Abstract

We investigate numerically the dynamical behavior in a semiconductor ring laser (SRL) subject to a periodic modulation of the injection current. By varying the amplitude and frequency of the modulation at a fixed bias current, different dynamical states including periodic, quasi-periodic, and chaotic states are found. At frequencies comparable to the relaxation oscillation frequency, the intensities of the counterpropagating modes of the SRLs may exhibit in-phase chaotic motion similar to single mode semiconductor lasers. However, antiphase chaotic oscillations in the modal intensities are observed for modulation frequencies significantly lower than the relaxation oscillations frequency. We show that this antiphase chaotic regime does not involve carrier dynamics and is a result of the underlying symmetry of the SRL. We derive a two-dimensional asymptotic model valid on time-scales longer than the relaxation oscillations, which reproduces the observed dynamical behavior. In a further simplification, we can link this reduced set of equations to Duffing-type oscillators.

© 2012 Optical Society of America

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  4. R. Lopez-Ruiz, G. B. Mindlin, C. Perez-Garcia, and J. R. Tredicce, “A mode–mode interaction for a CO2 laser with imperfect O(2) symmetry,” Phys. Rev. A 47, 500–509 (1993).
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    [CrossRef]
  48. M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
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2012 (2)

S. T. Kingni, J. H. Talla Mbé, and P. Woafo, “Nonlinear dynamics in VCSELs driven by a sinusoidally modulated current and Rössler oscillator,” Eur. Phys. J. Plus 127, 46–55 (2012).
[CrossRef]

X. Cai, Y.-L. D. Ho, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “Frequency-domain model of longitudinal mode interaction in semiconductor ring lasers,” IEEE J. Quantum Electron. 48, 406–418 (2012).
[CrossRef]

2010 (4)

J. H. Talla Mbé, K. S. Takougang, and P. Woafo, “Chaos and pulse packages in current-modulated VCSELs,” Phys. Scr. 81, 035002 (2010).
[CrossRef]

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–217 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
[CrossRef]

W. Coomans, S. Beri, G. Van der Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A 81, 033802 (2010).
[CrossRef]

2009 (4)

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “A phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[CrossRef]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[CrossRef]

A. R. Bahrampour, H. Rooholamini, L. Rahimi, and A. A. Askari, “An inhomogeneous theoretical model for analysis of PbSe quantum-dot-doped fiber amplifie,” Opt. Commun. 282, 4449–4454 (2009).
[CrossRef]

A. Mignot, G. Feugnet, S. Schwartz, I. Sagnes, A. Garnache, C. Fabre, and J.-P. Pocholle, “Single-frequency external-cavity semiconductor ring-laser gyroscope,” Opt. Lett. 34, 97–99 (2009).
[CrossRef]

2008 (7)

S. J. Chang, C. Y. Ni, Z. P. Wang, and Y. J. Chen, “A compact and low power consumption optical switch based on microrings,” IEEE Photon. Technol. Lett. 20, 1021–1023 (2008).
[CrossRef]

S. Sunada, S. Tamura, K. Inagaki, and T. Harayama, “Ring-laser gyroscope without the lock-in phenomenon,” Phys. Rev. A 78, 053822 (2008).
[CrossRef]

Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Integrated small-size semiconductor micro-ring laser with novel retro-reflector cavity,” IEEE Photon. Technol. Lett. 20, 99–101 (2008).
[CrossRef]

M. Waldow, T. Plotzing, M. Gottheil, M. Först, J. Bolten, T. Wahlbrink, and H. Kurz, “25 ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express 16, 7693–7702 (2008).
[CrossRef]

S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
[CrossRef]

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behavior of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[CrossRef]

S. Furst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

2007 (1)

A. Valle, M. Sciamanna, and K. Panajotov, “Nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers under current modulation,” Phys. Rev. E 76, 046206 (2007).
[CrossRef]

2005 (2)

G. Giuliani, R. Miglierina, M. Sorel, and A. Sciré, “Linewidth, autocorrelation and cross-correlation measurement of counter-propagating modes in GaAs-GaAlAs semiconductor ring lasers,” IEEE J. Sel. Top. Quantum. Electron. 11, 1187–1192(2005).
[CrossRef]

C. Born, M. Sorel, and S. Yu, “Controllable and stable mode selection in a semiconductor ring laser by injection locking,” IEEE J. Quantum Electron. 41, 261–271 (2005).
[CrossRef]

2004 (1)

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

2003 (3)

M. Sciamanna, A. Valle, P. Megret, M. Blondel, and K. Panajotov, “Nonlinear polarization dynamics in directly modulated vertical-cavity surface-emitting lasers,” Phys. Rev. E 68, 016207 (2003).
[CrossRef]

Y. Chembo and P. Woafo, “Stability analysis for the synchronization of semiconductor lasers with ultra-high frequency current modulation,” Phys. Lett. A 308, 381–390 (2003).
[CrossRef]

M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[CrossRef]

2002 (4)

A. Valle, L. Pesquera, S. I. Turovets, and J. M. Lopez, “Nonlinear dynamics of current-modulated vertical-cavity surface-emitting lasers,” Opt. Commun. 208, 173–182 (2002).
[CrossRef]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[CrossRef]

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

E. F. Manffra, I. L. Caldas, R. L. Viana, and H. J. Kalinowski, “Type-I intermittency and crisis-induced intermitency in a semiconductor laser under injection current modulation,” Nonlinear Dyn. 27, 185–195 (2002).
[CrossRef]

2000 (2)

L. Lachinova and W. Lu, “Pattern formation and competition in a nonlinear ring cavity,” J. Opt. B: Quant. Semiclass. Opt. 2, 393–398 (2000).
[CrossRef]

T. Numai, “Analysis of signal voltage in a semiconductor ring laser gyro,” IEEE J. Quantum Electron. 36, 1161–1167 (2000).
[CrossRef]

1997 (2)

D. Y. Tang and N. R. Heckenberg, “Anti-phase dynamics of a chaotic multimode laser,” Phys. Rev. A 56, 1050–1052 (1997).
[CrossRef]

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

1993 (3)

R. Lopez-Ruiz, G. B. Mindlin, C. Perez-Garcia, and J. R. Tredicce, “A mode–mode interaction for a CO2 laser with imperfect O(2) symmetry,” Phys. Rev. A 47, 500–509 (1993).
[CrossRef]

M. Sargent, “Theory of a multimode quasi-equilibrium semiconductor laser,” Phys. Rev. A 48, 717–726 (1993).
[CrossRef]

H.-F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 pm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29, 1668–1675 (1993).
[CrossRef]

1992 (2)

C. Etrich, P. Mandel, N. B. Abraham, and H. Zeghlache, “Dynamics of a two-mode semiconductor laser,” IEEE J. Quantum Electron. 28, 811–821 (1992).
[CrossRef]

E. D. Angelo, E. Izaguirre, G. Mindlin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

1990 (1)

T. Krauss, P. J. R. Laybourn, and J. S. Roberts, “CW operation of semiconductor ring lasers,” Electron. Lett. 26, 2095–2097 (1990).
[CrossRef]

1988 (2)

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497(1988).
[CrossRef]

Y. Hori, H. Serizawa, and H. Sato, “Chaos in a directly modulated semiconductor laser,” J. Opt. Soc. Am. B 5, 1128–1133 (1988).
[CrossRef]

1986 (1)

M. Tang and S. Wang, “Simulation studies of bifurcation and chaos in semiconductor lasers,” Appl. Phys. Lett. 48, 900–902 (1986).
[CrossRef]

1985 (2)

C. H. Lee, T.-H. Yoon, and S.-Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46, 95–97 (1985).
[CrossRef]

Y. C. Chen, H. G. Winful, and J. M. Liu, “Subharmonic bifurcations and irregular pulsing behavior of modulated semiconductor lasers,” Appl. Phys. Lett. 47, 208–210 (1985).
[CrossRef]

1980 (1)

A. Liao and S. Wang, “Semiconductor injection lasers with a circular resonator,” Appl. Phys. Lett. 36, 801–803 (1980).
[CrossRef]

1977 (1)

N. Matsumoto and K. Kumabe, “AlGaAs-GaAs semiconductor ring lasers,” Jpn. J. Appl. Phys. 16, 1395–1398 (1977).
[CrossRef]

1973 (1)

L. N. Menegozzi and W. E. Lamb, “Theory of a ring laser,” Phys. Rev. A 8, 2103–2125 (1973).
[CrossRef]

1965 (1)

F. Aronowitz, “Theory of a traveling-wave optical maser,” Phys. Rev. A 139, 635–646 (1965).
[CrossRef]

Abraham, N. B.

C. Etrich, P. Mandel, N. B. Abraham, and H. Zeghlache, “Dynamics of a two-mode semiconductor laser,” IEEE J. Quantum Electron. 28, 811–821 (1992).
[CrossRef]

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497(1988).
[CrossRef]

Angelo, E. D.

E. D. Angelo, E. Izaguirre, G. Mindlin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Aronowitz, F.

F. Aronowitz, “Theory of a traveling-wave optical maser,” Phys. Rev. A 139, 635–646 (1965).
[CrossRef]

Askari, A. A.

A. R. Bahrampour, H. Rooholamini, L. Rahimi, and A. A. Askari, “An inhomogeneous theoretical model for analysis of PbSe quantum-dot-doped fiber amplifie,” Opt. Commun. 282, 4449–4454 (2009).
[CrossRef]

Bahrampour, A. R.

A. R. Bahrampour, H. Rooholamini, L. Rahimi, and A. A. Askari, “An inhomogeneous theoretical model for analysis of PbSe quantum-dot-doped fiber amplifie,” Opt. Commun. 282, 4449–4454 (2009).
[CrossRef]

Ballantyne, J. M.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

Balle, S.

S. Furst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[CrossRef]

Beri, S.

W. Coomans, S. Beri, G. Van der Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A 81, 033802 (2010).
[CrossRef]

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–217 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
[CrossRef]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[CrossRef]

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “A phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[CrossRef]

S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
[CrossRef]

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Blondel, M.

M. Sciamanna, A. Valle, P. Megret, M. Blondel, and K. Panajotov, “Nonlinear polarization dynamics in directly modulated vertical-cavity surface-emitting lasers,” Phys. Rev. E 68, 016207 (2003).
[CrossRef]

Bolten, J.

Born, C.

C. Born, M. Sorel, and S. Yu, “Controllable and stable mode selection in a semiconductor ring laser by injection locking,” IEEE J. Quantum Electron. 41, 261–271 (2005).
[CrossRef]

Cai, X.

X. Cai, Y.-L. D. Ho, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “Frequency-domain model of longitudinal mode interaction in semiconductor ring lasers,” IEEE J. Quantum Electron. 48, 406–418 (2012).
[CrossRef]

X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

Caldas, I. L.

E. F. Manffra, I. L. Caldas, R. L. Viana, and H. J. Kalinowski, “Type-I intermittency and crisis-induced intermitency in a semiconductor laser under injection current modulation,” Nonlinear Dyn. 27, 185–195 (2002).
[CrossRef]

Chang, S. J.

S. J. Chang, C. Y. Ni, Z. P. Wang, and Y. J. Chen, “A compact and low power consumption optical switch based on microrings,” IEEE Photon. Technol. Lett. 20, 1021–1023 (2008).
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Y. Chembo and P. Woafo, “Stability analysis for the synchronization of semiconductor lasers with ultra-high frequency current modulation,” Phys. Lett. A 308, 381–390 (2003).
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Y. C. Chen, H. G. Winful, and J. M. Liu, “Subharmonic bifurcations and irregular pulsing behavior of modulated semiconductor lasers,” Appl. Phys. Lett. 47, 208–210 (1985).
[CrossRef]

Chen, Y. J.

S. J. Chang, C. Y. Ni, Z. P. Wang, and Y. J. Chen, “A compact and low power consumption optical switch based on microrings,” IEEE Photon. Technol. Lett. 20, 1021–1023 (2008).
[CrossRef]

Chi, N.

X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

Coomans, W.

W. Coomans, S. Beri, G. Van der Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A 81, 033802 (2010).
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Danckaert, J.

W. Coomans, S. Beri, G. Van der Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A 81, 033802 (2010).
[CrossRef]

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–217 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
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L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
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L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “A phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[CrossRef]

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behavior of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[CrossRef]

S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
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Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Integrated small-size semiconductor micro-ring laser with novel retro-reflector cavity,” IEEE Photon. Technol. Lett. 20, 99–101 (2008).
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M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

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M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Donati, S.

M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[CrossRef]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
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M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
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C. Etrich, P. Mandel, N. B. Abraham, and H. Zeghlache, “Dynamics of a two-mode semiconductor laser,” IEEE J. Quantum Electron. 28, 811–821 (1992).
[CrossRef]

Fabre, C.

Feugnet, G.

Först, M.

Furst, S.

S. Furst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
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Garnache, A.

Gelens, L.

W. Coomans, S. Beri, G. Van der Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A 81, 033802 (2010).
[CrossRef]

L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–217 (2010).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
[CrossRef]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[CrossRef]

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “A phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[CrossRef]

G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behavior of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
[CrossRef]

S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
[CrossRef]

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E. D. Angelo, E. Izaguirre, G. Mindlin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
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G. Giuliani, R. Miglierina, M. Sorel, and A. Sciré, “Linewidth, autocorrelation and cross-correlation measurement of counter-propagating modes in GaAs-GaAlAs semiconductor ring lasers,” IEEE J. Sel. Top. Quantum. Electron. 11, 1187–1192(2005).
[CrossRef]

M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[CrossRef]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
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S. Sunada, S. Tamura, K. Inagaki, and T. Harayama, “Ring-laser gyroscope without the lock-in phenomenon,” Phys. Rev. A 78, 053822 (2008).
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D. Y. Tang and N. R. Heckenberg, “Anti-phase dynamics of a chaotic multimode laser,” Phys. Rev. A 56, 1050–1052 (1997).
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M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Ho, Y.-L. D.

X. Cai, Y.-L. D. Ho, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “Frequency-domain model of longitudinal mode interaction in semiconductor ring lasers,” IEEE J. Quantum Electron. 48, 406–418 (2012).
[CrossRef]

Hoffer, L. M.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497(1988).
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J. Guckenheimer and P. Holmes, “Nonlinear oscillations, dynamical systems and bifurcations of vector fields,” 3rd ed. (Springer-Verlag, 1990).

Hori, Y.

Huyet, G.

E. D. Angelo, E. Izaguirre, G. Mindlin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Inagaki, K.

S. Sunada, S. Tamura, K. Inagaki, and T. Harayama, “Ring-laser gyroscope without the lock-in phenomenon,” Phys. Rev. A 78, 053822 (2008).
[CrossRef]

Izaguirre, E.

E. D. Angelo, E. Izaguirre, G. Mindlin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Jiang, P.

X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

Kalinowski, H. J.

E. F. Manffra, I. L. Caldas, R. L. Viana, and H. J. Kalinowski, “Type-I intermittency and crisis-induced intermitency in a semiconductor laser under injection current modulation,” Nonlinear Dyn. 27, 185–195 (2002).
[CrossRef]

Khoe, G. D.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

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S. T. Kingni, J. H. Talla Mbé, and P. Woafo, “Nonlinear dynamics in VCSELs driven by a sinusoidally modulated current and Rössler oscillator,” Eur. Phys. J. Plus 127, 46–55 (2012).
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T. Krauss, P. J. R. Laybourn, and J. S. Roberts, “CW operation of semiconductor ring lasers,” Electron. Lett. 26, 2095–2097 (1990).
[CrossRef]

Kumabe, K.

N. Matsumoto and K. Kumabe, “AlGaAs-GaAs semiconductor ring lasers,” Jpn. J. Appl. Phys. 16, 1395–1398 (1977).
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Kuznetsov, A.

A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed. (Springer-Verlag, 2004).

Lachinova, L.

L. Lachinova and W. Lu, “Pattern formation and competition in a nonlinear ring cavity,” J. Opt. B: Quant. Semiclass. Opt. 2, 393–398 (2000).
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Lamb, W. E.

L. N. Menegozzi and W. E. Lamb, “Theory of a ring laser,” Phys. Rev. A 8, 2103–2125 (1973).
[CrossRef]

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J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

Laybourn, J. P. R.

M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[CrossRef]

Laybourn, P. J. R.

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[CrossRef]

T. Krauss, P. J. R. Laybourn, and J. S. Roberts, “CW operation of semiconductor ring lasers,” Electron. Lett. 26, 2095–2097 (1990).
[CrossRef]

Leary, M. H.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

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C. H. Lee, T.-H. Yoon, and S.-Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46, 95–97 (1985).
[CrossRef]

Leijtens, X. J. M.

M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Liang, J. J.

J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Appl. Phys. Lett. 70, 1192–1194 (1997).
[CrossRef]

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A. Liao and S. Wang, “Semiconductor injection lasers with a circular resonator,” Appl. Phys. Lett. 36, 801–803 (1980).
[CrossRef]

Lippi, G. L.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497(1988).
[CrossRef]

Liu, H.-F.

H.-F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 pm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29, 1668–1675 (1993).
[CrossRef]

Liu, J. M.

Y. C. Chen, H. G. Winful, and J. M. Liu, “Subharmonic bifurcations and irregular pulsing behavior of modulated semiconductor lasers,” Appl. Phys. Lett. 47, 208–210 (1985).
[CrossRef]

Lopez, J. M.

A. Valle, L. Pesquera, S. I. Turovets, and J. M. Lopez, “Nonlinear dynamics of current-modulated vertical-cavity surface-emitting lasers,” Opt. Commun. 208, 173–182 (2002).
[CrossRef]

Lopez-Ruiz, R.

R. Lopez-Ruiz, G. B. Mindlin, C. Perez-Garcia, and J. R. Tredicce, “A mode–mode interaction for a CO2 laser with imperfect O(2) symmetry,” Phys. Rev. A 47, 500–509 (1993).
[CrossRef]

Lu, W.

L. Lachinova and W. Lu, “Pattern formation and competition in a nonlinear ring cavity,” J. Opt. B: Quant. Semiclass. Opt. 2, 393–398 (2000).
[CrossRef]

Mandel, P.

C. Etrich, P. Mandel, N. B. Abraham, and H. Zeghlache, “Dynamics of a two-mode semiconductor laser,” IEEE J. Quantum Electron. 28, 811–821 (1992).
[CrossRef]

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497(1988).
[CrossRef]

Manffra, E. F.

E. F. Manffra, I. L. Caldas, R. L. Viana, and H. J. Kalinowski, “Type-I intermittency and crisis-induced intermitency in a semiconductor laser under injection current modulation,” Nonlinear Dyn. 27, 185–195 (2002).
[CrossRef]

Mashall, L.

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
[CrossRef]

Matsumoto, N.

N. Matsumoto and K. Kumabe, “AlGaAs-GaAs semiconductor ring lasers,” Jpn. J. Appl. Phys. 16, 1395–1398 (1977).
[CrossRef]

Mayol, C.

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

Megret, P.

M. Sciamanna, A. Valle, P. Megret, M. Blondel, and K. Panajotov, “Nonlinear polarization dynamics in directly modulated vertical-cavity surface-emitting lasers,” Phys. Rev. E 68, 016207 (2003).
[CrossRef]

Mello, T.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497(1988).
[CrossRef]

Memon, M.

X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

Menegozzi, L. N.

L. N. Menegozzi and W. E. Lamb, “Theory of a ring laser,” Phys. Rev. A 8, 2103–2125 (1973).
[CrossRef]

Mestre, M.

S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
[CrossRef]

Mezosi, G.

X. Cai, Y.-L. D. Ho, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “Frequency-domain model of longitudinal mode interaction in semiconductor ring lasers,” IEEE J. Quantum Electron. 48, 406–418 (2012).
[CrossRef]

S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
[CrossRef]

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[CrossRef]

S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
[CrossRef]

X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

Miglierina, R.

G. Giuliani, R. Miglierina, M. Sorel, and A. Sciré, “Linewidth, autocorrelation and cross-correlation measurement of counter-propagating modes in GaAs-GaAlAs semiconductor ring lasers,” IEEE J. Sel. Top. Quantum. Electron. 11, 1187–1192(2005).
[CrossRef]

M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
[CrossRef]

M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
[CrossRef]

Mignot, A.

Mindlin, G.

E. D. Angelo, E. Izaguirre, G. Mindlin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Mindlin, G. B.

R. Lopez-Ruiz, G. B. Mindlin, C. Perez-Garcia, and J. R. Tredicce, “A mode–mode interaction for a CO2 laser with imperfect O(2) symmetry,” Phys. Rev. A 47, 500–509 (1993).
[CrossRef]

Mirasso, C. R.

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

Ngai, W. F.

H.-F. Liu and W. F. Ngai, “Nonlinear dynamics of a directly modulated 1.55 pm InGaAsP distributed feedback semiconductor laser,” IEEE J. Quantum Electron. 29, 1668–1675 (1993).
[CrossRef]

Ni, C. Y.

S. J. Chang, C. Y. Ni, Z. P. Wang, and Y. J. Chen, “A compact and low power consumption optical switch based on microrings,” IEEE Photon. Technol. Lett. 20, 1021–1023 (2008).
[CrossRef]

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T. Numai, “Analysis of signal voltage in a semiconductor ring laser gyro,” IEEE J. Quantum Electron. 36, 1161–1167 (2000).
[CrossRef]

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M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
[CrossRef]

Panajotov, K.

A. Valle, M. Sciamanna, and K. Panajotov, “Nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers under current modulation,” Phys. Rev. E 76, 046206 (2007).
[CrossRef]

M. Sciamanna, A. Valle, P. Megret, M. Blondel, and K. Panajotov, “Nonlinear polarization dynamics in directly modulated vertical-cavity surface-emitting lasers,” Phys. Rev. E 68, 016207 (2003).
[CrossRef]

Perez-Garcia, C.

R. Lopez-Ruiz, G. B. Mindlin, C. Perez-Garcia, and J. R. Tredicce, “A mode–mode interaction for a CO2 laser with imperfect O(2) symmetry,” Phys. Rev. A 47, 500–509 (1993).
[CrossRef]

Perez-Serrano, A.

S. Furst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
[CrossRef]

Pesquera, L.

C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
[CrossRef]

A. Valle, L. Pesquera, S. I. Turovets, and J. M. Lopez, “Nonlinear dynamics of current-modulated vertical-cavity surface-emitting lasers,” Opt. Commun. 208, 173–182 (2002).
[CrossRef]

Plotzing, T.

Pocholle, J.-P.

Rahimi, L.

A. R. Bahrampour, H. Rooholamini, L. Rahimi, and A. A. Askari, “An inhomogeneous theoretical model for analysis of PbSe quantum-dot-doped fiber amplifie,” Opt. Commun. 282, 4449–4454 (2009).
[CrossRef]

Roberts, J. S.

T. Krauss, P. J. R. Laybourn, and J. S. Roberts, “CW operation of semiconductor ring lasers,” Electron. Lett. 26, 2095–2097 (1990).
[CrossRef]

Rooholamini, H.

A. R. Bahrampour, H. Rooholamini, L. Rahimi, and A. A. Askari, “An inhomogeneous theoretical model for analysis of PbSe quantum-dot-doped fiber amplifie,” Opt. Commun. 282, 4449–4454 (2009).
[CrossRef]

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Sargent, M.

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A. Valle, M. Sciamanna, and K. Panajotov, “Nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers under current modulation,” Phys. Rev. E 76, 046206 (2007).
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S. Furst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
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S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
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M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
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G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behavior of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
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C. H. Lee, T.-H. Yoon, and S.-Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46, 95–97 (1985).
[CrossRef]

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M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
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M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S. Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature 432, 206–209 (2004).
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X. Cai, Y.-L. D. Ho, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “Frequency-domain model of longitudinal mode interaction in semiconductor ring lasers,” IEEE J. Quantum Electron. 48, 406–418 (2012).
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S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
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L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
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S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
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S. Furst, A. Perez-Serrano, A. Scire, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: experiment and theory,” Appl. Phys. Lett. 93, 251109 (2008).
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C. Born, M. Sorel, and S. Yu, “Controllable and stable mode selection in a semiconductor ring laser by injection locking,” IEEE J. Quantum Electron. 41, 261–271 (2005).
[CrossRef]

M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39, 1187–1195 (2003).
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M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. 27, 1992–1994 (2002).
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X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

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S. T. Kingni, J. H. Talla Mbé, and P. Woafo, “Nonlinear dynamics in VCSELs driven by a sinusoidally modulated current and Rössler oscillator,” Eur. Phys. J. Plus 127, 46–55 (2012).
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J. H. Talla Mbé, K. S. Takougang, and P. Woafo, “Chaos and pulse packages in current-modulated VCSELs,” Phys. Scr. 81, 035002 (2010).
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S. Sunada, S. Tamura, K. Inagaki, and T. Harayama, “Ring-laser gyroscope without the lock-in phenomenon,” Phys. Rev. A 78, 053822 (2008).
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M. Tang and S. Wang, “Simulation studies of bifurcation and chaos in semiconductor lasers,” Appl. Phys. Lett. 48, 900–902 (1986).
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G. Van der Sande, L. Gelens, P. Tassin, A. Scirè, and J. Danckaert, “Two-dimensional phase-space analysis and bifurcation study of the dynamical behavior of a semiconductor ring laser,” J. Phys. B 41, 095402 (2008).
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C. Mayol, R. Toral, C. R. Mirasso, S. I. Turovets, and L. Pesquera, “Theory of main resonances in directly modulated diode lasers,” IEEE J. Quantum Electron. 38, 260–269 (2002).
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R. Lopez-Ruiz, G. B. Mindlin, C. Perez-Garcia, and J. R. Tredicce, “A mode–mode interaction for a CO2 laser with imperfect O(2) symmetry,” Phys. Rev. A 47, 500–509 (1993).
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A. Valle, M. Sciamanna, and K. Panajotov, “Nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers under current modulation,” Phys. Rev. E 76, 046206 (2007).
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L. Gelens, S. Beri, G. Van der Sande, G. Verschaffelt, and J. Danckaert, “Multistable and excitable behavior in semiconductor ring lasers with broken Z2-symmetry,” Eur. Phys. J. D 58, 197–217 (2010).
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S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
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S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
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L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
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S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
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Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Integrated small-size semiconductor micro-ring laser with novel retro-reflector cavity,” IEEE Photon. Technol. Lett. 20, 99–101 (2008).
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X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

Wang, Z. P.

S. J. Chang, C. Y. Ni, Z. P. Wang, and Y. J. Chen, “A compact and low power consumption optical switch based on microrings,” IEEE Photon. Technol. Lett. 20, 1021–1023 (2008).
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S. T. Kingni, J. H. Talla Mbé, and P. Woafo, “Nonlinear dynamics in VCSELs driven by a sinusoidally modulated current and Rössler oscillator,” Eur. Phys. J. Plus 127, 46–55 (2012).
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J. H. Talla Mbé, K. S. Takougang, and P. Woafo, “Chaos and pulse packages in current-modulated VCSELs,” Phys. Scr. 81, 035002 (2010).
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Y. Chembo and P. Woafo, “Stability analysis for the synchronization of semiconductor lasers with ultra-high frequency current modulation,” Phys. Lett. A 308, 381–390 (2003).
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C. H. Lee, T.-H. Yoon, and S.-Y. Shin, “Period doubling and chaos in a directly modulated laser diode,” Appl. Phys. Lett. 46, 95–97 (1985).
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X. Cai, Y.-L. D. Ho, G. Mezosi, Z. Wang, M. Sorel, and S. Yu, “Frequency-domain model of longitudinal mode interaction in semiconductor ring lasers,” IEEE J. Quantum Electron. 48, 406–418 (2012).
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Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Integrated small-size semiconductor micro-ring laser with novel retro-reflector cavity,” IEEE Photon. Technol. Lett. 20, 99–101 (2008).
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C. Born, M. Sorel, and S. Yu, “Controllable and stable mode selection in a semiconductor ring laser by injection locking,” IEEE J. Quantum Electron. 41, 261–271 (2005).
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X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

Yuan, G.

Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Integrated small-size semiconductor micro-ring laser with novel retro-reflector cavity,” IEEE Photon. Technol. Lett. 20, 99–101 (2008).
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S. T. Kingni, J. H. Talla Mbé, and P. Woafo, “Nonlinear dynamics in VCSELs driven by a sinusoidally modulated current and Rössler oscillator,” Eur. Phys. J. Plus 127, 46–55 (2012).
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IEEE J. Sel. Top. Quantum. Electron. (1)

G. Giuliani, R. Miglierina, M. Sorel, and A. Sciré, “Linewidth, autocorrelation and cross-correlation measurement of counter-propagating modes in GaAs-GaAlAs semiconductor ring lasers,” IEEE J. Sel. Top. Quantum. Electron. 11, 1187–1192(2005).
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Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “Integrated small-size semiconductor micro-ring laser with novel retro-reflector cavity,” IEEE Photon. Technol. Lett. 20, 99–101 (2008).
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Nonlinear Dyn. (1)

E. F. Manffra, I. L. Caldas, R. L. Viana, and H. J. Kalinowski, “Type-I intermittency and crisis-induced intermitency in a semiconductor laser under injection current modulation,” Nonlinear Dyn. 27, 185–195 (2002).
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Opt. Express (1)

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Y. Chembo and P. Woafo, “Stability analysis for the synchronization of semiconductor lasers with ultra-high frequency current modulation,” Phys. Lett. A 308, 381–390 (2003).
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S. Beri, L. Mashall, L. Gelens, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Excitability in optical systems close to Z2-symmetry,” Phys. Lett. A 374, 739–743 (2010).
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Phys. Rev. A (8)

S. Sunada, S. Tamura, K. Inagaki, and T. Harayama, “Ring-laser gyroscope without the lock-in phenomenon,” Phys. Rev. A 78, 053822 (2008).
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H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, and T. Mello, “Bidirectional ring laser: stability analysis and time-dependent solutions,” Phys. Rev. A 37, 470–497(1988).
[CrossRef]

M. Sargent, “Theory of a multimode quasi-equilibrium semiconductor laser,” Phys. Rev. A 48, 717–726 (1993).
[CrossRef]

D. Y. Tang and N. R. Heckenberg, “Anti-phase dynamics of a chaotic multimode laser,” Phys. Rev. A 56, 1050–1052 (1997).
[CrossRef]

W. Coomans, S. Beri, G. Van der Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A 81, 033802 (2010).
[CrossRef]

Phys. Rev. E (3)

M. Sciamanna, A. Valle, P. Megret, M. Blondel, and K. Panajotov, “Nonlinear polarization dynamics in directly modulated vertical-cavity surface-emitting lasers,” Phys. Rev. E 68, 016207 (2003).
[CrossRef]

A. Valle, M. Sciamanna, and K. Panajotov, “Nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers under current modulation,” Phys. Rev. E 76, 046206 (2007).
[CrossRef]

L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “A phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E 79, 016213 (2009).
[CrossRef]

Phys. Rev. Lett. (3)

L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multistability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. 102, 193904 (2009).
[CrossRef]

S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Sciré, G. Mezosi, M. Sorel, and J. Danckaert, “Topological insight into the non-Arrhenius mode hopping of semiconductor ring lasers,” Phys. Rev. Lett. 101, 093903 (2008).
[CrossRef]

E. D. Angelo, E. Izaguirre, G. Mindlin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
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Phys. Scr. (1)

J. H. Talla Mbé, K. S. Takougang, and P. Woafo, “Chaos and pulse packages in current-modulated VCSELs,” Phys. Scr. 81, 035002 (2010).
[CrossRef]

Other (3)

X. Cai, G. Mezosi, M. Sorel, Z. Wang, N. Chi, P. Jiang, M. Memon, and S. Yu, “Direct modulation of bistable semiconductor ring lasers,” in CLEO/Europe and EQEC Conference Digest (OSA, 2011), paper CB2_5.

J. Guckenheimer and P. Holmes, “Nonlinear oscillations, dynamical systems and bifurcations of vector fields,” 3rd ed. (Springer-Verlag, 1990).

A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed. (Springer-Verlag, 2004).

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Figures (8)

Fig. 1.
Fig. 1.

The extremes of the modal intensity versus the dc bias injection current μdc (with μm=0.0 and fm=0.00MHz) for specific values of ϕk: (a) ϕk=0.2 and (b) ϕk1.5. The maxima (minima) of ICCW=|ECCW|2 are denoted by open black squares (circles). The maxima (minima) of ICW=|ECW|2 are denoted by gray crosses (dots). Bifurcation diagrams are obtained by scanning upwards and downwards in μdc. The roman numbers indicate different operating regimes.

Fig. 2.
Fig. 2.

Bifurcation diagrams of the modal intensities ICCW, ICW versus μm for specific values of fm: (a) fm=731.40MHz (fm/fRO|μdc=1.7040.866) and (b) fm=1.12GHz (fm/fRO|μdc=1.7041.326) with μdc=1.704 (corresponding to fRO|μdc=1.704845MHz). Black (gray) shaded areas indicate local maxima (minima).

Fig. 3.
Fig. 3.

In panel (1) we plot time traces of modal intensities ICCW (black), ICW (gray) while in panel (2), we depict the corresponding phase portraits modal intensity vs. carrier density for specific values of μm and fm in Fig. 2 with μdc=1.704 (corresponding to fRO|μdc=1.704845MHz): (a) μm=1.50 and fm=731.40MHz and (b) μm=1.58 and fm=1.12GHz.

Fig. 4.
Fig. 4.

Regions of dynamical behavior in the parameter space spanned by modulation frequency fm and amplitude μm for ϕk1.5 and specific values of μdc: (a) μdc=1.4 (in bi-A0 regime and corresponding to fRO|μdc=1.4637MHz) and (b) μdc=1.704 (in bis-Uni regime and corresponding to fRO|μdc=1.704845MHz). Periodic oscillations are in black, quasi-periodic oscillations are in yellow and chaos is in red.

Fig. 5.
Fig. 5.

Bifurcation diagrams of modal intensities ICCW, ICW versus μm for specific values of μdc and fm: (a) fm=110MHz and μdc=1.4 (corresponding to fRO|μdc=1.4637MHz) and (b) fm=240MHz and μdc=1.704 (corresponding to fRO|μdc=1.704845MHz). Black (gray) indicates local maxima (minima).

Fig. 6.
Fig. 6.

Temporal evolution of the term [μ(t)1.0](red) and time traces of modal intensities ICCW (black), ICW (gray) and the total intensity Icw+Iccw (green) inside the cavity of the SRL when operating in the periodic regime (a) μdc=1.704, μm=0.05, fm=240MHz and in the chaotic regime; (b) μdc=1.704, μm=0.30, fm=270MHz.

Fig. 7.
Fig. 7.

Time traces of modal intensities ICCW (black), ICW (gray) calculated using Eqs. (1) and (2), the relative modal intensity θ/π calculated using the reduced model and the corresponding phase portrait of different pulsing states obtained by projection of the numerical results from Eqs. (1)–(2) on the corresponding two-dimensional (θ,ψ) phase portraits of the asymptotic results from Eqs. (6)–(7) and the one obtained from the numerical integration of Eqs. (6)–(7) (gray): (a) period-1-oscillation; (b) period-3-oscillation and (c) quasi-periodicity states, where (μdc, μm, fm): (1.704,0.1,240 MHz), (1.4,0.25,110 MHz) and (1.4,0.05,110 MHz), respectively.

Fig. 8.
Fig. 8.

Time traces of modal intensities ICCW (black), ICW (gray) calculated using Eqs. (1) and (2) and the corresponding Poincaré sections (black dots) [obtained by projection of the numerical results from Eqs. (1)–(2) on the corresponding two-dimensional (θ,ψ)phase portraits of the asymptotic results from Eqs. (6)–(7)] together with the stable and instable manifolds of the saddle point solution of the reduced system Eqs. (6)–(7) (without current modulation at the specific value of the current) with (μdc, μm, fm): (a) (1.4,0.35,110 MHz) and (b) (1.704,0.233,240 MHz), respectively. The stable (unstable) manifold is indicated with light (dark) gray dots.

Equations (41)

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dEcw,ccwdt=κ(1+iα)[N(1s|Ecw,ccw|2c|Eccw,cw|2)1]Ecw,ccwkeiϕkEccw,cw
dNdt=γ[μNN(1s|Eccw|2c|Ecw|2)|Eccw|2N(1s|Ecw|2c|Eccw|2)|Ecw|2],
μ=μdc+μmsin(2πfmt),
ρ=γ/κ,N1=ρn,s=ρS,c=ρC,kκ=ρK,t=γKt,ω=2πfmγK,
|Eccw|2+|Ecw|2=(μdc1)+μmsin(2πfmt/γK)>0
dθdt=2sinϕksinψ+2cosϕkcosψsinθ+J(t)cosθsinθ
cosθdψdt=αJ(t)sinθcosθ+2cosϕksinψ+2sinϕkcosψsinθ,
x¨+δx˙βx+x3=f0sin(2πfmt),
V(x)=βx22+x44.
JTB2=±4cosϕkTB2
sin2(ϕkTB2)=1/2±α/2α2+1.
dθdt=2sinψ+Jcosθsinθ
dψdt=αJsinθ+2cosψtanθ.
dθdt=2(ϕϕ3/6+)+J(θ2θ3/3+)+H.O.T.
dϕdt=αJ(θθ3/6+)2(1ϕ2/2+)(θ+θ3/3+)+H.O.T.,
θ=ε2y,ϕ=εx,J=εI,α=υ/ε.
θ=εy,ϕ=ε2x,ε2=αJ2,J=ε3I,s=εt,
εdydt=2(xε2x3/6)+ε2Iy+H.O.T.
dxdt=ε(υI2)y+H.O.T.
d2xd2t=εIdxdt+2(υI2)xε2(υI2)3x3+H.O.T.
dyds=2x+ε2Iy+H.O.T.
dxdt=y(ε2+6)y3/6+ε2x2y+ε2(ε230)y5/15+H.O.T.
d2yd2s=2y2y3+ε2[Idydsy33+(dyds)2y2y52]+H.O.T.
12(dyds)2y2+y42=E,
Ecw,ccw=Qcw,ccwExp(iϕcw,ccw),
dQccwdt=κ[N(1sQccw2cQcw2)1]QccwkQcwcos(ψ+ϕk)
dQcwdt=κ[N(1sQcw2cQccw2)1]QcwkQccwcos(ψϕk)
dψdt=ακN(cs)(Qcw2Qccw2)+kQccwQcwsin(ψϕk)+kQcwQccwsin(ψ+ϕk)
dNdt=γ{[μdc+μmsin(2πfmt)]NN(1sQccw2cQcw2)Qccw2N(1sQcw2cQccw2)Qcw2},
ρ=γ/κ,N1=ρn,s=ρS,c=ρC,k/κ=ρK,t=γKt,ω=2πfm/Kγ,
KdQccwdt=(nSQccw2CQcw2)QccwKQcwcos(ψ+ϕk)
KdQcwdt=(nSQcw2CQccw2)QcwKQccwcos(ψϕk)
Kdψdt=α(CS)(Qcw2Qccw2)+KQccwQcwsin(ψϕk)+KQcwQccwsin(ψ+ϕk)
Qccw2+Qcw2=[μdc+μmsin(ωt)]1.
n(Qccw,ψ,t)=1μ(t)1{ωμm2cos(ωt)+S[Qccw4+Qcw4(Qccw,t)]+2CQccw2Qcw2(Qccw,t)+2KdQccwQcw(Qccw,t)cos(ψ)}
KdQccwdt=[n(Qccw,ψ,t)SQccw2CQcw2(Qccw,t)]QccwKQcw(Qccw,t)cos(ψ+ϕk)
Kdψdt=α(CS)[Qcw2(Qccw,t)Qccw2]+KQccwQcw(Qccw,t)sin(ψϕk)+KQcw(Qccw,t)Qccwsin(ψ+ϕk).
Qccw=[μdc+μmsin(ωt)]1cos[(θ+π/2)/2]
Qcw=[μdc+μmsin(ωt)]1sin[(θ+π/2)/2]
dθdt=2sinϕksinψ+2cosϕkcosψsinθ+J(t)cosθsinθ
cosθdψdt=αJ(t)sinθcosθ+2cosϕksinψ+2sinϕkcosψsinθ.

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