Abstract

A previously unknown mechanism of bistable behavior in lasers with single-mode cavities is proposed and analyzed. It is shown that if losses in a cavity exhibit nonmonotonic dependence on frequency, the equation for stationary lasing frequencies can have multiple solutions even in single-mode cavities. In such a case, a system can generate one of several lasing outputs characterized by different frequencies and intensities. All these potential lasing states are stable at the same pumping level, and the choice between them is determined by initial conditions. The latter can be, in principle, controlled by seeding pulses. This mechanism does not depend on such nonlinear effects responsible for most known types of bistability as saturable absorption or cross saturation. An example of a cavity structure, in which such a mechanism can be realized, is presented. Standard lasing equations fail to describe dynamical behavior of such systems; therefore a generalized approach treating dynamic of lasing frequency and intensity on equal footing is developed.

© 2012 Optical Society of America

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  1. C. M. Bowden, M. Ciftan, and H. R. Robl, Optical Bistability (Plenum, 1981).
  2. P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge University, 1997).
  3. K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Kluwer, 1999).
  4. N. N. Rosanov, Spatial Hysteresis and Optical Patterns(Springer, 2002).
  5. H. Kawaguchi, Bistabilities and Nonlinearities in Laser Diodes (Artech House, 1994).
  6. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).
  7. A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with 2-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
    [CrossRef]
  8. A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
    [CrossRef]
  9. A. J. Vanwonderen and L. G. Suttorp, “Dispersive optical bistability in a nonideal Fabry–Perot cavity. 1. Stability analysis of the Maxwell–Bloch equations,” Z. Phys. B 83, 135–142 (1991).
  10. T. Vivero, J. M. Rivas-Moscoso, A. P. Gonzalez-Marcos, and J. A. Martin-Pereda, “Dispersive optical bistability in quantum wells with logarithmic gain,” IEEE J. Quantum Electron. 46, 1184–1190 (2010).
    [CrossRef]
  11. H. A. Batarfi, “Dispersive switching in bistable models,” J. Nonlinear Opt. Phys. 17, 265–273 (2008).
    [CrossRef]
  12. S. T. Dembinski, A. Kossakowski, L. A. Lugiato, and P. Mandel, “Semi-classical and quantum-theory of bistability in lasers containing saturable absorbers 2,” Phys. Rev. A 18, 1145–1151 (1978).
    [CrossRef]
  13. L. A. Lugiato, P. Mandel, S. T. Dembinski, and A. Kossakowski, “Semi-classical and quantum theories of bistability in lasers containing saturable absorbers,” Phys. Rev. A 18, 238–254 (1978).
    [CrossRef]
  14. H. Kawaguchi, “Optical bistability and chaos in a semiconductor-laser with a saturable absorber,” Appl. Phys. Lett. 45, 1264–1266 (1984).
    [CrossRef]
  15. E. Arimondo, D. Dangoisse, C. Gabbanini, E. Menchi, and F. Papoff, “Dynamic behavior of bistability in a laser with a saturable absorber,” J. Opt. Soc. Am. B 4, 892–899 (1987).
    [CrossRef]
  16. J. M. Oh and D. H. Lee, “Strong optical bistability in a simple L-band tunable erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 40, 374–377 (2004).
    [CrossRef]
  17. L. Guidoni, R. Mannella, V. Isaia, P. Verkerk, and E. Arimondo, “Stochastic resonance in a laser with saturable absorber,” Nuovo Cimento D 17, 803–810 (1995).
    [CrossRef]
  18. S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a laser containing a saturable absorber,” Ann. Phys. 32, 63–65 (2007).
    [CrossRef]
  19. C. Masoller, M. Oria, and R. Vilaseca, “Modeling a semiconductor laser with an intracavity atomic absorber,” Phys. Rev. A 80, 013830 (2009).
    [CrossRef]
  20. S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” Appl. Phys. Lett. 87, 181102 (2005).
    [CrossRef]
  21. A. V. Naumenko, N. A. Loiko, and T. Ackemann, “Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation,” Phys. Rev. A 76, 023802 (2007).
    [CrossRef]
  22. B. Farias, T. P. de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett. 94, 173902 (2005).
    [CrossRef]
  23. H. E. Tureci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
    [CrossRef]
  24. L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
    [CrossRef]
  25. L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
    [CrossRef]
  26. M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
    [CrossRef]
  27. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics(Addison-Wesley, 1974).
  28. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
    [CrossRef]
  29. O. Zaitsev and L. Deych, “Diagrammatic semiclassical laser theory,” Phys. Rev. A 81, 023822 (2010).
    [CrossRef]
  30. A. E. Siegman, Lasers (University Science, 1986).
  31. H. E. Tureci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
    [CrossRef]
  32. H. E. Tureci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
    [CrossRef]
  33. H. E. Tureci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
    [CrossRef]
  34. K. Staliunas and V. J. Sánchez Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer, 2003).

2012 (1)

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[CrossRef]

2011 (1)

L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[CrossRef]

2010 (3)

O. Zaitsev and L. Deych, “Diagrammatic semiclassical laser theory,” Phys. Rev. A 81, 023822 (2010).
[CrossRef]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

T. Vivero, J. M. Rivas-Moscoso, A. P. Gonzalez-Marcos, and J. A. Martin-Pereda, “Dispersive optical bistability in quantum wells with logarithmic gain,” IEEE J. Quantum Electron. 46, 1184–1190 (2010).
[CrossRef]

2009 (2)

C. Masoller, M. Oria, and R. Vilaseca, “Modeling a semiconductor laser with an intracavity atomic absorber,” Phys. Rev. A 80, 013830 (2009).
[CrossRef]

H. E. Tureci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

2008 (2)

H. E. Tureci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef]

H. A. Batarfi, “Dispersive switching in bistable models,” J. Nonlinear Opt. Phys. 17, 265–273 (2008).
[CrossRef]

2007 (3)

A. V. Naumenko, N. A. Loiko, and T. Ackemann, “Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation,” Phys. Rev. A 76, 023802 (2007).
[CrossRef]

S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a laser containing a saturable absorber,” Ann. Phys. 32, 63–65 (2007).
[CrossRef]

H. E. Tureci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[CrossRef]

2006 (1)

H. E. Tureci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

2005 (2)

B. Farias, T. P. de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett. 94, 173902 (2005).
[CrossRef]

S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” Appl. Phys. Lett. 87, 181102 (2005).
[CrossRef]

2004 (1)

J. M. Oh and D. H. Lee, “Strong optical bistability in a simple L-band tunable erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 40, 374–377 (2004).
[CrossRef]

2003 (1)

A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
[CrossRef]

2000 (1)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[CrossRef]

1995 (1)

L. Guidoni, R. Mannella, V. Isaia, P. Verkerk, and E. Arimondo, “Stochastic resonance in a laser with saturable absorber,” Nuovo Cimento D 17, 803–810 (1995).
[CrossRef]

1991 (1)

A. J. Vanwonderen and L. G. Suttorp, “Dispersive optical bistability in a nonideal Fabry–Perot cavity. 1. Stability analysis of the Maxwell–Bloch equations,” Z. Phys. B 83, 135–142 (1991).

1987 (1)

1984 (1)

H. Kawaguchi, “Optical bistability and chaos in a semiconductor-laser with a saturable absorber,” Appl. Phys. Lett. 45, 1264–1266 (1984).
[CrossRef]

1983 (1)

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with 2-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[CrossRef]

1978 (2)

S. T. Dembinski, A. Kossakowski, L. A. Lugiato, and P. Mandel, “Semi-classical and quantum-theory of bistability in lasers containing saturable absorbers 2,” Phys. Rev. A 18, 1145–1151 (1978).
[CrossRef]

L. A. Lugiato, P. Mandel, S. T. Dembinski, and A. Kossakowski, “Semi-classical and quantum theories of bistability in lasers containing saturable absorbers,” Phys. Rev. A 18, 238–254 (1978).
[CrossRef]

Ackemann, T.

A. V. Naumenko, N. A. Loiko, and T. Ackemann, “Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation,” Phys. Rev. A 76, 023802 (2007).
[CrossRef]

Arimondo, E.

L. Guidoni, R. Mannella, V. Isaia, P. Verkerk, and E. Arimondo, “Stochastic resonance in a laser with saturable absorber,” Nuovo Cimento D 17, 803–810 (1995).
[CrossRef]

E. Arimondo, D. Dangoisse, C. Gabbanini, E. Menchi, and F. Papoff, “Dynamic behavior of bistability in a laser with a saturable absorber,” J. Opt. Soc. Am. B 4, 892–899 (1987).
[CrossRef]

Baba, T.

S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” Appl. Phys. Lett. 87, 181102 (2005).
[CrossRef]

Batarfi, H. A.

H. A. Batarfi, “Dispersive switching in bistable models,” J. Nonlinear Opt. Phys. 17, 265–273 (2008).
[CrossRef]

Boudoukha, H.

S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a laser containing a saturable absorber,” Ann. Phys. 32, 63–65 (2007).
[CrossRef]

Bowden, C. M.

C. M. Bowden, M. Ciftan, and H. R. Robl, Optical Bistability (Plenum, 1981).

Cerjan, A.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[CrossRef]

Chevrollier, M.

B. Farias, T. P. de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett. 94, 173902 (2005).
[CrossRef]

Chong, Y. D.

L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[CrossRef]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

Ciftan, M.

C. M. Bowden, M. Ciftan, and H. R. Robl, Optical Bistability (Plenum, 1981).

Collier, B.

H. E. Tureci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

Dangoisse, D.

de Silans, T. P.

B. Farias, T. P. de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett. 94, 173902 (2005).
[CrossRef]

Dembinski, S. T.

S. T. Dembinski, A. Kossakowski, L. A. Lugiato, and P. Mandel, “Semi-classical and quantum-theory of bistability in lasers containing saturable absorbers 2,” Phys. Rev. A 18, 1145–1151 (1978).
[CrossRef]

L. A. Lugiato, P. Mandel, S. T. Dembinski, and A. Kossakowski, “Semi-classical and quantum theories of bistability in lasers containing saturable absorbers,” Phys. Rev. A 18, 238–254 (1978).
[CrossRef]

Deych, L.

O. Zaitsev and L. Deych, “Diagrammatic semiclassical laser theory,” Phys. Rev. A 81, 023822 (2010).
[CrossRef]

Djabi, M.

S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a laser containing a saturable absorber,” Ann. Phys. 32, 63–65 (2007).
[CrossRef]

Djabi, S.

S. Djabi, H. Boudoukha, and M. Djabi, “Optical bistability in a laser containing a saturable absorber,” Ann. Phys. 32, 63–65 (2007).
[CrossRef]

Farias, B.

B. Farias, T. P. de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett. 94, 173902 (2005).
[CrossRef]

Gabbanini, C.

Ge, L.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[CrossRef]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[CrossRef]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

H. E. Tureci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

H. E. Tureci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef]

H. E. Tureci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, 1985).

Gonzalez-Marcos, A. P.

T. Vivero, J. M. Rivas-Moscoso, A. P. Gonzalez-Marcos, and J. A. Martin-Pereda, “Dispersive optical bistability in quantum wells with logarithmic gain,” IEEE J. Quantum Electron. 46, 1184–1190 (2010).
[CrossRef]

Guidoni, L.

L. Guidoni, R. Mannella, V. Isaia, P. Verkerk, and E. Arimondo, “Stochastic resonance in a laser with saturable absorber,” Nuovo Cimento D 17, 803–810 (1995).
[CrossRef]

Isaia, V.

L. Guidoni, R. Mannella, V. Isaia, P. Verkerk, and E. Arimondo, “Stochastic resonance in a laser with saturable absorber,” Nuovo Cimento D 17, 803–810 (1995).
[CrossRef]

Ishii, S.

S. Ishii and T. Baba, “Bistable lasing in twin microdisk photonic molecules,” Appl. Phys. Lett. 87, 181102 (2005).
[CrossRef]

Joshi, A.

A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
[CrossRef]

Kawaguchi, H.

H. Kawaguchi, “Optical bistability and chaos in a semiconductor-laser with a saturable absorber,” Appl. Phys. Lett. 45, 1264–1266 (1984).
[CrossRef]

H. Kawaguchi, Bistabilities and Nonlinearities in Laser Diodes (Artech House, 1994).

Kimble, H. J.

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with 2-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[CrossRef]

Kossakowski, A.

L. A. Lugiato, P. Mandel, S. T. Dembinski, and A. Kossakowski, “Semi-classical and quantum theories of bistability in lasers containing saturable absorbers,” Phys. Rev. A 18, 238–254 (1978).
[CrossRef]

S. T. Dembinski, A. Kossakowski, L. A. Lugiato, and P. Mandel, “Semi-classical and quantum-theory of bistability in lasers containing saturable absorbers 2,” Phys. Rev. A 18, 1145–1151 (1978).
[CrossRef]

Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics(Addison-Wesley, 1974).

Lee, D. H.

J. M. Oh and D. H. Lee, “Strong optical bistability in a simple L-band tunable erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 40, 374–377 (2004).
[CrossRef]

Liertzer, M.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[CrossRef]

Loiko, N. A.

A. V. Naumenko, N. A. Loiko, and T. Ackemann, “Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation,” Phys. Rev. A 76, 023802 (2007).
[CrossRef]

Lugiato, L. A.

S. T. Dembinski, A. Kossakowski, L. A. Lugiato, and P. Mandel, “Semi-classical and quantum-theory of bistability in lasers containing saturable absorbers 2,” Phys. Rev. A 18, 1145–1151 (1978).
[CrossRef]

L. A. Lugiato, P. Mandel, S. T. Dembinski, and A. Kossakowski, “Semi-classical and quantum theories of bistability in lasers containing saturable absorbers,” Phys. Rev. A 18, 238–254 (1978).
[CrossRef]

Mandel, P.

L. A. Lugiato, P. Mandel, S. T. Dembinski, and A. Kossakowski, “Semi-classical and quantum theories of bistability in lasers containing saturable absorbers,” Phys. Rev. A 18, 238–254 (1978).
[CrossRef]

S. T. Dembinski, A. Kossakowski, L. A. Lugiato, and P. Mandel, “Semi-classical and quantum-theory of bistability in lasers containing saturable absorbers 2,” Phys. Rev. A 18, 1145–1151 (1978).
[CrossRef]

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge University, 1997).

Mannella, R.

L. Guidoni, R. Mannella, V. Isaia, P. Verkerk, and E. Arimondo, “Stochastic resonance in a laser with saturable absorber,” Nuovo Cimento D 17, 803–810 (1995).
[CrossRef]

Martin-Pereda, J. A.

T. Vivero, J. M. Rivas-Moscoso, A. P. Gonzalez-Marcos, and J. A. Martin-Pereda, “Dispersive optical bistability in quantum wells with logarithmic gain,” IEEE J. Quantum Electron. 46, 1184–1190 (2010).
[CrossRef]

Masoller, C.

C. Masoller, M. Oria, and R. Vilaseca, “Modeling a semiconductor laser with an intracavity atomic absorber,” Phys. Rev. A 80, 013830 (2009).
[CrossRef]

Menchi, E.

Naumenko, A. V.

A. V. Naumenko, N. A. Loiko, and T. Ackemann, “Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation,” Phys. Rev. A 76, 023802 (2007).
[CrossRef]

Oh, J. M.

J. M. Oh and D. H. Lee, “Strong optical bistability in a simple L-band tunable erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 40, 374–377 (2004).
[CrossRef]

Oria, M.

C. Masoller, M. Oria, and R. Vilaseca, “Modeling a semiconductor laser with an intracavity atomic absorber,” Phys. Rev. A 80, 013830 (2009).
[CrossRef]

B. Farias, T. P. de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett. 94, 173902 (2005).
[CrossRef]

Orozco, L. A.

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with 2-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[CrossRef]

Otsuka, K.

K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Kluwer, 1999).

Papoff, F.

Rivas-Moscoso, J. M.

T. Vivero, J. M. Rivas-Moscoso, A. P. Gonzalez-Marcos, and J. A. Martin-Pereda, “Dispersive optical bistability in quantum wells with logarithmic gain,” IEEE J. Quantum Electron. 46, 1184–1190 (2010).
[CrossRef]

Robl, H. R.

C. M. Bowden, M. Ciftan, and H. R. Robl, Optical Bistability (Plenum, 1981).

Rosanov, N. N.

N. N. Rosanov, Spatial Hysteresis and Optical Patterns(Springer, 2002).

Rosenberger, A. T.

A. T. Rosenberger, L. A. Orozco, and H. J. Kimble, “Observation of absorptive bistability with 2-level atoms in a ring cavity,” Phys. Rev. A 28, 2569–2572 (1983).
[CrossRef]

Rotter, S.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[CrossRef]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[CrossRef]

H. E. Tureci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

H. E. Tureci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef]

Sánchez Morcillo, V. J.

K. Staliunas and V. J. Sánchez Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer, 2003).

Sargent, M.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics(Addison-Wesley, 1974).

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics(Addison-Wesley, 1974).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Staliunas, K.

K. Staliunas and V. J. Sánchez Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer, 2003).

Stone, A. D.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[CrossRef]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[CrossRef]

L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

H. E. Tureci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

H. E. Tureci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef]

H. E. Tureci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[CrossRef]

H. E. Tureci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

Suttorp, L. G.

A. J. Vanwonderen and L. G. Suttorp, “Dispersive optical bistability in a nonideal Fabry–Perot cavity. 1. Stability analysis of the Maxwell–Bloch equations,” Z. Phys. B 83, 135–142 (1991).

Tandy, R. J.

H. E. Tureci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

Tureci, H. E.

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[CrossRef]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[CrossRef]

H. E. Tureci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

H. E. Tureci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef]

H. E. Tureci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[CrossRef]

H. E. Tureci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

Vanwonderen, A. J.

A. J. Vanwonderen and L. G. Suttorp, “Dispersive optical bistability in a nonideal Fabry–Perot cavity. 1. Stability analysis of the Maxwell–Bloch equations,” Z. Phys. B 83, 135–142 (1991).

Verkerk, P.

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T. Vivero, J. M. Rivas-Moscoso, A. P. Gonzalez-Marcos, and J. A. Martin-Pereda, “Dispersive optical bistability in quantum wells with logarithmic gain,” IEEE J. Quantum Electron. 46, 1184–1190 (2010).
[CrossRef]

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A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
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[CrossRef]

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A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[CrossRef]

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J. M. Oh and D. H. Lee, “Strong optical bistability in a simple L-band tunable erbium-doped fiber ring laser,” IEEE J. Quantum Electron. 40, 374–377 (2004).
[CrossRef]

T. Vivero, J. M. Rivas-Moscoso, A. P. Gonzalez-Marcos, and J. A. Martin-Pereda, “Dispersive optical bistability in quantum wells with logarithmic gain,” IEEE J. Quantum Electron. 46, 1184–1190 (2010).
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Nonlinearity (1)

H. E. Tureci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009).
[CrossRef]

Nuovo Cimento D (1)

L. Guidoni, R. Mannella, V. Isaia, P. Verkerk, and E. Arimondo, “Stochastic resonance in a laser with saturable absorber,” Nuovo Cimento D 17, 803–810 (1995).
[CrossRef]

Phys. Rev. A (10)

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[CrossRef]

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L. Ge, Y. D. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” Phys. Rev. A 82, 063824 (2010).
[CrossRef]

L. Ge, Y. D. Chong, S. Rotter, H. E. Tureci, and A. D. Stone, “Unconventional modes in lasers with spatially varying gain and loss,” Phys. Rev. A 84, 023820 (2011).
[CrossRef]

O. Zaitsev and L. Deych, “Diagrammatic semiclassical laser theory,” Phys. Rev. A 81, 023822 (2010).
[CrossRef]

A. V. Naumenko, N. A. Loiko, and T. Ackemann, “Analysis of bistability conditions between lasing and nonlasing states for a vertical-cavity surface-emitting laser with frequency-selective optical feedback using an envelope approximation,” Phys. Rev. A 76, 023802 (2007).
[CrossRef]

H. E. Tureci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

H. E. Tureci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

B. Farias, T. P. de Silans, M. Chevrollier, and M. Oria, “Frequency bistability of a semiconductor laser under a frequency-dependent feedback,” Phys. Rev. Lett. 94, 173902 (2005).
[CrossRef]

M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Tureci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
[CrossRef]

A. Joshi and M. Xiao, “Optical multistability in three-level atoms inside an optical ring cavity,” Phys. Rev. Lett. 91, 143904 (2003).
[CrossRef]

Science (1)

H. E. Tureci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef]

Z. Phys. B (1)

A. J. Vanwonderen and L. G. Suttorp, “Dispersive optical bistability in a nonideal Fabry–Perot cavity. 1. Stability analysis of the Maxwell–Bloch equations,” Z. Phys. B 83, 135–142 (1991).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of a structure considered in the paper: two waveguides filled with a gain medium (shaded area) are coupled to each other via a passive ring resonator.

Fig. 2.
Fig. 2.

Left- and right-hand sides of Eq. (34) as functions of frequency normalized by the frequency of atomic transition, illustrating the existence of three stationary frequencies.

Fig. 3.
Fig. 3.

Dependence of the stationary frequencies and respective lasing intensities on the strength of pumping.

Fig. 4.
Fig. 4.

Temporal evolution of the emission intensity and frequency toward their stationary values for various pumping rates and initial conditions.

Equations (41)

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E1={A(+)eik(1)z+A()eik(1)z,z<LT(1)eik(1)z,z>L,E2={B(+)eik(2)z+B()eik(2)z,z<LT(2)eik(2)z,z>L,
A(+)=rA(),B(+)=rB(),
(a1,out+d1,out+)=(piκiκp)(a1,in+d1,in+),
(a2,outd2,out+)=(p*iκ*iκ*p*)(0d2,in+).
d2,in+=αeiΦ/2d1,out+,d1,in+=αeiΦ/2d2,out+,
d1,in=αeiΦ/2d2,out,d2,in=αeiΦ/2d1,out,
a2,out=α|κ|2eiΦ/21α2|p|2eiΦa1,in+.
B()eik(2)L=α|κ|2eiΦ/21α2|p|2eiΦA(+)eik(1)L.
A()eik(2)L=α|κ|2eiΦ/21α2|p|2eiΦB(+)eik(1)L.
e2i(k(1)+k(2))L=α2r2|κ|4|1α|p|2eiΦ|2.
(qm(1)+qm(2))L=πm.
(δm(1)+δm(2))L=12lnS(ω),
S=α2r2|κ|41+α4|p|42α2|p|2cos(2πωωr),
Bm()Am()=(1)meiψ,
ψ=Φ/2tan1|p|2α2sinΦ1|p|2α2cosΦ.
Um(1,2)(z)=reikm(1,2)z+eikm(1,2)z,U¯m(1,2)(z)=rei[km(1,2)]*z+ei[km(1,2)]*z.
0LUm(1)(z)[U¯n(1)(z)]*dz+L2LUm(2)(z)[U¯n(2)(z)]*dz=Nmδmn,
nw2c22E(1,2)t22E(1,2)z2=4πc22P(1,2)t2,
E(1,2)(z,t)=12m[|Am(1,2)(t)|eiΩmtiφm(1,2)Um(1,2)(z)+c.c.].
φm(t)=φ0+0tΩm(τ)dτ,
[2id|Am(1,2)|dt+|Am(1,2)|(ωm(1,2)dφm(1,2)dttdΩmdtΩm)]Um(1,2)(z)=4πΩmPm(1,2)(z,t),
ωm(1)dφm(1)dt=ωm(2)dφm(2)dt.
ωm(1,2)=νmiξm±12dψdt,
νm=mωB;ξm=ωB2πlnS(Ωm),
ωB=cπ2nwL,
2id|A|dt+|A|(ν+iξ12dψdttdΩdtΩ)=4πΩN02LP+(z,t)[U¯(z)]*dz,
dΩdt(t+π1+W(Ω)2ωr)+Ω=ν+PΩg(Ω)[ω0ΩγF1(I,Ω)F2(I,Ω)],
W(Ω)=2|p|2α2(cosΦ|p|2α2)12|p|2α2cosΦ+|p|4α4,
dIdt=ωB2πIlnS(Ω)+PΩg(Ω)I[F1(I,Ω)+ω0ΩγF2(I,Ω)],
I=|A|2|ς|222γγ
g(Ω)=γ2γ2+(ω0Ω)2,
F1(I,Ω)=Re(1N02LU(z)U¯*(z)1+Rdz);F2(I,Ω)=Im(1N02LU(z)U¯*(z)1+Rdz),
R=I|U(z)|2g(Ω),
Ων=PΩg(Ω)[ω0ΩγF1(I,Ω)F2(I,Ω)],lnS(Ω)=2πPΩg(Ω)ωB[F1(I,Ω)+ω0ΩγF2(I,Ω)].
lnS(Ω)=2πγωBΩνΩω0,
Pth=ωB2πΩg(Ω)lnS(Ω).
S(min)=(κ2αr1+α2p2)2,
Ωn(min)=(n12)ωr,n=1,2,3,
S(max)=(κ2αr1α2p2)2,
Ωn(max)=nωr,n=1,2,3
|lnSmin|<2πγωB|Ωn(min)νΩn(min)ω0|;|lnSmax|>2πγωB|Ωn(max)νΩn(max)ω0|.

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