Abstract

We have explored theoretically and experimentally the steady-state transverse light fields emitted from ring-shaped lasers, specifically, those from vertical-cavity surface-emitting lasers (VCSELs). We saw the switching on of patterns of increasing spatial frequencies as a function of pump parameter. Furthermore, we were able to identify the mechanism for such an evolution as geometrical modulational instability within the nonlinear cavity. Other mechanisms such as the conventional gain–loss balance had no effect on the ring configuration that was modeled. The experiments with annular VCSELs gave results that matched the theoretical predictions well, although other mechanisms not considered in our model, such as carrier diffusion, took place in the experimental devices. We conclude that the nonlinear mechanisms presented here can be regarded as limiting cases in the interpretation of more-complex functions, such as patterns, modes, and filamentation switching, in VCSELs.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
  28. J. Scheuer and B. A. Malomed, “Stable and chaotic solutions of the complex Ginzburg–Landau equation with periodical boundary conditions,” Physica D 161, 102–115 (2002).
    [CrossRef]
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    [CrossRef]

2002 (1)

J. Scheuer and B. A. Malomed, “Stable and chaotic solutions of the complex Ginzburg–Landau equation with periodical boundary conditions,” Physica D 161, 102–115 (2002).
[CrossRef]

2001 (1)

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

1999 (1)

J. Scheuer and M. Orenstein, “Optical vortices crystals—spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef] [PubMed]

1998 (2)

T. Sh. Misirpashaev and C. W. J. Beenakker, “Lasing threshold and mode competition in chaotic cavities,” Phys. Rev. A 57, 2041–2045 (1998).
[CrossRef]

L. Djaloshinsky and M. Orenstein, “Coupling of concentric semiconductor microring lasers,” Opt. Lett. 23, 364–366 (1998).
[CrossRef]

1997 (1)

V. N. Morozov, J. A. Neff, and Z. Haijun, “Analysis of vertical-cavity surface-emitting laser multimode behavior,” IEEE J. Quantum Electron. 33, 980–988 (1997).
[CrossRef]

1996 (3)

G. Huyet, C. Mathis, and J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

J. R. Maricante and G. P. Agrawal, “Nonlinear mechanisms of filamentation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 32, 590–596 (1996).
[CrossRef]

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

1995 (2)

B. J. Flanigan and J. E. Carroll, “Mode selection in complex-coupled semiconductor DFB lasers,” Electron. Lett. 31, 977–979 (1995).
[CrossRef]

G. Chen, “A comparative study on the thermal characteristics of vertical-cavity surface-emitting lasers,” J. Appl. Phys. 77, 4251–4258 (1995).
[CrossRef]

1994 (3)

W. J. Firth and A. J. Scroggie, “Spontaneous pattern formation in an absorptive system,” Europhys. Lett. 26, 521–523 (1994).
[CrossRef]

G. Huyet, C. Mathis, H. Grassi, J. R. Tredicce, and N. B. Abraham, “Regarding standing versus traveling waves in the transverse spatial patterns of homogeneously and inhomogeneously broadened lasers,” Opt. Commun. 111, 488–492 (1994).
[CrossRef]

D. L. Huffaker, D. G. Deppe, and T. J. Rogers, “Transverse mode behavior in native-oxide-defined low threshold vertical-cavity lasers,” Appl. Phys. Lett. 65, 1611–1613 (1994).
[CrossRef]

1993 (5)

J. W. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldern, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

Q. Feng, J. V. Moloney, and A. C. Newell, “Amplitude instabilities of transverse traveling waves in lasers,” Phys. Rev. Lett. 71, 1705–1708 (1993).
[CrossRef] [PubMed]

E. Pampaloni, S. Residori, and F. T. Arecchi, “Roll-hexagon transition in a Kerr-like experiment,” Europhys. Lett. 24, 647–652 (1993).
[CrossRef]

R. J. Lang, D. Mehuys, A. Hardy, K. D. Dzurko, and D. F. Welch, “Spatial evolution of filaments in broad area diode laser amplifiers,” Appl. Phys. Lett. 62, 1209–1211 (1993).
[CrossRef]

H. Adachihara, O. Hess, E. Abraham, P. Ru, and J. V. Moloney, “Spatiotemporal chaos in broad-area semiconductor lasers,” J. Opt. Soc. Am. B 10, 658–665 (1993).
[CrossRef]

1992 (3)

G. DG. D’ Alessandro and W. J. Firth, “Hexagonal spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537–548 (1992).
[CrossRef] [PubMed]

E. J. D’Angelo, E. Izaguirre, G. B. Mindin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

1991 (1)

G. L. Oppo, G. D’Alessandro, and W. J. Firth, “Spatiotemporal instabilities of lasers in models reduced via center manifold techniques,” Phys. Rev. A 44, 4712–4720 (1991).
[CrossRef] [PubMed]

1990 (2)

C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: the experimental side,” Phys. Rev. Lett. 65, 3124–3127 (1990).
[CrossRef] [PubMed]

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

Abraham, E.

Abraham, N. B.

G. Huyet, C. Mathis, H. Grassi, J. R. Tredicce, and N. B. Abraham, “Regarding standing versus traveling waves in the transverse spatial patterns of homogeneously and inhomogeneously broadened lasers,” Opt. Commun. 111, 488–492 (1994).
[CrossRef]

Adachihara, H.

Agrawal, G. P.

J. R. Maricante and G. P. Agrawal, “Nonlinear mechanisms of filamentation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 32, 590–596 (1996).
[CrossRef]

Alessandro, G. DG. D’

G. DG. D’ Alessandro and W. J. Firth, “Hexagonal spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537–548 (1992).
[CrossRef] [PubMed]

Arecchi, F. T.

E. Pampaloni, S. Residori, and F. T. Arecchi, “Roll-hexagon transition in a Kerr-like experiment,” Europhys. Lett. 24, 647–652 (1993).
[CrossRef]

Beenakker, C. W. J.

T. Sh. Misirpashaev and C. W. J. Beenakker, “Lasing threshold and mode competition in chaotic cavities,” Phys. Rev. A 57, 2041–2045 (1998).
[CrossRef]

Beva, G. P.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Brunner, M.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Carroll, J. E.

B. J. Flanigan and J. E. Carroll, “Mode selection in complex-coupled semiconductor DFB lasers,” Electron. Lett. 31, 977–979 (1995).
[CrossRef]

Chang-Hasnain, C.

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

Chen, G.

G. Chen, “A comparative study on the thermal characteristics of vertical-cavity surface-emitting lasers,” J. Appl. Phys. 77, 4251–4258 (1995).
[CrossRef]

Choquette, K. D.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Clausen, E.

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

Coldern, L. A.

J. W. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldern, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

Corzine, S. W.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

J. W. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldern, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

D’Alessandro, G.

G. L. Oppo, G. D’Alessandro, and W. J. Firth, “Spatiotemporal instabilities of lasers in models reduced via center manifold techniques,” Phys. Rev. A 44, 4712–4720 (1991).
[CrossRef] [PubMed]

D’Angelo, E. J.

E. J. D’Angelo, E. Izaguirre, G. B. Mindin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: the experimental side,” Phys. Rev. Lett. 65, 3124–3127 (1990).
[CrossRef] [PubMed]

Debernardi, P.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Degen, C.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Deppe, D. G.

D. L. Huffaker, D. G. Deppe, and T. J. Rogers, “Transverse mode behavior in native-oxide-defined low threshold vertical-cavity lasers,” Appl. Phys. Lett. 65, 1611–1613 (1994).
[CrossRef]

Djaloshinsky, L.

Dzurko, K. D.

R. J. Lang, D. Mehuys, A. Hardy, K. D. Dzurko, and D. F. Welch, “Spatial evolution of filaments in broad area diode laser amplifiers,” Appl. Phys. Lett. 62, 1209–1211 (1993).
[CrossRef]

Elsässen, W.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Feng, Q.

Q. Feng, J. V. Moloney, and A. C. Newell, “Amplitude instabilities of transverse traveling waves in lasers,” Phys. Rev. Lett. 71, 1705–1708 (1993).
[CrossRef] [PubMed]

Firth, W. J.

W. J. Firth and A. J. Scroggie, “Spontaneous pattern formation in an absorptive system,” Europhys. Lett. 26, 521–523 (1994).
[CrossRef]

G. DG. D’ Alessandro and W. J. Firth, “Hexagonal spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537–548 (1992).
[CrossRef] [PubMed]

G. L. Oppo, G. D’Alessandro, and W. J. Firth, “Spatiotemporal instabilities of lasers in models reduced via center manifold techniques,” Phys. Rev. A 44, 4712–4720 (1991).
[CrossRef] [PubMed]

Fischer, L.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Flanigan, B. J.

B. J. Flanigan and J. E. Carroll, “Mode selection in complex-coupled semiconductor DFB lasers,” Electron. Lett. 31, 977–979 (1995).
[CrossRef]

Florez, L. T.

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

Fratta, L.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Geels, R. S.

J. W. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldern, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

Gil, L.

E. J. D’Angelo, E. Izaguirre, G. B. Mindin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Grassi, H.

G. Huyet, C. Mathis, H. Grassi, J. R. Tredicce, and N. B. Abraham, “Regarding standing versus traveling waves in the transverse spatial patterns of homogeneously and inhomogeneously broadened lasers,” Opt. Commun. 111, 488–492 (1994).
[CrossRef]

Green, C.

C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: the experimental side,” Phys. Rev. Lett. 65, 3124–3127 (1990).
[CrossRef] [PubMed]

Gulden, K.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Hadley, G. R.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Haijun, Z.

V. N. Morozov, J. A. Neff, and Z. Haijun, “Analysis of vertical-cavity surface-emitting laser multimode behavior,” IEEE J. Quantum Electron. 33, 980–988 (1997).
[CrossRef]

Hardy, A.

R. J. Lang, D. Mehuys, A. Hardy, K. D. Dzurko, and D. F. Welch, “Spatial evolution of filaments in broad area diode laser amplifiers,” Appl. Phys. Lett. 62, 1209–1211 (1993).
[CrossRef]

Harison, J. P.

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

Hess, O.

Hövel, R.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Huffaker, D. L.

D. L. Huffaker, D. G. Deppe, and T. J. Rogers, “Transverse mode behavior in native-oxide-defined low threshold vertical-cavity lasers,” Appl. Phys. Lett. 65, 1611–1613 (1994).
[CrossRef]

Huyet, G.

G. Huyet, C. Mathis, and J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

G. Huyet, C. Mathis, H. Grassi, J. R. Tredicce, and N. B. Abraham, “Regarding standing versus traveling waves in the transverse spatial patterns of homogeneously and inhomogeneously broadened lasers,” Opt. Commun. 111, 488–492 (1994).
[CrossRef]

E. J. D’Angelo, E. Izaguirre, G. B. Mindin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Indik, R.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Izaguirre, E.

E. J. D’Angelo, E. Izaguirre, G. B. Mindin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Jakobsen, P. K.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Lang, R. J.

R. J. Lang, D. Mehuys, A. Hardy, K. D. Dzurko, and D. F. Welch, “Spatial evolution of filaments in broad area diode laser amplifiers,” Appl. Phys. Lett. 62, 1209–1211 (1993).
[CrossRef]

Lear, K. L.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Malomed, B. A.

J. Scheuer and B. A. Malomed, “Stable and chaotic solutions of the complex Ginzburg–Landau equation with periodical boundary conditions,” Physica D 161, 102–115 (2002).
[CrossRef]

Maricante, J. R.

J. R. Maricante and G. P. Agrawal, “Nonlinear mechanisms of filamentation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 32, 590–596 (1996).
[CrossRef]

Mathis, C.

G. Huyet, C. Mathis, and J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

G. Huyet, C. Mathis, H. Grassi, J. R. Tredicce, and N. B. Abraham, “Regarding standing versus traveling waves in the transverse spatial patterns of homogeneously and inhomogeneously broadened lasers,” Opt. Commun. 111, 488–492 (1994).
[CrossRef]

Mehuys, D.

R. J. Lang, D. Mehuys, A. Hardy, K. D. Dzurko, and D. F. Welch, “Spatial evolution of filaments in broad area diode laser amplifiers,” Appl. Phys. Lett. 62, 1209–1211 (1993).
[CrossRef]

Mindin, G. B.

E. J. D’Angelo, E. Izaguirre, G. B. Mindin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Mindlin, G. B.

C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: the experimental side,” Phys. Rev. Lett. 65, 3124–3127 (1990).
[CrossRef] [PubMed]

Misirpashaev, T. Sh.

T. Sh. Misirpashaev and C. W. J. Beenakker, “Lasing threshold and mode competition in chaotic cavities,” Phys. Rev. A 57, 2041–2045 (1998).
[CrossRef]

Moloney, J. V.

Q. Feng, J. V. Moloney, and A. C. Newell, “Amplitude instabilities of transverse traveling waves in lasers,” Phys. Rev. Lett. 71, 1705–1708 (1993).
[CrossRef] [PubMed]

H. Adachihara, O. Hess, E. Abraham, P. Ru, and J. V. Moloney, “Spatiotemporal chaos in broad-area semiconductor lasers,” J. Opt. Soc. Am. B 10, 658–665 (1993).
[CrossRef]

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Morozov, V. N.

V. N. Morozov, J. A. Neff, and Z. Haijun, “Analysis of vertical-cavity surface-emitting laser multimode behavior,” IEEE J. Quantum Electron. 33, 980–988 (1997).
[CrossRef]

Moser, M.

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

Neff, J. A.

V. N. Morozov, J. A. Neff, and Z. Haijun, “Analysis of vertical-cavity surface-emitting laser multimode behavior,” IEEE J. Quantum Electron. 33, 980–988 (1997).
[CrossRef]

Newell, A. C.

Q. Feng, J. V. Moloney, and A. C. Newell, “Amplitude instabilities of transverse traveling waves in lasers,” Phys. Rev. Lett. 71, 1705–1708 (1993).
[CrossRef] [PubMed]

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

Oppo, G. L.

G. L. Oppo, G. D’Alessandro, and W. J. Firth, “Spatiotemporal instabilities of lasers in models reduced via center manifold techniques,” Phys. Rev. A 44, 4712–4720 (1991).
[CrossRef] [PubMed]

Orenstein, M.

J. Scheuer and M. Orenstein, “Optical vortices crystals—spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef] [PubMed]

L. Djaloshinsky and M. Orenstein, “Coupling of concentric semiconductor microring lasers,” Opt. Lett. 23, 364–366 (1998).
[CrossRef]

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

Pampaloni, E.

E. Pampaloni, S. Residori, and F. T. Arecchi, “Roll-hexagon transition in a Kerr-like experiment,” Europhys. Lett. 24, 647–652 (1993).
[CrossRef]

Residori, S.

E. Pampaloni, S. Residori, and F. T. Arecchi, “Roll-hexagon transition in a Kerr-like experiment,” Europhys. Lett. 24, 647–652 (1993).
[CrossRef]

Rogers, T. J.

D. L. Huffaker, D. G. Deppe, and T. J. Rogers, “Transverse mode behavior in native-oxide-defined low threshold vertical-cavity lasers,” Appl. Phys. Lett. 65, 1611–1613 (1994).
[CrossRef]

Ru, P.

Scheuer, J.

J. Scheuer and B. A. Malomed, “Stable and chaotic solutions of the complex Ginzburg–Landau equation with periodical boundary conditions,” Physica D 161, 102–115 (2002).
[CrossRef]

J. Scheuer and M. Orenstein, “Optical vortices crystals—spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef] [PubMed]

Scott, J. W.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

J. W. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldern, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

Scroggie, A. J.

W. J. Firth and A. J. Scroggie, “Spontaneous pattern formation in an absorptive system,” Europhys. Lett. 26, 521–523 (1994).
[CrossRef]

Solari, H. G.

C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: the experimental side,” Phys. Rev. Lett. 65, 3124–3127 (1990).
[CrossRef] [PubMed]

Stoffel, N. G.

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

Tredicce, J. R.

G. Huyet, C. Mathis, and J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

G. Huyet, C. Mathis, H. Grassi, J. R. Tredicce, and N. B. Abraham, “Regarding standing versus traveling waves in the transverse spatial patterns of homogeneously and inhomogeneously broadened lasers,” Opt. Commun. 111, 488–492 (1994).
[CrossRef]

E. J. D’Angelo, E. Izaguirre, G. B. Mindin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: the experimental side,” Phys. Rev. Lett. 65, 3124–3127 (1990).
[CrossRef] [PubMed]

Von Lehmen, A. C.

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

Warren, M. E.

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

Welch, D. F.

R. J. Lang, D. Mehuys, A. Hardy, K. D. Dzurko, and D. F. Welch, “Spatial evolution of filaments in broad area diode laser amplifiers,” Appl. Phys. Lett. 62, 1209–1211 (1993).
[CrossRef]

Appl. Phys. Lett. (3)

D. L. Huffaker, D. G. Deppe, and T. J. Rogers, “Transverse mode behavior in native-oxide-defined low threshold vertical-cavity lasers,” Appl. Phys. Lett. 65, 1611–1613 (1994).
[CrossRef]

M. Orenstein, A. C. Von Lehmen, C. Chang-Hasnain, N. G. Stoffel, L. T. Florez, J. P. Harison, and E. Clausen, “Vertical cavity surface emitting InGaAs/GaAs lasers with planar lateral definition,” Appl. Phys. Lett. 56, 2384–2386 (1990).
[CrossRef]

R. J. Lang, D. Mehuys, A. Hardy, K. D. Dzurko, and D. F. Welch, “Spatial evolution of filaments in broad area diode laser amplifiers,” Appl. Phys. Lett. 62, 1209–1211 (1993).
[CrossRef]

Electron. Lett. (1)

B. J. Flanigan and J. E. Carroll, “Mode selection in complex-coupled semiconductor DFB lasers,” Electron. Lett. 31, 977–979 (1995).
[CrossRef]

Europhys. Lett. (2)

W. J. Firth and A. J. Scroggie, “Spontaneous pattern formation in an absorptive system,” Europhys. Lett. 26, 521–523 (1994).
[CrossRef]

E. Pampaloni, S. Residori, and F. T. Arecchi, “Roll-hexagon transition in a Kerr-like experiment,” Europhys. Lett. 24, 647–652 (1993).
[CrossRef]

IEEE J. Quantum Electron. (4)

J. R. Maricante and G. P. Agrawal, “Nonlinear mechanisms of filamentation in broad-area semiconductor lasers,” IEEE J. Quantum Electron. 32, 590–596 (1996).
[CrossRef]

V. N. Morozov, J. A. Neff, and Z. Haijun, “Analysis of vertical-cavity surface-emitting laser multimode behavior,” IEEE J. Quantum Electron. 33, 980–988 (1997).
[CrossRef]

G. R. Hadley, K. L. Lear, M. E. Warren, K. D. Choquette, J. W. Scott, and S. W. Corzine, “Comprehensive numerical modeling of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 607–616 (1996).
[CrossRef]

J. W. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldern, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE J. Quantum Electron. 29, 1295–1308 (1993).
[CrossRef]

J. Appl. Phys. (1)

G. Chen, “A comparative study on the thermal characteristics of vertical-cavity surface-emitting lasers,” J. Appl. Phys. 77, 4251–4258 (1995).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

G. Huyet, C. Mathis, and J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

G. Huyet, C. Mathis, H. Grassi, J. R. Tredicce, and N. B. Abraham, “Regarding standing versus traveling waves in the transverse spatial patterns of homogeneously and inhomogeneously broadened lasers,” Opt. Commun. 111, 488–492 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (5)

G. DG. D’ Alessandro and W. J. Firth, “Hexagonal spatial patterns for a Kerr slice with a feedback mirror,” Phys. Rev. A 46, 537–548 (1992).
[CrossRef] [PubMed]

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8137 (1992).
[CrossRef] [PubMed]

G. L. Oppo, G. D’Alessandro, and W. J. Firth, “Spatiotemporal instabilities of lasers in models reduced via center manifold techniques,” Phys. Rev. A 44, 4712–4720 (1991).
[CrossRef] [PubMed]

C. Degen, L. Fischer, W. Elsässen, L. Fratta, P. Debernardi, G. P. Beva, M. Brunner, R. Hövel, M. Moser, and K. Gulden, “Transverse modes in thermally detuned oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A 63, 023817 (2001).
[CrossRef]

T. Sh. Misirpashaev and C. W. J. Beenakker, “Lasing threshold and mode competition in chaotic cavities,” Phys. Rev. A 57, 2041–2045 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

C. Green, G. B. Mindlin, E. J. D’Angelo, H. G. Solari, and J. R. Tredicce, “Spontaneous symmetry breaking in a laser: the experimental side,” Phys. Rev. Lett. 65, 3124–3127 (1990).
[CrossRef] [PubMed]

E. J. D’Angelo, E. Izaguirre, G. B. Mindin, G. Huyet, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett. 68, 3702–3705 (1992).
[CrossRef]

Q. Feng, J. V. Moloney, and A. C. Newell, “Amplitude instabilities of transverse traveling waves in lasers,” Phys. Rev. Lett. 71, 1705–1708 (1993).
[CrossRef] [PubMed]

Physica D (1)

J. Scheuer and B. A. Malomed, “Stable and chaotic solutions of the complex Ginzburg–Landau equation with periodical boundary conditions,” Physica D 161, 102–115 (2002).
[CrossRef]

Science (1)

J. Scheuer and M. Orenstein, “Optical vortices crystals—spontaneous generation in nonlinear semiconductor microcavities,” Science 285, 230–233 (1999).
[CrossRef] [PubMed]

Other (4)

H. Li, T. L. Lucas, J. G. McInerney, and R. A. Morgan, “Transverse modes and patterns in electrically pumped vertical cavity surface-emitting semiconductor lasers,” in Conference on Lasers and Electro-Optics CLEO/Europe 1994 (Optical Society of America, Washington, D.C., 1994), pp. 197–198.

J. V. Moloney and A. C. Newell, “Spatio-temporal structures in wide aperture lasers,” in Nonlinear Dynamics and Spatial Complexity in Optical Systems, R. G. Harrison and J. S. Uppal, eds. (Institute of Physics, Bristol, UK, 1992), pp. 197–216.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

A. Valle, L. Pesquera, P. Rees, and A. Shore, “Transverse mode selection and light-current characteristics in index-guided VCSELs,” in Digest of the LEOS Summer Topical Meetings (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1999), pp. III53–III54.

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Figures (9)

Fig. 1
Fig. 1

Schematic of the structure of the VCSEL. DBR, distributed Bragg reflector.

Fig. 2
Fig. 2

VCSEL geometry.

Fig. 3
Fig. 3

Evolution of the electrical field at various pump rates: A, g0=0.01004; B, g0=0.01005; C, g0=0.01008; D, g0=0.01012. The VCSEL parameters are R=2, Isat=20, αtot=0.01, n0=1.45, L=185 µm, and λ=1 µm.

Fig. 4
Fig. 4

Comparison of numerical results (solid curves) and the superposition approximation (circles). A, g0=0.01008, B, g0=0.01012. The other parameters are as in Fig. 3. BPM, beam-propagation method.

Fig. 5
Fig. 5

MI gain curve for the m=1 mode with L=1, R=2, and γ=2π2.

Fig. 6
Fig. 6

Excitation of the first-order mode while the fundamental mode is lasing. Parameters are same as for Fig. 4.

Fig. 7
Fig. 7

Experimental setup.

Fig. 8
Fig. 8

Measured near-field patterns of ring-shaped VCSELs. Dimeters, 28 and 40 µm. A, quasi-uniform distribution; B, uniform distribution. n, number of lobes.

Fig. 9
Fig. 9

Number of intensity lobes as a function of injection current.

Equations (22)

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

Eˆ(x, r, z, t)=Re{E(x, z)R(r)exp[i(βz-ωt)]},
2E(x, z)x2+2iβE(x, z)z+[k02neff2(x)-β2]E(x, z)=0.
neff2(x)=n02+2n0ΓΔn(x, N)+in0k0[αtot-Γg(x, N)],
Δn(x, N)=-R2k0g(x, N),
2E(x, z)x2+2iβE(x, z)z-ik0n0[Γg(x, N)(1-iR)-αtot]E(x, z)+[k02n02-β2]E(x, z)=0.
E(x, z)z-i2k0n02E(x, z)x2-12[Γg(x, N)(1-iR)-αtot]E(x, z)=0.
N(x)-Ntr=Np-Ntr1+|E|2/Isat,
g(N, x)=Γα(Np-Ntr)1+|E|2/Isat=g0(Np)1+|E|2/Isat,
Ez-i2k0n02Ex2-12g0(1-iR)1+|E|2/Isat-αtotE=0.
Ez-i2Ex2-n0k0g0(1-iR)1+|E|2-αtotE=0.
E02=g0αtot-1,η=n0g0Rk0(1+E02)+km2,
km=m2π/(k0L),m=0, 1, 2 ,
E=A0 cos(2πm/Lx),
L=k0L,m=1, 2 ,.
z0L|E|2dx=0Lg01+|E|2|E|2dx-αtot 0L|E|2dx=0.
gm=0Lg01+|En|2|Em|2dx=g00Lcos2[(2π/L)mx+φ]dx1+A02 cos2[(2π/L)nx]=g0L21+q=1(-1)qA02q22q2qq+g0LL cos(2φ)q=1(-1)p+qA02(p+q)22(p+q)2(p+q)qm=pn0else.
E(x, z)=E0 exp[i(kmx-ηz)]×{1+μ(z)exp[i(Ω-km)x]+c.c.},
km<Ω<(km2+2γR)1/2,
γ=n0αtotk0g0(g0-αtot).
g0(m)=αtot1-[2π2(2m+1)/n0k0RL2αtot].
Ith(m)Ith(0)=1+2π2n0k0RL2Γα(2m+1).
Nmax=1+n0k0RL2αtot2π2.

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