Abstract

Experiments have yielded polarization self-modulation in vertical-cavity surface-emitting lasers (VCSELs) subject to a π/2 polarization-rotating optical feedback. The phenomenon has been simulated numerically, but its bifurcation has never been explained. We show that polarization self-modulation results from a Hopf bifurcation mechanism that can be analyzed in terms of the laser feedback parameters. Our analysis predicts other bifurcations for low values of the feedback rate, which explain why more-complex time-dependent outputs have been observed as alternatives to polarization self-modulation.

© 2002 Optical Society of America

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Errata

M. Sciamanna, F. Rogister, O. Deparis, P. Mégret, M. Blondel, and T. Erneux, "Bifurcation to polarization self-modulation in vertical-cavity surface-emitting lasers: errata," Opt. Lett. 27, 875-875 (2002)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-27-10-875

References

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1999

C. Masoller and N. B. Abraham, Appl. Phys. Lett. 74, 1078 (1999).
[CrossRef]

1998

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, Appl. Phys. Lett. 72, 2355 (1998).
[CrossRef]

C. L. Tang, J. Opt. B 10, R51 (1998).

1997

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, IEEE J. Quantum Electron. 33, 2231 (1997).
[CrossRef]

1996

A. K. Jansen van Doorn, M. P. Van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

P. M. Alsing, V. Kovanis, A. Gavrielides, and T. Erneux, Phys. Rev. A 53, 4429 (1996).
[CrossRef] [PubMed]

1995

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

1994

N. Badr, I. H. White, M. R. T. Tan, Y. M. Young, and S. Y. Wang, Electron. Lett. 30, 1227 (1994).
[CrossRef]

1993

S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, Appl. Phys. Lett. 63, 3545 (1993).
[CrossRef]

1992

J. Mork, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

1990

W. H. Loh, Y. Ozeki, and C. L. Tang, Appl. Phys. Lett. 56, 2613 (1990).
[CrossRef]

1987

Y. C. Chen and J. M. Liu, Appl. Phys. Lett. 50, 1406 (1987).
[CrossRef]

Abraham, N. B.

C. Masoller and N. B. Abraham, Appl. Phys. Lett. 74, 1078 (1999).
[CrossRef]

Alsing, P. M.

P. M. Alsing, V. Kovanis, A. Gavrielides, and T. Erneux, Phys. Rev. A 53, 4429 (1996).
[CrossRef] [PubMed]

Badr, N.

N. Badr, I. H. White, M. R. T. Tan, Y. M. Young, and S. Y. Wang, Electron. Lett. 30, 1227 (1994).
[CrossRef]

Besnard, P.

F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, IEEE J. Quantum Electron. 33, 2231 (1997).
[CrossRef]

Bretenaker, F.

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

Brunel, M.

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

Chares, M. L.

F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, IEEE J. Quantum Electron. 33, 2231 (1997).
[CrossRef]

Chen, Y. C.

Y. C. Chen and J. M. Liu, Appl. Phys. Lett. 50, 1406 (1987).
[CrossRef]

Choquette, K. D.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, Appl. Phys. Lett. 72, 2355 (1998).
[CrossRef]

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

Dagenais, M.

S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, Appl. Phys. Lett. 63, 3545 (1993).
[CrossRef]

Erneux, T.

P. M. Alsing, V. Kovanis, A. Gavrielides, and T. Erneux, Phys. Rev. A 53, 4429 (1996).
[CrossRef] [PubMed]

Feng, Q.

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

Gavrielides, A.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, Appl. Phys. Lett. 72, 2355 (1998).
[CrossRef]

P. M. Alsing, V. Kovanis, A. Gavrielides, and T. Erneux, Phys. Rev. A 53, 4429 (1996).
[CrossRef] [PubMed]

Hohl, A.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, Appl. Phys. Lett. 72, 2355 (1998).
[CrossRef]

Hou, H.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, Appl. Phys. Lett. 72, 2355 (1998).
[CrossRef]

Jansen van Doorn, A. K.

A. K. Jansen van Doorn, M. P. Van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

Jiang, S.

S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, Appl. Phys. Lett. 63, 3545 (1993).
[CrossRef]

Kojima, K.

S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, Appl. Phys. Lett. 63, 3545 (1993).
[CrossRef]

Kovanis, V.

P. M. Alsing, V. Kovanis, A. Gavrielides, and T. Erneux, Phys. Rev. A 53, 4429 (1996).
[CrossRef] [PubMed]

Langot, P.

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

Le Floch, A.

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

Li, H.

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, Appl. Phys. Lett. 72, 2355 (1998).
[CrossRef]

Liu, J. M.

Y. C. Chen and J. M. Liu, Appl. Phys. Lett. 50, 1406 (1987).
[CrossRef]

Loh, W. H.

W. H. Loh, Y. Ozeki, and C. L. Tang, Appl. Phys. Lett. 56, 2613 (1990).
[CrossRef]

Mark, J.

J. Mork, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Masoller, C.

C. Masoller and N. B. Abraham, Appl. Phys. Lett. 74, 1078 (1999).
[CrossRef]

Moloney, J. V.

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

Morgan, R. A.

S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, Appl. Phys. Lett. 63, 3545 (1993).
[CrossRef]

Mork, J.

J. Mork, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Ozeki, Y.

W. H. Loh, Y. Ozeki, and C. L. Tang, Appl. Phys. Lett. 56, 2613 (1990).
[CrossRef]

Pan, Z.

S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, Appl. Phys. Lett. 63, 3545 (1993).
[CrossRef]

Robert, F.

F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, IEEE J. Quantum Electron. 33, 2231 (1997).
[CrossRef]

Ropars, G.

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

San Miguel, M.

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

Stephan, G. M.

F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, IEEE J. Quantum Electron. 33, 2231 (1997).
[CrossRef]

Tan, M. R. T.

N. Badr, I. H. White, M. R. T. Tan, Y. M. Young, and S. Y. Wang, Electron. Lett. 30, 1227 (1994).
[CrossRef]

Tang, C. L.

C. L. Tang, J. Opt. B 10, R51 (1998).

W. H. Loh, Y. Ozeki, and C. L. Tang, Appl. Phys. Lett. 56, 2613 (1990).
[CrossRef]

Tromborg, B.

J. Mork, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

Vallet, M.

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

Van Exter, M. P.

A. K. Jansen van Doorn, M. P. Van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

Wang, S. Y.

N. Badr, I. H. White, M. R. T. Tan, Y. M. Young, and S. Y. Wang, Electron. Lett. 30, 1227 (1994).
[CrossRef]

White, I. H.

N. Badr, I. H. White, M. R. T. Tan, Y. M. Young, and S. Y. Wang, Electron. Lett. 30, 1227 (1994).
[CrossRef]

Woerdman, J. P.

A. K. Jansen van Doorn, M. P. Van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

Young, Y. M.

N. Badr, I. H. White, M. R. T. Tan, Y. M. Young, and S. Y. Wang, Electron. Lett. 30, 1227 (1994).
[CrossRef]

Appl. Phys. Lett.

A. K. Jansen van Doorn, M. P. Van Exter, and J. P. Woerdman, Appl. Phys. Lett. 69, 1041 (1996).
[CrossRef]

S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, Appl. Phys. Lett. 63, 3545 (1993).
[CrossRef]

H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, Appl. Phys. Lett. 72, 2355 (1998).
[CrossRef]

G. Ropars, P. Langot, M. Brunel, M. Vallet, F. Bretenaker, A. Le Floch, and K. D. Choquette, Appl. Phys. Lett. 70, 2661 (1997).
[CrossRef]

C. Masoller and N. B. Abraham, Appl. Phys. Lett. 74, 1078 (1999).
[CrossRef]

W. H. Loh, Y. Ozeki, and C. L. Tang, Appl. Phys. Lett. 56, 2613 (1990).
[CrossRef]

Y. C. Chen and J. M. Liu, Appl. Phys. Lett. 50, 1406 (1987).
[CrossRef]

Electron. Lett.

N. Badr, I. H. White, M. R. T. Tan, Y. M. Young, and S. Y. Wang, Electron. Lett. 30, 1227 (1994).
[CrossRef]

IEEE J. Quantum Electron.

F. Robert, P. Besnard, M. L. Chares, and G. M. Stephan, IEEE J. Quantum Electron. 33, 2231 (1997).
[CrossRef]

J. Mork, B. Tromborg, and J. Mark, IEEE J. Quantum Electron. 28, 93 (1992).
[CrossRef]

J. Opt. B

C. L. Tang, J. Opt. B 10, R51 (1998).

Phys. Rev. A

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

P. M. Alsing, V. Kovanis, A. Gavrielides, and T. Erneux, Phys. Rev. A 53, 4429 (1996).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Bifurcation diagram of the extrema of Ix versus η. The delay is θ=100, and the other parameters are specified in the text. The labels indicate specific dynamic states that are described in the text.

Fig. 2
Fig. 2

Time traces of Ix and Iy for η=0.04. The values of the other parameters are as in Fig. 1.

Fig. 3
Fig. 3

Same as in Fig. 1 but for θ=140.

Fig. 4
Fig. 4

(a), (b) Time traces of Ix and Iy for η=0.00564 and η=0.0068, respectively. The values of the other parameters are as in Fig. 3.

Equations (5)

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

dExds= 121+iα1+2ZFx-1Ex+ηEys-θexp-iϕf,
dExds= 121+iα1+2ZFx-1Ex+ηExs-θexp-iϕf,
TdZds=P-Z-1+2ZFxEx2+FyEy2,
Fx=1-xxEx2-P2-xyEy2-P2,
Fy=1-yxEx2-P2-yyEy2-P2.

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