Abstract

A split nanobeam cavity is theoretically designed and experimentally demonstrated. Compared with the traditional photonic crystal nanobeam cavities, it has an air-slot in its center. Through the longitudinal and lateral movement of half part of the cavity, the resonance wavelength and quality factor are tuned. Instead of achieving a cavity with a large tunable wavelength range, the proposed split nanobeam cavity demonstrates a considerable quality factor change but the resonance wavelength is hardly varied. Using a nanoelectromechanical system (NEMS) comb-drive actuator to control the longitudinal and lateral movement of the split nanobeam cavity, the experimentally-measured change of quality factor agrees well with the simulated value. Meanwhile, the variation range of resonance wavelength is smaller than the full width at half maximum of the resonance. The proposed structure may have potential application in Q-switched lasers.

© 2015 Optical Society of America

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2013 (2)

F. Tian, G. Zhou, F. S. Chau, J. Deng, and R. Akkipeddi, “Measurement of coupled cavities' optomechanical coupling coefficient using a nanoelectromechanical actuator,” Appl. Phys. Lett. 102(8), 081101 (2013).
[Crossref]

F. Tian, G. Zhou, Y. Du, F. S. Chau, J. Deng, S. L. Teo, and R. Akkipeddi, “Nanoelectromechanical-systems-controlled bistability of double-coupled photonic crystal cavities,” Opt. Lett. 38(17), 3394–3397 (2013).
[Crossref] [PubMed]

2012 (2)

F. Tian, G. Zhou, F. S. Chau, J. Deng, Y. Du, X. Tang, R. Akkipeddi, and Y. C. Loke, “Tuning of split-ladder cavity by its intrinsic nano-deformation,” Opt. Express 20(25), 27697–27707 (2012).
[Crossref] [PubMed]

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, “All optical reconfiguration of optomechanical filters,” Nat. Commun. 3, 846 (2012).
[Crossref] [PubMed]

2011 (4)

X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Nanomechanically tunable photonic crystal resonator utilizing triple-beam coupled nanocavities,” IEEE Photon. Technol. Lett. 23(18), 1310–1312 (2011).
[Crossref]

L. Midolo, P. J. van Veldhoven, M. A. Dundar, R. Notzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavites,” Appl. Phys. Lett. 98(21), 211120 (2011).
[Crossref]

X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Enhanced resonance tuning of photonic crystal nanocavities by integration of optimized near-field multitip nanoprobes,” J. Nanophotonics 5(1), 059503 (2011).
[Crossref]

M. Winger, T. D. Blasius, T. P. Mayer Alegre, A. H. Safavi-Naeini, S. Meenehan, J. Cohen, S. Stobbe, and O. Painter, “A chip-scale integrated cavity-electro-optomechanics platform,” Opt. Express 19(25), 24905–24921 (2011).
[PubMed]

2010 (6)

A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q twodimensional photonic crystal cavity,” Appl. Phys. Lett. 97(18), 181106 (2010).
[Crossref]

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4(7), 477–483 (2010).
[Crossref]

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[Crossref]

E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. G. Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO2 claddings and on air claddings,” Opt. Express 18(15), 15859–15869 (2010).
[PubMed]

X. Chew, G. Zhou, F. S. Chau, J. Deng, X. Tang, and Y. C. Loke, “Dynamic tuning of an optical resonator through MEMS-driven coupled photonic crystal nanocavities,” Opt. Lett. 35(15), 2517–2519 (2010).
[Crossref] [PubMed]

X. Chew, G. Zhou, H. Yu, F. S. Chau, J. Deng, Y. C. Loke, and X. Tang, “An in-plane nano-mechanics approach to achieve reversible resonance control of photonic crystal nanocavities,” Opt. Express 18(21), 22232–22244 (2010).
[Crossref] [PubMed]

2009 (9)

J. H. Wülbern, A. Petrov, and M. Eich, “Electro-optical modulator in a polymerinfiltrated silicon slotted photonic crystal waveguide heterostructure resonator,” Opt. Express 17(1), 304–313 (2009).
[Crossref] [PubMed]

J. H. Wülbern, A. Petrov, and M. Eich, “Electro-optical modulator in a polymerinfiltrated silicon slotted photonic crystal waveguide heterostructure resonator,” Opt. Express 17(1), 304–313 (2009).
[Crossref] [PubMed]

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009).
[Crossref] [PubMed]

M. Toishi, D. Englund, A. Faraon, and J. Vucković, “High-brightness single photon source from a quantum dot in a directional-emission nanocavity,” Opt. Express 17(17), 14618–14626 (2009).
[Crossref] [PubMed]

M. Brunstein, R. Braive, R. Hostein, A. Beveratos, I. Rober-Philip, I. Sagnes, T. J. Karle, A. M. Yacomotti, J. A. Levenson, V. Moreau, G. Tessier, and Y. De Wilde, “Thermo-optical dynamics in an optically pumped Photonic Crystal nano-cavity,” Opt. Express 17(19), 17118–17129 (2009).
[Crossref] [PubMed]

T. Tanabe, K. Nishiguchi, E. Kuramochi, and M. Notomi, “Low power and fast electro-optic silicon modulator with lateral p-i-n embedded photonic crystal nanocavity,” Opt. Express 17(25), 22505–22513 (2009).
[Crossref] [PubMed]

T. Tanabe, K. Nishiguchi, E. Kuramochi, and M. Notomi, “Low power and fast electro-optic silicon modulator with lateral p-i-n embedded photonic crystal nanocavity,” Opt. Express 17(25), 22505–22513 (2009).
[Crossref] [PubMed]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[Crossref]

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

2008 (4)

2007 (3)

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90(3), 031911 (2007).
[Crossref]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

Z. Qiang, W. Zhou, and R. A. Soref, “Optical add-drop filters based on photonic crystal ring resonators,” Opt. Express 15(4), 1823–1831 (2007).
[Crossref] [PubMed]

2006 (4)

2005 (4)

2003 (2)

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical diction,” Appl. Phys. Lett. 82(26), 4648 (2003).
[Crossref]

M. Qiu and B. Jaskorzynska, “Design of a channel drop filter in a two-dimensional triangular photonic crystal,” Appl. Phys. Lett. 83(6), 1074 (2003).
[Crossref]

2000 (1)

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
[Crossref]

1999 (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[Crossref] [PubMed]

1997 (1)

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[Crossref]

1996 (1)

R. Legtenberg, A. W. Groeneveld, and M. Elwenspoek, “Comb-drive actuators for large displacement,” J. Micromech. Microeng. 6(3), 320–329 (1996).
[Crossref]

Akkipeddi, R.

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q twodimensional photonic crystal cavity,” Appl. Phys. Lett. 97(18), 181106 (2010).
[Crossref]

Asano, T.

Atatüre, M.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

Badolato, A.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

Balet, L.

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

Barclay, P.

Beveratos, A.

Birner, A.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
[Crossref]

Blasius, T. D.

Bogaerts, W.

Braive, R.

Brunstein, M.

Buchler, B. C.

A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
[Crossref] [PubMed]

Bulu, I.

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, “All optical reconfiguration of optomechanical filters,” Nat. Commun. 3, 846 (2012).
[Crossref] [PubMed]

Busch, K.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
[Crossref]

Camacho, R.

Chan, J.

Chau, F. S.

F. Tian, G. Zhou, Y. Du, F. S. Chau, J. Deng, S. L. Teo, and R. Akkipeddi, “Nanoelectromechanical-systems-controlled bistability of double-coupled photonic crystal cavities,” Opt. Lett. 38(17), 3394–3397 (2013).
[Crossref] [PubMed]

F. Tian, G. Zhou, F. S. Chau, J. Deng, and R. Akkipeddi, “Measurement of coupled cavities' optomechanical coupling coefficient using a nanoelectromechanical actuator,” Appl. Phys. Lett. 102(8), 081101 (2013).
[Crossref]

F. Tian, G. Zhou, F. S. Chau, J. Deng, Y. Du, X. Tang, R. Akkipeddi, and Y. C. Loke, “Tuning of split-ladder cavity by its intrinsic nano-deformation,” Opt. Express 20(25), 27697–27707 (2012).
[Crossref] [PubMed]

X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Enhanced resonance tuning of photonic crystal nanocavities by integration of optimized near-field multitip nanoprobes,” J. Nanophotonics 5(1), 059503 (2011).
[Crossref]

X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Nanomechanically tunable photonic crystal resonator utilizing triple-beam coupled nanocavities,” IEEE Photon. Technol. Lett. 23(18), 1310–1312 (2011).
[Crossref]

X. Chew, G. Zhou, F. S. Chau, J. Deng, X. Tang, and Y. C. Loke, “Dynamic tuning of an optical resonator through MEMS-driven coupled photonic crystal nanocavities,” Opt. Lett. 35(15), 2517–2519 (2010).
[Crossref] [PubMed]

X. Chew, G. Zhou, H. Yu, F. S. Chau, J. Deng, Y. C. Loke, and X. Tang, “An in-plane nano-mechanics approach to achieve reversible resonance control of photonic crystal nanocavities,” Opt. Express 18(21), 22232–22244 (2010).
[Crossref] [PubMed]

Chew, X.

X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Nanomechanically tunable photonic crystal resonator utilizing triple-beam coupled nanocavities,” IEEE Photon. Technol. Lett. 23(18), 1310–1312 (2011).
[Crossref]

X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Enhanced resonance tuning of photonic crystal nanocavities by integration of optimized near-field multitip nanoprobes,” J. Nanophotonics 5(1), 059503 (2011).
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X. Chew, G. Zhou, H. Yu, F. S. Chau, J. Deng, Y. C. Loke, and X. Tang, “An in-plane nano-mechanics approach to achieve reversible resonance control of photonic crystal nanocavities,” Opt. Express 18(21), 22232–22244 (2010).
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X. Chew, G. Zhou, F. S. Chau, J. Deng, X. Tang, and Y. C. Loke, “Dynamic tuning of an optical resonator through MEMS-driven coupled photonic crystal nanocavities,” Opt. Lett. 35(15), 2517–2519 (2010).
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Cohen, J.

Dapkus, P. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
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de Ridder, R. M.

De Wilde, Y.

Deng, J.

F. Tian, G. Zhou, Y. Du, F. S. Chau, J. Deng, S. L. Teo, and R. Akkipeddi, “Nanoelectromechanical-systems-controlled bistability of double-coupled photonic crystal cavities,” Opt. Lett. 38(17), 3394–3397 (2013).
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F. Tian, G. Zhou, F. S. Chau, J. Deng, and R. Akkipeddi, “Measurement of coupled cavities' optomechanical coupling coefficient using a nanoelectromechanical actuator,” Appl. Phys. Lett. 102(8), 081101 (2013).
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F. Tian, G. Zhou, F. S. Chau, J. Deng, Y. Du, X. Tang, R. Akkipeddi, and Y. C. Loke, “Tuning of split-ladder cavity by its intrinsic nano-deformation,” Opt. Express 20(25), 27697–27707 (2012).
[Crossref] [PubMed]

X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Enhanced resonance tuning of photonic crystal nanocavities by integration of optimized near-field multitip nanoprobes,” J. Nanophotonics 5(1), 059503 (2011).
[Crossref]

X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Nanomechanically tunable photonic crystal resonator utilizing triple-beam coupled nanocavities,” IEEE Photon. Technol. Lett. 23(18), 1310–1312 (2011).
[Crossref]

X. Chew, G. Zhou, F. S. Chau, J. Deng, X. Tang, and Y. C. Loke, “Dynamic tuning of an optical resonator through MEMS-driven coupled photonic crystal nanocavities,” Opt. Lett. 35(15), 2517–2519 (2010).
[Crossref] [PubMed]

X. Chew, G. Zhou, H. Yu, F. S. Chau, J. Deng, Y. C. Loke, and X. Tang, “An in-plane nano-mechanics approach to achieve reversible resonance control of photonic crystal nanocavities,” Opt. Express 18(21), 22232–22244 (2010).
[Crossref] [PubMed]

Deotare, P. B.

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, “All optical reconfiguration of optomechanical filters,” Nat. Commun. 3, 846 (2012).
[Crossref] [PubMed]

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[Crossref]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[Crossref]

Du, Y.

Dundar, M. A.

L. Midolo, P. J. van Veldhoven, M. A. Dundar, R. Notzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavites,” Appl. Phys. Lett. 98(21), 211120 (2011).
[Crossref]

Eich, M.

Eichenfield, M.

Elwenspoek, M.

R. Legtenberg, A. W. Groeneveld, and M. Elwenspoek, “Comb-drive actuators for large displacement,” J. Micromech. Microeng. 6(3), 320–329 (1996).
[Crossref]

Englund, D.

Erickson, D.

Fält, S.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
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Fan, S.

Z. Wang and S. Fan, “Optical circulators in two-dimensional magneto-optical photonic crystals,” Opt. Lett. 30(15), 1989–1991 (2005).
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P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[Crossref]

Faraon, A.

Ferrera, J.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[Crossref]

Fiore, A.

L. Midolo, P. J. van Veldhoven, M. A. Dundar, R. Notzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavites,” Appl. Phys. Lett. 98(21), 211120 (2011).
[Crossref]

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

Forchel, A.

T. Sünner, T. Stichel, S. H. Kwon, T. W. Schlereth, S. Hofling, M. Kamp, and A. Forchel, “Photonic crystal cavity based gas sensor,” Appl. Phys. Lett. 92(26), 261112 (2008).
[Crossref]

Foresi, J. S.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[Crossref]

Francardi, M.

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

Frank, I. W.

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, “All optical reconfiguration of optomechanical filters,” Nat. Commun. 3, 846 (2012).
[Crossref] [PubMed]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[Crossref]

Gerace, D.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

Gerardino, A.

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

Gösele, U.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
[Crossref]

Groeneveld, A. W.

R. Legtenberg, A. W. Groeneveld, and M. Elwenspoek, “Comb-drive actuators for large displacement,” J. Micromech. Microeng. 6(3), 320–329 (1996).
[Crossref]

Gulde, S.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

Gurioli, M.

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

Hane, K.

K. Takahashi, Y. Kanamori, Y. Kokubun, and K. Hane, “A wavelength-selective add-drop switch using silicon microring resonator with a submicron-comb electrostatic actuator,” Opt. Express 16(19), 14421–14428 (2008).
[Crossref] [PubMed]

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90(3), 031911 (2007).
[Crossref]

K. Umemori, Y. Kanamori, and K. Hane, “Photonic crystal waveguide switch with a microelectromechanical actuator,” Appl. Phys. Lett. 89(2), 021102 (2006).
[Crossref]

Hennessy, K.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

Herzig, H. P.

Hofling, S.

T. Sünner, T. Stichel, S. H. Kwon, T. W. Schlereth, S. Hofling, M. Kamp, and A. Forchel, “Photonic crystal cavity based gas sensor,” Appl. Phys. Lett. 92(26), 261112 (2008).
[Crossref]

Hollink, A. J. F.

Hopman, W. C. L.

Hostein, R.

Hu, E. L.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

Ilic, R.

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, “All optical reconfiguration of optomechanical filters,” Nat. Commun. 3, 846 (2012).
[Crossref] [PubMed]

Imamoglu, A.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

Intonti, F.

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

Ippen, E. P.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[Crossref]

Jaskorzynska, B.

M. Qiu and B. Jaskorzynska, “Design of a channel drop filter in a two-dimensional triangular photonic crystal,” Appl. Phys. Lett. 83(6), 1074 (2003).
[Crossref]

Joannopoulos, J. D.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[Crossref]

John, S.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
[Crossref]

Kafesaki, M.

A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
[Crossref] [PubMed]

Kamp, M.

T. Sünner, T. Stichel, S. H. Kwon, T. W. Schlereth, S. Hofling, M. Kamp, and A. Forchel, “Photonic crystal cavity based gas sensor,” Appl. Phys. Lett. 92(26), 261112 (2008).
[Crossref]

Kanamori, Y.

K. Takahashi, Y. Kanamori, Y. Kokubun, and K. Hane, “A wavelength-selective add-drop switch using silicon microring resonator with a submicron-comb electrostatic actuator,” Opt. Express 16(19), 14421–14428 (2008).
[Crossref] [PubMed]

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90(3), 031911 (2007).
[Crossref]

K. Umemori, Y. Kanamori, and K. Hane, “Photonic crystal waveguide switch with a microelectromechanical actuator,” Appl. Phys. Lett. 89(2), 021102 (2006).
[Crossref]

Karle, T. J.

Kawasaki, K.

Khan, M.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[Crossref]

Kim, I.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[Crossref] [PubMed]

Kimerling, L. C.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[Crossref]

Kitani, T.

Y. Kanamori, T. Kitani, and K. Hane, “Control of guided resonance in a photonic crystal slab using microelectromechanical actuators,” Appl. Phys. Lett. 90(3), 031911 (2007).
[Crossref]

Koenderink, A. F.

A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
[Crossref] [PubMed]

Kokubun, Y.

Kuramochi, E.

Kwon, S. H.

T. Sünner, T. Stichel, S. H. Kwon, T. W. Schlereth, S. Hofling, M. Kamp, and A. Forchel, “Photonic crystal cavity based gas sensor,” Appl. Phys. Lett. 92(26), 261112 (2008).
[Crossref]

Lee, R. K.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[Crossref] [PubMed]

Legtenberg, R.

R. Legtenberg, A. W. Groeneveld, and M. Elwenspoek, “Comb-drive actuators for large displacement,” J. Micromech. Microeng. 6(3), 320–329 (1996).
[Crossref]

Lehmann, V.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
[Crossref]

Leonard, S. W.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
[Crossref]

Levenson, J. A.

Li, L. H.

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

Loke, Y. C.

Loncar, M.

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, “All optical reconfiguration of optomechanical filters,” Nat. Commun. 3, 846 (2012).
[Crossref] [PubMed]

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[Crossref]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[Crossref]

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical diction,” Appl. Phys. Lett. 82(26), 4648 (2003).
[Crossref]

Mandal, S.

Märki, I.

Matsuo, S.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4(7), 477–483 (2010).
[Crossref]

Mayer Alegre, T. P.

McCutcheon, M. W.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Loncar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[Crossref]

Meenehan, S.

Midolo, L.

L. Midolo, P. J. van Veldhoven, M. A. Dundar, R. Notzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavites,” Appl. Phys. Lett. 98(21), 211120 (2011).
[Crossref]

Mitsugi, S.

Mondia, J. P.

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
[Crossref]

Moreau, V.

Nishiguchi, K.

Noda, S.

Notomi, M.

Notzel, R.

L. Midolo, P. J. van Veldhoven, M. A. Dundar, R. Notzel, and A. Fiore, “Electromechanical wavelength tuning of double-membrane photonic crystal cavites,” Appl. Phys. Lett. 98(21), 211120 (2011).
[Crossref]

Nozaki, K.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4(7), 477–483 (2010).
[Crossref]

O’Brien, J. D.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[Crossref] [PubMed]

Painter, O.

Petrov, A.

Qiang, Z.

Qiu, M.

M. Qiu and B. Jaskorzynska, “Design of a channel drop filter in a two-dimensional triangular photonic crystal,” Appl. Phys. Lett. 83(6), 1074 (2003).
[Crossref]

Qiu, Y.

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical diction,” Appl. Phys. Lett. 82(26), 4648 (2003).
[Crossref]

Quan, Q.

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, “All optical reconfiguration of optomechanical filters,” Nat. Commun. 3, 846 (2012).
[Crossref] [PubMed]

Q. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010).
[Crossref]

Riboli, F.

F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
[Crossref]

Rober-Philip, I.

Roh, Y. G.

Safavi-Naeini, A. H.

M. Winger, T. D. Blasius, T. P. Mayer Alegre, A. H. Safavi-Naeini, S. Meenehan, J. Cohen, S. Stobbe, and O. Painter, “A chip-scale integrated cavity-electro-optomechanics platform,” Opt. Express 19(25), 24905–24921 (2011).
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A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q twodimensional photonic crystal cavity,” Appl. Phys. Lett. 97(18), 181106 (2010).
[Crossref]

Sagnes, I.

Salt, M.

Sandoghdar, V.

A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
[Crossref] [PubMed]

Sato, T.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4(7), 477–483 (2010).
[Crossref]

Scherer, A.

M. Lončar, A. Scherer, and Y. Qiu, “Photonic crystal laser sources for chemical diction,” Appl. Phys. Lett. 82(26), 4648 (2003).
[Crossref]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
[Crossref] [PubMed]

Schlereth, T. W.

T. Sünner, T. Stichel, S. H. Kwon, T. W. Schlereth, S. Hofling, M. Kamp, and A. Forchel, “Photonic crystal cavity based gas sensor,” Appl. Phys. Lett. 92(26), 261112 (2008).
[Crossref]

Shinya, A.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4(7), 477–483 (2010).
[Crossref]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. 30(19), 2575–2577 (2005).
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Smith, H. I.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
[Crossref]

Song, B. S.

Soref, R. A.

Srinivasan, K.

Steinmeyer, G.

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
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Stichel, T.

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Takahashi, K.

Tanabe, T.

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T. Sünner, T. Stichel, S. H. Kwon, T. W. Schlereth, S. Hofling, M. Kamp, and A. Forchel, “Photonic crystal cavity based gas sensor,” Appl. Phys. Lett. 92(26), 261112 (2008).
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F. Intonti, S. Vignolini, F. Riboli, M. Zani, D. S. Wiersma, L. Balet, L. H. Li, M. Francardi, A. Gerardino, A. Fiore, and M. Gurioli, “Tuning of photonic crystal cavities by controlled removal of locally infiltrated water,” Appl. Phys. Lett. 95(17), 173112 (2009).
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K. Umemori, Y. Kanamori, and K. Hane, “Photonic crystal waveguide switch with a microelectromechanical actuator,” Appl. Phys. Lett. 89(2), 021102 (2006).
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F. Tian, G. Zhou, F. S. Chau, J. Deng, and R. Akkipeddi, “Measurement of coupled cavities' optomechanical coupling coefficient using a nanoelectromechanical actuator,” Appl. Phys. Lett. 102(8), 081101 (2013).
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X. Chew, G. Zhou, F. S. Chau, and J. Deng, “Enhanced resonance tuning of photonic crystal nanocavities by integration of optimized near-field multitip nanoprobes,” J. Nanophotonics 5(1), 059503 (2011).
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Nat. Commun. (1)

P. B. Deotare, I. Bulu, I. W. Frank, Q. Quan, Y. Zhang, R. Ilic, and M. Loncar, “All optical reconfiguration of optomechanical filters,” Nat. Commun. 3, 846 (2012).
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Nat. Photonics (1)

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4(7), 477–483 (2010).
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Nature (2)

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445(7130), 896–899 (2007).
[Crossref] [PubMed]

P. R. Villeneuve, J. S. Foresi, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390(6656), 143–145 (1997).
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P. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13(3), 801–820 (2005).
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M. Winger, T. D. Blasius, T. P. Mayer Alegre, A. H. Safavi-Naeini, S. Meenehan, J. Cohen, S. Stobbe, and O. Painter, “A chip-scale integrated cavity-electro-optomechanics platform,” Opt. Express 19(25), 24905–24921 (2011).
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F. Tian, G. Zhou, F. S. Chau, J. Deng, Y. Du, X. Tang, R. Akkipeddi, and Y. C. Loke, “Tuning of split-ladder cavity by its intrinsic nano-deformation,” Opt. Express 20(25), 27697–27707 (2012).
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T. Uesugi, B. S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14(1), 377–386 (2006).
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I. Märki, M. Salt, and H. P. Herzig, “Tuning the resonance of a photonic crystal microcavity with an AFM probe,” Opt. Express 14(7), 2969–2978 (2006).
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M. Notomi, E. Kuramochi, and H. Taniyama, “Ultrahigh-Q nanocavity with 1D photonic gap,” Opt. Express 16(15), 11095–11102 (2008).
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J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystal optomechanical cavity,” Opt. Express 17(5), 3802–3817 (2009).
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M. Brunstein, R. Braive, R. Hostein, A. Beveratos, I. Rober-Philip, I. Sagnes, T. J. Karle, A. M. Yacomotti, J. A. Levenson, V. Moreau, G. Tessier, and Y. De Wilde, “Thermo-optical dynamics in an optically pumped Photonic Crystal nano-cavity,” Opt. Express 17(19), 17118–17129 (2009).
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T. Tanabe, K. Nishiguchi, E. Kuramochi, and M. Notomi, “Low power and fast electro-optic silicon modulator with lateral p-i-n embedded photonic crystal nanocavity,” Opt. Express 17(25), 22505–22513 (2009).
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Opt. Lett. (4)

Phys. Rev. B (1)

S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K. Busch, A. Birner, U. Gösele, and V. Lehmann, “Tunable two-dimenisional photonic crystals using liquid crystal infiltration,” Phys. Rev. B 61(4), 2389 (2000).
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A. F. Koenderink, M. Kafesaki, B. C. Buchler, and V. Sandoghdar, “Controlling the resonance of a photonic crystal microcavity by a near-field probe,” Phys. Rev. Lett. 95(15), 153904 (2005).
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Science (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-Gap defect mode laser,” Science 284(5421), 1819–1821 (1999).
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Figures (8)

Fig. 1
Fig. 1 (a) Global SEM image of electrically tunable split nanobeam cavity with NEMS comb-drive actuators. The light grey area shows the released region, which is about 1.5 µm undercut enabling the comb-drive to move in the z direction. (b) The magnified SEM image for split nanobeam cavity. The width of the nanobeam, air-slot and middle hole diameter are 700 nm, 80 nm and 224 nm, respectively. (c) The magnified SEM image showing the actuator’s critical dimensions. The width of comb-drive fingers is 195 nm; initial finger overlap is 490 nm; the air gap between two adjacent fingers is 205 nm; the finger number is 41 on each side. (d) The magnified SEM image for NEMS springs. The width and length of the flexible beams are 300 nm and 14 μm respectively; (e) The magnified SEM image showing an addition air-slot in the waveguide, which is introduced to enable the cavity to move in the z direction. The slot width is 60 nm. The slot may introduce some very low Q factor Fabry-Perot (FP) modes. However, these modes have negligible effect on the experimental results because the Q factor of the cavity under investigation is much higher by orders of magnitude.
Fig. 2
Fig. 2 (a) Global SEM image of the laterally tunable split nanobeam cavity with a NEMS comb-drive actuator. (b) A magnified SEM image showing the center of the cavity as indicated in the red dotted box in (a). The lower part of the cavity is movable in the x-direction.
Fig. 3
Fig. 3 Simulated modal profile of the (a) first mode, (b) second mode, (c) third mode, (d) forth mode, and (e) fifth mode.
Fig. 4
Fig. 4 (a) Simulated resonant wavelengths for the fundamental (red), third (blue) and fifth (black) order modes change as functions of slot width (ws) from 0.04µm to 0.12µm. (b) Simulated Q-factor of the fundamental (red), third (blue) and fifth (black) order modes versus slot width (ws) from 0.04µm to 0.12µm.
Fig. 5
Fig. 5 Simulated resonant wavelengths for the fundamental (a), third (c) and fifth (e) order modes change as functions of lateral movement from 0 µm to 0.05 µm at four different widths of air-slot, and the simulated Q factors of the fundamental (b), third (d) and fifth (f) order modes versus lateral movement from 0 µm to 0.05 µm at four different widths of air-slot.
Fig. 6
Fig. 6 Schematic of setup used to characterize the tunable split nanobeam cavity. TLS, tunable laser source; FPC, fiber polarization controller; OSA, optical spectrum analyzer.
Fig. 7
Fig. 7 (a) The red and black curves give the experimentally-measured transmission spectrum of the split nanobeam cavity shown in Fig. 1(b) at the applied voltage of 0 and 18V, respectively. (b) The red curve shows the measured cavity resonance wavelength as a function of applied voltage Va. (c) The red curve shows the measured Q factor versus applied voltage Va.
Fig. 8
Fig. 8 (a) The black and red curves give the experimentally-measured transmission spectrum of the split nanobeam cavity shown in Fig. 2(b) at the applied voltage of 0 and 18V, respectively. (b) The red curve shows the measured cavity resonance wavelength versus applied voltage Va. (c) The red curve shows the measured Q factor as a function of applied voltage Va.

Equations (2)

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r n = r 0 [ 1( r 0 r 39 r 0 ) ( n 39 ) 2 ],
x= F e k = ( nεt s V a 2 ) / ( 4Et w 3 L 3 ) = nε L 3 V a 2 4Es w 3 ,

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