Abstract

In this paper we discuss the force exerted by the field of an optical cavity on a polarizable dipole. We show that the modification of the cavity modes due to interaction with the dipole significantly alters the properties of the force. In particular, all components of the force are found to be non-conservative, and cannot, therefore, be derived from a potential energy. We also suggest a simple generalization of the standard formulas for the optical force on the dipole, which reproduces the results of calculations based on the Maxwell stress tensor.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  3. G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
    [CrossRef]
  4. A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
    [CrossRef]
  5. A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
    [CrossRef]
  6. S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
    [CrossRef]
  7. O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
    [CrossRef]
  8. A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
    [CrossRef]
  9. D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
    [CrossRef]
  10. D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
    [CrossRef] [PubMed]
  11. O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” N. J. Phys. 12, 033015 (2010).
    [CrossRef]
  12. P. F. Barker and M. N. Shneider, “Cavity cooling of an optically trapped nanoparticle,” Phys. Rev. A 81, 023826 (2010).
    [CrossRef]
  13. O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
    [CrossRef]
  14. Z.-q. Yin, T. Li, and M. Feng, “Three-dimensional cooling and detection of a nanosphere with a single cavity,” Phys. Rev. A 83, 013816 (2011).
    [CrossRef]
  15. T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
    [CrossRef]
  16. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172–1176 (2008).
    [CrossRef] [PubMed]
  17. A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
    [CrossRef]
  18. V. Braginsky and A. Manukin, Measurment of Weak Forces in Physics Experiments (University of Chicago Press, 1977).
  19. R. J. Schulze, C. Genes, and H. Ritsch, “Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity,” Phys. Rev. A 81, 063820 (2010).
    [CrossRef]
  20. M. Nieto-Vesperinas, P. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. Lond. A 362, 719–737 (2004).
    [CrossRef]
  21. J. Rubin and L. Deych, “Optical forces due to spherical microresonators and their manifestation in optically induced orbital motion of nanoparticles,” Phys. Rev. A 84, 023844 (2011).
    [CrossRef]
  22. S. Arnold, D. Keng, S. I. Shopova, S. Holler, W. Zurawsky, and F. Vollmer, “Whispering gallery mode carousel—a photonic mechanism for enhanced nanoparticle detection in biosensing,” Opt. Express 17, 6230–6238 (2009).
    [CrossRef] [PubMed]
  23. L. Landau, E. Lifshitz, and L. Pitaevski?, Electrodynamics of continuous media, Course of theoretical physics (Butterworth-Heinemann, 1984).
  24. V. Wong and M. A. Ratner, “Explicit computation of gradient and nongradient contributions to optical forces in the discrete-dipole approximation,” J. Opt. Soc. Am. B 23, 1801–1814 (2006).
    [CrossRef]
  25. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).
  26. L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A 80, 061805 (2009).
    [CrossRef]
  27. J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A 81, 053827 (2010).
    [CrossRef]
  28. J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82, 053819 (2010).
    [CrossRef]
  29. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (National Bureau of Standards, 1972).
  30. J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2010).
    [CrossRef]

2011 (4)

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

Z.-q. Yin, T. Li, and M. Feng, “Three-dimensional cooling and detection of a nanosphere with a single cavity,” Phys. Rev. A 83, 013816 (2011).
[CrossRef]

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[CrossRef]

J. Rubin and L. Deych, “Optical forces due to spherical microresonators and their manifestation in optically induced orbital motion of nanoparticles,” Phys. Rev. A 84, 023844 (2011).
[CrossRef]

2010 (8)

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” N. J. Phys. 12, 033015 (2010).
[CrossRef]

P. F. Barker and M. N. Shneider, “Cavity cooling of an optically trapped nanoparticle,” Phys. Rev. A 81, 023826 (2010).
[CrossRef]

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
[CrossRef]

R. J. Schulze, C. Genes, and H. Ritsch, “Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity,” Phys. Rev. A 81, 063820 (2010).
[CrossRef]

J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A 81, 053827 (2010).
[CrossRef]

J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82, 053819 (2010).
[CrossRef]

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2010).
[CrossRef]

2009 (6)

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A 80, 061805 (2009).
[CrossRef]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[CrossRef]

S. Arnold, D. Keng, S. I. Shopova, S. Holler, W. Zurawsky, and F. Vollmer, “Whispering gallery mode carousel—a photonic mechanism for enhanced nanoparticle detection in biosensing,” Opt. Express 17, 6230–6238 (2009).
[CrossRef] [PubMed]

2008 (4)

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

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

2006 (1)

2004 (1)

M. Nieto-Vesperinas, P. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. Lond. A 362, 719–737 (2004).
[CrossRef]

1986 (1)

1970 (1)

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

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (National Bureau of Standards, 1972).

Anetsberger, G.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Arcizet, O.

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Arnold, S.

Ashkin, A.

Aspelmeyer, M.

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

Barker, P. F.

P. F. Barker and M. N. Shneider, “Cavity cooling of an optically trapped nanoparticle,” Phys. Rev. A 81, 023826 (2010).
[CrossRef]

Bjorkholm, J. E.

Braginsky, V.

V. Braginsky and A. Manukin, Measurment of Weak Forces in Physics Experiments (University of Chicago Press, 1977).

Briant, T.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Chang, D. E.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

Chaumet, P.

M. Nieto-Vesperinas, P. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. Lond. A 362, 719–737 (2004).
[CrossRef]

Chen, J.

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2010).
[CrossRef]

Chu, S.

Cirac, J. I.

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” N. J. Phys. 12, 033015 (2010).
[CrossRef]

Clerk, A. A.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Cohadon, P.-F.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Crozier, K.

J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82, 053819 (2010).
[CrossRef]

Deych, L.

J. Rubin and L. Deych, “Optical forces due to spherical microresonators and their manifestation in optically induced orbital motion of nanoparticles,” Phys. Rev. A 84, 023844 (2011).
[CrossRef]

J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A 81, 053827 (2010).
[CrossRef]

L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A 80, 061805 (2009).
[CrossRef]

Dziedzic, J. M.

Feng, M.

Z.-q. Yin, T. Li, and M. Feng, “Three-dimensional cooling and detection of a nanosphere with a single cavity,” Phys. Rev. A 83, 013816 (2011).
[CrossRef]

Francais, O.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Genes, C.

R. J. Schulze, C. Genes, and H. Ritsch, “Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity,” Phys. Rev. A 81, 063820 (2010).
[CrossRef]

Girvin, S. M.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Groeblacher, S.

S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

Hammerer, K.

S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

Harris, J. G. E.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Heidmann, A.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Holler, S.

Hu, J.

J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82, 053819 (2010).
[CrossRef]

Jayich, A. M.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Juan, M. L.

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” N. J. Phys. 12, 033015 (2010).
[CrossRef]

Keng, D.

Kheifets, S.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[CrossRef]

Kiesel, N.

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

Kimble, H. J.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[CrossRef]

Kimerling, L. C.

J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82, 053819 (2010).
[CrossRef]

Kippenberg, T. J.

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
[CrossRef]

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

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

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Kotthaus, J. P.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

Landau, L.

L. Landau, E. Lifshitz, and L. Pitaevski?, Electrodynamics of continuous media, Course of theoretical physics (Butterworth-Heinemann, 1984).

Li, T.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[CrossRef]

Z.-q. Yin, T. Li, and M. Feng, “Three-dimensional cooling and detection of a nanosphere with a single cavity,” Phys. Rev. A 83, 013816 (2011).
[CrossRef]

Lifshitz, E.

L. Landau, E. Lifshitz, and L. Pitaevski?, Electrodynamics of continuous media, Course of theoretical physics (Butterworth-Heinemann, 1984).

Lin, S.

J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82, 053819 (2010).
[CrossRef]

Lin, Z.

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2010).
[CrossRef]

Liu, S.

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2010).
[CrossRef]

Mackowski, J.-M.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Manukin, A.

V. Braginsky and A. Manukin, Measurment of Weak Forces in Physics Experiments (University of Chicago Press, 1977).

Marquardt, F.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Michel, C.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

Molinelli, C.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Ng, J.

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2010).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, P. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. Lond. A 362, 719–737 (2004).
[CrossRef]

Painter, O.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

Papp, S. B.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[CrossRef]

Pflanzer, A. C.

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

Pinard, L.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Pitaevskii, L.

L. Landau, E. Lifshitz, and L. Pitaevski?, Electrodynamics of continuous media, Course of theoretical physics (Butterworth-Heinemann, 1984).

Quidant, R.

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” N. J. Phys. 12, 033015 (2010).
[CrossRef]

Rahmani, A.

M. Nieto-Vesperinas, P. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. Lond. A 362, 719–737 (2004).
[CrossRef]

Raizen, M. G.

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[CrossRef]

Ratner, M. A.

Regal, C. A.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[CrossRef]

Ritsch, H.

R. J. Schulze, C. Genes, and H. Ritsch, “Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity,” Phys. Rev. A 81, 063820 (2010).
[CrossRef]

Riviere, R.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Romero-Isart, O.

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” N. J. Phys. 12, 033015 (2010).
[CrossRef]

Rousseau, L.

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

Rubin, J.

J. Rubin and L. Deych, “Optical forces due to spherical microresonators and their manifestation in optically induced orbital motion of nanoparticles,” Phys. Rev. A 84, 023844 (2011).
[CrossRef]

L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A 80, 061805 (2009).
[CrossRef]

Rubin, J. T.

J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A 81, 053827 (2010).
[CrossRef]

Sankey, J. C.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Schliesser, A.

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
[CrossRef]

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

Schulze, R. J.

R. J. Schulze, C. Genes, and H. Ritsch, “Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity,” Phys. Rev. A 81, 063820 (2010).
[CrossRef]

Shneider, M. N.

P. F. Barker and M. N. Shneider, “Cavity cooling of an optically trapped nanoparticle,” Phys. Rev. A 81, 023826 (2010).
[CrossRef]

Shopova, S. I.

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (National Bureau of Standards, 1972).

Thompson, J. D.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

Unterreithmeier, Q. P.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

Vahala, K. J.

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

Vanner, M. R.

S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

Vollmer, F.

Weig, E. M.

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

Wilson, D. J.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[CrossRef]

Wong, V.

Yang, C.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Ye, J.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

Yin, Z.-q.

Z.-q. Yin, T. Li, and M. Feng, “Three-dimensional cooling and detection of a nanosphere with a single cavity,” Phys. Rev. A 83, 013816 (2011).
[CrossRef]

Zoller, P.

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

Zurawsky, W.

Zwickl, B. M.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Adv. At. Mol. Opt. Phys. (1)

A. Schliesser and T. J. Kippenberg, “Cavity optomechanics with whispering-gallery mode optical micro-resonators,” Adv. At. Mol. Opt. Phys. 58, 207–323 (2010).
[CrossRef]

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

N. J. Phys. (3)

O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superposition of living organisms,” N. J. Phys. 12, 033015 (2010).
[CrossRef]

O. Arcizet, C. Molinelli, T. Briant, P.-F. Cohadon, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais, and L. Rousseau, “Experimental optomechanics with silicon micromirrors,” N. J. Phys. 10, 125021 (2008).
[CrossRef]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” N. J. Phys. 10, 095008 (2008).
[CrossRef]

Nat. Phys. (4)

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[CrossRef]

A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger, and T. J. Kippenberg, “Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the heisenberg uncertainty limit,” Nat. Phys. 5, 509–514 (2009).
[CrossRef]

A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4, 415–419 (2008).
[CrossRef]

T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nat. Phys. 7, 527–530 (2011).
[CrossRef]

Nature (1)

S. Groeblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460, 724–727 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phil. Trans. R. Soc. Lond. A (1)

M. Nieto-Vesperinas, P. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. Lond. A 362, 719–737 (2004).
[CrossRef]

Phys. Rev. A (8)

J. Rubin and L. Deych, “Optical forces due to spherical microresonators and their manifestation in optically induced orbital motion of nanoparticles,” Phys. Rev. A 84, 023844 (2011).
[CrossRef]

L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A 80, 061805 (2009).
[CrossRef]

J. T. Rubin and L. Deych, “Ab initio theory of defect scattering in spherical whispering-gallery-mode resonators,” Phys. Rev. A 81, 053827 (2010).
[CrossRef]

J. Hu, S. Lin, L. C. Kimerling, and K. Crozier, “Optical trapping of dielectric nanoparticles in resonant cavities,” Phys. Rev. A 82, 053819 (2010).
[CrossRef]

R. J. Schulze, C. Genes, and H. Ritsch, “Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity,” Phys. Rev. A 81, 063820 (2010).
[CrossRef]

P. F. Barker and M. N. Shneider, “Cavity cooling of an optically trapped nanoparticle,” Phys. Rev. A 81, 023826 (2010).
[CrossRef]

O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Optically levitating dielectrics in the quantum regime: theory and protocols,” Phys. Rev. A 83, 013803 (2011).
[CrossRef]

Z.-q. Yin, T. Li, and M. Feng, “Three-dimensional cooling and detection of a nanosphere with a single cavity,” Phys. Rev. A 83, 013816 (2011).
[CrossRef]

Phys. Rev. E (1)

J. Chen, J. Ng, S. Liu, and Z. Lin, “Analytical calculation of axial optical force on a rayleigh particle illuminated by gaussian beams beyond the paraxial approximation,” Phys. Rev. E 80, 026607 (2010).
[CrossRef]

Phys. Rev. Lett. (2)

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

D. J. Wilson, C. A. Regal, S. B. Papp, and H. J. Kimble, “Cavity optomechanics with stoichiometric sin films,” Phys. Rev. Lett. 103, 207204 (2009).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010).
[CrossRef] [PubMed]

Science (1)

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

Other (4)

V. Braginsky and A. Manukin, Measurment of Weak Forces in Physics Experiments (University of Chicago Press, 1977).

L. Landau, E. Lifshitz, and L. Pitaevski?, Electrodynamics of continuous media, Course of theoretical physics (Butterworth-Heinemann, 1984).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, 2002).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (National Bureau of Standards, 1972).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Set up for evaluating the force on a dipole modeled as a system of equal and opposite charges. The distance between the charges will be taken to zero.

Fig. 2
Fig. 2

Coordinate systems used for evaluation of optical forces together with schematic presentation of the resonator, WGM, and the dipole. The axis always connects the center of the resonator and the point of observation. When using this coordinate system to calculate the forces, this axis passes through the center of the particle.

Equations (39)

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

u tot = 1 2 E D dV ,
u pol = u tot 1 2 ɛ 0 | 𝔈 | 2 dV
δ u pol = P δ 𝔈 dV .
δ u pol = δ r p ( P ) 𝔈 dV .
F = ( P ) 𝔈 dV .
F = ( p ) 𝔈 ,
F = 1 2 α 0 | 𝔈 | 2
F = [ ( p r ) E ] r = r p
F = 1 2 α r | E ( r , r p ) | 2 | r = r p .
F = 1 4 e [ α ] | 𝔈 | 2 + 1 2 m [ α ] ( ω e [ 𝔈 * × 𝔅 ] + m [ ( 𝔈 * ) 𝔈 ] ) .
α = 4 π ɛ 0 ( α 0 + 2 3 i k 3 α 0 2 ) ,
α 0 = R p 3 ( n p 2 1 ) / ( n p 2 + 2 ) ,
F = ˜ u + σ c g + σ c ɛ 0 2 ω m [ ( E * ˜ ) E ] ,
= i Γ l ( 0 ) / ( ω ω l ( 0 ) + i Γ l ( 0 ) )
X l , m ( θ , ϕ ) = m = l l D m , m ( l ) ( α , β , γ ) X l , m ( θ , ϕ )
D m , m ( l ) ( α , β , γ ) = e i ( m α + m γ ) d m , m ( l ) ( β ) ,
d m , l ( l ) ( β ) = ( 2 l ) ! ( l + m ) ! ( l m ) ! [ cos β 2 sin β 2 ] l [ cot ( β 2 ) ] m
E = E 0 h L ( 1 ) m a m X L , m
a m = D m , L ( L ) ( 0 , β , α ) = i e iL α d m , L ( L ) ( β ) y 0 + i
E = E 0 h L ( 1 ) ( k r ) L 4 π ( a 1 ξ ^ + + a 1 ξ ^ )
H = E 0 ɛ 0 μ 0 L 4 π [ i 2 L h L ( 1 ) ( k r ) k r a 0 r ^ + [ k r h L ( 1 ) ( k r ) ] k r ( a 1 ξ ^ + a 1 ξ ^ ) ]
d m , L ( L ) ( θ p ) 1 ( π L ) 1 / 4 e L 2 ( θ p π 2 ) 2 [ cot ( θ p 2 ) ] m
F ( g ) = 𝔉 1 y 0 2 + 1 ( [ n L ( k r p ) ] r ^ L k r p n L ( k r p ) θ ¯ θ ^ )
𝔉 = 1 4 π 3 / 2 e [ α ] | E 0 | 2 k L n L ( k r p ) e L θ ¯ 2
F ( s ) = 𝔉 p y 0 2 + 1 L k r p n L ( k r p ) ϕ ^
a m = i e iL ϕ p d m , l ( l ) ( θ p ) × { [ y 0 + i ] 1 m ± 1 Γ L ( 0 ) Γ p [ y + i ] 1 m = ± 1 }
δ ω L = e [ α ] k 3 6 π ɛ 0 Γ L ( 0 ) [ V L , 1 ( k r p ) ] 2 ; δ Γ L = p | δ ω L |
V L , m ( k r p ) = i ( 1 ) L + 1 3 2 m 2 L + 1 h L ( k r p ) ( 1 ) ( 1 ) L 3 L / 2 n L ( k r p ) .
F r ( p g ) = 1 4 ɛ 0 | E 0 | 2 L α 0 ( | a 1 | 2 + | a 1 | 2 ) d | h L ( 1 ) ( k r ) | 2 dr | r = r p
F θ ( p g ) = 1 4 ɛ 0 | E 0 | 2 | h L ( 1 ) ( k r p ) | 2 L 2 α 0 r p sin θ e [ a 1 ( a 0 * a 2 * ) a 1 ( a 0 * a 2 * ) ]
F ϕ ( p g ) = 1 4 ɛ 0 | E 0 | 2 | h L ( 1 ) ( k r p ) | 2 L 2 α 0 r p m [ a 1 ( a 0 * + a 2 * ) + a 1 ( a 0 * + a 2 * ) ]
F ( s ) = ɛ 0 | E 0 | 2 L 2 α 0 2 k 3 3 r | h L ( 1 ) ( k r ) | 2 { θ ^ m [ ( a 0 * ( a 1 a 1 ) ] + ϕ ^ e [ ( a 0 * ( a 1 + a 1 ) ] }
P = | E 0 | 2 ɛ 0 c 3 2 ω 2 ( | a 1 | 2 + | a 1 | 2 ) = | E 0 | 2 ɛ 0 c 3 ω 2 [ d 0 , L L ] 2 ( Γ L ( 0 ) Γ p ) 2 1 y 2 + 1 ,
N = P [ 2 Γ L ( 0 ) ] 1 h ¯ ω .
F r = N h ¯ d δ ω L d r p ,
d u pol d r p = N h ¯ d δ ω L d r p + d N d r p h ¯ δ ω L
F θ = 2 h ¯ N L Γ p r p Γ L ( 0 ) ( y 0 2 + 1 ) ( y 2 + 1 ) θ ¯ [ δ ω L ( 1 + y y 0 ) + δ Γ L ( y 0 y ) ]
F ϕ = 2 h ¯ N L Γ p r p Γ L ( 0 ) ( y 0 2 + 1 ) ( y 2 + 1 ) [ δ ω L ( y 0 y ) + δ Γ L ( 1 + y 0 y ) ]
y ( r 0 ) = 1 Γ p d δ ω L d r p ( 1 + y p )

Metrics