Abstract

We describe a method for integrating analytically, over a circular aperture, the emission from an oscillating dipole inside a dielectric sphere. The model is useful for investigating fluorescence, Raman, or other emission from molecules inside of spherical particles or droplets. The analysis is performed for two cases: (a) the dipole emits from a fixed orientation, and (b) the dipole emits from all orientations and the collected energy is summed. This second case models the collection of emission from a molecule that is excited repeatedly; after each excitation it rotates to a random orientation before emitting. These results are applicable to single-molecule detection techniques employing microdroplets and to other techniques for characterizing microparticles with luminescence or inelastic scattering.

© 1997 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. K. C. Ng, W. H. Whitten, S. Arnold, J. M. Ramsey, “Digital chemical analysis of dilute microdroplets,” Anal. Chem. 64, 2914–2919 (1992).
    [CrossRef]
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    [CrossRef]
  7. R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  23. S. C. Hill, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Collection of fluorescence from single molecules in microspheres: effects of illumination geometry,” Appl. Opt. 36, 4425–4437 (1997).
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    [CrossRef]
  36. E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [CrossRef]
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    [CrossRef]
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1997 (2)

S. C. Hill, G Videen, J. D. Pendleton, “Reciprocity method for obtaining the far fields generated by a source inside or near a microparticle,” J. Opt. Soc. Am. B14, 2522–2529 (1997), B8401.

S. C. Hill, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Collection of fluorescence from single molecules in microspheres: effects of illumination geometry,” Appl. Opt. 36, 4425–4437 (1997).
[CrossRef] [PubMed]

1996 (3)

1995 (4)

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

R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
[CrossRef]

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]

G. Videen, D. Ngo, P. Chylek, R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12, 922–928 (1995).
[CrossRef]

1994 (5)

1993 (3)

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

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

B. V. Bronk, M. J. Smith, S. Arnold, “Photon-correlation spectroscopy for small spherical inclusions in a micrometer-sized electrodynamically levitated droplet,” Opt. Lett. 18, 93–95 (1993).
[CrossRef] [PubMed]

1992 (4)

W. B. Whitten, J. M. Ramsey, “Photocount probability distributions for single fluorescent molecules,” Appl. Spectrosc. 46, 1587–1589 (1992).
[CrossRef]

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

K. C. Ng, W. H. Whitten, S. Arnold, J. M. Ramsey, “Digital chemical analysis of dilute microdroplets,” Anal. Chem. 64, 2914–2919 (1992).
[CrossRef]

1991 (3)

W. B. Whitten, J. M. Ramsey, “Single-molecule detection limits in levitated microdroplets,” Anal. Chem. 63, 1027–1031 (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]

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

1990 (1)

J.-Z. Zhang, G Chen, R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single liquid droplet: input laser linewidth effects,” J. Opt. Soc. Am. B7, 108–115 (1990); J. C. Swindal, D. H. Leach, R. K. Chang, K. Young, “Precession of morphology-dependent resonances in nonspherical liquid droplets,” Opt. Lett.18, 191–193 (1993). For other examples, see the review chapter by S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, Commack, N.Y., 1995), pp. 171–242.

1988 (1)

1987 (1)

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

1986 (1)

1983 (1)

1982 (1)

1980 (1)

1979 (1)

1977 (2)

1976 (1)

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]

1973 (1)

Alexander, D. R.

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

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]

Arfken, G

G Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, San Diego, 1985), pp. 198–200, 253, 678.

Arnold, S

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S Arnold, “Molecular fluorescence in a microcavity: solvation dynamics and single molecule detection,” in Optical Processes in Microcavities, in R. K. Chang, A. J. Campillo, eds., (World Scientific, Singapore, 1996), pp. 135–165.
[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]

B. V. Bronk, M. J. Smith, S. Arnold, “Photon-correlation spectroscopy for small spherical inclusions in a micrometer-sized electrodynamically levitated droplet,” Opt. Lett. 18, 93–95 (1993).
[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]

K. C. Ng, W. H. Whitten, S. Arnold, J. M. Ramsey, “Digital chemical analysis of dilute microdroplets,” Anal. Chem. 64, 2914–2919 (1992).
[CrossRef]

L. Folan, S. Arnold, “Determination of molecular orientation at the surface of an aerosol particle by morphology-dependent photoselection,” Opt. Lett. 13, 1–3 (1988).
[PubMed]

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Barber, P. W.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

D.-S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1702–1705 (1979).
[CrossRef]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Barnes, M. D.

S. C. Hill, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Collection of fluorescence from single molecules in microspheres: effects of illumination geometry,” Appl. Opt. 36, 4425–4437 (1997).
[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]

S. C. Hill, H. I. Saleheen, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Modeling fluorescence collection from single molecules in microspheres: effects of position, orientation, and frequency,” Appl. Opt. 35, 6278–6288 (1996).
[CrossRef] [PubMed]

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

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

M. D. Barnes, 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]

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S Arnold, “Molecular fluorescence in a microcavity: solvation dynamics and single molecule detection,” in Optical Processes in Microcavities, in R. K. Chang, A. J. Campillo, eds., (World Scientific, Singapore, 1996), pp. 135–165.
[CrossRef]

Barton, J. P.

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

Belousov, S. L.

S. L. Belousov, Tables of Normalized Associated Legendre Polynomials (Pergamon, New York, 1962).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–101.

Borghese, F.

Bronk, B. V.

Bruno, J. G.

R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
[CrossRef]

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, A. J. Campillo, “Cavity-modified spontaneous emission rates in liquid microdroplets,” Phys. Rev. A 45, 6756–6760 (1992).
[CrossRef] [PubMed]

Chang, R. K.

G. Chen, P. Nachman, R. G. Pinnick, S. C. Hill, R. K. Chang, “Conditional-firing aerosol-fluorescence spectrum analyzer for individual airborne particles with pulsed 266-nm laser excitation,” Opt. Lett. 21, 1307–1309 (1996).
[CrossRef] [PubMed]

J.-Z. Zhang, G Chen, R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single liquid droplet: input laser linewidth effects,” J. Opt. Soc. Am. B7, 108–115 (1990); J. C. Swindal, D. H. Leach, R. K. Chang, K. Young, “Precession of morphology-dependent resonances in nonspherical liquid droplets,” Opt. Lett.18, 191–193 (1993). For other examples, see the review chapter by S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, Commack, N.Y., 1995), pp. 171–242.

Chen, G

J.-Z. Zhang, G Chen, R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single liquid droplet: input laser linewidth effects,” J. Opt. Soc. Am. B7, 108–115 (1990); J. C. Swindal, D. H. Leach, R. K. Chang, K. Young, “Precession of morphology-dependent resonances in nonspherical liquid droplets,” Opt. Lett.18, 191–193 (1993). For other examples, see the review chapter by S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, Commack, N.Y., 1995), pp. 171–242.

Chen, G.

Chew, H.

D.-S. Wang, M. Kerker, H. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
[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]

Chu, W. P.

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]

Denti, P.

Druger, S.

Druger, S. D.

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Eversole, J. D.

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

Farmer, W. M.

Fernandez, G. L.

R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
[CrossRef]

Folan, L.

Folan, L. M.

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Fuller, K. A.

Giel, T. V.

Hill, S. C.

S. C. Hill, G Videen, J. D. Pendleton, “Reciprocity method for obtaining the far fields generated by a source inside or near a microparticle,” J. Opt. Soc. Am. B14, 2522–2529 (1997), B8401.

S. C. Hill, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Collection of fluorescence from single molecules in microspheres: effects of illumination geometry,” Appl. Opt. 36, 4425–4437 (1997).
[CrossRef] [PubMed]

G. Chen, P. Nachman, R. G. Pinnick, S. C. Hill, R. K. Chang, “Conditional-firing aerosol-fluorescence spectrum analyzer for individual airborne particles with pulsed 266-nm laser excitation,” Opt. Lett. 21, 1307–1309 (1996).
[CrossRef] [PubMed]

S. C. Hill, H. I. Saleheen, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Modeling fluorescence collection from single molecules in microspheres: effects of position, orientation, and frequency,” Appl. Opt. 35, 6278–6288 (1996).
[CrossRef] [PubMed]

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]

R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

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]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–101.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd. ed. (Wiley, New York, 1975), p. 746.

Kerker, M.

D.-S. Wang, M. Kerker, H. Chew, “Raman and fluorescent scattering by molecules embedded in dielectric spheroids,” Appl. Opt. 19, 2315–2328 (1980).
[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]

Khaled, E. E. M.

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

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]

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S Arnold, “Molecular fluorescence in a microcavity: solvation dynamics and single molecule detection,” in Optical Processes in Microcavities, in R. K. Chang, A. J. Campillo, eds., (World Scientific, Singapore, 1996), pp. 135–165.
[CrossRef]

Lin, H.-B.

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

Mayo, M. W.

R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
[CrossRef]

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]

Nachman, P.

G. Chen, P. Nachman, R. G. Pinnick, S. C. Hill, R. K. Chang, “Conditional-firing aerosol-fluorescence spectrum analyzer for individual airborne particles with pulsed 266-nm laser excitation,” Opt. Lett. 21, 1307–1309 (1996).
[CrossRef] [PubMed]

R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
[CrossRef]

Ng, K. C.

K. C. Ng, W. H. Whitten, S. Arnold, J. M. Ramsey, “Digital chemical analysis of dilute microdroplets,” Anal. Chem. 64, 2914–2919 (1992).
[CrossRef]

Ngo, D.

Pendleton, J. D.

S. C. Hill, G Videen, J. D. Pendleton, “Reciprocity method for obtaining the far fields generated by a source inside or near a microparticle,” J. Opt. Soc. Am. B14, 2522–2529 (1997), B8401.

R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
[CrossRef]

J. D. Pendleton, “Mie scattering into solid angles,” J. Opt. Soc. Am. 72, 1029–1033 (1982).
[CrossRef]

J. D. Pendleton, “A generalized Mie theory solution and its application to particle sizing interferometry,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn.1982), p. 90.

Pinnick, R. G.

Ramsey, J. M.

S. C. Hill, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Collection of fluorescence from single molecules in microspheres: effects of illumination geometry,” Appl. Opt. 36, 4425–4437 (1997).
[CrossRef] [PubMed]

S. C. Hill, H. I. Saleheen, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Modeling fluorescence collection from single molecules in microspheres: effects of position, orientation, and frequency,” Appl. Opt. 35, 6278–6288 (1996).
[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]

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

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

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

K. C. Ng, W. H. Whitten, S. Arnold, J. M. Ramsey, “Digital chemical analysis of dilute microdroplets,” Anal. Chem. 64, 2914–2919 (1992).
[CrossRef]

W. B. Whitten, J. M. Ramsey, “Photocount probability distributions for single fluorescent molecules,” Appl. Spectrosc. 46, 1587–1589 (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]

W. B. Whitten, J. M. Ramsey, “Single-molecule detection limits in levitated microdroplets,” Anal. Chem. 63, 1027–1031 (1991).
[CrossRef]

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S Arnold, “Molecular fluorescence in a microcavity: solvation dynamics and single molecule detection,” in Optical Processes in Microcavities, in R. K. Chang, A. J. Campillo, eds., (World Scientific, Singapore, 1996), pp. 135–165.
[CrossRef]

Robinson, D. M.

Saija, R.

Saleheen, H. I.

Smith, M. J.

Son, J. Y.

J. Y. Son, W. M. Farmer, T. V. Giel, “New optical geometry for the particle sizing interferometer,” Appl. Opt. 25, 4332–4337 (1986).
[CrossRef] [PubMed]

J. Y. Son, “Multiple methods for obtaining particle sizing distribution with a particle sizing interferometer,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn.1985).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), p. 401.

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, N.J., 1994), Chap. 10.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 34, 124.

Videen, G

S. C. Hill, G Videen, J. D. Pendleton, “Reciprocity method for obtaining the far fields generated by a source inside or near a microparticle,” J. Opt. Soc. Am. B14, 2522–2529 (1997), B8401.

Videen, G.

Wang, D.-S.

Whitten, W. B.

S. C. Hill, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Collection of fluorescence from single molecules in microspheres: effects of illumination geometry,” Appl. Opt. 36, 4425–4437 (1997).
[CrossRef] [PubMed]

S. C. Hill, H. I. Saleheen, M. D. Barnes, W. B. Whitten, J. M. Ramsey, “Modeling fluorescence collection from single molecules in microspheres: effects of position, orientation, and frequency,” Appl. Opt. 35, 6278–6288 (1996).
[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]

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

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

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

W. B. Whitten, J. M. Ramsey, “Photocount probability distributions for single fluorescent molecules,” Appl. Spectrosc. 46, 1587–1589 (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]

W. B. Whitten, J. M. Ramsey, “Single-molecule detection limits in levitated microdroplets,” Anal. Chem. 63, 1027–1031 (1991).
[CrossRef]

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S Arnold, “Molecular fluorescence in a microcavity: solvation dynamics and single molecule detection,” in Optical Processes in Microcavities, in R. K. Chang, A. J. Campillo, eds., (World Scientific, Singapore, 1996), pp. 135–165.
[CrossRef]

Whitten, W. H.

K. C. Ng, W. H. Whitten, S. Arnold, J. M. Ramsey, “Digital chemical analysis of dilute microdroplets,” Anal. Chem. 64, 2914–2919 (1992).
[CrossRef]

Wiscombe, W. J.

Zhang, J.-Z.

J.-Z. Zhang, G Chen, R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single liquid droplet: input laser linewidth effects,” J. Opt. Soc. Am. B7, 108–115 (1990); J. C. Swindal, D. H. Leach, R. K. Chang, K. Young, “Precession of morphology-dependent resonances in nonspherical liquid droplets,” Opt. Lett.18, 191–193 (1993). For other examples, see the review chapter by S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, Commack, N.Y., 1995), pp. 171–242.

Aerosol Sci. Technol. (1)

R. G. Pinnick, S. C. Hill, P. Nachman, J. D. Pendleton, G. L. Fernandez, M. W. Mayo, J. G. Bruno, “Fluorescence particle counter for detecting airborne bacteria and other biological particles,” Aerosol Sci. Technol. 23, 653–664 (1995).
[CrossRef]

Anal. Chem. (4)

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

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

W. B. Whitten, J. M. Ramsey, “Single-molecule detection limits in levitated microdroplets,” Anal. Chem. 63, 1027–1031 (1991).
[CrossRef]

K. C. Ng, W. H. Whitten, S. Arnold, J. M. Ramsey, “Digital chemical analysis of dilute microdroplets,” Anal. Chem. 64, 2914–2919 (1992).
[CrossRef]

Appl. Opt. (8)

Appl. Spectrosc. (1)

IEEE Trans. Antennas Propag. (1)

E. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, “Electromagnetic fields for an irregularly shaped, near-spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 69, 7973–7986 (1991).
[CrossRef]

J. Chem. Phys. (2)

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]

S. D. Druger, S. Arnold, L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

J. Colloid Interface Sci. (1)

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. (3)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (3)

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

S. C. Hill, G Videen, J. D. Pendleton, “Reciprocity method for obtaining the far fields generated by a source inside or near a microparticle,” J. Opt. Soc. Am. B14, 2522–2529 (1997), B8401.

J.-Z. Zhang, G Chen, R. K. Chang, “Pumping of stimulated Raman scattering by stimulated Brillouin scattering within a single liquid droplet: input laser linewidth effects,” J. Opt. Soc. Am. B7, 108–115 (1990); J. C. Swindal, D. H. Leach, R. K. Chang, K. Young, “Precession of morphology-dependent resonances in nonspherical liquid droplets,” Opt. Lett.18, 191–193 (1993). For other examples, see the review chapter by S. C. Hill, R. K. Chang, “Nonlinear optics in droplets,” in Studies in Classical and Quantum Nonlinear Optics, O. Keller, ed. (Nova Science, Commack, N.Y., 1995), pp. 171–242.

Opt. Lett. (4)

Phys. Rev. A (2)

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

Phys. Rev. Lett. (1)

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 (14)

M. D. Barnes, C.-Y. Kung, W. B. Whitten, J. M. Ramsey, S Arnold, “Molecular fluorescence in a microcavity: solvation dynamics and single molecule detection,” in Optical Processes in Microcavities, in R. K. Chang, A. J. Campillo, eds., (World Scientific, Singapore, 1996), pp. 135–165.
[CrossRef]

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (IEEE Press, Piscataway, N.J., 1994), Chap. 10.

The Green function shown here can be obtained following the approach of Tai in Ref. 37. Equation (1) differs from Tai’s in that it employs the complex vector spherical harmonics (see Appendix B).

J. D. Pendleton, “A generalized Mie theory solution and its application to particle sizing interferometry,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn.1982), p. 90.

G Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, San Diego, 1985), pp. 198–200, 253, 678.

S. L. Belousov, Tables of Normalized Associated Legendre Polynomials (Pergamon, New York, 1962).

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), p. 401.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 34, 124.

J. D. Jackson, Classical Electrodynamics, 2nd. ed. (Wiley, New York, 1975), p. 746.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964).

I. S. Gradshteyn, I. M. Ryzhik, eds., Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 1005, 1008.

J. Y. Son, “Multiple methods for obtaining particle sizing distribution with a particle sizing interferometer,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn.1985).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–101.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

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Equations (89)

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E ( r ) = ω 2 μ 2 G ( r , r d ) · p ( r d ) ,
G ( r , r d ) = i k 2 μ 2 4 π μ 1 n = 1 2 n ( n + 1 ) m = - n n ( - 1 ) m × [ c n M ¯ n m ( 3 ) ( k 2 r ) M ¯ n , - m ( 1 ) ( k 1 r d ) + d n N ¯ n m ( 3 ) ( k 2 r ) N ¯ n , - m ( 1 ) ( k 1 r d ) ] .
M ¯ n m q ( k r ) × [ r z n ( q ) ( ρ ) exp ( i m ϕ ) P ¯ n m ( cos θ ) ]
= i z n ( q ) ( ρ ) exp ( i m ϕ ) [ θ ^ π ¯ n m ( cos θ ) + i ϕ ^ τ ¯ n m ( cos θ ) ] ,
N ¯ n m ( q ) ( k r ) 1 k × M ¯ n m ( q ) ( k r )
= exp ( i m ϕ ) { r ^ n ( n + 1 ) z n ( q ) ( ρ ) ρ P ¯ n m ( cos θ ) + ( d / d ρ [ ρ z n ( q ) ( ρ ) ] ) ρ [ θ ^ τ ¯ n m ( cos θ ) + i ϕ ^ π ¯ n m ( cos θ ) ] } ,
π ¯ n m ( cos θ ) = m P ¯ n m ( cos θ ) sin θ ,
τ ¯ n m ( cos θ ) = ( d / d θ ) P ¯ n m ( cos θ ) ,
p ^ = j = 1 3 f j e ^ j ,
f 1 = sin β cos α ,
f 2 = sin β sin α ,
f 3 = cos β .
E ( r ) = E r e f n = 1 2 i n ( n + 1 ) m = - n n j = 1 3 f j [ c n , - m j M ¯ n m ( 3 ) ( k r ) ] + d n , - m j N ¯ n m ( 3 ) ( k r ) ,
E ref ( p 4 π 0 ) [ ( μ 2 ) 2 μ 0 μ 1 ] k 2 ( k 0 ) 2 ,
c n , - m j ( - 1 ) m c n e ^ j · M ¯ n , - m ( 1 ) ( k 1 r d ) ,
d n , - m j ( - 1 ) m d n e ^ j · N ¯ n , - m ( 1 ) ( k 1 r d ) .
Δ Ω = 0 2 π θ = θ min θ = θ max d Ω = 2 π [ cos ( θ min ) - cos ( θ max ) ] .
P = Δ Ω d Ω r 2 I ( r ) ,
I ( r ) = N 2 2 μ 2 c E ( r ) 2 .
P D ( k 2 ) 2 P I ref ,
I ref = N 2 2 μ 2 c ( E ref ) 2 ,
= ( 1 8 π ) ( 1 4 π ɛ 0 ) [ ( μ 2 ) 3 μ 0 ( μ 1 ) 2 ] c p 2 ( k 2 ) 3 ( k 0 ) 3 .
P D = Δ Ω d Ω ( k 2 ) 2 r 2 E ( r ) 2 ( E ref ) 2 .
[ M ¯ n m ( 3 ) ( k 2 r ) N ¯ n m ( 3 ) ( k 2 r ) ] = ( - i ) n exp ( i ρ 2 ) ρ 2 exp ( i m ϕ ) { θ ^ [ π ¯ n m ( cos θ ) τ ¯ n m ( cos θ ) ] + i ϕ ^ [ τ ¯ n m ( cos θ ) π ¯ n m ( cos θ ) ] } .
P D = 8 π n = 1 n = 1 m = - min ( n , n ) min ( n , n ) j = 1 3 j = 1 3 f j f j [ ( ψ n n , - m dot ) j j I ¯ n n m dot + ( ψ n n , - m cross ) j j I ¯ n n m c r o s s ] ,
( ψ n n , - m dot ) j j ( c n , - m j c n , - m j * + d n , - m j d n , - m j * ) ( - i ) n ( i ) n n ( n + 1 ) n ( n + 1 ) ,
( ψ n n , - m cross ) j j ( c n , - m j d n , - m j * + d n , - m j c n , - m j * ) ( - i ) n ( i ) n n ( n + 1 ) n ( n + 1 ) ,
I ¯ n n m cross = μ max μ min d μ [ π ¯ n m ( μ ) τ ¯ n m ( μ ) + τ ¯ n m ( μ ) π ¯ n m ( μ ) ] = [ - m P ¯ n m ( μ ) P ¯ n m ( μ ) ] μ max μ min ,
I ¯ n n m dot μ max μ min d μ [ π ¯ n m ( μ ) π ¯ n m ( μ ) + τ ¯ n m ( μ ) τ ¯ n m ( μ ) ] .
I ¯ n n m dot = { ( - 1 ) [ n ( n + 1 ) P ¯ n m ( μ ) ( 1 - μ 2 ) 1 / 2 τ ¯ n m ( μ ) - n ( n + 1 ) P ¯ n m ( μ ) ( 1 - μ 2 ) 1 / 2 τ ¯ n m ( μ ) ] [ n ( n + 1 ) - n ( n + 1 ) ] } μ max μ min .
I ¯ n n m dot = [ ( - 1 ) P ¯ n m ( μ ) ( 1 - μ 2 ) 1 / 2 τ ¯ n m ( μ ) ] μ max μ min + n ( n + 1 ) I ¯ n n m ,
I ¯ n n m μ max μ min d μ [ P ¯ n m ( μ ) ] 2 , = [ P ¯ n m ( μ ) ( 1 - μ 2 ) 1 / 2 P n , m - 1 ( μ ) [ ( n + m ) ( n - m + 1 ) ] 1 / 2 ] μ max μ min + I ¯ n n , m - 1 .
m = - M M F m = m = 0 M m 1 2 ( F m + F - m ) ,
I ¯ n n , - m dot = I ¯ n n m dot ,
I ¯ n n , - m cross = - I ¯ n n m cross .
P ¯ n , - m ( μ ) = ( - 1 ) m P ¯ n m ( μ ) ,
π ¯ n , - m ( μ ) = ( - 1 ) m + 1 π ¯ n m ( μ ) ,
τ ¯ n , - m ( μ ) = ( - 1 ) m τ ¯ n m ( μ ) ,
P D = 4 π n = 1 n = 1 m = 0 min ( n , n ) m × { I ¯ n n m dot j = 1 3 j = 1 3 f j f j [ ( ψ n n m dot ) j j + ( ψ n n , - m dot ) j j ] - I ¯ n n m cross j = 1 3 j = 1 3 f j f j [ ( ψ n n m cross ) j j - ( ψ n n , - m cross ) j j ] } .
j = 1 3 j = 1 3 f j f j ( ψ n n m dot ) j j = ( - i ) n ( i ) n n ( n + 1 ) n ( n + 1 ) × [ ( j f j c n m j ) ( j f j c n m j ) * + ( j f j d n m j ) ( j f j d n m j ) * ] ,
j = 1 3 j = 1 3 f j f j ( ψ n n m cross ) j j = ( - i ) n ( i ) n n ( n + 1 ) n ( n + 1 ) × [ ( j f j c n m j ) ( j f j d n m j ) * + ( j f j d n m j ) ( j f j c n m j ) * ] ,
j = 1 3 f j c n m j = ( - 1 ) m c n [ j = 1 3 f j e ^ j · M ¯ n m ( 1 ) ]
= ( - 1 ) m c n j n ( ρ d ) i exp ( i m ϕ d ) { [ sin β cos θ d × cos ( α - ϕ d ) - cos β sin θ d ] π ¯ n m ( cos θ d ) + sin β sin ( α - ϕ d ) i τ ¯ n m ( cos θ d ) } ,
j = 1 3 f j d n m j = ( - 1 ) m d n [ j = 1 3 f j e ^ j · N ¯ n m ( 1 ) ]
( - 1 ) m d n exp ( i m ϕ d ) ( [ sin β cos θ d × cos ( α - ϕ d ) - cos β sin θ d ] × n ( n + 1 ) j n ( ρ d ) ρ d P n m ( cos θ d ) + { [ sin β cos θ d cos ( α - ϕ d ) - cos β sin θ d ] τ ¯ n m ( cos θ d ) + sin β × sin ( α - ϕ d ) i π ¯ n m ( cos θ d ) } × { 1 ρ d d [ ρ d j n ( ρ d ) ] d ρ d } ) .
{ g ( α , β ) } 1 4 π α = 0 2 π d α β = 0 2 π d β ( sin β ) g ( α , β ) .
{ f j f j } = 1 3 δ j j ,
{ P D } = 4 π 3 n = 1 n = 1 m = 0 min ( n , n ) m { I ¯ n n m dot j = 1 3 [ ( ψ n n m dot ) j j + ( ψ n n , - m dot ) j j ] - I ¯ n n m cross j = 1 3 [ ( ψ n n m cross ) j j - ( ψ n n , - m cross ) j j ] } .
j = 1 3 ( ψ n n m dot ) j j = j = 1 3 ( ψ n n , - m dot ) j j = Ψ n n m dot ,
j = 1 3 ( ψ n n m cross ) j j = - j = 1 3 ( ψ n n , - m cross ) j j = Ψ n n m cross + ψ n n m cross * ,
Ψ n n m dot [ c n c n * M ¯ n m ( 1 ) ( k r d ) · M ¯ n m ( 1 ) * ( k r d ) + d n d n * N ¯ n m ( 1 ) ( k r d ) · N ¯ n m ( 1 ) * ( k r d ) ] × [ ( - i ) n ( i ) n n ( n + 1 ) n ( n + 1 ) ] ,
Ψ n n m cross [ c n d n * M ¯ n m ( 1 ) ( k r d ) · N ¯ n m ( 1 ) * ( k r d ) ] × [ ( - i ) n ( i ) n n ( n + 1 ) n ( n + 1 ) ] .
{ P D } = 8 π 3 n = 1 n = 1 m = 0 min ( n , n ) m { Ψ n n m dot I ¯ n n m dot - [ 2 Re ( Ψ n n m cross ) ] I ¯ n n m cross } .
{ P } = [ I ref ( k 2 ) 2 ] { P D } .
F ( r d , ω ) { P } { P } hom .
{ P } hom = [ I ref ( k 2 ) 2 ] ( 2 Δ Ω 3 )
F ( r d , ω ) = 4 π Δ Ω n = 1 n = 1 m = 0 min ( n , n ) m { Ψ n n m dot I ¯ n n m dot - [ 2 Re ( Ψ n n m cross ) ] I ¯ n n m cross } .
π ¯ n m ( μ ) m P ¯ n m ( μ ) ( 1 - μ 2 ) 1 / 2 ,
τ ¯ n m ( μ ) - ( 1 - μ 2 ) 1 / 2 ( d / d μ ) P ¯ n m ( μ ) ,
P ¯ n m ( μ ) A n m P n m ( μ ) ,
A n m [ ( 2 n + 1 ) ( n - m ) ! 2 ( n + m ) ! ] 1 / 2 ,
P n m ( μ ) = ( 1 - μ 2 ) m / 2 d m d μ m P n ( μ )
μ π ¯ n m ( μ ) = ( 1 2 ) { [ ( n - m + 1 ) ( n + m ) ] 1 / 2 P ¯ n , m - 1 ( μ ) + [ ( n + m + 1 ) ( n - m ) ] 1 / 2 P ¯ n , m + 1 ( μ ) } ,
τ ¯ n m ( μ ) = ( 1 2 ) { [ ( n - m + 1 ) ( n + m ) ] 1 / 2 P ¯ n , m - 1 ( μ ) - [ ( n + m + 1 ) ( n - m ) ] 1 / 2 P ¯ n , m + 1 ( μ ) } .
P ¯ n m ( 1 ) = [ ( 2 n + 1 ) 2 ] 1 / 2 δ m , 0 ,
π ¯ n m ( 1 ) = 1 2 [ ( 2 n + 1 ) n ( n + 1 ) 2 ] 1 / 2 ( δ m , 1 + δ m , - 1 ) ,
τ ¯ n m ( 1 ) = 1 2 [ ( 2 n + 1 ) n ( n + 1 ) 2 ] 1 / 2 ( δ m , 1 - δ m , - 1 ) .
A n = - b n ,             B n = - a n ,             C n = c n ,             D n = d n .
G ( r , r d ) = i k 2 μ 2 4 π μ 1 n = 1 2 n ( n + 1 ) m = 0 n m ( A n m ) 2 × σ = e , o [ c n M σ m n ( 3 ) ( k 2 r ) M σ m n ( 1 ) ( k 1 r d ) + d n N σ m n ( 3 ) ( k 2 r ) N σ m n ( 1 ) ( k 1 r d ) ] .
A n m M e m n ( q ) = 1 2 [ M ¯ n m ( q ) + ( - 1 ) m M ¯ n , - m ( q ) ] ,
A n m M o m n ( q ) = 1 2 i [ M ¯ n m ( q ) - ( - 1 ) m M ¯ n , - m ( q ) ] ,
A n m ( q ) N e m n ( q ) = 1 2 [ N ¯ n m ( q ) + ( - 1 ) m N ¯ n , - m ( q ) ] ,
A n m N o m n ( q ) = 1 2 i [ N ¯ n m ( q ) - ( - 1 ) m N ¯ n , - m ( q ) ] ,
G ( r , r d ) = i k 2 μ 2 4 π μ 1 n = 1 1 n ( n + 1 ) m = 0 m m ( - 1 ) m × { c n [ M ¯ n m ( 3 ) ( k 2 r ) M ¯ n , - m ( 1 ) ( k 1 r d ) + M ¯ n , - m ( 3 ) ( k 2 r ) M ¯ n m ( 1 ) ( k 1 r d ) ] + d n [ N ¯ n m ( 3 ) ( k 2 r ) N ¯ n , - m ( 1 ) ( k 1 r d ) + N ¯ n , - m ( 3 ) ( k 2 r ) N ¯ n m ( 1 ) ( k 1 r d ) ] } .
G ( r , r d ) = i k 2 μ 2 4 π μ 1 n = 1 2 n ( n + 1 ) m = - n m ( - 1 ) m × [ c n M ¯ n m ( 3 ) ( k 2 r ) M ¯ n , - m ( 1 ) ( k 1 r d ) + d n N ¯ n m ( 3 ) ( k 2 r ) N ¯ n , - m ( 1 ) ( k 1 r d ) ] .
P ¯ n , m - 1 ( μ ) = [ μ π ¯ n m ( μ ) + τ ¯ n m ( μ ) ] / [ ( n + m ) ( n - m + 1 ] 1 / 2 ,
d d μ [ ( 1 - μ 2 ) 1 / 2 τ ¯ n m ( μ ) ] = n ( n + 1 ) P ¯ n m ( μ ) - m π ¯ n m ( μ ) ( 1 - μ 2 ) 1 / 2 .
{ P D } hom Δ Ω = 4 π = 8 π 3 n = 1 m = 0 n m [ M ¯ n m ( 1 ) ( k 1 r d ) 2 + N ¯ n m ( 1 ) ( k 1 r d ) 2 ] n ( n + 1 ) .
M ¯ n m ( 1 ) ( k 1 r d ) 2 = [ j n ( ρ d ) ] 2 { [ π ¯ n m ( cos θ d ) ] 2 + [ τ ¯ n m ( cos θ d ) ] 2 } ,
N ¯ n m ( 1 ) ( k 1 r d ) 2 = n 2 ( n + 1 ) 2 [ j n ( ρ d ) ρ d ] 2 [ P ¯ n m ( cos θ d ) ] 2 + { ( d / d ρ d ) [ ρ d j n ( ρ d ) ] ρ d } 2 { [ π ¯ n m ( cos θ d ) ] 2 + [ π ¯ n m ( cos θ d ) ] 2 } .
m = 0 n m [ P ¯ n m ( cos θ d ) ] 2 = ( 2 n + 1 ) 2 ,
m = 0 n m { [ π ¯ n m ( cos θ d ) ] 2 + [ τ ¯ n m ( cos θ d ) ] 2 } = ( 2 n + 1 ) n ( n + 1 ) 2 ,
{ P d } hom Δ Ω = 4 π = 4 π 3 n = 1 ( 2 n + 1 ) ( [ j n ( ρ d ) ] 2 + n ( n + 1 ) [ j n ( ρ d ) ρ d ] 2 + { ( d / d ρ d ) [ ρ d j n ( ρ d ) ] ρ d } 2 ) .
exp ( i ρ d μ ) = n = 0 i n ( 2 n + 1 ) j n ( ρ d ) P n ( μ ) ,
n = 1 ( 2 n + 1 ) [ j n ( ρ d ) ] 2 = 1 - [ j 0 ( ρ d ) ] 2 .
n = 1 ( 2 n + 1 ) { ( d / d ρ d ) [ ρ d j n ( ρ d ) ] ϱ d } 2 = 1 3 + 1 ( ρ d ) 2 - { ( d / d p d ) [ ρ d j 0 ( ρ d ) ] ρ d } 2 .
n = 1 ( 2 n + 1 ) n ( n + 1 ) [ j n ( ρ d ) ρ d ] 2 = 2 3 .
{ P D } hom Δ Ω = 4 π = 8 π / 3.
{ P D } hom = ( 8 π 3 ) ( Δ Ω 4 π ) = ( 2 3 ) Δ Ω

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