Abstract

We present an experimental and a theoretical investigation of the unstable behavior of the laser-pumped short-cavity alexandrite laser. Its chaotic dynamics is achieved by an increase of the pump laser power. The route to chaos depends on the cavity length and the pump wavelength. The experimental results are explained by the theoretical model, which includes the host-lattice phonons’ dynamics in the standard laser equations. We conclude that the observed laser instabilities are due to the competition between the laser field and phonons in the host crystal. A linear-stability analysis yields the conclusion that the observed chaos has a homoclinic character.

© 2000 Optical Society of America

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    [CrossRef]
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1999

1998

1997

1996

R. Herrero, F. R. Pons, J. Farjas, F. Pi, and G. Orriols, “Homoclinic dynamics in experimental Shil’nikov attractors,” Phys. Rev. E 53, 5627–5636 (1996).
[CrossRef]

1993

R. Herrero, F. Boixader, J. Farjas, F. Pi, G. Orriols, and F. Rossler, “Rossler chaos in opto-thermal bistable devices,” Opt. Commun. 113, 324–334 (1993).
[CrossRef]

A. Argoul, J. Huth, P. Merzeau, A. Arneodo, and H. L. F. Swinney, “Experimental evidence for homoclinic chaos in electrochemical growth process,” Physica D 62, 170–185 (1993).
[CrossRef]

F. T. Arecchi, A. Lapucci, and R. Meucci, “Poincaré versus Boltzman in Shil’nikov phenomena,” Physica D 62, 186–191 (1993).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Homoclinic chaos in chemical systems,” Physica D 62, 134–169 (1993).
[CrossRef]

1992

T. Braun, J. A. Lisboa, and J. A. C. Gallas, “Evidence of homoclinic chaos in the plasma of a glow discharge,” Phys. Rev. Lett. 68, 2770–2773 (1992).
[CrossRef] [PubMed]

1991

1990

R. Scheps, B. M. Gately, J. F. Mayers, J. S. Krasinski, and D. F. Heller, “Alexandrite laser pumped by semiconductor laser,” Appl. Phys. Lett. 56, 2288–2290 (1990).
[CrossRef]

1988

F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feedback: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[CrossRef]

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with a saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[CrossRef] [PubMed]

M. R. Basset and J. L. Hudson, “Shil’nikov chaos during copper electrodissolution,” J. Phys. Chem. 92, 6963–6966 (1988).
[CrossRef]

1987

F. T. Arecchi, R. Meucci, and W. Gadomski, “Laser dynamics with competing instabilities,” Phys. Rev. Lett. 58, 2205–2208 (1987).
[CrossRef] [PubMed]

F. Argoul, A. Arneodo, and P. Richgetti, “Experimental evidence for homoclinic chaos in Belolusov–Zabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987).
[CrossRef]

1985

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[CrossRef]

1984

P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
[CrossRef]

1983

S. T. Lai and M. L. Shand, “High efficiency cw laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642–5644 (1983).
[CrossRef]

1980

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable cw alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120–121 (1980).
[CrossRef]

J. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302–1314 (1980).
[CrossRef]

1970

L. P. Shil’nikov, “A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle focus type,” Math. USSR Sbornik 10, 91–102 (1970).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, A. Lapucci, and R. Meucci, “Poincaré versus Boltzman in Shil’nikov phenomena,” Physica D 62, 186–191 (1993).
[CrossRef]

F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feedback: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[CrossRef]

F. T. Arecchi, R. Meucci, and W. Gadomski, “Laser dynamics with competing instabilities,” Phys. Rev. Lett. 58, 2205–2208 (1987).
[CrossRef] [PubMed]

Argoul, A.

A. Argoul, J. Huth, P. Merzeau, A. Arneodo, and H. L. F. Swinney, “Experimental evidence for homoclinic chaos in electrochemical growth process,” Physica D 62, 170–185 (1993).
[CrossRef]

Argoul, F.

F. Argoul, A. Arneodo, and P. Richgetti, “Experimental evidence for homoclinic chaos in Belolusov–Zabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987).
[CrossRef]

Arimondo, E.

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with a saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[CrossRef] [PubMed]

Arneodo, A.

A. Argoul, J. Huth, P. Merzeau, A. Arneodo, and H. L. F. Swinney, “Experimental evidence for homoclinic chaos in electrochemical growth process,” Physica D 62, 170–185 (1993).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Homoclinic chaos in chemical systems,” Physica D 62, 134–169 (1993).
[CrossRef]

F. Argoul, A. Arneodo, and P. Richgetti, “Experimental evidence for homoclinic chaos in Belolusov–Zabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[CrossRef]

Basset, M. R.

M. R. Basset and J. L. Hudson, “Shil’nikov chaos during copper electrodissolution,” J. Phys. Chem. 92, 6963–6966 (1988).
[CrossRef]

Boixader, F.

R. Herrero, F. Boixader, J. Farjas, F. Pi, G. Orriols, and F. Rossler, “Rossler chaos in opto-thermal bistable devices,” Opt. Commun. 113, 324–334 (1993).
[CrossRef]

Braun, T.

T. Braun, J. A. Lisboa, and J. A. C. Gallas, “Evidence of homoclinic chaos in the plasma of a glow discharge,” Phys. Rev. Lett. 68, 2770–2773 (1992).
[CrossRef] [PubMed]

Coullet, P. H.

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Homoclinic chaos in chemical systems,” Physica D 62, 134–169 (1993).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[CrossRef]

Dangoisse, D.

de Tomasi, F.

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with a saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[CrossRef] [PubMed]

Farjas, J.

R. Herrero, F. R. Pons, J. Farjas, F. Pi, and G. Orriols, “Homoclinic dynamics in experimental Shil’nikov attractors,” Phys. Rev. E 53, 5627–5636 (1996).
[CrossRef]

R. Herrero, F. Boixader, J. Farjas, F. Pi, G. Orriols, and F. Rossler, “Rossler chaos in opto-thermal bistable devices,” Opt. Commun. 113, 324–334 (1993).
[CrossRef]

Gadomski, W.

Gallas, J. A. C.

T. Braun, J. A. Lisboa, and J. A. C. Gallas, “Evidence of homoclinic chaos in the plasma of a glow discharge,” Phys. Rev. Lett. 68, 2770–2773 (1992).
[CrossRef] [PubMed]

Gately, B. M.

R. Scheps, B. M. Gately, J. F. Mayers, J. S. Krasinski, and D. F. Heller, “Alexandrite laser pumped by semiconductor laser,” Appl. Phys. Lett. 56, 2288–2290 (1990).
[CrossRef]

Glendinning, P.

P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
[CrossRef]

Heckenberg, N. R.

Heller, D. F.

R. Scheps, B. M. Gately, J. F. Mayers, J. S. Krasinski, and D. F. Heller, “Alexandrite laser pumped by semiconductor laser,” Appl. Phys. Lett. 56, 2288–2290 (1990).
[CrossRef]

Hennequin, D.

M. Lefranc, D. Hennequin, and D. Dangoisse, “Homoclinic chaos in a laser containing saturable absorber,” J. Opt. Soc. Am. B 8, 239–249 (1991).
[CrossRef]

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with a saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[CrossRef] [PubMed]

Herrero, R.

R. Herrero, F. R. Pons, J. Farjas, F. Pi, and G. Orriols, “Homoclinic dynamics in experimental Shil’nikov attractors,” Phys. Rev. E 53, 5627–5636 (1996).
[CrossRef]

R. Herrero, F. Boixader, J. Farjas, F. Pi, G. Orriols, and F. Rossler, “Rossler chaos in opto-thermal bistable devices,” Opt. Commun. 113, 324–334 (1993).
[CrossRef]

Hudson, J. L.

M. R. Basset and J. L. Hudson, “Shil’nikov chaos during copper electrodissolution,” J. Phys. Chem. 92, 6963–6966 (1988).
[CrossRef]

Huth, J.

A. Argoul, J. Huth, P. Merzeau, A. Arneodo, and H. L. F. Swinney, “Experimental evidence for homoclinic chaos in electrochemical growth process,” Physica D 62, 170–185 (1993).
[CrossRef]

Jenssen, H. P.

J. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302–1314 (1980).
[CrossRef]

Kellou, A.

Krasinski, J. S.

R. Scheps, B. M. Gately, J. F. Mayers, J. S. Krasinski, and D. F. Heller, “Alexandrite laser pumped by semiconductor laser,” Appl. Phys. Lett. 56, 2288–2290 (1990).
[CrossRef]

Lai, S. T.

S. T. Lai and M. L. Shand, “High efficiency cw laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642–5644 (1983).
[CrossRef]

Lapucci, A.

Lefranc, M.

Lisboa, J. A.

T. Braun, J. A. Lisboa, and J. A. C. Gallas, “Evidence of homoclinic chaos in the plasma of a glow discharge,” Phys. Rev. Lett. 68, 2770–2773 (1992).
[CrossRef] [PubMed]

Mancini, H.

Mayers, J. F.

R. Scheps, B. M. Gately, J. F. Mayers, J. S. Krasinski, and D. F. Heller, “Alexandrite laser pumped by semiconductor laser,” Appl. Phys. Lett. 56, 2288–2290 (1990).
[CrossRef]

Merzeau, P.

A. Argoul, J. Huth, P. Merzeau, A. Arneodo, and H. L. F. Swinney, “Experimental evidence for homoclinic chaos in electrochemical growth process,” Physica D 62, 170–185 (1993).
[CrossRef]

Meucci, R.

F. T. Arecchi, A. Lapucci, and R. Meucci, “Poincaré versus Boltzman in Shil’nikov phenomena,” Physica D 62, 186–191 (1993).
[CrossRef]

F. T. Arecchi, W. Gadomski, A. Lapucci, H. Mancini, R. Meucci, and J. A. Roversi, “Laser with feedback: an optical implementation of competing instabilities, Shil’nikov chaos, and transient fluctuation enhancement,” J. Opt. Soc. Am. B 5, 1153–1159 (1988).
[CrossRef]

F. T. Arecchi, R. Meucci, and W. Gadomski, “Laser dynamics with competing instabilities,” Phys. Rev. Lett. 58, 2205–2208 (1987).
[CrossRef] [PubMed]

Morris, R. C.

J. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302–1314 (1980).
[CrossRef]

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable cw alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120–121 (1980).
[CrossRef]

O’Dell, E. W.

J. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302–1314 (1980).
[CrossRef]

Orriols, G.

R. Herrero, F. R. Pons, J. Farjas, F. Pi, and G. Orriols, “Homoclinic dynamics in experimental Shil’nikov attractors,” Phys. Rev. E 53, 5627–5636 (1996).
[CrossRef]

R. Herrero, F. Boixader, J. Farjas, F. Pi, G. Orriols, and F. Rossler, “Rossler chaos in opto-thermal bistable devices,” Opt. Commun. 113, 324–334 (1993).
[CrossRef]

Peterson, O. G.

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable cw alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120–121 (1980).
[CrossRef]

J. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302–1314 (1980).
[CrossRef]

Pi, F.

R. Herrero, F. R. Pons, J. Farjas, F. Pi, and G. Orriols, “Homoclinic dynamics in experimental Shil’nikov attractors,” Phys. Rev. E 53, 5627–5636 (1996).
[CrossRef]

R. Herrero, F. Boixader, J. Farjas, F. Pi, G. Orriols, and F. Rossler, “Rossler chaos in opto-thermal bistable devices,” Opt. Commun. 113, 324–334 (1993).
[CrossRef]

Pons, F. R.

R. Herrero, F. R. Pons, J. Farjas, F. Pi, and G. Orriols, “Homoclinic dynamics in experimental Shil’nikov attractors,” Phys. Rev. E 53, 5627–5636 (1996).
[CrossRef]

Ratajska-Gadomska, B.

Richgetti, P.

F. Argoul, A. Arneodo, and P. Richgetti, “Experimental evidence for homoclinic chaos in Belolusov–Zabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987).
[CrossRef]

Rossler, F.

R. Herrero, F. Boixader, J. Farjas, F. Pi, G. Orriols, and F. Rossler, “Rossler chaos in opto-thermal bistable devices,” Opt. Commun. 113, 324–334 (1993).
[CrossRef]

Roversi, J. A.

Sanchez, F.

Scheps, R.

R. Scheps, B. M. Gately, J. F. Mayers, J. S. Krasinski, and D. F. Heller, “Alexandrite laser pumped by semiconductor laser,” Appl. Phys. Lett. 56, 2288–2290 (1990).
[CrossRef]

Shand, M. L.

S. T. Lai and M. L. Shand, “High efficiency cw laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642–5644 (1983).
[CrossRef]

Shil’nikov, L. P.

L. P. Shil’nikov, “A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle focus type,” Math. USSR Sbornik 10, 91–102 (1970).
[CrossRef]

Sparrow, C.

P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
[CrossRef]

Spiegel, E. A.

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Homoclinic chaos in chemical systems,” Physica D 62, 134–169 (1993).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[CrossRef]

Swinney, H. L. F.

A. Argoul, J. Huth, P. Merzeau, A. Arneodo, and H. L. F. Swinney, “Experimental evidence for homoclinic chaos in electrochemical growth process,” Physica D 62, 170–185 (1993).
[CrossRef]

Tang, D. Y.

Tresser, C.

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Homoclinic chaos in chemical systems,” Physica D 62, 134–169 (1993).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[CrossRef]

Walling, J.

J. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302–1314 (1980).
[CrossRef]

Walling, J. C.

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable cw alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120–121 (1980).
[CrossRef]

Zambon, B.

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with a saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[CrossRef] [PubMed]

Appl. Phys. Lett.

R. Scheps, B. M. Gately, J. F. Mayers, J. S. Krasinski, and D. F. Heller, “Alexandrite laser pumped by semiconductor laser,” Appl. Phys. Lett. 56, 2288–2290 (1990).
[CrossRef]

IEEE J. Quantum Electron.

J. C. Walling, O. G. Peterson, and R. C. Morris, “Tunable cw alexandrite laser,” IEEE J. Quantum Electron. QE-16, 120–121 (1980).
[CrossRef]

J. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302–1314 (1980).
[CrossRef]

J. Appl. Phys.

S. T. Lai and M. L. Shand, “High efficiency cw laser-pumped tunable alexandrite laser,” J. Appl. Phys. 54, 5642–5644 (1983).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem.

M. R. Basset and J. L. Hudson, “Shil’nikov chaos during copper electrodissolution,” J. Phys. Chem. 92, 6963–6966 (1988).
[CrossRef]

J. Stat. Phys.

P. Glendinning and C. Sparrow, “Local and global behavior near homoclinic orbits,” J. Stat. Phys. 35, 645–696 (1984).
[CrossRef]

Math. USSR Sbornik

L. P. Shil’nikov, “A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle focus type,” Math. USSR Sbornik 10, 91–102 (1970).
[CrossRef]

Opt. Commun.

R. Herrero, F. Boixader, J. Farjas, F. Pi, G. Orriols, and F. Rossler, “Rossler chaos in opto-thermal bistable devices,” Opt. Commun. 113, 324–334 (1993).
[CrossRef]

Phys. Lett. A

F. Argoul, A. Arneodo, and P. Richgetti, “Experimental evidence for homoclinic chaos in Belolusov–Zabotinskii reaction,” Phys. Lett. A 120, 269–275 (1987).
[CrossRef]

Phys. Rev. A

D. Hennequin, F. de Tomasi, B. Zambon, and E. Arimondo, “Homoclinic orbits and cycles in the instabilities of a laser with a saturable absorber,” Phys. Rev. A 37, 2243–2246 (1988).
[CrossRef] [PubMed]

Phys. Rev. E

R. Herrero, F. R. Pons, J. Farjas, F. Pi, and G. Orriols, “Homoclinic dynamics in experimental Shil’nikov attractors,” Phys. Rev. E 53, 5627–5636 (1996).
[CrossRef]

Phys. Rev. Lett.

T. Braun, J. A. Lisboa, and J. A. C. Gallas, “Evidence of homoclinic chaos in the plasma of a glow discharge,” Phys. Rev. Lett. 68, 2770–2773 (1992).
[CrossRef] [PubMed]

F. T. Arecchi, R. Meucci, and W. Gadomski, “Laser dynamics with competing instabilities,” Phys. Rev. Lett. 58, 2205–2208 (1987).
[CrossRef] [PubMed]

Physica D

F. T. Arecchi, A. Lapucci, and R. Meucci, “Poincaré versus Boltzman in Shil’nikov phenomena,” Physica D 62, 186–191 (1993).
[CrossRef]

A. Argoul, J. Huth, P. Merzeau, A. Arneodo, and H. L. F. Swinney, “Experimental evidence for homoclinic chaos in electrochemical growth process,” Physica D 62, 170–185 (1993).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Homoclinic chaos in chemical systems,” Physica D 62, 134–169 (1993).
[CrossRef]

A. Arneodo, P. H. Coullet, E. A. Spiegel, and C. Tresser, “Asymptotic chaos,” Physica D 14, 327–347 (1985).
[CrossRef]

Other

R. Englman, Non-Radiative Decay of Ions and Molecules in Solids (North-Holland, Amsterdam, 1979).

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Figures (11)

Fig. 1
Fig. 1

Scheme of the experimental setup: BS’s, beam splitters; PD, photodiode; EOM, electro-optic modulator; L1, lens; M1, flat mirror; M2, concave mirror; AlX, alexandrite crystal; Pd T, photodiode; G, grating; TM, toroidal mirror; CCD, charge-coupled-device diode array.

Fig. 2
Fig. 2

Output of the Ar-ion pumped alexandrite laser (the route to chaos through intermittency): (a) regular stable pulsations at a pump power of 300 mW; (b) partly chaotic pulsations at a pump power of 350 mW; (c) chaotic pulsations at a pump power of 420 mW.

Fig. 3
Fig. 3

Output of the Ar-ion pumped alexandrite laser (the route to chaos through bifurcation): (a) the f/2 bifurcation at a pump power of 380 mW; (b) the periodic f/3 orbit at a pump power of 430 mW; (c) the f/6 orbit at 500 mW; (d) chaotic pulsations at a pump power of 440 mW.

Fig. 4
Fig. 4

Output of the Ar-ion and Kr-ion pumped alexandrite laser with f/12 bifurcation.

Fig. 5
Fig. 5

Scheme of the laser energy levels.

Fig. 6
Fig. 6

Numerical solutions of Eqs. (1) for the photon number: (a) regular pulsations; (b) f/2 bifurcation with p=4pT; (c) f/4 bifurcation with p=4.4pT; (d) periodic, f/6 window with p=6pT; and (e) chaotic pulsations with p=5.2pT. Cavity losses are 4.5×107s-1.

Fig. 7
Fig. 7

Stationary solutions for phonon number and photon number versus pump intensity for three values of parameter f. Pump power equal to 1 corresponds to pT=2×10-6.

Fig. 8
Fig. 8

Pump-power dependence of eigenvalues that correspond to the stationary solutions: (a) real eigenvalue λ1, (b) real eigenvalue λ2, (c) real part α of the complex eigenvalues λ3 and λ4, and (d) imaginary part of the complex eigenvalues λ3 and λ4. Pump power equal to 1 corresponds to pT=2×10-6.

Fig. 9
Fig. 9

Attractor corresponding to the case of chaotic pulsations of the laser output shown in Fig. 6(d).

Fig. 10
Fig. 10

Numerical solutions of Eqs. (1) for the photon number and phonon number, for cavity losses 6×107s-1.

Fig. 11
Fig. 11

Intensity of the laser output versus laser-cavity losses. The straight line of stationary solutions splits the plot into the upper and the lower branches that correspond to the maxima and the minima of laser pulses, respectively (see Fig. 6).

Equations (26)

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ddtn3=-Γ3n3+K3WW+K30n0,
ddtW=-ΓWW-2CIW1+CI/ΓP+KW3n3+KW0n0,
ddtI=-2κI+2CI1+CI/ΓP+BN/ΓPW,
ddtN=-ΓP(N-N(0))-KN3n3+KNWW+KN0n0,
ΓW=(2γ4+BN+AI0)/2,Γ3=γ3+AI0/2+3BN/2,KW3=3BN/2-AI0/2+γ4-γ3,K3W=(BN-AI0)/2,KW0=(AI0-BN-2γ˜4)/2,K30=(AI0+BN)/2,KN3=(3BN+fAI0)/2,KNW=(BN-fAI0)/2.KN0=(fAI0+BN)/2,
f=ΔE53-ωMΔE43,
I¯=0;
W¯=bN¯(p-γ¯3-γ¯4)-γ¯3γ¯4bN¯(2p+γ¯3+γ¯4)+γ¯4(p+γ¯3)n0,
n¯3=p(γ¯4+bN¯)bN¯(2p+γ¯3+γ¯4)+γ¯4(p+γ¯3)n0,
N¯=N(0)+n0p2p+γ¯3+γ¯4fγ¯3+(f-1)γ¯4+γ¯4[(fγ¯3-(f-1)γ¯4)p+γ¯3γ¯4]bN(0)(2p+γ¯3+γ¯4)+bn0p[fγ¯3+(f-1)γ¯4],
γ¯3=γ3Γ,γ¯4=γ4Γ,p=AI0Γ,b=BΓ,c˜=CΓ,k=κΓ.
I¯=(2γ¯3+p+3bN¯)(N¯-N(0))-n0[γ¯3(fp+bN¯)+(f-1)bNp¯]k[γ3(bN¯-fp)-2(f-1)bNp¯]-1c˜(1-bN¯),
W¯=(2γ¯3+p+3bN¯)(N¯-N(0))-n0[γ¯3(fp+bN¯)+(f-1)bNp¯]γ¯3(bN¯-fp)-2(f-1)bNp¯,
n¯3=(bN¯-p)(N¯-N(0))-(f-1)n0bNp¯γ¯3(bN¯-fp)-2(f-1)bNp¯,
W¯=κC(1-bN¯).
N¯T=N(0)+13γ¯3n0+κC+pT(f-1)n0-2κC-1+γ¯3pTfn0-κCγ¯3(n0+κ/C)+pT(f-1)n0-2κC-1-3N(0).
n0bN¯T-κC(2bN¯T+γ¯4)pT
=n0+κC[γ¯4bN¯T+γ¯3(bN¯T+γ4)].
pT=γ¯3n0+κCn0-2κC,
δn3=n3-n¯3,δN=N-N¯,
δI=I-I¯,δW=W-W¯.
Δt=MˆΔ,
Δ=δn3δNδIΔW.
Mˆ
=Γ2-(γ¯3+3bN¯+p)b(n0+W¯-3n¯3)0bN¯-p-(3bN¯+fp)b(n0+W¯-3n¯3)-10bN¯-fp0-4bc˜IW¯(1+c˜I¯+bN¯)24-k+c˜W¯1+c˜I¯+bN¯-c˜IW¯(1+c˜I¯+bN¯)24c˜I¯1+cI¯¯+bN¯γ¯4-γ¯3+3bN¯-p-b(n0+W¯-3n¯3)-4c˜W¯(1+c˜I¯)2-γ¯4+bN¯+p+2c˜I¯(1+c˜I¯)2.
λ4=2kc˜W¯k(1+bN¯)-1,

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