Abstract

We present calculations of fluorescence from single molecules (modeled as damped oscillating dipoles) inside a dielectric sphere. For an excited molecule at an arbitrary position within the sphere we calculate the fluorescence intensity collected by an objective in some well-defined detection geometry. We find that, for the cases we model, integration over the emission linewidth of the molecule is essential for obtaining representative results. Effects such as dipole position and orientation, numerical aperture of the collection objective, sphere size, emission wavelength, and linewidth are examined. These results are applicable to single-molecule detection techniques employing microdroplets.

© 1996 Optical Society of America

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  1. M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Detecting single molecules in liquids,” Anal. Chem. 67, 418A–423A (1995) and references cited therein.
    [CrossRef]
  2. M. D. Barnes, K. C. Ng, W. B. Whitten, J. M. Ramsey, “Detection of single rhodamine 6G molecules in levitated microdroplets,” Anal. Chem. 65, 2360–2365 (1993).
    [CrossRef]
  3. For a review, see S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Nonlinear Optics, O. Keller, ed. (Nova, Commack, New York, 1995), pp. 171–242.
  4. H. M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Evaporation and condensation rates of liquid droplets deduced from structure resonances in the fluorescence spectra,” Opt. Lett. 9, 273–275 (1984).
    [CrossRef] [PubMed]
  5. M. F. Buehler, T. M. Allen, E. J. Davis, “Microparticle Raman spectroscopy of multicomponent aerosols,” J. Colloid Interface Sci. 146, 79–89 (1991).
    [CrossRef]
  6. A. Serpenguzel, G. Chen, R. K. Chang, “Stimulated Raman scattering of aqueous droplets containing ions: concentration and size determination,” Partic. Sci. Technol. 8, 1–10 (1990).
  7. L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
    [CrossRef]
  8. H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A. 13, 396–404 (1976).
    [CrossRef]
  9. S. Druger, P. J. McNulty, “Radiation patterns of fluorescence from molecules embedded in small particles: general case,” Appl. Opt. 22, 75–82 (1983).
    [CrossRef] [PubMed]
  10. D. S. Benincasa, P. W. Barber, J. Z. Zhang, W.-F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
    [CrossRef] [PubMed]
  11. J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1549 (1991).
    [CrossRef]
  12. D. Q. Chowdury, P. W. Barber, S. C. Hill, “Energy density distribution inside large nonabsorbing spheres via Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
    [CrossRef]
  13. S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 1–61.
  14. P. Chylek, “Resonance structure of Mie scattering: distances between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990).
    [CrossRef]
  15. R. Fuchs, K. L. Kliewar, “Optical modes of vibration in an ionic crystal sphere,” J. Opt. Soc. Am. 58, 319–330 (1968).
    [CrossRef]
  16. M. D. Barnes, W. B. Whitten, S. Arnold, J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
    [CrossRef]
  17. M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Enhanced fluorescence yields from cavity quantum electrodynamic effects in microdroplets,” J. Opt. Soc. Am. B 11, 1297–1304 (1994).
    [CrossRef]
  18. H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance,” Phys. Rev. A 37, 1597 (1988);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
    [CrossRef] [PubMed]
  19. H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987);H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
    [CrossRef] [PubMed]
  20. H.-B. Lin, J. D. Eversole, C. D. Merrit, A. J. Campillo, “Cavity-modified spontaneous emission rates in liquid micro-droplets,” Phys. Rev. A 45, 6756–6760 (1992).
    [CrossRef] [PubMed]
  21. For a comprehensive review, see R. K. Chang, A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996), Chap. 1.
  22. P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
    [CrossRef] [PubMed]
  23. M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
    [CrossRef] [PubMed]
  24. The most commonly used solvent in single-molecule detection experiments is water where the rotational diffusion time is of the order of a few picoseconds. Because the excited state lifetime is ∼3 ns, an orientational average is clearly appropriate. In these calculations, we also use the value of 1.34 as the refractive index of water.
  25. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 116–117.
  26. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, New York, 1995), pp. 20–28 and Chap. 7.
  27. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), pp. 79–91, 189–192.
    [CrossRef]
  28. S. C. Hill, H. I. Saleheen, K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
    [CrossRef]
  29. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), pp. 150–153.

1996

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
[CrossRef] [PubMed]

1995

S. C. Hill, H. I. Saleheen, K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
[CrossRef]

M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Detecting single molecules in liquids,” Anal. Chem. 67, 418A–423A (1995) and references cited therein.
[CrossRef]

1994

1993

M. D. Barnes, K. C. Ng, W. B. Whitten, J. M. Ramsey, “Detection of single rhodamine 6G molecules in levitated microdroplets,” Anal. Chem. 65, 2360–2365 (1993).
[CrossRef]

1992

H.-B. Lin, J. D. Eversole, C. D. Merrit, A. J. Campillo, “Cavity-modified spontaneous emission rates in liquid micro-droplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

D. Q. Chowdury, P. W. Barber, S. C. Hill, “Energy density distribution inside large nonabsorbing spheres via Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[CrossRef]

M. D. Barnes, W. B. Whitten, S. Arnold, J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[CrossRef]

1991

J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1549 (1991).
[CrossRef]

M. F. Buehler, T. M. Allen, E. J. Davis, “Microparticle Raman spectroscopy of multicomponent aerosols,” J. Colloid Interface Sci. 146, 79–89 (1991).
[CrossRef]

1990

A. Serpenguzel, G. Chen, R. K. Chang, “Stimulated Raman scattering of aqueous droplets containing ions: concentration and size determination,” Partic. Sci. Technol. 8, 1–10 (1990).

P. Chylek, “Resonance structure of Mie scattering: distances between resonances,” J. Opt. Soc. Am. A 7, 1609–1613 (1990).
[CrossRef]

1989

P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
[CrossRef] [PubMed]

1988

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance,” Phys. Rev. A 37, 1597 (1988);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef] [PubMed]

1987

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987);H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

D. S. Benincasa, P. W. Barber, J. Z. Zhang, W.-F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
[CrossRef] [PubMed]

1985

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

1984

1983

1976

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A. 13, 396–404 (1976).
[CrossRef]

1968

Allen, T. M.

M. F. Buehler, T. M. Allen, E. J. Davis, “Microparticle Raman spectroscopy of multicomponent aerosols,” J. Colloid Interface Sci. 146, 79–89 (1991).
[CrossRef]

Arnold, S.

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
[CrossRef] [PubMed]

M. D. Barnes, W. B. Whitten, S. Arnold, J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[CrossRef]

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Barber, P. W.

Barnes, M. D.

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
[CrossRef] [PubMed]

M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Detecting single molecules in liquids,” Anal. Chem. 67, 418A–423A (1995) and references cited therein.
[CrossRef]

M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Enhanced fluorescence yields from cavity quantum electrodynamic effects in microdroplets,” J. Opt. Soc. Am. B 11, 1297–1304 (1994).
[CrossRef]

M. D. Barnes, K. C. Ng, W. B. Whitten, J. M. Ramsey, “Detection of single rhodamine 6G molecules in levitated microdroplets,” Anal. Chem. 65, 2360–2365 (1993).
[CrossRef]

M. D. Barnes, W. B. Whitten, S. Arnold, J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[CrossRef]

Becker, P. C.

P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
[CrossRef] [PubMed]

Benincasa, D. S.

Benner, R. E.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 1–61.

Bigot, J. Y.

P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
[CrossRef] [PubMed]

Brito Cruz, C. H.

P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
[CrossRef] [PubMed]

Buehler, M. F.

M. F. Buehler, T. M. Allen, E. J. Davis, “Microparticle Raman spectroscopy of multicomponent aerosols,” J. Colloid Interface Sci. 146, 79–89 (1991).
[CrossRef]

Campillo, A. J.

H.-B. Lin, J. D. Eversole, C. D. Merrit, A. J. Campillo, “Cavity-modified spontaneous emission rates in liquid micro-droplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Chang, R. K.

A. Serpenguzel, G. Chen, R. K. Chang, “Stimulated Raman scattering of aqueous droplets containing ions: concentration and size determination,” Partic. Sci. Technol. 8, 1–10 (1990).

D. S. Benincasa, P. W. Barber, J. Z. Zhang, W.-F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
[CrossRef] [PubMed]

H. M. Tzeng, K. F. Wall, M. B. Long, R. K. Chang, “Evaporation and condensation rates of liquid droplets deduced from structure resonances in the fluorescence spectra,” Opt. Lett. 9, 273–275 (1984).
[CrossRef] [PubMed]

For a review, see S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Nonlinear Optics, O. Keller, ed. (Nova, Commack, New York, 1995), pp. 171–242.

Chen, G.

A. Serpenguzel, G. Chen, R. K. Chang, “Stimulated Raman scattering of aqueous droplets containing ions: concentration and size determination,” Partic. Sci. Technol. 8, 1–10 (1990).

Chew, H.

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987);H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A. 13, 396–404 (1976).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, New York, 1995), pp. 20–28 and Chap. 7.

Chowdury, D. Q.

Chylek, P.

Davis, E. J.

M. F. Buehler, T. M. Allen, E. J. Davis, “Microparticle Raman spectroscopy of multicomponent aerosols,” J. Colloid Interface Sci. 146, 79–89 (1991).
[CrossRef]

Druger, S.

Druger, S. D.

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Eversole, J. D.

H.-B. Lin, J. D. Eversole, C. D. Merrit, A. J. Campillo, “Cavity-modified spontaneous emission rates in liquid micro-droplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Folan, L. M.

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Fork, R. L.

P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
[CrossRef] [PubMed]

Fragnito, H. L.

P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
[CrossRef] [PubMed]

Fuchs, R.

Fuller, K. A.

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 116–117.

Hill, S. C.

S. C. Hill, H. I. Saleheen, K. A. Fuller, “Volume current method for modeling light scattering by inhomogeneously perturbed spheres,” J. Opt. Soc. Am. A 12, 905–915 (1995).
[CrossRef]

D. Q. Chowdury, P. W. Barber, S. C. Hill, “Energy density distribution inside large nonabsorbing spheres via Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[CrossRef]

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 1–61.

For a review, see S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Nonlinear Optics, O. Keller, ed. (Nova, Commack, New York, 1995), pp. 171–242.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), pp. 79–91, 189–192.
[CrossRef]

Holler, S.

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
[CrossRef] [PubMed]

Hovenac, E. A.

Hsieh, W.-F.

Kerker, M.

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A. 13, 396–404 (1976).
[CrossRef]

Kliewar, K. L.

Kung, C.-Y.

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
[CrossRef] [PubMed]

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance,” Phys. Rev. A 37, 1597 (1988);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef] [PubMed]

Leung, P. T.

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance,” Phys. Rev. A 37, 1597 (1988);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef] [PubMed]

Lin, H.-B.

H.-B. Lin, J. D. Eversole, C. D. Merrit, A. J. Campillo, “Cavity-modified spontaneous emission rates in liquid micro-droplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Lock, J. A.

Long, M. B.

McNulty, P. J.

S. Druger, P. J. McNulty, “Radiation patterns of fluorescence from molecules embedded in small particles: general case,” Appl. Opt. 22, 75–82 (1983).
[CrossRef] [PubMed]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A. 13, 396–404 (1976).
[CrossRef]

Merrit, C. D.

H.-B. Lin, J. D. Eversole, C. D. Merrit, A. J. Campillo, “Cavity-modified spontaneous emission rates in liquid micro-droplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Ng, K. C.

M. D. Barnes, K. C. Ng, W. B. Whitten, J. M. Ramsey, “Detection of single rhodamine 6G molecules in levitated microdroplets,” Anal. Chem. 65, 2360–2365 (1993).
[CrossRef]

Ramsey, J. M.

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
[CrossRef] [PubMed]

M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Detecting single molecules in liquids,” Anal. Chem. 67, 418A–423A (1995) and references cited therein.
[CrossRef]

M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Enhanced fluorescence yields from cavity quantum electrodynamic effects in microdroplets,” J. Opt. Soc. Am. B 11, 1297–1304 (1994).
[CrossRef]

M. D. Barnes, K. C. Ng, W. B. Whitten, J. M. Ramsey, “Detection of single rhodamine 6G molecules in levitated microdroplets,” Anal. Chem. 65, 2360–2365 (1993).
[CrossRef]

M. D. Barnes, W. B. Whitten, S. Arnold, J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[CrossRef]

Saleheen, H. I.

Serpenguzel, A.

A. Serpenguzel, G. Chen, R. K. Chang, “Stimulated Raman scattering of aqueous droplets containing ions: concentration and size determination,” Partic. Sci. Technol. 8, 1–10 (1990).

Shank, C. V.

P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
[CrossRef] [PubMed]

Tzeng, H. M.

Wall, K. F.

Whitten, W. B.

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
[CrossRef] [PubMed]

M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Detecting single molecules in liquids,” Anal. Chem. 67, 418A–423A (1995) and references cited therein.
[CrossRef]

M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Enhanced fluorescence yields from cavity quantum electrodynamic effects in microdroplets,” J. Opt. Soc. Am. B 11, 1297–1304 (1994).
[CrossRef]

M. D. Barnes, K. C. Ng, W. B. Whitten, J. M. Ramsey, “Detection of single rhodamine 6G molecules in levitated microdroplets,” Anal. Chem. 65, 2360–2365 (1993).
[CrossRef]

M. D. Barnes, W. B. Whitten, S. Arnold, J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), pp. 150–153.

Young, K.

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance,” Phys. Rev. A 37, 1597 (1988);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef] [PubMed]

Zhang, J. Z.

Anal. Chem.

M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Detecting single molecules in liquids,” Anal. Chem. 67, 418A–423A (1995) and references cited therein.
[CrossRef]

M. D. Barnes, K. C. Ng, W. B. Whitten, J. M. Ramsey, “Detection of single rhodamine 6G molecules in levitated microdroplets,” Anal. Chem. 65, 2360–2365 (1993).
[CrossRef]

Appl. Opt.

Chem. Phys. Lett.

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

J. Chem. Phys.

M. D. Barnes, W. B. Whitten, S. Arnold, J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[CrossRef]

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987);H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

J. Colloid Interface Sci.

M. F. Buehler, T. M. Allen, E. J. Davis, “Microparticle Raman spectroscopy of multicomponent aerosols,” J. Colloid Interface Sci. 146, 79–89 (1991).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Partic. Sci. Technol.

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Phys. Rev. A

H. M. Lai, P. T. Leung, K. Young, “Electromagnetic decay into a narrow resonance,” Phys. Rev. A 37, 1597 (1988);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987);S. C. Ching, H. M. Lai, K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef] [PubMed]

H.-B. Lin, J. D. Eversole, C. D. Merrit, A. J. Campillo, “Cavity-modified spontaneous emission rates in liquid micro-droplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Phys. Rev. A.

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[CrossRef]

Phys. Rev. Lett.

P. C. Becker, H. L. Fragnito, J. Y. Bigot, C. H. Brito Cruz, R. L. Fork, C. V. Shank, “Femtosecond photon echoes from molecules in solution,” Phys. Rev. Lett. 63, 505–508 (1989).
[CrossRef] [PubMed]

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S. Arnold, S. Holler, “Fluorescence of oriented molecules in a microcavity,” Phys. Rev. Lett. 76, 3931–3934 (1996).
[CrossRef] [PubMed]

Other

The most commonly used solvent in single-molecule detection experiments is water where the rotational diffusion time is of the order of a few picoseconds. Because the excited state lifetime is ∼3 ns, an orientational average is clearly appropriate. In these calculations, we also use the value of 1.34 as the refractive index of water.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), pp. 116–117.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, New York, 1995), pp. 20–28 and Chap. 7.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990), pp. 79–91, 189–192.
[CrossRef]

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), pp. 150–153.

For a review, see S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Nonlinear Optics, O. Keller, ed. (Nova, Commack, New York, 1995), pp. 171–242.

For a comprehensive review, see R. K. Chang, A. J. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996), Chap. 1.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 1–61.

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Figures (10)

Fig. 1
Fig. 1

Geometry for collection of fluorescence from a dipole inside a sphere. The dipole may have any orientation and may be at any point inside the sphere. Because the lens is centered on the z axis, only points in the half-plane, x > 0, y = 0, need be computed; dipoles at other positions can be obtained by symmetry.

Fig. 2
Fig. 2

Normalized fluorescence collected from a dipole emitting from different positions inside a droplet (indicated by the normalized source coordinates x′/a and z′/a where a is the droplet radius). The intensity is normalized by the intensity collected from a dipole with the same orientation emitting in free space. The lens is centered on the +z axis. The diameter of the sphere, 2 μm; NA, 10−6; wavelength, 0.6 μm; refractive index, 1.34. In (a) the dipoles are oriented in the y-direction and in (b) the dipoles are x-directed.

Fig. 3
Fig. 3

Normalized power collected when the dipole is randomly oriented and the NA is small (10−6). Diameter, 2 μm; NA, 10−6; wavelength, 0.6 μm; refractive index, 1.34.

Fig. 4
Fig. 4

Same as Fig. 3 except NA = 0.5.

Fig. 5
Fig. 5

Fluorescence collected from randomly oriented dipoles at different positions inside a 5-μm-diameter sphere. NA is 0.5. (a) The wavelength is 0.60454259 μm (on the TE30,1 resonance of the sphere). (b) The wavelength is 0.6 μm (not on a resonance).

Fig. 6
Fig. 6

Relative transition rates, F (r) = Γ c (r), for a 5-μm-diameter sphere having a refractive index of 1.34. (a) The wavelength is 0.60454259 μm (on the TE30,1 resonance); (b) The wavelength is 0.6 μm (off resonance). In each, the dashed curve is for a dipole polarized radially and the solid curve is for dipoles polarized transverse to the radial direction.

Fig. 7
Fig. 7

Fluorescence collected from randomly oriented dipoles at different positions inside a sphere (x′/a, z′/a) = (0.0, −0.9), (0.0, −0.98), (0.0, −0.3), (0.6, −0.6), (0.7, −0.7) as labeled. The fluorescence is shown as a function of the emission frequency. Diameter, 10 μm; refractive index, 1.34; NA, 0.5.

Fig. 8
Fig. 8

Normalized fluorescence collected from a randomly oriented dipole emitting inside a 5-μm-diameter sphere as in Fig. 5(a) but normalized also by the transition rate.

Fig. 9
Fig. 9

Fluorescence collected as a function of position inside the sphere for randomly oriented dipoles emitting with a center frequency of 16666.67 cm−1 and a linewidth of 100 cm−1. Diameter, 10 μm; refractive index, 1.34.

Fig. 10
Fig. 10

Fluorescence collected as a function of x′/a inside the sphere at a constant, z′/a = 0.5, for randomly oriented dipoles emitting with a linewidth of 100 cm−1 and three different center frequencies, two on resonances, and one halfway between the resonances. (a) The droplet has a refractive index of 1.4746 and a diameter of 3 μm. (b) The refractive index is 1.34 and the diameter is 4 μm. (a) The solid curve is for the dipole emission centered on the TE18,1 resonance frequency (15921.3 cm−1), which has a Q of 320. The curve of solid dots is for the dipole emission centered on the TM18,1 resonance frequency (16337.2 cm−1), which has a Q of 200. The dashed curve is for the dipole emission centered halfway between the TE18,1 and TM18,1 resonance frequencies (16127.7 cm−1). (b) The solid curve is for the dipole emission centered on the TE23,1 resonance frequency (16206.3 cm−1), which has a Q of 170. The curve of solid dots is for the dipole emission centered on the TM23,1 resonance frequency (16477.7 cm−1) which has a Q of 110.The dashed curve is for the dipole emission centered halfway between the TE18,1 and TM18,1 resonance frequencies (16341.2 cm−1).

Equations (29)

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F ( r , ω ) = power collected from a unit-amplitude diode at r inside a sphere power collected from a unit-amplitude diode in free space ,
F ( r ) = 0 L ( ω , ω o , Δ ω ) F u ( r , ω ) d ω 0 L ( ω , ω o , Δ ω ) F dip u ( r , ω ) d ω ,
p ( r ) = frequency-integrated power collected from a dipole at r inside a sphere frequency-integrated total power emitted from a dipole at r inside a sphere .
p ( r ) F ( r ) solid angle subtended by the lens 4 π .
F ( r ) F 4 π ( r )
E T ( m k r ) = E H ( m k r ) + E iG ( m k r ) .
E H ( m k r ) = ν = 1 D ν [ c ν H M ν 3 ( m k r ) + d ν H N ν 3 ( m k r ) ] ,
D mn = m ( 2 n + 1 ) ( n m ) ! 4 n ( n + 1 ) ( n + m ) ! ,
c ν H = i ω 2 μ k m / π V P ( r ) M ν 1 ( m k r ) d υ ,
d ν H = i ω 2 μ k m / π V P ( r ) N ν 1 ( m k r ) d υ ,
E iG ( m k r ) = ν = 1 [ c ν i G M ν 1 ( m k r ) + d ν i G N ν 1 ( m k r ) ] ,
E sG ( k r ) = ν = 1 D ν [ f ν G M ν 3 ( k r ) + g ν G N ν 3 ( k r ) ] ,
c ν iG = c ν H D ν h n ( 1 ) ( m x ) [ x h n ( 1 ) ( x ) ] [ m x h n ( 1 ) ( m x ) ] h n ( 1 ) ( x ) j n ( m x ) [ x h n ( 1 ) ( x ) ] [ m x j n ( m x ) ] h n ( 1 ) ( x ) ,
d ν iG = d ν H D ν m 2 h n ( 1 ) ( m x ) [ x h n ( 1 ) ( x ) ] [ m x h n ( 1 ) ( m x ) ] h n ( 1 ) ( x ) m 2 j n ( m x ) [ x h n ( 1 ) ( x ) ] [ m x j n ( m x ) ] h n ( 1 ) ( x ) ;
f ν G = c ν H i / ( m x ) j n ( m x ) [ x h n ( 1 ) ( x ) ] [ m x j n ( m x ) ] h n ( 1 ) ( x ) ,
g ν G = d ν H i / x m 2 j n ( m x ) [ x h n ( 1 ) ( x ) ] [ m x j n ( m x ) ] h n ( 1 ) ( x ) ,
F u ( r , ω ) = 1 2 μ c Ω lens | E sG ( k r l ) | 2 r l 2 d Ω ,
E s ( k r l ) = F s exp ( ik r l ) r l .
F u ( r , ω ) = 1 2 k 2 μ c Ω lens | k F s ( k r l ) | 2 d Ω .
R ( r , ω ) = F u ( r , ω ) / ω .
μ m = V P ( r ) d υ .
μ m ( t ) = μ m 0 exp [ ( Δ ω o / 2 ) t ] cos ( ω o t ) .
L ( ω , ω o , Δ ω ) = Δ ω o / 2 π ( ω ω o ) 2 + ( Δ ω o / 2 ) 2 .
F u ( r ) = 0 L ( ω , ω o , Δ ω ) F u ( r , ω ) d ω ,
F u ( r , ω ) = A b + i = 1 N res A i L ( ω = ω i , ω i , Δ ω i ) L ( ω , ω i , Δ ω i ) ,
F u ( r ) = A b + i = 1 N res A i Δ ω i ( Δ ω o + Δ ω i ) / 4 ( ω o ω i ) 2 + ( Δ ω o 2 + Δ ω i 2 ) 2 .
Γ c ( r ) = γ c ( r ) γ ,
Γ c ( r ) = total fluorescence collected from the dipole at r in the sphere fluorescence collected from the dipole in free the sphere .
F 4 π ( r ) = Γ c ( r ) .

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