Abstract

A numerical study on Mamyshev signal regeneration realized on silicon photonic wires is reported. Unlike fiber-optics Mamyshev regenerators employing cross-phase modulation, silicon photonic wires have to include two-photon absorption and the two-photon-absorption-induced free-carrier effect. By well adjusting time delay between the co-propagating signal and clock pulses, both cross-phase modulation and free-carrier dispersion could induce nonlinear wavelength shift, which is essential for signal recovery in the Mamyshev regeneration scheme. A simulation result shows the quality factor of signal eye diagram improved by more than 4 dB for Return-to-Zero signals with pulse width 10 ps, peak power 6.5 W, and operation speed 10 Gbit/s through a 1-cm silicon photonic wire.

©2010 Optical Society of America

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References

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  1. O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
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2009 (1)

2007 (5)

2006 (2)

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

E. Dulkeith, Y. A. Vlasov, X. Chen, N. C. Panoiu, and R. M. Osgood., “Self-phase-modulation in submicron silicon-on-insulator photonic wires,” Opt. Express 14(12), 5524–5534 (2006).
[Crossref] [PubMed]

2005 (1)

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, “All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber,” IEEE Photon. Technol. Lett. 17(2), 423–425 (2005).
[Crossref]

2004 (1)

2003 (3)

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15(7), 957–959 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “All-optical signal regeneration: from first principles to a 40 Gbit/s system demonstration,” C. R. Phys. 4(1), 163–173 (2003).
[Crossref]

2001 (1)

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13(2), 142–144 (2001).
[Crossref]

1999 (1)

T. Miyazaki, T. Otani, N. Edagawa, M. Suzuki, and S. Yamamoto, ““Novel optical-regenerator using electroabsorption modulators,” IEICE Transactions on Electronics,” E 82C(8), 1414–1419 (1999).

1987 (1)

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

Agrawal, G. P.

Balmefrezol, E.

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “All-optical signal regeneration: from first principles to a 40 Gbit/s system demonstration,” C. R. Phys. 4(1), 163–173 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
[Crossref]

Bennett, B. R.

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

Brindel, P.

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “All-optical signal regeneration: from first principles to a 40 Gbit/s system demonstration,” C. R. Phys. 4(1), 163–173 (2003).
[Crossref]

Centanni, J. C.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15(7), 957–959 (2003).
[Crossref]

Chen, X.

Chen, X. G.

Chraplyvy, A. R.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15(7), 957–959 (2003).
[Crossref]

Ciaramella, E.

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13(2), 142–144 (2001).
[Crossref]

Curti, F.

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13(2), 142–144 (2001).
[Crossref]

Dadap, J. I.

Dulkeith, E.

Edagawa, N.

T. Miyazaki, T. Otani, N. Edagawa, M. Suzuki, and S. Yamamoto, ““Novel optical-regenerator using electroabsorption modulators,” IEICE Transactions on Electronics,” E 82C(8), 1414–1419 (1999).

Eggleton, B. J.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

Foster, M. A.

Fu, L.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

Fukuda, H.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Gaeta, A. L.

Geraghty, D. F.

Hsieh, I. W.

Inokawa, H.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Itabashi, S.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Jopson, R. M.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15(7), 957–959 (2003).
[Crossref]

Kikuchi, K.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, “All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber,” IEEE Photon. Technol. Lett. 17(2), 423–425 (2005).
[Crossref]

Kuramochi, E.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Lavigne, B.

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “All-optical signal regeneration: from first principles to a 40 Gbit/s system demonstration,” C. R. Phys. 4(1), 163–173 (2003).
[Crossref]

Leclerc, O.

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “All-optical signal regeneration: from first principles to a 40 Gbit/s system demonstration,” C. R. Phys. 4(1), 163–173 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
[Crossref]

Lin, Q.

Lipson, M.

Littler, I. C. M.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

Luther-Davies, B.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

McKinstrie, C. J.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15(7), 957–959 (2003).
[Crossref]

McNab, S. J.

Miyazaki, T.

T. Miyazaki, T. Otani, N. Edagawa, M. Suzuki, and S. Yamamoto, ““Novel optical-regenerator using electroabsorption modulators,” IEICE Transactions on Electronics,” E 82C(8), 1414–1419 (1999).

Moss, D. J.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

Murphy, T. E.

Nishiguchi, K.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Notomi, M.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Osgood, J. R. M.

Osgood, R. M.

Otani, T.

T. Miyazaki, T. Otani, N. Edagawa, M. Suzuki, and S. Yamamoto, ““Novel optical-regenerator using electroabsorption modulators,” IEICE Transactions on Electronics,” E 82C(8), 1414–1419 (1999).

Ozeki, Y.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, “All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber,” IEEE Photon. Technol. Lett. 17(2), 423–425 (2005).
[Crossref]

Painter, O. J.

Panoiu, N. C.

Pierre, L.

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “All-optical signal regeneration: from first principles to a 40 Gbit/s system demonstration,” C. R. Phys. 4(1), 163–173 (2003).
[Crossref]

Radic, S.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, and A. R. Chraplyvy, “All-optical regeneration in one- and two-pump parametric amplifiers using highly nonlinear optical fiber,” IEEE Photon. Technol. Lett. 15(7), 957–959 (2003).
[Crossref]

Rochette, M.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

Rouvillain, D.

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “All-optical signal regeneration: from first principles to a 40 Gbit/s system demonstration,” C. R. Phys. 4(1), 163–173 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
[Crossref]

Salem, R.

Seguineau, F.

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “Optical regeneration at 40 Gb/s and beyond,” J. Lightwave Technol. 21(11), 2779–2790 (2003).
[Crossref]

O. Leclerc, B. Lavigne, E. Balmefrezol, P. Brindel, L. Pierre, D. Rouvillain, and F. Seguineau, “All-optical signal regeneration: from first principles to a 40 Gbit/s system demonstration,” C. R. Phys. 4(1), 163–173 (2003).
[Crossref]

Shinojima, H.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Shinya, A.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Shokooh-Saremi, M.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

Soref, R. A.

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

Suzuki, J.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, “All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber,” IEEE Photon. Technol. Lett. 17(2), 423–425 (2005).
[Crossref]

Suzuki, M.

T. Miyazaki, T. Otani, N. Edagawa, M. Suzuki, and S. Yamamoto, ““Novel optical-regenerator using electroabsorption modulators,” IEICE Transactions on Electronics,” E 82C(8), 1414–1419 (1999).

Ta’eed, V. G.

V. G. Ta’eed, M. Shokooh-Saremi, L. Fu, I. C. M. Littler, D. J. Moss, M. Rochette, B. J. Eggleton, Yinlan Ruan, and B. Luther-Davies, “Self-phase modulation-based integrated optical regeneration in chalcogenide waveguides,” IEEE J. Sel. Top. Quantum Electron. 12(3), 360–370 (2006).
[Crossref]

Taira, K.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, “All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber,” IEEE Photon. Technol. Lett. 17(2), 423–425 (2005).
[Crossref]

Tanabe, T.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Tanemura, T.

J. Suzuki, T. Tanemura, K. Taira, Y. Ozeki, and K. Kikuchi, “All-optical regenerator using wavelength shift induced by cross-phase modulation in highly nonlinear dispersion-shifted fiber,” IEEE Photon. Technol. Lett. 17(2), 423–425 (2005).
[Crossref]

Trillo, S.

E. Ciaramella, F. Curti, and S. Trillo, “All-optical signal reshaping by means of four-wave mixing in optical fibers,” IEEE Photon. Technol. Lett. 13(2), 142–144 (2001).
[Crossref]

Tsuchizawa, T.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Turner, A. C.

Vlasov, Y. A.

Watanabe, T.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Yamada, K.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90(3), 031115 (2007).
[Crossref]

Yamamoto, S.

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Figures (9)

Fig. 1
Fig. 1 System diagram of Mamyshev 3R regeneration using a silicon photonic wire as the nonlinear medium

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Fig. 2
Fig. 2 (a) Free-carrier density varying across a pulse in time domain for different free-carrier lifetimes and (b) peak free-carrier absorption coefficient as a function of free-carrier lifetime

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Fig. 3
Fig. 3 (a) Illustration of a clock pulse synchronizing with a signal pulse at the (i) leading edge, (ii) central peak, and (iii) trailing edge. (b) Asymmetric spectral broadening of signal pulses after passing through a 1-cm silicon photonic wire

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Fig. 4
Fig. 4 Output clock pulses in time and frequency domains. In case (i), the clock spectrum extends to long wavelength but shifts to short wavelength in case (ii) and (iii).

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Fig. 5
Fig. 5 (a) Illustration of three zones where clock pulses are placed. In Zone I, XPM and FCD cause clock pulses red- and blue-shifted respectively. In Zone II, both XPM and FCD induce a blue shift, and in Zone III, XPM results in a blue shift but FCD incurs a red shift. (b) The optimal time delay to get maximal wavelength shift varied by free-carrier lifetime. (c) The maximal wavelength shift versus free-carrier lifetime.

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Fig. 6
Fig. 6 (a) Launched signal and clock pulses at signal logic ‘1’ and ‘0’ (time domain), (b) the corresponding clock spectra after propagation in the silicon photonic wire, and (c) normalized clock transmittance versus the peak power of signal for different free-carrier lifetimes.

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Fig. 7
Fig. 7 (a) Optical eye diagrams of input signal and output clock pulses at two operation speeds, 4 Gbit/s and 20 Gbit/s respectively. (b) Q-factors of the input and output eye diagrams versus operation speed.

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Fig. 8
Fig. 8 Q-factors of input and output eye diagrams versus OSNR for different free-carrier lifetimes.

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Fig. 9
Fig. 9 Q-factor improvement as functions free carrier lifetime with different duty cycles.

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Tables (1)

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Table 1 Simulation parameters for a silicon photonic wire

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Equations (5)

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∂ A s ∂ z + β 1 s ∂ A s ∂ t + i 2 β 2 s ∂ 2 A s ∂ t 2 = − 1 2 α A s + i ( r s s | A s | 2 A s + 2 r s c | A c | 2 A s ) − 1 2 β T A e f f ( | A s | 2 + 2 | A c | 2 ) A s − 1 2 α f s A s + i 2 π λ s Δ n A s
∂ A c ∂ z + β 1 c ∂ A c ∂ t + i 2 β 2 c ∂ 2 A c ∂ t 2 = − 1 2 α A c + i ( r c c | A c | 2 A c + 2 r c s | A s | 2 A c ) − 1 2 β T A e f f ( | A c | 2 + 2 | A s | 2 ) A c − 1 2 α f c A c + i 2 π λ c Δ n A c
f i j = ∬ | F i ( x , y ) | 2 | F j ( x , y ) | 2 d x d y ∬ | F i ( x , y ) | 2 d x d y ∬ | F j ( x , y ) | 2 d x d y
d Δ N h , e ( t ) d t = β T 2 h υ s I s 2 ( t ) + β T 2 h υ c I c 2 ( t )   + 2 β T h υ I s ( t ) ⋅ I c ( t )    - Δ N h , e ( t ) τ ,
α f = − [ 8.5 × 10 − 18 ( λ s , c 1.55 ) 2 Δ N e + 6 × 10 − 18 ( λ s , c 1.55 ) 2 Δ N h ] Δ n = − [ 8.8 × 10 − 22 ( λ s , c 1.55 ) 2 Δ N e + 6 × 10 − 18 ( λ s , c 1.55 ) 2 Δ N h 0.8 ]

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