Abstract

We demonstrate that polarization control in combination with the geometrical shape affects wave chaos phenomena in vertical-cavity surface-emitting lasers (VCSELs). In a grating-polarization-stabilized VCSEL, we demonstrate wave chaos in one polarization mode and a transition from wave chaos to regular behavior in the other orthogonal polarization mode. We quantify the amount of chaoticity via the Brody nearest-neighbor distribution. Comparing with a reference VCSEL without a surface-grating, we explain these results and their injection current dependence by the surface-grating-induced mode selection and a change in the effective resonator shape.

© 2011 Optical Society of America

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  1. T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
    [CrossRef] [PubMed]
  2. K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
    [CrossRef] [PubMed]
  3. S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
    [CrossRef]
  4. H.-J. Stöckmann and J. Stein, Phys. Rev. Lett. 64, 2215(1990).
    [CrossRef] [PubMed]
  5. J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, IEEE Photon. Technol. Lett. 17, 1593 (2005).
    [CrossRef]
  6. C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
    [CrossRef]
  7. A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
    [CrossRef]
  8. M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
    [CrossRef]
  9. P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
    [CrossRef]
  10. L. Reichl, The Transition to Chaos (Springer, 2004).
  11. H.-J. Stöckmann, Quantum Chaos—An Introduction, (Cambridge U. Press, 1999).
  12. P. J. Richens and M. V. Berry, Physica D 2, 495 (1981).
    [CrossRef]
  13. S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
    [CrossRef]
  14. J. W. Goodman, Statistical Optics (Wiley, 2000).
  15. C. Degen, I. Fischer, and W. Elsäßer, Opt. Express 5, 38(1999).
    [CrossRef] [PubMed]

2011

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
[CrossRef]

2010

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
[CrossRef]

2007

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

2006

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
[CrossRef]

2005

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, IEEE Photon. Technol. Lett. 17, 1593 (2005).
[CrossRef]

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

2002

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

1999

1990

H.-J. Stöckmann and J. Stein, Phys. Rev. Lett. 64, 2215(1990).
[CrossRef] [PubMed]

1981

P. J. Richens and M. V. Berry, Physica D 2, 495 (1981).
[CrossRef]

Ackemann, T.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

Arteaga, M. A.

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
[CrossRef]

Bava, G. P.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

Becker, K.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

Berry, M. V.

P. J. Richens and M. V. Berry, Physica D 2, 495 (1981).
[CrossRef]

Bittner, S.

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

Blazek, M.

A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
[CrossRef]

Bogomolny, E.

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

Chen, Y. F.

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Debernardi, P.

A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, IEEE Photon. Technol. Lett. 17, 1593 (2005).
[CrossRef]

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

Degen, C.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

C. Degen, I. Fischer, and W. Elsäßer, Opt. Express 5, 38(1999).
[CrossRef] [PubMed]

Dietz, B.

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

Elsäßer, W.

A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
[CrossRef]

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

C. Degen, I. Fischer, and W. Elsäßer, Opt. Express 5, 38(1999).
[CrossRef] [PubMed]

Fischer, I.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

C. Degen, I. Fischer, and W. Elsäßer, Opt. Express 5, 38(1999).
[CrossRef] [PubMed]

Fuchs, C.

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

Fukushima, T.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
[CrossRef]

Gensty, T.

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Haglund, A.

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

Harayama, T.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
[CrossRef]

Hentschel, M.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
[CrossRef]

Huang, K. F.

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Jalics, C.

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, IEEE Photon. Technol. Lett. 17, 1593 (2005).
[CrossRef]

Lai, H. C.

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Lan, Y. P.

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Larsson, A.

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

Michalzik, R.

A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
[CrossRef]

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, IEEE Photon. Technol. Lett. 17, 1593 (2005).
[CrossRef]

Miski-Oglu, M.

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

Molitor, A.

A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
[CrossRef]

Narimanov, E. E.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
[CrossRef]

Oria Iriarte, P.

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

Ostermann, J. M.

A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
[CrossRef]

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
[CrossRef]

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, IEEE Photon. Technol. Lett. 17, 1593 (2005).
[CrossRef]

Panajotov, K.

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
[CrossRef]

Reichl, L.

L. Reichl, The Transition to Chaos (Springer, 2004).

Richens, P. J.

P. J. Richens and M. V. Berry, Physica D 2, 495 (1981).
[CrossRef]

Richter, A.

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

Schäfer, F.

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

Shinohara, S.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
[CrossRef]

Sondermann, M.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

Stein, J.

H.-J. Stöckmann and J. Stein, Phys. Rev. Lett. 64, 2215(1990).
[CrossRef] [PubMed]

Stöckmann, H.-J.

H.-J. Stöckmann and J. Stein, Phys. Rev. Lett. 64, 2215(1990).
[CrossRef] [PubMed]

H.-J. Stöckmann, Quantum Chaos—An Introduction, (Cambridge U. Press, 1999).

Sunada, S.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
[CrossRef]

Thienpont, H.

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
[CrossRef]

Unold, H. J.

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
[CrossRef]

IEEE J. Quantum Electron.

C. Fuchs, T. Gensty, P. Debernardi, G. P. Bava, J. M. Ostermann, R. Michalzik, A. Haglund, A. Larsson, and W. Elsäßer, IEEE J. Quantum Electron. 43, 1227 (2007).
[CrossRef]

A. Molitor, M. Blazek, J. M. Ostermann, R. Michalzik, P. Debernardi, and W. Elsäßer, IEEE J. Quantum Electron. 46, 554 (2010).
[CrossRef]

M. A. Arteaga, H. J. Unold, J. M. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

P. Debernardi, J. M. Ostermann, M. Sondermann, T. Ackemann, G. P. Bava, and R. Michalzik, IEEE J. Sel. Top. Quantum Electron. 13, 1340 (2007).
[CrossRef]

IEEE Photon. Technol. Lett.

J. M. Ostermann, P. Debernardi, C. Jalics, and R. Michalzik, IEEE Photon. Technol. Lett. 17, 1593 (2005).
[CrossRef]

Opt. Express

Phys. Rev. A

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. E. Narimanov, Phys. Rev. A 83, 053837(2011).
[CrossRef]

Phys. Rev. E

S. Bittner, E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, Phys. Rev. E 81, 066215(2010).
[CrossRef]

Phys. Rev. Lett.

H.-J. Stöckmann and J. Stein, Phys. Rev. Lett. 64, 2215(1990).
[CrossRef] [PubMed]

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, Phys. Rev. Lett. 94, 233901(2005).
[CrossRef] [PubMed]

K. F. Huang, Y. F. Chen, H. C. Lai, and Y. P. Lan, Phys. Rev. Lett. 89, 224102 (2002).
[CrossRef] [PubMed]

Physica D

P. J. Richens and M. V. Berry, Physica D 2, 495 (1981).
[CrossRef]

Other

J. W. Goodman, Statistical Optics (Wiley, 2000).

L. Reichl, The Transition to Chaos (Springer, 2004).

H.-J. Stöckmann, Quantum Chaos—An Introduction, (Cambridge U. Press, 1999).

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Figures (3)

Fig. 1
Fig. 1

Left scale: Total power (solid curve), P-polarization power (solid triangles) and S-polarization power (open squares) together with the corresponding OPSR (right scale) of a surface-grating VCSEL (dashed–dotted curve) and a reference VCSEL (dotted curve).

Fig. 2
Fig. 2

(a) Probability density distribution P ( s ) (histogram bins) as a function of the spacing interval s for the S polarization at 12 mA together with the Poisson (solid curve) and Wigner distributions (dashed curve), respectively. Inset: part of an optical spectrum with labeled resonances. (b) Nearest-neighbor spacing in the integrated form I ( s ) as a function of s. Experimental data (black squares) and Poisson (solid curve), Wigner (dashed curve), and Brody distributions with a Brody parameter of w = 0.4 (crosses).

Fig. 3
Fig. 3

Brody parameter w for the P polarization (solid triangles), the S polarization (open squares) of a surface-grating, and a reference VCSEL (dotted curve) as a function of pump current. The two horizontal lines depict the limiting cases of pure regular (solid curve) and pure chaotic behavior (dashed curve). Inset: orientation of both orthogonal polarizations in relation to the grating grooves.

Equations (3)

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

s n = E n E n 1 / E n E n 1 .
I ( s ) = 0 s P ( s ) d s .
I Brody ( s ) = 1 exp ( Γ ( ( w + 2 ) / ( w + 1 ) ) ( w + 1 ) * s ( w + 1 ) ) .

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