Abstract

We consider the lateral optical force between a resonator and a waveguide, and study the possibility of an equilibrium that occurs solely from the optical force in such system. We prove analytically that a single-resonance system cannot give such an equilibrium in the resonator-waveguide force. We then show that two-resonance systems can provide such an equilibrium. We provide an intuitive way to predict the existence of an equilibrium, and give numerical examples.

© 2013 OSA

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References

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  1. D. Van Thourhout and J. Roels, “Optomechnical device actuation through the optical gradient force,” Nat. Photonics4, 211–217 (2010).
    [CrossRef]
  2. P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express18, 14439–14453 (2010).
    [CrossRef] [PubMed]
  3. V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express17, 21897–21909 (2009).
    [CrossRef] [PubMed]
  4. W.H.P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating light-waves in parallel 3D waveguides,” Opt. Express7, 16032–16037 (2009).
    [CrossRef]
  5. J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
    [CrossRef] [PubMed]
  6. W. H. P. Pernice, M. Li, D. Garcia-Sanchez, and H. X. Tang, “Analysis of short range forces in opto-mechanical devices with a nanogap,” Opt. Express18, 12615–12621 (2010).
    [CrossRef] [PubMed]
  7. 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, 3042–3044 (2005).
    [CrossRef] [PubMed]
  8. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3, 464–468 (2009).
    [CrossRef]
  9. V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett.98, 241104 (2011).
    [CrossRef]
  10. A. Einat and U. Levy, “Analysis of the optical force in the micro ring resonator,” Opt. Express19, 20405–20419 (2011).
    [CrossRef] [PubMed]
  11. V. Intaraprasonk and S. Fan, “Enhancing the waveguide-resonator optical force with an all-optical on-chip analog of electromagnetically induced transparency,” Phys. Rev. A86, 063833 (2012).
    [CrossRef]
  12. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1, 416–422 (2007).
    [CrossRef]
  13. M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett.103, 223901 (2009).
    [CrossRef]
  14. M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators,” Opt. Express13, 8286–8295 (2005).
    [CrossRef] [PubMed]
  15. P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics1, 658–665 (2007).
    [CrossRef]
  16. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3, 478–483 (2009).
    [CrossRef]
  17. G. S. Wiederhecker, L. Chen, A. Gondarenk, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 633–636 (2009).
    [CrossRef] [PubMed]
  18. G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express19, 2782–2790 (2011).
    [CrossRef] [PubMed]
  19. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express15, 17172–17205 (2007).
    [CrossRef] [PubMed]
  20. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555 (2009).
    [CrossRef] [PubMed]
  21. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, 1984).
  22. P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express17, 18116–18135 (2009).
    [CrossRef] [PubMed]
  23. www.comsol.com

2012 (1)

V. Intaraprasonk and S. Fan, “Enhancing the waveguide-resonator optical force with an all-optical on-chip analog of electromagnetically induced transparency,” Phys. Rev. A86, 063833 (2012).
[CrossRef]

2011 (3)

2010 (3)

2009 (9)

W.H.P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating light-waves in parallel 3D waveguides,” Opt. Express7, 16032–16037 (2009).
[CrossRef]

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3, 464–468 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett.103, 223901 (2009).
[CrossRef]

P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express17, 18116–18135 (2009).
[CrossRef] [PubMed]

V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express17, 21897–21909 (2009).
[CrossRef] [PubMed]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3, 478–483 (2009).
[CrossRef]

G. S. Wiederhecker, L. Chen, A. Gondarenk, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 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,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

2007 (3)

T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express15, 17172–17205 (2007).
[CrossRef] [PubMed]

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics1, 658–665 (2007).
[CrossRef]

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1, 416–422 (2007).
[CrossRef]

2005 (2)

Baets, R.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[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,” Nature459, 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,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

Chen, L.

G. S. Wiederhecker, L. Chen, A. Gondarenk, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 633–636 (2009).
[CrossRef] [PubMed]

Davids, P.

De Vlaminck, I.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[CrossRef] [PubMed]

Eichenfield, M.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1, 416–422 (2007).
[CrossRef]

Einat, A.

Fan, S.

V. Intaraprasonk and S. Fan, “Enhancing the waveguide-resonator optical force with an all-optical on-chip analog of electromagnetically induced transparency,” Phys. Rev. A86, 063833 (2012).
[CrossRef]

V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett.98, 241104 (2011).
[CrossRef]

V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express17, 21897–21909 (2009).
[CrossRef] [PubMed]

Fong, K. Y.

W.H.P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating light-waves in parallel 3D waveguides,” Opt. Express7, 16032–16037 (2009).
[CrossRef]

Garcia-Sanchez, D.

Gondarenk, A.

G. S. Wiederhecker, L. Chen, A. Gondarenk, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 633–636 (2009).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, 1984).

Ibanescu, M.

Intaraprasonk, V.

V. Intaraprasonk and S. Fan, “Enhancing the waveguide-resonator optical force with an all-optical on-chip analog of electromagnetically induced transparency,” Phys. Rev. A86, 063833 (2012).
[CrossRef]

V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett.98, 241104 (2011).
[CrossRef]

Ippen, E. P.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics1, 658–665 (2007).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kippenberg, T. J.

Lagae, L.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[CrossRef] [PubMed]

Lee, S.

Levy, U.

Li, M.

W. H. P. Pernice, M. Li, D. Garcia-Sanchez, and H. X. Tang, “Analysis of short range forces in opto-mechanical devices with a nanogap,” Opt. Express18, 12615–12621 (2010).
[CrossRef] [PubMed]

W.H.P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating light-waves in parallel 3D waveguides,” Opt. Express7, 16032–16037 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3, 464–468 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett.103, 223901 (2009).
[CrossRef]

Lin, Q.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3, 478–483 (2009).
[CrossRef]

Lipson, M.

G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express19, 2782–2790 (2011).
[CrossRef] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenk, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 633–636 (2009).
[CrossRef] [PubMed]

Liu, V.

Loncar, M.

Maes, B.

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[CrossRef] [PubMed]

Manipatruni, S.

Michael, C. P.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1, 416–422 (2007).
[CrossRef]

Painter, O.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3, 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,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1, 416–422 (2007).
[CrossRef]

Perahia, R.

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1, 416–422 (2007).
[CrossRef]

Pernice, W. H. P.

W. H. P. Pernice, M. Li, D. Garcia-Sanchez, and H. X. Tang, “Analysis of short range forces in opto-mechanical devices with a nanogap,” Opt. Express18, 12615–12621 (2010).
[CrossRef] [PubMed]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3, 464–468 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett.103, 223901 (2009).
[CrossRef]

Pernice, W.H.P.

W.H.P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating light-waves in parallel 3D waveguides,” Opt. Express7, 16032–16037 (2009).
[CrossRef]

Popovic, M. A.

P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express17, 18116–18135 (2009).
[CrossRef] [PubMed]

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics1, 658–665 (2007).
[CrossRef]

Povinelli, M.

Povinelli, M. L.

Rakich, P. T.

Roels, J.

D. Van Thourhout and J. Roels, “Optomechnical device actuation through the optical gradient force,” Nat. Photonics4, 211–217 (2010).
[CrossRef]

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[CrossRef] [PubMed]

Rosenberg, J.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3, 478–483 (2009).
[CrossRef]

Smythe, E. J.

Soljacic, M.

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics1, 658–665 (2007).
[CrossRef]

Tang, H. X.

W. H. P. Pernice, M. Li, D. Garcia-Sanchez, and H. X. Tang, “Analysis of short range forces in opto-mechanical devices with a nanogap,” Opt. Express18, 12615–12621 (2010).
[CrossRef] [PubMed]

W.H.P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating light-waves in parallel 3D waveguides,” Opt. Express7, 16032–16037 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3, 464–468 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett.103, 223901 (2009).
[CrossRef]

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express15, 17172–17205 (2007).
[CrossRef] [PubMed]

Van Thourhout, D.

D. Van Thourhout and J. Roels, “Optomechnical device actuation through the optical gradient force,” Nat. Photonics4, 211–217 (2010).
[CrossRef]

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[CrossRef] [PubMed]

Wang, Z.

Wiederhecker, G. S.

G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express19, 2782–2790 (2011).
[CrossRef] [PubMed]

G. S. Wiederhecker, L. Chen, A. Gondarenk, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 633–636 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett.98, 241104 (2011).
[CrossRef]

Nat. Nanotechnol. (1)

J. Roels, I. De Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol.4, 510–513 (2009).
[CrossRef] [PubMed]

Nat. Photonics (5)

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics3, 464–468 (2009).
[CrossRef]

D. Van Thourhout and J. Roels, “Optomechnical device actuation through the optical gradient force,” Nat. Photonics4, 211–217 (2010).
[CrossRef]

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics1, 416–422 (2007).
[CrossRef]

P. T. Rakich, M. A. Popovic, M. Soljacic, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics1, 658–665 (2007).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics3, 478–483 (2009).
[CrossRef]

Nature (2)

G. S. Wiederhecker, L. Chen, A. Gondarenk, and M. Lipson, “Controlling photonic structures using optical forces,” Nature462, 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,” Nature459, 550–555 (2009).
[CrossRef] [PubMed]

Opt. Express (9)

T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express15, 17172–17205 (2007).
[CrossRef] [PubMed]

P. T. Rakich, M. A. Popovic, and Z. Wang, “General treatment of optical forces and potentials in mechanically variable photonic systems,” Opt. Express17, 18116–18135 (2009).
[CrossRef] [PubMed]

V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express17, 21897–21909 (2009).
[CrossRef] [PubMed]

W. H. P. Pernice, M. Li, D. Garcia-Sanchez, and H. X. Tang, “Analysis of short range forces in opto-mechanical devices with a nanogap,” Opt. Express18, 12615–12621 (2010).
[CrossRef] [PubMed]

P. T. Rakich, P. Davids, and Z. Wang, “Tailoring optical forces in waveguides through radiation pressure and electrostrictive forces,” Opt. Express18, 14439–14453 (2010).
[CrossRef] [PubMed]

G. S. Wiederhecker, S. Manipatruni, S. Lee, and M. Lipson, “Broadband tuning of optomechanical cavities,” Opt. Express19, 2782–2790 (2011).
[CrossRef] [PubMed]

A. Einat and U. Levy, “Analysis of the optical force in the micro ring resonator,” Opt. Express19, 20405–20419 (2011).
[CrossRef] [PubMed]

W.H.P. Pernice, M. Li, K. Y. Fong, and H. X. Tang, “Modeling of the optical force between propagating light-waves in parallel 3D waveguides,” Opt. Express7, 16032–16037 (2009).
[CrossRef]

M. L. Povinelli, S. G. Johnson, M. Loncar, M. Ibanescu, E. J. Smythe, F. Capasso, and J. D. Joannopoulos, “High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery-mode resonators,” Opt. Express13, 8286–8295 (2005).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (1)

V. Intaraprasonk and S. Fan, “Enhancing the waveguide-resonator optical force with an all-optical on-chip analog of electromagnetically induced transparency,” Phys. Rev. A86, 063833 (2012).
[CrossRef]

Phys. Rev. Lett. (1)

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett.103, 223901 (2009).
[CrossRef]

Other (2)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, 1984).

www.comsol.com

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic for a travelling-wave ring resonator. The ring and the waveguide are 0.2μm wide. The radius of the semi-circular part of the resonator is 1.5μm and the straight section is 1.1μm long. The waveguide and the resonator have permittivity of 12.1 and are surrounded by air with permittivity of 1. The red color shows the local light intensity (proportional to the square of the electric field). (b) Schematic of a general resonator-waveguide system. The resonator has a single resonance at ω0. The incident light enters the input waveguide on the port marked with a big green arrow. The coupling rate between the resonator and the input waveguide is γe. The resonator has an intrinsic loss rate of γ0.

Fig. 2
Fig. 2

The force lineshapes for three regimes of gom. The dashed lines are at a smaller waveguide-resonator distance d than the solid lines. Notice that the force vanishes at the same frequency for different d.

Fig. 3
Fig. 3

Simulation result for the system in Fig. 1(a). (The length unit a = 1μm.) (a) T as a function of ω for d = 0.2μm (dash-dotted line), d = 0.25μm (dashed), and d = 0.3μm (solid). (b) Fitted values of γ0 (solid), γe (dashed) and |ω0ω| (dash-dotted) as a function of d. (c) F as a function of ω for the same d as (a). (d) The sign of F as a function of ω and d (shaded means attractive, white means repulsive). The dashed line is ω0 as a function of d.

Fig. 4
Fig. 4

(a) Force lineshape for a system with two gom< 0 resonances. The top plot is at smaller d than the bottom plot. The dashed lines are the contributions from each resonance. The solid lines are the total force. The frequency range between the zeros of force is shaded. (b) Same as (a) but with two gom> 0 resonances.

Fig. 5
Fig. 5

(a) and (b) the schematic of the ring resonator in Fig. 1(b) with a small circular bump (radius of 0.06μm). The red color shows the light intensity of the two modes. (c) T and (d) F as a function of ω for d = 0.20μm (dashed line), and d = 0.32μm (solid). (e) F as a function of d at ω = ωe≡ 0.655 × 2πc/a. (f) The sign of F as a function of ω and d (shaded means attractive, white means repulsive). The bracket denotes the frequency range where an optical equilibrium occurs.

Fig. 6
Fig. 6

The schematic of the two-mode ring resonator with an inner radius of 1.80μm and an outer radius of 2.15μm. The red color shows the light intensity. (a) the first-order mode. (b) the second-order mode. (c) The ratio of the energy inside the resonator to the input power (E) as a function of ω at d = 0.18μm. Dark green arrows denote the first-order modes. Light orange arrows denote the second-order modes.

Fig. 7
Fig. 7

F for a system in Fig. 6 near the frequency of 0.665c/a. (a) F as a function of ω for d = 0.06μm (dashed line), and d = 0.1μm (solid). (b) The sign of F as a function of ω and d (shaded means attractive, white means repulsive). ωe = 0.655 × 2πc/a. The bracket denotes the frequency range where an optical equilibrium occurs.

Fig. 8
Fig. 8

F for a system in Fig. 6 near the frequency of 0.645c/a. (a) F as a function of ω for d = 0.06μm (dashed line), and d = 0.1μm (solid). (b) The sign of F as a function of ω and d (shaded means attractive, white means repulsive).

Equations (20)

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η = e i θ 1 2 γ e / t r i Δ + γ 0 + γ e
t = e i θ 2 i Δ + γ 0 γ e i Δ + γ 0 + γ e
F = 1 ω i P i d ϕ i
F = P out ω d ϕ out P loss ω d ϕ loss .
P out = P in T
ϕ out = arg ( t )
P loss = P in ( 1 T )
ϕ loss = arg ( η )
F = 2 P in ω ( γ om Δ Δ 2 + ( γ 0 + γ e ) 2 ) 2 P in ω ( g om γ e Δ 2 + ( γ 0 + γ e ) 2 )
g om = d ω 0
γ om = d γ e .
γ e = Γ exp ( κ d )
ω 0 = ω + Ω exp ( κ d )
γ om = κ Γ exp ( κ d ) = κ γ e
g om = κ Ω exp ( κ d ) .
ω z = ω .
γ e = Γ exp ( κ d f 2 + z 2 )
ω 0 = ω + Ω exp ( κ d f 2 + z 2 )
F z = 2 P in ω γ e Δ 2 + ( γ 0 + γ e ) 2 ( ω ω ) κ z d f 2 + z 2 .
F = i = 1 2 2 P i ω ( γ om , i Δ i + g om , i γ e , i Δ i 2 + ( γ 0 , i + γ e , i ) 2 )

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