Abstract

Experimental near-forward-scattering diagrams obtained with one particle in optical levitation are recorded and compared with scattering diagrams computed by using the generalized Lorenz–Mie theory. Comparisons concern the particular case of an off-axis location of the particle. Agreement between experimental and computed diagrams is found to be satisfactory.

© 1992 Optical Society of America

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  1. F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser Doppler Anemometry (Academic, London, 1981).
  2. W. D. Bachalo, M. J. Houser, “Phase Doppler spray analyzer for simultaneous measurements of the drop size and velocity distribution,” Opt. Eng. 23, 583–590 (1984).
  3. A. Naqwi, F. Durst, X.-Z. Liu, “An extended phase-Doppler system for characterization of multiphase flows,” in Proceedings of the Fifth International Symposium on Application of Laser Techniques to Fluid Mechanics, (Instituto Superior Tecnico, Lisbon, 1990), paper 24–4.
  4. S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 107–120.
  5. G. Gréhan, G. Gouesbet, “Simultaneous measurements of the velocities and sizes of particles in flows using a combined system incorporating a top-hat beam system,” Appl. Opt. 25, 3527–3538 (1986).
    [CrossRef] [PubMed]
  6. F. Corbin, G. Grehan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
    [CrossRef]
  7. M. Maeda, K. Hishida, “Application of top-hat laser beam to particle sizing in LDV system,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 431–441.
  8. G. Gréhan, B. Maheu, G. Gouesbet, “Diffusion de la lumiére par une sphére dans le cas d’un faisceau d’extension finie −2. Theorie de Lorenz–Mie généralisée: application a la granulométrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
    [CrossRef]
  9. J. P. Chevaillier, J. Fabre, P. Hamelin, “Forward scattered light intensities by a sphere located anywhere in a Gaussian beam,” Appl. Opt. 25, 1222–1225 (1986).
    [CrossRef] [PubMed]
  10. W. D. Bachalo, S. V. Sankar, “Analysis of the light scattering interferometry for a sphere larger than the light wavelength,” in Proceedings of the Fourth International Symposium on Applications of Laser Anemometry to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1990), pp. 1–8.
  11. B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
    [CrossRef]
  12. G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  13. G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.
  14. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  15. P. Hamelin, “Application de la diffusion lumineuse à la métrologie des particules en écoulement diphasique dispersé,” thèse (Institut National Polytechnique de Toulouse, Toulouse, France, 1986).
  16. J. P. Chevaillier, J. Fabre, G. Gréhan, G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on the axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
    [CrossRef] [PubMed]
  17. G. Gréhan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
    [CrossRef] [PubMed]
  18. B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: Numerical results and application to optical sizing,” Part. Part. Syst. Charact. 4, 141–146 (1987).
    [CrossRef]
  19. A. Ashkin, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
    [CrossRef] [PubMed]
  20. R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of single levitated droplets by Raman spectroscopy,” Langmuir ACS J. Surf. Colloids 1, 365–367 (1985).
    [CrossRef]
  21. N. Y. Misconi, J. P. Olivier, K. F. Ratcliff, E. T. Rusk, W. X. Wang, “Light scattering by laser levitated particles,” Appl. Opt. 29, 2276–2281 (1990).
    [CrossRef] [PubMed]
  22. G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
    [CrossRef]
  23. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  24. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  25. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beam,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  26. G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
    [CrossRef]
  27. G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. 7, 998–1007 (1990).
    [CrossRef]
  28. G. Gréhan, G. Gouesbet, “Scattering of a laser beam by one particle: behaviour of gnm coefficients,” in Proceedings of the Third International Aerosol Conference, S. Masuda, K. Takahashi, eds. (Pergamon, Oxford, 1990), pp. 273–276.
  29. G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
    [CrossRef]
  30. F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
    [CrossRef] [PubMed]
  31. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156 (1970).
    [CrossRef]
  32. A. Ashkin, J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24, 586–588 (1974).
    [CrossRef]
  33. G. Roosen, “La léitation optique de sphères,” Can. J. Phys. 57, 1260–1279 (1979).
    [CrossRef]
  34. G. Roosen, B. Delaunay, C. Imbert, “Etude de la pression de radiation exercée par un faisceau lumineux sur une sphère réfringente,” J. Opt. (Paris) 8, 181–187 (1977).
    [CrossRef]
  35. A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz-Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
    [CrossRef] [PubMed]
  36. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  37. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  38. P. W. Barber, R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).
  39. R. T. Killinger, R. H. Zerull, “Effects of shape and orientation to be considered for optical particle sizing,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G.Gréhan Gréhan, eds. (Plenum, New York, 1988), pp. 419–429.

1991 (1)

F. Corbin, G. Grehan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

1990 (3)

1989 (2)

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beam,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

1988 (6)

G. Gréhan, B. Maheu, G. Gouesbet, “Diffusion de la lumiére par une sphére dans le cas d’un faisceau d’extension finie −2. Theorie de Lorenz–Mie généralisée: application a la granulométrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

1987 (1)

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: Numerical results and application to optical sizing,” Part. Part. Syst. Charact. 4, 141–146 (1987).
[CrossRef]

1986 (3)

1985 (2)

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of single levitated droplets by Raman spectroscopy,” Langmuir ACS J. Surf. Colloids 1, 365–367 (1985).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

1984 (2)

W. D. Bachalo, M. J. Houser, “Phase Doppler spray analyzer for simultaneous measurements of the drop size and velocity distribution,” Opt. Eng. 23, 583–590 (1984).

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

1981 (1)

1980 (2)

1979 (1)

G. Roosen, “La léitation optique de sphères,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

1977 (1)

G. Roosen, B. Delaunay, C. Imbert, “Etude de la pression de radiation exercée par un faisceau lumineux sur une sphère réfringente,” J. Opt. (Paris) 8, 181–187 (1977).
[CrossRef]

1974 (1)

A. Ashkin, J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24, 586–588 (1974).
[CrossRef]

1970 (1)

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

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Al-Chalabi, S. A. M.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 107–120.

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Allano, D.

Ashkin, A.

A. Ashkin, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24, 586–588 (1974).
[CrossRef]

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

Bachalo, W. D.

W. D. Bachalo, M. J. Houser, “Phase Doppler spray analyzer for simultaneous measurements of the drop size and velocity distribution,” Opt. Eng. 23, 583–590 (1984).

W. D. Bachalo, S. V. Sankar, “Analysis of the light scattering interferometry for a sphere larger than the light wavelength,” in Proceedings of the Fourth International Symposium on Applications of Laser Anemometry to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1990), pp. 1–8.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Chevaillier, J. P.

Corbin, F.

F. Corbin, G. Grehan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

Delaunay, B.

G. Roosen, B. Delaunay, C. Imbert, “Etude de la pression de radiation exercée par un faisceau lumineux sur une sphère réfringente,” J. Opt. (Paris) 8, 181–187 (1977).
[CrossRef]

Durst, F.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser Doppler Anemometry (Academic, London, 1981).

A. Naqwi, F. Durst, X.-Z. Liu, “An extended phase-Doppler system for characterization of multiphase flows,” in Proceedings of the Fifth International Symposium on Application of Laser Techniques to Fluid Mechanics, (Instituto Superior Tecnico, Lisbon, 1990), paper 24–4.

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24, 586–588 (1974).
[CrossRef]

Fabre, J.

Gouesbet, G.

F. Corbin, G. Grehan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. 7, 998–1007 (1990).
[CrossRef]

J. P. Chevaillier, J. Fabre, G. Gréhan, G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on the axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beam,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Diffusion de la lumiére par une sphére dans le cas d’un faisceau d’extension finie −2. Theorie de Lorenz–Mie généralisée: application a la granulométrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: Numerical results and application to optical sizing,” Part. Part. Syst. Charact. 4, 141–146 (1987).
[CrossRef]

G. Gréhan, G. Gouesbet, “Simultaneous measurements of the velocities and sizes of particles in flows using a combined system incorporating a top-hat beam system,” Appl. Opt. 25, 3527–3538 (1986).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz-Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Scattering of a laser beam by one particle: behaviour of gnm coefficients,” in Proceedings of the Third International Aerosol Conference, S. Masuda, K. Takahashi, eds. (Pergamon, Oxford, 1990), pp. 273–276.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

Grehan, G.

F. Corbin, G. Grehan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

Gréhan, G.

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. 7, 998–1007 (1990).
[CrossRef]

J. P. Chevaillier, J. Fabre, G. Gréhan, G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on the axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beam,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Diffusion de la lumiére par une sphére dans le cas d’un faisceau d’extension finie −2. Theorie de Lorenz–Mie généralisée: application a la granulométrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: Numerical results and application to optical sizing,” Part. Part. Syst. Charact. 4, 141–146 (1987).
[CrossRef]

G. Gréhan, G. Gouesbet, “Simultaneous measurements of the velocities and sizes of particles in flows using a combined system incorporating a top-hat beam system,” Appl. Opt. 25, 3527–3538 (1986).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz-Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Scattering of a laser beam by one particle: behaviour of gnm coefficients,” in Proceedings of the Third International Aerosol Conference, S. Masuda, K. Takahashi, eds. (Pergamon, Oxford, 1990), pp. 273–276.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

Hamelin, P.

J. P. Chevaillier, J. Fabre, P. Hamelin, “Forward scattered light intensities by a sphere located anywhere in a Gaussian beam,” Appl. Opt. 25, 1222–1225 (1986).
[CrossRef] [PubMed]

P. Hamelin, “Application de la diffusion lumineuse à la métrologie des particules en écoulement diphasique dispersé,” thèse (Institut National Polytechnique de Toulouse, Toulouse, France, 1986).

Hardalupas, Y.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 107–120.

Hishida, K.

M. Maeda, K. Hishida, “Application of top-hat laser beam to particle sizing in LDV system,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 431–441.

Houser, M. J.

W. D. Bachalo, M. J. Houser, “Phase Doppler spray analyzer for simultaneous measurements of the drop size and velocity distribution,” Opt. Eng. 23, 583–590 (1984).

Imbert, C.

G. Roosen, B. Delaunay, C. Imbert, “Etude de la pression de radiation exercée par un faisceau lumineux sur une sphère réfringente,” J. Opt. (Paris) 8, 181–187 (1977).
[CrossRef]

Jones, A. R.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 107–120.

Killinger, R. T.

R. T. Killinger, R. H. Zerull, “Effects of shape and orientation to be considered for optical particle sizing,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G.Gréhan Gréhan, eds. (Plenum, New York, 1988), pp. 419–429.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Lettieri, T. R.

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of single levitated droplets by Raman spectroscopy,” Langmuir ACS J. Surf. Colloids 1, 365–367 (1985).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Liu, X.-Z.

A. Naqwi, F. Durst, X.-Z. Liu, “An extended phase-Doppler system for characterization of multiphase flows,” in Proceedings of the Fifth International Symposium on Application of Laser Techniques to Fluid Mechanics, (Instituto Superior Tecnico, Lisbon, 1990), paper 24–4.

Maeda, M.

M. Maeda, K. Hishida, “Application of top-hat laser beam to particle sizing in LDV system,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 431–441.

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. 7, 998–1007 (1990).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beam,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Diffusion de la lumiére par une sphére dans le cas d’un faisceau d’extension finie −2. Theorie de Lorenz–Mie généralisée: application a la granulométrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: Numerical results and application to optical sizing,” Part. Part. Syst. Charact. 4, 141–146 (1987).
[CrossRef]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

Melling, A.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser Doppler Anemometry (Academic, London, 1981).

Misconi, N. Y.

Naqwi, A.

A. Naqwi, F. Durst, X.-Z. Liu, “An extended phase-Doppler system for characterization of multiphase flows,” in Proceedings of the Fifth International Symposium on Application of Laser Techniques to Fluid Mechanics, (Instituto Superior Tecnico, Lisbon, 1990), paper 24–4.

Olivier, J. P.

Preston, R. E.

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of single levitated droplets by Raman spectroscopy,” Langmuir ACS J. Surf. Colloids 1, 365–367 (1985).
[CrossRef]

Ratcliff, K. F.

Roosen, G.

G. Roosen, “La léitation optique de sphères,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

G. Roosen, B. Delaunay, C. Imbert, “Etude de la pression de radiation exercée par un faisceau lumineux sur une sphère réfringente,” J. Opt. (Paris) 8, 181–187 (1977).
[CrossRef]

Rusk, E. T.

Sankar, S. V.

W. D. Bachalo, S. V. Sankar, “Analysis of the light scattering interferometry for a sphere larger than the light wavelength,” in Proceedings of the Fourth International Symposium on Applications of Laser Anemometry to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1990), pp. 1–8.

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Semerjian, H. G.

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of single levitated droplets by Raman spectroscopy,” Langmuir ACS J. Surf. Colloids 1, 365–367 (1985).
[CrossRef]

Slimani, F.

Taylor, A. M. K. P.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 107–120.

Ungut, A.

Wang, W. X.

Whitelaw, J. H.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser Doppler Anemometry (Academic, London, 1981).

Zerull, R. H.

R. T. Killinger, R. H. Zerull, “Effects of shape and orientation to be considered for optical particle sizing,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G.Gréhan Gréhan, eds. (Plenum, New York, 1988), pp. 419–429.

Appl. Opt. (10)

A. Ashkin, J. M. Dziedzic, “Observation of light scattering from nonspherical particles using optical levitation,” Appl. Opt. 19, 660–668 (1980).
[CrossRef] [PubMed]

F. Slimani, G. Gréhan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef] [PubMed]

J. P. Chevaillier, J. Fabre, P. Hamelin, “Forward scattered light intensities by a sphere located anywhere in a Gaussian beam,” Appl. Opt. 25, 1222–1225 (1986).
[CrossRef] [PubMed]

N. Y. Misconi, J. P. Olivier, K. F. Ratcliff, E. T. Rusk, W. X. Wang, “Light scattering by laser levitated particles,” Appl. Opt. 29, 2276–2281 (1990).
[CrossRef] [PubMed]

J. P. Chevaillier, J. Fabre, G. Gréhan, G. Gouesbet, “Comparison of diffraction theory and generalized Lorenz–Mie theory for a sphere located on the axis of a laser beam,” Appl. Opt. 29, 1293–1298 (1990).
[CrossRef] [PubMed]

A. Ungut, G. Gréhan, G. Gouesbet, “Comparisons between geometrical optics and Lorenz-Mie theory,” Appl. Opt. 20, 2911–2918 (1981).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Simultaneous measurements of the velocities and sizes of particles in flows using a combined system incorporating a top-hat beam system,” Appl. Opt. 25, 3527–3538 (1986).
[CrossRef] [PubMed]

G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gréhan, G. Gouesbet, “Optical levitation of a single particle to study the theory of the quasi-elastic scattering of light,” Appl. Opt. 19, 2485–2487 (1980).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

A. Ashkin, J. M. Dziedzic, “Stability of optical levitation by radiation pressure,” Appl. Phys. Lett. 24, 586–588 (1974).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Can. J. Phys. (1)

G. Roosen, “La léitation optique de sphères,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

J. Aerosol Sci. (1)

G. Gréhan, B. Maheu, G. Gouesbet, “Diffusion de la lumiére par une sphére dans le cas d’un faisceau d’extension finie −2. Theorie de Lorenz–Mie généralisée: application a la granulométrie optique,” J. Aerosol Sci. 19, 55–64 (1988).
[CrossRef]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

J. Opt. (Paris) (5)

B. Maheu, G. Gouesbet, G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. (Paris) 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnm in the generalized Lorenz–Mie theory using finite series,” J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “On the generalized Lorenz–Mie theory: first attempt to design a localized approximation to the computation of the coefficients gnm,” J. Opt. (Paris) 20, 31–43 (1989).
[CrossRef]

G. Roosen, B. Delaunay, C. Imbert, “Etude de la pression de radiation exercée par un faisceau lumineux sur une sphère réfringente,” J. Opt. (Paris) 8, 181–187 (1977).
[CrossRef]

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnm in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. 7, 998–1007 (1990).
[CrossRef]

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

Langmuir ACS J. Surf. Colloids (1)

R. E. Preston, T. R. Lettieri, H. G. Semerjian, “Characterization of single levitated droplets by Raman spectroscopy,” Langmuir ACS J. Surf. Colloids 1, 365–367 (1985).
[CrossRef]

Opt. Commun. (1)

B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beam,” Opt. Commun. 70, 259–262 (1989).
[CrossRef]

Opt. Eng. (1)

W. D. Bachalo, M. J. Houser, “Phase Doppler spray analyzer for simultaneous measurements of the drop size and velocity distribution,” Opt. Eng. 23, 583–590 (1984).

Part. Part. Syst. Charact. (2)

F. Corbin, G. Grehan, G. Gouesbet, “Top-hat beam technique: improvements and application to bubble measurements,” Part. Part. Syst. Charact. 8, 222–228 (1991).
[CrossRef]

B. Maheu, G. Gréhan, G. Gouesbet, “Laser beam scattering by individual spherical particles: Numerical results and application to optical sizing,” Part. Part. Syst. Charact. 4, 141–146 (1987).
[CrossRef]

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (10)

P. W. Barber, R. K. Chang, eds., Optical Effects Associated with Small Particles (World Scientific, Singapore, 1988).

R. T. Killinger, R. H. Zerull, “Effects of shape and orientation to be considered for optical particle sizing,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G.Gréhan Gréhan, eds. (Plenum, New York, 1988), pp. 419–429.

F. Durst, A. Melling, J. H. Whitelaw, Principles and Practice of Laser Doppler Anemometry (Academic, London, 1981).

G. Gréhan, G. Gouesbet, “Scattering of a laser beam by one particle: behaviour of gnm coefficients,” in Proceedings of the Third International Aerosol Conference, S. Masuda, K. Takahashi, eds. (Pergamon, Oxford, 1990), pp. 273–276.

M. Maeda, K. Hishida, “Application of top-hat laser beam to particle sizing in LDV system,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 431–441.

A. Naqwi, F. Durst, X.-Z. Liu, “An extended phase-Doppler system for characterization of multiphase flows,” in Proceedings of the Fifth International Symposium on Application of Laser Techniques to Fluid Mechanics, (Instituto Superior Tecnico, Lisbon, 1990), paper 24–4.

S. A. M. Al-Chalabi, Y. Hardalupas, A. R. Jones, A. M. K. P. Taylor, “Calculation of calibration curves for the phase Doppler technique: comparison between Mie theory and geometrical optics,” in Optical Particle Sizing: Theory and Practice, G. Gouesbet, G. Gréhan, eds. (Plenum, New York, 1988), pp. 107–120.

P. Hamelin, “Application de la diffusion lumineuse à la métrologie des particules en écoulement diphasique dispersé,” thèse (Institut National Polytechnique de Toulouse, Toulouse, France, 1986).

W. D. Bachalo, S. V. Sankar, “Analysis of the light scattering interferometry for a sphere larger than the light wavelength,” in Proceedings of the Fourth International Symposium on Applications of Laser Anemometry to Fluid Mechanics (Instituto Superior Tecnico, Lisbon, 1990), pp. 1–8.

G. Gouesbet, G. Gréhan, B. Maheu, “Generalized Lorenz-Mie theory and applications to optical sizing,” in Combustion Measurements, N. Chigier, ed. (Hemisphere, New York, 1991), pp. 339–384.

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Figures (16)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

One-beam experimental setup.

Fig. 3
Fig. 3

Two-beam experimental setup.

Fig. 4
Fig. 4

Computed scattering diagrams for one particle at different locations along the beam axis.

Fig. 5
Fig. 5

Sensitivity to the particle diameter (z0 = 1 mm, 2ω0 = 10 μm, 2ωlocal = 32.22 μm, M = 1.5 + 0.0i).

Fig. 6
Fig. 6

Sensitivity to the beam-waist size (z0 = 1 mm, d = 20 μm, 2ωlocal = 32.22 μm, M = 1.5 + 0.0i).

Fig. 7
Fig. 7

Sensitivity to the complex refractive index (z0 = 1 mm, d = 20 μm, 2ω0 = 10 μm, 2ωlocal = 32.22 μm)

Fig. 8
Fig. 8

One-beam setup results. A laser beam (wavelength 0.5145 μm, beam waist diameter 13.2 μm) is used to levitate a glass particle (diameter 19.5 μm, complex refractive index 1.5–0.0i). In agreement with measurements the particle location is assumed to be x0 = 1.5 μm, y0 = 0 μm, and z0 = 0.55 mm. Thus the local beam waist (the diameter of the plane that contains the particle center) is 30.32 μm: continuous curve, GLMT computation; dashed curve, experimental data; dotted curve, LMT computation.

Fig. 9
Fig. 9

Two-beams setup results. A blue laser beam (wavelength 0.4765 μm, beam waist diameter 9.2 μm) impinges upon a glass particle (diameter, 18.0 μm, complex refractive index 1.5–0.0i) located at x0 = 1.0 μm, y0 = 0.0 μm, and z0 = 0.85 mm (2ωlocal = 56.80 μm): continuous curve, GLMT computation; dashed curve, experimental data; dotted curve, LMT computation.

Fig. 10
Fig. 10

Two-beam setup results. A blue laser beam (wavelength, 0.4765 μm; beam waist diameter, 9.2 μm) impinges upon a glass particle (diameter 18.0 μm, complex refractive index 1.5–0.0i) located at x0 = −1.1 μm, y0 = 0.0 μm, and z0 = 0.80 mm (2ωlocal = 53.56 μm). Continuous curves, GLMT computation; dashed curve, experimental data; dotted curves, LMT computation.

Fig. 11
Fig. 11

Two-beam setup results. A blue laser beam (wavelength, 0.4765 μm; beam waist diameter, 9.2 μm) impinges upon a glass particle (diameter 18.0 μm, complex refractive index 1.5–0.0i) located at x0 = 2.0 μm, y0 = 0.0 μm, and z0 = 0.85 mm (2ωlocal = 56.80 μm). Continuous curves, GLMT computation; dashed curve, experimental data; dotted curves, LMT computation.

Fig. 12
Fig. 12

Two-beam setup results. A blue laser beam (wavelength, 0.4765 μm; beam waist diameter, 9.2 μm) impinges upon a glass particle (diameter 18.0 μm, complex refractive index 1.5–0.0i) located at x0 = −3.0 μm, y0 = 0.0 μm, and z0 = 0.90 mm (2ωlocal = 60.06 μm). Continuous curves, GLMT computation; dashed curve, experimental data; dotted curves, LMT computation.

Fig. 13
Fig. 13

Two-beam setup results. A blue laser beam (wavelength, 0.4765 μm; beam waist diameter, 9.2 μm) impinges upon a glass particle (diameter 18.0 μm, complex refractive index 1.5–0.0i) located at x0 = −4.6 μm, y0 = 0.0 μm, and z0 = 0.80 mm (2ωlocal = 53.56 μm). Continuous curves, GLMT computation; dashed curve, experimental data; dotted curves, LMT computation.

Fig. 14
Fig. 14

Two-beam setup results. A blue laser beam (wavelength, 0.4765 μm; beam waist diameter, 9.2 μm) impinges upon a glass particle (diameter 18.0 μm, complex refractive index 1.5–0.0i) located at x0 = 6.5 μm, y0 = 0.0 μm, and z0 = 0.90 mm (2ωlocal = 60.06 μm). Continuous curves, GLMT computation; dashed curve, experimental data; dotted curves, LMT computation.

Fig. 15
Fig. 15

Two-beam setup results. A blue laser beam (wavelength, 0.4765 μm; beam waist diameter, 9.2 μm) impinges upon a glass particle (diameter 18.0 μm, complex refractive index 1.5–0.0i) located at x0 = −6.5 μm, y0 = 0.0 μm, and z0 = 0.75 mm (2ωlocal = 50.30 μm). Continuous curves, GLMT computation; dashed curve, experimental data; dotted curves, LMT computation.

Fig. 16
Fig. 16

Two-beam setup results. A blue laser beam (wavelength, 0.4765 μm; beam waist diameter, 9.2 μm) impinges upon a glass particle (diameter 18.0 μm, complex refractive index 1.5–0.0i) located at x0 = −9.0 μm, y0 = 0.0 μm, and z0 = 0.80 mm (2ωlocal = 53.56 μm). Continuous curves, GLMT computation; dashed curve, experimental data; dotted curves, LMT computation.

Equations (7)

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E θ = i E 0 k r exp ( - i k r ) n = 1 m = - n n 2 n + 1 n ( n + 1 ) × [ a n g n , TM m τ n m ( cos θ ) + i m b n g n , TE m Π n m ( cos θ ) ] exp ( i m φ ) ,
E φ = - E 0 k r exp ( - i k r ) n = 1 m = - n n 2 n + 1 n ( n + 1 ) × [ m a n g n , TM m Π n m ( cos θ ) + i b n g n , TE m τ n m ( cos θ ) ] exp ( i m φ ) ,
H θ = - H 0 E 0 E φ ,
H φ = H 0 E 0 E θ ,
τ n m ( cos θ ) = d d θ P n m ( cos θ ) ,
Π n m ( cos θ ) = P n m ( cos θ ) sin θ ,
X ^ 0 = X 0 ω 0 , Y ^ 0 = Y 0 ω 0 , Z ^ 0 = Z 0 l ,

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