Abstract

Based on a linear chain composed of a central semiconductor laser and two outer semiconductor lasers, chaos synchronization and bidirectional communication between two outer lasers have been investigated under the case that the central laser and the two outer lasers are coupled mutually, whereas there exists no coupling between the two outer lasers. The simulation results show that high-quality and stable isochronal synchronization between the two outer lasers can be achieved, while the cross-correlation coefficients between the two outer lasers and the central laser are very low under proper operation condition. Based on the high performance chaos synchronization between the two outer lasers, message bidirectional transmissions of bit rates up to 20 Gbit/s can be realized through adopting a novel decoding scheme which is different from that based on chaos pass filtering effect. Furthermore, the security of bidirectional communication is also analyzed.

© 2011 OSA

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    [Crossref]
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    [Crossref] [PubMed]
  29. R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066202 (2008).
    [Crossref] [PubMed]
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    [Crossref]
  33. J. M. Liu, H. F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit rates,” IEEE J. Quantum Electron. 38(9), 1184–1196 (2002).
    [Crossref]
  34. T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
    [Crossref] [PubMed]
  35. J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. 40(6), 640–650 (2004).
    [Crossref]
  36. D. Kanakidis, A. Argyris, and D. Syvridis, “Performance characterization of high-bit-rate optical chaotic communication systems in a back-to-back configuration,” J. Lightwave Technol. 21(3), 750–758 (2003).
    [Crossref]
  37. R. Mislovaty, E. Klein, I. Kanter, and W. Kinzel, “Public channel cryptography by synchronization of neural networks and chaotic maps,” Phys. Rev. Lett. 91(11), 118701 (2003).
    [Crossref] [PubMed]

2011 (3)

A. Jafari, H. Sedghi, Kh. Mabhouti, and S. Behnia, “Slave-master dynamics of semiconductor laser with short external cavity,” Opt. Commun. 284(12), 3018–3029 (2011).
[Crossref]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

T. Deng, G. Q. Xia, Z. M. Wu, X. D. Lin, and J. G. Wu, “Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents,” Opt. Express 19(9), 8762–8773 (2011).
[Crossref] [PubMed]

2010 (3)

N. Jiang, W. Pan, L. Yan, B. Luo, W. L. Zhang, S. Y. Xiang, L. Yang, and D. Zheng, “Chaos synchronization and communication in mutually coupled semiconductor lasers driven by a third laser,” J. Lightwave Technol. 28(13), 1978–1986 (2010).
[Crossref]

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

V. Annovazzi-Lodi, G. Aromataris, M. Benedetti, and S. Merlo, “Private message transmission by common driving of two chaotic lasers,” IEEE J. Quantum Electron. 46(2), 258–264 (2010).
[Crossref]

2009 (5)

2008 (2)

W. L. Zhang, W. Pan, B. Luo, X. H. Zou, M. Y. Wang, and Z. Zhou, “Chaos synchronization communication using extremely unsymmetrical bidirectional injections,” Opt. Lett. 33(3), 237–239 (2008).
[Crossref] [PubMed]

R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066202 (2008).
[Crossref] [PubMed]

2007 (3)

2006 (2)

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

2005 (5)

M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback,” IEEE J. Quantum Electron. 41(11), 1333–1340 (2005).
[Crossref]

E. Klein, R. Mislovaty, I. Kanter, and W. Kinzel, “Public-channel cryptography using chaos synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016214 (2005).
[Crossref] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005).
[Crossref] [PubMed]

R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005).
[Crossref]

2004 (3)

J. Paul, M. W. Lee, and K. A. Shore, “Effect of chaos pass filtering on message decoding quality using chaotic external-cavity laser diodes,” Opt. Lett. 29(21), 2497–2499 (2004).
[Crossref] [PubMed]

J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. 40(6), 640–650 (2004).
[Crossref]

A. Uchida, R. McAllister, and R. Roy, “Consistency of nonlinear system response to complex drive signals,” Phys. Rev. Lett. 93(24), 244102 (2004).
[Crossref] [PubMed]

2003 (3)

C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003).
[Crossref] [PubMed]

D. Kanakidis, A. Argyris, and D. Syvridis, “Performance characterization of high-bit-rate optical chaotic communication systems in a back-to-back configuration,” J. Lightwave Technol. 21(3), 750–758 (2003).
[Crossref]

R. Mislovaty, E. Klein, I. Kanter, and W. Kinzel, “Public channel cryptography by synchronization of neural networks and chaotic maps,” Phys. Rev. Lett. 91(11), 118701 (2003).
[Crossref] [PubMed]

2002 (1)

J. M. Liu, H. F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit rates,” IEEE J. Quantum Electron. 38(9), 1184–1196 (2002).
[Crossref]

2001 (2)

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

S. Tang and J. M. Liu, “Message encoding-decoding at 2.5 Gbits/s through synchronization of chaotic pulsing semiconductor lasers,” Opt. Lett. 26(23), 1843–1845 (2001).
[Crossref] [PubMed]

1998 (2)

R. V. Jensen, “Synchronization of randomly driven nonlinear oscillators,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(6), R6907–R6910 (1998).
[Crossref]

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref] [PubMed]

1994 (1)

I. Fischer, O. Hess, W. Elsaβer, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

1990 (1)

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
[Crossref] [PubMed]

Allaria, E.

C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003).
[Crossref] [PubMed]

Annovazzi-Lodi, V.

V. Annovazzi-Lodi, G. Aromataris, M. Benedetti, and S. Merlo, “Private message transmission by common driving of two chaotic lasers,” IEEE J. Quantum Electron. 46(2), 258–264 (2010).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Arecchi, F. T.

C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003).
[Crossref] [PubMed]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

D. Kanakidis, A. Argyris, and D. Syvridis, “Performance characterization of high-bit-rate optical chaotic communication systems in a back-to-back configuration,” J. Lightwave Technol. 21(3), 750–758 (2003).
[Crossref]

Ariizumi, H.

Aromataris, G.

V. Annovazzi-Lodi, G. Aromataris, M. Benedetti, and S. Merlo, “Private message transmission by common driving of two chaotic lasers,” IEEE J. Quantum Electron. 46(2), 258–264 (2010).
[Crossref]

Aviad, Y.

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

Behnia, S.

A. Jafari, H. Sedghi, Kh. Mabhouti, and S. Behnia, “Slave-master dynamics of semiconductor laser with short external cavity,” Opt. Commun. 284(12), 3018–3029 (2011).
[Crossref]

Benedetti, M.

V. Annovazzi-Lodi, G. Aromataris, M. Benedetti, and S. Merlo, “Private message transmission by common driving of two chaotic lasers,” IEEE J. Quantum Electron. 46(2), 258–264 (2010).
[Crossref]

Boccaletti, S.

C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003).
[Crossref] [PubMed]

Buldú, J. M.

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. 40(6), 640–650 (2004).
[Crossref]

Butkovski, M.

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

Cao, L. P.

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
[Crossref] [PubMed]

Chen, H. F.

M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback,” IEEE J. Quantum Electron. 41(11), 1333–1340 (2005).
[Crossref]

J. M. Liu, H. F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit rates,” IEEE J. Quantum Electron. 38(9), 1184–1196 (2002).
[Crossref]

Chen, J. G.

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Chiang, M. C.

M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback,” IEEE J. Quantum Electron. 41(11), 1333–1340 (2005).
[Crossref]

Colet, P.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005).
[Crossref]

Daudén, J.

R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005).
[Crossref]

Davis, P.

Deng, T.

T. Deng, G. Q. Xia, Z. M. Wu, X. D. Lin, and J. G. Wu, “Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents,” Opt. Express 19(9), 8762–8773 (2011).
[Crossref] [PubMed]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Elsaßer, W.

I. Fischer, O. Hess, W. Elsaβer, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

Elsässer, W.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

Englert, A.

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

Feng, G. Y.

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

Fischer, I.

R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066202 (2008).
[Crossref] [PubMed]

R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007).
[Crossref] [PubMed]

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

I. Fischer, O. Hess, W. Elsaβer, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

García-Ojalvo, J.

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. 40(6), 640–650 (2004).
[Crossref]

Göbel, E.

I. Fischer, O. Hess, W. Elsaβer, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

Goto, S. I.

Gross, N.

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

Heil, T.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

Hess, O.

I. Fischer, O. Hess, W. Elsaβer, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

Hong, Y. H.

Jafari, A.

A. Jafari, H. Sedghi, Kh. Mabhouti, and S. Behnia, “Slave-master dynamics of semiconductor laser with short external cavity,” Opt. Commun. 284(12), 3018–3029 (2011).
[Crossref]

Jensen, R. V.

R. V. Jensen, “Synchronization of randomly driven nonlinear oscillators,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(6), R6907–R6910 (1998).
[Crossref]

Jiang, N.

Kanakidis, D.

Kanter, I.

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

E. Klein, R. Mislovaty, I. Kanter, and W. Kinzel, “Public-channel cryptography using chaos synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016214 (2005).
[Crossref] [PubMed]

R. Mislovaty, E. Klein, I. Kanter, and W. Kinzel, “Public channel cryptography by synchronization of neural networks and chaotic maps,” Phys. Rev. Lett. 91(11), 118701 (2003).
[Crossref] [PubMed]

Khaykovich, L.

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

Kinzel, W.

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

E. Klein, R. Mislovaty, I. Kanter, and W. Kinzel, “Public-channel cryptography using chaos synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016214 (2005).
[Crossref] [PubMed]

R. Mislovaty, E. Klein, I. Kanter, and W. Kinzel, “Public channel cryptography by synchronization of neural networks and chaotic maps,” Phys. Rev. Lett. 91(11), 118701 (2003).
[Crossref] [PubMed]

Klein, E.

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

E. Klein, R. Mislovaty, I. Kanter, and W. Kinzel, “Public-channel cryptography using chaos synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016214 (2005).
[Crossref] [PubMed]

R. Mislovaty, E. Klein, I. Kanter, and W. Kinzel, “Public channel cryptography by synchronization of neural networks and chaotic maps,” Phys. Rev. Lett. 91(11), 118701 (2003).
[Crossref] [PubMed]

Kopelowitz, E.

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

Kurths, J.

C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003).
[Crossref] [PubMed]

Larger, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Lee, M. W.

Li, M.

Lin, X. D.

T. Deng, G. Q. Xia, Z. M. Wu, X. D. Lin, and J. G. Wu, “Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents,” Opt. Express 19(9), 8762–8773 (2011).
[Crossref] [PubMed]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Liu, J.

Liu, J. M.

M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback,” IEEE J. Quantum Electron. 41(11), 1333–1340 (2005).
[Crossref]

J. M. Liu, H. F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit rates,” IEEE J. Quantum Electron. 38(9), 1184–1196 (2002).
[Crossref]

S. Tang and J. M. Liu, “Message encoding-decoding at 2.5 Gbits/s through synchronization of chaotic pulsing semiconductor lasers,” Opt. Lett. 26(23), 1843–1845 (2001).
[Crossref] [PubMed]

Luo, B.

Mabhouti, Kh.

A. Jafari, H. Sedghi, Kh. Mabhouti, and S. Behnia, “Slave-master dynamics of semiconductor laser with short external cavity,” Opt. Commun. 284(12), 3018–3029 (2011).
[Crossref]

McAllister, R.

A. Uchida, R. McAllister, and R. Roy, “Consistency of nonlinear system response to complex drive signals,” Phys. Rev. Lett. 93(24), 244102 (2004).
[Crossref] [PubMed]

Merlo, S.

V. Annovazzi-Lodi, G. Aromataris, M. Benedetti, and S. Merlo, “Private message transmission by common driving of two chaotic lasers,” IEEE J. Quantum Electron. 46(2), 258–264 (2010).
[Crossref]

Meucci, R.

C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003).
[Crossref] [PubMed]

Mirasso, C. R.

R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066202 (2008).
[Crossref] [PubMed]

R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007).
[Crossref] [PubMed]

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

Mislovaty, R.

E. Klein, R. Mislovaty, I. Kanter, and W. Kinzel, “Public-channel cryptography using chaos synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016214 (2005).
[Crossref] [PubMed]

R. Mislovaty, E. Klein, I. Kanter, and W. Kinzel, “Public channel cryptography by synchronization of neural networks and chaotic maps,” Phys. Rev. Lett. 91(11), 118701 (2003).
[Crossref] [PubMed]

Mulet, J.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

Muramatsu, J.

Oowada, I.

Pan, W.

Paul, J.

Pecora, L. M.

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
[Crossref] [PubMed]

Peil, M.

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Reidler, I.

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

Rosenbluh, M.

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

Roy, R.

B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
[Crossref] [PubMed]

A. Uchida, R. McAllister, and R. Roy, “Consistency of nonlinear system response to complex drive signals,” Phys. Rev. Lett. 93(24), 244102 (2004).
[Crossref] [PubMed]

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref] [PubMed]

Sedghi, H.

A. Jafari, H. Sedghi, Kh. Mabhouti, and S. Behnia, “Slave-master dynamics of semiconductor laser with short external cavity,” Opt. Commun. 284(12), 3018–3029 (2011).
[Crossref]

Shore, K. A.

Spencer, P. S.

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

D. Kanakidis, A. Argyris, and D. Syvridis, “Performance characterization of high-bit-rate optical chaotic communication systems in a back-to-back configuration,” J. Lightwave Technol. 21(3), 750–758 (2003).
[Crossref]

Tang, S.

J. M. Liu, H. F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit rates,” IEEE J. Quantum Electron. 38(9), 1184–1196 (2002).
[Crossref]

S. Tang and J. M. Liu, “Message encoding-decoding at 2.5 Gbits/s through synchronization of chaotic pulsing semiconductor lasers,” Opt. Lett. 26(23), 1843–1845 (2001).
[Crossref] [PubMed]

Tang, X.

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

Toral, R.

R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005).
[Crossref]

Torrent, M. C.

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. 40(6), 640–650 (2004).
[Crossref]

Uchida, A.

VanWiggeren, G. D.

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref] [PubMed]

Vicente, R.

R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066202 (2008).
[Crossref] [PubMed]

R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007).
[Crossref] [PubMed]

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005).
[Crossref]

Wang, M. Y.

Wu, J. G.

Wu, Z. M.

Xia, G. Q.

Xiang, S. Y.

Yamamoto, T.

Yan, L.

Yang, L.

Yip, H.

Yoshimori, S.

Yoshimura, K.

Zhang, W. L.

Zheng, D.

Zhou, B. B.

B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
[Crossref] [PubMed]

Zhou, C. S.

C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003).
[Crossref] [PubMed]

Zhou, Z.

Zigzag, M.

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

Zou, X. H.

IEEE J. Quantum Electron. (5)

R. Vicente, J. Daudén, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41(4), 541–548 (2005).
[Crossref]

M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback,” IEEE J. Quantum Electron. 41(11), 1333–1340 (2005).
[Crossref]

V. Annovazzi-Lodi, G. Aromataris, M. Benedetti, and S. Merlo, “Private message transmission by common driving of two chaotic lasers,” IEEE J. Quantum Electron. 46(2), 258–264 (2010).
[Crossref]

J. M. Liu, H. F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit rates,” IEEE J. Quantum Electron. 38(9), 1184–1196 (2002).
[Crossref]

J. M. Buldú, J. García-Ojalvo, and M. C. Torrent, “Multimode synchronization and communication using unidirectionally coupled semiconductor lasers,” IEEE J. Quantum Electron. 40(6), 640–650 (2004).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

J. Lightwave Technol. (3)

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Opt. Commun. (2)

A. Jafari, H. Sedghi, Kh. Mabhouti, and S. Behnia, “Slave-master dynamics of semiconductor laser with short external cavity,” Opt. Commun. 284(12), 3018–3029 (2011).
[Crossref]

T. Deng, G. Q. Xia, L. P. Cao, J. G. Chen, X. D. Lin, and Z. M. Wu, “Bidirectional chaos synchronization and communication in semiconductor lasers with optoelectronic feedback,” Opt. Commun. 282(11), 2243–2249 (2009).
[Crossref]

Opt. Express (6)

Opt. Lett. (4)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (5)

B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
[Crossref] [PubMed]

E. Klein, R. Mislovaty, I. Kanter, and W. Kinzel, “Public-channel cryptography using chaos synchronization,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016214 (2005).
[Crossref] [PubMed]

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

R. Vicente, I. Fischer, and C. R. Mirasso, “Synchronization properties of three delay-coupled semiconductor lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(6), 066202 (2008).
[Crossref] [PubMed]

C. S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F. T. Arecchi, “Constructive effects of noise in homoclinic chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67(6), 066220 (2003).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

R. V. Jensen, “Synchronization of randomly driven nonlinear oscillators,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(6), R6907–R6910 (1998).
[Crossref]

Phys. Rev. Lett. (7)

A. Uchida, R. McAllister, and R. Roy, “Consistency of nonlinear system response to complex drive signals,” Phys. Rev. Lett. 93(24), 244102 (2004).
[Crossref] [PubMed]

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, C. R. Mirasso, M. C. Torrent, and J. García-Ojalvo, “Zero-lag long-range synchronization via dynamical relaying,” Phys. Rev. Lett. 97(12), 123902 (2006).
[Crossref] [PubMed]

A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104(11), 114102 (2010).
[Crossref] [PubMed]

R. Mislovaty, E. Klein, I. Kanter, and W. Kinzel, “Public channel cryptography by synchronization of neural networks and chaotic maps,” Phys. Rev. Lett. 91(11), 118701 (2003).
[Crossref] [PubMed]

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001).
[Crossref] [PubMed]

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824 (1990).
[Crossref] [PubMed]

I. Fischer, O. Hess, W. Elsaβer, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

Science (1)

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998).
[Crossref] [PubMed]

Other (1)

J. Ohtsubo, Semiconductor Lasers, Stability, Instability and Chaos, 2nd ed. (Springer-Verlag, 2008).

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Figures (9)

Fig. 1
Fig. 1

Schematic of system setup. OSL: outer semiconductor laser; CSL: central semiconductor laser; M: mirror; BS: beam splitter; OI: optical isolator; PD: photodetector; F: communication channel; m: message.

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Fig. 2
Fig. 2

Maximum of C1C (a) and C12 (b) as a function of K1C and KC1.

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Fig. 3
Fig. 3

Time series (left volume), power spectrum (middle volume) and cross-correlation coefficient (right volume).

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Fig. 4
Fig. 4

(a) Maximum of the cross-correlation coefficient between arbitrary pairs of three SLs as a function of the coupling time mismatch Δτ; (b)-(d) Cross-correlation function between arbitrary pairs of three SLs under τc1 = τ1c = 3ns and τc2 = τ2c = 5ns.

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Fig. 5
Fig. 5

Maximums of cross-correlation coefficient versus three typical internal parameter mismatches between arbitrary two SLs, where (a) is for OSL1 and OSL2; (b) is for OSL1 and CSL; and (c) is for OSL2 and CSL. The circle, triangular, and square denote mismatched ε, g and α, respectively.

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Fig. 6
Fig. 6

Maximum of cross-correlation coefficient versus three typical internal parameter mismatches between CSL and OSLs, where (a) is for OSL1 and OSL2, (b) is for OSL1 and CSL, and (c) is for OSL2 and CSL. The circle, triangular and square denote mismatched ε, g and α, respectively.

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Fig. 7
Fig. 7

Illustrations of bidirectional message transmission. (a1)-(b1): the original messages; (a2)-(b2): chaotic time series with messages; (a3)-(b3): message difference; (a4)-(b4): recovered messages; (c) and (d) are the power spectra corresponding to the time series shown in (a2) and (b2), respectively.

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Fig. 8
Fig. 8

(a): Q-factor as a function of bit rate of message; (b)-(d): eye diagrams of message with 5Gbits/s, 10Gbits/s and 15Gbits/s, respectively.

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Fig. 9
Fig. 9

The signals extracted from two different communication channels by an eavesdropper, where (a) is for the case that both of channels include messages; and (b) is for the case that one channel includes message but the other channel does not.

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Equations (8)

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d E 1,2 ( t ) dt = 1 2 ( 1+i α 1,2 )( G 1,2 − 1 τ P ) E 1,2 ( t ) + K C1,C2 E C ( t− τ C1,C2 )exp( −i2π f C τ C1,C2 )exp( i2πΔ f C1,C2 t )+ 2β N 1,2 χ 1,2
d N 1,2 ( t ) dt = I e − N 1,2 ( t ) τ e − G 1,2 | E 1,2 ( t ) | 2
d E C ( t ) dt = 1 2 ( 1+i α C )( G C − 1 τ P ) E C ( t )+ K 1C E 1 ( t− τ 1C )exp( −i2π f 1 τ 1C )exp( −i2πΔ f C1 t ) + K 2C E 2 ( t− τ 2C )exp( −i2π f 2 τ 2C )exp( −i2πΔ f C2 t )+ 2β N C χ C
d N C ( t ) dt = I e − N C ( t ) τ e − G C | E C ( t ) | 2
C ij (Δt)= 〈[ P i (t)−〈 P i 〉][ P j (t+Δt)−〈 P j 〉]〉 {〈 [ P i (t)−〈 P i 〉] 2 〉〈 [ P j (t)−〈 P j 〉] 2 〉} 1/2
Q= 〈 P 1 〉−〈 P 0 〉 σ 1 + σ 0
Δε= ε 1 − ε 2 ε1 , Δα= α 1 − α 2 α1 , Δg= g 1 − g 2 g1
Δ ε ′ = ε 1 − ε c ε 1 , Δ α ′ = α 1 − α c α 1 , Δ g ′ = g 1 − g c g 1

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