Abstract

The full scattering pattern from a sphere in the vicinity of a plane surface is calculated through an approach based on the expansion of the electromagnetic field in terms of vector multipole fields and on the imposition of the boundary conditions. Our approach does not invoke any approximation but can easily incorporate the simplifying assumptions of Bobbert and Vlieger [Physica (Utrecht) 137A, 202 (1986)] and of Johnson [J. Opt. Soc. Am. A 13, 326 (1996)], whose results are compared with ours. Real progress is achieved, since, unlike the previous theories but in agreement with the available experimental data, a nonvanishing field is allowed to propagate along the surface, even when the latter is nonperfectly reflecting.

© 1997 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9, pp. 119–121; Chap. 15, pp. 297–301; Chap. 16, pp. 329–334.
  2. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–242 (1986).
  3. B. R. Johnson, “Calculation of light scattering from a spherical particle on a surface by multipole expansion method,” J. Opt. Soc. Am. A 13, 326–337 (1996).
    [CrossRef]
  4. H. Yousif, “Light scattering from parallel tilted fibers,” Ph.D. dissertation (Department of Physics, University of Arizona, Tucson, Ariz., 1987).
  5. G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991);J. Opt. Soc. Am. A9, 844–845 (erratum) (1992).
    [CrossRef]
  6. I. V. Lindell, A. H. Sihvola, K. O. Muinonen, P. Barber, “Scattering by a small object close to an interface. I. Exact-image theory formulation,” J. Opt. Soc. Am. A 8, 472–476 (1991).
    [CrossRef]
  7. K. O. Muinonen, A. H. Sihvola, I. V. Lindell, K. A. Lumme, “Scattering by a small object close to an interface. II. Study of backscattering,” J. Opt. Soc. Am. A 8, 477–482 (1991).
    [CrossRef]
  8. I. V. Lindell, E. Alanen, “Exact image theory for Sommerfeld half-space problem. Part III: General formulation,” IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
    [CrossRef]
  9. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
    [CrossRef]
  10. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
    [CrossRef]
  11. F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
    [CrossRef]
  12. E. Fucile, F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “General reflection rule for electromagnetic multipole fields on a plane interface,” IEEE Trans. Antennas Propag. 45, 868–875 (1997).
    [CrossRef]
  13. P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  14. F. Borghese, P. Denti, R. Saija, E. Fucile, O. I. Sindoni, “Optical properties of model anisotropic particles on or near a perfectly reflecting surface,” J. Opt. Soc. Am. A 12, 530–540 (1995).
    [CrossRef]
  15. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, pp. 432–435; Chap. 16, pp. 744–747.
  16. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, p. 1.
  17. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
    [CrossRef]
  18. M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 5, pp. 232–238.
  19. F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento 81, 29–50 (1984).
    [CrossRef]
  20. E. Fucile, F. Borghese, P. Denti, R. Saija, “Effect of an electrostatic field on the optical properties of a cloud of dielectric particles,” Appl. Opt. 34, 4552–4562 (1995).
    [CrossRef] [PubMed]
  21. G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” inLasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
    [CrossRef]
  22. H. S. Lee, S. Chac, Y. Ye, D. Y. H. Pui, G. L. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol. Sci. Technol. 14, 177–192 (1991).
    [CrossRef]
  23. P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica (Utrecht) 137A, 243–257 (1986).
  24. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Sec. 25.4.45, p. 890.
  25. E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 16–31.

1997 (1)

E. Fucile, F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “General reflection rule for electromagnetic multipole fields on a plane interface,” IEEE Trans. Antennas Propag. 45, 868–875 (1997).
[CrossRef]

1996 (1)

1995 (2)

1991 (4)

1987 (1)

1986 (2)

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica (Utrecht) 137A, 243–257 (1986).

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–242 (1986).

1984 (4)

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

I. V. Lindell, E. Alanen, “Exact image theory for Sommerfeld half-space problem. Part III: General formulation,” IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento 81, 29–50 (1984).
[CrossRef]

1980 (1)

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

1971 (1)

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Sec. 25.4.45, p. 890.

Alanen, E.

I. V. Lindell, E. Alanen, “Exact image theory for Sommerfeld half-space problem. Part III: General formulation,” IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
[CrossRef]

Barber, P.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica (Utrecht) 137A, 243–257 (1986).

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–242 (1986).

Borghese, F.

E. Fucile, F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “General reflection rule for electromagnetic multipole fields on a plane interface,” IEEE Trans. Antennas Propag. 45, 868–875 (1997).
[CrossRef]

F. Borghese, P. Denti, R. Saija, E. Fucile, O. I. Sindoni, “Optical properties of model anisotropic particles on or near a perfectly reflecting surface,” J. Opt. Soc. Am. A 12, 530–540 (1995).
[CrossRef]

E. Fucile, F. Borghese, P. Denti, R. Saija, “Effect of an electrostatic field on the optical properties of a cloud of dielectric particles,” Appl. Opt. 34, 4552–4562 (1995).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento 81, 29–50 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

Chac, S.

H. S. Lee, S. Chac, Y. Ye, D. Y. H. Pui, G. L. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol. Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Denti, P.

E. Fucile, F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “General reflection rule for electromagnetic multipole fields on a plane interface,” IEEE Trans. Antennas Propag. 45, 868–875 (1997).
[CrossRef]

E. Fucile, F. Borghese, P. Denti, R. Saija, “Effect of an electrostatic field on the optical properties of a cloud of dielectric particles,” Appl. Opt. 34, 4552–4562 (1995).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, E. Fucile, O. I. Sindoni, “Optical properties of model anisotropic particles on or near a perfectly reflecting surface,” J. Opt. Soc. Am. A 12, 530–540 (1995).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento 81, 29–50 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

Fucile, E.

Galbraith, L. K.

G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” inLasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
[CrossRef]

Greef, R.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica (Utrecht) 137A, 243–257 (1986).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, pp. 432–435; Chap. 16, pp. 744–747.

Johnson, B. R.

Kerker, M.

M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 5, pp. 232–238.

Lee, H. S.

H. S. Lee, S. Chac, Y. Ye, D. Y. H. Pui, G. L. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol. Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Lindell, I. V.

Lumme, K. A.

Muinonen, K. O.

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, p. 1.

Pui, D. Y. H.

H. S. Lee, S. Chac, Y. Ye, D. Y. H. Pui, G. L. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol. Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Rose, E. M.

E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 16–31.

Saija, R.

Sihvola, A. H.

Sindoni, O. I.

E. Fucile, F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “General reflection rule for electromagnetic multipole fields on a plane interface,” IEEE Trans. Antennas Propag. 45, 868–875 (1997).
[CrossRef]

F. Borghese, P. Denti, R. Saija, E. Fucile, O. I. Sindoni, “Optical properties of model anisotropic particles on or near a perfectly reflecting surface,” J. Opt. Soc. Am. A 12, 530–540 (1995).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento 81, 29–50 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Sec. 25.4.45, p. 890.

Toscano, G.

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Extinction coefficients for a random dispersion of small stratified spheres and a random dispersion of their binary aggregates,” J. Opt. Soc. Am. A 4, 1984–1991 (1987).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Use of group theory for the description of electromagnetic scattering from molecular systems,” J. Opt. Soc. Am. A 1, 183–191 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento 81, 29–50 (1984).
[CrossRef]

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9, pp. 119–121; Chap. 15, pp. 297–301; Chap. 16, pp. 329–334.

Vaughan, D. K.

G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” inLasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
[CrossRef]

Videen, G.

Vlieger, J.

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica (Utrecht) 137A, 243–257 (1986).

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–242 (1986).

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Wojcik, G. L.

H. S. Lee, S. Chac, Y. Ye, D. Y. H. Pui, G. L. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol. Sci. Technol. 14, 177–192 (1991).
[CrossRef]

G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” inLasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
[CrossRef]

Ye, Y.

H. S. Lee, S. Chac, Y. Ye, D. Y. H. Pui, G. L. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol. Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Yousif, H.

H. Yousif, “Light scattering from parallel tilted fibers,” Ph.D. dissertation (Department of Physics, University of Arizona, Tucson, Ariz., 1987).

Aerosol Sci. Technol. (1)

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Multiple electromagnetic scattering from a cluster of spheres. I. Theory,” Aerosol Sci. Technol. 3, 227–235 (1984).
[CrossRef]

Aerosol. Sci. Technol. (1)

H. S. Lee, S. Chac, Y. Ye, D. Y. H. Pui, G. L. Wojcik, “Theoretical and experimental particle size response of wafer surface scanners,” Aerosol. Sci. Technol. 14, 177–192 (1991).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (2)

I. V. Lindell, E. Alanen, “Exact image theory for Sommerfeld half-space problem. Part III: General formulation,” IEEE Trans. Antennas Propag. AP-32, 841–847 (1984).
[CrossRef]

E. Fucile, F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “General reflection rule for electromagnetic multipole fields on a plane interface,” IEEE Trans. Antennas Propag. 45, 868–875 (1997).
[CrossRef]

J. Math. Phys. (1)

F. Borghese, P. Denti, G. Toscano, O. I. Sindoni, “An addition theorem for vector Helmholtz harmonics,” J. Math. Phys. 21, 2754–2755 (1980).
[CrossRef]

J. Opt. Soc. Am. A (7)

Nuovo Cimento (1)

F. Borghese, P. Denti, R. Saija, G. Toscano, O. I. Sindoni, “Macroscopic optical constants of a cloud of randomly oriented nonspherical scatterers,” Nuovo Cimento 81, 29–50 (1984).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Physica (Utrecht) (2)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–242 (1986).

P. A. Bobbert, J. Vlieger, R. Greef, “Light reflection from a substrate sparsely seeded with spheres. Comparison with ellipsometric experiment,” Physica (Utrecht) 137A, 243–257 (1986).

Other (8)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Sec. 25.4.45, p. 890.

E. M. Rose, Multipole Fields (Wiley, New York, 1956), Chap. 2, pp. 16–31.

G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” inLasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. SPIE774, 21–31 (1987).
[CrossRef]

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 9, pp. 119–121; Chap. 15, pp. 297–301; Chap. 16, pp. 329–334.

H. Yousif, “Light scattering from parallel tilted fibers,” Ph.D. dissertation (Department of Physics, University of Arizona, Tucson, Ariz., 1987).

M. Kerker, The Scattering of Light (Academic, New York, 1969), Chap. 5, pp. 232–238.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9, pp. 432–435; Chap. 16, pp. 744–747.

R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), Chap. 1, p. 1.

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Figures (5)

Fig. 1
Fig. 1

Sketch of the geometry that we adopted in the present study.

Fig. 2
Fig. 2

Comparison of the results of our theory with those yielded by approximations B0, B1, J0, and J1 (see the text) for a sphere of radius ρ=0.27 μm and refractive index n0 =1.59 in contact with the surface and illuminated by light of wavelength λ=0.6328 μm. The sphere is embedded in vacuo, n=1, and the refractive index of the medium beyond the surface is n =3.8. We report, in square micrometers and for direction of incidence normal to the surface, the quantities J1,1 =r2I11/I01 in (a) and J2,2=r2I22/I02 in (b), where I0η is the intensity of the incident wave; I11 and I22 are the intensities that would be observed for parallel and for perpendicular polarization, respectively.

Fig. 3
Fig. 3

Same as Fig. 2, but for a sphere of radius ρ=0.38 μm.

Fig. 4
Fig. 4

Comparison of the results of our theory with those of approximations B0, B1, J0, and J1 for a small sphere, with polarizability α, whose distance a from the surface is such that ka =π/2. The quantity that is actually reported is J=(I1 +I2)/(2α2k4), where I1 and I2 are the normalized intensities that would be observed for parallel and for perpendicular polarization, respectively. The angle of incidence is held fixed at θI =-45°.

Fig. 5
Fig. 5

Full scattering pattern from a sphere of radius ρ =126.0 nm and refractive index n0=3 in contact with the surface. The quantity that is actually reported is J=r2Iϕϕ/I0ϕ in square micrometers, where I 0ϕ is the intensity of the incident light whose wavelength is λ=0.6283 μm and whose angle of incidence is ϕI=225°. In (a) the surface is perfectly reflecting (n=); the refractive index of the medium beyond the surface is n=9.0 in (b) and n=1.3 in (c). In all figures n=1.

Equations (63)

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

Eext=EI+ER+ES+ERS,
EI=E0e^I exp(ikIr),
ER=E0e^R exp(ikRr),
u^I1×u^I2=k^I,u^R1×u^R2=k^R.
EI=E0 η (e^Iu^Iη)u^Iη exp(ikIr),
ER=E0 η (e^Ru^Rη)u^Rη exp(ikRr),
E0e^Ru^Rη=E0Fη(θI)e^Iu^Iη.
F1(θI)=n2 cos θI-βn2 cos θI+β,F2(θI)=cos θI-βcos θI+β,
β=[(n2-1)+cos2 θI]1/2.
ER=E0 η (e^Iu^Iη)Fη(θI)u^Rη exp(ikRr).
EI=exp(ikIR) η E0η plm Jlm(p)(r, nk)×Wlm(p)(u^Iη, k^I),
ER=exp(ikRR) η Fη(θI)E0η×plm Jlm(p)(r, nk)Wlm(p)(u^Rη, k^R),
EηS=E0η plm Hlm(p)(r, nk)Aηlm(p),
EηRS=E0η plmpl Jlm(p)(r, nk)Fl,l;m(p,p)Aηlm(p),
iBηext=1k ×Eηext.
Eηint=E0η plm Jlm(p)(r, n0k)Cηlm(p),
iBηint=1k ×Eηint.
pl(M-1)l,l;m(p,p)Aηlm(p)=-Wηlm(p),
(M-1)l,l;m(p, p)=(R-1)l(p)δppδll+Fl,l;m(p, p),
Wηlm(p)=exp(ikIR)Wlm(p)(u^Iη, k^I)+Fη exp(ikRR)Wlm(p)(u^Rη, k^R),
Rl(p)=(1+n¯ δp1)ul(n0kρ)ul(nkρ)-(1+n¯ δp2)ul(n0kρ)ul(nkρ)(1+n¯ δp1)ul(n0kρ)wl(nkρ)-(1+n¯ δp2)ul(n0kρ)wl(nkρ),
n¯=n0/n-1,ul(x)=xjl(x),wl(x)=xhl(x).
EηRS=E0ηplmHlm(p)(r, nk)A¯ηlm(p),
A¯ηlm(p)=plal,l;m(p, p)Aηlm(p).
Eηobs=E0η plmHlm(p)(r, nk)Aηlm(p),
Aηlm(p)=pl[Jl,l;m(p, p)(azˆ, nk)Aηlm(p)+Jl,l;m(p, p)(-azˆ, nk)A¯ηlm(p)].
Jlmlm(p, p)(±azˆ, nk)=Jl,l;m(p, p)(±azˆ, nk)δmm.
Eηobs=exp(inkr)r E0ηfη,
fη=1nk plm(-i)p+lZlm(p)(rˆ)Aηlm(p).
Iηη=1r2 |E0ηfηη|2 =1r2 I0η|fηη|2,
fηη=fηu^Oη=-i4πnk plmWlm(p)*(u^Oη, k^O)Aηlm(p).
exp(ikR)Wlm(p)(uˆ, kˆ)=plmJlmlm(p, p)(R, nk)×Wlm(p)(uˆ, kˆ),
exp(-ikR)Wlm(p)*(uˆ, kˆ)=plmWlm(p)*(uˆ, kˆ)×Jlmlm(p, p)(-R, nk),
fηη=-i4πnk plm Wlm(p)*(u^Oη, k^O)[exp(ikOzˆa)Aηlm(p)+exp(-ikOzˆa)A¯ηlm(p)].
Aηlm(p)=-plMl,l;m(p, p)Wηlm(p),
Wηlm(p)=plJl,l;m(p, p)(-azˆ, nk)WEηlm(p),
WEηlm(p)=Wlm(p)(u^Iη, k^I)[1+(-)η+p+l+mFη(θI)].
fηη=i4πnk ppllmWlm(p)*(u^Oη, k^O)Sl,l;m(p, p)WEηlm(p),
Sl,l;m(p, p)=qLqLJl,L;m(p,q)(azˆ, nk)ML,L;m(q,q)+Jl,L;m(p,q)(-azˆ, nk)qLaL,L;m(q, q)ML, L;m(q, q)×JL,l;m(q, p)(-azˆ, nk)
plplmJlm(p)(r, nk)Hl,l;m(p, p)(-2azˆ, nk)A¯ηlm(p)=plmplJlm(p)(r, nk)Fl,l;m(p, p)Aηlm(p),
A¯ηlm(p)=(-)η+p+l+mAηlm(p)Fη(θ=0°),
Eθ/Eϕ=tan ψ exp(-iΔ),
E=E0uˆ exp(iKr)=E0 plm Wlm(p)(uˆ, Kˆ)Jlm(p)(r, K),
Jlm(1)(r, K)=jl(Kr)Xlm(rˆ),
Jlm(2)(r, K)=1K ×jl(Kr)Xlm(rˆ)
Wlm(p)(uˆ, Kˆ)=4πip+l-1(-)m+1Zl,-m(p)(Kˆ)uˆ
Xlm(rˆ)=[l(l+1)]-1/2LYlm(rˆ),
Zlm(1)(Kˆ)=Xlm(Kˆ),Zlm(2)(Kˆ)=Xlm(Kˆ)×Kˆ.
u^RηZlm(p)(k^R)=(-)η+p+l+mu^IηZlm(p)(k^I)
Wηlm(p)(u^Rη, k^R)=(-)η+p+l+mWηlm(p)(u^Iη, kI).
Hlm(p)(r, nk)=(-i)p+l-12π D η [u^ηZlm(p)(kˆ)]u^η×exp(ikr)exp(-ikR)dkˆ,
HRlm(p)=(-i)p+l-12π D η Fη(θk)[u^ηZlm(p)(kˆ)]u^Rη×exp(ikRr)exp(-ikR)dkˆ.
HRlm(p)=(-i)p+l-12π D η Fη(θk)[u^ηZlm(p)(kˆ)]u^Rη×exp(ikRr)exp[i(kR-k)R]dkˆ.
Flmlm(p, p)=2ip-p+l-l(-)m+1 D η Fη(θk)×[u^ηZlm(p)(kˆ)][u^RηZl,-m(p)(k^R)]×exp(2inka cos θk)dkˆ.
Flmlm(p, p)=Fl,l;m(p, p)δmm,
Fl,l;-m(p, p)=Fl,l;m(p, p),
exp[i(kR-k)R]=exp(2inka cos θk)
HRlm(p)=(-i)p+l-1(-)p+l+m-12π D η(-)η+1Fη(θk)
×[u^RηZlm(p)(k^R)]u^Rη exp(ikRr)dkˆ,
Fη=(-)η-1,
HRlm(p)=plm Hlm(p)(r, nk)almlm(p, p).
HRlm(p)=plmplm Jlm(p)(r, nk)×Hlmlm(p, p)(R, nk)almlm(p, p),
almlm(p, p)=pl (H-1)l,l;m(p, p)Fl,l;m(p, p)=al,l;m(p, p)δmm,

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