Abstract

Generalized Lorenz–Mie theory describes electromagnetic scattering of an arbitrary light beam by a spherical particle. The computationally most expensive feature of the theory is the evaluation of the beam-shape coefficients, which give the decomposition of the incident light beam into partial waves. The so-called localized approximation to these coefficients for a focused Gaussian beam is an analytical function whose use greatly simplifies Gaussian-beam scattering calculations. A mathematical justification and physical interpretation of the localized approximation is presented for on-axis beams.

© 1994 Optical Society of America

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References

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  1. G. Gouesbet, G. Gréhan, “Sur la généralisation de la théorie de Lorenz–Mie,”J. Opt. (Paris) 13, 97–103 (1982).
    [CrossRef]
  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]
  3. 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]
  4. 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]
  5. 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]
  6. E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
    [CrossRef] [PubMed]
  7. E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,”IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [CrossRef]
  8. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.
  9. B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
    [CrossRef] [PubMed]
  10. G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
    [CrossRef] [PubMed]
  11. S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
    [CrossRef]
  12. 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]
  13. G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnmin the generalized Lorenz–Mie theory using finite series,”J. Opt. (Paris) 19, 35–48 (1988).
    [CrossRef]
  14. G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
    [CrossRef]
  15. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Secs. 12.31 and 12.33.
  16. B. Maheu, G. Gréhan, G. Gouesbet, “Ray localization in Gaussian beams,” Opt. Commun. 70, 259–262 (1989).
    [CrossRef]
  17. G. Gouesbet, 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. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  19. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.175).
  20. A. Wunsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  21. G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
    [CrossRef]
  22. 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]
  23. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
    [CrossRef]
  24. J. P. Barton, D. R. Alexander, 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]
  25. S. C. Hill, Army Research Laboratory/BED, White Sands Missile Range, New Mexico 88002 (personal communication, December1992).
  26. M. V. Klein, Optics (Wiley, New York, 1970), Fig. 9.19.
  27. M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
    [CrossRef]

1994 (1)

G. Gouesbet, 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]

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]

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,”IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

1992 (2)

1990 (1)

1989 (5)

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

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

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 fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, 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 (4)

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnmin the generalized Lorenz–Mie theory using finite series,”J. Opt. (Paris) 19, 35–48 (1988).
[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]

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]

1987 (1)

1986 (1)

1985 (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, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (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]

1979 (1)

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

1972 (1)

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
[CrossRef]

Alexander, D. R.

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

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 fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

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

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]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.175).

Barber, P. W.

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,”IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[CrossRef] [PubMed]

Barton, J. P.

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

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, “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]

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]

Berry, M. V.

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
[CrossRef]

Chowdhury, D. Q.

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Davis, L. W.

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

Gouesbet, G.

G. Gouesbet, 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]

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

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

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnmin the generalized Lorenz–Mie theory using finite series,”J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

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]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

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, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (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.

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

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

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnmin the generalized Lorenz–Mie theory using finite series,”J. Opt. (Paris) 19, 35–48 (1988).
[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. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

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, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

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

Hill, S. C.

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,”IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

E. E. M. Khaled, S. C. Hill, P. W. Barber, D. Q. Chowdhury, “Near-resonance excitation of dielectric spheres with plane waves and off-axis Gaussian beams,” Appl. Opt. 31, 1166–1169 (1992).
[CrossRef] [PubMed]

S. C. Hill, Army Research Laboratory/BED, White Sands Missile Range, New Mexico 88002 (personal communication, December1992).

Khaled, E. E.

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,”IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

Khaled, E. E. M.

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970), Fig. 9.19.

Lock, J. A.

G. Gouesbet, 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, “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]

Maheu, B.

G. Gouesbet, G. Gréhan, B. Maheu, “Localized interpretation to compute all the coefficients gnmin the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998–1007 (1990).
[CrossRef]

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

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnmin the generalized Lorenz–Mie theory using finite series,”J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Computations of the gncoefficients in the generalized Lorenz–Mie theory using three different methods,” Appl. Opt. 27, 4874–4883 (1988).
[CrossRef] [PubMed]

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]

B. Maheu, G. Gréhan, G. Gouesbet, “Generalized Lorenz–Mie theory: first exact values and comparisons with the localized approximation,” Appl. Opt. 26, 23–25 (1987).
[CrossRef] [PubMed]

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, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Mount, K. E.

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
[CrossRef]

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

Schaub, S. A.

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

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

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (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]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Secs. 12.31 and 12.33.

Wunsche, A.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

E. E. Khaled, S. C. Hill, P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,”IEEE Trans. Antennas Propag. 41, 295–303 (1993).
[CrossRef]

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]

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 fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

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

J. Opt. (Paris) (4)

G. Gouesbet, B. Maheu, G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,”J. Opt. (Paris) 16, 239–247 (1985).
[CrossRef]

G. Gouesbet, G. Gréhan, B. Maheu, “Expressions to compute the coefficients gnmin the generalized Lorenz–Mie theory using finite series,”J. Opt. (Paris) 19, 35–48 (1988).
[CrossRef]

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

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]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. A. (1)

G. Gouesbet, 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]

Opt. Commun. (1)

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

Phys. Rev. A (1)

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

Rep. Prog. Phys. (1)

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), Sec. 6.
[CrossRef]

Other (5)

S. C. Hill, Army Research Laboratory/BED, White Sands Missile Range, New Mexico 88002 (personal communication, December1992).

M. V. Klein, Optics (Wiley, New York, 1970), Fig. 9.19.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), Eq. (11.175).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), Secs. 12.31 and 12.33.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory, 3rd ed. (Addison-Wesley, Reading, Mass., 1979), Sec. 17.5.

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Figures (1)

Fig. 1
Fig. 1

Magnitude of the beam-profile functions of Eqs. (91)(96) as a function of r for λ = 0.6328 μm and for six values of s. For (a)–(d) the localized beam profiles (circles), the modified localized beam profiles (triangles), and the Barton fifth-order beam profiles (solid curves) are indistinguishable. For (a)–(d), F1(r) and a Gaussian function are indistinguishable.

Tables (4)

Tables Icon

Table 1 Terms in the Series Expansion of gn in Powers of s That Are Independent of R for zf = 0

Tables Icon

Table 2 Average of the Magnitude of the Deviation of the Ratio Filoc/Fis from Unity in Parts per 106 for i = 1, 2, 3a

Tables Icon

Table 3 Average of the Magnitude of the Deviation of the Ratio Fimloc/Fis From Unity in Parts per 106 for i = 1, 2, 3a

Tables Icon

Table 4 Actual rms Half-Width of the Focal Waist of the Localized Beam, the Modified Localized Beam, and the Barton Symmetrized Fifth-Order Beam Approximation

Equations (97)

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k = 2 π n λ = ω n c ,
2 ψ inc + k 2 ψ inc = 0.
ψ inc TE ( R , θ , ϕ ) = - E 0 n = 1 m = - n n ( - i ) n 2 n + 1 n ( n + 1 ) ( g n m ) TE × j n ( R ) P n m ( cos θ ) exp ( i m ϕ ) , ψ inc TM ( R , θ , ϕ ) = - E 0 n = 1 m = - n n ( - i ) n 2 n + 1 n ( n + 1 ) ( g n m ) TM × j n ( R ) P n m ( cos θ ) exp ( i m ϕ ) ,
R k r ,
( g n m ) TE = - 1 4 π ( i n - 1 ) R j n ( R ) ( n - m ) ! ( n + m ) ! × 0 π sin θ d θ 0 2 π d ϕ P n m ( cos θ ) × exp ( - i m ϕ ) c B rad inc ( R , θ , ϕ ) n E 0 , ( g n m ) TM = - 1 4 π ( i n - 1 ) R j n ( R ) ( n - m ) ! ( n + m ) ! × 0 π sin θ d θ 0 2 π d ϕ P n m ( cos θ ) × exp ( - i m ϕ ) E rad inc ( R , θ , ϕ ) E 0 .
( g n m ) TE = - 1 2 π 2 ( i n - 1 ) ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! × 0 R d R 0 π sin θ d θ 0 2 π d ϕ j n ( R ) × P n m ( cos θ ) exp ( - i m ϕ ) c B rad inc ( R , θ , ϕ ) n E 0 , ( g n m ) TM = - 1 2 π 2 ( i n - 1 ) ( 2 n + 1 ) ( n - m ) ! ( n + m ) ! × 0 R d R 0 π sin θ d θ 0 2 π d ϕ j n ( R ) × P n m ( cos θ ) exp ( - i m ϕ ) E rad inc ( R , θ , ϕ ) E 0 ,
lim r I ( r , θ , ϕ ) = n E 0 2 2 μ 0 c 1 R 2 [ S 1 ( θ , ϕ ) 2 + S 2 ( θ , ϕ ) 2 ] ,
S 1 ( θ , ϕ ) = n = 1 m = - n n 2 n + 1 n ( n + 1 ) [ ( g n m ) TM a n m π n m ( θ ) + i ( g n m ) TE b n τ n m ( θ ) ] exp ( i m ϕ ) , S 2 ( θ , ϕ ) = n = 1 m = - n n 2 n + 1 n ( n + 1 ) [ i ( g n m ) TE b n m π n m ( θ ) + ( g n m ) TM a n τ n m ( θ ) ] exp ( i m ϕ ) ,
π n m ( θ ) = 1 sin θ P n m ( cos θ ) , τ n m ( θ ) = d d θ P n m ( cos θ ) ,
E rad inc ( R , θ , ϕ ) = E 0 exp ( - i R cos θ ) f e ( R , θ ) sin θ cos ϕ , B rad inc ( R , θ , ϕ ) = B 0 exp ( - i R cos θ ) f b ( R , θ ) sin θ sin ϕ , B 0 = n E 0 / c .
( g n m ) TE = / 2 i ( g n ) b δ m , ± 1 ,             ( g n m ) TM = ½ ( g n ) e δ m , ± 1 ,
( g n ) e = - ½ ( i n - 1 ) R j n ( R ) 1 n ( n + 1 ) 0 π sin 2 θ d θ f e ( R , θ ) × exp ( - i R cos θ ) P n 1 ( cos θ ) , ( g n ) b = - ½ ( i n - 1 ) R j n ( R ) 1 n ( n + 1 ) 0 π sin 2 θ d θ f b ( R , θ ) × exp ( - i R cos θ ) P n 1 ( cos θ )
( g n ) e = - 1 π ( i n - 1 ) 2 n + 1 n ( n + 1 ) 0 R d R 0 π sin 2 θ d θ f e ( R , θ ) × j n ( R ) exp ( - i R cos θ ) P n 1 ( cos θ ) , ( g n ) b = - 1 π ( i n - 1 ) 2 n + 1 n ( n + 1 ) 0 R d R 0 π sin 2 θ d θ f b ( R , θ ) × j n ( R ) exp ( - i R cos θ ) P n 1 ( cos θ )
S 1 ( θ , ϕ ) = sin ϕ n = 1 2 n + 1 n ( n + 1 ) × [ ( g n ) e a n π n 1 ( θ ) + ( g n ) b b n τ n 1 ( θ ) ] , S 2 ( θ , ϕ ) = cos ϕ n = 1 2 n + 1 n ( n + 1 ) × [ ( g n ) e a n τ n 1 ( θ ) + ( g n ) b b n π n 1 ( θ ) ] .
s = 1 k w 0
A ( r , t ) = A ( r ) u ^ x exp ( i ω t ) .
2 A ( r ) + k 2 A ( r ) = 0.
A ( r ) = i E 0 c k exp ( - i k z ) α ( r ) .
ξ x w 0 ,
η y w 0 ,
ζ s z w 0 ,
2 α ξ 2 + 2 α η 2 + s 2 2 α ζ 2 - 2 i α 2 ζ = 0.
α ( ξ , η , ζ ) = D 0 exp ( - ν D 0 ) [ 1 + s 2 ( 2 D 0 - ν 2 D 0 3 ) + s 4 ( 6 D 0 2 - 3 ν 2 D 0 4 - 2 ν 3 D 0 5 + ½ ν 4 D 0 6 ) + O ( s 6 ) ] ,
ν = ξ 2 + η 2 ,
D 0 = ( 1 - 2 i ζ ) - 1 .
E ( r , t ) = c [ ( A + s 2 2 A ξ 2 ) u ^ x + s 2 2 A ξ u ^ y + s 3 2 A ξ ζ u ^ z ] × exp ( i ω t ) ,
B ( r , t ) = 1 w 0 ( s A ζ u ^ y - A η u ^ z ) exp ( i ω t ) .
E ( r , t ) = E 0 exp ( - i k z + i ω t ) D 0 exp ( - ν D 0 ) × ( e x u ^ x + e y u ^ y + 2 i D 0 ξ e z u ^ z ) ,
B ( r , t ) = B 0 exp ( - i k z + i ω t ) D 0 exp ( - ν D 0 ) × ( b x u ^ x + b y u ^ y + 2 i D 0 η b z u ^ z ) ,
E rad ( r , θ , ϕ ) = E x sin θ cos ϕ + E y sin θ sin ϕ + E z cos θ ,
B rad ( r , θ , ϕ ) = B x sin θ cos ϕ + B y sin θ sin ϕ + B z cos θ .
f e ( R , θ ) = D 0 exp ( - s 2 R 2 D 0 sin 2 θ ) h e ( R , θ ) ,
f b ( R , θ ) = D 0 exp ( - s 2 R 2 D 0 sin 2 θ ) h b ( R , θ ) ,
α D 1 ( r ) = D 0 exp ( - ν D 0 ) .
e x MC 1 = 1 ,             e y MC 1 = 0 ,             e z MC 1 = s ,
b x MC 1 = 0 ,             b y MC 1 = 1 ,             b z MC 1 = s ,
h MC 1 h e MCI = h b MC 1 = 1.
α D 3 ( r ) = D 0     exp ( - ν D 0 ) [ 1 + s 2 ( 2 D 0 - ν 2 D 0 3 ) ] .
e x MC 3 = 1 + s 2 ( 4 ξ 2 D 0 2 - ν 2 D 0 3 ) , e y MC 3 = s 2 ( 4 ξ η D 0 2 ) , e z MC 3 = s + s 3 ( - 2 D 0 + 4 ν D 0 2 - ν 2 D 0 3 ) ,
b x MC 3 = 0 , b y MC 3 = 1 + s 2 ( 2 ν D 0 2 - ν 2 D 0 3 ) , b z MC 3 = s + s 3 ( 2 D 0 + 2 ν D 0 2 - ν 2 D 0 3 ) ,
h MC 3 h e MC 3 = h b MC 3 = 1 + 2 i s 2 R D 0 cos θ .
α D 5 ( r ) = D 0 exp ( - ν D 0 ) [ 1 + s 2 ( 2 D 0 - ν 2 D 0 3 ) + s 4 ( 6 D 0 2 - 3 ν 2 D 0 4 - 2 ν 3 D 0 5 + ½ ν 4 D 0 6 ) ] .
e x MC 5 = 1 + s 2 ( 4 ξ 2 D 0 2 - ν 2 D 0 3 ) + s 4 ( 2 D 0 2 - 4 ν D 0 3 - ν 2 D 0 4 + 16 ξ 2 ν D 0 4 - 2 ν 3 D 0 5 - 4 ξ 2 ν 2 D 0 5 + ½ ν 4 D 0 6 ) , e y MC 5 = s 2 ( 4 ξ η D 0 2 ) + s 4 ( 4 ξ η D 0 2 ) ( 4 ν D 0 2 - ν 2 D 0 3 ) , e z MC 5 = s + s 3 ( - 2 D 0 + 4 ν D 0 2 - ν 2 D 0 3 ) + s 5 ( - 6 D 0 2 - 6 ν D 0 3 + 17 ν 2 D 0 4 - 6 ν 3 D 0 5 + ½ ν 4 D 0 6 ) ,
b x MC 5 = 0 , b y MC 5 = 1 + s 2 ( 2 ν D 0 2 - ν 2 D 0 3 ) + s 4 ( - 2 D 0 2 + 4 ν D 0 3 + 5 ν 2 D 0 4 - 4 ν 3 D 0 5 + ½ ν 4 D 0 6 ) , b z MC 5 = s + s 3 ( 2 D 0 + 2 ν D 0 2 - ν 2 D 0 3 ) + s 5 ( 6 D 0 2 + 6 ν D 0 3 + 3 ν 2 D 0 4 - 4 ν 3 D 0 5 + ½ ν 4 D 0 6 ) ,
h e MC 5 = 1 + s 2 D 0 ( 2 i R cos θ ) + s 4 D 0 2 ( 2 - 4 i R cos θ + 4 R 2 sin 2 θ ) , h b MC 5 = 1 + s 2 D 0 ( 2 i R cos θ ) + s 4 D 0 2 ( - 2 + 4 i R cos θ + 2 R 2 sin 2 θ ) ,
h L 1 h e L 1 = h b L 1 = 1 + s 2 D 0 ( 2 i R cos θ ) = D 0
h e L 3 = s 4 D 0 2 ( - 4 i R cos θ ) + D 0 ( 1 + 4 s 4 D 0 2 R 2 sin 2 θ - s 6 D 0 3 R 4 sin 4 θ ) , h b L 3 = s 4 D 0 2 ( 4 i R cos θ ) + D 0 ( 1 + 2 s 4 D 0 2 R 2 sin 2 θ - s 6 D 0 3 R 4 sin 4 θ )
h e L 5 = s 4 D 0 2 ( 2 - 4 i R cos θ ) + s 6 D 0 3 ( - 12 i R cos θ - 4 R 2 sin 2 θ ) + s 8 D 0 4 ( - 12 i R 3 sin 2 θ cos θ - 2 R 4 sin 4 θ ) + D 0 ( 1 + 4 s 4 D 0 2 R 2 sin 2 θ - s 6 D 0 3 R 4 sin 4 θ + 17 s 8 D 0 4 R 4 sin 4 θ - 6 s 10 D 0 5 R 6 sin 6 θ + ½ s 12 D 0 6 R 8 sin 8 θ ) , h b L 5 = s 4 D 0 2 ( - 2 + 4 i R cos θ ) + s 6 D 0 3 ( 12 i R cos θ + 4 R 2 sin 2 θ ) + s 8 D 0 4 ( 12 i R 3 sin 2 θ cos θ + 2 R 4 sin 4 θ ) + D 0 ( 1 + 2 s 4 D 0 2 R 2 sin 2 θ - s 6 D 0 3 R 4 sin 4 θ + 3 s 8 D 0 4 R 4 sin 4 θ - 4 s 10 D 0 5 R 6 sin 6 θ + ½ s 12 D 0 6 R 8 sin 8 θ )
e x = b y ,             e y = b x ,             e z = b z ,
e x B 1 = b y B 1 = 1 , e y B 1 = b x B 1 = 0 , e z B 1 = b z B 1 = s
e x B 3 = b y B 3 = 1 + s 2 ( 2 ξ 2 D 0 2 + ν D 0 2 - ν 2 D 0 3 ) , e y B 3 = b x B 3 = s 2 ( 2 ξ η D 0 2 ) , e z B 3 = b z B 3 = s + s 3 ( 3 ν D 0 2 - v 2 D 0 3 )
e x B 5 = b y B 5 = 1 + s 2 ( 2 ξ 2 D 0 2 + ν D 0 2 - ν 2 D 0 3 ) + s 4 ( 2 ν 2 D 0 4 + 8 ξ 2 ν D 0 4 - 3 ν 3 D 0 5 - 2 ξ 2 ν 2 D 0 5 + ½ ν 4 D 0 6 ) , e y B 5 = b x B 5 = s 2 ( 2 ξ η D 0 2 ) + s 4 ( 2 ξ η D 0 2 ) ( 4 ν D 0 2 - ν 2 D 0 3 ) , e z B 5 = b z B 5 = s + s 3 ( 3 ν D 0 2 - ν 2 D 0 3 ) + s 5 ( 10 ν 2 D 0 4 - 5 ν 3 D 0 5 + ½ ν 4 D 0 6 )
h B 1 h e B 1 = h b B 1 = D 0 ,
h B 3 h e B 3 = h b B 3 = D 0 ( 1 + 3 s 4 D 0 2 R 2 sin 2 θ - s 6 D 0 3 R 4 sin 4 θ ) ,
h B 5 h e B 5 = h b B 5 = D 0 ( 1 + 3 s 4 D 0 2 R 2 sin 2 θ - s 6 D 0 3 R 4 sin 4 θ + 10 s 8 D 0 4 R 4 sin 4 θ - 5 s 10 D 0 5 R 6 sin 6 θ + ½ s 12 D 0 6 R 8 sin 8 θ ) .
z z - z f
D 0 D = ( 1 + 2 i s z f w 0 - 2 i ζ ) - 1
k r n + 1 / 2 , θ π / 2.
( g n ) loc = f loc exp ( i k z f ) = [ 1 + 2 i s ( z f / w 0 ) ] - 1 exp ( i k z f ) × exp [ - s 2 ( n + 1 / 2 ) 2 1 + 2 i s ( z f / w 0 ) ] , f loc = f MC 1 ( R = n + 1 / 2 , θ = π / 2 ) .
f MC 1 ( R , θ ) = 1 + s 2 ( 2 i R cos θ - R 2 sin 2 θ ) + s 4 ( 1 / 2 R 4 sin 4 θ - 4 R 2 cos 2 θ - 4 i R 3 sin 2 θ cos θ ) + s 6 ( - 8 i R 3 cos 3 θ + 12 R 4 sin 2 θ cos 2 θ + 3 i R 5 sin 4 θ cos θ - 1 / 6 R 6 sin 6 θ ) + O ( s 8 ) .
0 π sin 2 θ d θ T ( θ ) exp ( - i R cos θ ) P l 1 ( cos θ ) ,
T ( θ ) = 1 , cos θ , cos 2 θ , sin 2 θ , cos 3 θ , sin 2 θ cos θ , sin 2 θ cos 2 θ , sin 4 θ , sin 4 θ cos θ , sin 6 θ ,
0 π sin 2 θ d θ exp ( - i R cos θ ) P n 1 ( cos θ ) = - 2 ( - i ) n - 1 n ( n + 1 ) G n ( R ) ,
G n ( R ) j n ( R ) R .
R 2 G n + 4 R G n + R 2 G n = ( n - 1 ) ( n + 2 ) G n ,
( g n MC 1 ) e = ( g n MC 1 ) b = g n MC 1 = 1 - s 2 ( n - 1 ) ( n + 2 ) + 2 s 2 R G n ( R ) G n ( R ) + O ( s 4 )
g n MC 3 = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 ( n - 3 ) ( n - 1 ) ( n + 2 ) ( n + 4 ) + 12 s 4 R G n ( R ) G n ( R ) + O ( s 6 ) .
( g n MC 5 ) e = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 [ ( n - 1 ) 2 ( n + 2 ) 2 - 2 ( n - 1 ) ( n + 2 ) + 4 ] + NCT [ s 6 R G n ( R ) G n ( R ) , s 6 R 2 ] , ( g n MC 5 ) b = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 [ ( n - 1 ) 2 ( n + 2 ) 2 - 6 ( n - 1 ) ( n + 2 ) - 4 ] + NCT [ s 6 R G n ( R ) G n ( R ) , s 6 R 2 ] ,
g n L 1 = 1 - s 2 ( n - 1 ) ( n + 2 ) + NCT [ s 4 G n ( R ) G n ( R ) ] ,
( g n L 3 ) e = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 ( n - 1 ) ( n + 2 ) × ( n 2 + n - 4 ) + NCT [ s 6 R G n ( R ) G n ( R ) ] , ( g n L 3 ) b = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 ( n - 1 ) ( n + 2 ) × ( n 2 + n - 8 ) + NCT [ s 6 R G n ( R ) G n ( R ) ] ,
( g n L 5 ) e = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 [ ( n - 1 ) 2 ( n + 2 ) 2 - 2 ( n - 1 ) ( n + 2 ) + 4 ] - s 6 6 ( n - 2 ) ( n - 1 ) ( n + 2 ) ( n + 3 ) ( n 2 + n - 6 ) + NCT [ s 8 R G n ( R ) G n ( R ) ] , ( g n L 5 ) b = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 [ ( n - 1 ) 2 ( n + 2 ) 2 - 6 ( n - 1 ) ( n + 2 ) - 4 ] - s 6 6 ( n - 2 ) ( n - 1 ) ( n + 2 ) ( n + 3 ) ( n 2 + n - 18 ) + NCT [ s 8 R G n ( R ) G n ( R ) ] ,
g n B 1 = 1 - s 2 ( n - 1 ) ( n + 2 ) + NCT [ s 4 R G n ( R ) G n ( R ) ] .
g n B 3 = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 ( n - 2 ) ( n - 1 ) ( n + 2 ) × ( n + 3 ) - s 6 6 ( n - 3 ) ( n - 2 ) ( n - 1 ) ( n + 2 ) ( n + 3 ) × ( n + 4 ) + NCT [ s 8 R G n ( R ) G n ( R ) , s 8 R 2 ] ,
g n B 5 = 1 - s 2 ( n - 1 ) ( n + 2 ) + s 4 2 ( n - 2 ) ( n - 1 ) × ( n + 2 ) ( n + 3 ) - s 6 6 ( n - 3 ) ( n - 2 ) ( n - 1 ) × ( n + 2 ) ( n + 3 ) ( n + 4 ) + s 8 24 ( n - 4 ) ( n - 3 ) × ( n - 2 ) ( n - 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) ( n + 5 ) - s 10 120 ( n - 5 ) ( n - 4 ) ( n - 3 ) ( n - 2 ) × ( n - 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) ( n + 5 ) ( n + 6 ) + NCT [ s 12 R G n ( R ) G n ( R ) , s 12 R 2 , s 12 R 3 G n ( R ) G n ( R ) ] .
g n B k = l = 0 k ( - 1 ) l s 2 l l ! ( n - 1 ) ! ( n - 1 - l ) ! ( n + 1 + l ) ! ( n + 1 ) ! + NCT ( s 2 k + 2 ) ,
g n B k ( z f ) = j = 0 j + 2 l = 2 k + 1 l = 0 ( - 2 i s z f w 0 ) j ( - 1 ) l s 2 l ( l + j ) ! l ! j ! 1 l ! × ( n - 1 ) ! ( n - 1 - l ) ! ( n + 1 + l ) ! ( n + 1 ) ! exp ( i k z f ) + NCT ( s 2 k + 2 )
( g n ) loc = exp [ - s 2 ( n + 1 / 2 ) 2 ] ,
g n B 5 ( 0 ) = l = 0 5 ( - 1 ) l s 2 l l ! ( n - 1 ) ! ( n - 1 - l ) ! ( n + 1 + l ) ! ( n + 1 ) ! .
( n - 1 - q ) ( n + 2 + q ) = n 2 + n - ( 2 + 3 q + q 2 ) ,
( n + 1 / 2 ) 2 = n 2 + n + 1 / 4 ,
g n B 5 ( 0 ) l = 0 5 1 l ! [ - s 2 ( n + 1 / 2 ) 2 ] l exp [ - s 2 ( n + 1 / 2 ) 2 ] ,
g n B 5 ( z f ) = l = 0 j + 2 l = 11 j = 0 ( - 2 i s z f w 0 ) j ( l + j ) ! l ! j ! [ - s 2 ( n + 1 / 2 ) 2 ] l l ! × exp ( i k z f ) .
( 1 + ) - p = 1 - p + p ( p + 1 ) 2 ! 2 - p ( p + 1 ) ( p + 2 ) 3 ! 3 + = j = 0 ( p + j - 1 ) ! ( p - 1 ) ! j ! ( - ) j ,
( g n ) loc = 1 1 + 2 i s z f w 0 l = 0 [ - s 2 ( n + 1 / 2 ) 2 1 + 2 i s z f w 0 ] l 1 l ! exp ( i k z f ) = l = 0 1 ( 1 + 2 i s z f w 0 ) l + 1 [ - s 2 ( n + 1 / 2 ) 2 ] l l ! exp ( i k z f ) = j = 0 l = 0 ( l + j ) ! l ! j ! ( - 2 i s z f w 0 ) j [ - s 2 ( n + 1 / 2 ) 2 ] l l ! × exp ( i k z f ) ,
( g n ) m . loc = f m . loc exp ( i k z f ) = ( 1 + 2 i s z f w 0 ) - 1 exp ( i k z f ) × exp [ - s 2 ( n - 1 ) ( n + 2 ) 1 + 2 i s z f w 0 ] f m . loc = f MC 1 { R = [ ( n - 1 ) ( n + 2 ) ] 1 / 2 , θ = π / 2 } .
E x = F 1 - F 2 sin 2 ϕ , c B x / n = F 2 sin ϕ cos ϕ , E y = F 2 sin ϕ cos ϕ , c B y / n = F 1 - F 2 cos 2 ϕ , E z = F 3 cos ϕ , c B z / n = F 3 sin ϕ ,
F 1 = G 1 sin θ + G 2 cos θ , F 2 = G 1 sin θ + G 2 cos θ - G 3 , F 3 = G 1 cos θ - G 2 sin θ ,
G 1 = - i R sin θ n = 1 ( - i ) n ( 2 n + 1 ) g n J n ( R ) R π n 1 ( θ ) , G 2 = - 1 R n = 1 ( - i ) n 2 n + 1 n ( n + 1 ) g n [ J n ( R ) π n 1 ( θ ) + i J n ( R ) τ n 1 ( θ ) ] , G 3 = - 1 R n = 1 ( - i ) n 2 n + 1 n ( n + 1 ) g n [ J n ( R ) τ n 1 ( θ ) + i J n ( R ) π n 1 ( θ ) ] ,
J n ( R ) R j n ( R ) .
g n S ( z f ) = j = 0 l = 0 ( - 2 i s z f w 0 ) j ( - 1 ) l s 2 l ( l + j ) ! l ! j ! 1 l ! × ( n - 1 ) ! ( n - 1 - l ) ! ( n + 1 + l ) ! ( n + 1 ) ! exp ( i k z f ) .
F 1 ( R , π 2 ) = n = 1 ( 2 n + 1 ) g n j n ( R ) R | π n 1 ( π 2 ) | ,
F 2 ( R , π 2 ) = n = 1 ( ( n + 1 ) g n + 2 - ( 2 n + 3 ) g n + 1 + ( n + 2 ) g n ( n + 1 ) ) j n + 1 ( R ) × | π n 1 ( π 2 ) | ,
F 3 ( R , π 2 ) = - 3 2 ( g 2 - g 1 ) j 1 ( R ) + n = 1 [ - ( n + 2 ) ( n + 4 ) g n + 3 + ( 2 n + 5 ) g n + 2 + ( n + 1 ) ( n + 3 ) g n + 1 ( n + 1 ) ( n + 3 ) ] j n + 2 ( R ) | π n 1 ( π 2 ) | .
F 1 B 5 ( R , π / 2 ) = ( 1 + 3 s 4 R 2 - s 6 R 4 + 10 s 8 R 4 - 5 s 10 R 6 + ½ s 12 R 8 ) exp ( - s 2 R 2 ) ,
F 2 B 5 ( R , π / 2 ) = ( 2 s 4 R 2 + 8 s 8 R 4 - 2 s 10 R 6 ) exp ( - s 2 R 2 ) ,
F 3 B 5 ( R , π / 2 ) = ( 2 s 2 R + 6 s 6 R 3 - 2 s 8 R 5 + 20 s 10 R 5 - 5 s 12 R 7 + s 14 R 9 ) exp ( - s 2 R 2 ) .
k w 0 rms = [ 2 3 0 R 4 ( E x 2 + E y 2 + E z 2 ) 1 / 2 d R 0 R 2 ( E x 2 + E y 2 + E z 2 ) 1 / 2 d R ] 1 / 2 ,

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