Abstract

Based on the recent results in the generalized Lorenz–Mie theory, solutions for scattering problems of a sphere with an eccentrically located spherical inclusion illuminated by an arbitrary shaped electromagnetic beam in an arbitrary orientation are obtained. Particular attention is paid to the description and application of an arbitrary shaped beam in an arbitrary orientation to the scattering problem under study. The theoretical formalism is implemented in a homemade computer program written in FORTRAN. Numerical results concerning spatial distributions of both internal and external fields are displayed in different formats in order to properly display exemplifying results. More specifically, as an example, we consider the case of a focused fundamental Gaussian beam (TEM00 mode) illuminating a glass sphere (having a real refractive index equal to 1.50) with an eccentrically located spherical water inclusion (having a real refractive index equal to 1.33). Displayed results are for various parameters of the incident electromagnetic beam (incident orientation, beam waist radius, location of the beam waist center) and of the scatterer system (location of the inclusion inside the host sphere and relative diameter of the inclusion to the host sphere).

© 2011 Optical Society of America

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  1. G. Mie, “Beiträge zur optik trüben medien speziell kolloidaler metalösungen,” Ann. Phys. 25, 377–452 (1908).
    [CrossRef]
  2. J. A. Lock and G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transfer 110, 800–807 (2009).
    [CrossRef]
  3. G. Gouesbet, “Generalized Lorenz–Mie theories, the third decade: a perspective,” J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
    [CrossRef]
  4. G. Gouesbet, B. Maheu, and 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]
  5. B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatter in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
    [CrossRef]
  6. G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz–Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
    [CrossRef]
  7. L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
    [CrossRef]
  8. J. P. Barton, D. R. Alexander, and 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]
  9. G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
    [CrossRef] [PubMed]
  10. G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821–837 (2000).
  11. Y. P. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621–6629 (2003).
    [CrossRef] [PubMed]
  12. F. Xu, K. F. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrary oriented, located, and shaped beam scattered by homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007).
    [CrossRef]
  13. K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
    [CrossRef]
  14. L. Méès, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results,” Appl. Opt. 38, 1867–1876 (1999).
    [CrossRef]
  15. G. Gouesbet and L. Méès, “Generalized Lorenz–Mie theory for infinitely long elliptical cylinders,” J. Opt. Soc. Am. A 16, 1333–1341 (1999).
    [CrossRef]
  16. J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  17. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  18. Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients of arbitrary shaped beam in oblique illumination,” Opt. Express 15, 735–746 (2007).
    [CrossRef] [PubMed]
  19. Y. P. Han, Y. Zhang, H. Y. Zhang, and G. X. Han, “Scattering of typical particles by beam shape in oblique illumination,” J. Quant. Spectrosc. Radiat. Transfer 110, 1375–1381 (2009).
    [CrossRef]
  20. G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. I. General formulation,” Opt. Commun. 283, 3218–3225 (2010).
    [CrossRef]
  21. J. J. Wang, G. Gouesbet, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. II. Axisymmetric beams,” Opt. Commun. 283, 3226–3234 (2010).
    [CrossRef]
  22. G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. III. Special Euler angles,” Opt. Commun. 283, 3235–3243 (2010).
    [CrossRef]
  23. G. Gouesbet, J. J. Wang, Y. P. Han, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. IV. Plane waves,” Opt. Commun. 283, 3244–3254 (2010).
    [CrossRef]
  24. G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. V. Localized beam models,” Opt. Commun. (to be published).
  25. J. G. Fikioris and N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. A 69, 1359–1366 (1979).
    [CrossRef]
  26. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing several spherical inclusions,” Appl. Opt. 33, 484–493 (1994).
    [CrossRef] [PubMed]
  27. K. A. Fuller, “Morphology-dependent resonances in eccentrically stratified sphere,” Opt. Lett. 19, 1272–1274 (1994).
    [CrossRef] [PubMed]
  28. G. Videen, D. Ngo, P. Chylek, and R. G. Pinnick, “Light scattering from a sphere with an irregular inclusion,” J. Opt. Soc. Am. A 12, 922–928 (1995).
    [CrossRef]
  29. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources: Theory and Programs (Springer, 2006).
  30. S. M. Hasheminejad and Y. Mirzaei, “Exact 3D elasticity solution for free vibrations of eccentric hollow sphere,” J. Sound Vib. (to be published).
  31. D. R. Secker, P. H. Kaye, R. S. Greenaway, E. Hirst, D. L. Bartley, and G. Videen, “Light scattering from deformed droplets and droplets with inclusions. I. Experimental results,” Appl. Opt. 39, 5023–5030 (2000).
    [CrossRef]
  32. G. Videen, W. Sun, Q. Fu, D. R. Secker, R. S. Greenaway, P. H. Kaye, E. Hirst, and D. Bartley, “Light scattering from deformed droplets and droplets with inclusions. II. Theoretical treatment,” Appl. Opt. 39, 5031–5039 (2000).
    [CrossRef]
  33. N. Riefler, R. Schuh, and T. Wriedt, “Investigation of a measurement technique to estimate concentration and size of inclusions in droplets,” Meas. Sci. Technol. 18, 2209–2218 (2007).
    [CrossRef]
  34. A. A. Riziq, M. Trainic, C. Erlick, E. Segre, and Y. Rudich, “Extinction efficiencies of coated absorbing aerosols measured by cavity ring down aerosol spectrometry,” Atmos. Chem. Phys. 8, 1823–1833 (2008).
    [CrossRef]
  35. D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112(1996).
    [CrossRef]
  36. G. X. Han, Y. P. Han, J. Y. Liu, and Y. Zhang, “Scattering of an eccentric sphere arbitrarily located in a shaped beam,” J. Opt. Soc. Am. B 25, 2064–2072 (2008).
    [CrossRef]
  37. B. Yan, X. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A Pure Appl. Opt. 11, 015705 (2009).
    [CrossRef]
  38. S. Saengkaew, G. Godard, J. B. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
    [CrossRef]
  39. G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located inclusion, and optical chaos,” Part. Part. Syst. Charact. 18, 190–195 (2001).
    [CrossRef]
  40. G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Periodic orbits in Hamiltonian chaos of the annular billiard,” Phys. Rev. E 65, 016212 (2001).
    [CrossRef]
  41. G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Morphology-dependent resonances and/or whispering gallery modes for a two-dimensional dielectric cavity with an eccentrically located circular inclusion, a Hamiltonian point of view with Hamiltonian (optical) chaos,” Opt. Commun. 201, 223–242(2002).
    [CrossRef]
  42. P. T. Leung, S. W. Ng, and K. M. Pang, “Morphology-dependent resonances in dielectric spheres with many tiny inclusions,” Opt. Lett. 27, 1749–1751 (2002).
    [CrossRef]
  43. V. S. C. M. Rao, Gupta, and S. Dutta, “Broken azimuthal degeneracy with whispering gallery modes of microspheres,” J. Opt. A Pure Appl. Opt. 7, 279–285 (2005).
    [CrossRef]
  44. G. Gouesbet, “T-matrix formulation and generalized Lorenz–Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521(2010).
    [CrossRef]
  45. S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).
  46. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–44 (1962).
  47. P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A (Amsterdam) 137, 209–241 (1986).
    [CrossRef]
  48. D. W. Mackowski, “Analysis of radiative scattering from multiple sphere configurations,” Proc. R. Soc. Lond. 433, 599–614 (1991).
    [CrossRef]
  49. G. Gouesbet, C. Letellier, K. F. Ren, and G. Gréhan, “Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542(1996).
    [CrossRef] [PubMed]
  50. G. Gouesbet, G. Gréhan, and B. Maheu, “Expressions to compute the coefficients gmn in the generalized Lorenz–Mie theory using finite series,” J. Opt. 19, 35 (1988).
    [CrossRef]
  51. K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
    [CrossRef]
  52. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz–Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
    [CrossRef] [PubMed]
  53. H. Y. Zhang and Y. P. Han, “Addition theorem for the spherical vector wave functions and its application to the beam shape coefficients,” J. Opt. Soc. Am. B 25, 255–260 (2008).
    [CrossRef]
  54. G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
    [CrossRef]
  55. J. A. Lock, “An improved Gaussian beam scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
    [CrossRef] [PubMed]
  56. T. Wriedt, “The website maintained by Thomas Wriedt,” http://www.scattport.org.
  57. M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 2000).
  58. R. Schuh and T. Wriedt, “Computer programs for light scattering by particles with inclusions,” J. Quant. Spectrosc. Radiat. Transfer 70, 715–723 (2001).
    [CrossRef]
  59. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods, Advanced Series in Applied Physics (World Scientific, 1990), Vol. 2.
    [CrossRef]
  60. J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
    [CrossRef]

2010

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. I. General formulation,” Opt. Commun. 283, 3218–3225 (2010).
[CrossRef]

J. J. Wang, G. Gouesbet, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. II. Axisymmetric beams,” Opt. Commun. 283, 3226–3234 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. III. Special Euler angles,” Opt. Commun. 283, 3235–3243 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, Y. P. Han, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. IV. Plane waves,” Opt. Commun. 283, 3244–3254 (2010).
[CrossRef]

G. Gouesbet, “T-matrix formulation and generalized Lorenz–Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521(2010).
[CrossRef]

2009

B. Yan, X. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A Pure Appl. Opt. 11, 015705 (2009).
[CrossRef]

S. Saengkaew, G. Godard, J. B. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[CrossRef]

Y. P. Han, Y. Zhang, H. Y. Zhang, and G. X. Han, “Scattering of typical particles by beam shape in oblique illumination,” J. Quant. Spectrosc. Radiat. Transfer 110, 1375–1381 (2009).
[CrossRef]

J. A. Lock and G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transfer 110, 800–807 (2009).
[CrossRef]

G. Gouesbet, “Generalized Lorenz–Mie theories, the third decade: a perspective,” J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
[CrossRef]

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

2008

2007

2005

V. S. C. M. Rao, Gupta, and S. Dutta, “Broken azimuthal degeneracy with whispering gallery modes of microspheres,” J. Opt. A Pure Appl. Opt. 7, 279–285 (2005).
[CrossRef]

2003

2002

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Morphology-dependent resonances and/or whispering gallery modes for a two-dimensional dielectric cavity with an eccentrically located circular inclusion, a Hamiltonian point of view with Hamiltonian (optical) chaos,” Opt. Commun. 201, 223–242(2002).
[CrossRef]

P. T. Leung, S. W. Ng, and K. M. Pang, “Morphology-dependent resonances in dielectric spheres with many tiny inclusions,” Opt. Lett. 27, 1749–1751 (2002).
[CrossRef]

2001

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located inclusion, and optical chaos,” Part. Part. Syst. Charact. 18, 190–195 (2001).
[CrossRef]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Periodic orbits in Hamiltonian chaos of the annular billiard,” Phys. Rev. E 65, 016212 (2001).
[CrossRef]

R. Schuh and T. Wriedt, “Computer programs for light scattering by particles with inclusions,” J. Quant. Spectrosc. Radiat. Transfer 70, 715–723 (2001).
[CrossRef]

2000

1999

1998

1997

1996

D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112(1996).
[CrossRef]

G. Gouesbet, C. Letellier, K. F. Ren, and G. Gréhan, “Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542(1996).
[CrossRef] [PubMed]

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
[CrossRef]

1995

1994

1991

J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering from multiple sphere configurations,” Proc. R. Soc. Lond. 433, 599–614 (1991).
[CrossRef]

1988

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

J. P. Barton, D. R. Alexander, and 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, B. Maheu, and 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]

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

1986

P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A (Amsterdam) 137, 209–241 (1986).
[CrossRef]

1979

J. G. Fikioris and N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. A 69, 1359–1366 (1979).
[CrossRef]

1962

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–44 (1962).

1961

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

1908

G. Mie, “Beiträge zur optik trüben medien speziell kolloidaler metalösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and 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]

Barber, P. W.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods, Advanced Series in Applied Physics (World Scientific, 1990), Vol. 2.
[CrossRef]

Bartley, D.

Bartley, D. L.

Barton, J. P.

J. P. Barton, D. R. Alexander, and 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]

Blaisot, J. B.

S. Saengkaew, G. Godard, J. B. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[CrossRef]

Bobbert, P. A.

P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A (Amsterdam) 137, 209–241 (1986).
[CrossRef]

Borghese, F.

Cai, X.

Chylek, P.

Chýlek, P.

D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112(1996).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–44 (1962).

Denti, P.

Doicu, A.

Dutta, S.

V. S. C. M. Rao, Gupta, and S. Dutta, “Broken azimuthal degeneracy with whispering gallery modes of microspheres,” J. Opt. A Pure Appl. Opt. 7, 279–285 (2005).
[CrossRef]

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources: Theory and Programs (Springer, 2006).

Erlick, C.

A. A. Riziq, M. Trainic, C. Erlick, E. Segre, and Y. Rudich, “Extinction efficiencies of coated absorbing aerosols measured by cavity ring down aerosol spectrometry,” Atmos. Chem. Phys. 8, 1823–1833 (2008).
[CrossRef]

Fikioris, J. G.

J. G. Fikioris and N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. A 69, 1359–1366 (1979).
[CrossRef]

Fu, Q.

Fuller, K. A.

Godard, G.

S. Saengkaew, G. Godard, J. B. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[CrossRef]

Gouesbet, G.

G. Gouesbet, “T-matrix formulation and generalized Lorenz–Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521(2010).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. III. Special Euler angles,” Opt. Commun. 283, 3235–3243 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, Y. P. Han, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. IV. Plane waves,” Opt. Commun. 283, 3244–3254 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. I. General formulation,” Opt. Commun. 283, 3218–3225 (2010).
[CrossRef]

J. J. Wang, G. Gouesbet, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. II. Axisymmetric beams,” Opt. Commun. 283, 3226–3234 (2010).
[CrossRef]

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

J. A. Lock and G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transfer 110, 800–807 (2009).
[CrossRef]

G. Gouesbet, “Generalized Lorenz–Mie theories, the third decade: a perspective,” J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
[CrossRef]

F. Xu, K. F. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrary oriented, located, and shaped beam scattered by homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Morphology-dependent resonances and/or whispering gallery modes for a two-dimensional dielectric cavity with an eccentrically located circular inclusion, a Hamiltonian point of view with Hamiltonian (optical) chaos,” Opt. Commun. 201, 223–242(2002).
[CrossRef]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located inclusion, and optical chaos,” Part. Part. Syst. Charact. 18, 190–195 (2001).
[CrossRef]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Periodic orbits in Hamiltonian chaos of the annular billiard,” Phys. Rev. E 65, 016212 (2001).
[CrossRef]

G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821–837 (2000).

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz–Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
[CrossRef]

L. Méès, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results,” Appl. Opt. 38, 1867–1876 (1999).
[CrossRef]

G. Gouesbet and L. Méès, “Generalized Lorenz–Mie theory for infinitely long elliptical cylinders,” J. Opt. Soc. Am. A 16, 1333–1341 (1999).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[CrossRef]

G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
[CrossRef] [PubMed]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

G. Gouesbet, C. Letellier, K. F. Ren, and G. Gréhan, “Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542(1996).
[CrossRef] [PubMed]

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[CrossRef]

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

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

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. V. Localized beam models,” Opt. Commun. (to be published).

Greenaway, R. S.

Gréhan, G.

G. Gouesbet, J. J. Wang, Y. P. Han, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. IV. Plane waves,” Opt. Commun. 283, 3244–3254 (2010).
[CrossRef]

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

S. Saengkaew, G. Godard, J. B. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[CrossRef]

F. Xu, K. F. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrary oriented, located, and shaped beam scattered by homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Morphology-dependent resonances and/or whispering gallery modes for a two-dimensional dielectric cavity with an eccentrically located circular inclusion, a Hamiltonian point of view with Hamiltonian (optical) chaos,” Opt. Commun. 201, 223–242(2002).
[CrossRef]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located inclusion, and optical chaos,” Part. Part. Syst. Charact. 18, 190–195 (2001).
[CrossRef]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Periodic orbits in Hamiltonian chaos of the annular billiard,” Phys. Rev. E 65, 016212 (2001).
[CrossRef]

G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821–837 (2000).

L. Méès, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results,” Appl. Opt. 38, 1867–1876 (1999).
[CrossRef]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

G. Gouesbet, C. Letellier, K. F. Ren, and G. Gréhan, “Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542(1996).
[CrossRef] [PubMed]

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

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

G. Gouesbet, B. Maheu, and 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, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. V. Localized beam models,” Opt. Commun. (to be published).

Gupta,

V. S. C. M. Rao, Gupta, and S. Dutta, “Broken azimuthal degeneracy with whispering gallery modes of microspheres,” J. Opt. A Pure Appl. Opt. 7, 279–285 (2005).
[CrossRef]

Han, G. X.

Han, X.

B. Yan, X. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A Pure Appl. Opt. 11, 015705 (2009).
[CrossRef]

Han, Y. P.

J. J. Wang, G. Gouesbet, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. II. Axisymmetric beams,” Opt. Commun. 283, 3226–3234 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. I. General formulation,” Opt. Commun. 283, 3218–3225 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. III. Special Euler angles,” Opt. Commun. 283, 3235–3243 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, Y. P. Han, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. IV. Plane waves,” Opt. Commun. 283, 3244–3254 (2010).
[CrossRef]

Y. P. Han, Y. Zhang, H. Y. Zhang, and G. X. Han, “Scattering of typical particles by beam shape in oblique illumination,” J. Quant. Spectrosc. Radiat. Transfer 110, 1375–1381 (2009).
[CrossRef]

G. X. Han, Y. P. Han, J. Y. Liu, and Y. Zhang, “Scattering of an eccentric sphere arbitrarily located in a shaped beam,” J. Opt. Soc. Am. B 25, 2064–2072 (2008).
[CrossRef]

H. Y. Zhang and Y. P. Han, “Addition theorem for the spherical vector wave functions and its application to the beam shape coefficients,” J. Opt. Soc. Am. B 25, 255–260 (2008).
[CrossRef]

Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients of arbitrary shaped beam in oblique illumination,” Opt. Express 15, 735–746 (2007).
[CrossRef] [PubMed]

Y. P. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

Hasheminejad, S. M.

S. M. Hasheminejad and Y. Mirzaei, “Exact 3D elasticity solution for free vibrations of eccentric hollow sphere,” J. Sound Vib. (to be published).

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods, Advanced Series in Applied Physics (World Scientific, 1990), Vol. 2.
[CrossRef]

Hirst, E.

Hovenac, E. A.

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 2000).

Kaye, P. H.

Letellier, C.

Leung, P. T.

Liu, J. Y.

Lock, J. A.

Mackowski, D. W.

D. W. Mackowski, “Analysis of radiative scattering from multiple sphere configurations,” Proc. R. Soc. Lond. 433, 599–614 (1991).
[CrossRef]

Maheu, B.

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

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

G. Gouesbet, B. Maheu, and 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]

Méès, L.

Meunier-Guttin-Cluzel, S.

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Morphology-dependent resonances and/or whispering gallery modes for a two-dimensional dielectric cavity with an eccentrically located circular inclusion, a Hamiltonian point of view with Hamiltonian (optical) chaos,” Opt. Commun. 201, 223–242(2002).
[CrossRef]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Periodic orbits in Hamiltonian chaos of the annular billiard,” Phys. Rev. E 65, 016212 (2001).
[CrossRef]

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located inclusion, and optical chaos,” Part. Part. Syst. Charact. 18, 190–195 (2001).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur optik trüben medien speziell kolloidaler metalösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Mirzaei, Y.

S. M. Hasheminejad and Y. Mirzaei, “Exact 3D elasticity solution for free vibrations of eccentric hollow sphere,” J. Sound Vib. (to be published).

Mishchenko, M. I.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 2000).

Ng, S. W.

Ngo, D.

D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112(1996).
[CrossRef]

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

Pang, K. M.

Pinnick, R. G.

Rao, V. S. C. M.

V. S. C. M. Rao, Gupta, and S. Dutta, “Broken azimuthal degeneracy with whispering gallery modes of microspheres,” J. Opt. A Pure Appl. Opt. 7, 279–285 (2005).
[CrossRef]

Ren, K. F.

Riefler, N.

N. Riefler, R. Schuh, and T. Wriedt, “Investigation of a measurement technique to estimate concentration and size of inclusions in droplets,” Meas. Sci. Technol. 18, 2209–2218 (2007).
[CrossRef]

Riziq, A. A.

A. A. Riziq, M. Trainic, C. Erlick, E. Segre, and Y. Rudich, “Extinction efficiencies of coated absorbing aerosols measured by cavity ring down aerosol spectrometry,” Atmos. Chem. Phys. 8, 1823–1833 (2008).
[CrossRef]

Rudich, Y.

A. A. Riziq, M. Trainic, C. Erlick, E. Segre, and Y. Rudich, “Extinction efficiencies of coated absorbing aerosols measured by cavity ring down aerosol spectrometry,” Atmos. Chem. Phys. 8, 1823–1833 (2008).
[CrossRef]

Saengkaew, S.

S. Saengkaew, G. Godard, J. B. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[CrossRef]

Saija, R.

Schaub, S. A.

J. P. Barton, D. R. Alexander, and 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]

Schuh, R.

N. Riefler, R. Schuh, and T. Wriedt, “Investigation of a measurement technique to estimate concentration and size of inclusions in droplets,” Meas. Sci. Technol. 18, 2209–2218 (2007).
[CrossRef]

R. Schuh and T. Wriedt, “Computer programs for light scattering by particles with inclusions,” J. Quant. Spectrosc. Radiat. Transfer 70, 715–723 (2001).
[CrossRef]

Secker, D. R.

Segre, E.

A. A. Riziq, M. Trainic, C. Erlick, E. Segre, and Y. Rudich, “Extinction efficiencies of coated absorbing aerosols measured by cavity ring down aerosol spectrometry,” Atmos. Chem. Phys. 8, 1823–1833 (2008).
[CrossRef]

Sindoni, O. I.

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

Sun, W.

Trainic, M.

A. A. Riziq, M. Trainic, C. Erlick, E. Segre, and Y. Rudich, “Extinction efficiencies of coated absorbing aerosols measured by cavity ring down aerosol spectrometry,” Atmos. Chem. Phys. 8, 1823–1833 (2008).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 2000).

Uzunoglu, N. K.

J. G. Fikioris and N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. A 69, 1359–1366 (1979).
[CrossRef]

Videen, G.

Vlieger, J.

P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A (Amsterdam) 137, 209–241 (1986).
[CrossRef]

Wang, J. J.

G. Gouesbet, J. J. Wang, Y. P. Han, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. IV. Plane waves,” Opt. Commun. 283, 3244–3254 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. III. Special Euler angles,” Opt. Commun. 283, 3235–3243 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. I. General formulation,” Opt. Commun. 283, 3218–3225 (2010).
[CrossRef]

J. J. Wang, G. Gouesbet, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. II. Axisymmetric beams,” Opt. Commun. 283, 3226–3234 (2010).
[CrossRef]

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. V. Localized beam models,” Opt. Commun. (to be published).

Wriedt, T.

N. Riefler, R. Schuh, and T. Wriedt, “Investigation of a measurement technique to estimate concentration and size of inclusions in droplets,” Meas. Sci. Technol. 18, 2209–2218 (2007).
[CrossRef]

R. Schuh and T. Wriedt, “Computer programs for light scattering by particles with inclusions,” J. Quant. Spectrosc. Radiat. Transfer 70, 715–723 (2001).
[CrossRef]

A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz–Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
[CrossRef] [PubMed]

T. Wriedt, “The website maintained by Thomas Wriedt,” http://www.scattport.org.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources: Theory and Programs (Springer, 2006).

Xu, F.

Yan, B.

B. Yan, X. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A Pure Appl. Opt. 11, 015705 (2009).
[CrossRef]

Zhang, H. Y.

Zhang, Y.

Y. P. Han, Y. Zhang, H. Y. Zhang, and G. X. Han, “Scattering of typical particles by beam shape in oblique illumination,” J. Quant. Spectrosc. Radiat. Transfer 110, 1375–1381 (2009).
[CrossRef]

G. X. Han, Y. P. Han, J. Y. Liu, and Y. Zhang, “Scattering of an eccentric sphere arbitrarily located in a shaped beam,” J. Opt. Soc. Am. B 25, 2064–2072 (2008).
[CrossRef]

Ann. Phys.

G. Mie, “Beiträge zur optik trüben medien speziell kolloidaler metalösungen,” Ann. Phys. 25, 377–452 (1908).
[CrossRef]

Appl. Opt.

G. Gouesbet, “Interaction between an infinite cylinder and an arbitrary-shaped beam,” Appl. Opt. 36, 4292–4304 (1997).
[CrossRef] [PubMed]

L. Méès, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: numerical results,” Appl. Opt. 38, 1867–1876 (1999).
[CrossRef]

Y. P. Han, G. Gréhan, and G. Gouesbet, “Generalized Lorenz–Mie theory for a spheroidal particle with off-axis Gaussian-beam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing several spherical inclusions,” Appl. Opt. 33, 484–493 (1994).
[CrossRef] [PubMed]

D. R. Secker, P. H. Kaye, R. S. Greenaway, E. Hirst, D. L. Bartley, and G. Videen, “Light scattering from deformed droplets and droplets with inclusions. I. Experimental results,” Appl. Opt. 39, 5023–5030 (2000).
[CrossRef]

G. Videen, W. Sun, Q. Fu, D. R. Secker, R. S. Greenaway, P. H. Kaye, E. Hirst, and D. Bartley, “Light scattering from deformed droplets and droplets with inclusions. II. Theoretical treatment,” Appl. Opt. 39, 5031–5039 (2000).
[CrossRef]

G. Gouesbet, C. Letellier, K. F. Ren, and G. Gréhan, “Discussion of two quadrature methods of evaluating beam-shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542(1996).
[CrossRef] [PubMed]

K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral localized approximation in generalized Lorenz–Mie theory,” Appl. Opt. 37, 4218–4225 (1998).
[CrossRef]

A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz–Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
[CrossRef] [PubMed]

J. A. Lock, “An improved Gaussian beam scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
[CrossRef] [PubMed]

Atmos. Chem. Phys.

A. A. Riziq, M. Trainic, C. Erlick, E. Segre, and Y. Rudich, “Extinction efficiencies of coated absorbing aerosols measured by cavity ring down aerosol spectrometry,” Atmos. Chem. Phys. 8, 1823–1833 (2008).
[CrossRef]

Comput. Phys. Commun.

D. Ngo, G. Videen, and P. Chýlek, “A FORTRAN code for the scattering of EM waves by a sphere with a nonconcentric spherical inclusion,” Comput. Phys. Commun. 99, 94–112(1996).
[CrossRef]

Exp. Fluids

S. Saengkaew, G. Godard, J. B. Blaisot, and G. Gréhan, “Experimental analysis of global rainbow technique: sensitivity of temperature and size distribution measurements to non-spherical droplets,” Exp. Fluids 47, 839–848 (2009).
[CrossRef]

J. Appl. Phys.

J. P. Barton, D. R. Alexander, and 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. Mod. Opt.

G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located spherical inclusion,” J. Mod. Opt. 47, 821–837 (2000).

J. Opt.

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

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
[CrossRef]

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

J. Opt. A Pure Appl. Opt.

V. S. C. M. Rao, Gupta, and S. Dutta, “Broken azimuthal degeneracy with whispering gallery modes of microspheres,” J. Opt. A Pure Appl. Opt. 7, 279–285 (2005).
[CrossRef]

B. Yan, X. Han, and K. F. Ren, “Scattering of a shaped beam by a spherical particle with an eccentric spherical inclusion,” J. Opt. A Pure Appl. Opt. 11, 015705 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. G. Fikioris and N. K. Uzunoglu, “Scattering from an eccentrically stratified dielectric sphere,” J. Opt. Soc. Am. A 69, 1359–1366 (1979).
[CrossRef]

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

F. Xu, K. F. Ren, G. Gouesbet, G. Gréhan, and X. Cai, “Generalized Lorenz–Mie theory for an arbitrary oriented, located, and shaped beam scattered by homogeneous spheroid,” J. Opt. Soc. Am. A 24, 119–131 (2007).
[CrossRef]

K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz–Mie theory: formulation and numerical results,” J. Opt. Soc. Am. A 14, 3014–3025 (1997).
[CrossRef]

G. Gouesbet, “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz–Mie theory for spheres,” J. Opt. Soc. Am. A 16, 1641–1650 (1999).
[CrossRef]

G. Gouesbet, B. Maheu, and 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 and L. Méès, “Generalized Lorenz–Mie theory for infinitely long elliptical cylinders,” J. Opt. Soc. Am. A 16, 1333–1341 (1999).
[CrossRef]

J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
[CrossRef]

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[CrossRef]

J. A. Lock and E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
[CrossRef]

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transfer

Y. P. Han, Y. Zhang, H. Y. Zhang, and G. X. Han, “Scattering of typical particles by beam shape in oblique illumination,” J. Quant. Spectrosc. Radiat. Transfer 110, 1375–1381 (2009).
[CrossRef]

J. A. Lock and G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transfer 110, 800–807 (2009).
[CrossRef]

G. Gouesbet, “Generalized Lorenz–Mie theories, the third decade: a perspective,” J. Quant. Spectrosc. Radiat. Transfer 110, 1223–1238 (2009).
[CrossRef]

R. Schuh and T. Wriedt, “Computer programs for light scattering by particles with inclusions,” J. Quant. Spectrosc. Radiat. Transfer 70, 715–723 (2001).
[CrossRef]

Meas. Sci. Technol.

N. Riefler, R. Schuh, and T. Wriedt, “Investigation of a measurement technique to estimate concentration and size of inclusions in droplets,” Meas. Sci. Technol. 18, 2209–2218 (2007).
[CrossRef]

Opt. Commun.

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Morphology-dependent resonances and/or whispering gallery modes for a two-dimensional dielectric cavity with an eccentrically located circular inclusion, a Hamiltonian point of view with Hamiltonian (optical) chaos,” Opt. Commun. 201, 223–242(2002).
[CrossRef]

L. Méès, G. Gouesbet, and G. Gréhan, “Transient internal and scattered fields from a multi-layered sphere illuminated by a pulsed laser,” Opt. Commun. 282, 4189–4193 (2009).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. I. General formulation,” Opt. Commun. 283, 3218–3225 (2010).
[CrossRef]

J. J. Wang, G. Gouesbet, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. II. Axisymmetric beams,” Opt. Commun. 283, 3226–3234 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. III. Special Euler angles,” Opt. Commun. 283, 3235–3243 (2010).
[CrossRef]

G. Gouesbet, J. J. Wang, Y. P. Han, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. IV. Plane waves,” Opt. Commun. 283, 3244–3254 (2010).
[CrossRef]

G. Gouesbet, “T-matrix formulation and generalized Lorenz–Mie theories in spherical coordinates,” Opt. Commun. 283, 517–521(2010).
[CrossRef]

Opt. Express

Opt. Lett.

Part. Part. Syst. Charact.

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Generalized Lorenz–Mie theory for a sphere with an eccentrically located inclusion, and optical chaos,” Part. Part. Syst. Charact. 18, 190–195 (2001).
[CrossRef]

Phys. Rev. E

G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Gréhan, “Periodic orbits in Hamiltonian chaos of the annular billiard,” Phys. Rev. E 65, 016212 (2001).
[CrossRef]

Physica A (Amsterdam)

P. A. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A (Amsterdam) 137, 209–241 (1986).
[CrossRef]

Proc. R. Soc. Lond.

D. W. Mackowski, “Analysis of radiative scattering from multiple sphere configurations,” Proc. R. Soc. Lond. 433, 599–614 (1991).
[CrossRef]

Quart. Appl. Math.

S. Stein, “Addition theorems for spherical wave functions,” Quart. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Quart. Appl. Math. 20, 33–44 (1962).

Other

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods, Advanced Series in Applied Physics (World Scientific, 1990), Vol. 2.
[CrossRef]

T. Wriedt, “The website maintained by Thomas Wriedt,” http://www.scattport.org.

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, 2000).

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources: Theory and Programs (Springer, 2006).

S. M. Hasheminejad and Y. Mirzaei, “Exact 3D elasticity solution for free vibrations of eccentric hollow sphere,” J. Sound Vib. (to be published).

G. Gouesbet, J. A. Lock, J. J. Wang, and G. Gréhan, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. V. Localized beam models,” Opt. Commun. (to be published).

Supplementary Material (2)

» Media 1: AVI (3061 KB)     
» Media 2: AVI (3082 KB)     

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Figures (8)

Fig. 1
Fig. 1

Scattering geometry of the problem under study.

Fig. 2
Fig. 2

Comparison of scattered intensity distribution between the result obtained from our code and Fig. 6 published in [36].

Fig. 3
Fig. 3

Scattered field distribution in the far zone. (a) Single-frame excerpt from Media 1 for cases of plane wave illumination. (b) Single-frame excerpt from Media 2 for cases of focused Gaussian beam illumination. The vertical axis in the movie is the zenith angle θ in degrees, and the horizontal axis is the azimuthal angle φ in degrees..

Fig. 4
Fig. 4

Normalized source function for external and internal fields along the z axis with center–center separation distance d as the parameter. The left column [(a), (b), (c)] is for cases of plane wave illumination; the right column [(d), (e), (f)] is for cases of focused Gaussian beam illumination.

Fig. 5
Fig. 5

Normalized source function for external and internal field over the x z plane with center–center separation distance d as the para meter. The left column [(a), (b), (c)] is for cases of plane wave illumination; the right column [(d), (e), (f)] is for cases of focused Gaussian beam illumination.

Fig. 6
Fig. 6

Illustration of localization of partial waves in a geometric optics point of view.

Fig. 7
Fig. 7

Normalized source function distribution for external and internal fields over the x z plane with Euler angles α = γ = 0.0 ° and β as a parameter..

Fig. 8
Fig. 8

Normalized source function distribution for external and internal field over the x z plane with location of the Gaussian beam waist center as a parameter. The Gaussian beam propagates along the x axis from left to right..

Equations (57)

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x 2 = x 1 , y 2 = y 1 , z 2 = z 1 d .
M n m ( j ) = ( 1 ) m [ i m π ˜ n m ( cos θ ) i θ τ ˜ n m ( cos θ ) i φ ] z n ( k r ) exp ( i m φ ) ,
N n m ( j ) = ( 1 ) m { n ( n + 1 ) k r z n ( k r ) P ˜ n m ( cos θ ) i r + 1 k r [ d d r r z n ( k r ) ] τ ˜ n m ( cos θ ) i θ + 1 k r [ d d r r z n ( k r ) ] i m π ˜ n m ( cos θ ) i φ } exp ( i m φ ) ,
π ˜ n m ( cos θ ) = P ˜ n m ( cos θ ) sin θ ; τ ˜ n m ( cos θ ) = d d θ P ˜ n m ( cos θ ) .
P ˜ n m ( cos θ ) = c n m P n m ( cos θ ) ,
c n m = ( 1 ) m 2 n + 1 2 ( n m ) ! ( n + m ) ! ,
P n m ( cos θ ) = ( 1 ) m ( sin θ ) m d m P n ( cos θ ) ( d cos θ ) m .
E inc = n = 1 m = n + n a n m M n m ( 1 ) ( k 0 r 1 ) + b n m N n m ( 1 ) ( k 0 r 1 ) ,
E sca = n = 1 m = n + n c n m M n m ( 4 ) ( k 0 r 1 ) + d n m N n m ( 4 ) ( k 0 r 1 ) .
E int 1 = n = 1 m = n + n e n m M n m ( 3 ) ( k 1 r 1 ) + f n m N n m ( 3 ) ( k 1 r 1 ) + v n m M n m ( 4 ) ( k 1 r 1 ) + h n m N n m ( 4 ) ( k 1 r 1 ) .
E int 1 = n = 1 m = n + n r n m M n m ( 3 ) ( k 1 r 2 ) + s n m N n m ( 3 ) ( k 1 r 2 ) + t n m M n m ( 4 ) ( k 1 r 2 ) + u n m N n m ( 4 ) ( k 1 r 2 ) ,
E int 2 = n = 1 m = n + n p n m M n m ( 1 ) ( k 2 r 2 ) + q n m N n m ( 1 ) ( k 2 r 2 ) ,
a n m = i k c n p w ( 1 ) m ( 1 ) m | m | 2 ( n m ) ! ( n | m | ) ! g ˜ n , TE m c n m ,
b n m = k c n p w ( 1 ) m ( 1 ) m | m | 2 ( n m ) ! ( n | m | ) ! g ˜ n , TM m c n m ,
c n p w = 1 k ( i ) n + 1 2 n + 1 n ( n + 1 ) .
g ˜ n , X m = μ n m s = n n H s n m μ n s g n , X s ,
μ n m = ( 1 ) m ( 1 ) m | m | 2 ( n | m | ) ! ( n m ) ! ,
H s n m = ( 1 ) n + s ( n m ) ! ( n s ) ! e i m γ e i s α σ ( 1 ) σ ( n + s n m σ ) ( n s σ ) ( cos β 2 ) 2 σ + m + s ( sin β 2 ) 2 n 2 σ m s ,
S = | E | 2 / | E 0 | 2 ,
E r int 1 = n = 1 m = n + n ( 1 ) m [ s n m h n ( 1 ) ( k 1 r 2 ) + u n m h n ( 2 ) ( k 1 r 2 ) ] n ( n + 1 ) k 1 r 2 P ˜ n m ( cos θ ) exp ( i m φ ) ,
E θ int 1 = n = 1 m = n + n ( 1 ) m { [ r n m h n ( 1 ) ( k 1 r 2 ) + t n m h n ( 2 ) ( k 1 r 2 ) ] i m π ˜ n m ( cos θ ) + 1 k 1 r 2 [ s n m d ( r 2 h n ( 1 ) ( k 1 r 2 ) ) d r 2 + u n m d ( r 2 h n ( 2 ) ( k 1 r 2 ) ) d r 2 ] τ ˜ n m ( cos θ ) } exp ( i m φ ) ,
E φ int 1 = n = 1 m = n + n ( 1 ) m { [ r n m h n ( 1 ) ( k 1 r 2 ) t n m h n ( 2 ) ( k 1 r 2 ) ] τ ˜ n m ( cos θ ) + 1 k 1 r 2 [ s n m d ( r 2 h n ( 1 ) ( k 1 r 2 ) ) d r 2 + u n m d ( r 2 h n ( 2 ) ( k 1 r 2 ) ) d r 2 ] i m π ˜ n m ( cos θ ) } exp ( i m φ ) .
E r int 1 = n = 1 m = n + n ( 1 ) m q n m j n ( k 2 r 2 ) k 2 r 2 n ( n + 1 ) P ˜ n m ( cos θ ) exp ( i m φ ) ,
E θ int 1 = n = 1 m = n + n ( 1 ) m { p n m j n ( k 2 r 2 ) i m π ˜ n m ( cos θ ) + q n m 1 k 2 r 2 d ( r 2 j n ( k 2 r 2 ) ) d r 2 τ ˜ n m ( cos θ ) } exp ( i m φ ) ,
E φ int 1 = n = 1 m = n + n ( 1 ) m { p n m j n ( k 2 r 2 ) τ ˜ n m ( cos θ ) + q n m 1 k 2 r 2 d ( r 2 j n ( k 2 r 2 ) ) d r 2 i m π ˜ n m ( cos θ ) } exp ( i m φ ) .
E r sca = n = 1 m = n + n ( 1 ) m d n m h n ( 2 ) ( k 0 r 1 ) k 0 r 1 n ( n + 1 ) P ˜ n m ( cos θ ) exp ( i m φ ) ,
E θ sca = n = 1 m = n + n ( 1 ) m { c n m h n ( 2 ) ( k 0 r 1 ) i m π ˜ n m ( cos θ ) + d n m 1 k 0 r 1 d ( r 1 h n ( 2 ) ( k 0 r 1 ) ) d r 1 τ ˜ n m ( cos θ ) } exp ( i m φ ) ,
E φ sca = n = 1 m = n + n ( 1 ) m { c n m h n ( 2 ) ( k 0 r 1 ) τ ˜ n m ( cos θ ) + d n m 1 k 0 r 1 d ( r 1 h n ( 2 ) ( k 0 r 1 ) ) d r 1 i m π ˜ n m ( cos θ ) } exp ( i m φ ) .
h n ( 2 ) ( k r ) i n + 1 e i k r k r , d h n ( 2 ) ( k r ) d ( k r ) i n e i k r k r .
E θ sca = i e i k 0 r 1 k 0 r 1 n = 1 m = n + n [ c n m m π ˜ n m ( cos θ ) d n m τ ˜ n m ( cos θ ) ] i n + 1 ( 1 ) m exp ( i m φ ) ,
E φ sca = e i k 0 r 1 k 0 r 1 n = 1 m = n + n [ c n m τ ˜ n m ( cos θ ) + d n m m π ˜ n m ( cos θ ) ] i n + 1 ( 1 ) m exp ( i m φ ) .
( I θ I φ ) = λ 2 4 π 2 r 1 2 ( | S 2 | 2 | S 1 | 2 ) ,
S 2 = n = 1 m = n + n [ c n m m π ˜ n m ( cos θ ) d n m τ ˜ n m ( cos θ ) ] i n + 1 ( 1 ) m exp ( i m φ ) ,
S 1 = n = 1 m = n + n [ c n m τ ˜ n m ( cos θ ) d n m m π ˜ n m ( cos θ ) ] i n + 1 ( 1 ) m exp ( i m φ ) .
H s n m = [ ( n m ) ! ( n + s ) ! ( n + m ) ! ( n s ) ! ] 1 / 2 e i m γ e i s α d m s n ,
M n m ( i ) ( k r ) = n = 1 m = n n A m n m n ( k r 0 ) M n m ( i ) ( k r 1 ) + B m n m n ( k r 0 ) N n m ( i ) ( k r 1 ) ,
N n m ( i ) ( k r ) = n = 1 m = n n B m n m n ( k r 0 ) M n m ( i ) ( k r 1 ) + A m n m n ( k r 0 ) N n m ( i ) ( k r 1 ) .
M n m ( i ) ( k r ) = n = 1 A n n m ( k r 0 ) M n m ( i ) ( k r 1 ) + B n n m ( k r 0 ) N n m ( i ) ( k r 1 ) ,
N n m ( i ) ( k r ) = n = 1 B n n m ( k r 0 ) M n m ( i ) ( k r 1 ) + A n n m ( k r 0 ) N n m ( i ) ( k r 1 ) .
A m n m n ( k z 0 ) = C m n m n ( k z 0 ) + k z 0 n + 1 ( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) C m n + 1 m n ( k z 0 ) + k z 0 n ( n m ) ( n + m ) ( 2 n + 1 ) ( 2 n 1 ) C m n 1 m n ( k z 0 ) ,
B m n m n ( k z 0 ) = j k z 0 m n ( n + 1 ) C m n m n ( k z 0 ) .
A m n m n ( k z 0 ) = A m n m n ( k z 0 ) , B m n m n ( k z 0 ) = B m n m n ( k z 0 ) ,
A m n m n ( k z 0 ) = A m n m n ( k z 0 ) , B m n m n ( k z 0 ) = B m n m n ( k z 0 ) .
( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) C m n m n + 1 ( k z 0 ) ( n m ) ( n + m ) ( 2 n 1 ) ( 2 n + 1 ) C m n m n 1 ( k z 0 ) = ( n m ) ( n + m ) ( 2 n 1 ) ( 2 n + 1 ) C m n 1 m n ( k z 0 ) ( n m + 1 ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) C m n + 1 m n ( k z 0 ) ,
( n m 1 ) ( n m + 1 ) ( 2 n 1 ) ( 2 n + 1 ) C m n m n 1 ( k z 0 ) + ( n + m ) ( n + m + 1 ) ( 2 n + 1 ) ( 2 n + 3 ) C m n m n + 1 ( k z 0 ) = ( n + m 1 ) ( n + m ) ( 2 n 1 ) ( 2 n + 1 ) C m 1 n 1 m 1 n ( k z 0 ) + ( n m + 1 ) ( n m + 2 ) ( 2 n + 1 ) ( 2 n + 3 ) C m 1 n + 1 m 1 n ( k z 0 ) .
C 0 n 00 , ( 1 ) ( k z 0 ) = ( 1 ) n 2 n + 1 j n ( k z 0 ) for regular VSWFs translation, C 0 n 00 , ( 3 ) ( k z 0 ) = ( 1 ) n 2 n + 1 h n ( 1 ) ( k z 0 ) for radiating VSWFs translation .
d m s n + 1 ( β ) = 1 n ( n + 1 ) 2 m 2 ( n + 1 ) 2 s 2 { ( 2 n + 1 ) [ n ( n + 1 ) x m s ] d m s n ( β ) ( n + 1 ) n 2 m 2 n 2 s 2 d m s n 1 ( β ) } , n n min .
d m s n min 1 ( β ) = 0 ,
d m s n min ( β ) = ξ m s 2 n min [ ( 2 n min ) ! ( | m s | ) ! | m + s | ! ] 1 / 2 ( 1 x ) | m s | / 2 ( 1 + x ) | m + s | / 2 ,
ξ m s = { 1 if     s m ( 1 ) m s if     s < m ,
n min = max ( | m | , | s | ) , x = cos β .
d m s n ( β ) = ( 1 ) m + s d m , s n ( β ) = d s , m n ( β ) ,
d m s n ( π β ) = ( 1 ) n s d m s n ( β ) = ( 1 ) n m d m , s n ( β ) .
d m 0 n ( β ) = ( n m ) ! ( n + m ) ! P n m ( cos β ) ,
m π ˜ n m ( cos β ) = ( 1 ) m 2 n + 1 2 n ( n + 1 ) 2 [ d m 1 n ( β ) + d m 1 n ( β ) ] ,
τ ˜ n m ( cos β ) = ( 1 ) m 2 n + 1 2 n ( n + 1 ) 2 [ d m 1 n ( β ) d m 1 n ( β ) ] ,
P ˜ n m ( cos β ) = ( 1 ) m 2 n + 1 2 d m 0 n ( β ) .

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