Abstract

Optically induced thermal and free carrier nonlinearities in silicon micro-ring resonator influence their behavior. They can be either deleterious by making them instable and by driving their resonances out of the designed wavelengths, or enabler of different applications. Among the most interesting one, there are optical bistability and self induced oscillations. These lead to all optical logic, signal modulation, optical memories and applications in neural networks. Here, we theoretically and experimentally demonstrate that when many resonators are coupled together, thermal and free carrier nonlinearities induce also chaos. The chaotic dynamics are deeply analyzed using experimentally reconstructed phase space trajectories and the tool of Lyapunov exponents.

© 2014 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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2014 (1)

J. Petráček, Y. Ekşioğlu, and A. Sterkhova, “Simulation of self-pulsing in kerr-nonlinear coupled ring resonators,” Opt. Commun. 318, 147–152 (2014).
[CrossRef]

2012 (3)

2011 (2)

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous kerr response,” Phys. Rev. A 84, 053816 (2011).
[CrossRef]

M. Mancinelli, R. Guider, M. Masi, P. Bettotti, M. Vanacharla, J. Fedeli, and L. Pavesi, “Optical characterization of a SCISSOR device,” Opt. Express 19, 13664–13674 (2011).
[CrossRef] [PubMed]

2009 (1)

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

2008 (2)

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[CrossRef] [PubMed]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

2007 (2)

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75, 063833 (2007).
[CrossRef]

Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15, 924–929 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (4)

2004 (1)

1999 (1)

1997 (1)

L. F. Shampine and M. W. Reichelt, “The matlab ode suite,” SIAM journal on scientific computing 18, 1–22 (1997).
[CrossRef]

1985 (1)

J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Reviews of Modern Physics 57, 617 (1985).
[CrossRef]

Almeida, V. R.

Amano, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Argyris, A.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[CrossRef] [PubMed]

Armaroli, A.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous kerr response,” Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Baets, R.

Barclay, P. E.

Bellanca, G.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous kerr response,” Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Bettotti, P.

Bienstman, P.

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A 86, 063808 (2012).
[CrossRef]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
[CrossRef] [PubMed]

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Bogaerts, W.

Bogris, A.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[CrossRef] [PubMed]

Borselli, M.

Cao, T.

Cecconi, F.

M. Cencini, F. Cecconi, and A. Vulpiani, Chaos: From Simple Models to Complex Systems (World Scientific Publishing, 2010).

Cencini, M.

M. Cencini, F. Cecconi, and A. Vulpiani, Chaos: From Simple Models to Complex Systems (World Scientific Publishing, 2010).

Chen, S.

Chen, Y. L.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75, 063833 (2007).
[CrossRef]

Chin, M.-K.

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-d microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005).
[CrossRef]

Chlouverakis, K.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[CrossRef] [PubMed]

Combrié, S.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous kerr response,” Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Dambre, J.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
[CrossRef] [PubMed]

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A 86, 063808 (2012).
[CrossRef]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Darmawan, S.

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-d microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005).
[CrossRef]

Davis, P.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

de Rossi, A.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous kerr response,” Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Dumon, P.

Eckmann, J.-P.

J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Reviews of Modern Physics 57, 617 (1985).
[CrossRef]

Eksioglu, Y.

J. Petráček, Y. Ekşioğlu, and A. Sterkhova, “Simulation of self-pulsing in kerr-nonlinear coupled ring resonators,” Opt. Commun. 318, 147–152 (2014).
[CrossRef]

Fedeli, J.

Fei, Y.

Fiers, M.

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A 86, 063808 (2012).
[CrossRef]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
[CrossRef] [PubMed]

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Guider, R.

Guo, G. C.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75, 063833 (2007).
[CrossRef]

Hamacher, M.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics, vol. 1 (Prentice-Hall1984).

Hirano, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Inoue, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Jiang, W.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75, 063833 (2007).
[CrossRef]

Johnson, T.

Johnson, T. J.

Kumar, A.A.

A.A. Kumar, Fundamentals of Digital Circuits (Prentice Hall, 2012).

Kumar, R.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
[CrossRef] [PubMed]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Kurashige, T.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Landobasa, Y. M.

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-d microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005).
[CrossRef]

Lee, R. K.

Lipson, M.

Maes, B.

B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

Malaguti, S.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous kerr response,” Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Mancinelli, M.

M. Mancinelli, R. Guider, M. Masi, P. Bettotti, M. Vanacharla, J. Fedeli, and L. Pavesi, “Optical characterization of a SCISSOR device,” Opt. Express 19, 13664–13674 (2011).
[CrossRef] [PubMed]

M. Mancinelli, PhD Thesis, “Linear and non linear coupling effects in sequence of microresonators,” http://eprints-phd.biblio.unitn.it/1050/ (2013).

Masi, M.

Mechet, P.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
[CrossRef] [PubMed]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Morthier, G.

Mortier, G.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Naito, S.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Oowada, I.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Painter, O.

Pavesi, L.

Petrácek, J.

J. Petráček, Y. Ekşioğlu, and A. Sterkhova, “Simulation of self-pulsing in kerr-nonlinear coupled ring resonators,” Opt. Commun. 318, 147–152 (2014).
[CrossRef]

Priem, G.

Reichelt, M. W.

L. F. Shampine and M. W. Reichelt, “The matlab ode suite,” SIAM journal on scientific computing 18, 1–22 (1997).
[CrossRef]

Ruelle, D.

J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Reviews of Modern Physics 57, 617 (1985).
[CrossRef]

Scherer, A.

Schneider, B.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Schrauwen, B.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
[CrossRef] [PubMed]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Shampine, L. F.

L. F. Shampine and M. W. Reichelt, “The matlab ode suite,” SIAM journal on scientific computing 18, 1–22 (1997).
[CrossRef]

Shiki, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Someya, H.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Spuesens, T.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
[CrossRef] [PubMed]

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Srinivasan, K.

Sterkhova, A.

J. Petráček, Y. Ekşioğlu, and A. Sterkhova, “Simulation of self-pulsing in kerr-nonlinear coupled ring resonators,” Opt. Commun. 318, 147–152 (2014).
[CrossRef]

Syvridis, D.

A. Argyris, M. Hamacher, K. Chlouverakis, A. Bogris, and D. Syvridis, “Photonic integrated device for chaos applications in communications,” Phys. Rev. Lett. 100, 194101 (2008).
[CrossRef] [PubMed]

Takens, F.

F. Takens, “Detecting strange attractors in turbulence,” in Dynamical systems and turbulence, Warwick 1980, (Springer, 1981), pp. 366–381.
[CrossRef]

Trillo, S.

A. Armaroli, S. Malaguti, G. Bellanca, S. Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous kerr response,” Phys. Rev. A 84, 053816 (2011).
[CrossRef]

Uchida, A.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Van Thourhout, D.

Van Vaerenbergh, T.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
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T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Vanacharla, M.

Vandoorne, K.

T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Mortier, K. Vandoorne, B. Schneider, B. Schrauwen, J. Dambre, and P. Bienstman, “Self-pulsation and excitability mechanism in silicon-on-insulator microrings,” in Asia Communications and Photonics Conference, J. Opt. Soc. Am. (2012).

Vulpiani, A.

M. Cencini, F. Cecconi, and A. Vulpiani, Chaos: From Simple Models to Complex Systems (World Scientific Publishing, 2010).

Xiao, Y. F.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75, 063833 (2007).
[CrossRef]

Xu, Q.

Xu, Y.

Yariv, A.

Yoshimori, S.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Yoshimura, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Zhang, L.

Zou, X. B.

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75, 063833 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. M. Landobasa, S. Darmawan, and M.-K. Chin, “Matrix analysis of 2-d microresonator lattice optical filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005).
[CrossRef]

Nature Photon. (1)

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nature Photon. 2, 728–732 (2008).
[CrossRef]

Opt. Commun. (1)

J. Petráček, Y. Ekşioğlu, and A. Sterkhova, “Simulation of self-pulsing in kerr-nonlinear coupled ring resonators,” Opt. Commun. 318, 147–152 (2014).
[CrossRef]

Opt. Express (8)

P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005).
[CrossRef] [PubMed]

M. Borselli, T. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13, 1515–1530 (2005).
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G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in silicon-on-insulator ring resonator structures,” Opt. Express 13, 9623–9628 (2005).
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T. J. Johnson, M. Borselli, and O. Painter, “Self-induced optical modulation of the transmission through a high-Q silicon microdisk resonator,” Opt. Express 14, 817–831 (2006).
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Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15, 924–929 (2007).
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M. Mancinelli, R. Guider, M. Masi, P. Bettotti, M. Vanacharla, J. Fedeli, and L. Pavesi, “Optical characterization of a SCISSOR device,” Opt. Express 19, 13664–13674 (2011).
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S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express 20, 7454–7468 (2012).
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T. Van Vaerenbergh, M. Fiers, P. Mechet, T. Spuesens, R. Kumar, G. Morthier, B. Schrauwen, J. Dambre, and P. Bienstman, “Cascadable excitability in microrings,” Opt. Express 20, 20292–20308 (2012).
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Opt. Lett. (2)

Phys. Rev. A (4)

Y. F. Xiao, X. B. Zou, W. Jiang, Y. L. Chen, and G. C. Guo, “Analog to multiple electromagnetically induced transparency in all-optical drop-filter systems,” Phys. Rev. A 75, 063833 (2007).
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B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
[CrossRef]

T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A 86, 063808 (2012).
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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Sketch of the device geometry. The rings have a nominal radius of R = 7 μm and are separated by a distance L = 22μm. They are evanescently coupled to the waveguides by means of a 300nm gap. The SCISSOR is excited from the Input (In) port. Dropped and transmitted signals are recorded in the Drop (D) and Through (T) ports, respectively. (b) Schematic of the experimental setup.

Fig. 2
Fig. 2

(a) Top scattered light from the SCISSOR associated to the three waveforms shown in (b–d), respectively. A schematic of the device geometry (white line on the scattering images) illustrates the position of the rings in the chain. (b–d) Drop signal waveforms at different input powers (Pin, coupled in the waveguide) and wavelength (λp) combinations -(b) Pin = 20mW λp = 1543.225nm, (c) Pin = 13mW λp = 1543.170nm, (d) Pin = 14.5mW λp = 1543.990nm - showing six (b), three (c) and two (d) hot resonators.

Fig. 3
Fig. 3

(a) Frequency distributions of the periodic and the aperiodic Drop signal waveforms shown in Fig. 2(d) and in Fig. 2(b) respectively. Red bars are for 21mW of incident power and 1543.225nm of input pump wavelength, while green bars for 21mW and 1543.420nm. (b) Time evolution of two periodic (aperiodic) signals, indicated as run 1 and 2, which start from slightly different initial conditions. Input powers and wavelenght positions are the same as in a. (c) Reconstructed phase spaces of the system for the periodic and the aperiodic outputs in (b). The reconstruction makes use of the Taken’s theorem with m = 2 and τ = 8ns. (d) Cross correlation between the two periodic (aperiodic) Drop signal waveforms in (b).

Fig. 4
Fig. 4

(a) Experimental (black) and simulated (red) low power (< mW) spectral response of the SCISSOR in the 1542.5 – 1545nm range. The resonance positions of each ring in the SCISSOR are indicated with a vertical blue line. (b) Experimental phase space density as a function of the input power and wavelength. (c) Simulated phase space density as a function of the input power and wavelength. Density is computed as in (b) using simulated Drop signals. Colored dots indicate the combinations of input power and wavelength at which the Lyapunov exponents, reported in Fig. 6, are computed. Dots coordinates are (21.0mW, 1543.11nm) for the red one, (17.8mW, 1543.91nm) for the green one and (13.3mW, 1543.59nm) for the blue one.

Fig. 5
Fig. 5

Comparison between simulation and experiment for an input power of 14.5mW and input wavelength 1543.99nm.

Fig. 6
Fig. 6

System stability analysis through Lyapunov Exponents (LE). At each combination of input power and wavelength position, indicated with colored dots in Fig. 4(c), the spectrum of Lyapunov exponents of the SCISSOR is computed. As the time evolves, curves in each panel converge to the values of such exponents.Only the largest six LE are plotted in each panel. The Drop signal waveforms (not shown) is constant in the case of the blue dot, periodic for the green one and chaotic for the red one.

Fig. 7
Fig. 7

(a) Sketch of the device geometry. By proceeding from left to right, the three rings have resonance wavelenght λ1 = λ3 + δλ, λ2 = λ3 − 0.1nm and λ3 = 1542.92nm respectively. The input signal wavelength is set to λp = λ3 + 0.075nm. Vertical red lines show the input signal and the central ring resonance wavelengths, respectively.The ratio between the resonator separation and its perimeter is 0.5, which ensures the maximum feedback between the cavities. Each cavity has a linewidth of 0.8nm. (b) Phase space density plot as a function of the resonance detuning δλ of the first ring and of the input power. The diagram has been calculated using the same procedure as the one in Fig. 4(b). Contour lines separate the regions where the largest Lyapunov exponent turns from negative (labelled Stable) to null (labelled SP, for Self Pulsing regime), null to positive (labelled Chaos) and vice versa.

Fig. 8
Fig. 8

(a) Cavity interactions: each ring in the chain stores an internal energy |ai|2. The ith upper waveguide carries a power P i h while the lower one P i l. Powers are fed to the cavity with a rate ki. The cavity loss rate rate 1 τ i is due to waveguide loading, linear absorption, two photon absorption and free carrier absorption [2]. (b) A differential temperature ΔT is achieved as the result of the balance between the absorbed power Pabs and the heat which is dissipated into the external oxide with a rate γ t h = 1 τ t h. A differential free carrier population ΔN is obtained by generating electron-hole pairs at a rate GTPA through two photon absorption, and by recombining them with a rate γ f c = 1 τ f c. Thermal and free carrier effects shift the resonance position λ0 (black dashed curve in the central inset) by ΔλT) and ΔλN) respectively [2]. The detuning with respect to the input wavelength λp is the sum of these shifts.

Tables (1)

Tables Icon

Table 1 Values of the simulation parameters. The design value of the resonator radius R0 is perturbed by a quantity ΔRi because of fabrication errors.

Equations (5)

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d a i ( t ) d t = [ j ( ω p ω 0 i ( 1 + Δ ω ( Δ T i ( t ) ) + Δ ω ( Δ N i ( t ) ) ) 1 τ i ( a i ( t ) ) ] a i ( t ) + j k i ( P i l ( t ) + P i h ( t ) )
d Δ T i ( t ) d t = γ f h Δ T i ( t ) + P abs , i ( a i ( t ) , Δ N i ( t ) )
d Δ N i ( t ) d t = γ f c Δ N i ( t ) + G TPA , i ( a i ( t ) )
P i l ( t ) = e j ϕ L ( P i 1 l ( t ) + j k i 1 a i 1 ( t ) )
P i h ( t ) = e j ϕ L ( P i + 1 h ( t ) + j k i + 1 a i + 1 ( t ) )

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