In 1990 Ott, Grebogi and Yorke described an attractive method (OGY) whereby small time-dependent perturbation applied to a chaotic system allowed to stabilize unstable periodic orbits[1]. This method is applicable to experimental situations in which a priori analytical knowledge of the system is not available[2,3]. Their method assumes the dynamics of the system can be represented as arising from a nonlinear map (e.g., a return map). The iterates are then given by Xn+1 = F(Xn,p), where p is some accessible parameter of the system. To control chaotic dynamics one only needs to learn the local dynamics around the desired unstable periodic orbit (e.g., a fixed point Xn=XF) on the nonlinear map : especially, the derivatives with respect to p of the orbit location. When the motion is near the periodic orbit(Xn#XF), small appropriate temporal perturbations of the control parameter p allow to hold the motion on its unstable periodic orbits.
{"title":"Controlling Laser Chaos","authors":"P. Glorieux","doi":"10.1002/3527607455.CH19","DOIUrl":"https://doi.org/10.1002/3527607455.CH19","url":null,"abstract":"In 1990 Ott, Grebogi and Yorke described an attractive method (OGY) whereby small time-dependent perturbation applied to a chaotic system allowed to stabilize unstable periodic orbits[1]. This method is applicable to experimental situations in which a priori analytical knowledge of the system is not available[2,3]. Their method assumes the dynamics of the system can be represented as arising from a nonlinear map (e.g., a return map). The iterates are then given by Xn+1 = F(Xn,p), where p is some accessible parameter of the system. To control chaotic dynamics one only needs to learn the local dynamics around the desired unstable periodic orbit (e.g., a fixed point Xn=XF) on the nonlinear map : especially, the derivatives with respect to p of the orbit location. When the motion is near the periodic orbit(Xn#XF), small appropriate temporal perturbations of the control parameter p allow to hold the motion on its unstable periodic orbits.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126955908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-03-01DOI: 10.1016/0030-4018(93)90631-E
K. Shaw
{"title":"Observation of Chaos in Off-Bragg Photorefractive Four-wave Mixing","authors":"K. Shaw","doi":"10.1016/0030-4018(93)90631-E","DOIUrl":"https://doi.org/10.1016/0030-4018(93)90631-E","url":null,"abstract":"","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124618370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-10-01DOI: 10.1080/09500349214552081
G. Harkness, W. Firth
In recent years there has been much research into lasers using solid state materials such as Nd:YAG, Nd:YLF and LNP as their active media [1]. Microchip solid state lasers use a thin slab of these materials in a short cavity to ensure single longitudinal mode operation. They may be pumped by a diode laser and they produce single transverse, single longitudinal mode output over a large range of pump powers. It is useful to be able to model the spatial and temporal behaviour of these lasers with a view to design optimisation.
{"title":"Transverse Modes Of Microchip Solid State Lasers","authors":"G. Harkness, W. Firth","doi":"10.1080/09500349214552081","DOIUrl":"https://doi.org/10.1080/09500349214552081","url":null,"abstract":"In recent years there has been much research into lasers using solid state materials such as Nd:YAG, Nd:YLF and LNP as their active media [1]. Microchip solid state lasers use a thin slab of these materials in a short cavity to ensure single longitudinal mode operation. They may be pumped by a diode laser and they produce single transverse, single longitudinal mode output over a large range of pump powers. It is useful to be able to model the spatial and temporal behaviour of these lasers with a view to design optimisation.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121069887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The transverse modulational instability (TMI) of two counterpropagating light waves in finite Kerr media is modeled by a pair of coupled nonlinear Schroedinger equations. This model predicts that transverse modulations in the wave amplitudes can be either convectively or absolutely unstable, in both self-focusing and self-defocusing media. In general, both frequency- and wavenumber-shifted sideband modes can grow if the wave intensities are unequal.
{"title":"Transverse Modulational Instability of Counterpropagating Light Waves","authors":"G. Luther, C. McKinstrie","doi":"10.1364/JOSAB.9.001047","DOIUrl":"https://doi.org/10.1364/JOSAB.9.001047","url":null,"abstract":"The transverse modulational instability (TMI) of two counterpropagating light waves in finite Kerr media is modeled by a pair of coupled nonlinear Schroedinger equations. This model predicts that transverse modulations in the wave amplitudes can be either convectively or absolutely unstable, in both self-focusing and self-defocusing media. In general, both frequency- and wavenumber-shifted sideband modes can grow if the wave intensities are unequal.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130445070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1991-03-01DOI: 10.1142/9789814503372_0004
F. Argoul, A. Arneodo, P. Richetti
The Belousov-Zhabotinskii reaction has revealed most of the well-known scenarios to chaos including period-doubling, intermittency, quasiperiodicity, frequency locking, fractal torus …. However, although the data have been shown to display unambiguous features of deterministic chaos, the understanding of the nature and the origin of the observed behavior has been incomplete. In 1976, Rössler suggested an intuitive interpretation to explain chemical chaos. His feeling was that nonperiodic wandering trajectories might arise in chemical systems from a pleated slow manifold (Fig. 1a), if the flow on the lower surface of the pleat had the property of returning trajectories to a small neighborhood of an unstable focus lying on the upper surface. In this communication, we intend to revisit the terminology introduced by Rössler of “spiral-type”, “screw-type” and “funnel-type” strange attractors in terms of chaotic orbits that occur in nearly homoclinic conditions. According to a theorem by Shil’nikov, there exist uncountably many nonperiodic trajectories in systems which display a homoclinic orbit biasymptotic to a saddle-focus O, providing the following condition is fulfilled: ρ/λ < 1, where the eigenvalue of O are (−λ, ρ ± iω). This subset of chaotic trajectories is actually in one to one correspondance with a shift automorphism with an infinite number of symbols. Since homoclinic orbits are structurally unstable objects which lie on codimension-one hypersurfaces in the constraint space, one can reasonably hope to cross these hypersurfaces when following a one-parameter path. The bifurcation structure encountered near homoclinicity involves infinite sequences of saddle-node and period-doubling bifurcations. The aim of this paper is to provide numerical and experimental evidences for Shil’nikov homoclinic chaos in nonequilibrium chemical systems.
{"title":"Symbolic dynamics in the Belousov-Zhabotinskii reaction: from Rössler’s intuition to experimental evidence for Shil’nikov homoclinic chaos","authors":"F. Argoul, A. Arneodo, P. Richetti","doi":"10.1142/9789814503372_0004","DOIUrl":"https://doi.org/10.1142/9789814503372_0004","url":null,"abstract":"The Belousov-Zhabotinskii reaction has revealed most of the well-known scenarios to chaos including period-doubling, intermittency, quasiperiodicity, frequency locking, fractal torus …. However, although the data have been shown to display unambiguous features of deterministic chaos, the understanding of the nature and the origin of the observed behavior has been incomplete. In 1976, Rössler suggested an intuitive interpretation to explain chemical chaos. His feeling was that nonperiodic wandering trajectories might arise in chemical systems from a pleated slow manifold (Fig. 1a), if the flow on the lower surface of the pleat had the property of returning trajectories to a small neighborhood of an unstable focus lying on the upper surface. In this communication, we intend to revisit the terminology introduced by Rössler of “spiral-type”, “screw-type” and “funnel-type” strange attractors in terms of chaotic orbits that occur in nearly homoclinic conditions. According to a theorem by Shil’nikov, there exist uncountably many nonperiodic trajectories in systems which display a homoclinic orbit biasymptotic to a saddle-focus O, providing the following condition is fulfilled: ρ/λ < 1, where the eigenvalue of O are (−λ, ρ ± iω). This subset of chaotic trajectories is actually in one to one correspondance with a shift automorphism with an infinite number of symbols. Since homoclinic orbits are structurally unstable objects which lie on codimension-one hypersurfaces in the constraint space, one can reasonably hope to cross these hypersurfaces when following a one-parameter path. The bifurcation structure encountered near homoclinicity involves infinite sequences of saddle-node and period-doubling bifurcations. The aim of this paper is to provide numerical and experimental evidences for Shil’nikov homoclinic chaos in nonequilibrium chemical systems.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126582491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sustained relaxation oscillations and irregular spiking have been observed in many periodically modulated lasers [2]. These observations have been substantiated numerically by recent studies of the laser rate equations [3,4]. In this paper, we propose a new asymptotic analysis of the laser equations which assumes that the laser oscillations correspond to relaxation oscillations. We identify a large parameter and construct these periodic solutions using singular perturbation techniques. We obtain the equations for the Poincare map and determine the first period doubling bifurcation.
{"title":"New Asymptotic Theory for the Periodically Forced Laser","authors":"T. Erneux, I. Schwartz","doi":"10.21236/ADA232633","DOIUrl":"https://doi.org/10.21236/ADA232633","url":null,"abstract":"Sustained relaxation oscillations and irregular spiking have been observed in many periodically modulated lasers [2]. These observations have been substantiated numerically by recent studies of the laser rate equations [3,4]. In this paper, we propose a new asymptotic analysis of the laser equations which assumes that the laser oscillations correspond to relaxation oscillations. We identify a large parameter and construct these periodic solutions using singular perturbation techniques. We obtain the equations for the Poincare map and determine the first period doubling bifurcation.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"239 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121948822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1991-02-01DOI: 10.1088/0954-8998/3/1/003
P. Alsing, H. Carmichael
The interaction between two-state atoms and the modes of an optical cavity has been studied extensively over many years; in particular, in connection with lasers and optical bistability. These are dissipative systems, and it has been usual in their study to assume that dissipation rates are, in some sense, large compared with the coupling strength between the atoms and the field. Recent work has shown that interesting new dynamical behavior occurs when this is not so. For example, spontaneous emission linewidths can be altered,1,2 and if the coupling is very strong, an emission line can be split into a pair of resonances identified with the normal- mode frequencies of coupled oscillators describing the atomic polarization and the cavity field.3.4 Altered linewidths and normal-mode splitting are features of weak excitation; both are explained by a coupled damped harmonic oscillator model. In this paper we report new results obtained for strong excitation.
{"title":"Spontaneous Dressed-State Polarization of a Coupled Atom and Cavity Mode","authors":"P. Alsing, H. Carmichael","doi":"10.1088/0954-8998/3/1/003","DOIUrl":"https://doi.org/10.1088/0954-8998/3/1/003","url":null,"abstract":"The interaction between two-state atoms and the modes of an optical cavity has been studied extensively over many years; in particular, in connection with lasers and optical bistability. These are dissipative systems, and it has been usual in their study to assume that dissipation rates are, in some sense, large compared with the coupling strength between the atoms and the field. Recent work has shown that interesting new dynamical behavior occurs when this is not so. For example, spontaneous emission linewidths can be altered,1,2 and if the coupling is very strong, an emission line can be split into a pair of resonances identified with the normal- mode frequencies of coupled oscillators describing the atomic polarization and the cavity field.3.4 Altered linewidths and normal-mode splitting are features of weak excitation; both are explained by a coupled damped harmonic oscillator model. In this paper we report new results obtained for strong excitation.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1991-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131113145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
B. Sfez, J. Oudar, J. Michel, R. Kuszelewicz, R. Azoulay
Monolithic bistable microcavities with a GaAs/Al0.3Ga0.7As multiple quantum well active layer and AlAs/Al0.1Ga0.9As Bragg reflectors have been fabricated by metalorganic vapor phase epitaxy. The design of the whole structure is such that a good cavity finesse and a high contrast in the reflective mode are simultaneously obtained. This results in a bistability power threshold of less than 3 mW at 838 nm and a contrast ratio as high as 30:1. A new method is proposed, which allows to measure the nonlinear refractive index in a quasi-continuous regime, and in the operating conditions for bistability. The nonlinear index is shown to saturate at higher power, which evidences the need of a good cavity finesse for such bistable devices. Memory effect is then demonstrated: the sample can stay in either of its two stable states for more than 1 ms without spontaneously switching, and for input pulses as short as 60 ns. Finally we describe two-beam experiments with a new experimental configuration, also in reflection, which allows to mix the two input beams and extract the output beam with an ideal 100% efficiency.
{"title":"High-Contrast Multiple Quantum-Well Optical Bistable Device with Integrated Bragg Reflectors","authors":"B. Sfez, J. Oudar, J. Michel, R. Kuszelewicz, R. Azoulay","doi":"10.1063/1.103679","DOIUrl":"https://doi.org/10.1063/1.103679","url":null,"abstract":"Monolithic bistable microcavities with a GaAs/Al0.3Ga0.7As multiple quantum well active layer and AlAs/Al0.1Ga0.9As Bragg reflectors have been fabricated by metalorganic vapor phase epitaxy. The design of the whole structure is such that a good cavity finesse and a high contrast in the reflective mode are simultaneously obtained. This results in a bistability power threshold of less than 3 mW at 838 nm and a contrast ratio as high as 30:1. A new method is proposed, which allows to measure the nonlinear refractive index in a quasi-continuous regime, and in the operating conditions for bistability. The nonlinear index is shown to saturate at higher power, which evidences the need of a good cavity finesse for such bistable devices. Memory effect is then demonstrated: the sample can stay in either of its two stable states for more than 1 ms without spontaneously switching, and for input pulses as short as 60 ns. Finally we describe two-beam experiments with a new experimental configuration, also in reflection, which allows to mix the two input beams and extract the output beam with an ideal 100% efficiency.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123124339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1990-05-21DOI: 10.1364/nldos.1990.tdsls66
G. Bava, P. Debernardi, L. Lugiato
We formulate a set of dynamical equations, which govern the dynamical evolution of optically bistable systems based on Multiple Quantum Well Structures, in conditions of quasi-resonance with an excitonic line. The steady state diagrams indicate the possibility of bistability in this system.
{"title":"Dynamic Model for Optical Bistability in Multiple Quantum-Well Structures","authors":"G. Bava, P. Debernardi, L. Lugiato","doi":"10.1364/nldos.1990.tdsls66","DOIUrl":"https://doi.org/10.1364/nldos.1990.tdsls66","url":null,"abstract":"We formulate a set of dynamical equations, which govern the dynamical evolution of optically bistable systems based on Multiple Quantum Well Structures, in conditions of quasi-resonance with an excitonic line. The steady state diagrams indicate the possibility of bistability in this system.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1990-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133807679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1364/nldos.1990.tdsls47
L. Rivlin
The Routh-Hurwitz analysis of the operation of semiconductor lasers shows all types of dynamic behavior that are essentially in accord with the majority of experimental results.
劳斯-赫维茨对半导体激光器运行的分析显示了与大多数实验结果基本一致的所有类型的动态行为。
{"title":"Pulsations, Bistability, and Hysteresis in Semiconductor Injection Lasers","authors":"L. Rivlin","doi":"10.1364/nldos.1990.tdsls47","DOIUrl":"https://doi.org/10.1364/nldos.1990.tdsls47","url":null,"abstract":"The Routh-Hurwitz analysis of the operation of semiconductor lasers shows all types of dynamic behavior that are essentially in accord with the majority of experimental results.","PeriodicalId":441335,"journal":{"name":"Nonlinear Dynamics in Optical Systems","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115617954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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