Abstract

We demonstrate gradient optical forces in metal-dielectric hybrid plasmonic waveguides (HPWG) for the first time. The magnitude of optical force is quantified through excitation of the nanomechanical vibration of the suspended waveguides. Integrated Mach-Zehnder interferometry is utilized to transduce the mechanical motion and characterize the propagation loss of the HPWG. Compared with theory, the experimental results have confirmed the optical force enhancement, but also suggested a significantly higher optical loss in HPWG. The excessive loss is attributed to metal surface roughness and other non-idealities in the device fabrication process.

©2013 Optical Society of America

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    [Crossref]

2012 (4)

H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun 3, 1091 (2012).
[Crossref] [PubMed]

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14(9), 095014 (2012).
[Crossref]

M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. 51, 06FC05 (2012).
[Crossref]

2011 (7)

K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express 19(16), 15098–15108 (2011).
[Crossref] [PubMed]

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6(11), 726–732 (2011).
[Crossref] [PubMed]

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano 5(4), 2831–2838 (2011).
[Crossref] [PubMed]

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

2010 (3)

M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010).
[Crossref] [PubMed]

C. Huang and L. Zhu, “Enhanced optical forces in 2D hybrid and plasmonic waveguides,” Opt. Lett. 35(10), 1563–1565 (2010).
[Crossref] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

2009 (9)

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325(5940), 594–597 (2009).
[Crossref] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[Crossref] [PubMed]

P. T. Rakich, M. A. Popović, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express 17(20), 18116–18135 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[Crossref] [PubMed]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[Crossref]

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

F. Marquardt and S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

2008 (4)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. 93(11), 113110 (2008).
[Crossref]

2006 (1)

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97(24), 243905 (2006).
[Crossref] [PubMed]

2005 (2)

M. L. Povinelli, M. Loncar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30(22), 3042–3044 (2005).
[Crossref] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

1997 (1)

M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. 81(6), 2480 (1997).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Aksyuk, V.

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Atwater, H. A.

M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. 93(11), 113110 (2008).
[Crossref]

Baehr-Jones, T.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Bagheri, M.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6(11), 726–732 (2011).
[Crossref] [PubMed]

Bartal, G.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010).
[Crossref] [PubMed]

Bergh, H. S.

M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. 81(6), 2480 (1997).
[Crossref]

Børkje, K.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

Brown, D. E.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

Capasso, F.

Chan, J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Chen, Y.

H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun 3, 1091 (2012).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Davanço, M.

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Del’Haye, P.

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97(24), 243905 (2006).
[Crossref] [PubMed]

Deléglise, S.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

Dew, S.

M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. 51, 06FC05 (2012).
[Crossref]

Eichenfield, M.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

Fito, T.

M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. 51, 06FC05 (2012).
[Crossref]

Fong, K. Y.

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14(9), 095014 (2012).
[Crossref]

K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express 19(16), 15098–15108 (2011).
[Crossref] [PubMed]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Girvin, S.

F. Marquardt and S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Girvin, S. M.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

Gondarenko, A.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

Hiller, J. M.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

Hochberg, M.

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Hua, J.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

Huang, C.

Ibanescu, M.

Ishikawa, A.

X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano 5(4), 2831–2838 (2011).
[Crossref] [PubMed]

Joannopoulos, J. D.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Johnson, S. G.

Kimball, C. W.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

Kippenberg, T. J.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97(24), 243905 (2006).
[Crossref] [PubMed]

Koshelev, K.

M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. 51, 06FC05 (2012).
[Crossref]

Kuttge, M.

M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. 93(11), 113110 (2008).
[Crossref]

Lai, J.

M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. 81(6), 2480 (1997).
[Crossref]

Lezec, H. J.

M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. 93(11), 113110 (2008).
[Crossref]

Li, H.

H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun 3, 1091 (2012).
[Crossref] [PubMed]

Li, M.

H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun 3, 1091 (2012).
[Crossref] [PubMed]

K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express 19(16), 15098–15108 (2011).
[Crossref] [PubMed]

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6(11), 726–732 (2011).
[Crossref] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Lin, Q.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[Crossref]

Lindquist, N. C.

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325(5940), 594–597 (2009).
[Crossref] [PubMed]

Lipson, M.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Liu, M.

M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010).
[Crossref] [PubMed]

Liu, Y.

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010).
[Crossref] [PubMed]

Loncar, M.

Majumdar, A.

M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. 81(6), 2480 (1997).
[Crossref]

Marquardt, F.

F. Marquardt and S. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

McFarland, E.

M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. 81(6), 2480 (1997).
[Crossref]

Miao, H.

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Mohammad, M. A.

M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. 51, 06FC05 (2012).
[Crossref]

Nagpal, P.

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325(5940), 594–597 (2009).
[Crossref] [PubMed]

Noh, J.

H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun 3, 1091 (2012).
[Crossref] [PubMed]

Nooshi, N.

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97(24), 243905 (2006).
[Crossref] [PubMed]

Norris, D. J.

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325(5940), 594–597 (2009).
[Crossref] [PubMed]

Nunnenkamp, A.

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

Oh, S.-H.

P. Nagpal, N. C. Lindquist, S.-H. Oh, and D. J. Norris, “Ultrasmooth patterned metals for plasmonics and metamaterials,” Science 325(5940), 594–597 (2009).
[Crossref] [PubMed]

Oulton, R. F.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Painter, O.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[Crossref]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

Pearson, J.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

Pernice, W. H. P.

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14(9), 095014 (2012).
[Crossref]

K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express 19(16), 15098–15108 (2011).
[Crossref] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Pernice, W. P. H.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6(11), 726–732 (2011).
[Crossref] [PubMed]

Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Polman, A.

M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. 93(11), 113110 (2008).
[Crossref]

Poot, M.

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6(11), 726–732 (2011).
[Crossref] [PubMed]

Popovic, M. A.

Povinelli, M. L.

Rakher, M. T.

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Rakich, P. T.

Rosenberg, J.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
[Crossref]

Salapaka, M. V.

M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. 81(6), 2480 (1997).
[Crossref]

Schliesser, A.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97(24), 243905 (2006).
[Crossref] [PubMed]

Schuck, C.

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14(9), 095014 (2012).
[Crossref]

Smythe, E. J.

Sorger, V. J.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Srinivasan, K.

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Stepanova, M.

M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. 51, 06FC05 (2012).
[Crossref]

Sun, X.

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14(9), 095014 (2012).
[Crossref]

Tadesse, S.

H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun 3, 1091 (2012).
[Crossref] [PubMed]

Tang, H. X.

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14(9), 095014 (2012).
[Crossref]

K. Y. Fong, W. H. P. Pernice, M. Li, and H. X. Tang, “Tunable optical coupler controlled by optical gradient forces,” Opt. Express 19(16), 15098–15108 (2011).
[Crossref] [PubMed]

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6(11), 726–732 (2011).
[Crossref] [PubMed]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17(3), 1806–1816 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3(8), 464–468 (2009).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459(7246), 550–555 (2009).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiation pressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 97(24), 243905 (2006).
[Crossref] [PubMed]

Verhagen, E.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

Verhoeven, J.

M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. 93(11), 113110 (2008).
[Crossref]

Vesseur, E. J. R.

M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. 93(11), 113110 (2008).
[Crossref]

Vlasko-Vlasov, V. K.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

Wang, Y.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

Wang, Z.

Weis, S.

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482(7383), 63–67 (2012).
[Crossref] [PubMed]

Welp, U.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

Wiederhecker, G. S.

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
[Crossref] [PubMed]

Xiong, C.

C. Xiong, W. H. P. Pernice, X. Sun, C. Schuck, K. Y. Fong, and H. X. Tang, “Aluminum nitride as a new material for chip-scale optomechanics and nonlinear optics,” New J. Phys. 14(9), 095014 (2012).
[Crossref]

M. Li, W. H. P. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456(7221), 480–484 (2008).
[Crossref] [PubMed]

Yang, X.

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano 5(4), 2831–2838 (2011).
[Crossref] [PubMed]

Ye, Z.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

Yin, L.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

Yin, X.

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano 5(4), 2831–2838 (2011).
[Crossref] [PubMed]

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

Zentgraf, T.

M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010).
[Crossref] [PubMed]

Zhang, X.

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano 5(4), 2831–2838 (2011).
[Crossref] [PubMed]

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

Zheng, D. A. Z.

M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. 51, 06FC05 (2012).
[Crossref]

Zhu, L.

ACS Nano (1)

X. Yang, A. Ishikawa, X. Yin, and X. Zhang, “Hybrid photonic-plasmonic crystal nanocavities,” ACS Nano 5(4), 2831–2838 (2011).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

M. Kuttge, E. J. R. Vesseur, J. Verhoeven, H. J. Lezec, H. A. Atwater, and A. Polman, “Loss mechanisms of surface plasmon polaritons on gold probed by cathodoluminescence imaging spectroscopy,” Appl. Phys. Lett. 93(11), 113110 (2008).
[Crossref]

J. Appl. Phys. (1)

M. V. Salapaka, H. S. Bergh, J. Lai, A. Majumdar, and E. McFarland, “Multi-mode noise analysis of cantilevers for scanning probe microscopy,” J. Appl. Phys. 81(6), 2480 (1997).
[Crossref]

Jpn. J. Appl. Phys. (1)

M. A. Mohammad, K. Koshelev, T. Fito, D. A. Z. Zheng, M. Stepanova, and S. Dew, “Study of development processes for ZEP-520 as a high-resolution positive and negative tone electron beam lithography resist,” Jpn. J. Appl. Phys. 51, 06FC05 (2012).
[Crossref]

Nano Lett. (3)

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Hiller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5(7), 1399–1402 (2005).
[Crossref] [PubMed]

X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
[Crossref] [PubMed]

K. Srinivasan, H. Miao, M. T. Rakher, M. Davanço, and V. Aksyuk, “Optomechanical transduction of an integrated silicon cantilever probe using a microdisk resonator,” Nano Lett. 11(2), 791–797 (2011).
[Crossref] [PubMed]

Nat Commun (1)

H. Li, Y. Chen, J. Noh, S. Tadesse, and M. Li, “Multichannel cavity optomechanics for all-optical amplification of radio frequency signals,” Nat Commun 3, 1091 (2012).
[Crossref] [PubMed]

Nat. Commun. (1)

V. J. Sorger, Z. Ye, R. F. Oulton, Y. Wang, G. Bartal, X. Yin, and X. Zhang, “Experimental demonstration of low-loss optical waveguiding at deep sub-wavelength scales,” Nat. Commun. 2, 331 (2011).
[Crossref]

Nat. Nanotechnol. (3)

M. Liu, T. Zentgraf, Y. Liu, G. Bartal, and X. Zhang, “Light-driven nanoscale plasmonic motors,” Nat. Nanotechnol. 5(8), 570–573 (2010).
[Crossref] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnol. 4(6), 377–382 (2009).
[Crossref] [PubMed]

M. Bagheri, M. Poot, M. Li, W. P. H. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6(11), 726–732 (2011).
[Crossref] [PubMed]

Nat. Photonics (4)

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J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3(8), 478–483 (2009).
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D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
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Nature (4)

G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature 462(7273), 633–636 (2009).
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Figures (7)

Fig. 1
Fig. 1 HPWG structure, optical modes and optical force comparison. (a) HPWG structure illustration. (b) and (c) Transverse electric field component of the HPM0 and HPM1, respectively, with the overlaid green curves showing the cross-sectional profile of the transverse electric field. (d) Comparison of the dependence of the total optical force on the waveguide length in different waveguide structures from literatures and this work. Solid lines represent experimental work while the other line styles represent theoretical calculation.

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Fig. 2
Fig. 2 2D FDTD simulation of the mode evolution between fundamental TE SWM and HPM. (a) The overview of the simulated HPWG structure overlaid by the transverse electric field component. The red arrows indicate where the SWM enters and exits. (b) A close-up view of the enhanced transverse electric field component in the nano gap. (c) Mode evolution from SWM to HPM, and then back to SWM again. The red curves show the cross sections of the transverse electric field component in the evolving mode.

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Fig. 3
Fig. 3 HPWG local optical force and mechanical properties. (a) Local optical force and beam mechanical in-plane mode profile. (b) Comparison of the excitation of the mechanical modes by different force distributions.

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Fig. 4
Fig. 4 Fabrication process diagram of the HPWG device. All of the EBL processes are done with Vistec EBPG 5000 + system, with which 20 nm alignment precision can be routinely achieved.

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Fig. 5
Fig. 5 Images of the fabricated HPWG device. (a) Optical microscope image showing the device overview. (b) Optical microscope image showing the HPWG with gold and suspended Si waveguide. (c) SEM image showing the HPWG with gold and suspended Si waveguide. (d) SEM image showing the gap in the HPWG.

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Fig. 6
Fig. 6 Optical characteristics of the fabricated HPWGs with 450 nm wide Si waveguides. The measured HPWG loss is about 30 times higher than theoretical calculation. (a) MZI transmission spectra showing deceasing extinction ratio as the length of the HPWG is increased. The gap is 100 nm. (b) Experimental and theoretical results of the waveguide loss as a function of HPWG length. The gap is 100 nm. (c) Experimental and theoretical results of the waveguide loss normalized to unit HPWG length as a function of the HPWG gap.

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Fig. 7
Fig. 7 Optomechanical measurements of the HPWG. (a) Thermomechanical noise measurement for transduction calibration. In this case, G1 = 4.82 V/nm. (b) Normalized driven responses from 3 representative devices, showing peaks at their respective resonance frequencies. The length and gap size of each device are labeled nearby its resonance peak. (c) The measured normalized local optical forces plotted against the gap width, for two different Si waveguide widths in the HPWG. The symbols are experimental results while the dotted lines are simulation results.

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Tables (1)

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Table 1 Definitions of the terms of optical forces

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Equations (19)

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f o ( ω o ,q )= 1 c ∂Re[ n eff ( ω o ,q ) ] ∂q .
F n = ∫ 0 L | p( x ) |dx P( 0 ) = p n ∫ 0 L e −αx dx = p n α ( 1− e −αL ).
EI ∂ 4 u( x,t ) ∂ x 4 +ρA ∂ 2 u( x,t ) ∂ t 2 =0,
u( x,t )= ∑ j=1 ∞ C j sin( ω j t+ δ j ) ϕ j ( x ) , with
ϕ j ( x )=[ sin( λ j L )−sinh( λ j L ) ][ cos( λ j x )−cosh( λ j x ) ] −[ sin( λ j x )−sinh( λ j x ) ][ cos( λ j L )−cosh( λ j L ) ],
cos( λ j L )cosh( λ j L )=1.
( λ j L ) 4 = ω j 2 ρA L 4 EI .
u( x,t )= ∑ j=1 ∞ ϕ j ( x ) q j ( t ) .
m j q ¨ j ( t )+ k j m j Q j q ˙ j ( t )+ k j q j ( t )= p j ( t ).
p j ( t )= ∫ 0 L p( x,t ) ϕ j ( x )dx .
m j =ρA ∫ 0 L ( ϕ j ) 2 dx and k j =EI ∫ 0 L ( ϕ ′ ′ j ) 2 dx .
N j = p j ( t ) ∫ 0 L | p( x,t ) |dx ​ 1 L ∫ 0 L ϕ j 2 ( x )dx .
S q j q j ( ω )= k B T ω j 2 Q j k j ( ω− ω j ) 2 + ω j 2 4 Q j 2 ,
p j ( t )= ∫ 0 L p( 0,t ) e −αx ϕ j ( x )dx = p n P( 0,t ) ∫ 0 L e −αx ϕ j ( x )dx ,
∫ e −αx ϕ j ( x )dx = α 3 ϕ j ( x )+ α 2 ϕ ′ j ( x )+α ϕ ′ ′ j ( x )+ ϕ ′ ′ ′ j ( x ) λ j 4 − α 4 e −αx .
H j ( ω )= q ˜ j ( ω ) p ˜ j ( ω ) = 1 − ω 2 m j +iω k j m j Q j + k j ,
| H j ( ω j ) |= | q ˜ j ( ω j ) | | p ˜ j ( ω j ) | = Q j k j .
p n = 1 ∫ 0 L e −αx ϕ j ( x )dx k j Q j | q ˜ j ( ω j ) | | P ˜ ( 0, ω j ) | .
α=− 20 L log 10 ( ER −1 ER +1 ) [ dB/m ].

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