Abstract

We present an analytical formalism for the treatment of the forces and potentials induced by light in mechanically variable photonic systems (or optomechanically variable systems) consisting of linear media. Through energy and photon-number conservation, we show that knowledge of the phase and the amplitude response of an optomechanically variable system, and its dependence on the mechanical coordinate of interest, is sufficient to compute the forces produced by light. This formalism not only offers a simple analytical alternative to computationally intensive Maxwell stress-tensor methods, but also greatly simplifies the analysis of mechanically variable photonic systems driven by multiple external laser sources. Furthermore, we show, through this formalism, that a scalar optical potential can be derived in terms of the phase and amplitude response of an arbitrary optomechanically variable one-port system and in generalized optomechanically variable multi-port systems, provided that their optical response is variable through a single mechanical degree of freedom. With these simplifications, well-established theories of optical filter synthesis could be extended to allow for the synthesis of complex optical force and potential profiles, independent of the construction of the underlying device or its field distribution.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  6. M. Bhattacharya, and P. Meystre, "Trapping and cooling a mirror to its quantum mechanical ground state," Phys. Rev. Lett. 99, 073601 (2007).
    [CrossRef] [PubMed]
  7. M. Bhattacharya and P. Meystre, "Multiple membrane cavity optomechanics," Phys. Rev. A 78, 041801 (2008).
    [CrossRef]
  8. T. Carmon, H. Rokhsari, L. Yang, T. Kippenberg, and K. Vahala, "Temporal Behavior of Radiation-pressureinduced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 223902 (2005).
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    [CrossRef] [PubMed]
  12. M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, "Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces," Nat. Photonics 1, 416-422 (2007).
    [CrossRef]
  13. 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. Photonics 1, 658-665 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
  18. 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, 1806-1816 (2009).
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  25. W. Suh, M. F. Yanik, O. Solgaard, and S. H. Fan, "Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs," Appl. Phys. Lett. 82 (13), 1999-2001 (2003).
    [CrossRef]
  26. W. Suh, Z. Wang, and S. H. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
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    [CrossRef] [PubMed]
  34. G. T. Moore, "Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity," J. Math. Phys. 11, 2679-2691 (1970).
    [CrossRef]
  35. F. Gires, and P. Tournois "Interferometre utilisable pour la compression d’impulsions lumineuses modulees en frequence," C. R. Acad. Sci. Paris 25861126115 (1964).
  36. P. T. Rakich, M. A. Popovic, M. R. Watts, T. Barwicz, H. I. Smith, and E. P. Ippen, "Ultrawide tuning of photonic microcavities via evanescent field perturbation," Opt. Lett. 31, 1241-1243 (2006).
    [CrossRef] [PubMed]

2009 (3)

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, "A picogram- and nanometre-scale photoniccrystal optomechanical cavity," Nature 459, 550-555 (2009).
[CrossRef] [PubMed]

M. A. Popovic and P. T. Rakich, "Optonanomechanical self-adaptive photonic devices based on light forces: a path to robust high-index-contrast nanophotonic circuits," Proc. SPIE 7219, 72190A (Feb. 10, 2009)
[CrossRef]

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, 1806-1816 (2009).
[CrossRef] [PubMed]

2008 (2)

M. Bhattacharya and P. Meystre, "Multiple membrane cavity optomechanics," Phys. Rev. A 78, 041801 (2008).
[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, 480-484 (2008).
[CrossRef] [PubMed]

2007 (4)

M. Bhattacharya, and P. Meystre, "Trapping and cooling a mirror to its quantum mechanical ground state," Phys. Rev. Lett. 99, 073601 (2007).
[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, 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. Photonics 1, 658-665 (2007).
[CrossRef]

A. Mizrahi and L. Schchter, "Two-slab all-optical spring," Opt. Lett. 32, 692-694 (2007).
[CrossRef] [PubMed]

2006 (2)

P. T. Rakich, M. A. Popovic, M. R. Watts, T. Barwicz, H. I. Smith, and E. P. Ippen, "Ultrawide tuning of photonic microcavities via evanescent field perturbation," Opt. Lett. 31, 1241-1243 (2006).
[CrossRef] [PubMed]

M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, "Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs," Phys. Rev. Lett. 97, 023903 (2006).
[CrossRef] [PubMed]

2005 (5)

2004 (1)

W. Suh, Z. Wang, and S. H. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
[CrossRef]

2003 (1)

W. Suh, M. F. Yanik, O. Solgaard, and S. H. Fan, "Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs," Appl. Phys. Lett. 82 (13), 1999-2001 (2003).
[CrossRef]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

1994 (1)

C. K. Law, "Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium," Phys. Rev. A 49, 433-437 (1994).
[CrossRef] [PubMed]

1985 (2)

A. Ashkin, and J. M. Dziedzic, "Observation of radiation pressure trapping of particles by alternating lightbeams," Phys. Rev. Lett. 54, 1245-1248 (1985).
[CrossRef] [PubMed]

P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, "Theory of radiation pressure driven interferometers," J. Opt. Soc. Am. B 2, 1830-1840 (1985).
[CrossRef]

1983 (1)

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, "Optical bistability and mirror confinement induced by radiation pressure," Phys. Rev. Lett. 51, 1550-1553 (1983).
[CrossRef]

1978 (1)

A. Ashkin, "Trapping of atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
[CrossRef]

1970 (2)

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

G. T. Moore, "Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity," J. Math. Phys. 11, 2679-2691 (1970).
[CrossRef]

1964 (1)

F. Gires, and P. Tournois "Interferometre utilisable pour la compression d’impulsions lumineuses modulees en frequence," C. R. Acad. Sci. Paris 25861126115 (1964).

Ashkin, A.

A. Ashkin, and J. M. Dziedzic, "Observation of radiation pressure trapping of particles by alternating lightbeams," Phys. Rev. Lett. 54, 1245-1248 (1985).
[CrossRef] [PubMed]

A. Ashkin, "Trapping of atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
[CrossRef]

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 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, 480-484 (2008).
[CrossRef] [PubMed]

Barwicz, T.

Bhattacharya, M.

M. Bhattacharya and P. Meystre, "Multiple membrane cavity optomechanics," Phys. Rev. A 78, 041801 (2008).
[CrossRef]

M. Bhattacharya, and P. Meystre, "Trapping and cooling a mirror to its quantum mechanical ground state," Phys. Rev. Lett. 99, 073601 (2007).
[CrossRef] [PubMed]

Camacho, R.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, "A picogram- and nanometre-scale photoniccrystal optomechanical cavity," Nature 459, 550-555 (2009).
[CrossRef] [PubMed]

Capasso, F.

Carmon, T.

T. Carmon, H. Rokhsari, L. Yang, T. Kippenberg, and K. Vahala, "Temporal Behavior of Radiation-pressureinduced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 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, 033901 (2005).
[CrossRef] [PubMed]

Chan, J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, "A picogram- and nanometre-scale photoniccrystal optomechanical cavity," Nature 459, 550-555 (2009).
[CrossRef] [PubMed]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Dorsel, A.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, "Optical bistability and mirror confinement induced by radiation pressure," Phys. Rev. Lett. 51, 1550-1553 (1983).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, and J. M. Dziedzic, "Observation of radiation pressure trapping of particles by alternating lightbeams," Phys. Rev. Lett. 54, 1245-1248 (1985).
[CrossRef] [PubMed]

Eichenfield, M.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, "A picogram- and nanometre-scale photoniccrystal optomechanical cavity," Nature 459, 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. Photonics 1, 416-422 (2007).
[CrossRef]

Fan, S. H.

W. Suh, Z. Wang, and S. H. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
[CrossRef]

W. Suh, M. F. Yanik, O. Solgaard, and S. H. Fan, "Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs," Appl. Phys. Lett. 82 (13), 1999-2001 (2003).
[CrossRef]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Gires, F.

F. Gires, and P. Tournois "Interferometre utilisable pour la compression d’impulsions lumineuses modulees en frequence," C. R. Acad. Sci. Paris 25861126115 (1964).

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

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, 480-484 (2008).
[CrossRef] [PubMed]

Ibanescu, M.

Ippen, E. P.

Joannopoulos, J. D.

Johnson, S. G.

Kippenberg, T.

T. Carmon, H. Rokhsari, L. Yang, T. Kippenberg, and K. Vahala, "Temporal Behavior of Radiation-pressureinduced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef] [PubMed]

Kippenberg, T. J.

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, 033901 (2005).
[CrossRef] [PubMed]

Kuramochi, E.

M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, "Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs," Phys. Rev. Lett. 97, 023903 (2006).
[CrossRef] [PubMed]

Laine, J. P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Law, C. K.

C. K. Law, "Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium," Phys. Rev. A 49, 433-437 (1994).
[CrossRef] [PubMed]

Li, M.

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, 1806-1816 (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, 480-484 (2008).
[CrossRef] [PubMed]

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

Loncar, M.

McCullen, J. D.

P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, "Theory of radiation pressure driven interferometers," J. Opt. Soc. Am. B 2, 1830-1840 (1985).
[CrossRef]

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, "Optical bistability and mirror confinement induced by radiation pressure," Phys. Rev. Lett. 51, 1550-1553 (1983).
[CrossRef]

Meystre, P.

M. Bhattacharya and P. Meystre, "Multiple membrane cavity optomechanics," Phys. Rev. A 78, 041801 (2008).
[CrossRef]

M. Bhattacharya, and P. Meystre, "Trapping and cooling a mirror to its quantum mechanical ground state," Phys. Rev. Lett. 99, 073601 (2007).
[CrossRef] [PubMed]

P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, "Theory of radiation pressure driven interferometers," J. Opt. Soc. Am. B 2, 1830-1840 (1985).
[CrossRef]

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, "Optical bistability and mirror confinement induced by radiation pressure," Phys. Rev. Lett. 51, 1550-1553 (1983).
[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, 416-422 (2007).
[CrossRef]

Mitsugi, S.

M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, "Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs," Phys. Rev. Lett. 97, 023903 (2006).
[CrossRef] [PubMed]

Mizrahi, A.

Moore, G. T.

G. T. Moore, "Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity," J. Math. Phys. 11, 2679-2691 (1970).
[CrossRef]

Notomi, M.

M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, "Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs," Phys. Rev. Lett. 97, 023903 (2006).
[CrossRef] [PubMed]

Painter, O.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, "A picogram- and nanometre-scale photoniccrystal optomechanical cavity," Nature 459, 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. Photonics 1, 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, 416-422 (2007).
[CrossRef]

Pernice, W. H. P.

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, 1806-1816 (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, 480-484 (2008).
[CrossRef] [PubMed]

Popovic, M. A.

Povinelli, M. L.

Rakich, P. T.

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. Photonics 1, 658-665 (2007).
[CrossRef]

P. T. Rakich, M. A. Popovic, M. R. Watts, T. Barwicz, H. I. Smith, and E. P. Ippen, "Ultrawide tuning of photonic microcavities via evanescent field perturbation," Opt. Lett. 31, 1241-1243 (2006).
[CrossRef] [PubMed]

Rokhsari, H.

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, 033901 (2005).
[CrossRef] [PubMed]

T. Carmon, H. Rokhsari, L. Yang, T. Kippenberg, and K. Vahala, "Temporal Behavior of Radiation-pressureinduced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef] [PubMed]

Schachter, L.

Schchter, L.

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, 033901 (2005).
[CrossRef] [PubMed]

Smith, H. I.

Smythe, E. J.

Solgaard, O.

W. Suh, M. F. Yanik, O. Solgaard, and S. H. Fan, "Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs," Appl. Phys. Lett. 82 (13), 1999-2001 (2003).
[CrossRef]

Suh, W.

W. Suh, Z. Wang, and S. H. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
[CrossRef]

W. Suh, M. F. Yanik, O. Solgaard, and S. H. Fan, "Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs," Appl. Phys. Lett. 82 (13), 1999-2001 (2003).
[CrossRef]

Tang, H. X.

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, 1806-1816 (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, 480-484 (2008).
[CrossRef] [PubMed]

Taniyama, H.

M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, "Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs," Phys. Rev. Lett. 97, 023903 (2006).
[CrossRef] [PubMed]

Tournois, P.

F. Gires, and P. Tournois "Interferometre utilisable pour la compression d’impulsions lumineuses modulees en frequence," C. R. Acad. Sci. Paris 25861126115 (1964).

Vahala, K.

T. Carmon, H. Rokhsari, L. Yang, T. Kippenberg, and K. Vahala, "Temporal Behavior of Radiation-pressureinduced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef] [PubMed]

Vahala, K. J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, "A picogram- and nanometre-scale photoniccrystal optomechanical cavity," Nature 459, 550-555 (2009).
[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, 033901 (2005).
[CrossRef] [PubMed]

Vignes, E.

P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, "Theory of radiation pressure driven interferometers," J. Opt. Soc. Am. B 2, 1830-1840 (1985).
[CrossRef]

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, "Optical bistability and mirror confinement induced by radiation pressure," Phys. Rev. Lett. 51, 1550-1553 (1983).
[CrossRef]

Walther, H.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, "Optical bistability and mirror confinement induced by radiation pressure," Phys. Rev. Lett. 51, 1550-1553 (1983).
[CrossRef]

Wang, Z.

W. Suh, Z. Wang, and S. H. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
[CrossRef]

Watts, M. R.

Wright, E. M.

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, 480-484 (2008).
[CrossRef] [PubMed]

Yang, L.

T. Carmon, H. Rokhsari, L. Yang, T. Kippenberg, and K. Vahala, "Temporal Behavior of Radiation-pressureinduced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 223902 (2005).
[CrossRef] [PubMed]

Yanik, M. F.

W. Suh, M. F. Yanik, O. Solgaard, and S. H. Fan, "Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs," Appl. Phys. Lett. 82 (13), 1999-2001 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

W. Suh, M. F. Yanik, O. Solgaard, and S. H. Fan, "Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs," Appl. Phys. Lett. 82 (13), 1999-2001 (2003).
[CrossRef]

C. R. Acad. Sci. Paris (1)

F. Gires, and P. Tournois "Interferometre utilisable pour la compression d’impulsions lumineuses modulees en frequence," C. R. Acad. Sci. Paris 25861126115 (1964).

IEEE J. Quantum Electron. (1)

W. Suh, Z. Wang, and S. H. Fan, "Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities," IEEE J. Quantum Electron. 40, 1511-1518 (2004).
[CrossRef]

J. Lightwave Technol. (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, "Microring resonator channel dropping filters," J. Lightwave Technol. 15, 998-1005 (1997).
[CrossRef]

J. Math. Phys. (1)

G. T. Moore, "Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity," J. Math. Phys. 11, 2679-2691 (1970).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Photonics (2)

M. Eichenfield, C. P. Michael, R. Perahia, and O. Painter, "Actuation of micro-optomechanical systems via cavity-enhanced optical dipole forces," Nat. Photonics 1, 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. Photonics 1, 658-665 (2007).
[CrossRef]

Nature (2)

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, "A picogram- and nanometre-scale photoniccrystal optomechanical cavity," Nature 459, 550-555 (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, 480-484 (2008).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (2)

C. K. Law, "Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium," Phys. Rev. A 49, 433-437 (1994).
[CrossRef] [PubMed]

M. Bhattacharya and P. Meystre, "Multiple membrane cavity optomechanics," Phys. Rev. A 78, 041801 (2008).
[CrossRef]

Phys. Rev. Lett. (8)

T. Carmon, H. Rokhsari, L. Yang, T. Kippenberg, and K. Vahala, "Temporal Behavior of Radiation-pressureinduced vibrations of an optical microcavity phonon mode," Phys. Rev. Lett. 94, 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, 033901 (2005).
[CrossRef] [PubMed]

M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, "Optomechanical wavelength and energy conversion in high-Q double-layer cavities of photonic crystal slabs," Phys. Rev. Lett. 97, 023903 (2006).
[CrossRef] [PubMed]

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

A. Ashkin, "Trapping of atoms by resonance radiation pressure," Phys. Rev. Lett. 40, 729-732 (1978).
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A. Ashkin, and J. M. Dziedzic, "Observation of radiation pressure trapping of particles by alternating lightbeams," Phys. Rev. Lett. 54, 1245-1248 (1985).
[CrossRef] [PubMed]

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[CrossRef]

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M. A. Popovic and P. T. Rakich, "Optonanomechanical self-adaptive photonic devices based on light forces: a path to robust high-index-contrast nanophotonic circuits," Proc. SPIE 7219, 72190A (Feb. 10, 2009)
[CrossRef]

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H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

M. A. Popovic, Theory and design of high-index-contrast microphotonic circuits, Ph.D. Thesis (MIT Archives, Cambridge, 2008)

C. K. Madsen, C. K. and J. H. Zhao, Optical filter design and analysis: a signal processing approach (Wiley, New York, 1999).

W. Greiner, Classical mechanics: systems of particles and Hamiltonian dynamics (Springer, New York, 2003).

H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics (Addison Wesley, San Francisco, 2002).

D. J. Griffiths, Intoduction to Quantum Mechanics (Prentice-Hall, Englewood Cliffs, NJ, 1995).

P. Penfield and H. A. Haus, Electrodynamics of moving media (MIT Press, Massachusetts, 1967)

R. Loudon, The Quantum Theory of Light (Oxford Science Publications, 2000)

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Figures (9)

Fig. 1.
Fig. 1.

(a) Schematic showing a generic open photonic system that can be mechanically varied through displacement of q. Optical power flows into and out of system. The response of the device, S̃(ω,q), is also a function of frequency ω. (b) An example of a mechanically-variable open system in the form of an ideal lossless Gires-Tournois interferometer with a fixed partial mirror M1 and a movable mirror M2. Here, q is taken to be the separation of mirror M2 from M1.

Fig. 2.
Fig. 2.

Schematic showing generic optomechanically-variable open-system within a closed surface forming the boundary to the volume, V. This system can be seen as a reflectionless one-port system. Optical power flows into and out of system (no power is reflected). Here q represents the mechanical degree of freedom of the system, which impacts the optical response of the system in some manner. Through the motion of q against internally generated optically-induced forces, work can be done on the electromagnetic field.

Fig. 3.
Fig. 3.

Schematic showing lossless optomechanically variable open system consisting of linear media. Optical power flows into and out of system. Here q represents a generalized coordinate which changes the response of the device, S̃(ω,q). Here it is assumed that this is a reflectionless system.

Fig. 4.
Fig. 4.

(a) Sketch of a generic lossless and linear optomechanically-variable open-system with N inputs and N outputs. Light of constant intensity, frequency and phase flows into system. Here q represents a generalized coordinate which changes the response of the device, which can be expressed in terms of a scattering matrix of the form S̃ l,m (ω,q). (b) A lossless Fabry-Perot interferometer, which serves as a specific example of a multi-port system of this form.

Fig. 5.
Fig. 5.

(a) Diagram of an optomechanically variable dual-waveguide system within a closed surface of length L. (b–d) and (e–f) show two different optomechanically variable waveguide geometries which might be examined in a nearly identical manner. (b–d) are schematics of the waveguide cross-section and mode-profiles for the coupled dual-waveguide system treated in ref [10], and (d–e) is a schematic representation of a waveguide mode whose effective index is modified through evanescent perturbation by a uniform dielectric body (treated in ref [18]). In the latter system it is assumed that n 1>n 2.

Fig. 6.
Fig. 6.

Shows the computed forces (pN/µm/mW) generated by a symmetric waveguide mode on the dual waveguide system. Forces computed by RTOF method (dashed line) and Maxwell stress-tensor method (circles) are over-layed, revealing perfect agreement. Inset shows and intensity-map of the Ex electric-field component computed with a full-vectorial mode-solver.

Fig. 7.
Fig. 7.

(a) and (b) are schematics of the same lossless Gires-Tournois interferometer. In both diagrams, the generalized coordinate q is taken to be the separation between mirrors M 1 and M 2. (a) Shows the incident (Ei ) and exiting (Eo ) field amplitudes, while (b) shows the internal fields impinging on (E + in ) and receeding from (E - in ) mirror M 2.

Fig. 8.
Fig. 8.

Examples of single- and multi-port systems which are analytically treatable with RTOF method. (a) and (b) show an all-pass ring resonator which is optomechanically tunable via dielectric perturbation, after Ref. [36]. (c) and (d) show the analagous optomechanically variable geometry for a photonic cyrstal defect cavity. (e) shows a tunable resonant structure utilizing photonic crystal guided-slab resonances (see Ref. [25].

Fig. 9.
Fig. 9.

(a) Schematic showing generic optomechanically-variable closed system. Here q represents a generalized coordinate through which work can be done on the system, changing the electromagnetic energy within. (b) A specific example of a closed system in the form of an idealized and lossless electromagnetically-closed cavity. Here, the generalized coordinate q is taken to be the mirror separation.

Equations (42)

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V S · n ̂ d a + U in t = V J · E d v .
V S · n ̂ d a + U i n t = F o p t · q ˙ .
( P 0 ( t ) P i ) + U i n t = W t .
Φ o ( t ) = Φ i d N d t .
e x p [ i ( ω t ) ] e x p [ i ( ω t ϕ ( q , ω ) ) ] .
e x p [ i ( ω t ) ] e x p [ i ( ω t ψ ( t , ω ) ) ] .
dW dt = Φ i [ ω ( t ) ω ] + d d t · [ N ( t ) · ω i n ( t ) ] d N d t · ω ( t ) .
Δ W = t i = 0 t f = Δ t + T dW dt . d t
= Φ t i t f δ ω ( t ) · d t
t i t f d N d t · δ ω ( t ) d t
+ [ N ( t ) · ( ω i n ( t ) ω ) ] t i t f .
Δ W = Φ [ ψ f ψ i ]
+ t i t f d N d t · d ψ d t d t
+ [ N ( t ) · ( ω i n ( t ) ω ) ] t i t f .
Δ W = Φ [ ϕ f ϕ i ]
+ t i t f d N d t · d t d t .
Δ W = Φ [ ϕ f ϕ i ]
+ d ϕ d q d N q d q ( Δ q ) 2 t i t f ( d f d t ) 2 d t + H . O . T .
F q | q i = [ d W d q ] q i = Φ · [ d ϕ d q ] q i .
F q ( q ) = Φ · · d ϕ ( q ) d q
Δ W ( q ) = Φ . q o q d ϕ dq . dq = Φ . [ ϕ ( q o ) ϕ ( q ) ] .
U o p t ( q , ω ) = Φ . ϕ ( q , ω ) ,
U o p t t o t ( q ) = l = 1 N U o p t ( q , ω l ) = l = 1 N Φ l · ϕ ( q , ω l ) .
F q = d U o p t d q = · k Φ o , k ( q ) · d ϕ o , k d q .
U o p t ( q ) = · [ k Φ o , k ( q ) · d ϕ o , k ( q ) d q ] · d q .
b ˜ l = m S ˜ l , m ( ω , q ) a ˜ m ,
ϕ o , k ( ω , q ) = tan -1 ( I m ( b ˜ k ) R e ( b ˜ k ) ) .
ϕ ( ω , q ) = ω c · n e f f ( ω , q ) · L .
F q o ( ω , q ) = L · P c n e f f q .
ϕ ( ω , q ) = tan -1 [ ( 1 r 2 ) sin ( δ ) 2 r ( r 2 + 1 ) cos ( δ ) ] .
Φ i = p i ω = | E i | 2 2 μ o c · A ω .
F q ( ω , q ) A = E 2 μ o c 2 [ ( 1 r 2 ) 2 r · cos [ 2 q ( ω c ) ] ( r 2 + 1 ) ] .
F = S T · d a .
E i n ( Z ) = x ̂ [ E i n e i k z E i n e i k z ]
B i n ( z ) = y ̂ [ E i n e i k z + E i n e i k z ] c .
F z A = 1 2 T z , z = 1 2 ( 2 · E i n 2 μ o c 2 ) = 2 · P i n c · A .
P i n ( ω , q ) A = E 2 2 μ o c [ ( 1 r 2 ) 2 r · cos [ 2 q ( ω c ) ] ( r 2 + 1 ) ] .
F q ( ω , q ) A = E 2 μ o c 2 [ ( 1 r 2 ) 2 r · cos [ 2 q ( ω c ) ] ( r 2 + 1 ) ] .
F c ( q ) N = N d ω o ( q ) d q .
N = Φ · τ g = P ω L ν g = P ω P L n g c .
ω q = ω n g n e f f q .
F q c ( ω , q ) = L · P c n e f f q .

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