Abstract

We consider the properties of the non-degenerate two frequency regime of oscillation of the semi-linear photorefractive oscillator and analyze its relation with the mirrorless oscillation. We consider the oscillator with or without a frequency shifted feedback by a vibrating mirror. This study shows that these two apparently different phenomena are closely related. We conclude from the obtained results that the two frequency oscillation can be considered as a perturbation of the mirrorless oscillation.

© 2010 Optical Society of America

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  1. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
    [CrossRef] [PubMed]
  2. S. R. Liu and G. Indebetouw, “Periodic and chaotic spatiotemporal states in a phase-conjugate resonator using a photorefractive BaTiO3 phase-conjugate mirror,” J. Opt. Soc. Am. B 9, 1507–1520 (1992).
    [CrossRef]
  3. K. Staliunas, M. F. H. Tarroja, G. Slekys, C. O. Weiss, and L. Dambly, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
    [CrossRef] [PubMed]
  4. C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998).
    [CrossRef]
  5. J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
    [CrossRef]
  6. B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. QE-25, 550–569 (1989).
    [CrossRef]
  7. B. Fischer, “Theory of self-frequency detuning of oscillations by wave mixing in photorefractive crystals,” Opt. Lett. 11, 236–238 (1986).
    [CrossRef] [PubMed]
  8. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
    [CrossRef]
  9. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
    [CrossRef] [PubMed]
  10. J. Feinberg and R. Hellwarth, “Phase-conjugating mirror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980).
    [CrossRef] [PubMed]
  11. R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Strong lowering of the mirrorless optical oscillation threshold by angular mismatches for nonlocal photorefractive nonlinearity,” Opt. Lett. 33, 2773–2775 (2008).
    [CrossRef] [PubMed]
  12. R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Effects of angular pump mismatch for the semi-linear oscillator,” Appl. Phys. B 99, 163–172 (2010).
    [CrossRef]
  13. L. C. Foster, M. D. Ewy, and C. B. Crumly, “Laser mode locking by an external Doppler cell,” Appl. Phys. Lett. 6, 6–8 (1965).
    [CrossRef]
  14. L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
    [CrossRef]
  15. R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Semilinear coherent optical oscillator with frequency shifted feedback,” Opt. Express 15, 17136–17145 (2007).
    [CrossRef] [PubMed]
  16. N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection gratings in electrooptic crystals,” Appl. Phys. B 41, 259–263 (1986).
    [CrossRef]
  17. C. Denz, J. Golz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
    [CrossRef]
  18. R. Rebhi, P. Mathey, M. Grapinet, H.-R. Jauslin, and S. Odoulov, “Phase mismatch effects on the dynamics of a semilinear photorefractive oscillator with reflection-type gratings,” Appl. Phys. B 91, 583–589 (2008).
    [CrossRef]

2010 (1)

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Effects of angular pump mismatch for the semi-linear oscillator,” Appl. Phys. B 99, 163–172 (2010).
[CrossRef]

2008 (2)

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Strong lowering of the mirrorless optical oscillation threshold by angular mismatches for nonlocal photorefractive nonlinearity,” Opt. Lett. 33, 2773–2775 (2008).
[CrossRef] [PubMed]

R. Rebhi, P. Mathey, M. Grapinet, H.-R. Jauslin, and S. Odoulov, “Phase mismatch effects on the dynamics of a semilinear photorefractive oscillator with reflection-type gratings,” Appl. Phys. B 91, 583–589 (2008).
[CrossRef]

2007 (1)

2004 (1)

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[CrossRef]

1998 (1)

1995 (1)

K. Staliunas, M. F. H. Tarroja, G. Slekys, C. O. Weiss, and L. Dambly, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

1992 (1)

1990 (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

1989 (2)

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. QE-25, 550–569 (1989).
[CrossRef]

C. Denz, J. Golz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

1986 (2)

B. Fischer, “Theory of self-frequency detuning of oscillations by wave mixing in photorefractive crystals,” Opt. Lett. 11, 236–238 (1986).
[CrossRef] [PubMed]

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection gratings in electrooptic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

1982 (1)

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

1980 (1)

1977 (1)

1965 (1)

L. C. Foster, M. D. Ewy, and C. B. Crumly, “Laser mode locking by an external Doppler cell,” Appl. Phys. Lett. 6, 6–8 (1965).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Bergmann, K.

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[CrossRef]

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Crumly, C. B.

L. C. Foster, M. D. Ewy, and C. B. Crumly, “Laser mode locking by an external Doppler cell,” Appl. Phys. Lett. 6, 6–8 (1965).
[CrossRef]

Dambly, L.

K. Staliunas, M. F. H. Tarroja, G. Slekys, C. O. Weiss, and L. Dambly, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Denz, C.

C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998).
[CrossRef]

C. Denz, J. Golz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

Ewy, M. D.

L. C. Foster, M. D. Ewy, and C. B. Crumly, “Laser mode locking by an external Doppler cell,” Appl. Phys. Lett. 6, 6–8 (1965).
[CrossRef]

Feinberg, J.

Fischer, B.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. QE-25, 550–569 (1989).
[CrossRef]

B. Fischer, “Theory of self-frequency detuning of oscillations by wave mixing in photorefractive crystals,” Opt. Lett. 11, 236–238 (1986).
[CrossRef] [PubMed]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Foster, L. C.

L. C. Foster, M. D. Ewy, and C. B. Crumly, “Laser mode locking by an external Doppler cell,” Appl. Phys. Lett. 6, 6–8 (1965).
[CrossRef]

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Golz, J.

C. Denz, J. Golz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

Grapinet, M.

R. Rebhi, P. Mathey, M. Grapinet, H.-R. Jauslin, and S. Odoulov, “Phase mismatch effects on the dynamics of a semilinear photorefractive oscillator with reflection-type gratings,” Appl. Phys. B 91, 583–589 (2008).
[CrossRef]

Hellwarth, R.

Honda, T.

Indebetouw, G.

Jauslin, H. -R.

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Effects of angular pump mismatch for the semi-linear oscillator,” Appl. Phys. B 99, 163–172 (2010).
[CrossRef]

R. Rebhi, P. Mathey, M. Grapinet, H.-R. Jauslin, and S. Odoulov, “Phase mismatch effects on the dynamics of a semilinear photorefractive oscillator with reflection-type gratings,” Appl. Phys. B 91, 583–589 (2008).
[CrossRef]

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Strong lowering of the mirrorless optical oscillation threshold by angular mismatches for nonlocal photorefractive nonlinearity,” Opt. Lett. 33, 2773–2775 (2008).
[CrossRef] [PubMed]

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Semilinear coherent optical oscillator with frequency shifted feedback,” Opt. Express 15, 17136–17145 (2007).
[CrossRef] [PubMed]

Kukhtarev, N. V.

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection gratings in electrooptic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Liu, S. R.

Mathey, P.

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Effects of angular pump mismatch for the semi-linear oscillator,” Appl. Phys. B 99, 163–172 (2010).
[CrossRef]

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Strong lowering of the mirrorless optical oscillation threshold by angular mismatches for nonlocal photorefractive nonlinearity,” Opt. Lett. 33, 2773–2775 (2008).
[CrossRef] [PubMed]

R. Rebhi, P. Mathey, M. Grapinet, H.-R. Jauslin, and S. Odoulov, “Phase mismatch effects on the dynamics of a semilinear photorefractive oscillator with reflection-type gratings,” Appl. Phys. B 91, 583–589 (2008).
[CrossRef]

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Semilinear coherent optical oscillator with frequency shifted feedback,” Opt. Express 15, 17136–17145 (2007).
[CrossRef] [PubMed]

Odoulov, S.

R. Rebhi, P. Mathey, M. Grapinet, H.-R. Jauslin, and S. Odoulov, “Phase mismatch effects on the dynamics of a semilinear photorefractive oscillator with reflection-type gratings,” Appl. Phys. B 91, 583–589 (2008).
[CrossRef]

Pepper, D. M.

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Rebhi, R.

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Effects of angular pump mismatch for the semi-linear oscillator,” Appl. Phys. B 99, 163–172 (2010).
[CrossRef]

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Strong lowering of the mirrorless optical oscillation threshold by angular mismatches for nonlocal photorefractive nonlinearity,” Opt. Lett. 33, 2773–2775 (2008).
[CrossRef] [PubMed]

R. Rebhi, P. Mathey, M. Grapinet, H.-R. Jauslin, and S. Odoulov, “Phase mismatch effects on the dynamics of a semilinear photorefractive oscillator with reflection-type gratings,” Appl. Phys. B 91, 583–589 (2008).
[CrossRef]

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Semilinear coherent optical oscillator with frequency shifted feedback,” Opt. Express 15, 17136–17145 (2007).
[CrossRef] [PubMed]

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

Ringhofer, K. H.

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection gratings in electrooptic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Schwab, M.

Sedlatschek, M.

Semenets, T. I.

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection gratings in electrooptic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Shore, B. W.

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[CrossRef]

Slekys, G.

K. Staliunas, M. F. H. Tarroja, G. Slekys, C. O. Weiss, and L. Dambly, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Staliunas, K.

K. Staliunas, M. F. H. Tarroja, G. Slekys, C. O. Weiss, and L. Dambly, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Sternklar, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. QE-25, 550–569 (1989).
[CrossRef]

Sturman, B.

Tarroja, M. F. H.

K. Staliunas, M. F. H. Tarroja, G. Slekys, C. O. Weiss, and L. Dambly, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Tomberger, G.

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection gratings in electrooptic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Tschudi, T.

C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15, 2057–2064 (1998).
[CrossRef]

C. Denz, J. Golz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

Weiss, C. O.

K. Staliunas, M. F. H. Tarroja, G. Slekys, C. O. Weiss, and L. Dambly, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Weiss, S.

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. QE-25, 550–569 (1989).
[CrossRef]

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Yariv, A.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
[CrossRef] [PubMed]

Yatsenko, L. P.

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[CrossRef]

Appl. Phys. B (3)

R. Rebhi, P. Mathey, H.-R. Jauslin, and B. Sturman, “Effects of angular pump mismatch for the semi-linear oscillator,” Appl. Phys. B 99, 163–172 (2010).
[CrossRef]

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection gratings in electrooptic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

R. Rebhi, P. Mathey, M. Grapinet, H.-R. Jauslin, and S. Odoulov, “Phase mismatch effects on the dynamics of a semilinear photorefractive oscillator with reflection-type gratings,” Appl. Phys. B 91, 583–589 (2008).
[CrossRef]

Appl. Phys. Lett. (2)

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

L. C. Foster, M. D. Ewy, and C. B. Crumly, “Laser mode locking by an external Doppler cell,” Appl. Phys. Lett. 6, 6–8 (1965).
[CrossRef]

IEEE J. Quantum Electron. (2)

B. Fischer, S. Sternklar, and S. Weiss, “Photorefractive oscillators,” IEEE J. Quantum Electron. QE-25, 550–569 (1989).
[CrossRef]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. QE-20, 12–30 (1984).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

C. Denz, J. Golz, and T. Tschudi, “Enhanced four-wave mixing in photorefractive BaTiO3 by use of tilted pump waves,” Opt. Commun. 72, 129–134 (1989).
[CrossRef]

L. P. Yatsenko, B. W. Shore, and K. Bergmann, “Theory of a frequency-shifted feedback laser,” Opt. Commun. 236, 183–202 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

K. Staliunas, M. F. H. Tarroja, G. Slekys, C. O. Weiss, and L. Dambly, “Analogy between photorefractive oscillators and class-A lasers,” Phys. Rev. A 51, 4140–4151 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Experimental evidence of chaotic itinerancy and spatiotemporal chaos in optics,” Phys. Rev. Lett. 65, 2531–2534 (1990).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Scheme of the semi-linear oscillator: 1 and 2 are the pump waves, 3 and 4 are the oscillation waves, PRC is the photorefractive crystal, M is the feedback mirror which can be made to vibrate by a piezoelectric component to produce a frequency shift in the feedback wave.

Fig. 2
Fig. 2

Oscillation intensity versus the coupling strength and the phase mismatch calculated for a pump ratio r = 2 and classical mirror reflectivities (a) R = 0.4 , (b) R = 0 .

Fig. 3
Fig. 3

Coupling strength at threshold versus the phase mismatch for the mirrorless oscillation ( R = 0 ) and the oscillation of the semi-linear oscillator, with R = 1 % . The pump ratio is r = 5 . The colors of the curves are related with the frequency detuning plotted in the same colors in Fig. 4.

Fig. 4
Fig. 4

Frequency detuning at threshold versus the phase mismatch, for the mirrorless oscillation ( R = 0 ) and the oscillation of the semi-linear oscillator ( R = 1 % ) . The pump ratio is r = 5 . The colors of the curves are related with the coupling strength threshold plotted in same colors in Fig. 3.

Fig. 5
Fig. 5

Phase diagrams constructed from the threshold analysis for a feedback mirror reflectivity R = 0.1 and different values of the phase mismatch. The branches (1, ±2, and M) correspond, respectively, to the one frequency oscillation, the two frequency oscillation, and the mirrorless oscillation.

Fig. 6
Fig. 6

Phase diagrams calculated for Δ = 5 and R = 0.001 .

Fig. 7
Fig. 7

Contours of the oscillation intensity as a function of ( Ω M τ , Δ ) for (a) the semi-linear oscillator with parameters γ = 4.8 , r = 2 , R = 0.3 , in which the areas in gray correspond to a two frequency oscillation and (b) the mirrorless oscillation (independent of the excitation frequency).

Fig. 8
Fig. 8

Frequency diagram of the forced oscillator numerically calculated for γ = 4.5 , Δ = 2.8 , r = 5 , R = 0.1 . Horizontal blue line: the frequency detuning relative to the mirrorless oscillation.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

A 1 z = ν eff   exp ( i Δ z 2 ) A 4 ,
A 2 z = ν eff   exp ( i Δ z 2 ) A 3 ,
A 3 z = ν eff   exp ( i Δ z 2 ) A 2 ,
A 4 z = ν eff   exp ( i Δ z 2 ) A 1 ,
τ ν eff t + ν eff = γ I 0 [ A 1 A 4   exp ( i Δ z 2 ) + A 2 A 3   exp ( i Δ z 2 ) ] .
A 1 ( z = 0 , t ) = A 1 ( 0 ) = const ,
A 2 ( z = , t ) = A 2 ( ) = const ,
A 3 ( z = 0 , t ) = 0 ,
A 4 ( z = , t ) = R A 3 ( z = , t ) exp ( i Ω M t ) ,

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