Abstract

The chaotic unpredictability of vertical-cavity surface-emitting lasers (VCSELs) with polarization-rotated optical feedback is evaluated quantitatively via the normalized permutation entropy H. The effect of polarization mode competition (PMC) is explored. For weak PMC, the H for X-polarization mode, Y-polarization mode, and total output are close to each other, while for strong PMC, the H for the total output is the highest one, which indicates that the permutation entropy is an effective tool for quantifying chaotic unpredictability of VCSELs and provides valuable information for choosing the proper chaotic carrier.

© 2011 Optical Society of America

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2010 (3)

S. Xiang, W. Pan, L. Yan, B. Luo, N. Jiang, K. Wen, X. Zou, and L. Yang, J. Opt. Soc. Am. B 27, 476 (2010).
[CrossRef]

M. C. Soriano, L. Zunino, O. A. Rosso, and C. R. Mirasso, Proc. SPIE 7720, 77202G (2010).
[CrossRef]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[CrossRef]

2009 (2)

2008 (2)

2007 (2)

2006 (1)

2005 (3)

R. Vicente, J. Daudén, P. Colet, and R. Toral, IEEE J. Quantum Electron. 41, 541 (2005).
[CrossRef]

R. Ju, P. S. Spencer, and K. A. Shore, IEEE J. Quantum Electron. 41, 1461 (2005).
[CrossRef]

C. Masoller and M. S. Torre, IEEE J. Quantum Electron. 41, 483 (2005).
[CrossRef]

2003 (3)

M. Sciamanna, K. Panajotov, H. Thienpont, I. Veretennicoff, P. Mégret, and M. Blondel, Opt. Lett. 28, 1543 (2003).
[CrossRef] [PubMed]

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

M. T. Martin, A. Plastino, and O. A. Rosso, Phys. Lett. A 311, 126 (2003).
[CrossRef]

2002 (2)

C. Bandt, G. Keller, and B. Pompe, Nonlinearity 15, 1595 (2002).
[CrossRef]

C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
[CrossRef] [PubMed]

1997 (1)

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

1995 (1)

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

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

Bandt, C.

C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
[CrossRef] [PubMed]

C. Bandt, G. Keller, and B. Pompe, Nonlinearity 15, 1595 (2002).
[CrossRef]

Blondel, M.

Colet, P.

R. Vicente, J. Daudén, P. Colet, and R. Toral, IEEE J. Quantum Electron. 41, 541 (2005).
[CrossRef]

Cuenot, J. B.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Daudén, J.

R. Vicente, J. Daudén, P. Colet, and R. Toral, IEEE J. Quantum Electron. 41, 541 (2005).
[CrossRef]

Feng, Q.

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

Fischer, I.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[CrossRef]

Gatare, I.

Goedgebuer, J. P.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Hong, Y.

Jiang, N.

Ju, R.

R. Ju, P. S. Spencer, and K. A. Shore, IEEE J. Quantum Electron. 41, 1461 (2005).
[CrossRef]

Kane, D. M.

Keller, G.

C. Bandt, G. Keller, and B. Pompe, Nonlinearity 15, 1595 (2002).
[CrossRef]

Larger, L.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Lee, M. W.

Levy, P.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Liu, J.

Locquet, A.

Luo, B.

Martin, M. T.

M. T. Martin, A. Plastino, and O. A. Rosso, Phys. Lett. A 311, 126 (2003).
[CrossRef]

Martin-Regalado, J.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

Masoller, C.

Mégret, P.

Miguel, M. San

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

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

Mirasso, C. R.

M. C. Soriano, L. Zunino, O. A. Rosso, and C. R. Mirasso, Proc. SPIE 7720, 77202G (2010).
[CrossRef]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[CrossRef]

O. A. Rosso, R. Vicente, and C. R. Mirasso, Phys. Lett. A 372, 1018 (2008).
[CrossRef]

Moloney, J. V.

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

Pan, W.

Panajotov, K.

Paul, J.

Plastino, A.

M. T. Martin, A. Plastino, and O. A. Rosso, Phys. Lett. A 311, 126 (2003).
[CrossRef]

Pompe, B.

C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
[CrossRef] [PubMed]

C. Bandt, G. Keller, and B. Pompe, Nonlinearity 15, 1595 (2002).
[CrossRef]

Prati, F.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

Rhodes, W. T.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Rosso, O. A.

M. C. Soriano, L. Zunino, O. A. Rosso, and C. R. Mirasso, Proc. SPIE 7720, 77202G (2010).
[CrossRef]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[CrossRef]

O. A. Rosso, R. Vicente, and C. R. Mirasso, Phys. Lett. A 372, 1018 (2008).
[CrossRef]

M. T. Martin, A. Plastino, and O. A. Rosso, Phys. Lett. A 311, 126 (2003).
[CrossRef]

Sciamanna, M.

Shore, K. A.

Soriano, M. C.

M. C. Soriano, L. Zunino, O. A. Rosso, and C. R. Mirasso, Proc. SPIE 7720, 77202G (2010).
[CrossRef]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[CrossRef]

Spencer, P. S.

Thienpont, H.

Toomey, J. P.

Toral, R.

R. Vicente, J. Daudén, P. Colet, and R. Toral, IEEE J. Quantum Electron. 41, 541 (2005).
[CrossRef]

Torre, M. S.

C. Masoller and M. S. Torre, IEEE J. Quantum Electron. 41, 483 (2005).
[CrossRef]

Udaltsov, V. S.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Veretennicoff, I.

Vicente, R.

O. A. Rosso, R. Vicente, and C. R. Mirasso, Phys. Lett. A 372, 1018 (2008).
[CrossRef]

R. Vicente, J. Daudén, P. Colet, and R. Toral, IEEE J. Quantum Electron. 41, 541 (2005).
[CrossRef]

Wen, K.

Wu, Z. M.

Xia, G. Q.

Xiang, S.

Yan, L.

Yang, L.

Zou, X.

Zunino, L.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[CrossRef]

M. C. Soriano, L. Zunino, O. A. Rosso, and C. R. Mirasso, Proc. SPIE 7720, 77202G (2010).
[CrossRef]

IEEE J. Quantum Electron. (4)

R. Ju, P. S. Spencer, and K. A. Shore, IEEE J. Quantum Electron. 41, 1461 (2005).
[CrossRef]

C. Masoller and M. S. Torre, IEEE J. Quantum Electron. 41, 483 (2005).
[CrossRef]

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, IEEE J. Quantum Electron. 33, 765 (1997).
[CrossRef]

R. Vicente, J. Daudén, P. Colet, and R. Toral, IEEE J. Quantum Electron. 41, 541 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (2)

Nonlinearity (1)

C. Bandt, G. Keller, and B. Pompe, Nonlinearity 15, 1595 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (3)

M. T. Martin, A. Plastino, and O. A. Rosso, Phys. Lett. A 311, 126 (2003).
[CrossRef]

O. A. Rosso, R. Vicente, and C. R. Mirasso, Phys. Lett. A 372, 1018 (2008).
[CrossRef]

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, Phys. Lett. A 308, 54 (2003).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. E (1)

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, Phys. Rev. E 82, 046212 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002).
[CrossRef] [PubMed]

Proc. SPIE (1)

M. C. Soriano, L. Zunino, O. A. Rosso, and C. R. Mirasso, Proc. SPIE 7720, 77202G (2010).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

PE as a function of k pr , (a) τ pr = 3 ns , (b) τ pr = 5 ns , (c) τ pr = 7 ns . The solid curves with circles, stars, and triangles correspond to H X , H Y , and H T , respectively.

Fig. 2
Fig. 2

Outputs in the time domain for a VCSEL with k pr = 10 ns 1 and τ pr = 5 ns (left column) and with k pr = 25 ns 1 and τ pr = 7 ns (right column), (a) and (d) XP mode, (b) and (e) YP mode, (c) and (f) total output.

Fig. 3
Fig. 3

2D maps of PE in the parameter space of k pr and τ pr , (a) H X , (b) H Y , (c) H T .

Fig. 4
Fig. 4

2D maps of PE in the parameter space of k pr and μ for a VCSEL with τ pr = 5 ns , (a) H X , (b) H Y , (c) H T .

Equations (5)

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d E x , y d t = k ( 1 + i α ) [ ( N 1 ) E x , y ± i n E y , x ] ( γ a + i γ p ) E x , y + β sp ξ x , y + k pr E y , x ( t τ pr ) e i ω τ pr ,
d N d t = γ N [ μ N ( 1 + | E x | 2 + | E y | 2 ) + i n ( E x E y * E y E x * ) ] ,
d n d t = γ s n γ N [ n ( | E x | 2 + | E y | 2 ) + i N ( E y E x * E x E y * ) ] ,
p ( π ) = # { t | t T d + 1 ; X t     has type π } T d + 1 ,
h [ P ] = p ( π ) log p ( π ) .

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