Abstract

We present a new optoelectronic architecture intended for chaotic optical intensity generation. The principle relies on an electro-optic nonlinear delay dynamics, where the nonlinearity originates from an integrated four-wave optical interferometer, involving two independent electro-optic modulation inputs. Consequently, the setup involves both two-dimensional nonlinearity and dual-delay feedback dynamics, which results in enhanced chaos complexity of particular interest in chaos encryption schemes. The generated chaos observed with large feedback gains has a bandwidth ranging from 30kHz to 13GHz and is confirmed by numerical simulations of the proposed dynamical model and bifurcation diagram calculation.

© 2011 Optical Society of America

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  1. J. P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
    [CrossRef]
  2. G. D. Van Wiggeren and R. Roy, Science 279, 1198 (1998).
    [CrossRef]
  3. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
    [CrossRef]
  4. K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
    [CrossRef] [PubMed]
  5. Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, Phys. Rev. Lett. 95, 203903 (2005).
    [CrossRef] [PubMed]
  6. K. Ikeda, Opt. Commun. 30, 257 (1979).
    [CrossRef]
  7. J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002).
    [CrossRef]
  8. T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
    [CrossRef]
  9. M. Nourine, M. Peil, and L. Larger, in C. R. Rencontres du Non Linéaire (2009), Vol.  12, 149.
  10. J. D. Farmer, Physica D 4, 366 (1982).
    [CrossRef]
  11. Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, IEEE J. Quantum Electron. 41, 156 (2005).
    [CrossRef]

2010 (1)

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

2006 (1)

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

2005 (3)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, Phys. Rev. Lett. 95, 203903 (2005).
[CrossRef] [PubMed]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, IEEE J. Quantum Electron. 41, 156 (2005).
[CrossRef]

2002 (1)

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002).
[CrossRef]

1998 (2)

J. P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

G. D. Van Wiggeren and R. Roy, Science 279, 1198 (1998).
[CrossRef]

1982 (1)

J. D. Farmer, Physica D 4, 366 (1982).
[CrossRef]

1979 (1)

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

Achiam, Y.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Adamczyk, O.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Bhandare, S.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Callan, K. E.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Chen, C.-C.

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002).
[CrossRef]

Colet, P.

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, Phys. Rev. Lett. 95, 203903 (2005).
[CrossRef] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, IEEE J. Quantum Electron. 41, 156 (2005).
[CrossRef]

Farmer, J. D.

J. D. Farmer, Physica D 4, 366 (1982).
[CrossRef]

Fischer, I.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Gao, Z.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Garcia-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Gastaud, N.

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, Phys. Rev. Lett. 95, 203903 (2005).
[CrossRef] [PubMed]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, IEEE J. Quantum Electron. 41, 156 (2005).
[CrossRef]

Gauthier, D. J.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Goedgebuer, J. P.

J. P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

Goedgebuer, J.-P.

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002).
[CrossRef]

Hoffmann, S.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Ibrahim, S.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Ikeda, K.

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

Illing, L.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Kouomou, Y. C.

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, Phys. Rev. Lett. 95, 203903 (2005).
[CrossRef] [PubMed]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, IEEE J. Quantum Electron. 41, 156 (2005).
[CrossRef]

Larger, L.

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, IEEE J. Quantum Electron. 41, 156 (2005).
[CrossRef]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, Phys. Rev. Lett. 95, 203903 (2005).
[CrossRef] [PubMed]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002).
[CrossRef]

J. P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

M. Nourine, M. Peil, and L. Larger, in C. R. Rencontres du Non Linéaire (2009), Vol.  12, 149.

Levy, P.

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002).
[CrossRef]

Mirasso, C. R.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Noé, R.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Nourine, M.

M. Nourine, M. Peil, and L. Larger, in C. R. Rencontres du Non Linéaire (2009), Vol.  12, 149.

Peil, M.

M. Nourine, M. Peil, and L. Larger, in C. R. Rencontres du Non Linéaire (2009), Vol.  12, 149.

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Peveling, R.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Pfau, T.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Porrmann, M.

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

Porte, H.

J. P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

Rhodes, W. T.

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002).
[CrossRef]

Roy, R.

G. D. Van Wiggeren and R. Roy, Science 279, 1198 (1998).
[CrossRef]

Schöll, E.

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Shore, K. A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Van Wiggeren, G. D.

G. D. Van Wiggeren and R. Roy, Science 279, 1198 (1998).
[CrossRef]

Electron. Lett. (1)

T. Pfau, S. Hoffmann, R. Peveling, S. Ibrahim, O. Adamczyk, M. Porrmann, S. Bhandare, R. Noé, and Y. Achiam, Electron. Lett. 42, 1175 (2006).
[CrossRef]

IEEE J. Quantum Electron. (2)

J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, IEEE J. Quantum Electron. 38, 1178 (2002).
[CrossRef]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, IEEE J. Quantum Electron. 41, 156 (2005).
[CrossRef]

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature 437, 343 (2005).
[CrossRef]

Opt. Commun. (1)

K. Ikeda, Opt. Commun. 30, 257 (1979).
[CrossRef]

Phys. Rev. Lett. (3)

K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, Phys. Rev. Lett. 104, 113901 (2010).
[CrossRef] [PubMed]

Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, Phys. Rev. Lett. 95, 203903 (2005).
[CrossRef] [PubMed]

J. P. Goedgebuer, L. Larger, and H. Porte, Phys. Rev. Lett. 80, 2249 (1998).
[CrossRef]

Physica D (1)

J. D. Farmer, Physica D 4, 366 (1982).
[CrossRef]

Science (1)

G. D. Van Wiggeren and R. Roy, Science 279, 1198 (1998).
[CrossRef]

Other (1)

M. Nourine, M. Peil, and L. Larger, in C. R. Rencontres du Non Linéaire (2009), Vol.  12, 149.

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Figures (4)

Fig. 1
Fig. 1

QPSK modulator: (a) device architecture, (b) physical model of the four-wave interferometer, and (c) plot of the 2D nonlinear transfer function.

Fig. 2
Fig. 2

Experimental setup. LD, laser diode; QPSK-M, quadrature phase-shift-keying modulator; RF Amp, radio-frequency amplifier; PD, photodiode; SMF, single-mode fiber; OC, optical coupler; Att, variable attenuator.

Fig. 3
Fig. 3

Bifurcation diagrams [(a) experiment and (b) numerics] of the probability density function (PDF) of the solution trajectory x 1 ( t ) , as β 1 is increased, with ϕ 1 = 1.59 rad , ϕ 2 = 0.39 rad , ϕ 3 = 0.11 rad , and β 2 = 1.1 . (c) Corresponding entropy Ω 1 (numerics; dashed curve, single loop entropy, i.e., for β 2 = 0 ).

Fig. 4
Fig. 4

(a) Experimental time series for β 1 = 5.0 , and amplitude probability distribution (right plot; linear, lower axis; log scale, upper axis) [other parameters identical as in Fig. 3a]. (b) Corresponding RF power density spectrum ( 1 MHz resolution).

Equations (4)

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

P out ( t ) = | E out ( t ) | 2 = P in · F [ v 1 ( t ) , v 2 ( t ) ] .
E out ( t ) = E in ( t ) 2 × { 1 + e i φ 1 ( t ) + [ 1 + e i φ 2 ( t ) ] e i ϕ 3 } ,
F [ v 1 ( t ) , v 2 ( t ) ] = 1 2 { cos 2 ϕ 3 + cos 2 [ φ 1 φ 2 ϕ 3 ] + 2 cos ϕ 3 cos [ φ 1 φ 2 ϕ 3 ] cos [ φ 1 + φ 2 ] } ,
x k + τ k d x k d t + 1 θ k t 0 t x k ( ξ ) d ξ = β k F [ x 1 ( t T k ) , x 2 ( t T k ) ] ,

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