Abstract

Bistability (BS) and self-pulsation (SP) phenomena in silicon microring resonators (MRR) with intense CW light injection are studied. Several nonlinear optical effects including Kerr effect, two-photon absorption, free carrier absorption and free carrier dispersion are taken into account. The threshold optical intensity of BS and SP is derived from the coupled mode theory and a linear stability analysis method. The influences of MRR’s parameters (carrier lifetime, linear loss and radius) and light injection conditions (input power, wavelength detuning) on the characteristics of SP (modulation depth and oscillating frequency) are analyzed and discussed. It is shown that, SP occurs only if the carrier lifetime ranges from several ps to several-hundred ps and the input light intensity is higher than 106W/cm2. The modulation depth of SP can be as large as 8dB and the associated oscillating frequency is in the range from several GHz to beyond 10 GHz.

© 2012 OSA

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    [CrossRef]

2012

P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
[CrossRef]

2011

J. Petráček, A. Sterkhova, J. Luksch, “Numerical scheme for simulation of self-pulsing and chaos in coupled microring resonators,” Microw. Opt. Technol. Lett. 53(10), 2238–2242 (2011).
[CrossRef]

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A 83(5), 051802 (2011).
[CrossRef]

V. Grigoriev, F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A 83(4), 043816 (2011).
[CrossRef]

2010

V. Grigoriev, F. Biancalana, “Bistability, multistability and non-reciprocal light propagation in Thue–Morse multilayered structures,” New J. Phys. 12(5), 053041 (2010).
[CrossRef]

J. Leuthold, C. Koos, W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
[CrossRef]

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18(4), 3582–3591 (2010).
[CrossRef] [PubMed]

2009

A. D. Rossi, M. Lauritano, S. Combrié, Q. V. Tran, C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A 79(4), 043818 (2009).
[CrossRef]

B. Maes, M. Fiers, P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80(3), 033805 (2009).
[CrossRef]

2008

2007

M. Först, J. Niehusmann, T. Plötzing, J. Bolten, T. Wahlbrink, C. Moormann, H. Kurz, “High-speed all-optical switching in ion-implanted silicon-on-insulator microring resonators,” Opt. Lett. 32(14), 2046–2048 (2007).
[CrossRef] [PubMed]

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
[CrossRef]

Q. Lin, O. J. Painter, G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15(25), 16604–16644 (2007).
[CrossRef] [PubMed]

A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, C. D. Angelis, “Self-pulsing and bistability in nonlinear Bragg gratings,” J. Opt. Soc. Am. B 24(9), 2229–2237 (2007).
[CrossRef]

R. Dekker, N. Usechak, M. Först, A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
[CrossRef]

2006

Q. Xu, M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006).
[CrossRef] [PubMed]

G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, R. Baets, “Nonlinear effects in ultrasmall Silicon-on-Insulator ring resonators,” Proc. SPIE 6183, 61831A, 61831A-8 (2006).
[CrossRef]

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, O. Stafsudd, “Prospects for silicon mid-IR raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
[CrossRef]

2005

2002

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

1997

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

1987

R. A. Soref, B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

1986

R. G. Harrison, D. J. Biswas, “Chaos in light,” Nature 321(6068), 394–401 (1986).
[CrossRef]

1982

K. Ikeda, O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. 48(9), 617–620 (1982).
[CrossRef]

1981

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46(7), 474–477 (1981).
[CrossRef]

1980

K. Ikeda, H. Daido, O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45(9), 709–712 (1980).
[CrossRef]

1979

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979).
[CrossRef]

Absil, P. P.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Agrawal, G. P.

Akimoto, O.

K. Ikeda, O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. 48(9), 617–620 (1982).
[CrossRef]

K. Ikeda, H. Daido, O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45(9), 709–712 (1980).
[CrossRef]

Angelis, C. D.

Baets, R.

G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, R. Baets, “Nonlinear effects in ultrasmall Silicon-on-Insulator ring resonators,” Proc. SPIE 6183, 61831A, 61831A-8 (2006).
[CrossRef]

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005).
[CrossRef] [PubMed]

Bellanca, G.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A 83(5), 051802 (2011).
[CrossRef]

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
[CrossRef]

A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, C. D. Angelis, “Self-pulsing and bistability in nonlinear Bragg gratings,” J. Opt. Soc. Am. B 24(9), 2229–2237 (2007).
[CrossRef]

Bennett, B. R.

R. A. Soref, B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Biancalana, F.

V. Grigoriev, F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A 83(4), 043816 (2011).
[CrossRef]

V. Grigoriev, F. Biancalana, “Bistability, multistability and non-reciprocal light propagation in Thue–Morse multilayered structures,” New J. Phys. 12(5), 053041 (2010).
[CrossRef]

Bienstman, P.

B. Maes, M. Fiers, P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80(3), 033805 (2009).
[CrossRef]

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
[CrossRef]

Biswas, D. J.

R. G. Harrison, D. J. Biswas, “Chaos in light,” Nature 321(6068), 394–401 (1986).
[CrossRef]

Bogaerts, W.

G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, R. Baets, “Nonlinear effects in ultrasmall Silicon-on-Insulator ring resonators,” Proc. SPIE 6183, 61831A, 61831A-8 (2006).
[CrossRef]

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005).
[CrossRef] [PubMed]

Bolten, J.

Calzada, J. A.

P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
[CrossRef]

Cáp, J.

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
[CrossRef]

Chamorro-Posada, P.

P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
[CrossRef]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Combrié, S.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A 83(5), 051802 (2011).
[CrossRef]

A. D. Rossi, M. Lauritano, S. Combrié, Q. V. Tran, C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A 79(4), 043818 (2009).
[CrossRef]

Conforti, M.

Daido, H.

K. Ikeda, H. Daido, O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45(9), 709–712 (1980).
[CrossRef]

de Rossi, A.

S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A 83(5), 051802 (2011).
[CrossRef]

Dekker, R.

R. Dekker, N. Usechak, M. Först, A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
[CrossRef]

Dimitropoulos, D.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, O. Stafsudd, “Prospects for silicon mid-IR raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
[CrossRef]

Driessen, A.

R. Dekker, N. Usechak, M. Först, A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
[CrossRef]

Dumon, P.

G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, R. Baets, “Nonlinear effects in ultrasmall Silicon-on-Insulator ring resonators,” Proc. SPIE 6183, 61831A, 61831A-8 (2006).
[CrossRef]

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005).
[CrossRef] [PubMed]

Durán, A.

P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
[CrossRef]

Fathpour, S.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, O. Stafsudd, “Prospects for silicon mid-IR raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
[CrossRef]

Fiers, M.

B. Maes, M. Fiers, P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80(3), 033805 (2009).
[CrossRef]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Först, M.

Foster, M. A.

Freude, W.

J. Leuthold, C. Koos, W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4(8), 535–544 (2010).
[CrossRef]

Gaeta, A. L.

García-Escartín, J. C.

P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
[CrossRef]

Gibbs, H. M.

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46(7), 474–477 (1981).
[CrossRef]

Gottheil, M.

Grigoriev, V.

V. Grigoriev, F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A 83(4), 043816 (2011).
[CrossRef]

V. Grigoriev, F. Biancalana, “Bistability, multistability and non-reciprocal light propagation in Thue–Morse multilayered structures,” New J. Phys. 12(5), 053041 (2010).
[CrossRef]

Grover, R.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Harrison, R. G.

R. G. Harrison, D. J. Biswas, “Chaos in light,” Nature 321(6068), 394–401 (1986).
[CrossRef]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Ho, P. T.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Hopf, F. A.

H. M. Gibbs, F. A. Hopf, D. L. Kaplan, R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46(7), 474–477 (1981).
[CrossRef]

Husko, C.

A. D. Rossi, M. Lauritano, S. Combrié, Q. V. Tran, C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A 79(4), 043818 (2009).
[CrossRef]

Ibrahim, T. A.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

Ikeda, K.

K. Ikeda, O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. 48(9), 617–620 (1982).
[CrossRef]

K. Ikeda, H. Daido, O. Akimoto, “Optical turbulence: chaotic behavior of transmitted light from a ring cavity,” Phys. Rev. Lett. 45(9), 709–712 (1980).
[CrossRef]

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979).
[CrossRef]

Jalali, B.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, O. Stafsudd, “Prospects for silicon mid-IR raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
[CrossRef]

Johnson, F. G.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
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H. M. Gibbs, F. A. Hopf, D. L. Kaplan, R. L. Shoemaker, “Observation of chaos in optical bistability,” Phys. Rev. Lett. 46(7), 474–477 (1981).
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B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
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A. D. Rossi, M. Lauritano, S. Combrié, Q. V. Tran, C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A 79(4), 043818 (2009).
[CrossRef]

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
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H. K. Tsang, Y. Liu, “Nonlinear optical properties of silicon waveguides,” Semicond. Sci. Technol. 23(6), 064007 (2008).
[CrossRef]

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Luksch, J.

J. Petráček, A. Sterkhova, J. Luksch, “Numerical scheme for simulation of self-pulsing and chaos in coupled microring resonators,” Microw. Opt. Technol. Lett. 53(10), 2238–2242 (2011).
[CrossRef]

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B. Maes, M. Fiers, P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80(3), 033805 (2009).
[CrossRef]

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
[CrossRef]

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S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A 83(5), 051802 (2011).
[CrossRef]

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P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
[CrossRef]

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F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
[CrossRef]

Moormann, C.

Morichetti, F.

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
[CrossRef]

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G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, R. Baets, “Nonlinear effects in ultrasmall Silicon-on-Insulator ring resonators,” Proc. SPIE 6183, 61831A, 61831A-8 (2006).
[CrossRef]

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005).
[CrossRef] [PubMed]

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Palencia, C.

P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
[CrossRef]

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J. Petráček, A. Sterkhova, J. Luksch, “Numerical scheme for simulation of self-pulsing and chaos in coupled microring resonators,” Microw. Opt. Technol. Lett. 53(10), 2238–2242 (2011).
[CrossRef]

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
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Poitras, C. B.

Priem, G.

F. Morichetti, A. Melloni, J. Čáp, J. Petráćek, P. Bienstman, G. Priem, B. Maes, M. Lauritano, G. Bellanca, “Self-phase modulation in slow-wave structures: A comparative numerical analysis,” Opt. Quantum Electron. 38(9-11), 761–780 (2007).
[CrossRef]

G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, R. Baets, “Nonlinear effects in ultrasmall Silicon-on-Insulator ring resonators,” Proc. SPIE 6183, 61831A, 61831A-8 (2006).
[CrossRef]

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005).
[CrossRef] [PubMed]

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[CrossRef]

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A. D. Rossi, M. Lauritano, S. Combrié, Q. V. Tran, C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A 79(4), 043818 (2009).
[CrossRef]

Salem, R.

Sánchez-Curto, J.

P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
[CrossRef]

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[CrossRef]

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B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, O. Stafsudd, “Prospects for silicon mid-IR raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
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[CrossRef]

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J. Petráček, A. Sterkhova, J. Luksch, “Numerical scheme for simulation of self-pulsing and chaos in coupled microring resonators,” Microw. Opt. Technol. Lett. 53(10), 2238–2242 (2011).
[CrossRef]

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G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, R. Baets, “Nonlinear effects in ultrasmall Silicon-on-Insulator ring resonators,” Proc. SPIE 6183, 61831A, 61831A-8 (2006).
[CrossRef]

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A. D. Rossi, M. Lauritano, S. Combrié, Q. V. Tran, C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A 79(4), 043818 (2009).
[CrossRef]

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S. Malaguti, G. Bellanca, A. de Rossi, S. Combrié, S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A 83(5), 051802 (2011).
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[CrossRef]

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V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
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Wahlbrink, T.

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[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002).
[CrossRef]

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, O. Stafsudd, “Prospects for silicon mid-IR raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
[CrossRef]

J. Lightwave Technol.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

J. Opt.

P. Chamorro-Posada, P. Martin-Ramos, J. Sánchez-Curto, J. C. García-Escartín, J. A. Calzada, C. Palencia, A. Durán, “Nonlinear Bloch modes, optical switching and Bragg solitons in tightly coupled micro-ring resonator chains,” J. Opt. 14(1), 015205 (2012).
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J. Phys. D Appl. Phys.

R. Dekker, N. Usechak, M. Först, A. Driessen, “Ultrafast nonlinear all-optical processes in silicon-on-insulator waveguides,” J. Phys. D Appl. Phys. 40(14), R249–R271 (2007).
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J. Petráček, A. Sterkhova, J. Luksch, “Numerical scheme for simulation of self-pulsing and chaos in coupled microring resonators,” Microw. Opt. Technol. Lett. 53(10), 2238–2242 (2011).
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A. D. Rossi, M. Lauritano, S. Combrié, Q. V. Tran, C. Husko, “Interplay of plasma-induced and fast thermal nonlinearities in a GaAs-based photonic crystal nanocavity,” Phys. Rev. A 79(4), 043818 (2009).
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Figures (10)

Fig. 1
Fig. 1

Configuration of an all-pass microring resonator.

Fig. 2
Fig. 2

Left (a): Steady state response E versus input normalized power P with (plotted in blue) and without (plotted in red) the Kerr effect, respectively. Right (b): Real eigenvalue (solid curves) and the real part of the two conjugate complex eigenvalues (bold dashed curves) that with (plotted in blue) and without (plotted in red) the Kerr effect, respectively. The dotted lines indicate BS threshold values; while the fine dashed lines indicate the start and end threshold values of the SP (both of them are plotted in two colors for the cases of with/without the Kerr effect.) Note that the above analyses are carried out for the carrier lifetime of 150ps and the input light wavelength detuning of −0.05nm (if not other specified, the carrier lifetime and the input light wavelength detuning take these values in the following simulations).

Fig. 3
Fig. 3

Output light intensities in time domain with different input intensities (a), (c), (e), (g) and their associated phase loops (b), (d), (f), (h). The phase loop of the output light is normalized according to the input light.

Fig. 4
Fig. 4

Modulation depth of SP versus input light intensity. The modulation depth of SP is defined as the difference between the maximum and minimum output light intensity divided by their sum. The two red dashed lines indicate SP threshold values, denoted as I i n S P + and I i n S P .

Fig. 5
Fig. 5

SP frequency versus input light intensity (in logarithmic coordinates).

Fig. 6
Fig. 6

Threshold value of the input light intensity for BS and SP versus carrier lifetime. SP has a cut-off carrier lifetime of about 260ps, while BS has a more wider carrier lifetime range.

Fig. 7
Fig. 7

Threshold input light intensity regions of BS and SP versus input wavelength detuning and different carrier lifetimes 100ps (red lines), 150ps (blue lines) and 200ps (green lines), respectively. Solid curves are SP threshold lines, whereas dashed and dashed-dot curves refer to BS threshold lines.

Fig. 8
Fig. 8

Optical spectra of the output light for different input light intensities. This picture illustrates the relationship between the relative output intensity (the output light intensity divided by the input light intensity in dB) and the input wavelength detuning in different input light intensities. Inset: Output light intensity versus input light intensity when omitting SP. When the input light is 1 × 106W/cm2 and the wavelength detuning is −0.03nm, there are two output states shown in this figure, which fits well with the black curves. Note that the carrier lifetime in this case is 100 ps.

Fig. 9
Fig. 9

Optical spectra at different time points (plotted in different colors). These time points are sampled evenly within one SP oscillating circle. (a): the normalized dimensionless energy and carrier density in the MRR at initial state are both 0. (b): the normalized dimensionless energy and carrier density in the MRR at initial state are 1 and 1.5, respectively. This figure is obtained by calculating the output intensity at the fixed time of each wavelength detuning. In (a), there are four fixed time points from 22.4 ns to 22.55 ns. In (b), there are nine points from 22.4 ns to 22.625 ns with 0.025 ns time interval. The waveguide dispersion and material dispersion are omitted due to the narrow wavelength detuning.

Fig. 10
Fig. 10

Input light intensity regions for BS ( I i n B S + , I i n B S ) and SP ( I i n S P + , I i n S P ) versus the linear loss of the MRR. The input light wavelength detuning is chosen to be –Δ3dB. The carrier lifetime is fixed at 100ps.

Equations (10)

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u t = { i ( ω 0 ω L ) i ω 0 n 0 [ n 2 I c n 0 V K e r r | u | 2 ( σ r 1 N + σ r 2 N 0.8 ) ] ( ω 0 2 Q L + β 2 c 2 2 n 0 2 | u | 2 V T P A + σ F C A N c 2 n 0 ) } u + Γ c P i n ,
N t = β 2 2 ω I 2 N τ c a r = c 2 β 2 n 0 2 2 ω 0 V T P A V c a r | u | 4 N τ c a r .
b o u t p u t = P i n Γ c u ,
a t = P + i δ a i n K e r r _ N | a | 2 a + i ( n + σ F C D n 0.8 ) a ( 1 + γ F C A n ) a α T P A | a | 2 a ,
n t = | a | 4 n τ .
N 0 = τ | A | 4 = τ E 2 ,
P = E [ ( 1 + γ F C A τ E 2 + α T P A E ) 2 + ( δ n K e r r _ N E + τ E 2 + σ F C D τ 0.8 E 1.6 ) 2 ] .
d ε / d t = M ε ,
M = ( M 11 M 12 M 13 M 21 M 22 M 23 M 31 M 32 M 33 ) ,
M 11 = i δ i n K e r r _ N 2 | A | 2 + i ( N + σ F C D N 0 0.8 ) ( 1 + γ F C A N 0 ) 2 α T P A | A | 2 , M 12 = i n K e r r _ N A 2 α T P A A 2 , M 13 = i ( 1 + 0.8 σ F C D N 0 0.2 ) A γ F C A A , M 21 = M 12 * , M 22 = M 11 * , M 23 = M 13 * , M 31 = 2 | A | 2 A * , M 32 = M 31 * , M 33 = 1 / τ .

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