Abstract

The optical injection dynamics of weakly index-guided vertical-cavity surface-emitting lasers (VCSELs) under single and multiple transverse modes are studied numerically. We consider the spatial dependences of the model and eliminate the spatial variables by integrating the spatially dependent rate equations over the cross section of the VCSELs' active region. Based on the simplified equations, a visual Simulink model is established. In the single-mode case, rich dynamics are observed. In the multimode case, when all modes are injected equally they exhibit similar injection-locking behaviors; however, if only one type of these modes is injected, all modes with parallel polarization are enhanced, while the other orthogonally polarized modes are suppressed greatly. It is shown further that the frequencies of those suppressed modes cannot be locked as in the case of the enhanced modes, although there are certain frequency shifts.

© 2006 Optical Society of America

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  1. N. R. Lang, "Injection locking properties of a semiconductor laser," IEEE J. Quantum Electron. 18, 976-983 (1982).
    [CrossRef]
  2. Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
    [CrossRef]
  3. L. Goldberg, H. F. Taylor, J. F. Weller, and D. M. Bloom, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
    [CrossRef]
  4. J. Wang, M. K. Haldar, L. Li, and F. V. C. Mendis, "Enhancement of modulation bandwidth of laser diodes by injection locking," IEEE Photon. Technol. Lett. 8, 34-36 (1996).
    [CrossRef]
  5. T. B. Simpson and J. M. Liu, "Enhanced modulation bandwidth in injection-locked semiconductor lasers," IEEE Photonics Technol. Lett. 9, 1322-1324 (1997).
    [CrossRef]
  6. G. Yabre, "Effect of relatively strong light injection on the chirp-to-power ration and the 3dB bandwidth of directly modulated semiconductor lasers," J. Lightwave Technol. 14, 2367-2373 (1996).
    [CrossRef]
  7. J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, "Modulation bandwidth, noise, and stability of as semiconductor laser subject to strong injection locking," IEEE Photonics Technol. Lett. 9, 1325-1327 (1997).
    [CrossRef]
  8. S. Arahira, S. Kutsuzawa, and Y. Ogawa, "Extreme timing jitter reduction of a passively mode-locked laser diode by optical pulse injection," IEEE J. Quantum Electron. 35, 1805-1811 (1999).
    [CrossRef]
  9. S. Mohrdiek, H. Burhard, and H. Walter, "Chirp reduction of directly modulated semiconductor lasers at 10Gb/s by strong cw light injection," J. Lightwave Technol. 12, 418-424 (1994).
    [CrossRef]
  10. J. Luo, M. Osinski, and J. G. McInerney, "Side-mode injection locking of semiconductor lasers," IEE Proc.: Optoelectron. 136, 33-37 (1989).
    [CrossRef]
  11. V. Annovazzi-Lodi, S. Donati, and M. Manna, "Chaos and locking in a semiconductor laser due to external injection," IEEE J. Quantum Electron. 30, 1537-1541 (1994).
    [CrossRef]
  12. S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental observation of antiphase polarization dynamics in VCSELs," Opt. Commun. 202, 145-154 (2002).
    [CrossRef]
  13. T. B. Simpson, "Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection," Opt. Commun. 215, 135-151 (2003).
    [CrossRef]
  14. V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, "Dynamic behavior and locking of a semiconductor laser subject to external injection," IEEE J. Quantum Electron. 34, 2350-2357 (1998).
    [CrossRef]
  15. H. Li, T. L. Lucas, J. G. Mcinerney, M. W. Wright, and R. A. Morgan, "Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection," IEEE J. Quantum Electron. 32, 227-235 (1996).
    [CrossRef]
  16. S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "VCSEL polarization control by optical injection," J. Lightwave Technol. 21, 2395-2404 (2003).
    [CrossRef]
  17. S. F. Yu, "Theoretical analysis of polarization bistability in vertical cavity surface emitting semiconductor lasers," J. Lightwave Technol. 15, 1032-1041 (1997).
    [CrossRef]
  18. Y. Hong, P. S. Spencer, S. Bandyopadhyay, P. Rees, and K. A. Shore, "Polarization-resolved chaos and instabilities in a vertical cavity surface emitting laser subject to optical injection," Opt. Commun. 216, 185-190 (2003).
    [CrossRef]
  19. S. F. Yu, P. Shum, and N. Q. Ngo, "Performance of optical chaotic communication systems using multimode vertical cavity surface emitting lasers," Opt. Commun. 200, 143-152 (2001).
    [CrossRef]
  20. N. Fujiwara, Y. Takiguchi, and J. Ohtsubo, "Observation of the synchronization of chaos in mutually injected vertical-cavity surface-emitting semiconductor lasers," Opt. Lett. 28, 1677-1679 (2003).
    [CrossRef] [PubMed]
  21. P. S. Spencer and C. R. Mirasso, "Analysis of optical chaos synchronization in frequency-detuned external-cavity VCSELs," IEEE J. Quantum Electron. 35, 803-809 (1999).
    [CrossRef]
  22. J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, "Effect of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers," Quantum Semiclassic. Opt. 9, 737-747 (1997).
    [CrossRef]
  23. Y. Hong, P. S. Spencer, P. Rees, and K. A. Shore, "Optical injection dynamics of two-mode vertical cavity surface-emitting semiconductor lasers," IEEE J. Quantum Electron. 38, 274-278 (2002).
    [CrossRef]
  24. M. S. Torre, C. Masoller, and K. A. Shore, "Numerical study of optical injection dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 40, 25-30 (2004).
    [CrossRef]
  25. P. V. Mena, J. J. Morikuni, S. M. Kang, A. V. Harton, and K. W. Wyatt, "A comprehensive circuit-level model of vertical-cavity surface-emitting lasers," J. Lightwave Technol. 17, 2612-2632 (1999).
    [CrossRef]
  26. A. Valle, J. Sarma, and K. A. Shore, "Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes," IEEE J. Quantum Electron. 31, 1423-1430 (1995).
    [CrossRef]
  27. H. B. Zhang, G. Mrozynski, A. Wallrabenstein, and J. Schrage, "Analysis of transverse mode competition of VCSELs based on a spatially independent model," IEEE J. Quantum Electron. 40, 18-23 (2004).
    [CrossRef]
  28. J. J. Morikuni, P. V. Mena, A. V. Harton, K. W. Wyatt, and S.-M. Kang, "Spatially independent VCSEL models for the simulation of diffusive turn-off transients," J. Lightwave Technol. 17, 95-100 (1999).
    [CrossRef]
  29. S. F. Yu, "Nonlinear dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 35, 332-341 (1999).
    [CrossRef]
  30. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).
  31. A. Gavrielides, V. Kovanis, and T. Erneux, "Analytical stability boundaries for a semiconductor laser subject to optical injection," Opt. Commun. 136, 253-256 (1997).
    [CrossRef]
  32. T. Erneux, V. Kovanis, A. Gavielides, and P. M. Alsing, "Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection," Phys. Rev. A 53, 4372-4380 (1996).
    [CrossRef] [PubMed]
  33. J. Y. Law and G. P. Agrawal, "Effect of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers," IEEE J. Sel. Top. Quantum Electron. 3, 353-358 (1997).
    [CrossRef]

2004 (2)

H. B. Zhang, G. Mrozynski, A. Wallrabenstein, and J. Schrage, "Analysis of transverse mode competition of VCSELs based on a spatially independent model," IEEE J. Quantum Electron. 40, 18-23 (2004).
[CrossRef]

M. S. Torre, C. Masoller, and K. A. Shore, "Numerical study of optical injection dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 40, 25-30 (2004).
[CrossRef]

2003 (4)

N. Fujiwara, Y. Takiguchi, and J. Ohtsubo, "Observation of the synchronization of chaos in mutually injected vertical-cavity surface-emitting semiconductor lasers," Opt. Lett. 28, 1677-1679 (2003).
[CrossRef] [PubMed]

T. B. Simpson, "Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection," Opt. Commun. 215, 135-151 (2003).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "VCSEL polarization control by optical injection," J. Lightwave Technol. 21, 2395-2404 (2003).
[CrossRef]

Y. Hong, P. S. Spencer, S. Bandyopadhyay, P. Rees, and K. A. Shore, "Polarization-resolved chaos and instabilities in a vertical cavity surface emitting laser subject to optical injection," Opt. Commun. 216, 185-190 (2003).
[CrossRef]

2002 (2)

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental observation of antiphase polarization dynamics in VCSELs," Opt. Commun. 202, 145-154 (2002).
[CrossRef]

Y. Hong, P. S. Spencer, P. Rees, and K. A. Shore, "Optical injection dynamics of two-mode vertical cavity surface-emitting semiconductor lasers," IEEE J. Quantum Electron. 38, 274-278 (2002).
[CrossRef]

2001 (1)

S. F. Yu, P. Shum, and N. Q. Ngo, "Performance of optical chaotic communication systems using multimode vertical cavity surface emitting lasers," Opt. Commun. 200, 143-152 (2001).
[CrossRef]

1999 (5)

S. Arahira, S. Kutsuzawa, and Y. Ogawa, "Extreme timing jitter reduction of a passively mode-locked laser diode by optical pulse injection," IEEE J. Quantum Electron. 35, 1805-1811 (1999).
[CrossRef]

P. S. Spencer and C. R. Mirasso, "Analysis of optical chaos synchronization in frequency-detuned external-cavity VCSELs," IEEE J. Quantum Electron. 35, 803-809 (1999).
[CrossRef]

J. J. Morikuni, P. V. Mena, A. V. Harton, K. W. Wyatt, and S.-M. Kang, "Spatially independent VCSEL models for the simulation of diffusive turn-off transients," J. Lightwave Technol. 17, 95-100 (1999).
[CrossRef]

S. F. Yu, "Nonlinear dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 35, 332-341 (1999).
[CrossRef]

P. V. Mena, J. J. Morikuni, S. M. Kang, A. V. Harton, and K. W. Wyatt, "A comprehensive circuit-level model of vertical-cavity surface-emitting lasers," J. Lightwave Technol. 17, 2612-2632 (1999).
[CrossRef]

1998 (1)

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, "Dynamic behavior and locking of a semiconductor laser subject to external injection," IEEE J. Quantum Electron. 34, 2350-2357 (1998).
[CrossRef]

1997 (6)

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, "Modulation bandwidth, noise, and stability of as semiconductor laser subject to strong injection locking," IEEE Photonics Technol. Lett. 9, 1325-1327 (1997).
[CrossRef]

S. F. Yu, "Theoretical analysis of polarization bistability in vertical cavity surface emitting semiconductor lasers," J. Lightwave Technol. 15, 1032-1041 (1997).
[CrossRef]

T. B. Simpson and J. M. Liu, "Enhanced modulation bandwidth in injection-locked semiconductor lasers," IEEE Photonics Technol. Lett. 9, 1322-1324 (1997).
[CrossRef]

J. Y. Law and G. P. Agrawal, "Effect of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers," IEEE J. Sel. Top. Quantum Electron. 3, 353-358 (1997).
[CrossRef]

A. Gavrielides, V. Kovanis, and T. Erneux, "Analytical stability boundaries for a semiconductor laser subject to optical injection," Opt. Commun. 136, 253-256 (1997).
[CrossRef]

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, "Effect of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers," Quantum Semiclassic. Opt. 9, 737-747 (1997).
[CrossRef]

1996 (5)

T. Erneux, V. Kovanis, A. Gavielides, and P. M. Alsing, "Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection," Phys. Rev. A 53, 4372-4380 (1996).
[CrossRef] [PubMed]

G. Yabre, "Effect of relatively strong light injection on the chirp-to-power ration and the 3dB bandwidth of directly modulated semiconductor lasers," J. Lightwave Technol. 14, 2367-2373 (1996).
[CrossRef]

Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
[CrossRef]

J. Wang, M. K. Haldar, L. Li, and F. V. C. Mendis, "Enhancement of modulation bandwidth of laser diodes by injection locking," IEEE Photon. Technol. Lett. 8, 34-36 (1996).
[CrossRef]

H. Li, T. L. Lucas, J. G. Mcinerney, M. W. Wright, and R. A. Morgan, "Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection," IEEE J. Quantum Electron. 32, 227-235 (1996).
[CrossRef]

1995 (1)

A. Valle, J. Sarma, and K. A. Shore, "Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes," IEEE J. Quantum Electron. 31, 1423-1430 (1995).
[CrossRef]

1994 (2)

V. Annovazzi-Lodi, S. Donati, and M. Manna, "Chaos and locking in a semiconductor laser due to external injection," IEEE J. Quantum Electron. 30, 1537-1541 (1994).
[CrossRef]

S. Mohrdiek, H. Burhard, and H. Walter, "Chirp reduction of directly modulated semiconductor lasers at 10Gb/s by strong cw light injection," J. Lightwave Technol. 12, 418-424 (1994).
[CrossRef]

1989 (1)

J. Luo, M. Osinski, and J. G. McInerney, "Side-mode injection locking of semiconductor lasers," IEE Proc.: Optoelectron. 136, 33-37 (1989).
[CrossRef]

1983 (1)

L. Goldberg, H. F. Taylor, J. F. Weller, and D. M. Bloom, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[CrossRef]

1982 (1)

N. R. Lang, "Injection locking properties of a semiconductor laser," IEEE J. Quantum Electron. 18, 976-983 (1982).
[CrossRef]

Agrawal, G. P.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, "Effect of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers," Quantum Semiclassic. Opt. 9, 737-747 (1997).
[CrossRef]

J. Y. Law and G. P. Agrawal, "Effect of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers," IEEE J. Sel. Top. Quantum Electron. 3, 353-358 (1997).
[CrossRef]

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).

Ahmed, Z.

Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
[CrossRef]

Alsing, P. M.

T. Erneux, V. Kovanis, A. Gavielides, and P. M. Alsing, "Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection," Phys. Rev. A 53, 4372-4380 (1996).
[CrossRef] [PubMed]

Annovazzi-Lodi, V.

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, "Dynamic behavior and locking of a semiconductor laser subject to external injection," IEEE J. Quantum Electron. 34, 2350-2357 (1998).
[CrossRef]

V. Annovazzi-Lodi, S. Donati, and M. Manna, "Chaos and locking in a semiconductor laser due to external injection," IEEE J. Quantum Electron. 30, 1537-1541 (1994).
[CrossRef]

Arahira, S.

S. Arahira, S. Kutsuzawa, and Y. Ogawa, "Extreme timing jitter reduction of a passively mode-locked laser diode by optical pulse injection," IEEE J. Quantum Electron. 35, 1805-1811 (1999).
[CrossRef]

Bandyopadhyay, S.

Y. Hong, P. S. Spencer, S. Bandyopadhyay, P. Rees, and K. A. Shore, "Polarization-resolved chaos and instabilities in a vertical cavity surface emitting laser subject to optical injection," Opt. Commun. 216, 185-190 (2003).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "VCSEL polarization control by optical injection," J. Lightwave Technol. 21, 2395-2404 (2003).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental observation of antiphase polarization dynamics in VCSELs," Opt. Commun. 202, 145-154 (2002).
[CrossRef]

Bloom, D. M.

L. Goldberg, H. F. Taylor, J. F. Weller, and D. M. Bloom, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[CrossRef]

Burhard, H.

S. Mohrdiek, H. Burhard, and H. Walter, "Chirp reduction of directly modulated semiconductor lasers at 10Gb/s by strong cw light injection," J. Lightwave Technol. 12, 418-424 (1994).
[CrossRef]

Chen, H. F.

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, "Modulation bandwidth, noise, and stability of as semiconductor laser subject to strong injection locking," IEEE Photonics Technol. Lett. 9, 1325-1327 (1997).
[CrossRef]

Donati, S.

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, "Dynamic behavior and locking of a semiconductor laser subject to external injection," IEEE J. Quantum Electron. 34, 2350-2357 (1998).
[CrossRef]

V. Annovazzi-Lodi, S. Donati, and M. Manna, "Chaos and locking in a semiconductor laser due to external injection," IEEE J. Quantum Electron. 30, 1537-1541 (1994).
[CrossRef]

Dutta, N. K.

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).

Erneux, T.

A. Gavrielides, V. Kovanis, and T. Erneux, "Analytical stability boundaries for a semiconductor laser subject to optical injection," Opt. Commun. 136, 253-256 (1997).
[CrossRef]

T. Erneux, V. Kovanis, A. Gavielides, and P. M. Alsing, "Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection," Phys. Rev. A 53, 4372-4380 (1996).
[CrossRef] [PubMed]

Fujiwara, N.

Gavielides, A.

T. Erneux, V. Kovanis, A. Gavielides, and P. M. Alsing, "Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection," Phys. Rev. A 53, 4372-4380 (1996).
[CrossRef] [PubMed]

Gavrielides, A.

A. Gavrielides, V. Kovanis, and T. Erneux, "Analytical stability boundaries for a semiconductor laser subject to optical injection," Opt. Commun. 136, 253-256 (1997).
[CrossRef]

Goldberg, L.

L. Goldberg, H. F. Taylor, J. F. Weller, and D. M. Bloom, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[CrossRef]

Haldar, M. K.

J. Wang, M. K. Haldar, L. Li, and F. V. C. Mendis, "Enhancement of modulation bandwidth of laser diodes by injection locking," IEEE Photon. Technol. Lett. 8, 34-36 (1996).
[CrossRef]

Harton, A. V.

Hong, Y.

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "VCSEL polarization control by optical injection," J. Lightwave Technol. 21, 2395-2404 (2003).
[CrossRef]

Y. Hong, P. S. Spencer, S. Bandyopadhyay, P. Rees, and K. A. Shore, "Polarization-resolved chaos and instabilities in a vertical cavity surface emitting laser subject to optical injection," Opt. Commun. 216, 185-190 (2003).
[CrossRef]

Y. Hong, P. S. Spencer, P. Rees, and K. A. Shore, "Optical injection dynamics of two-mode vertical cavity surface-emitting semiconductor lasers," IEEE J. Quantum Electron. 38, 274-278 (2002).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental observation of antiphase polarization dynamics in VCSELs," Opt. Commun. 202, 145-154 (2002).
[CrossRef]

Kang, S. M.

Kang, S.-M.

Kim, D. Y.

Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
[CrossRef]

Kovanis, V.

A. Gavrielides, V. Kovanis, and T. Erneux, "Analytical stability boundaries for a semiconductor laser subject to optical injection," Opt. Commun. 136, 253-256 (1997).
[CrossRef]

T. Erneux, V. Kovanis, A. Gavielides, and P. M. Alsing, "Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection," Phys. Rev. A 53, 4372-4380 (1996).
[CrossRef] [PubMed]

Kutsuzawa, S.

S. Arahira, S. Kutsuzawa, and Y. Ogawa, "Extreme timing jitter reduction of a passively mode-locked laser diode by optical pulse injection," IEEE J. Quantum Electron. 35, 1805-1811 (1999).
[CrossRef]

Lang, N. R.

N. R. Lang, "Injection locking properties of a semiconductor laser," IEEE J. Quantum Electron. 18, 976-983 (1982).
[CrossRef]

Law, J. Y.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, "Effect of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers," Quantum Semiclassic. Opt. 9, 737-747 (1997).
[CrossRef]

J. Y. Law and G. P. Agrawal, "Effect of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers," IEEE J. Sel. Top. Quantum Electron. 3, 353-358 (1997).
[CrossRef]

Li, H.

H. Li, T. L. Lucas, J. G. Mcinerney, M. W. Wright, and R. A. Morgan, "Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection," IEEE J. Quantum Electron. 32, 227-235 (1996).
[CrossRef]

Li, L.

J. Wang, M. K. Haldar, L. Li, and F. V. C. Mendis, "Enhancement of modulation bandwidth of laser diodes by injection locking," IEEE Photon. Technol. Lett. 8, 34-36 (1996).
[CrossRef]

Liu, H. F.

Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
[CrossRef]

Liu, J. M.

T. B. Simpson and J. M. Liu, "Enhanced modulation bandwidth in injection-locked semiconductor lasers," IEEE Photonics Technol. Lett. 9, 1322-1324 (1997).
[CrossRef]

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, "Modulation bandwidth, noise, and stability of as semiconductor laser subject to strong injection locking," IEEE Photonics Technol. Lett. 9, 1325-1327 (1997).
[CrossRef]

Lucas, T. L.

H. Li, T. L. Lucas, J. G. Mcinerney, M. W. Wright, and R. A. Morgan, "Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection," IEEE J. Quantum Electron. 32, 227-235 (1996).
[CrossRef]

Luo, J.

J. Luo, M. Osinski, and J. G. McInerney, "Side-mode injection locking of semiconductor lasers," IEE Proc.: Optoelectron. 136, 33-37 (1989).
[CrossRef]

Manna, M.

V. Annovazzi-Lodi, S. Donati, and M. Manna, "Chaos and locking in a semiconductor laser due to external injection," IEEE J. Quantum Electron. 30, 1537-1541 (1994).
[CrossRef]

Masoller, C.

M. S. Torre, C. Masoller, and K. A. Shore, "Numerical study of optical injection dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 40, 25-30 (2004).
[CrossRef]

Mcinerney, J. G.

H. Li, T. L. Lucas, J. G. Mcinerney, M. W. Wright, and R. A. Morgan, "Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection," IEEE J. Quantum Electron. 32, 227-235 (1996).
[CrossRef]

J. Luo, M. Osinski, and J. G. McInerney, "Side-mode injection locking of semiconductor lasers," IEE Proc.: Optoelectron. 136, 33-37 (1989).
[CrossRef]

Mena, P. V.

Mendis, F. V. C.

J. Wang, M. K. Haldar, L. Li, and F. V. C. Mendis, "Enhancement of modulation bandwidth of laser diodes by injection locking," IEEE Photon. Technol. Lett. 8, 34-36 (1996).
[CrossRef]

Meng, X. J.

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, "Modulation bandwidth, noise, and stability of as semiconductor laser subject to strong injection locking," IEEE Photonics Technol. Lett. 9, 1325-1327 (1997).
[CrossRef]

Mirasso, C. R.

P. S. Spencer and C. R. Mirasso, "Analysis of optical chaos synchronization in frequency-detuned external-cavity VCSELs," IEEE J. Quantum Electron. 35, 803-809 (1999).
[CrossRef]

Mohrdiek, S.

S. Mohrdiek, H. Burhard, and H. Walter, "Chirp reduction of directly modulated semiconductor lasers at 10Gb/s by strong cw light injection," J. Lightwave Technol. 12, 418-424 (1994).
[CrossRef]

Morgan, R. A.

H. Li, T. L. Lucas, J. G. Mcinerney, M. W. Wright, and R. A. Morgan, "Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection," IEEE J. Quantum Electron. 32, 227-235 (1996).
[CrossRef]

Morikuni, J. J.

Mrozynski, G.

H. B. Zhang, G. Mrozynski, A. Wallrabenstein, and J. Schrage, "Analysis of transverse mode competition of VCSELs based on a spatially independent model," IEEE J. Quantum Electron. 40, 18-23 (2004).
[CrossRef]

Ngo, N. Q.

S. F. Yu, P. Shum, and N. Q. Ngo, "Performance of optical chaotic communication systems using multimode vertical cavity surface emitting lasers," Opt. Commun. 200, 143-152 (2001).
[CrossRef]

Novak, D.

Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
[CrossRef]

Ogawa, Y.

S. Arahira, S. Kutsuzawa, and Y. Ogawa, "Extreme timing jitter reduction of a passively mode-locked laser diode by optical pulse injection," IEEE J. Quantum Electron. 35, 1805-1811 (1999).
[CrossRef]

Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
[CrossRef]

Ohtsubo, J.

Osinski, M.

J. Luo, M. Osinski, and J. G. McInerney, "Side-mode injection locking of semiconductor lasers," IEE Proc.: Optoelectron. 136, 33-37 (1989).
[CrossRef]

Pelusi, M. D.

Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
[CrossRef]

Rees, P.

Y. Hong, P. S. Spencer, S. Bandyopadhyay, P. Rees, and K. A. Shore, "Polarization-resolved chaos and instabilities in a vertical cavity surface emitting laser subject to optical injection," Opt. Commun. 216, 185-190 (2003).
[CrossRef]

Y. Hong, P. S. Spencer, P. Rees, and K. A. Shore, "Optical injection dynamics of two-mode vertical cavity surface-emitting semiconductor lasers," IEEE J. Quantum Electron. 38, 274-278 (2002).
[CrossRef]

Sarma, J.

A. Valle, J. Sarma, and K. A. Shore, "Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes," IEEE J. Quantum Electron. 31, 1423-1430 (1995).
[CrossRef]

Schrage, J.

H. B. Zhang, G. Mrozynski, A. Wallrabenstein, and J. Schrage, "Analysis of transverse mode competition of VCSELs based on a spatially independent model," IEEE J. Quantum Electron. 40, 18-23 (2004).
[CrossRef]

Scire, A.

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, "Dynamic behavior and locking of a semiconductor laser subject to external injection," IEEE J. Quantum Electron. 34, 2350-2357 (1998).
[CrossRef]

Shore, K. A.

M. S. Torre, C. Masoller, and K. A. Shore, "Numerical study of optical injection dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 40, 25-30 (2004).
[CrossRef]

Y. Hong, P. S. Spencer, S. Bandyopadhyay, P. Rees, and K. A. Shore, "Polarization-resolved chaos and instabilities in a vertical cavity surface emitting laser subject to optical injection," Opt. Commun. 216, 185-190 (2003).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "VCSEL polarization control by optical injection," J. Lightwave Technol. 21, 2395-2404 (2003).
[CrossRef]

Y. Hong, P. S. Spencer, P. Rees, and K. A. Shore, "Optical injection dynamics of two-mode vertical cavity surface-emitting semiconductor lasers," IEEE J. Quantum Electron. 38, 274-278 (2002).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental observation of antiphase polarization dynamics in VCSELs," Opt. Commun. 202, 145-154 (2002).
[CrossRef]

A. Valle, J. Sarma, and K. A. Shore, "Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes," IEEE J. Quantum Electron. 31, 1423-1430 (1995).
[CrossRef]

Shum, P.

S. F. Yu, P. Shum, and N. Q. Ngo, "Performance of optical chaotic communication systems using multimode vertical cavity surface emitting lasers," Opt. Commun. 200, 143-152 (2001).
[CrossRef]

Simpson, T. B.

T. B. Simpson, "Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection," Opt. Commun. 215, 135-151 (2003).
[CrossRef]

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, "Modulation bandwidth, noise, and stability of as semiconductor laser subject to strong injection locking," IEEE Photonics Technol. Lett. 9, 1325-1327 (1997).
[CrossRef]

T. B. Simpson and J. M. Liu, "Enhanced modulation bandwidth in injection-locked semiconductor lasers," IEEE Photonics Technol. Lett. 9, 1322-1324 (1997).
[CrossRef]

Sorel, M.

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, "Dynamic behavior and locking of a semiconductor laser subject to external injection," IEEE J. Quantum Electron. 34, 2350-2357 (1998).
[CrossRef]

Spencer, P. S.

Y. Hong, P. S. Spencer, S. Bandyopadhyay, P. Rees, and K. A. Shore, "Polarization-resolved chaos and instabilities in a vertical cavity surface emitting laser subject to optical injection," Opt. Commun. 216, 185-190 (2003).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "VCSEL polarization control by optical injection," J. Lightwave Technol. 21, 2395-2404 (2003).
[CrossRef]

Y. Hong, P. S. Spencer, P. Rees, and K. A. Shore, "Optical injection dynamics of two-mode vertical cavity surface-emitting semiconductor lasers," IEEE J. Quantum Electron. 38, 274-278 (2002).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental observation of antiphase polarization dynamics in VCSELs," Opt. Commun. 202, 145-154 (2002).
[CrossRef]

P. S. Spencer and C. R. Mirasso, "Analysis of optical chaos synchronization in frequency-detuned external-cavity VCSELs," IEEE J. Quantum Electron. 35, 803-809 (1999).
[CrossRef]

Takiguchi, Y.

Taylor, H. F.

L. Goldberg, H. F. Taylor, J. F. Weller, and D. M. Bloom, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[CrossRef]

Torre, M. S.

M. S. Torre, C. Masoller, and K. A. Shore, "Numerical study of optical injection dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 40, 25-30 (2004).
[CrossRef]

Valle, A.

A. Valle, J. Sarma, and K. A. Shore, "Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes," IEEE J. Quantum Electron. 31, 1423-1430 (1995).
[CrossRef]

van Tartwijk, G. H. M.

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, "Effect of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers," Quantum Semiclassic. Opt. 9, 737-747 (1997).
[CrossRef]

Wallrabenstein, A.

H. B. Zhang, G. Mrozynski, A. Wallrabenstein, and J. Schrage, "Analysis of transverse mode competition of VCSELs based on a spatially independent model," IEEE J. Quantum Electron. 40, 18-23 (2004).
[CrossRef]

Walter, H.

S. Mohrdiek, H. Burhard, and H. Walter, "Chirp reduction of directly modulated semiconductor lasers at 10Gb/s by strong cw light injection," J. Lightwave Technol. 12, 418-424 (1994).
[CrossRef]

Wang, J.

J. Wang, M. K. Haldar, L. Li, and F. V. C. Mendis, "Enhancement of modulation bandwidth of laser diodes by injection locking," IEEE Photon. Technol. Lett. 8, 34-36 (1996).
[CrossRef]

Weller, J. F.

L. Goldberg, H. F. Taylor, J. F. Weller, and D. M. Bloom, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[CrossRef]

Wright, M. W.

H. Li, T. L. Lucas, J. G. Mcinerney, M. W. Wright, and R. A. Morgan, "Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection," IEEE J. Quantum Electron. 32, 227-235 (1996).
[CrossRef]

Wyatt, K. W.

Yabre, G.

G. Yabre, "Effect of relatively strong light injection on the chirp-to-power ration and the 3dB bandwidth of directly modulated semiconductor lasers," J. Lightwave Technol. 14, 2367-2373 (1996).
[CrossRef]

Yu, S. F.

S. F. Yu, P. Shum, and N. Q. Ngo, "Performance of optical chaotic communication systems using multimode vertical cavity surface emitting lasers," Opt. Commun. 200, 143-152 (2001).
[CrossRef]

S. F. Yu, "Nonlinear dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 35, 332-341 (1999).
[CrossRef]

S. F. Yu, "Theoretical analysis of polarization bistability in vertical cavity surface emitting semiconductor lasers," J. Lightwave Technol. 15, 1032-1041 (1997).
[CrossRef]

Zhang, H. B.

H. B. Zhang, G. Mrozynski, A. Wallrabenstein, and J. Schrage, "Analysis of transverse mode competition of VCSELs based on a spatially independent model," IEEE J. Quantum Electron. 40, 18-23 (2004).
[CrossRef]

Electron. Lett. (1)

L. Goldberg, H. F. Taylor, J. F. Weller, and D. M. Bloom, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[CrossRef]

IEE Proc.: Optoelectron. (1)

J. Luo, M. Osinski, and J. G. McInerney, "Side-mode injection locking of semiconductor lasers," IEE Proc.: Optoelectron. 136, 33-37 (1989).
[CrossRef]

IEEE J. Quantum Electron. (11)

V. Annovazzi-Lodi, S. Donati, and M. Manna, "Chaos and locking in a semiconductor laser due to external injection," IEEE J. Quantum Electron. 30, 1537-1541 (1994).
[CrossRef]

V. Annovazzi-Lodi, A. Scire, M. Sorel, and S. Donati, "Dynamic behavior and locking of a semiconductor laser subject to external injection," IEEE J. Quantum Electron. 34, 2350-2357 (1998).
[CrossRef]

H. Li, T. L. Lucas, J. G. Mcinerney, M. W. Wright, and R. A. Morgan, "Injection locking dynamics of vertical cavity semiconductor lasers under conventional and phase conjugate injection," IEEE J. Quantum Electron. 32, 227-235 (1996).
[CrossRef]

N. R. Lang, "Injection locking properties of a semiconductor laser," IEEE J. Quantum Electron. 18, 976-983 (1982).
[CrossRef]

S. Arahira, S. Kutsuzawa, and Y. Ogawa, "Extreme timing jitter reduction of a passively mode-locked laser diode by optical pulse injection," IEEE J. Quantum Electron. 35, 1805-1811 (1999).
[CrossRef]

P. S. Spencer and C. R. Mirasso, "Analysis of optical chaos synchronization in frequency-detuned external-cavity VCSELs," IEEE J. Quantum Electron. 35, 803-809 (1999).
[CrossRef]

Y. Hong, P. S. Spencer, P. Rees, and K. A. Shore, "Optical injection dynamics of two-mode vertical cavity surface-emitting semiconductor lasers," IEEE J. Quantum Electron. 38, 274-278 (2002).
[CrossRef]

M. S. Torre, C. Masoller, and K. A. Shore, "Numerical study of optical injection dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 40, 25-30 (2004).
[CrossRef]

A. Valle, J. Sarma, and K. A. Shore, "Spatial holeburning effects on the dynamics of vertical cavity surface-emitting laser diodes," IEEE J. Quantum Electron. 31, 1423-1430 (1995).
[CrossRef]

H. B. Zhang, G. Mrozynski, A. Wallrabenstein, and J. Schrage, "Analysis of transverse mode competition of VCSELs based on a spatially independent model," IEEE J. Quantum Electron. 40, 18-23 (2004).
[CrossRef]

S. F. Yu, "Nonlinear dynamics of vertical-cavity surface-emitting lasers," IEEE J. Quantum Electron. 35, 332-341 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Y. Law and G. P. Agrawal, "Effect of optical feedback on static and dynamic characteristics of vertical-cavity surface-emitting lasers," IEEE J. Sel. Top. Quantum Electron. 3, 353-358 (1997).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

Z. Ahmed, H. F. Liu, D. Novak, Y. Ogawa, M. D. Pelusi, and D. Y. Kim, "Locking characteristics of a passively mode-locked monolithic DBR laser stabilized by optical injection," IEEE Photon. Technol. Lett. 8, 37-39 (1996).
[CrossRef]

J. Wang, M. K. Haldar, L. Li, and F. V. C. Mendis, "Enhancement of modulation bandwidth of laser diodes by injection locking," IEEE Photon. Technol. Lett. 8, 34-36 (1996).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

T. B. Simpson and J. M. Liu, "Enhanced modulation bandwidth in injection-locked semiconductor lasers," IEEE Photonics Technol. Lett. 9, 1322-1324 (1997).
[CrossRef]

J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, "Modulation bandwidth, noise, and stability of as semiconductor laser subject to strong injection locking," IEEE Photonics Technol. Lett. 9, 1325-1327 (1997).
[CrossRef]

J. Lightwave Technol. (6)

J. J. Morikuni, P. V. Mena, A. V. Harton, K. W. Wyatt, and S.-M. Kang, "Spatially independent VCSEL models for the simulation of diffusive turn-off transients," J. Lightwave Technol. 17, 95-100 (1999).
[CrossRef]

P. V. Mena, J. J. Morikuni, S. M. Kang, A. V. Harton, and K. W. Wyatt, "A comprehensive circuit-level model of vertical-cavity surface-emitting lasers," J. Lightwave Technol. 17, 2612-2632 (1999).
[CrossRef]

G. Yabre, "Effect of relatively strong light injection on the chirp-to-power ration and the 3dB bandwidth of directly modulated semiconductor lasers," J. Lightwave Technol. 14, 2367-2373 (1996).
[CrossRef]

S. Mohrdiek, H. Burhard, and H. Walter, "Chirp reduction of directly modulated semiconductor lasers at 10Gb/s by strong cw light injection," J. Lightwave Technol. 12, 418-424 (1994).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "VCSEL polarization control by optical injection," J. Lightwave Technol. 21, 2395-2404 (2003).
[CrossRef]

S. F. Yu, "Theoretical analysis of polarization bistability in vertical cavity surface emitting semiconductor lasers," J. Lightwave Technol. 15, 1032-1041 (1997).
[CrossRef]

Opt. Commun. (5)

Y. Hong, P. S. Spencer, S. Bandyopadhyay, P. Rees, and K. A. Shore, "Polarization-resolved chaos and instabilities in a vertical cavity surface emitting laser subject to optical injection," Opt. Commun. 216, 185-190 (2003).
[CrossRef]

S. F. Yu, P. Shum, and N. Q. Ngo, "Performance of optical chaotic communication systems using multimode vertical cavity surface emitting lasers," Opt. Commun. 200, 143-152 (2001).
[CrossRef]

S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, "Experimental observation of antiphase polarization dynamics in VCSELs," Opt. Commun. 202, 145-154 (2002).
[CrossRef]

T. B. Simpson, "Mapping the nonlinear dynamics of a distributed feedback semiconductor laser subject to external optical injection," Opt. Commun. 215, 135-151 (2003).
[CrossRef]

A. Gavrielides, V. Kovanis, and T. Erneux, "Analytical stability boundaries for a semiconductor laser subject to optical injection," Opt. Commun. 136, 253-256 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

T. Erneux, V. Kovanis, A. Gavielides, and P. M. Alsing, "Mechanism for period-doubling bifurcation in a semiconductor laser subject to optical injection," Phys. Rev. A 53, 4372-4380 (1996).
[CrossRef] [PubMed]

Quantum Semiclassic. Opt. (1)

J. Y. Law, G. H. M. van Tartwijk, and G. P. Agrawal, "Effect of transverse-mode competition on the injection dynamics of vertical-cavity surface-emitting lasers," Quantum Semiclassic. Opt. 9, 737-747 (1997).
[CrossRef]

Other (1)

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).

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Figures (11)

Fig. 1
Fig. 1

Simulink model for the multi-transverse-mode VCSELs subject to external optical injection. Module A is the rate equations for carrier density, N 0 and N 1 ; modules B, C, and D are the photon and the phase rate equations of the LP 01 , LP 11 , and LP 02 modes, respectively.

Fig. 2
Fig. 2

Radial intensity profiles of the LP 01 , LP 11 , and LP 02 modes. The insets are their corresponding three-dimensional views and near-field images.

Fig. 3
Fig. 3

Light-current characteristics (a) and temporal traces (b) of modal power: (a), (b) R d = 1.86 μ m ; (b) I = 1.15 mA .

Fig. 4
Fig. 4

Bifurcation diagram of LP 01 modal power versus injection strength k, where the laser is assumed to operate with the fundamental transverse mode (the LP 01 mode). Here the frequency detuning between the master and the slave is neglected ( Δ υ 1 = 0 ) . The inset is the bifurcation diagram when the injection strength is relatively weak.

Fig. 5
Fig. 5

Bifurcation diagram of modal power versus k, where all modes are injected equally. The frequency of the injection field is chosen to coincide with the LP 01 mode ( Δ υ 1 = 0 ) .

Fig. 6
Fig. 6

(a)–(c) Temporal traces of modal power; (d)–(f) frequency chirp, where k = 0.8 × 10 3 and Δ υ 1 = 0 .

Fig. 7
Fig. 7

(a), (b) Temporal traces of modal power; (c), (d) frequency chirp, where (a), (c) k = 3 × 10 3 ; (b), (d) k = 8 × 10 3 . Solid, dashed, and dot traces, LP 01 , LP 11 , and LP 02 modes, respectively.

Fig. 8
Fig. 8

Optical spectra of transverse modes for (a) the injection signal, (b) the free-running case, and (c) the mode-locked case; (c) k = 8 × 10 3 and Δ υ 1 = 0 .

Fig. 9
Fig. 9

Bifurcation diagram of modal power versus k, where the injection field is chosen as the X-polarized LP 01 mode. The frequency of the injection field is chosen to coincide with the LP 01 mode ( Δ υ 1 = 0 ) .

Fig. 10
Fig. 10

Bifurcation diagram of modal power versus k, where the injection field is chosen as the Y-polarized LP 11 mode. The frequency of the injection field is chosen to coincide with the LP 11 mode ( Δ υ 2 = 0 ) .

Fig. 11
Fig. 11

Optical spectra of LP 01 , LP 11 , and LP 02 modes when the laser is locked by the optical injection: left column, k = 7 × 10 3 , Δ υ 1 = 0 ; middle column, k = 1.6 × 10 2 , Δ υ 2 = 0 ; right column, k = 7 × 10 3 , Δ υ 3 = 0 .

Tables (1)

Tables Icon

Table 1 Parameters in the Simulation

Equations (25)

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

S mn ( r , φ , t ) t = [ T G mn ( r , t ) 1 τ p ] S mn ( r , φ , t ) + β sp B N 2 ( r , t ) + 2 1 τ in [ S mn ( r , φ , t ) S inj ( r , φ ) ] 1 2 cos [ θ mn ( r , φ , t ) ] ,
N ( r , t ) t = η j j ( r , t ) q n w d w N ( r , t ) τ e + L eff τ e 1 r r [ r N ( r , t ) r ] m n G mn ( r , t ) S mn ( r , φ , t ) ,
Φ mn ( r , φ , t ) t = 1 2 β c [ T G mn ( r , t ) 1 τ p ] 1 τ in [ S inj ( r , φ ) S mn ( r , φ , t ) ] 1 2 sin [ θ mn ( r , φ , t ) ] ,
G mn ( r , t ) = V g g 0 [ N ( r , t ) N T ] 1 + ε NL S mn ( t ) ,
θ mn ( r , φ , t ) = Φ mn ( r , φ , t ) + 2 π Δ υ mn t ,
S mn ( r , φ , t ) = S mn ( t ) ψ mn ( r , φ ) ,
S 0 ( r , φ ) = S 0 ψ inj ( r , φ ) ,
ψ mn ( r , φ ) = c mn [ J m ( u mn r R core ) J m ( u mn ) ] 2 cos 2 ( m φ ) r < R core ,
ψ mn ( r , φ ) = c mn [ K m ( w mn r R core ) K m ( w mn ) ] 2 cos 2 ( m φ ) r > R core ,
u mn J m + 1 ( u mn ) J m ( u mn ) = w mn K m + 1 ( w mn ) K m ( w mn ) ,
1 π R core 2 0 2 π 0 R core ψ mn ( r , φ ) r d r d φ = 1 .
N ( r , t ) = N 0 ( t ) i = 0 N i ( t ) J 0 ( σ i r R core ) .
d N 0 ( t ) d t = η j j 0 ( t ) q n w d w ξ 0 N 0 ( t ) τ e m n V g g 0 1 + ε NL S mn ( t ) [ N 0 ( t ) N T N 1 ( t ) h 1 mn ] S mn ( t ) ,
d N 1 ( t ) d t = η j j 0 ( t ) q n w d w ξ 1 + m n V g g 0 1 + ε NL S mn ( t ) { [ N 0 ( t ) N T ] h 2 mn N 1 ( t ) h 3 mn } S mn ( t ) N 1 ( t ) τ e [ 1 + ( σ 1 L eff R core ) 2 ] ,
d S mn ( t ) d t = Γ V g g 0 1 + ε NL S mn ( t ) [ N 0 ( t ) N T N 1 ( t ) h 1 mn ] S mn ( t ) S mn ( t ) τ p + 2 k ζ 1 mn [ S mn ( t ) S 0 ] 1 2 cos [ θ mn ( t ) ] ,
d Φ mn ( t ) d t = 1 2 β c { Γ V g g 0 1 + ε NL S mn ( t ) [ N 0 ( t ) N T N 1 ( t ) h 4 ] 1 τ p } k ζ 2 mn [ S 0 S mn ( t ) ] 1 2 sin [ θ mn ( t ) ] .
ξ i = 1 π R core 2 J 0 2 ( σ i ) 0 2 π 0 R core J 0 ( σ i r R core ) j ( r ) r d r d φ ,
h 1 mn = 1 π R core 2 0 2 π 0 R core ψ mn ( r , φ ) J 0 ( σ i r R core ) r d r d φ ,
h 2 mn = 1 π R core 2 J 0 2 ( σ 1 ) 0 2 π 0 R core ψ mn ( r , φ ) J 0 ( σ 1 r R core ) r d r d φ ,
h 3 mn = 1 π R core 2 J 0 2 ( σ 1 ) 0 2 π 0 R core ψ mn ( r , φ ) J 0 2 ( σ 1 r R core ) r d r d φ ,
h 4 = 1 π R core 2 0 2 π 0 R core J 0 ( σ 1 r R core ) r d r d φ ,
ζ 1 mn = c 1 π R core 2 0 2 π 0 R core [ Ψ mn ( r , φ ) ψ inj ( r , φ ) ] 1 2 r d r d φ ,
ζ 2 mn = c 1 π R core 2 0 2 π 0 R core [ ψ inj ( r , φ ) ψ mn ( r , φ ) ] 1 2 r d r d φ ,
P mn = 1 2 ν g h υ mn ( 1 R 2 ) π R core 2 Γ S mn ( t ) ,
δ υ mn = 1 2 π d Φ mn ( t ) d t ,

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