Abstract

It is shown that when an electromagnetic wave with some degree of amplitude rolloff in the transverse direction is scattered by a spherical particle, the optical theorem is not valid. For such shaped beams the extinction cross section may be written as an infinite series in powers of the reciprocal of the beam width. The imaginary part of the forward-scattering amplitude is shown to be the first term in this series. Two approximations to the extinction cross section are presented for the special case of Gaussian-beam scattering. The first one is based on the dominance of diffraction in the forward direction for w0a, where w0 is the beam half-width and a is the target particle radius. The second approximation, valid for w0a, is based on transmission-compensating field interference.

© 1995 Optical Society of America

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References

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  1. A. Messiah, Quantum Mechanics (Wiley, New York, 1966), Vol. 2, pp. 866–867.
  2. E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 505.
  3. R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
    [CrossRef]
  4. A. Messiah, Quantum Mechanics (Wiley, New York, 1966), Vol. 1, pp. 372–376.
  5. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  6. F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the generalized Lorenz–Mie theory,” Appl. Opt. 31, 2942–2951 (1992).
    [CrossRef] [PubMed]
  7. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass.1979), pp. 372–378.
  8. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  9. G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [CrossRef]
  10. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  11. G. Gréhan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  12. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  13. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.
  15. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 45.
  16. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 101.
  17. Ref. 15, p. 49.
  18. Ref. 16, pp. 69–76.
  19. G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
    [CrossRef]
  20. G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef] [PubMed]
  21. J. A. Lock, “The contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  22. J. A. Lock, G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beam,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  23. Higher-order Davis beam models that accurately describe tightly focused Gaussian beams are given in Ref. 13 and in J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  24. J. T. Hodges, G. Gouesbet, G. Gréhan, C. Presser, “Forward scattering of a Gaussian beam by a nonabsorbing sphere,” Appl. Opt. 34, 2120–2132 (1995).
    [CrossRef] [PubMed]
  25. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  26. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  27. J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
    [CrossRef]
  28. An alternative derivation of the WBA under the less restrictive assumption al+ bl= 1 will be given by one of us (G. Gouesbet) elsewhere.
  29. S. I. Rubinow, T. T. Wu, “First correction to the geometric-optics cross section from cylinders and spheres,” J. Appl. Phys. 27, 1032–1039 (1956).
    [CrossRef]
  30. T. T. Wu, “High-frequency scattering,” Phys. Rev. 104, 1201–1212 (1956).
    [CrossRef]
  31. P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic behavior of the Mie-scattering amplitude,” J. Opt. Soc. Am. A 3, 256–258 (1986).
    [CrossRef]
  32. H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
    [CrossRef]
  33. L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
    [CrossRef]
  34. J. A. Lock, “Interpretation of extinction in Gaussian beam scattering,” J. Opt. Soc. Am. A 12, 929–938 (1995).
    [CrossRef]
  35. P. Chýlek, J. Zhan, “Interference structure of the Mie extinction cross section,” J. Opt. Soc. Am. A 6, 1846–1851 (1989).
    [CrossRef]
  36. J. A. Lock, L. Yang, “Interference between diffraction and transmission in the Mie extinction efficiency,” J. Opt. Soc. Am. A 8, 1132–1134 (1991).
    [CrossRef]
  37. Ref. 14, pp. 183–191.
  38. P. Chýlek, J. D. Klett, “Absorption and scattering of electromagnetic radiation by prismatic columns: anomalous diffraction approximation,” J. Opt. Soc. Am. A 8, 1713–1720 (1991).
    [CrossRef]

1995 (3)

1994 (1)

1993 (2)

J. A. Lock, “The contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
[CrossRef]

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

1992 (1)

1991 (2)

1989 (3)

P. Chýlek, J. Zhan, “Interference structure of the Mie extinction cross section,” J. Opt. Soc. Am. A 6, 1846–1851 (1989).
[CrossRef]

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

Higher-order Davis beam models that accurately describe tightly focused Gaussian beams are given in Ref. 13 and in J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

1988 (3)

1986 (2)

1985 (1)

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

1982 (1)

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

1980 (1)

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

1976 (1)

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[CrossRef]

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

1966 (1)

1956 (2)

S. I. Rubinow, T. T. Wu, “First correction to the geometric-optics cross section from cylinders and spheres,” J. Appl. Phys. 27, 1032–1039 (1956).
[CrossRef]

T. T. Wu, “High-frequency scattering,” Phys. Rev. 104, 1201–1212 (1956).
[CrossRef]

1949 (1)

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

Alexander, D. R.

Higher-order Davis beam models that accurately describe tightly focused Gaussian beams are given in Ref. 13 and in J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

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

Attard, P.

Barton, J. P.

Higher-order Davis beam models that accurately describe tightly focused Gaussian beams are given in Ref. 13 and in J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

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

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 101.

Box, M. A.

Brillouin, L.

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

Bryant, G.

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass.1979), pp. 372–378.

Chýlek, P.

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Gouesbet, G.

J. T. Hodges, G. Gouesbet, G. Gréhan, C. Presser, “Forward scattering of a Gaussian beam by a nonabsorbing sphere,” Appl. Opt. 34, 2120–2132 (1995).
[CrossRef] [PubMed]

G. Gouesbet, J. A. Lock, G. Gréhan, “Partial wave representations of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
[CrossRef] [PubMed]

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

F. Guilloteau, G. Gréhan, G. Gouesbet, “Optical levitation experiments to assess the validity of the generalized Lorenz–Mie theory,” Appl. Opt. 31, 2942–2951 (1992).
[CrossRef] [PubMed]

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

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

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

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

Gréhan, G.

Guilloteau, F.

Hodges, J. T.

Hovenac, E. A.

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 101.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 45.

Klett, J. D.

Kogelnik, H.

Li, T.

Lock, J. A.

Maheu, B.

McKellar, B. H. J.

Merzbacher, E.

E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 505.

Messiah, A.

A. Messiah, Quantum Mechanics (Wiley, New York, 1966), Vol. 2, pp. 866–867.

A. Messiah, Quantum Mechanics (Wiley, New York, 1966), Vol. 1, pp. 372–376.

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass.1979), pp. 372–378.

Newton, R. G.

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

Presser, C.

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass.1979), pp. 372–378.

Rubinow, S. I.

S. I. Rubinow, T. T. Wu, “First correction to the geometric-optics cross section from cylinders and spheres,” J. Appl. Phys. 27, 1032–1039 (1956).
[CrossRef]

Schaub, S. A.

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

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.

Wiscombe, W. J.

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Wu, T. T.

S. I. Rubinow, T. T. Wu, “First correction to the geometric-optics cross section from cylinders and spheres,” J. Appl. Phys. 27, 1032–1039 (1956).
[CrossRef]

T. T. Wu, “High-frequency scattering,” Phys. Rev. 104, 1201–1212 (1956).
[CrossRef]

Yang, L.

Zhan, J.

Am. J. Phys. (2)

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[CrossRef]

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

Appl. Opt. (5)

J. Appl. Phys. (4)

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

S. I. Rubinow, T. T. Wu, “First correction to the geometric-optics cross section from cylinders and spheres,” J. Appl. Phys. 27, 1032–1039 (1956).
[CrossRef]

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

Higher-order Davis beam models that accurately describe tightly focused Gaussian beams are given in Ref. 13 and in J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. Math. Phys. (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

J. Opt. (Paris) (2)

G. Gouesbet, G. Gréhan, B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism,” J. Opt. (Paris) 16, 83–93 (1985).
[CrossRef]

G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,” J. Opt. (Paris) 13, 97–103 (1982).
[CrossRef]

J. Opt. Soc. Am. A (9)

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

J. A. Lock, “The contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
[CrossRef]

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

J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
[CrossRef]

J. A. Lock, “Interpretation of extinction in Gaussian beam scattering,” J. Opt. Soc. Am. A 12, 929–938 (1995).
[CrossRef]

P. Chýlek, J. Zhan, “Interference structure of the Mie extinction cross section,” J. Opt. Soc. Am. A 6, 1846–1851 (1989).
[CrossRef]

J. A. Lock, L. Yang, “Interference between diffraction and transmission in the Mie extinction efficiency,” J. Opt. Soc. Am. A 8, 1132–1134 (1991).
[CrossRef]

P. Attard, M. A. Box, G. Bryant, B. H. J. McKellar, “Asymptotic behavior of the Mie-scattering amplitude,” J. Opt. Soc. Am. A 3, 256–258 (1986).
[CrossRef]

P. Chýlek, J. D. Klett, “Absorption and scattering of electromagnetic radiation by prismatic columns: anomalous diffraction approximation,” J. Opt. Soc. Am. A 8, 1713–1720 (1991).
[CrossRef]

Opt. Commun. (1)

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

Phys. Rev. (1)

T. T. Wu, “High-frequency scattering,” Phys. Rev. 104, 1201–1212 (1956).
[CrossRef]

Phys. Rev. A (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

H. M. Nussenzveig, W. J. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Other (11)

Ref. 14, pp. 183–191.

An alternative derivation of the WBA under the less restrictive assumption al+ bl= 1 will be given by one of us (G. Gouesbet) elsewhere.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 123.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 45.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 101.

Ref. 15, p. 49.

Ref. 16, pp. 69–76.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass.1979), pp. 372–378.

A. Messiah, Quantum Mechanics (Wiley, New York, 1966), Vol. 1, pp. 372–376.

A. Messiah, Quantum Mechanics (Wiley, New York, 1966), Vol. 2, pp. 866–867.

E. Merzbacher, Quantum Mechanics, 2nd ed. (Wiley, New York, 1970), p. 505.

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Figures (1)

Fig. 1
Fig. 1

Extinction efficiency as a function of the nonabsorbing target particle radius for 50 μm ≤ a ≤ 55 μm, n = 1.333 + 0i, and λ = 0.6328 and for (a) w0 = 250 μm, (b) w0 = 100 μm, (c) w0 = 50 μm, (d) w0 = 25 μm, and (e) w0 = 10 μm. The exact extinction cross section is given by Eq. (23). The WBA is given by relation (37), and the NBA is given by Eqs. (45) and (46). The WBA accurately approximates ext for w0/a ≳ 1.5, and the NBA accurately represents ext for w0/a ≲ 1.0.

Tables (1)

Tables Icon

Table 1 Number of Terms jmax in Eq. (27) for 1 Part in 106 Agreement with Eq. (23) for λ = 0.6328 μm, a = 50 μm, and n = 1.333

Equations (46)

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σ ext = 4 π k 2 Re [ S ( 0 ) ] ,
k = 2 π λ .
E beam ( r , t ) = r × ψ TE + i c ω × ( r × ψ TM ) , B beam ( r , t ) = i ω × ( r × ψ TE ) + 1 c r × ψ TM .
ψ TE ( r , t ) ψ TM ( r , t ) } = E 0 exp ( i ω t ) l = 1 i l ( 2 l + 1 ) l ( l + 1 ) g l j l ( k r ) P l 1 ( cos θ ) × { sin ϕ cos ϕ .
g l ( i ) l 1 2 k r j l ( k r ) 1 l ( l + 1 ) 0 π sin 2 θ d θ f ( k r , θ ) × exp ( i k r cos θ ) P l 1 ( cos θ ) ,
E beam radial = E 0 exp ( i k r cos θ ) f ( k r , θ ) sin θ cos ϕ , B beam radial = E 0 c exp ( i k r cos θ ) f ( k r , θ ) sin θ sin ϕ .
g l = 1
E ( r , t ) = E 0 exp [ i ( k z ω t ) ] u ̂ x , B ( r , t ) = E 0 c exp [ i ( k z ω t ) ] u ̂ y .
g l = exp [ s 2 ( l + 1 / 2 ) 2 ] ,
s = 1 k w 0 ,
E beam Davis = E 0 D exp [ i ( k z ω t ) ] exp [ ( x 2 + y 2 ) / w 0 2 D ] × ( u ̂ x 2 i s x w 0 D u ̂ z ) , B beam Davis = E 0 c D exp [ i ( k z ω t ) ] exp [ ( x 2 + y 2 ) / w 0 2 D ] × ( u ̂ y 2 i s y w 0 D u ̂ z ) ,
D = 1 + 2 i s z w 0 .
E scatt ( r , t ) = i E 0 k r exp [ i ( k r ω t ) ] [ S 2 ( θ ) ( cos ϕ ) ] u ̂ θ + S 1 ( θ ) ( sin ϕ ) u ̂ ϕ ] , B scatt ( r , t ) = i E 0 c k r exp [ i ( k r ω t ) ] [ S 1 ( θ ) ( sin ϕ ) ] u ̂ θ S 2 ( θ ) ( cos ϕ ) u ̂ ϕ ] ,
S 1 ( θ ) = l = 1 2 l + 1 l ( l + 1 ) g l [ a l π l ( θ ) + b l τ l ( θ ) ] , S 2 ( θ ) = l = 1 2 l + 1 l ( l + 1 ) g l [ a l τ l ( θ ) + b l π l ( θ ) ] .
π l ( θ ) = 1 sin θ P l 1 ( cos θ ) , τ l ( θ ) = d d θ P l 1 ( cos θ ) ,
x = 2 π a λ ,
E total = E beam + E scatt , B total = B beam + B scatt .
σ abs = c r 2 E 0 2 0 π sin θ d θ 0 2 π d ϕ Re ( E total * × B total ) ,
σ scatt = c r 2 E 0 2 0 π sin θ d θ 0 2 π d ϕ Re ( E scatt * × B scatt ) ,
σ ext = c r 2 E 0 2 0 π sin θ d θ 0 2 π d ϕ Re ( E beam * × B scatt + E scatt * × B beam ) ,
σ ext = σ scatt + σ abs .
σ scatt = 4 π k 2 l = 1 ( l + 1 / 2 ) | g l | 2 ( | a l | 2 + | b l | 2 ) ,
σ ext = 4 π k 2 l = 1 ( l + 1 / 2 ) | g l | 2 Re ( a l + b l ) .
π l ( 0 ) = τ l ( 0 ) = l ( l + 1 ) 2 .
S ( 0 ) S 1 ( 0 ) = S 2 ( 0 ) = l = 1 ( l + 1 / 2 ) g l ( a l + b l ) .
S ( j ) ( 0 ) = l = 1 ( l + 1 / 2 ) j g l ( a l + b l ) .
σ ext = 4 π k 2 j = 0 ( 1 ) j j ! s 2 j Re [ S 2 j + 1 ( 0 ) ] .
I total ( 0 ) = Re ( E total θ * B total ϕ E total ϕ * B total θ 2 μ 0 ) = E 0 2 2 μ 0 c 1 k 2 r 2 [ k 4 w 0 4 4 k 2 w 0 2 Re [ S ( 0 ) ] + | S ( 0 ) | 2 ] .
a l diffraction = B l diffraction = 1 / 2
I diffracted ( 0 ) = E 0 2 2 μ 0 c k 2 r 2 ( k 4 w 0 4 4 ) exp ( 2 a 2 / w 0 2 ) .
ext = σ ext σ inc .
σ inc = π a 2 ,
σ inc 0 a ρ d ρ 0 2 π d ξ exp ( 2 ρ 2 / w 0 2 ) = π w 0 2 2 [ 1 exp ( 2 a 2 / w 0 2 ) ] .
ext 2 ,
σ ext π w 0 2 [ 1 exp ( 2 a 2 / w 0 2 ) ] = π w 0 2 [ 1 2 μ 0 c k 2 r 2 E 0 2 ( 4 k 4 w 0 4 ) I diffraction ( 0 ) ] .
I total ( 0 ) I diffraction ( 0 ) .
σ ext 4 π k 2 Re [ S ( 0 ) ] { 1 s 2 | S ( 0 ) | 2 Re [ S ( 0 ) ] }
S ( 0 ) x 2 2 + 0.49805 ( 1 + 3 i ) x 4 / 3 + 2 x n 2 ( n + 1 ) ( n + 1 ) 2 × exp [ 2 i x ( n 1 ) 3 π i / 2 ] .
scatt 2 + 1.9922 x 2 / 3 8 x n 2 ( n 1 ) ( n + 1 ) 2 × sin [ 2 ( n 1 ) x ] .
S transmission ( θ ) = 2 n 2 x ( n 1 ) ( n + 1 ) 2 exp [ 2 i x ( n 1 ) 3 i π / 2 ] × exp [ i x θ 2 4 ( n n 1 ) ] G ( θ ) ,
G ( θ ) = exp ( γ θ 2 )
γ = 4 a 2 w 0 2 n 4 ( n 1 ) 2 ( n + 1 ) 4 ·
S compensating ( θ ) k 2 w 0 2 2 exp ( θ 2 / 4 s 2 ) .
E scatt = E transmission + E compensating ,
scatt = 2.0 4 x n 2 ( n 1 ) ( n + 1 ) 2 [ ( 1 4 s 2 + γ ) 2 + x 2 n 2 16 ( n 1 ) 2 ] 1 / 2 sin [ 2 x ( n 1 ) + η ] ,
tan η = x n 4 ( n 1 ) ( 1 4 s 2 + γ ) 1 .

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