Abstract

We report on experimental demonstration of all-optical switching in a silicon-on-insulator photonic wire nanocavity operating at telecom wavelengths. The switching is performed with a control pulse energy as low as ~ 0.1 pJ on a cavity device that presents very high signal transmission, an ultra-high quality-factor, almost diffraction-limited modal volume and a footprint of only 5 μm2. High-speed modulation of the cavity mode is achieved by means of optical injection of free carriers using a nanosecond pulsed laser. Experimental results are interpreted by means of finite-difference time-domain simulations. The possibility of using this device as a logic gate is also demonstrated.

© 2010 Optical Society of America

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  24. M. McCutcheon, G. Rieger, I. Cheung, J. Young, D. Dalacu, S. Frederick, P. Poole, G. Aers, and R. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221,110 (2005).
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    [CrossRef]
  26. P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  34. K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
    [CrossRef]
  35. M. Waldow, T. Plotzing, M. Gottheil, M. Forst, J. Bolten, T. Wahlbrink, and H. Kurz, "25 ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator," Opt. Express 16(11), 7693-7702 (2008).
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  36. K. Sokolowski-Tinten and D. von der Linde, "Generation of dense electron-hole plasmas in silicon," Phys. Rev. B 61, 2643-2650 (2000).
    [CrossRef]
  37. T. G. Euser and W. L. Vos, "Spatial homogeneity of optically switched semiconductor photonic crystals and of bulk semiconductors," J. Appl. Phys. 97, 043102 (2005).
    [CrossRef]

2009 (4)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic Release of Trapped Light from an Ultrahigh-Q Nanocavity via Adiabatic Frequency Tuning," Phys. Rev. Lett. 102, 043907 (2009)
[CrossRef] [PubMed]

M. Galli, S. Portalupi, M. Belotti, L. Andreani, L. O’Faolain, and T. Krauss, "Light scattering and Fano resonances in high-Q photonic crystal nanocavities," Appl. Phys. Lett. 94, 071101 (2009).
[CrossRef]

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

K. Preston, S. Manipatruni, A. Gondarenko, C. Poitras, and M. Lipson, "Deposited silicon high-speed integrated electro-optic modulator," Opt. Express 17(7), 5118-5124 (2009).
[CrossRef] [PubMed]

2008 (6)

2007 (8)

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, "12.5 Gbit/s carrier-injection-based silicon microring silicon modulators," Opt. Express 15(2), 430-436 (2007).
[CrossRef] [PubMed]

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, "High-speed optical modulation based on carrier depletion in a silicon waveguide," Opt. Express 15(2), 660-668 (2007).
[CrossRef] [PubMed]

Q. Xu and M. Lipson, "All-optical logic based on silicon micro-ring resonators," Opt. Express 15(3), 924-929 (2007).
[CrossRef] [PubMed]

B. Schmidt, Q. Xu, J. Shakya, S. Manipatruni, and M. Lipson, "Compact electro-optic modulator on silicon-oninsulator substrates using cavities with ultra-small modal volumes," Opt. Express 15, 3140-3148 (2007).
[CrossRef] [PubMed]

M. Forst, J. Niehusmann, T. Plotzing, J. Bolten, T. Wahlbrink, C. Moormann, and H. Kurz, "High-speed alloptical switching in ion-implanted silicon-on-insulator microring resonators," Opt. Lett. 32, 2046-2048 (2007).
[CrossRef] [PubMed]

S. Manipatruni, Q. Xu, and M. Lipson, "PINIP based high-speed high-extinction ratio micron-size silicon electrooptic modulator," Opt. Express 15(20), 13035-13042 (2007).
[CrossRef] [PubMed]

P. Velha, E. Picard, T. Charvolin, E. Hadji, J. Rodier, P. Lalanne, and D. Peyrade, "Ultra-High Q/V Fabry-Perot microcavity on SOI substrate," Opt. Express 15, 16090-16096 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (7)

M. McCutcheon, G. Rieger, I. Cheung, J. Young, D. Dalacu, S. Frederick, P. Poole, G. Aers, and R. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221,110 (2005).
[CrossRef]

T. G. Euser and W. L. Vos, "Spatial homogeneity of optically switched semiconductor photonic crystals and of bulk semiconductors," J. Appl. Phys. 97, 043102 (2005).
[CrossRef]

C. Sauvan, G. Lecamp, P. Lalanne, and J. Hugonin, "Modal-reflectivity enhancement by geometry tuning in Photonic Crystal microcavities," Opt. Express 13(1), 245-255 (2005).
[CrossRef] [PubMed]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

Y. Vlasov, M. O’Boyle, H. Hamann, and S. McNab, "Active control of slow light on a chip with photonic crystal waveguides." Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator." Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

2004 (4)

V. Almeida, C. Barrios, R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

H. Chong and R. De La Rue, "Tuning of photonic crystal waveguide microcavity by thermo-optic effect," IEEE Photon. Technol. Lett. 16(6), 1528-1530 (2004).
[CrossRef]

M. Soljacic and J. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals." Nat. Mater. 3(4), 211-219 (2004).
[CrossRef] [PubMed]

S.F. Preble, V. Almeida, and M. Lipson, "Optically controlled photonic crystal nanocavity in silicon," Proc. SPIE 5511, 10-17 (2004).
[CrossRef]

2003 (1)

P. Lalanne and J. Hugonin, "Bloch-wave engineering for high-Q, small-V microcavities," IEEE J. Quantum Electron. 39(11), 1430-1438 (2003).
[CrossRef]

2002 (1)

S. Leonard, H. van Driel, J. Schilling, and R. Wehrspohn, "Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection," Phys. Rev. B 66, 161102 (2002).
[CrossRef]

2000 (1)

K. Sokolowski-Tinten and D. von der Linde, "Generation of dense electron-hole plasmas in silicon," Phys. Rev. B 61, 2643-2650 (2000).
[CrossRef]

1997 (1)

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

1987 (2)

F. Doany, D. Grischkowsky, and C. Chi, "Carrier lifetime versus ion-implantation dose in silicon on sapphire," Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

R. Soref and B. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. 23(1), 123-129 (1987).
[CrossRef]

Almeida, V.

V. Almeida, C. Barrios, R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

S.F. Preble, V. Almeida, and M. Lipson, "Optically controlled photonic crystal nanocavity in silicon," Proc. SPIE 5511, 10-17 (2004).
[CrossRef]

Andreani, L.

M. Galli, S. Portalupi, M. Belotti, L. Andreani, L. O’Faolain, and T. Krauss, "Light scattering and Fano resonances in high-Q photonic crystal nanocavities," Appl. Phys. Lett. 94, 071101 (2009).
[CrossRef]

Barrios, C.

V. Almeida, C. Barrios, R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Belotti, M.

M. Galli, S. Portalupi, M. Belotti, L. Andreani, L. O’Faolain, and T. Krauss, "Light scattering and Fano resonances in high-Q photonic crystal nanocavities," Appl. Phys. Lett. 94, 071101 (2009).
[CrossRef]

M. Belotti, J. Galisteo Lopez, S. De Angelis, M. Galli, I. Maksymov, L. Andreani, D. Peyrade, and Y. Chen, "Alloptical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities," Opt. Express 16(15), 11624-11636 (2008).
[PubMed]

Bennett, B.

R. Soref and B. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. 23(1), 123-129 (1987).
[CrossRef]

Bolten, J.

Charvolin, T.

Chetrit, Y.

Cheung, I.

M. McCutcheon, G. Rieger, I. Cheung, J. Young, D. Dalacu, S. Frederick, P. Poole, G. Aers, and R. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221,110 (2005).
[CrossRef]

Chi, C.

F. Doany, D. Grischkowsky, and C. Chi, "Carrier lifetime versus ion-implantation dose in silicon on sapphire," Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

Chong, H.

A.R. Md Zain, M. Gnan, H. Chong, M. Sorel, and R. De La Rue, "Tapered Photonic Crystal Microcavities Embedded in Photonic Wire Waveguides with large Resonance Quality-Factory and High Transmission," IEEE Photonics Technol. Lett. 20(1), 6-8 (2008).
[CrossRef]

H. Chong and R. De La Rue, "Tuning of photonic crystal waveguide microcavity by thermo-optic effect," IEEE Photon. Technol. Lett. 16(6), 1528-1530 (2004).
[CrossRef]

Ciftcioglu, B.

Dalacu, D.

M. McCutcheon, G. Rieger, I. Cheung, J. Young, D. Dalacu, S. Frederick, P. Poole, G. Aers, and R. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221,110 (2005).
[CrossRef]

De La Rue, R.

A.R. Md Zain, M. Gnan, H. Chong, M. Sorel, and R. De La Rue, "Tapered Photonic Crystal Microcavities Embedded in Photonic Wire Waveguides with large Resonance Quality-Factory and High Transmission," IEEE Photonics Technol. Lett. 20(1), 6-8 (2008).
[CrossRef]

A.R. Md Zain, N. Johnson, M. Sorel, and R. De La Rue, "Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI)," Opt. Express 16(16), 12084-12089 (2008).
[CrossRef]

H. Chong and R. De La Rue, "Tuning of photonic crystal waveguide microcavity by thermo-optic effect," IEEE Photon. Technol. Lett. 16(6), 1528-1530 (2004).
[CrossRef]

Deotare, P.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

Doany, F.

F. Doany, D. Grischkowsky, and C. Chi, "Carrier lifetime versus ion-implantation dose in silicon on sapphire," Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

Dong, P.

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

El Melhaoui, L.

Euser, T. G.

T. G. Euser and W. L. Vos, "Spatial homogeneity of optically switched semiconductor photonic crystals and of bulk semiconductors," J. Appl. Phys. 97, 043102 (2005).
[CrossRef]

F¨orst, M.

Fan, S.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Fedeli, J.

Ferrera, J.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Foresi, J.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Forst, M.

Frank, I.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

Fukuda, H.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

Galisteo, J.

Galli, M.

M. Galli, S. Portalupi, M. Belotti, L. Andreani, L. O’Faolain, and T. Krauss, "Light scattering and Fano resonances in high-Q photonic crystal nanocavities," Appl. Phys. Lett. 94, 071101 (2009).
[CrossRef]

Gnan, M.

A.R. Md Zain, M. Gnan, H. Chong, M. Sorel, and R. De La Rue, "Tapered Photonic Crystal Microcavities Embedded in Photonic Wire Waveguides with large Resonance Quality-Factory and High Transmission," IEEE Photonics Technol. Lett. 20(1), 6-8 (2008).
[CrossRef]

Gondarenko, A.

Gottheil, M.

Grischkowsky, D.

F. Doany, D. Grischkowsky, and C. Chi, "Carrier lifetime versus ion-implantation dose in silicon on sapphire," Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

Hadji, E.

Hamann, H.

Y. Vlasov, M. O’Boyle, H. Hamann, and S. McNab, "Active control of slow light on a chip with photonic crystal waveguides." Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Herzig, H.

Hugonin, J.

C. Sauvan, G. Lecamp, P. Lalanne, and J. Hugonin, "Modal-reflectivity enhancement by geometry tuning in Photonic Crystal microcavities," Opt. Express 13(1), 245-255 (2005).
[CrossRef] [PubMed]

P. Lalanne and J. Hugonin, "Bloch-wave engineering for high-Q, small-V microcavities," IEEE J. Quantum Electron. 39(11), 1430-1438 (2003).
[CrossRef]

Inokawa, H.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

Ippen, E.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Izhaky, N.

Joannopoulos, J.

M. Soljacic and J. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals." Nat. Mater. 3(4), 211-219 (2004).
[CrossRef] [PubMed]

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Johnson, N.

Khan, M.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

Kimerling, L.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Kira, G.

Krauss, T.

M. Galli, S. Portalupi, M. Belotti, L. Andreani, L. O’Faolain, and T. Krauss, "Light scattering and Fano resonances in high-Q photonic crystal nanocavities," Appl. Phys. Lett. 94, 071101 (2009).
[CrossRef]

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic Release of Trapped Light from an Ultrahigh-Q Nanocavity via Adiabatic Frequency Tuning," Phys. Rev. Lett. 102, 043907 (2009)
[CrossRef] [PubMed]

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

Kurz, H.

Lalanne, P.

Lecamp, G.

Leonard, S.

S. Leonard, H. van Driel, J. Schilling, and R. Wehrspohn, "Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection," Phys. Rev. B 66, 161102 (2002).
[CrossRef]

Liao, L.

Lipson, M.

K. Preston, S. Manipatruni, A. Gondarenko, C. Poitras, and M. Lipson, "Deposited silicon high-speed integrated electro-optic modulator," Opt. Express 17(7), 5118-5124 (2009).
[CrossRef] [PubMed]

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, "12.5 Gbit/s carrier-injection-based silicon microring silicon modulators," Opt. Express 15(2), 430-436 (2007).
[CrossRef] [PubMed]

Q. Xu and M. Lipson, "All-optical logic based on silicon micro-ring resonators," Opt. Express 15(3), 924-929 (2007).
[CrossRef] [PubMed]

S. Manipatruni, Q. Xu, and M. Lipson, "PINIP based high-speed high-extinction ratio micron-size silicon electrooptic modulator," Opt. Express 15(20), 13035-13042 (2007).
[CrossRef] [PubMed]

B. Schmidt, Q. Xu, J. Shakya, S. Manipatruni, and M. Lipson, "Compact electro-optic modulator on silicon-oninsulator substrates using cavities with ultra-small modal volumes," Opt. Express 15, 3140-3148 (2007).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator." Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

V. Almeida, C. Barrios, R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

S.F. Preble, V. Almeida, and M. Lipson, "Optically controlled photonic crystal nanocavity in silicon," Proc. SPIE 5511, 10-17 (2004).
[CrossRef]

Liu, A.

Loncar, M.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

Lyan, P.

Manipatruni, S.

Marki, I.

McCutcheon, M.

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
[CrossRef]

M. McCutcheon, G. Rieger, I. Cheung, J. Young, D. Dalacu, S. Frederick, P. Poole, G. Aers, and R. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221,110 (2005).
[CrossRef]

McNab, S.

Y. Vlasov, M. O’Boyle, H. Hamann, and S. McNab, "Active control of slow light on a chip with photonic crystal waveguides." Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Md Zain, A.R.

A.R. Md Zain, M. Gnan, H. Chong, M. Sorel, and R. De La Rue, "Tapered Photonic Crystal Microcavities Embedded in Photonic Wire Waveguides with large Resonance Quality-Factory and High Transmission," IEEE Photonics Technol. Lett. 20(1), 6-8 (2008).
[CrossRef]

A.R. Md Zain, N. Johnson, M. Sorel, and R. De La Rue, "Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI)," Opt. Express 16(16), 12084-12089 (2008).
[CrossRef]

Mitsugi, S.

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

Moormann, C.

Nguyen, H.

Niehusmann, J.

Nishiguchi, K.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic Release of Trapped Light from an Ultrahigh-Q Nanocavity via Adiabatic Frequency Tuning," Phys. Rev. Lett. 102, 043907 (2009)
[CrossRef] [PubMed]

T. Tanabe, H. Taniyama, and M. Notomi, "Carrier Diffusion and Recombination in Photonic Crystal Nanocavity Optical Switches," J. Lightwave Technol.,  26(11), 1396-1403 (2008).
[CrossRef]

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

O’Boyle, M.

Y. Vlasov, M. O’Boyle, H. Hamann, and S. McNab, "Active control of slow light on a chip with photonic crystal waveguides." Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

O’Faolain, L.

M. Galli, S. Portalupi, M. Belotti, L. Andreani, L. O’Faolain, and T. Krauss, "Light scattering and Fano resonances in high-Q photonic crystal nanocavities," Appl. Phys. Lett. 94, 071101 (2009).
[CrossRef]

Panepucci, R.

V. Almeida, C. Barrios, R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Paniccia, M.

Peyrade, D.

Picard, E.

Plotzing, T.

Poitras, C.

Portalupi, S.

M. Galli, S. Portalupi, M. Belotti, L. Andreani, L. O’Faolain, and T. Krauss, "Light scattering and Fano resonances in high-Q photonic crystal nanocavities," Appl. Phys. Lett. 94, 071101 (2009).
[CrossRef]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator." Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

Preble, S.F.

S.F. Preble, V. Almeida, and M. Lipson, "Optically controlled photonic crystal nanocavity in silicon," Proc. SPIE 5511, 10-17 (2004).
[CrossRef]

Preston, K.

K. Preston, S. Manipatruni, A. Gondarenko, C. Poitras, and M. Lipson, "Deposited silicon high-speed integrated electro-optic modulator," Opt. Express 17(7), 5118-5124 (2009).
[CrossRef] [PubMed]

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

Rieger, G.

M. McCutcheon, G. Rieger, I. Cheung, J. Young, D. Dalacu, S. Frederick, P. Poole, G. Aers, and R. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221,110 (2005).
[CrossRef]

Rodier, J.

Rubin, D.

Salt, M.

Sauvan, C.

Schilling, J.

S. Leonard, H. van Driel, J. Schilling, and R. Wehrspohn, "Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection," Phys. Rev. B 66, 161102 (2002).
[CrossRef]

Schmidt, B.

Shakya, J.

Shinya, A.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

Smith, H.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Sokolowski-Tinten, K.

K. Sokolowski-Tinten and D. von der Linde, "Generation of dense electron-hole plasmas in silicon," Phys. Rev. B 61, 2643-2650 (2000).
[CrossRef]

Soljacic, M.

M. Soljacic and J. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals." Nat. Mater. 3(4), 211-219 (2004).
[CrossRef] [PubMed]

Soref, R.

R. Soref and B. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. 23(1), 123-129 (1987).
[CrossRef]

Sorel, M.

A.R. Md Zain, M. Gnan, H. Chong, M. Sorel, and R. De La Rue, "Tapered Photonic Crystal Microcavities Embedded in Photonic Wire Waveguides with large Resonance Quality-Factory and High Transmission," IEEE Photonics Technol. Lett. 20(1), 6-8 (2008).
[CrossRef]

A.R. Md Zain, N. Johnson, M. Sorel, and R. De La Rue, "Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI)," Opt. Express 16(16), 12084-12089 (2008).
[CrossRef]

Stanley, R.

Steinmeyer, G.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic Release of Trapped Light from an Ultrahigh-Q Nanocavity via Adiabatic Frequency Tuning," Phys. Rev. Lett. 102, 043907 (2009)
[CrossRef] [PubMed]

T. Tanabe, H. Taniyama, and M. Notomi, "Carrier Diffusion and Recombination in Photonic Crystal Nanocavity Optical Switches," J. Lightwave Technol.,  26(11), 1396-1403 (2008).
[CrossRef]

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic Release of Trapped Light from an Ultrahigh-Q Nanocavity via Adiabatic Frequency Tuning," Phys. Rev. Lett. 102, 043907 (2009)
[CrossRef] [PubMed]

T. Tanabe, H. Taniyama, and M. Notomi, "Carrier Diffusion and Recombination in Photonic Crystal Nanocavity Optical Switches," J. Lightwave Technol.,  26(11), 1396-1403 (2008).
[CrossRef]

Thoen, E.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Tsuchizawa, T.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

van Driel, H.

S. Leonard, H. van Driel, J. Schilling, and R. Wehrspohn, "Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection," Phys. Rev. B 66, 161102 (2002).
[CrossRef]

Velha, P.

Villeneuve, P.

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Vlasov, Y.

Y. Vlasov, M. O’Boyle, H. Hamann, and S. McNab, "Active control of slow light on a chip with photonic crystal waveguides." Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

von der Linde, D.

K. Sokolowski-Tinten and D. von der Linde, "Generation of dense electron-hole plasmas in silicon," Phys. Rev. B 61, 2643-2650 (2000).
[CrossRef]

Vos, W. L.

T. G. Euser and W. L. Vos, "Spatial homogeneity of optically switched semiconductor photonic crystals and of bulk semiconductors," J. Appl. Phys. 97, 043102 (2005).
[CrossRef]

Wahlbrink, T.

Waldow, M.

Watanabe, T.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

Wehrspohn, R.

S. Leonard, H. van Driel, J. Schilling, and R. Wehrspohn, "Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection," Phys. Rev. B 66, 161102 (2002).
[CrossRef]

Xu, Q.

Yamada, K.

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

Young, J.

M. McCutcheon, G. Rieger, I. Cheung, J. Young, D. Dalacu, S. Frederick, P. Poole, G. Aers, and R. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221,110 (2005).
[CrossRef]

Appl. Phys. Lett. (7)

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda,  et al., "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

M. McCutcheon, G. Rieger, I. Cheung, J. Young, D. Dalacu, S. Frederick, P. Poole, G. Aers, and R. Williams, "Resonant scattering and second-harmonic spectroscopy of planar photonic crystal microcavities," Appl. Phys. Lett. 87, 221,110 (2005).
[CrossRef]

M. Galli, S. Portalupi, M. Belotti, L. Andreani, L. O’Faolain, and T. Krauss, "Light scattering and Fano resonances in high-Q photonic crystal nanocavities," Appl. Phys. Lett. 94, 071101 (2009).
[CrossRef]

P. Deotare, M. McCutcheon, I. Frank, M. Khan, and M. Loncar, "High quality factor photonic crystal nanobeam cavities," Appl. Phys. Lett. 94, 121106 (2009).
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F. Doany, D. Grischkowsky, and C. Chi, "Carrier lifetime versus ion-implantation dose in silicon on sapphire," Appl. Phys. Lett. 50, 460 (1987).
[CrossRef]

K. Preston, P. Dong, B. Schmidt, and M. Lipson, "High-speed all-optical modulation using polycrystalline silicon microring resonators," Appl. Phys. Lett. 92, 151104 (2008).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. Soref and B. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. 23(1), 123-129 (1987).
[CrossRef]

P. Lalanne and J. Hugonin, "Bloch-wave engineering for high-Q, small-V microcavities," IEEE J. Quantum Electron. 39(11), 1430-1438 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. Chong and R. De La Rue, "Tuning of photonic crystal waveguide microcavity by thermo-optic effect," IEEE Photon. Technol. Lett. 16(6), 1528-1530 (2004).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

A.R. Md Zain, M. Gnan, H. Chong, M. Sorel, and R. De La Rue, "Tapered Photonic Crystal Microcavities Embedded in Photonic Wire Waveguides with large Resonance Quality-Factory and High Transmission," IEEE Photonics Technol. Lett. 20(1), 6-8 (2008).
[CrossRef]

J. Appl. Phys. (1)

T. G. Euser and W. L. Vos, "Spatial homogeneity of optically switched semiconductor photonic crystals and of bulk semiconductors," J. Appl. Phys. 97, 043102 (2005).
[CrossRef]

J. Lightwave Technol. (1)

Nat. Mater. (1)

M. Soljacic and J. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals." Nat. Mater. 3(4), 211-219 (2004).
[CrossRef] [PubMed]

Nature (4)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator." Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

V. Almeida, C. Barrios, R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

J. Foresi, P. Villeneuve, J. Ferrera, E. Thoen, G. Steinmeyer, S. Fan, J. Joannopoulos, L. Kimerling, H. Smith, and E. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143 (1997).
[CrossRef]

Y. Vlasov, M. O’Boyle, H. Hamann, and S. McNab, "Active control of slow light on a chip with photonic crystal waveguides." Nature 438, 65-69 (2005).
[CrossRef] [PubMed]

Opt. Express (12)

M. Belotti, J. Galisteo Lopez, S. De Angelis, M. Galli, I. Maksymov, L. Andreani, D. Peyrade, and Y. Chen, "Alloptical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities," Opt. Express 16(15), 11624-11636 (2008).
[PubMed]

A.R. Md Zain, N. Johnson, M. Sorel, and R. De La Rue, "Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI)," Opt. Express 16(16), 12084-12089 (2008).
[CrossRef]

K. Preston, S. Manipatruni, A. Gondarenko, C. Poitras, and M. Lipson, "Deposited silicon high-speed integrated electro-optic modulator," Opt. Express 17(7), 5118-5124 (2009).
[CrossRef] [PubMed]

S. Manipatruni, Q. Xu, and M. Lipson, "PINIP based high-speed high-extinction ratio micron-size silicon electrooptic modulator," Opt. Express 15(20), 13035-13042 (2007).
[CrossRef] [PubMed]

P. Velha, E. Picard, T. Charvolin, E. Hadji, J. Rodier, P. Lalanne, and D. Peyrade, "Ultra-High Q/V Fabry-Perot microcavity on SOI substrate," Opt. Express 15, 16090-16096 (2007).
[CrossRef] [PubMed]

M. Waldow, T. Plotzing, M. Gottheil, M. Forst, J. Bolten, T. Wahlbrink, and H. Kurz, "25 ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator," Opt. Express 16(11), 7693-7702 (2008).
[CrossRef] [PubMed]

C. Sauvan, G. Lecamp, P. Lalanne, and J. Hugonin, "Modal-reflectivity enhancement by geometry tuning in Photonic Crystal microcavities," Opt. Express 13(1), 245-255 (2005).
[CrossRef] [PubMed]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 13, 2678-2687 (2005).
[CrossRef] [PubMed]

Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, "12.5 Gbit/s carrier-injection-based silicon microring silicon modulators," Opt. Express 15(2), 430-436 (2007).
[CrossRef] [PubMed]

A. Liu, L. Liao, D. Rubin, H. Nguyen, B. Ciftcioglu, Y. Chetrit, N. Izhaky, and M. Paniccia, "High-speed optical modulation based on carrier depletion in a silicon waveguide," Opt. Express 15(2), 660-668 (2007).
[CrossRef] [PubMed]

Q. Xu and M. Lipson, "All-optical logic based on silicon micro-ring resonators," Opt. Express 15(3), 924-929 (2007).
[CrossRef] [PubMed]

B. Schmidt, Q. Xu, J. Shakya, S. Manipatruni, and M. Lipson, "Compact electro-optic modulator on silicon-oninsulator substrates using cavities with ultra-small modal volumes," Opt. Express 15, 3140-3148 (2007).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Rev. B (2)

K. Sokolowski-Tinten and D. von der Linde, "Generation of dense electron-hole plasmas in silicon," Phys. Rev. B 61, 2643-2650 (2000).
[CrossRef]

S. Leonard, H. van Driel, J. Schilling, and R. Wehrspohn, "Ultrafast band-edge tuning of a two-dimensional silicon photonic crystal via free-carrier injection," Phys. Rev. B 66, 161102 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, "Dynamic Release of Trapped Light from an Ultrahigh-Q Nanocavity via Adiabatic Frequency Tuning," Phys. Rev. Lett. 102, 043907 (2009)
[CrossRef] [PubMed]

Proc. SPIE (1)

S.F. Preble, V. Almeida, and M. Lipson, "Optically controlled photonic crystal nanocavity in silicon," Proc. SPIE 5511, 10-17 (2004).
[CrossRef]

Other (1)

D. Miller, "Rationale and challenges for optical interconnects to electronic chips," Proceedings of IEEE (IEEE, 2000), pp. 728-749.
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Sketch and scanning electron micrograph (SEM) image of the tapered PhC cavity embedded in a PhW waveguide with two mirrors composed by periodically spaced holes and aperiodic taper region inside and outside the cavity. The cavity in SEM picture has 6 holes in each mirror spaced by 350 nm and identical aperiodic taper region inside and outside. The spacings between adjacent holes are 300, 315, and 325 nm respectively. The corresponding hole radii are 65, 80, 85 nm. Finally, we also demonstrate the possibility of performing various gate operations by carefully tuning the probe wavelength with respect to the cavity resonance.

Fig. 2.
Fig. 2.

Simulated transmission spectra of the device without (black curve) and with (red curve) the pump pulse. The simulated blue-shift is 0.12 nm. In the right inset, a broad-band spectrum is given, showing the cavity resonance within the 1D photonic band gap.

Fig. 3.
Fig. 3.

The calculated field intensities at resonance (1502.3 nm) and off resonance (1500 nm) are shown for the |Ey |2 and |Hz |2 component, respectively. Notice the change in scales from off to on resonance.

Fig. 4.
Fig. 4.

Transmission spectrum for the sample with 6 holes mirror, 3 holes taper inside and outside the cavity, cavity length of (a) 400 nm and (b) 425 nm, respectively. (a) The spectrum shows a resonance for the wavelength 1483.16 nm with a Q-factor of approximately 89,300. (b) The spectrum shows a strong resonance with a Q-factor of about 63,000 for the wavelength of 1502.3 nm.

Fig. 5.
Fig. 5.

Resonant scattering spectrum of the sample with 6 holes mirror, 3 holes taper inside and outside the cavity, cavity length of 400 nm. The red curve is the Fano lineshape best-fit.

Fig. 6.
Fig. 6.

Time evolution of switching logic operation: (a) NOT operation: the output is high (on-condition) when the control level is low and it switches to low (off-condition) when the control level becomes high. (b) AND operation: the output is low (off-condition) when the control level is low, and it is high (on-condition) when the control is high. (c) Pump intensity as a function of time. The pulse width is 2.5 ns. (d) Scheme of the applied probe and control beam.

Fig. 7.
Fig. 7.

(a) Time resolved evolution of the transmission spectrum when a pump with 2.5 ns duration and 2 pJ energy is applied. The normalized intensity color scale is in linear units. (b) Extracted shift Δλ of the resonance wavelength as a function of time.

Equations (4)

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N eh = I ( z ) τ pump ħ ω pump [ α + 1 2 β I ( z ) ] dz
ε ( ω ) = ε B ( ω ) + Δ ε eh ( ω ) = ε B ( ω ) ( ω p ω ) 2 1 1 + i ω τ D
ω p = N eh e 2 ε 0 m e m opt
n = n 0 e 2 2 n 0 ε 0 m opt m e ω 2 N eh .

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