Abstract

We extend the response theory of optical forces to general electromagnetic systems which can be treated as multi-port systems with multiple mechanical degrees of freedom. We demonstrate a fundamental link between the scattering properties of an optical system to its ability to produce conservative or non-conservative optical forces. Through the exploration of two nontrivial two-port systems, including an analytical Fabry-Perot interferometer and a more complex particle-in-a-waveguide structure, we show perfect agreement between the response theory and numerical first-principle calculations. We show that new insights into the origins of optical forces from the response theory provide clear means of understanding conservative and non-conservative forces in a regime where traditional gradient force picture fails.

© 2011 OSA

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2011 (2)

K. Dholakia and T. ?ižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

E. R. Shanblatt and D. G. Grier, “Extended and knotted optical traps in three dimensions,” Opt. Express 19(7), 5833–5838 (2011).
[CrossRef] [PubMed]

2010 (2)

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[CrossRef]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

2009 (8)

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]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103(10), 103601 (2009).
[CrossRef] [PubMed]

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, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

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

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 010401 (2009).
[CrossRef] [PubMed]

2008 (5)

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[CrossRef] [PubMed]

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[CrossRef]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[CrossRef] [PubMed]

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]

2007 (3)

P. T. Rakich, M. A. Popovi?, M. Solja?i?, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 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. Photonics 1(7), 416–422 (2007).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[CrossRef] [PubMed]

2005 (6)

2004 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

2002 (1)

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[CrossRef]

2000 (1)

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).
[CrossRef]

1997 (1)

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94(10), 4853–4860 (1997).
[CrossRef] [PubMed]

1992 (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[CrossRef] [PubMed]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[CrossRef] [PubMed]

Ashkin, A.

A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).
[CrossRef]

A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94(10), 4853–4860 (1997).
[CrossRef] [PubMed]

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[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]

Camacho, R.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

Camacho, R. M.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

Capasso, F.

Carmon, T.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

Chan, J.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

Chang, D.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

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]

Cižmár, T.

K. Dholakia and T. ?ižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

Darby, E.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 010401 (2009).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia and T. ?ižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[CrossRef]

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[CrossRef]

Eichenfield, M.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (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. Photonics 1(7), 416–422 (2007).
[CrossRef]

Erickson, D.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

Fan, S.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[CrossRef]

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]

Grier, D. G.

E. R. Shanblatt and D. G. Grier, “Extended and knotted optical traps in three dimensions,” Opt. Express 19(7), 5833–5838 (2011).
[CrossRef] [PubMed]

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 010401 (2009).
[CrossRef] [PubMed]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[CrossRef] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[CrossRef] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Grosberg, A. Y.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 010401 (2009).
[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]

Ibanescu, M.

Ippen, E. P.

P. T. Rakich, M. A. Popovi?, M. Solja?i?, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Jiang, X.

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103(10), 103601 (2009).
[CrossRef] [PubMed]

Joannopoulos, J. D.

Johnson, S. G.

Kippenberg, T. J.

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

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

Klug, M.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

Li, M.

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]

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, J.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 010401 (2009).
[CrossRef] [PubMed]

Lin, Q.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103(10), 103601 (2009).
[CrossRef] [PubMed]

Lipson, M.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

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

Loncar, M.

Lonèar, M.

Mansuripur, M.

Mazilu, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[CrossRef]

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. Photonics 1(7), 416–422 (2007).
[CrossRef]

Moloney, J. V.

Moore, S. D.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

Painter, O.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103(10), 103601 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (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. Photonics 1(7), 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. Photonics 1(7), 416–422 (2007).
[CrossRef]

Pernice, W. H. P.

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]

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]

Popovic, M. A.

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

P. T. Rakich, M. A. Popovi?, M. Solja?i?, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Povinelli, M. L.

Rakich, P. T.

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

P. T. Rakich, M. A. Popovi?, M. Solja?i?, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Roels, J.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[CrossRef]

Roichman, Y.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[CrossRef] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[CrossRef] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[CrossRef] [PubMed]

Rokhsari, H.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

Rosenberg, J.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103(10), 103601 (2009).
[CrossRef] [PubMed]

Scherer, A.

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

Schmidt, B. S.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

Shanblatt, E. R.

Smythe, E. J.

Soljacic, M.

P. T. Rakich, M. A. Popovi?, M. Solja?i?, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Stolarski, A.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[CrossRef] [PubMed]

Sun, B.

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 010401 (2009).
[CrossRef] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[CrossRef] [PubMed]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[CrossRef] [PubMed]

Tang, H. X.

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.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103(10), 103601 (2009).
[CrossRef] [PubMed]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (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]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

Van Thourhout, D.

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[CrossRef]

Wang, Z.

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.

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, A. H. J.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

Yang, L.

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

Zakharian, A. R.

Biophys. J. (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992).
[CrossRef] [PubMed]

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A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. 6(6), 841–856 (2000).
[CrossRef]

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M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[CrossRef]

Nat. Nanotechnol. (1)

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]

Nat. Photonics (6)

P. T. Rakich, M. A. Popovi?, M. Solja?i?, and E. P. Ippen, “Trapping, corralling and spectral bonding of optical resonances through optically induced potentials,” Nat. Photonics 1(11), 658–665 (2007).
[CrossRef]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4(4), 236–242 (2010).
[CrossRef]

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, “Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces,” Nat. Photonics 1(7), 416–422 (2007).
[CrossRef]

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]

D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4(4), 211–217 (2010).
[CrossRef]

K. Dholakia and T. ?ižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[CrossRef]

Nature (5)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

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, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462(7269), 78–82 (2009).
[CrossRef] [PubMed]

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[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]

Opt. Express (7)

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[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

B. Sun, J. Lin, E. Darby, A. Y. Grosberg, and D. G. Grier, “Brownian vortexes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(1), 010401 (2009).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101(12), 128301 (2008).
[CrossRef] [PubMed]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[CrossRef] [PubMed]

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103(10), 103601 (2009).
[CrossRef] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, “Temporal behavior of radiation-pressure-induced vibrations of an optical microcavity phonon mode,” Phys. Rev. Lett. 94(22), 223902 (2005).
[CrossRef] [PubMed]

T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity,” Phys. Rev. Lett. 95(3), 033901 (2005).
[CrossRef] [PubMed]

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A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. 94(10), 4853–4860 (1997).
[CrossRef] [PubMed]

Science (1)

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

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Figures (6)

Fig. 1
Fig. 1

Schematic of four main types of mechanically-variable optical systems: (a) a single-port system with a single degree of freedom (DOF); (b) a single-port system with n independent DOFs; (c) an M-port system with a single DOF; and (d) an M-port system with n independent DOFs.

Fig. 2
Fig. 2

Schematic of the field amplitudes in a lossless Fabry-Perot interferometer in vacuum. The incident wave enters the interferometer at normal angle from the left. The scattering matrix of the entire interferometer is taken at the two fixed reference planes. The cavity length l0 is the driving degrees of freedom and is marked in blue.

Fig. 3
Fig. 3

(a) Schematic of a high-index scattering cylinder (green) in a single-mode parallel-plate waveguide, with a width of a. The red arrow represents the incident direction. The blue and red regions are Ez fields with large positive and negative values. (b) Top view of the structure, where the blue and red regions are Ez fields. (c) Intensity distribution of the incident optical fields at power Pincident. After scattering, the transmitted field has a power of Pt and a phase of ϕ t , while the reflected field has a power of Pr and a phase of ϕ r . (d) The gradient force distribution derived from the intensity distribution. The scattering force follows the lateral distribution of the intensity plotted in (e).

Fig. 4
Fig. 4

The axial optical force (a) and the lateral force (b) are calculated from the Maxwell stress tensor (circles) for a silicon cylinder of radius 0.05a and from RTOF method (blue curve), in excellent agreement. The incident frequency is 1.36(c/2a) in both cases. Both Faxial (d) and Flateral (e and f) profiles exhibit large frequency dependence. Faxial only contains contributions from the reflection, while Flateral profile can be decomposed into the contributions from the reflection and from the transmission. All force profiles are plotted in a range slightly smaller than the full width of the waveguide, due to the finite size of the cylinder. The incident power is 1W/m in all cases.

Fig. 5
Fig. 5

Calculated axial forces and lateral forces as a function of the normalized frequency of the incidence and the location of the cylinder y. Three cases of cylinder radius, 0.05a, 0.075a and 0.125a, are compared. In all cases, only left half of the waveguide is shown, since with respect to the center, Faxial is symmetric and Flateral is antisymmetric. The color scale is chosen to compare the force amplitude and the maximum value (out-of-scale in the dark red region) of Flateral is calculated to be 14.8nN/a and 18.6nN/a for r = 0.075a and 0.125a respectively. The incident power is 1W/m in all cases. The dash lines in (a) and (b) indicate the frequencies discussed in Fig. 4. The gray regions in (e), (f), (h) and (i) are areas where the cylinder cannot move into because of its finite size.

Fig. 6
Fig. 6

The optical force fields (black arrows) of the two cases illustrated in Fig. 5. Colored areas represent the curl of the force field along the out-of-plane z direction. When the cylinder takes the trajectory indicated by the white contour on the left in panel (a), optical forces do work to the cylinder, because the total curl within the contour does not vanish. Only special trajectories inside which the positive curl region cancel the negative curl region, such as the white contour on the right, do not perform work onto the cylinder. (c) Schematic of the origin of nonconservative optical forces on a reflector. The grey rectangle represents a reflector, and the black arrows illustrate the directions and the amplitudes of the optical forces.

Tables (1)

Tables Icon

Table 1 Optical forces of mechanically-variable systems categorized by the number of ports and the number of degrees of freedom

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

F q (ω)= P i ω ϕ(q,ω) q .
F k = P i ω · ϕ({ q n },ω) q k ,
F= P i ω · q ϕ(q,ω).
F q = 1 ω · j=1 M P j o (q,ω)· ϕ j o (q,ω) q .
F k = 1 ω · j=1 M P j o ( { q n },ω )· ϕ j o ( { q n },ω ) q k ,
F= 1 ω j=1 M P j o ( q,ω )· q ϕ j o ( q,ω ).
q ×F(q,ω)= 1 ω j=1 M q ×[ P j (q,ω) q ϕ j (q,ω) ]= 1 ω j=1 M q P j (q,ω)× q ϕ j (q,ω),
F 1 = | a 1 | 2 c 2 r 2 ( r 2 2cosδ+1 ) 12 r 2 cosδ+ r 4 , F 2 = | a 1 | 2 c 2 r 2 ( 1 r 2 ) 12 r 2 cosδ+ r 4 .
F 1 = | a 1 | 2 c 2 r 2 ( r 2 2cosδ+1 ) 12 r 2 cosδ+ r 4 .
F 2 = | a 1 | 2 c 2 r 2 ( 1 r 2 ) 12 r 2 cosδ+ r 4 .
F axial (y)= P r (y) ω ϕ r (x,y) x + P t (y) ω ϕ t (y) x = P r (y) ω ϕ r (x,y) x = P r (y) ω 2 k x
F lateral (y)= P r (y) ω ϕ r (x,y) y + P t (y) ω ϕ t (y) y
×F= 1 ω d P r dy d ϕ 1 dx z ^ = 2 k x ω d P r dy z ^ .

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