Abstract

We develop a new and numerically efficient formalism to describe the general problem of the scattering and absorption of light by a spherical metal or dielectric particle illuminated by a tightly focused beam. The theory is based on (i) the generalized Mie theory equations, (ii) the plane-wave decomposition of the converging light beam, and (iii) the expansion of a plane wave in terms of vector spherical harmonics. The predictions of the model are illustrated in the case of silver nanoparticles. The results are compared with the Mie theory in the local approximation. Finally, some effects related to the convergence of the beam are analyzed in the context of experiments based on the spatial modulation spectroscopy technique.

© 2008 Optical Society of America

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  1. C.Dupas, Ph.Houdy, and M.Lahmani, eds., Nanosciences, Nanotechnologies and Nanophysics (Springer, 2006).
  2. R.Kelsall, I.W.Hamley, and M.Geoghegan, eds., Nanoscale Science and Technology (Wiley, 2006).
  3. S. Schultz, D. R. Smith, J. J. Mock, and D. A. Schultz, "Single-target molecule detection with nonbleaching multicolour optical immunolabels," Proc. Natl. Acad. Sci. USA 97, 996-1001 (2000).
    [CrossRef] [PubMed]
  4. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).
  5. E. Cottancin, G. Celep, J. Lermé, M. Pellarin, J. R. Huntzinger, J. L. Vialle, and M. Broyer, "Optical properties of noble metal clusters as a function of the size: comparison between experiments and a semi-quantal theory," Theor. Chem. Acc. 116, 514-523 (2006).
    [CrossRef]
  6. T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, "Surface-plasmon resonance in single metallic nanoparticles," Phys. Rev. Lett. 80, 4249-4252 (1998).
    [CrossRef]
  7. M. Perner, T. Klar, S. Grosse, U. Lemmer, G. von Plessen, W. Spirkl, and J. Feldmann, "Homogeneous line widths of surface plasmons in gold nanoparticles measured by femtosecond pump-and-probe and near-field optical spectroscopy," J. Lumin. 76, 181-184 (1998).
    [CrossRef]
  8. A. A. Mikhailovsky, M. A. Petruska, M. I. Stockman, and V. I. Klimov, "Broadband near-field interference spectroscopy of metal nanoparticles using a femtosecond white-light continuum," Opt. Lett. 28, 1686-1688 (2003).
    [CrossRef] [PubMed]
  9. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002).
    [CrossRef] [PubMed]
  10. J. J. Mock, D. R. Smith, and S. Schultz, "Local refractive index dependence of plasmon resonance spectra from individual nanoparticles," Nano Lett. 3, 485-491 (2003).
    [CrossRef]
  11. J. Prikulis, F. Svedberg, M. Käll, J. Enger, K. Ramser, M. Goksör, and D. Hanstorp, "Optical spectroscopy of single trapped metal nanoparticles in solution," Nano Lett. 4, 115-118 (2004).
    [CrossRef]
  12. A. Curry, G. Nusz, A. Chilkoti, and A. Wax, "Substrate effect on refractive index dependence of plasmon resonance for individual silver nanoparticles observed using darkfield micro-spectroscopy," Opt. Express 13, 2668-2677 (2005).
    [CrossRef] [PubMed]
  13. C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z. H. Chan, J. P. Spatz, and M. Möller, "Spectroscopy of single metallic nanoparticles using total internal reflection microscopy," Appl. Phys. Lett. 77, 2949-2951 (2000).
    [CrossRef]
  14. O. L. Muskens, N. Del Fatti, F. Vallée, J. R. Huntzinger, P. Billaud, and M. Broyer, "Single metal nanoparticle absorption spectroscopy and optical characterization," Appl. Phys. Lett. 88, 063109 (2006).
    [CrossRef]
  15. P. Billaud, J. R. Huntzinger, E. Cottancin, J. Lermé, M. Pellarin, L. Arnaud, M. Broyer, N. Del Fatti, and F. Vallée, "Optical extinction spectroscopy of single silver nanoparticles," Eur. Phys. J. D 43, 271-274 (2007).
    [CrossRef]
  16. T. Kalkbrenner, U. Hakanson, and V. Sandoghdar, "Tomographic plasmon spectroscopy of a single gold nanoparticle," Nano Lett. 4, 2309-2314 (2004).
    [CrossRef]
  17. T. Itoh, T. Asahi, and H. Masuhara, "Femtosecond light scattering spectroscopy of single gold nanoparticles," Appl. Phys. Lett. 79, 1667-1669 (2001).
    [CrossRef]
  18. P. Li, K. Shi, and Z. Liu, "Manipulation and spectroscopy of a single particle by use of white-light optical tweezers," Opt. Lett. 30, 156-158 (2005).
    [CrossRef] [PubMed]
  19. K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, "Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy," Phys. Rev. Lett. 93, 037401 (2004).
    [CrossRef] [PubMed]
  20. O. L. Muskens, N. Del Fatti, and F. Vallée, "Femtosecond response of a single metal nanoparticle," Nano Lett. 6, 552-556 (2006).
    [CrossRef] [PubMed]
  21. Y. H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, "Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
    [CrossRef]
  22. N. Nilius, N. Ernst, and H. J. Freund, "Photon emission spectroscopy of individual oxide-supported silver clusters in a scanning tunneling microscope," Phys. Rev. Lett. 84, 3994-3997 (2000).
    [CrossRef] [PubMed]
  23. D. Boyer, P. Tamarat, A. Maali, B. Lounis, and M. Orrit, "Photothermal imaging of nanometer-sized metal particles among scatterers," Science 297, 1160-1163 (2002).
    [CrossRef] [PubMed]
  24. S. Berciaud, L. Cognet, G. A. Blab, and B. Lounis, "Photothermal heterodyne imaging of individual nonfluorescent nanoclusters and nanocrystals," Phys. Rev. Lett. 93, 257402 (2004).
    [CrossRef]
  25. F. Stietz, J. Bosbach, T. Wenzel, T. Vartanyan, A. Goldmann, and F. Träger, "Decay times of surface plasmon excitation in metal nanoparticles by persistent spectral hole burning," Phys. Rev. Lett. 84, 5644-5647 (2000).
    [CrossRef] [PubMed]
  26. T. Ziegler, C. Hendrich, F. Hubenthal, T. Vartanyan, and F. Träger, "Dephasing times of surface plasmon excitation in Au nanoparticles determined by persistent spectral hole burning," Chem. Phys. Lett. 386, 319-324 (2004).
    [CrossRef]
  27. H. Hövel, S. Fritz, A. Hilger, and U. Kreibig, "Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping," Phys. Rev. B 48, 18178-18188 (1993).
    [CrossRef]
  28. L. V. Lorenz, "Lysvevægelsen i og uden for en af plane lysbølger belyst kugle," K. Dan. Vidensk. Selsk. Forth. 6, 1-62 (1890).
  29. G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908).
    [CrossRef]
  30. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  31. F. Moreno, F. Gonzales, and J. M. Saiz, "Plasmon spectroscopy of metallic nanoparticles above flat dielectric substrates," Opt. Lett. 31, 1902-1904 (2006).
    [CrossRef] [PubMed]
  32. T. Götz, W. Hoheisel, M. Vollmer, and F. Träger, "Characterization of large supported metal clusters by optical spectroscopy," Z. Phys. D 33, 133-141 (1995).
    [CrossRef]
  33. 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).
    [CrossRef]
  34. B. R. Johnson, "Light scattering from a spherical particle on a conducting plane: I. Normal incidence: errata," J. Opt. Soc. Am. A 10, 766 (1993).
    [CrossRef]
  35. B. R. Johnson, "Calculation of light scattering from a spherical particle on a surface by the multipole expansion method," J. Opt. Soc. Am. A 13, 326-337 (1996).
    [CrossRef]
  36. A. Arbouet, D. Christofilos, N. Del Fatti, F. Vallée, J. R. Huntzinger, L. Arnaud, P. Billaud, and M. Broyer, "Direct measurement of the single-metal-cluster optical absorption," Phys. Rev. Lett. 93, 127401 (2004).
    [CrossRef] [PubMed]
  37. M. Born and E. Wolf, Principles of Optics, 7th. ed. (Cambridge U. Press, 1999).
  38. L. W. Casperson, C. Yeh, and W. F. Yeung, "Single particle scattering with focused laser beam," Appl. Opt. 16, 1104-1107 (1977).
    [PubMed]
  39. F. T. Gucker and J. J. Egan, "Measurement of the angular variation of light scattering from single aerosol droplets," J. Colloid Sci. 16, 68-84 (1961).
    [CrossRef]
  40. H. Chew, M. Kerker, and D. D. Cooke, "Light scattering in converging beam," Opt. Lett. 1, 138-140 (1977).
    [CrossRef] [PubMed]
  41. H. Chew, M. Kerker, and D. D. Cooke, "Electromagnetic scattering by a dielectric sphere in a diverging radiation field," Phys. Rev. A 16, 320-323 (1977).
    [CrossRef]
  42. J. P. Chevaillier, J. Fabre, and P. Hamelin, "Forward scattering light intensities by a sphere located anywhere in a Gaussian beam," Appl. Opt. 25, 1222-1225 (1986).
    [CrossRef] [PubMed]
  43. W. C. Tsai and R. J. Pogorzelski, "Eigenfunction solution of the scattering of beam radiation fields by spherical objects," J. Opt. Soc. Am. 65, 1457-1463 (1975).
    [CrossRef]
  44. W. G. Tam and R. Corriveau, "Scattering of electromagnetic beams by spherical objects," J. Opt. Soc. Am. 68, 763-767 (1978).
    [CrossRef]
  45. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
    [CrossRef]
  46. L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  47. In the Davis's approach the electromagnetic field (E,H), derived from a linearly polarized vector potential A, is expressed as a series upon increasing powers of the beam-confinement parameter s=(kω0)−1(s⩽1), where k is the wave vector and ω0 the beam-waist radius.
  48. For instance, in the lowest-order paraxial approximation, the electric field is linearly polarized along a transverse direction and does not satisfy div(E)=0. See, for instance, W. H. Carter, "Electromagnetic field of a Gaussian beam with an elliptic cross section," J. Opt. Soc. Am. 62, 1195-1201 (1972).
    [CrossRef]
  49. G. Gouesbet, G. Grehan, and B. Maheu, "Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism," J. Opt. 16, 83-93 (1985).
    [CrossRef]
  50. 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]
  51. J. S. Kim and S. S. Lee, "Scattering of laser beams and the optical potential well for a homogeneous sphere," J. Opt. Soc. Am. 73, 303-312 (1983).
    [CrossRef]
  52. S. O. Park and S. S. Lee, "Forward far-field pattern of a laser beam scattered by a water-suspended homogeneous sphere trapped by a focused laser beam," J. Opt. Soc. Am. A 4, 417-422 (1987).
    [CrossRef]
  53. 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]
  54. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
    [CrossRef]
  55. J. A. Lock, "Interpretation of extinction in Gaussian-beam scattering," J. Opt. Soc. Am. A 12, 929-938 (1995).
    [CrossRef]
  56. J. T. Hodges, G. Gréhan, G. Gouesbet, and C. Presser, "Forward scattering of a Gaussian beam by a nonabsorbing sphere," Appl. Opt. 34, 2120-2132 (1995).
    [CrossRef] [PubMed]
  57. L. Novotny, R. D. Grober, and K. Karrai, "Reflected image of a strongly focused spot," Opt. Lett. 26, 789-791 (2001).
    [CrossRef]
  58. A. Rohrbach and E. H. K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91, 5475-5488 (2002).
    [CrossRef]
  59. P. Török, P. D. Higdon, R. Juskaitis, and T. Wilson, "Optimising the image contrast of conventional and confocal microscopes imaging finite sized spherical gold scatterers," Opt. Commun. 155, 335-341 (1998).
    [CrossRef]
  60. P. Li, K. Shi, and Z. Liu, "Optical scattering spectroscopy by using tightly focused supercontinuum," Opt. Express 13, 9039 (2005). Equation in this paper has been corrected ([cos(θ)]1/2 instead of [cos(θ)]−1/2).
    [CrossRef] [PubMed]
  61. Strictly, the amplitude E0 should be corrected to take into account the (weak) energy losses due to reflexion and absorption within the lens. If these losses does not depend noticeably on the azimuthal angle phiv, these effects can be easily included in our modeling by modifying the weighting factor [cos(θ)]1/2 appropriately.
  62. The center of the Cartesian frame defining the vector spherical harmonics has to be chosen at the particle center. For a given origin, note also that the expansion-coefficient sets of the various fields depend on the orientation of the Cartesian frame.
  63. Our method, applied in the context of highly convergent light beams, is basically similar to the Bromwich approach used in the context of paraxial Gaussian beams .
  64. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).
  65. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in a aplanatic system." Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
    [CrossRef]
  66. A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. 138, B1561-B1565 (1965).
    [CrossRef]
  67. The factor [cos(θ)]1/2 is necessary to ensure energy conservation, as the lens, which obeys the sine condition, converts the incoming plane phase front to a spherical phase front in the image space (see for the vector case). This factor is usually set equal to unity since small angles are implicitly assumed in scalar theory.
  68. For small u=kρsin(θ) value the Bessel functions Jn>0(u) are close to zero. Moreover, the leading θ-function factor in the integrals In is the one entering the I0 expression. The electric field is thus accurately calculated in retaining only the main term KI0ex≈Escalar.
  69. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier Science, 1991).
  70. When the properties of the associated Legendre polynomials are used, the integral involving the function πn,1(θ) in Eq. is found to be equal to (θmax)2(2n+1)1/2/4 (only terms with the lowest order in θmax are kept in the calculations). An identical result is obtained for the integral involving the function τn,1(θ). The field amplitude at the focus, given by Eq. , is equal to Kπ(θmax)2. The plane-wave expansion Eq. in Appendix is thus recovered.
  71. K. T. Kim, "Symmetry relations of the translation coefficients of the spherical scalar and vector multipole fields," Prog. Electromagn. Res. 48, 45-66 (2004).
    [CrossRef]
  72. Y. Xu, "Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories," J. Comput. Phys. 139, 137-165 (1998).
    [CrossRef]
  73. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985 and 1991), Vols. I and II.
  74. In the present context the sine condition implies that the emergent ray meets the focal sphere (sphere of centre O and radius equal to f) in the image space at the same height at which the corresponding ray in the object space entered the objective lens; see , and , p. 180. This property is illustrated in Fig. (the gray sphere--in fact a small portion of it--is assumed to be of radius f).
  75. In general a0/2 is small as compared with the second term in the denominator of Eq. .
  76. All the calculations have been performed with a standard PC (Intel Pentium 4 processor with 3.2 GHz clock speed, computational code written in the C language). Typically the spectra or error parameter curves (as a function of λ) displayed in the manuscript involve roughly Nλ=280 wavelengths and a maximum order Nmax in the expansion of the fields equal to 15 [n=1,2,3,...,Nmax in Eqs. ]. Actually this large Nmax value is not necessary for particles of radius equal to or smaller than 100 nm. Besides the Lorenz-Mie coefficients [Eqs. ] the present theory requires the Bessel functions Jm(x=kρPsin(θ)) and Legendre polynomials Pnm(θ) to be computed for each θ value in the integral of Eq. . Typically 800-1200 θ values are retained, depending on the parameter kρP in the argument of the Bessel function Jm (and also on kzP in the exponential factor), though this quite large number is unnecessary in most cases of interest. Thanks to efficient recurrence relations obeyed by the Bessel functions and the Legendre polynomials , a spectrum is obtained within 3-4 s for an arbitrary particle location (ρP≠0; zP≠0). This time obviously depends on the Nλ, Nmax, and θ step values that are selected. This high speed is rooted in the simple mathematical ingredients (standard analytical functions) involved in the theory.

2007 (1)

P. Billaud, J. R. Huntzinger, E. Cottancin, J. Lermé, M. Pellarin, L. Arnaud, M. Broyer, N. Del Fatti, and F. Vallée, "Optical extinction spectroscopy of single silver nanoparticles," Eur. Phys. J. D 43, 271-274 (2007).
[CrossRef]

2006 (4)

O. L. Muskens, N. Del Fatti, F. Vallée, J. R. Huntzinger, P. Billaud, and M. Broyer, "Single metal nanoparticle absorption spectroscopy and optical characterization," Appl. Phys. Lett. 88, 063109 (2006).
[CrossRef]

O. L. Muskens, N. Del Fatti, and F. Vallée, "Femtosecond response of a single metal nanoparticle," Nano Lett. 6, 552-556 (2006).
[CrossRef] [PubMed]

E. Cottancin, G. Celep, J. Lermé, M. Pellarin, J. R. Huntzinger, J. L. Vialle, and M. Broyer, "Optical properties of noble metal clusters as a function of the size: comparison between experiments and a semi-quantal theory," Theor. Chem. Acc. 116, 514-523 (2006).
[CrossRef]

F. Moreno, F. Gonzales, and J. M. Saiz, "Plasmon spectroscopy of metallic nanoparticles above flat dielectric substrates," Opt. Lett. 31, 1902-1904 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (7)

K. T. Kim, "Symmetry relations of the translation coefficients of the spherical scalar and vector multipole fields," Prog. Electromagn. Res. 48, 45-66 (2004).
[CrossRef]

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, "Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy," Phys. Rev. Lett. 93, 037401 (2004).
[CrossRef] [PubMed]

J. Prikulis, F. Svedberg, M. Käll, J. Enger, K. Ramser, M. Goksör, and D. Hanstorp, "Optical spectroscopy of single trapped metal nanoparticles in solution," Nano Lett. 4, 115-118 (2004).
[CrossRef]

T. Kalkbrenner, U. Hakanson, and V. Sandoghdar, "Tomographic plasmon spectroscopy of a single gold nanoparticle," Nano Lett. 4, 2309-2314 (2004).
[CrossRef]

S. Berciaud, L. Cognet, G. A. Blab, and B. Lounis, "Photothermal heterodyne imaging of individual nonfluorescent nanoclusters and nanocrystals," Phys. Rev. Lett. 93, 257402 (2004).
[CrossRef]

T. Ziegler, C. Hendrich, F. Hubenthal, T. Vartanyan, and F. Träger, "Dephasing times of surface plasmon excitation in Au nanoparticles determined by persistent spectral hole burning," Chem. Phys. Lett. 386, 319-324 (2004).
[CrossRef]

A. Arbouet, D. Christofilos, N. Del Fatti, F. Vallée, J. R. Huntzinger, L. Arnaud, P. Billaud, and M. Broyer, "Direct measurement of the single-metal-cluster optical absorption," Phys. Rev. Lett. 93, 127401 (2004).
[CrossRef] [PubMed]

2003 (2)

J. J. Mock, D. R. Smith, and S. Schultz, "Local refractive index dependence of plasmon resonance spectra from individual nanoparticles," Nano Lett. 3, 485-491 (2003).
[CrossRef]

A. A. Mikhailovsky, M. A. Petruska, M. I. Stockman, and V. I. Klimov, "Broadband near-field interference spectroscopy of metal nanoparticles using a femtosecond white-light continuum," Opt. Lett. 28, 1686-1688 (2003).
[CrossRef] [PubMed]

2002 (3)

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

D. Boyer, P. Tamarat, A. Maali, B. Lounis, and M. Orrit, "Photothermal imaging of nanometer-sized metal particles among scatterers," Science 297, 1160-1163 (2002).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91, 5475-5488 (2002).
[CrossRef]

2001 (3)

L. Novotny, R. D. Grober, and K. Karrai, "Reflected image of a strongly focused spot," Opt. Lett. 26, 789-791 (2001).
[CrossRef]

T. Itoh, T. Asahi, and H. Masuhara, "Femtosecond light scattering spectroscopy of single gold nanoparticles," Appl. Phys. Lett. 79, 1667-1669 (2001).
[CrossRef]

Y. H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, "Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

2000 (4)

N. Nilius, N. Ernst, and H. J. Freund, "Photon emission spectroscopy of individual oxide-supported silver clusters in a scanning tunneling microscope," Phys. Rev. Lett. 84, 3994-3997 (2000).
[CrossRef] [PubMed]

S. Schultz, D. R. Smith, J. J. Mock, and D. A. Schultz, "Single-target molecule detection with nonbleaching multicolour optical immunolabels," Proc. Natl. Acad. Sci. USA 97, 996-1001 (2000).
[CrossRef] [PubMed]

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z. H. Chan, J. P. Spatz, and M. Möller, "Spectroscopy of single metallic nanoparticles using total internal reflection microscopy," Appl. Phys. Lett. 77, 2949-2951 (2000).
[CrossRef]

F. Stietz, J. Bosbach, T. Wenzel, T. Vartanyan, A. Goldmann, and F. Träger, "Decay times of surface plasmon excitation in metal nanoparticles by persistent spectral hole burning," Phys. Rev. Lett. 84, 5644-5647 (2000).
[CrossRef] [PubMed]

1998 (4)

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, "Surface-plasmon resonance in single metallic nanoparticles," Phys. Rev. Lett. 80, 4249-4252 (1998).
[CrossRef]

M. Perner, T. Klar, S. Grosse, U. Lemmer, G. von Plessen, W. Spirkl, and J. Feldmann, "Homogeneous line widths of surface plasmons in gold nanoparticles measured by femtosecond pump-and-probe and near-field optical spectroscopy," J. Lumin. 76, 181-184 (1998).
[CrossRef]

Y. Xu, "Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories," J. Comput. Phys. 139, 137-165 (1998).
[CrossRef]

P. Török, P. D. Higdon, R. Juskaitis, and T. Wilson, "Optimising the image contrast of conventional and confocal microscopes imaging finite sized spherical gold scatterers," Opt. Commun. 155, 335-341 (1998).
[CrossRef]

1996 (1)

1995 (3)

1993 (2)

H. Hövel, S. Fritz, A. Hilger, and U. Kreibig, "Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping," Phys. Rev. B 48, 18178-18188 (1993).
[CrossRef]

B. R. Johnson, "Light scattering from a spherical particle on a conducting plane: I. Normal incidence: errata," J. Opt. Soc. Am. A 10, 766 (1993).
[CrossRef]

1992 (1)

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

1988 (2)

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]

1987 (1)

1986 (1)

1985 (1)

G. Gouesbet, G. Grehan, and B. Maheu, "Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism," J. Opt. 16, 83-93 (1985).
[CrossRef]

1983 (1)

1979 (1)

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

1978 (1)

1977 (3)

1975 (2)

1972 (1)

1965 (1)

A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

1961 (1)

F. T. Gucker and J. J. Egan, "Measurement of the angular variation of light scattering from single aerosol droplets," J. Colloid Sci. 16, 68-84 (1961).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in a aplanatic system." Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

1908 (1)

G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908).
[CrossRef]

1890 (1)

L. V. Lorenz, "Lysvevægelsen i og uden for en af plane lysbølger belyst kugle," K. Dan. Vidensk. Selsk. Forth. 6, 1-62 (1890).

Ann. Phys. (1)

G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen," Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (3)

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z. H. Chan, J. P. Spatz, and M. Möller, "Spectroscopy of single metallic nanoparticles using total internal reflection microscopy," Appl. Phys. Lett. 77, 2949-2951 (2000).
[CrossRef]

O. L. Muskens, N. Del Fatti, F. Vallée, J. R. Huntzinger, P. Billaud, and M. Broyer, "Single metal nanoparticle absorption spectroscopy and optical characterization," Appl. Phys. Lett. 88, 063109 (2006).
[CrossRef]

T. Itoh, T. Asahi, and H. Masuhara, "Femtosecond light scattering spectroscopy of single gold nanoparticles," Appl. Phys. Lett. 79, 1667-1669 (2001).
[CrossRef]

Chem. Phys. Lett. (1)

T. Ziegler, C. Hendrich, F. Hubenthal, T. Vartanyan, and F. Träger, "Dephasing times of surface plasmon excitation in Au nanoparticles determined by persistent spectral hole burning," Chem. Phys. Lett. 386, 319-324 (2004).
[CrossRef]

Eur. Phys. J. D (1)

P. Billaud, J. R. Huntzinger, E. Cottancin, J. Lermé, M. Pellarin, L. Arnaud, M. Broyer, N. Del Fatti, and F. Vallée, "Optical extinction spectroscopy of single silver nanoparticles," Eur. Phys. J. D 43, 271-274 (2007).
[CrossRef]

J. Appl. Phys. (3)

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. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989).
[CrossRef]

A. Rohrbach and E. H. K. Stelzer, "Three-dimensional position detection of optically trapped dielectric particles," J. Appl. Phys. 91, 5475-5488 (2002).
[CrossRef]

J. Colloid Sci. (1)

F. T. Gucker and J. J. Egan, "Measurement of the angular variation of light scattering from single aerosol droplets," J. Colloid Sci. 16, 68-84 (1961).
[CrossRef]

J. Comput. Phys. (1)

Y. Xu, "Efficient evaluation of vector translation coefficients in multiparticle light-scattering theories," J. Comput. Phys. 139, 137-165 (1998).
[CrossRef]

J. Lumin. (1)

M. Perner, T. Klar, S. Grosse, U. Lemmer, G. von Plessen, W. Spirkl, and J. Feldmann, "Homogeneous line widths of surface plasmons in gold nanoparticles measured by femtosecond pump-and-probe and near-field optical spectroscopy," J. Lumin. 76, 181-184 (1998).
[CrossRef]

J. Opt. (1)

G. Gouesbet, G. Grehan, and B. Maheu, "Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism," J. Opt. 16, 83-93 (1985).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (6)

J. Phys. Chem. B (1)

Y. H. Liau, A. N. Unterreiner, Q. Chang, and N. F. Scherer, "Ultrafast dephasing of single nanoparticles studied by two-pulse second-order interferometry," J. Phys. Chem. B 105, 2135-2142 (2001).
[CrossRef]

K. Dan. Vidensk. Selsk. Forth. (1)

L. V. Lorenz, "Lysvevægelsen i og uden for en af plane lysbølger belyst kugle," K. Dan. Vidensk. Selsk. Forth. 6, 1-62 (1890).

Nano Lett. (4)

O. L. Muskens, N. Del Fatti, and F. Vallée, "Femtosecond response of a single metal nanoparticle," Nano Lett. 6, 552-556 (2006).
[CrossRef] [PubMed]

T. Kalkbrenner, U. Hakanson, and V. Sandoghdar, "Tomographic plasmon spectroscopy of a single gold nanoparticle," Nano Lett. 4, 2309-2314 (2004).
[CrossRef]

J. J. Mock, D. R. Smith, and S. Schultz, "Local refractive index dependence of plasmon resonance spectra from individual nanoparticles," Nano Lett. 3, 485-491 (2003).
[CrossRef]

J. Prikulis, F. Svedberg, M. Käll, J. Enger, K. Ramser, M. Goksör, and D. Hanstorp, "Optical spectroscopy of single trapped metal nanoparticles in solution," Nano Lett. 4, 115-118 (2004).
[CrossRef]

Opt. Commun. (1)

P. Török, P. D. Higdon, R. Juskaitis, and T. Wilson, "Optimising the image contrast of conventional and confocal microscopes imaging finite sized spherical gold scatterers," Opt. Commun. 155, 335-341 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. (1)

A. Boivin and E. Wolf, "Electromagnetic field in the neighborhood of the focus of a coherent beam," Phys. Rev. 138, B1561-B1565 (1965).
[CrossRef]

Phys. Rev. A (3)

H. Chew, M. Kerker, and D. D. Cooke, "Electromagnetic scattering by a dielectric sphere in a diverging radiation field," Phys. Rev. A 16, 320-323 (1977).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Phys. Rev. B (1)

H. Hövel, S. Fritz, A. Hilger, and U. Kreibig, "Width of cluster plasmon resonances: bulk dielectric functions and chemical interface damping," Phys. Rev. B 48, 18178-18188 (1993).
[CrossRef]

Phys. Rev. Lett. (7)

N. Nilius, N. Ernst, and H. J. Freund, "Photon emission spectroscopy of individual oxide-supported silver clusters in a scanning tunneling microscope," Phys. Rev. Lett. 84, 3994-3997 (2000).
[CrossRef] [PubMed]

S. Berciaud, L. Cognet, G. A. Blab, and B. Lounis, "Photothermal heterodyne imaging of individual nonfluorescent nanoclusters and nanocrystals," Phys. Rev. Lett. 93, 257402 (2004).
[CrossRef]

F. Stietz, J. Bosbach, T. Wenzel, T. Vartanyan, A. Goldmann, and F. Träger, "Decay times of surface plasmon excitation in metal nanoparticles by persistent spectral hole burning," Phys. Rev. Lett. 84, 5644-5647 (2000).
[CrossRef] [PubMed]

A. Arbouet, D. Christofilos, N. Del Fatti, F. Vallée, J. R. Huntzinger, L. Arnaud, P. Billaud, and M. Broyer, "Direct measurement of the single-metal-cluster optical absorption," Phys. Rev. Lett. 93, 127401 (2004).
[CrossRef] [PubMed]

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002).
[CrossRef] [PubMed]

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, "Surface-plasmon resonance in single metallic nanoparticles," Phys. Rev. Lett. 80, 4249-4252 (1998).
[CrossRef]

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, "Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy," Phys. Rev. Lett. 93, 037401 (2004).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

S. Schultz, D. R. Smith, J. J. Mock, and D. A. Schultz, "Single-target molecule detection with nonbleaching multicolour optical immunolabels," Proc. Natl. Acad. Sci. USA 97, 996-1001 (2000).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in a aplanatic system." Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Prog. Electromagn. Res. (1)

K. T. Kim, "Symmetry relations of the translation coefficients of the spherical scalar and vector multipole fields," Prog. Electromagn. Res. 48, 45-66 (2004).
[CrossRef]

Science (1)

D. Boyer, P. Tamarat, A. Maali, B. Lounis, and M. Orrit, "Photothermal imaging of nanometer-sized metal particles among scatterers," Science 297, 1160-1163 (2002).
[CrossRef] [PubMed]

Theor. Chem. Acc. (1)

E. Cottancin, G. Celep, J. Lermé, M. Pellarin, J. R. Huntzinger, J. L. Vialle, and M. Broyer, "Optical properties of noble metal clusters as a function of the size: comparison between experiments and a semi-quantal theory," Theor. Chem. Acc. 116, 514-523 (2006).
[CrossRef]

Z. Phys. D (1)

T. Götz, W. Hoheisel, M. Vollmer, and F. Träger, "Characterization of large supported metal clusters by optical spectroscopy," Z. Phys. D 33, 133-141 (1995).
[CrossRef]

Other (18)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

M. Born and E. Wolf, Principles of Optics, 7th. ed. (Cambridge U. Press, 1999).

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

C.Dupas, Ph.Houdy, and M.Lahmani, eds., Nanosciences, Nanotechnologies and Nanophysics (Springer, 2006).

R.Kelsall, I.W.Hamley, and M.Geoghegan, eds., Nanoscale Science and Technology (Wiley, 2006).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985 and 1991), Vols. I and II.

In the present context the sine condition implies that the emergent ray meets the focal sphere (sphere of centre O and radius equal to f) in the image space at the same height at which the corresponding ray in the object space entered the objective lens; see , and , p. 180. This property is illustrated in Fig. (the gray sphere--in fact a small portion of it--is assumed to be of radius f).

In general a0/2 is small as compared with the second term in the denominator of Eq. .

All the calculations have been performed with a standard PC (Intel Pentium 4 processor with 3.2 GHz clock speed, computational code written in the C language). Typically the spectra or error parameter curves (as a function of λ) displayed in the manuscript involve roughly Nλ=280 wavelengths and a maximum order Nmax in the expansion of the fields equal to 15 [n=1,2,3,...,Nmax in Eqs. ]. Actually this large Nmax value is not necessary for particles of radius equal to or smaller than 100 nm. Besides the Lorenz-Mie coefficients [Eqs. ] the present theory requires the Bessel functions Jm(x=kρPsin(θ)) and Legendre polynomials Pnm(θ) to be computed for each θ value in the integral of Eq. . Typically 800-1200 θ values are retained, depending on the parameter kρP in the argument of the Bessel function Jm (and also on kzP in the exponential factor), though this quite large number is unnecessary in most cases of interest. Thanks to efficient recurrence relations obeyed by the Bessel functions and the Legendre polynomials , a spectrum is obtained within 3-4 s for an arbitrary particle location (ρP≠0; zP≠0). This time obviously depends on the Nλ, Nmax, and θ step values that are selected. This high speed is rooted in the simple mathematical ingredients (standard analytical functions) involved in the theory.

The factor [cos(θ)]1/2 is necessary to ensure energy conservation, as the lens, which obeys the sine condition, converts the incoming plane phase front to a spherical phase front in the image space (see for the vector case). This factor is usually set equal to unity since small angles are implicitly assumed in scalar theory.

For small u=kρsin(θ) value the Bessel functions Jn>0(u) are close to zero. Moreover, the leading θ-function factor in the integrals In is the one entering the I0 expression. The electric field is thus accurately calculated in retaining only the main term KI0ex≈Escalar.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, Field Representation and Introduction to Scattering (Elsevier Science, 1991).

When the properties of the associated Legendre polynomials are used, the integral involving the function πn,1(θ) in Eq. is found to be equal to (θmax)2(2n+1)1/2/4 (only terms with the lowest order in θmax are kept in the calculations). An identical result is obtained for the integral involving the function τn,1(θ). The field amplitude at the focus, given by Eq. , is equal to Kπ(θmax)2. The plane-wave expansion Eq. in Appendix is thus recovered.

Strictly, the amplitude E0 should be corrected to take into account the (weak) energy losses due to reflexion and absorption within the lens. If these losses does not depend noticeably on the azimuthal angle phiv, these effects can be easily included in our modeling by modifying the weighting factor [cos(θ)]1/2 appropriately.

The center of the Cartesian frame defining the vector spherical harmonics has to be chosen at the particle center. For a given origin, note also that the expansion-coefficient sets of the various fields depend on the orientation of the Cartesian frame.

Our method, applied in the context of highly convergent light beams, is basically similar to the Bromwich approach used in the context of paraxial Gaussian beams .

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).

In the Davis's approach the electromagnetic field (E,H), derived from a linearly polarized vector potential A, is expressed as a series upon increasing powers of the beam-confinement parameter s=(kω0)−1(s⩽1), where k is the wave vector and ω0 the beam-waist radius.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of a typical single-particle experimental setup. An incoming monochromatic plane wave, linearly polarized along the x-axis direction, is focused onto a spherical nanoparticle by a perfect high-numerical-aperture objective lens. The light beam emerging from the objective lens is an axially symmetric cone (maximum polar angle θ max ). The origin of the laboratory coordinate system O x y z is taken at the focus. The transmitted light is collected by a second objective lens; r P is the position vector of the spherical particle.

Fig. 2
Fig. 2

Normalized E x component of the electric field along the z axis ( x = y = 0 ) for three different wavelengths ( E y = E z = 0 along the z axis and E x ( z ) = E x ( z ) ). The E x component is normalized relative to its value E foc [Eq. (11)] at the focus O ( x = y = z = 0 ) .

Fig. 3
Fig. 3

Normalized E x and E z components of the electric field along (a), (b) the x axis ( y = z = 0 ) and (c), (d) an axis parallel to the x axis ( y = 0 , z = 500 nm ), for four different wavelengths λ = 200 , 400, 600, 800 nm ( E y = 0 along both axes). The E x and E z components are normalized relative to E foc [Eq. (11)], the magnitude of the electric field at the focus O ( x = y = z = 0 ) .

Fig. 4
Fig. 4

Upper panels, absorption (black curves) and scattering (gray dashed curves and black dotted curves for silver and Si O 2 , respectively) cross sections calculated within the Mie theory for silver and Si O 2 glass particles of radii R = 20 , 50, 100 nm , as a function of the wavelength. The scattering spectrum for a Si O 2 sphere of radius R = 20 nm , which is very weak, is not shown. Lower panels, corresponding error parameters Δ [Eq. (41)] for absorption (thick black curves), scattering (grey dashed and black dotted curves for silver and Si O 2 , respectively), and extinction (thin black curves). The particle is located at the focus O. Inset, lower right-hand figure, wavelength-dependent real ( n ) and imaginary ( k ) indexes of the complex refractive index of silver ( N Ag = n + i k ) [73]. In the spectral range displayed the real refractive index of Si O 2 glass is almost constant and is of the order of 1.5.

Fig. 5
Fig. 5

Error parameters Δ [Eq. (41)] for absorption (thick black curves), scattering (gray dashed curves), and extinction (thin black curves), for a silver particle of radius R = 50 nm located in the focal plane, on the y axis ( z P = 0 , ϕ P = π 2 ), as a function of the wavelength. The particle–focus distance ρ P is indicated.

Fig. 6
Fig. 6

Correlation existing between the local field intensity E ( r P ) 2 and the error parameter Δ [Eq. (41)] for a silver particle of radius R = 50 nm located in the focal plane, on the x axis ( z P = 0 , ϕ P = 0 ), as a function of the particle–focus distance ρ P , for various wavelengths in the visible–NUV spectral range. Upper panels, normalized intensity E ( ρ P ) E foc 2 along the x axis [ E foc is given in Eq. (11)]. Lower panels, corresponding error parameters Δ [Eq. (41)] for absorption (thick black curves), scattering (gray dashed curves), and extinction (thin black curves).

Fig. 7
Fig. 7

Anisotropy parameters a i = Y i X i [see Eq. (37)] characterizing the optical response of a silver particle of radius R = 50 nm located in the focal plane ( z P = 0 ) , for (b) wavelengths λ = 200 nm and (d) λ = 600 nm , as a function of the particle–focus distance ρ P . Black curves, absorption; gray dashed curves, scattering. (a), (c), Anisotropy parameters a E = Y E X E characterizing the field intensity [Eq. (43)] for (a) λ = 200 nm and (c) 600 nm .

Fig. 8
Fig. 8

Error parameters Δ i conv for absorption (thick black curves), scattering (gray dashed curves), and extinction (thin black curves), for a silver particle of radius R = 50 nm located in the focal plane, on the x axis ( z P = 0 , ϕ P = 0 ), as a function of the particle–focus distance ρ P , for wavelengths (b) λ = 200 nm and (d) λ = 600 nm . (a), (c), Normalized intensity ( E ( ρ P ) E foc 2 ) conv along the x axis [ E foc is given in Eq. (11)]. The particle response and the intensity distribution have been calculated assuming a pinhole image radius R c equal to 150 nm (see Subsection 3.B).

Fig. 9
Fig. 9

(b), (d) Anisotropy parameters a i conv = Y i conv X i conv [see Eq. (45)] characterizing the optical response of a silver particle of radius R = 50 nm located in the focal plane ( z P = 0 ) , for wavelengths (b) λ = 200 nm and (d) λ = 600 nm , as a function of the coordinate ρ P (distance from the z axis). The black curves correspond to absorption, the gray dashed curves to scattering. (a), (c) Anisotropy parameters a E conv = Y E conv X E conv characterizing the field intensity [see Eq. (46)] for (a) λ = 200 nm and (c) λ = 600 nm as a function of the coordinate ρ. The particle response and the intensity distribution have been calculated in assuming a pinhole image radius R c equal to 150 nm (see Subsection 3.B). In (a) and (c) gray dashed a E ( ρ P ) curves are results for R c = 250 nm .

Fig. 10
Fig. 10

Modeling of a SMS experiment for a silver particle of radius R = 50 nm oscillating along the ϕ = π 4 direction in the focal plane ( z P = 0 ) (see Subsection 3.B). The four curves in each figure correspond to different oscillatory motions [Eq. (48)]: r 0 = 400 nm , A = 300 nm (thin solid black curve), A = 400 nm (thick solid black curve), A = 500 nm (gray curve), A = 600 nm (dashed gray curve). (a) Normalized experimental signal S as a function of the wavelength. (b) Error parameter Δ SMS . (c) Anisotropy parameter a SMS .

Fig. 11
Fig. 11

Modeling of a SMS experiment for a silver particle of radius R = 50 nm oscillating along the ϕ = π 4 direction in the focal plane ( z P = 0 ) (see Subsection 3.B). The motion parameters are r 0 = A = 400 nm [Eq. (48)]. The three curves in each figure correspond to different pinhole image radii: R c = 0 (thick black curve), R c = 150 nm (gray dashed curve), R c = 250 nm (thin black curve). (a) Normalized experimental signal S as a function of the wavelength. (b) Error parameter Δ SMS . (c) Anisotropy parameter a SMS .

Equations (91)

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E ( r ) = E ( k ) exp ( i k r ) d k x d k y = 0 2 π θ min θ max E ( k ) exp ( i k r ) k 2 sin ( θ ) cos ( θ ) d θ d ϕ .
E ( k ) = i λ 0 ( 2 π ) 2 n 1 cos ( θ ) f exp ( i k f ) E f ( θ , ϕ ) ,
E f ( θ , ϕ ) = E 0 cos ( θ ) { [ cos ( θ ) cos 2 ( ϕ ) + sin 2 ( ϕ ) ] e x + [ cos ( θ ) 1 ] cos ( ϕ ) sin ( ϕ ) e y sin ( θ ) cos ( ϕ ) e z } .
E x ( z , ρ , ϕ M ) = K [ I 0 + I 2 cos ( 2 ϕ M ) ] ,
E y ( z , ρ , ϕ M ) = K I 2 sin ( 2 ϕ M ) ,
E z ( z , ρ , ϕ M ) = 2 i K I 1 cos ( ϕ M ) ,
I 0 ( ρ , z ) = θ min θ max d θ sin ( θ ) cos ( θ ) [ 1 + cos ( θ ) ] J 0 [ k ρ sin ( θ ) ] exp [ i k z cos ( θ ) ] ,
I 1 ( ρ , z ) = θ min θ max d θ sin ( θ ) 2 cos ( θ ) J 1 [ k ρ sin ( θ ) ] exp [ i k z cos ( θ ) ] ,
I 2 ( ρ , z ) = θ min θ max d θ sin ( θ ) cos ( θ ) [ 1 cos ( θ ) ] J 2 [ k ρ sin ( θ ) ] exp [ i k z cos ( θ ) ] ,
K = ( i k 2 ) f exp ( i k f ) E 0 .
E foc = K θ min θ max d θ sin ( θ ) cos ( θ ) [ 1 + cos ( θ ) ] e x .
E scalar ( r ) = 2 K θ min θ max d θ sin ( θ ) cos ( θ ) J 0 [ k ρ sin ( θ ) ] exp [ i k z cos ( θ ) ] e x .
E scalar ( r ) K sin ( θ max ) 2 2 J 1 [ k ρ sin ( θ max ) ] k ρ sin ( θ max ) exp ( i k z ) e x = K ( R f ) 2 2 J 1 [ k R ρ f ] [ k R ρ f ] exp ( i k z ) e x ,
E f ( θ , ϕ ) = E 0 cos ( θ ) [ cos ( ϕ ) e θ sin ( ϕ ) e ϕ ] .
E ( r ) = K π θ min θ max 0 2 π d ϕ d θ sin ( θ ) cos ( θ ) [ cos ( ϕ ) e θ sin ( ϕ ) e ϕ ] exp ( i k r ) .
exp ( i k r ) e θ = n = 1 m = n n [ e n m ( p ) M n m + u n m ( p ) N n m ] ,
exp ( i k r ) e ϕ = n = 1 m = n n [ e n m ( s ) M n m + u n m ( s ) N n m ] ,
e n m ( p ) = i n 1 π n m ( θ ) exp ( i m ϕ ) ,
u n m ( p ) = i n 1 τ n m ( θ ) exp ( i m ϕ ) ,
e n m ( s ) = i n τ n m ( θ ) exp ( i m ϕ ) ,
u n m ( s ) = i n π n m ( θ ) exp ( i m ϕ ) .
E ( r ) = n = 1 m = n n [ p n m M n m + q n m N n m ] ,
p n m = K i n 1 π θ min θ max 0 2 π d θ d ϕ sin ( θ ) cos ( θ ) × [ cos ( ϕ ) π n m ( θ ) + i sin ( ϕ ) τ n m ( θ ) ] exp ( i m ϕ ) ,
q n m = K i n 1 π θ min θ max 0 2 π d θ d ϕ sin ( θ ) cos ( θ ) × [ cos ( ϕ ) τ n m ( θ ) + i sin ( ϕ ) π n m ( θ ) ] exp ( i m ϕ ) .
E ( r ) = n = 1 p n , 1 [ M n , 1 ( r ) + M n , 1 ( r ) + N n , 1 ( r ) N n , 1 ( r ) ] ,
p n , 1 = K i n 1 θ min θ max d θ sin ( θ ) cos ( θ ) [ π n , 1 ( θ ) + τ n , 1 ( θ ) ] .
E ( r ) = K π θ min θ max 0 2 π d θ d ϕ sin ( θ ) cos ( θ ) [ cos ( ϕ ) e θ sin ( ϕ ) e ϕ ] exp ( i k r ) exp ( i k r P ) ,
p n m = K i n 1 π θ min θ max 0 2 π d θ d ϕ sin ( θ ) cos ( θ ) [ cos ( ϕ ) π n m ( θ ) + i sin ( ϕ ) τ n m ( θ ) ] exp ( i m ϕ ) exp [ i k ( θ , ϕ ) r P ] ,
q n m = K i n 1 π θ min θ max 0 2 π d θ d ϕ sin ( θ ) cos ( θ ) [ cos ( ϕ ) τ n m ( θ ) + i sin ( ϕ ) π n m ( θ ) ] exp ( i m ϕ ) exp [ i k ( θ , ϕ ) r P ] .
k ( θ , ϕ ) r P = k ρ P sin ( θ ) cos ( ϕ ϕ P ) + k z P cos ( θ ) .
I ( m , α , ϕ P ) = 0 2 π d ϕ g ( ϕ ) exp ( i m ϕ ) exp [ i α cos ( ϕ ϕ P ) ] ,
p n m = K i n m θ min θ max d θ sin ( θ ) cos ( θ ) exp [ i k z P cos ( θ ) ] × [ C m ( r P , θ ) π n m ( θ ) + D m ( r P , θ ) τ n m ( θ ) ] ,
C m ( r P , θ ) = exp [ i ( 1 m ) ϕ P ] J 1 m [ k ρ P sin ( θ ) ] exp [ i ( 1 + m ) ϕ P ] J 1 m [ k ρ P sin ( θ ) ] ,
D m ( r P , θ ) = exp [ i ( 1 m ) ϕ P ] J 1 m [ k ρ P sin ( θ ) ] + exp [ i ( 1 + m ) ϕ P ] J 1 m [ k ρ P sin ( θ ) ] .
p n , m 2 + p n , m 2 = K 2 [ X n , m ( ρ P , z P ) + Y n , m ( ρ P , z P ) cos ( 2 ϕ P ) ] ,
q n , m 2 + q n , m 2 = K 2 [ X n , m ( ρ P , z P ) Y n , m ( ρ P , z P ) cos ( 2 ϕ P ) ] .
W ext ( ρ P , z P , ϕ P ) = 2 π K 2 k ω μ 0 [ X ext ( ρ P , z P ) + Y ext ( ρ P , z P ) cos ( 2 ϕ P ) ] ,
X ext = n = 1 [ Re ( α n ) + Re ( β n ) ] m = 0 n X n , m ,
Y ext = n = 1 [ Re ( α n ) Re ( β n ) ] m = 0 n Y n , m .
W i Mie ( r P ) = I inc local ( r P ) C i Mie = k 2 ω μ 0 E ( r P ) 2 C i Mie ,
Δ i = W i W i Mie W i Mie = 1 C i Mie [ W i I inc local C i Mie ] ,
E ( r ) 2 = K 2 [ X E ( ρ , z ) + Y E ( ρ , z ) cos ( 2 ϕ ) ] .
a E = Y E X E = 2 [ Re ( I 0 I 2 * ) + I 1 2 ] I 0 2 + I 2 2 + 2 I 1 2 .
W i conv ( ρ P , z P , ϕ P ) = D W i ( ρ , z P , ϕ ) f c ( r r P ) ρ d ρ d ϕ ,
W ext conv ( ρ P , z P , ϕ P ) = 2 π K 2 k ω μ 0 [ X ext conv ( ρ P , z P , R c ) + Y ext conv ( ρ P , z P , R c ) cos ( 2 ϕ P ) ] .
( E ( r ) 2 ) conv = K 2 [ X E conv ( ρ , z , R c ) + Y E conv ( ρ , z , R c ) cos ( 2 ϕ ) ] .
W tr = P inc W ext + W s coll ,
ρ P ( t ) = r 0 + A cos ( 2 π ν 0 t ) .
W tr ( t ) = P inc + W 0 [ X ext conv ( ρ P ( t ) , z P , R c ) + Y ext conv ( ρ P ( t ) , z P , R c ) cos ( 2 ϕ P ) ] ,
P inc = I inc π f 2 [ sin 2 ( θ max ) sin 2 ( θ min ) ] = W 0 [ sin 2 ( θ max ) sin 2 ( θ min ) ] ,
S = a 1 a 0 2 + [ sin 2 ( θ max ) sin 2 ( θ min ) ] ,
a k = 2 ν 0 T 0 2 T 0 2 Z ( t ) cos ( 2 π k ν 0 ) d t .
W ext Mie ( t ) = k 2 4 π C ext W 0 ( E [ r P ( t ) ] / K 2 ) conv ,
S = b 1 C ext ( b 0 2 ) C ext ( 4 π k 2 ) [ sin 2 ( θ max ) sin 2 ( θ min ) ] ,
C ext = 4 π k 2 [ sin 2 ( θ max ) sin 2 ( θ min ) ] S ( b 0 2 ) S b 1 .
M n m ( r , k ) = K n m × [ r z n ( k r ) P n m [ cos ( θ ) ] exp ( i m ϕ ) ] ,
N n m ( r , k ) = 1 k × M n m ( r ) ,
K n m = [ ( 2 n + 1 ) n ( n + 1 ) ( n m ) ! ( n + m ) ! ] 1 2 .
M n m ( r , k ) = z n ( k r ) X n m ( θ , ϕ ) ,
N n m ( r , k ) = 1 k r { r [ r z n ( k r ) ] Y n m ( θ , ϕ ) + z n ( k r ) Z n m ( θ , ϕ ) } ,
X n m ( θ , ϕ ) = [ i π n m ( θ ) e θ τ n m ( θ ) e ϕ ] exp ( i m ϕ ) ,
Y n m ( θ , ϕ ) = [ τ n m ( θ ) e θ + i π n m ( θ ) e ϕ ] exp ( i m ϕ ) ,
Z n m ( θ , ϕ ) = [ n ( n + 1 ) K n m P n m [ cos ( θ ) ] e r ] exp ( i m ϕ ) .
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 ) ] .
0 2 π d ϕ 0 π d θ sin ( θ ) X n , m ( θ , ϕ ) X n , m * ( θ , ϕ ) = 4 π δ n , n δ m , m ,
0 2 π d ϕ 0 π d θ sin ( θ ) Y n , m ( θ , ϕ ) Y n , m * ( θ , ϕ ) = 4 π δ n , n δ m , m ,
0 2 π d ϕ 0 π d θ sin ( θ ) Z n , m ( θ , ϕ ) Z n , m * ( θ , ϕ ) = 4 π n ( n + 1 ) δ n , n δ m , m .
M e n m B ( r , k ) = × [ r z n ( k r ) P n m [ cos ( θ ) ] cos ( m ϕ ) ] ,
N e n m B ( r , k ) = 1 k × M e n m B ( r ) ,
M o n m B ( r , k ) = × [ r z n ( k r ) P n m [ cos ( θ ) ] sin ( m ϕ ) ] ,
N o n m B ( r , k ) = 1 k × M o n m B ( r ) ,
M e n m B = 1 2 K n m [ M n m + ( 1 ) m M n , m ] ,
M o n m B = i 1 2 K n m [ M n m ( 1 ) m M n , m ] ,
N e n m B = 1 2 K n m [ N n m + ( 1 ) m N n , m ] ,
N o n m B = i 1 2 K n m [ N n m ( 1 ) m N n , m ] .
E inc ( r ) = n = 1 m = n n [ p n m M n m ( r , k ) + q n m N n m ( r , k ) ] ,
E s ( r ) = n = 1 m = n n [ e n m M n m + ( r , k ) + f n m N n m + ( r , k ) ] ,
E int ( r ) = n = 1 m = n n [ c n m M n m ( r , k 1 ) + d n m N n m ( r , k 1 ) ] ,
e n , m = j n ( ρ 1 ) [ ρ j n ( ρ ) ] j n ( ρ ) [ ρ 1 j n ( ρ 1 ) ] h n + ( ρ ) [ ρ 1 j n ( ρ 1 ) ] j n ( ρ 1 ) [ ρ h n + ( ρ ) ] p n , m = α n p n , m ,
f n , m = j n ( ρ ) [ ρ 1 j n ( ρ 1 ) ] m 1 2 j n ( ρ 1 ) [ ρ j n ( ρ ) ] m 1 2 j n ( ρ 1 ) [ ρ h n + ( ρ ) ] h n + ( ρ ) [ ρ 1 j n ( ρ 1 ) ] q n , m = β n q n , m ,
c n , m = h n + ( ρ ) [ ρ j n ( ρ ) ] j n ( ρ ) [ ρ h n + ( ρ ) ] h n + ( ρ ) [ ρ 1 j n ( ρ 1 ) ] j n ( ρ 1 ) [ ρ h n + ( ρ ) ] p n , m = γ n p n , m ,
d n , m = m 1 j n ( ρ ) [ ρ h n + ( ρ ) ] m 1 h n + ( ρ ) [ ρ j n ( ρ ) ] m 1 2 j n ( ρ 1 ) [ ρ h n + ( ρ ) ] h n + ( ρ ) [ ρ 1 j n ( ρ 1 ) ] q n , m = δ n q n , m ,
W s = 2 π k ω μ 0 n , m [ e n , m 2 + f n , m 2 ] = 2 π k ω μ 0 n , m [ α n 2 p n m 2 + β n 2 q n m 2 ] ,
W ext = 2 π k ω μ 0 n , m [ Re ( p n m e n m * + q n m f n m * ) ] = 2 π k ω μ 0 n m [ Re ( α n ) p n m 2 + Re ( β n ) q n m 2 ] ,
W abs = W ext W s .
E 0 exp ( i k r ) e x = E 0 n = 1 [ ( 2 n + 1 ) 1 2 2 i n + 1 ] × [ M n , 1 + M n , 1 + N n , 1 N n , 1 ]
I inc = E 0 2 k 2 ω μ 0 .
C s = 4 π k 2 n , m [ α n 2 p n m 2 + β n 2 q n m 2 ] ,
C ext = 4 π k 2 n , m [ Re ( α n ) p n m 2 + Re ( β n ) q n m 2 ] ,
C abs = C ext C s ,

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