C. Bracikowski, G. James and R. Roy, Phys. Rev. Lett. 6.5 (1990) 1749. [2] [3] [4] [5]
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[ 191 P. Wazen and G.L. Bourdet, IEEE. 3. Qu~tum Ekctron. 27 11991) 152. [20] W. Klische and CO. Weiss, Phys. Rev. A 31 (1985) 4049. [21] N.B. Abraham and C.O. Weiss, Optics Comm. 68 (1988) 437. 1221 D.Y. Tang. CO. Weiss, E. RoldtIn and G.J. de Valckcel, Optics Comm. 89 ( 1992) 47. [23] D.Y. Tang, M.Y. Li, J.T. Males. N.R. Heckenberg and C.O. Weiss, Phys Rev. A 52 (1995) 717. [24] D.Y. Tang, J. Pujol and CO. Weiss, Phys. Rev. A 44 f I99 I) RX. 1251Tin Win, M.Y. LI, J.T. Males, N.R. Heckenberg and C.O. Weiss, Optics Comm. 103 (199.1) 479. [26] S. Singh and L. Mandei, Phys. Rev. A 20 ( 1979) 2459. 1271R. Roy and L. Mandel, Optics Comm. 34 it980) 133. [ZS] P. Lett and L. Mandel, J. Opt. Sot. Am. B 2 (1985) 1615. (291 H. Zeghlache, P. Mandel, N.B. Abraham, L.M. Hoffer, G.L. Lippi and T. Mello. Phys. Rev. A 37 (1988) 470. [301 L.M. Hoffer and N.B. Abraham, Optics Comm. 74 (1989) 261.