Abstract

Self-pulsing in semiconductor lasers containing a saturable absorber is well documented. We propose an alternative mechanism leading to self-pulsing in vertical-cavity surface-emitting lasers. We specifically consider waveguiding effects and determine the conditions for self-pulsing. Our theoretical and numerical results agree with experimental observations reported by Willemsen et al. [Appl. Phys. Lett. 77, 3514 (2000)].

© 2004 Optical Society of America

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  1. T. W. Carr and T. Erneux, IEEE J. Quantum Electron. 37, 1171 (2001), and references therein.
    [CrossRef]
  2. P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
    [CrossRef]
  3. C. Juang, T. M. Hwang, J. Juang, and W. W. Lin, IEEE J. Quantum Electron. 36, 300 (2000).
    [CrossRef]
  4. M. Yamada and T. Higashi, IEEE J. Quantum Electron. 27, 380 (1991).
    [CrossRef]
  5. M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
    [CrossRef]
  6. A. Scirè, J. Mulet, C. R. Mirasso, and M. San Miguel, Opt. Lett. 27, 391 (2002).
    [CrossRef]
  7. M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
    [CrossRef] [PubMed]
  8. A. Scirè, J. Mulet, C. R. Mirasso, J. Danckaert, and M. San Miguel, Phys. Rev. Lett. 90, 113901 (2003).
    [CrossRef]
  9. J. Buus, IEEE Photon. Technol. Lett. 6, 1179 (1994).
    [CrossRef]
  10. J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
    [CrossRef]
  11. J. Mulet and S. Balle, Phys. Rev. A 66, 053802 (2002).
    [CrossRef]
  12. C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985).
  13. The intensity-dependent confinement factor Γ can be found by solving n22p2u3+2n0n2V2 pu2+V2n02-na2u-n02-na2=0 for u=1-Γ, where V is the normalized frequency and na (n0) represents the refractive index in the unpumped (core) region. We found that α≈2n0n2/V4Γ0n02-na2, while β≈Γ0α/Γsat-Γ0, with Γsat being the saturated confinement factor.
  14. M. Peeters, MODEL: The Modest Ordinary Differential Equation Library, C++ framework, available at http://model.sourceforge.net (2000–2003).
  15. K. Panajotov, G. Van der Sande, H. Thienpont, and I. Veretennicoff have submitted a paper called “Self-pulsation in VCSELs as a result of the interplay between carrier-induced antiguiding and build-in index guiding” to J. Opt. Soc. Am. B.

2003 (1)

A. Scirè, J. Mulet, C. R. Mirasso, J. Danckaert, and M. San Miguel, Phys. Rev. Lett. 90, 113901 (2003).
[CrossRef]

2002 (3)

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
[CrossRef]

J. Mulet and S. Balle, Phys. Rev. A 66, 053802 (2002).
[CrossRef]

A. Scirè, J. Mulet, C. R. Mirasso, and M. San Miguel, Opt. Lett. 27, 391 (2002).
[CrossRef]

2001 (1)

T. W. Carr and T. Erneux, IEEE J. Quantum Electron. 37, 1171 (2001), and references therein.
[CrossRef]

2000 (2)

C. Juang, T. M. Hwang, J. Juang, and W. W. Lin, IEEE J. Quantum Electron. 36, 300 (2000).
[CrossRef]

M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
[CrossRef]

1999 (1)

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

1995 (1)

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

1994 (1)

J. Buus, IEEE Photon. Technol. Lett. 6, 1179 (1994).
[CrossRef]

1991 (1)

M. Yamada and T. Higashi, IEEE J. Quantum Electron. 27, 380 (1991).
[CrossRef]

Albert, J.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
[CrossRef]

Balle, S.

J. Mulet and S. Balle, Phys. Rev. A 66, 053802 (2002).
[CrossRef]

Brunner, M.

M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
[CrossRef]

Buus, J.

J. Buus, IEEE Photon. Technol. Lett. 6, 1179 (1994).
[CrossRef]

Carr, T. W.

T. W. Carr and T. Erneux, IEEE J. Quantum Electron. 37, 1171 (2001), and references therein.
[CrossRef]

Danckaert, J.

A. Scirè, J. Mulet, C. R. Mirasso, J. Danckaert, and M. San Miguel, Phys. Rev. Lett. 90, 113901 (2003).
[CrossRef]

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
[CrossRef]

Erneux, T.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
[CrossRef]

T. W. Carr and T. Erneux, IEEE J. Quantum Electron. 37, 1171 (2001), and references therein.
[CrossRef]

Feng, Q.

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

Hegasky, J.

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

Higashi, T.

M. Yamada and T. Higashi, IEEE J. Quantum Electron. 27, 380 (1991).
[CrossRef]

Hövel, R. H.

M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
[CrossRef]

Hwang, T. M.

C. Juang, T. M. Hwang, J. Juang, and W. W. Lin, IEEE J. Quantum Electron. 36, 300 (2000).
[CrossRef]

Juang, C.

C. Juang, T. M. Hwang, J. Juang, and W. W. Lin, IEEE J. Quantum Electron. 36, 300 (2000).
[CrossRef]

Juang, J.

C. Juang, T. M. Hwang, J. Juang, and W. W. Lin, IEEE J. Quantum Electron. 36, 300 (2000).
[CrossRef]

Landais, P.

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

Lin, W. W.

C. Juang, T. M. Hwang, J. Juang, and W. W. Lin, IEEE J. Quantum Electron. 36, 300 (2000).
[CrossRef]

Lynch, S.

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

McEvoy, P.

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

Mirasso, C. R.

A. Scirè, J. Mulet, C. R. Mirasso, J. Danckaert, and M. San Miguel, Phys. Rev. Lett. 90, 113901 (2003).
[CrossRef]

A. Scirè, J. Mulet, C. R. Mirasso, and M. San Miguel, Opt. Lett. 27, 391 (2002).
[CrossRef]

Moloney, J. V.

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

Mulet, J.

A. Scirè, J. Mulet, C. R. Mirasso, J. Danckaert, and M. San Miguel, Phys. Rev. Lett. 90, 113901 (2003).
[CrossRef]

A. Scirè, J. Mulet, C. R. Mirasso, and M. San Miguel, Opt. Lett. 27, 391 (2002).
[CrossRef]

J. Mulet and S. Balle, Phys. Rev. A 66, 053802 (2002).
[CrossRef]

Nagler, B.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
[CrossRef]

O’Gorman, J.

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

Panajotov, K.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
[CrossRef]

K. Panajotov, G. Van der Sande, H. Thienpont, and I. Veretennicoff have submitted a paper called “Self-pulsation in VCSELs as a result of the interplay between carrier-induced antiguiding and build-in index guiding” to J. Opt. Soc. Am. B.

Peeters, M.

M. Peeters, MODEL: The Modest Ordinary Differential Equation Library, C++ framework, available at http://model.sourceforge.net (2000–2003).

Pesquera, L.

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

Rees, P.

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

San Miguel, M.

A. Scirè, J. Mulet, C. R. Mirasso, J. Danckaert, and M. San Miguel, Phys. Rev. Lett. 90, 113901 (2003).
[CrossRef]

A. Scirè, J. Mulet, C. R. Mirasso, and M. San Miguel, Opt. Lett. 27, 391 (2002).
[CrossRef]

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

Scirè, A.

A. Scirè, J. Mulet, C. R. Mirasso, J. Danckaert, and M. San Miguel, Phys. Rev. Lett. 90, 113901 (2003).
[CrossRef]

A. Scirè, J. Mulet, C. R. Mirasso, and M. San Miguel, Opt. Lett. 27, 391 (2002).
[CrossRef]

Thienpont, H.

K. Panajotov, G. Van der Sande, H. Thienpont, and I. Veretennicoff have submitted a paper called “Self-pulsation in VCSELs as a result of the interplay between carrier-induced antiguiding and build-in index guiding” to J. Opt. Soc. Am. B.

Valle, A.

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

van de Nes, A. S.

M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
[CrossRef]

Van der Sande, G.

K. Panajotov, G. Van der Sande, H. Thienpont, and I. Veretennicoff have submitted a paper called “Self-pulsation in VCSELs as a result of the interplay between carrier-induced antiguiding and build-in index guiding” to J. Opt. Soc. Am. B.

van Exter, M. P.

M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
[CrossRef]

Vassallo, C.

C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985).

Veretennicoff, I.

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
[CrossRef]

K. Panajotov, G. Van der Sande, H. Thienpont, and I. Veretennicoff have submitted a paper called “Self-pulsation in VCSELs as a result of the interplay between carrier-induced antiguiding and build-in index guiding” to J. Opt. Soc. Am. B.

Willemsen, M. B.

M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
[CrossRef]

Woerdman, J. P.

M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
[CrossRef]

Yamada, M.

M. Yamada and T. Higashi, IEEE J. Quantum Electron. 27, 380 (1991).
[CrossRef]

Appl. Phys. Lett. (1)

M. B. Willemsen, A. S. van de Nes, M. P. van Exter, J. P. Woerdman, M. Brunner, and R. H. Hövel, Appl. Phys. Lett. 77, 3514 (2000).
[CrossRef]

IEEE J. Quantum Electron. (4)

T. W. Carr and T. Erneux, IEEE J. Quantum Electron. 37, 1171 (2001), and references therein.
[CrossRef]

P. Rees, P. McEvoy, A. Valle, J. O’Gorman, S. Lynch, P. Landais, L. Pesquera, and J. Hegasky, IEEE J. Quantum Electron. 35, 221 (1999).
[CrossRef]

C. Juang, T. M. Hwang, J. Juang, and W. W. Lin, IEEE J. Quantum Electron. 36, 300 (2000).
[CrossRef]

M. Yamada and T. Higashi, IEEE J. Quantum Electron. 27, 380 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Buus, IEEE Photon. Technol. Lett. 6, 1179 (1994).
[CrossRef]

Opt. Commun. (1)

J. Danckaert, B. Nagler, J. Albert, K. Panajotov, I. Veretennicoff, and T. Erneux, Opt. Commun. 201, 129 (2002).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

M. San Miguel, Q. Feng, and J. V. Moloney, Phys. Rev. A 52, 1728 (1995).
[CrossRef] [PubMed]

J. Mulet and S. Balle, Phys. Rev. A 66, 053802 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

A. Scirè, J. Mulet, C. R. Mirasso, J. Danckaert, and M. San Miguel, Phys. Rev. Lett. 90, 113901 (2003).
[CrossRef]

Other (4)

C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985).

The intensity-dependent confinement factor Γ can be found by solving n22p2u3+2n0n2V2 pu2+V2n02-na2u-n02-na2=0 for u=1-Γ, where V is the normalized frequency and na (n0) represents the refractive index in the unpumped (core) region. We found that α≈2n0n2/V4Γ0n02-na2, while β≈Γ0α/Γsat-Γ0, with Γsat being the saturated confinement factor.

M. Peeters, MODEL: The Modest Ordinary Differential Equation Library, C++ framework, available at http://model.sourceforge.net (2000–2003).

K. Panajotov, G. Van der Sande, H. Thienpont, and I. Veretennicoff have submitted a paper called “Self-pulsation in VCSELs as a result of the interplay between carrier-induced antiguiding and build-in index guiding” to J. Opt. Soc. Am. B.

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Figures (3)

Fig. 1
Fig. 1

Hopf bifurcation points as a function of α and for different values of β (ρ=10-3, g=7.0, Γ0=0.6).

Fig. 2
Fig. 2

Bifurcation diagram of Eqs. (1) and (2) with injected current j as the bifurcation parameter (ρ=10-3, g=7.0, α=0.015, β=2.0, Γ0=0.6, βc=-5.0).

Fig. 3
Fig. 3

(a)–(c) Optical spectra (OS) as calculated from Eqs. (1) and (2) for increasing current. (d)–(f) RIN spectra as calculated from Eqs. (1) and (2) for increasing current. From left to right: j=0.27, j=0.45, j=0.48) (ρ=10-3, g=7.0, α=0.015, β=2.0, Γ0=0.6, βc=-5.0, Rsp=10-4.

Equations (7)

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

ρdEdt=1+iβc2gΓE2n-1E+F˜E,
dndt=j-n-ΓE2nE2,
Γ=Γ01+αE21+βE2,
np=1gΓp,
jp=1gΓ01+αp/1+βp+pg.
pdΓpdp-ρΓp-ρpΓ2p=0.
α2ρΓ0p31+βp2-αp1+βp2-ρp1+2pΓ01+βp+ρ1+pΓ0=0.

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