Abstract

The multipole expansion method for calculating light scattering from a spherical particle on or near a smooth surface is reviewed and presented with special attention to symmetry for reduction of the computational effort. Formulas for the scattering amplitude and the differential scattering cross section are presented. In addition, useful new formulas are derived for the absorption and extinction cross sections. The theory is exact in the limiting case in which the substrate is a perfect conductor, but an approximation is needed to handle problems with dielectric substrates. As a test of the approximation, a series of calculations for particles on dielectric substrates is carried out. The results of these tests are presented and compared with previously published calculations for the same problems.

© 1996 Optical Society of America

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References

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  1. B. R. Johnson, “Light scattering from a spherical particle on a conducting plane. I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992); J. Opt. Soc. Am. A 10, 766 (erratum) (1993).
    [CrossRef]
  2. 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. A 9, 844–845 (erratum) (1992).
    [CrossRef]
  3. D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 19, 4019–4026 (1988).
    [CrossRef]
  4. E. J. Bawolek, E. D. Hirleman, “Light scattering by submicron spherical particles on semiconductor surfaces,” in Particles on Surfaces 3: Detection, Adhesion, and Removal, K. L. Mittal, ed. ( Plenum, New York, 1991), pp. 91–105.
  5. K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987).
    [CrossRef] [PubMed]
  6. G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
    [CrossRef]
  7. 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]
  8. M. A. Taubenblatt, T. K. Tran, “Calculation of light scattering from particles and structures on a surface by the coupled-dipole method,” J. Opt. Soc. Am. A 10, 912–919 (1993).
    [CrossRef]
  9. 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]
  10. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica (Utrecht) 137A, 209–242 (1986).
  11. G. Videen, “Light scattering from a sphere behind a surface,” J. Opt. Soc. Am. A 10, 110–117 (1993).
    [CrossRef]
  12. G. Videen, M. G. Turner, V. J. Iafelice, W. S. Bickel, W. L. Wolfe, “Scattering from a small sphere near a surface,” J. Opt. Soc. Am. A 10, 118–126 (1993).
    [CrossRef]
  13. R. P. Young, “Low scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
    [CrossRef]
  14. B. R. Johnson, G. S. Arnold, “Radiation scattering from particulate-contaminated mirrors,” Aerospace Tech. Rep. ATR-94(7281)-1 (The Aerospace Corporation, El Segundo, Calif., 1994).
  15. P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 1. Theory and experiment for polystyrene spheres and λ 3 0.6328 μm,” Opt. Eng. 30, 1746–1756 (1992).
    [CrossRef]
  16. P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 2. Theory and experiment for dust and λ 3 0.6328 μm,” Opt. Eng. 30, 1757–1763 (1992).
    [CrossRef]
  17. P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 3. Theory and experiment for dust and λ = 10.6 μm,” Opt. Eng. 30, 1764–1774 (1992).
    [CrossRef]
  18. P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 4. Properties of scatter from dust for visible to far-infrared wavelengths,” Opt. Eng. 30, 1775–1784 (1992).
    [CrossRef]
  19. B. R. Johnson, “Exact calculation of light scattering from a particle on a mirror,” in Optical System Contamination: Effects, Measurement, Control III, A. P. Glassford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1754, 72–83 (1992).
    [CrossRef]
  20. P. Lilienfeld, “Optical detection of particle contamination on surfaces: a review,” Aerosol Sci. Technol. 5, 145–165 (1986).
    [CrossRef]
  21. B. R. Johnson, “Morphology-dependent resonances of a dielectric sphere on a conducting plane,” J. Opt. Soc. Am. Å 11, 2055–2064 (1994).
  22. H. Yousif, “Light scattering from parallel tilted fibers,” Ph.D. dissertation (Department of Physics, University of Arizona, Tucson, Ariz., 1987).
  23. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 9.5 and 9.25.
  24. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Sec. 3.4.
  25. I. V. Lindell, A. H. Sihovola, K. O. Muinonen, P. W. 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]
  26. J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
    [CrossRef]
  27. M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (Technology Press, M.I.T., Cambridge, Mass., 1959).
  28. K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
    [CrossRef]

1994 (1)

B. R. Johnson, “Morphology-dependent resonances of a dielectric sphere on a conducting plane,” J. Opt. Soc. Am. Å 11, 2055–2064 (1994).

1993 (3)

1992 (5)

B. R. Johnson, “Light scattering from a spherical particle on a conducting plane. I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992); J. Opt. Soc. Am. A 10, 766 (erratum) (1993).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 1. Theory and experiment for polystyrene spheres and λ 3 0.6328 μm,” Opt. Eng. 30, 1746–1756 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 2. Theory and experiment for dust and λ 3 0.6328 μm,” Opt. Eng. 30, 1757–1763 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 3. Theory and experiment for dust and λ = 10.6 μm,” Opt. Eng. 30, 1764–1774 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 4. Properties of scatter from dust for visible to far-infrared wavelengths,” Opt. Eng. 30, 1775–1784 (1992).
[CrossRef]

1991 (4)

1988 (1)

D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 19, 4019–4026 (1988).
[CrossRef]

1987 (1)

1986 (2)

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

P. Lilienfeld, “Optical detection of particle contamination on surfaces: a review,” Aerosol Sci. Technol. 5, 145–165 (1986).
[CrossRef]

1976 (1)

R. P. Young, “Low scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
[CrossRef]

1975 (1)

K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
[CrossRef]

1971 (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Arnold, G. S.

B. R. Johnson, G. S. Arnold, “Radiation scattering from particulate-contaminated mirrors,” Aerospace Tech. Rep. ATR-94(7281)-1 (The Aerospace Corporation, El Segundo, Calif., 1994).

Barber, P. W.

Bawolek, E. J.

E. J. Bawolek, E. D. Hirleman, “Light scattering by submicron spherical particles on semiconductor surfaces,” in Particles on Surfaces 3: Detection, Adhesion, and Removal, K. L. Mittal, ed. ( Plenum, New York, 1991), pp. 91–105.

Bickel, W. S.

Bivens, R.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (Technology Press, M.I.T., Cambridge, Mass., 1959).

Bobbert, P. A.

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

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Sec. 3.4.

Bruning, J. H.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[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]

Galbraith, L. K.

G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
[CrossRef]

Gordon, R. G.

K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
[CrossRef]

Hirleman, E. D.

D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 19, 4019–4026 (1988).
[CrossRef]

E. J. Bawolek, E. D. Hirleman, “Light scattering by submicron spherical particles on semiconductor surfaces,” in Particles on Surfaces 3: Detection, Adhesion, and Removal, K. L. Mittal, ed. ( Plenum, New York, 1991), pp. 91–105.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Sec. 3.4.

Iafelice, V. J.

Johnson, B. R.

B. R. Johnson, “Morphology-dependent resonances of a dielectric sphere on a conducting plane,” J. Opt. Soc. Am. Å 11, 2055–2064 (1994).

B. R. Johnson, “Light scattering from a spherical particle on a conducting plane. I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992); J. Opt. Soc. Am. A 10, 766 (erratum) (1993).
[CrossRef]

B. R. Johnson, “Exact calculation of light scattering from a particle on a mirror,” in Optical System Contamination: Effects, Measurement, Control III, A. P. Glassford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1754, 72–83 (1992).
[CrossRef]

B. R. Johnson, G. S. Arnold, “Radiation scattering from particulate-contaminated mirrors,” Aerospace Tech. Rep. ATR-94(7281)-1 (The Aerospace Corporation, El Segundo, Calif., 1994).

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]

Lilienfeld, P.

P. Lilienfeld, “Optical detection of particle contamination on surfaces: a review,” Aerosol Sci. Technol. 5, 145–165 (1986).
[CrossRef]

Lindell, I. V.

Lo, Y. T.

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Lumme, K. A.

Metropolis, N.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (Technology Press, M.I.T., Cambridge, Mass., 1959).

Muinonen, K. O.

Nahm, K. B.

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]

Rotenberg, M. R.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (Technology Press, M.I.T., Cambridge, Mass., 1959).

Schulten, K.

K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
[CrossRef]

Sihovola, A. H.

Sihvola, A. H.

Spyak, P. R.

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 3. Theory and experiment for dust and λ = 10.6 μm,” Opt. Eng. 30, 1764–1774 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 4. Properties of scatter from dust for visible to far-infrared wavelengths,” Opt. Eng. 30, 1775–1784 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 1. Theory and experiment for polystyrene spheres and λ 3 0.6328 μm,” Opt. Eng. 30, 1746–1756 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 2. Theory and experiment for dust and λ 3 0.6328 μm,” Opt. Eng. 30, 1757–1763 (1992).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 9.5 and 9.25.

Taubenblatt, M. A.

Tran, T. K.

Turner, M. G.

Vaughan, D. K.

G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
[CrossRef]

Videen, G.

Vlieger, J.

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

Weber, D. C.

D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 19, 4019–4026 (1988).
[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,” in Lasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
[CrossRef]

Wolfe, W. L.

G. Videen, M. G. Turner, V. J. Iafelice, W. S. Bickel, W. L. Wolfe, “Scattering from a small sphere near a surface,” J. Opt. Soc. Am. A 10, 118–126 (1993).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 2. Theory and experiment for dust and λ 3 0.6328 μm,” Opt. Eng. 30, 1757–1763 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 1. Theory and experiment for polystyrene spheres and λ 3 0.6328 μm,” Opt. Eng. 30, 1746–1756 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 4. Properties of scatter from dust for visible to far-infrared wavelengths,” Opt. Eng. 30, 1775–1784 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 3. Theory and experiment for dust and λ = 10.6 μm,” Opt. Eng. 30, 1764–1774 (1992).
[CrossRef]

K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987).
[CrossRef] [PubMed]

Wooten, J. K.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (Technology Press, M.I.T., Cambridge, Mass., 1959).

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]

Young, R. P.

R. P. Young, “Low scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
[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. (2)

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]

P. Lilienfeld, “Optical detection of particle contamination on surfaces: a review,” Aerosol Sci. Technol. 5, 145–165 (1986).
[CrossRef]

Appl. Opt. (2)

K. B. Nahm, W. L. Wolfe, “Light scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987).
[CrossRef] [PubMed]

D. C. Weber, E. D. Hirleman, “Light scattering signatures of individual spheres on optically smooth conducting surfaces,” Appl. Opt. 19, 4019–4026 (1988).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. H. Bruning, Y. T. Lo, “Multiple scattering of EM waves by spheres. Part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

J. Math. Phys. (1)

K. Schulten, R. G. Gordon, “Exact recursive evaluation of 3-jand 6-jcoefficients for quantum mechanical coupling of angular momenta,” J. Math. Phys. 16, 1961–1970 (1975).
[CrossRef]

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

J. Opt. Soc. Am. Å (1)

B. R. Johnson, “Morphology-dependent resonances of a dielectric sphere on a conducting plane,” J. Opt. Soc. Am. Å 11, 2055–2064 (1994).

Opt. Eng. (5)

R. P. Young, “Low scatter mirror degradation by particle contamination,” Opt. Eng. 15, 516–520 (1976).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 1. Theory and experiment for polystyrene spheres and λ 3 0.6328 μm,” Opt. Eng. 30, 1746–1756 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 2. Theory and experiment for dust and λ 3 0.6328 μm,” Opt. Eng. 30, 1757–1763 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 3. Theory and experiment for dust and λ = 10.6 μm,” Opt. Eng. 30, 1764–1774 (1992).
[CrossRef]

P. R. Spyak, W. L. Wolfe, “Scatter from particulate-contaminated mirrors. Part 4. Properties of scatter from dust for visible to far-infrared wavelengths,” Opt. Eng. 30, 1775–1784 (1992).
[CrossRef]

Physica (Utrecht) (1)

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

Other (8)

E. J. Bawolek, E. D. Hirleman, “Light scattering by submicron spherical particles on semiconductor surfaces,” in Particles on Surfaces 3: Detection, Adhesion, and Removal, K. L. Mittal, ed. ( Plenum, New York, 1991), pp. 91–105.

B. R. Johnson, “Exact calculation of light scattering from a particle on a mirror,” in Optical System Contamination: Effects, Measurement, Control III, A. P. Glassford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1754, 72–83 (1992).
[CrossRef]

B. R. Johnson, G. S. Arnold, “Radiation scattering from particulate-contaminated mirrors,” Aerospace Tech. Rep. ATR-94(7281)-1 (The Aerospace Corporation, El Segundo, Calif., 1994).

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

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Sec. 9.5 and 9.25.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), Sec. 3.4.

M. R. Rotenberg, R. Bivens, N. Metropolis, J. K. Wooten, The 3-j and 6-j Symbols (Technology Press, M.I.T., Cambridge, Mass., 1959).

G. L. Wojcik, D. K. Vaughan, L. K. Galbraith, “Calculation of light scatter from structures on silicon surfaces,” in Lasers in Microlithography, J. S. Batchelder, D. J. Ehrlich, J. Y. Tsao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.774, 21–31 (1987).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram showing the geometry of the sphere–substrate system and representative light rays from the incident, reflected, and scattered radiation fields.

Fig. 2
Fig. 2

Comparison of the multipole expansion calculation (present study) with the exact image solution (Muinonen et al.) for scattering in the xz and yz planes for a small spherical particle on a substrate with a dielectric constant = 2.4. The scattering angle is measured from the surface normal. The incident beam is in the xz plane at an angle of −45°.

Fig. 3
Fig. 3

Comparison of the multipole expansion calculation (present study) with the finite-difference solution of Maxwell’s equations and experimental measurements (Wojcik) for light scattering from a 0.54 μm diameter polystyrene sphere on silicon. The incident beam is normal to the surface with a wavelength of 0.6328 μm. Scattering is measured from the surface normal, and the differential cross section is measured in square micrometers.

Fig. 4
Fig. 4

Same as Fig. 3, except that the particle diameter is 0.76 μm.

Fig. 5
Fig. 5

Same as Fig. 3, except that the particle diameter is 1.6 μm.

Fig. 6
Fig. 6

Comparison of the multipole expansion calculation (present study) with the coupled-dipole solution (Taubenblatt and Tran) for light scattering from a 0.3 μm diameter polystyrene sphere on silicon. The angle in incidence is −65°, the wavelength is 0.6328 μm, and scattering is measured from the surface normal.

Fig. 7
Fig. 7

Same as Fig. 6, except that the particle is Si3N4 and the diameter is 0.45 μm.

Fig. 8
Fig. 8

Comparison of the differential cross sections calculated by the present method for light scattering from a 1.0 μm diameter polystyrene sphere on a perfectly reflecting substrate and an aluminum substrate; the wavelength is 0.6328 μm. The incident beam is −30° from the normal, and the scattering angle is measured relative to the normal and in the plane of incidence.

Equations (71)

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

M n , m ( j ) = z n ( j ) ( k r ) exp ( i m ϕ ) X n , m ( θ ) , N n , m ( j ) = exp ( i m ϕ ) k r { r [ r z n ( j ) ( k r ) ] Y n , m ( θ ) + z n ( j ) ( k r ) Z n , m ( θ ) } ,
X n , m ( θ ) = i π n , m ( θ ) e ^ θ - τ n , m ( θ ) e ^ ϕ , Y n , m ( θ ) = τ n , m ( θ ) e ^ θ + i π n , m ( θ ) e ^ ϕ , Z n , m ( θ ) = n ( n + 1 ) K n , m P n m ( cos θ ) e ^ r ,
π n , m ( θ ) = K n , m m sin ( θ ) P n m ( cos θ ) , τ n , m ( θ ) = K n , m θ P n m ( cos θ ) ,
K n , m = [ 2 n + 1 n ( n + 1 ) ( n - m ) ! ( n + m ) ! ] 1 / 2 .
π n , m ( π - θ ) = ( - 1 ) n + m π n , m ( θ ) , τ n , m ( π - θ ) = - ( - 1 ) n + m τ n , m ( θ ) ;
π n , - m ( θ ) = - ( - 1 ) m π n , m ( θ ) , τ n , - m ( θ ) = ( - 1 ) m τ n , m ( θ ) .
E = exp ( i k z ) ( E p e ^ x + E s e ^ y ) ,
E = exp [ i k ( z cos α + x sin α ) ] { E p [ ( cos α ) e ^ x - ( sin α ) e ^ z ] + E s e ^ y } .
E = m = - n = μ [ p n , m M n , m ( 1 ) ( r ) + q n , m N n , m ( 1 ) ( r ) ] ,
p n , m = - i n + 1 [ E p π n , m ( α ) - i E s τ n , m ( α ) ] , q n , m = - i n + 1 [ E p τ n , m ( α ) - i E s π n , m ( α ) ] ,
μ = max ( 1 , m ) .
p n , m ( inc ) = - i n + 1 ( - 1 ) n + m [ E p π n , m ( β ) + i E s τ n , m ( β ) ] , q n , m ( inc ) = i n + 1 ( - 1 ) n + m [ E p τ n , m ( β ) + i E s π n , m ( β ) ] .
R s ( β ) = cos β - ( N sub 2 - sin 2 β ) 1 / 2 cos β + ( N sub 2 - sin 2 β ) 1 / 2 , R p ( β ) = N sub 2 cos β - ( N sub 2 - sin 2 β ) 1 / 2 N sub 2 cos β + ( N sub 2 - sin 2 β ) 1 / 2 ,
p n , m ( ref ) = - i n + 1 [ R p ( β ) E p π n , m ( β ) - i R s ( β ) E s τ n , m ( β ) ] , q n , m ( ref ) = - i n + 1 [ R p ( β ) E p τ n , m ( β ) - i R s ( β ) E s π n , m ( β ) ] .
σ = exp ( - i k d cos β )
E ir = m = - n = μ [ P n , m M n , m ( 1 ) ( r 2 ) + Q n , m N n , m ( 1 ) ( r 2 ) ] ,
P n , m = [ σ p n , m ( inc ) + σ * p n , m ( ref ) ] , Q n , m = [ σ q n , m ( inc ) + σ * q n , m ( ref ) ] .
P n , m ( s ) = - i n [ σ * R s ( β ) - σ ( - 1 ) n + m ] τ n , m ( β ) , Q n , m ( s ) = - i n [ σ * R s ( β ) + σ ( - 1 ) n + m ] π n , m ( β ) ;
P n , m ( p ) = - i n + 1 [ σ * R p ( β ) + σ ( - 1 ) n + m ] π n , m ( β ) , Q n , m ( p ) = - i n + 1 [ σ * R p ( β ) - σ ( - 1 ) n + m ] τ n , m ( β ) .
P n , - m ( s ) = ( - 1 ) m P n , m ( s ) ,             Q n , - m ( s ) = - ( - 1 ) m Q n , m ( s ) , P n , - m ( p ) = - ( - 1 ) m P n , m ( p ) ,             Q n , - m ( p ) = ( - 1 ) m Q n , m ( p ) .
E 2 scat = m = - n = μ [ a n , m M n , m ( 3 ) ( r 2 ) + b n , m N n , m ( 3 ) ( r 2 ) ] ,
E int = m = - n = μ [ c n , m M n , m ( 3 ) ( r 1 ) + d n , m N n , m ( 3 ) ( r 1 ) ] .
c n , m = ( - 1 ) n + m a n , m ,             d n , m = - ( - 1 ) n + m b n , m .
c n , m = R ( - 1 ) n + m a n , m ,             d n , m = - R ( - 1 ) n + m b n , m ,
R = R p ( 0 ) = - R s ( 0 ) = N sub - 1 N sub + 1 .
M n , m ( 3 ) ( r 1 ) = n = 1 [ A ˜ n , n ( m ) M n , m ( 1 ) ( r 2 ) + B ˜ n , n ( m ) N n , m ( 1 ) ( r 2 ) ] , N n , m ( 3 ) ( r 1 ) = n = 1 [ A ˜ n , n ( m ) N n , m ( 1 ) ( r 2 ) + B ˜ n , n ( m ) M n , m ( 1 ) ( r 2 ) ] .
E ( j ) = m = - n = μ [ α n , m ( j ) M n , m ( 1 ) + a n , m ( j ) M n , m ( 3 ) + β n , m ( j ) N n , m ( 1 ) + b n , m ( j ) N n , m ( 3 ) ] ,
α n , m ( j ) = P n , m ( j ) + n = μ [ A ˜ n , n ( m ) c n , m ( j ) + B ˜ n , n ( m ) d n , m ( j ) ] , β n , m ( j ) = Q n , m ( j ) + n = μ [ B ˜ n , n ( m ) c n , m ( j ) + A ˜ n , n ( m ) d n , m ( j ) ]
α n , m ( j ) = a n , m ( j ) / u n ,             β n , m ( j ) = b n , m ( j ) / v n ,
u n = - ψ n ( x ) ψ n ( N sph x ) - N sph ψ n ( N sph x ) ψ n ( x ) ξ n ( x ) ψ n ( N sph x ) - N sph ψ n ( N sph x ) ξ n ( x ) , v n = - ψ n ( x ) ψ n ( N sph x ) - N sph ψ n ( N sph x ) ψ n ( x ) ξ n ( x ) ψ n ( N sph x ) - N sph ψ n ( N sph x ) ξ n ( x ) ,
a n , m ( j ) = u n { P n , m ( j ) + R n = μ [ A n , n ( m ) a n , m ( j ) - B n , n ( m ) b n , m ( j ) ] } , b n , m ( j ) = v n { Q n , m ( j ) + R n = μ [ B n , n ( m ) a n , m ( j ) - A n , n ( m ) b n , m ( j ) ] } ,
A n , n ( m ) = ( - 1 ) m + n A ˜ n , n ( m ) ,             B n , n ( m ) = ( - 1 ) m + n B ˜ n , n ( m ) .
( [ u - 1 0 0 - v - 1 ] - R [ A ( m ) - B ( m ) - B ( m ) A ( m ) ] ) [ a m ( s ) a m ( p ) b m ( s ) b m ( p ) ] = [ P m ( s ) P m ( p ) - Q m ( s ) - Q m ( p ) ] .
a - m ( p ) = - ( - 1 ) m a m ( p ) ,             b - m ( p ) = ( - 1 ) m b m ( p ) , a - m ( s ) = ( - 1 ) m a m ( s ) ,             b - m ( s ) = - ( - 1 ) m b m ( s ) ,
E scat = E 2 scat ( r 2 ) + E 1 scat ( r 1 ) ,
M n , m ( 3 ) = ( - i ) n exp ( i k r ) i k r X n , m ( θ ) exp ( i m ϕ ) , N n , m ( 3 ) = i ( - i ) n exp ( i k r ) i k r Y n , m ( θ ) exp ( i m ϕ ) .
E 2 scat ( j ) ( r 2 ) = exp ( i k r 2 ) i k r 2 m = - N N exp ( i m ϕ 2 ) × n = μ N ( - i ) n [ a n , m ( j ) X n , m ( θ 2 ) + i b n , m ( j ) Y n , m ( θ 2 ) ] .
r 1 = r + d cos θ , r 2 = r - d cos θ , θ 1 = θ 2 = θ , ϕ 1 = ϕ 2 = ϕ .
E 2 scat ( j ) ( r ) = exp ( i k r ) i k r exp ( - i k d cos θ ) × m = - N N exp ( i m ϕ ) [ Θ 2 m ( j ) ( θ ) e ^ θ + Φ 2 m ( j ) ( θ ) e ^ ϕ ] ,
Θ 2 m ( j ) ( θ ) = - n = μ N ( - i ) n + 1 [ a n , m ( j ) π n , m ( θ ) + b n , m ( j ) τ n , m ( θ ) ] Φ 2 m ( j ) ( θ ) = - i n = μ N ( - i ) n + 1 [ a n , m ( j ) τ n , m ( θ ) + b n , m ( j ) π n , m ( θ ) ] .
E 1 scat ( j ) ( r ) = exp ( i k r ) i k r exp ( i k d cos θ ) × m = - N N exp ( i m ϕ ) [ Θ 1 m ( j ) ( θ ) e ^ θ + Φ 1 m ( j ) ( θ ) e ^ ϕ ] ,
Θ 1 m ( j ) ( θ ) = - R p n = μ N ( - i ) n + 1 [ a n , m ( j ) π n , m ( π - θ ) + b n , m ( j ) τ n , m ( π - θ ) ] , Φ 1 m ( j ) ( θ ) = - i R s n = μ N ( - i ) n + 1 [ a n , m ( j ) τ n , m ( π - θ ) × b n , m ( j ) π n , m ( π - θ ) ] .
Θ m ( j ) ( θ ) = Θ 1 m ( j ) ( θ ) exp ( i k d cos θ ) + Θ 2 m ( j ) ( θ ) × exp ( - i k d cos θ ) , Φ m ( j ) ( θ ) = Φ 1 m ( j ) ( θ ) exp ( i k d cos θ ) + Φ 2 m ( j ) ( θ ) × exp ( - i k d cos θ ) .
Θ - m ( p ) ( θ ) = Θ m ( p ) ( θ ) ,             Φ - m ( p ) ( θ ) = - Φ m ( p ) , Θ - m ( s ) ( θ ) = - Θ m ( s ) ( θ ) ,             Φ - m ( s ) ( θ ) = Φ m ( s ) .
E scat ( j ) ( r , θ , ϕ ) = S ( j ) ( θ , ϕ ) exp ( i k r ) i k r ,
S θ ( j ) ( θ , ϕ ) = m = - N N exp ( i m ϕ ) Θ m ( j ) ( θ ) , S ϕ ( j ) ( θ , ϕ ) = m = - N N exp ( i m ϕ ) Φ m ( j ) ( θ ) .
E scat = F ( θ , ϕ ) exp ( i k r ) i k r ,
( F θ F ϕ ) = [ S θ ( p ) S θ ( s ) S ϕ ( p ) S ϕ ( s ) ] ( E p E s ) ,
σ ( θ , ϕ ) = F ( θ , ϕ ) 2 k 2 = 1 k 2 [ F θ ( θ , ϕ ) 2 + F ϕ ( θ , ϕ ) 2 ] .
σ ( θ , ϕ ) = 1 2 k 2 [ S θ ( p ) ( θ , ϕ ) 2 + S ϕ ( p ) ( θ , ϕ ) 2 + S θ ( s ) ( θ , ϕ ) 2 + S ϕ ( s ) ( θ , ϕ ) 2 ] .
C abs = - 4 π k 2 n = 1 N m = - N N { p n , m 2 [ Re ( u n ) + u n 2 ] + q n , m 2 [ Re ( v n ) + v n 2 ] } ,
C abs ( j ) = - 4 π k 2 n = 1 N m = - N N { | a n , m ( j ) u n | 2 [ Re ( u n ) + u n 2 ] + | b n , m ( j ) v n | 2 [ Re ( v n ) + v n 2 ] } .
C abs = E p 2 C abs ( p ) + E s 2 C abs ( s ) .
C abs = 0.5 [ C abs ( p ) + C abs ( s ) ] .
C ext = U exp / I 0 ,
C ext = - 4 π k 2 Re [ E R * · F ( β , 0 ) ] ,
E ref = E R exp ( i k z ) ,
E R = R p ( β ) E p e ^ x + R s ( β ) E s e ^ y .
F ( β , 0 ) = F θ ( β , 0 ) e ^ x + F ϕ ( β , 0 ) e ^ y .
C ext = - 4 π k 2 Re { [ R p ( β ) E p ] * F θ ( β , 0 ) + [ R s ( β ) E s ] * F ϕ ( β , 0 ) } .
C ext = 2 π k 2 { Re [ R p * ( β ) S θ ( p ) ( β , 0 ) ] + Re [ R s * ( β ) S ϕ ( s ) ( β , 0 ) ] } .
C sca = 0 2 π 0 π / 2 σ ( θ , ϕ ) ( sin θ ) d θ d ϕ .
C sca = C ext - C abs .
S ( θ , ϕ ) = ( s - 1 s + 2 ) - 2 σ ( θ , ϕ ) k 4 a 6 .
c n , m = ( - 1 ) n + m n [ R 1 n , n ( m ) a n , m + R 2 n , n ( m ) b n , m ] , d n , m = - ( - 1 ) n + m n [ R 3 n , n ( m ) a n , m + R 4 n , n ( m ) b n , m ] .
A n , n ( m ) = C n , n v i - v [ n ( n + 1 ) + n ( n + 1 ) - v ( v + 1 ) ] × a ( m ; n , n , v ) h v ( 1 ) ( 2 k d ) ,
B n , n ( m ) = C n , n ( 4 i k d ) v i - v a ( m ; n , n , v ) h v ( 1 ) ( 2 k d ) ,
C n , n = ( - 1 ) n + n 1 2 [ 2 n + 1 n ( n + 1 ) 2 n + 1 n ( n + 1 ) ] 1 / 2 .
a ( m ; n , n , v ) = ( 2 v + 1 ) [ n n v 0 0 0 ] [ n n v m - m 0 ]
A n , n ( m ) = A n , n ( m ) ,             B n , n ( m ) = B n , n ( m ) ,
A n , m ( - m ) = A n , m ( m ) ,             B n , m ( - m ) = - B n , m ( m ) .

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