Abstract

The angular spectrum of the Davis fifth-order linearly polarized, dual, and symmetrized fields of a focused Gaussian laser beam is obtained. Since the original Davis fields are not an exact solution of the vector wave equation and Maxwell’s equations, a beam remodeling procedure within the angular spectrum is described that produces an exact solution. The spherical multipole beam shape coefficients of the remodeled beam are then obtained, and it is shown that in the weak focusing limit they simplify to the localized model Gaussian beam shape coefficients for both on-axis and off-axis beams. The angular spectrum method is then applied to a transversely confined electromagnetic beam with arbitrary profile in the focal plane, and to a general zero-order Bessel beam.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Gouesbet and G. Gréhan, “General Lorenz–Mie theory in the strict sense, and other GLMTs,” in Generalized Lorenz–Mie Theory (Springer, 2011), pp. 37–88.
  2. H. C. van de Hulst, “Mie’s formal solution,” in Light Scattering by Small Particles (Dover, 1981), pp. 119–126.
  3. J. A. Lock, “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]
  4. J. A. Lock, “Improved Gaussian beam-scattering algorithm,” Appl. Opt. 34, 559–570 (1995).
    [CrossRef]
  5. 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]
  6. 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]
  7. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrarily shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
    [CrossRef]
  8. P. C. Clemmow, “Plane wave representation,” in The Plane Wave Spectrum Representation of Electromagnetic Fields(Pergamon, 1966), pp. 11–38.
  9. J. W. Goodman, “The angular spectrum of plane waves,” in Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48–51.
  10. W. H. Carter, “Bandlimited angular spectrum approximation to a scalar dipole field,” Opt. Commun. 2, 142–148 (1970).
    [CrossRef]
  11. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201(1972).
    [CrossRef]
  12. A. Dociu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
    [CrossRef]
  13. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
    [CrossRef]
  14. S. Colak, C. Yeh, and L. W. Casperson, “Scattering of focused beams by tenuous particles,” Appl. Opt. 18, 294–302 (1979).
    [CrossRef]
  15. C. Yeh, S. Colak, and P. W. Barber, “Scattering by sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
    [CrossRef]
  16. E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propag. 41, 295–303 (1993).
    [CrossRef]
  17. W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, P. S. J. Russell, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Phys. OptarXiv:1003.2392v3.
  18. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef]
  19. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  20. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  21. L. W. Casperson and C. Yeh, “Rayleigh–Debye scattering with focused laser beams,” Appl. Opt. 17, 1637–1643 (1978).
    [CrossRef]
  22. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  23. J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 (1994).
    [CrossRef]
  24. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  25. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz–Mie theory. 1. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532–2544 (2004).
    [CrossRef]
  26. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  27. J. A. Lock, “Beam shape coefficients of the most general focused Gaussian laser beam for light scattering applications,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
    [CrossRef]
  28. M. E. Rose, “Dual fields,” in Multipole Fields (Wiley, 1955), p. 9.
  29. S. A. Schaub, J. P. Barton, and 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]
  30. A. Rohrbach and E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002).
    [CrossRef]
  31. G. Goertzel, “Angular correlation of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
    [CrossRef]
  32. M. E. Rose, “Formulation of the problem,” in Multipole Fields (Wiley, 1955), p. 73.
  33. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres. Part 1-multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
    [CrossRef]
  34. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994).
    [CrossRef]
  35. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A 11, 3251–3260 (1994).
    [CrossRef]
  36. J. J. Wang, G. Gouesbet, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems. II. Axisymmetric beams,” Opt. Commun. 283, 3226–3234 (2010).
    [CrossRef]
  37. G. Arfken, “Bessel functions of the first kind, Jν(x),” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 587, Eq. (11.1.16b).
  38. J. A. Lock, “Partial-wave expansions of angular spectra of plane waves,” J. Opt. Soc. Am. A 23, 2803–2809 (2006).
    [CrossRef]
  39. I. S. Gradshteyn and I. M. Ryzhik, “Combinations of Bessel functions, exponentials, and powers,” in Table of Integrals, Series, and Products (Academic, 1965), p. 718, Eq. (6.633.2).
  40. G. Arfken, “Orthoginality,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 594, Eq. (11.59).
  41. G. Gouesbet, J. A. Lock, and G. Gréhan, “Partial-wave representations of laser beams for use in light-scattering calculations,” Appl. Opt. 34, 2133–2143 (1995).
    [CrossRef]
  42. H. C. van de Hulst, “The localization principle,” in Light Scattering by Small Particles (Dover, 1981), pp. 208–209.
  43. A. Doicu and T. Wreidt, “Computation of the beam shape coefficients in the generalized Lorenz–Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36, 2971–2978 (1997).
    [CrossRef]
  44. H. Zhang and Y. Han, “Addition theorem for the vector spherical wave functions and its application to the beam shape coefficients,” J. Opt. Soc. Am. B 25, 255–260 (2008).
    [CrossRef]
  45. L. Boyde, K. J. Chalut, and J. Guck, “Exact analytical expansion of an off-axis Gaussian laser beam using the translation theorems for the vector spherical harmonics,” Appl. Opt. 50, 1023–1033 (2011).
    [CrossRef]
  46. B. Stout, B. Rolly, M. Fall, J. Hazart, and N. Bonod, “Laser-particle interactions in shaped beams: beam power normalization,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
    [CrossRef]
  47. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
    [CrossRef]
  48. T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 1–23 (2006).
    [CrossRef]
  49. J. M. Taylor and G. D. Love, “Multipole expansion of Bessel and Gaussian beams for Mie scattering calculations,” J. Opt. Soc. Am. A 26, 278–282 (2009).
    [CrossRef]
  50. J. Chen, J. Ng, P. Wang, and Z. Lin, “Analytical partial wave expansion of vector Bessel beam and its application to optical binding,” Opt. Lett. 35, 1674–1676 (2010).
    [CrossRef]
  51. L. A. Ambrosio and H. E. Hernández-Figueroa, “Integral localized approximation description of ordinary Bessel beams and applications to optical trapping forces,” Biomed. Opt. Express 2, 1893–1906 (2011).
    [CrossRef]
  52. G. Gouesbet and J. A. Lock, “List of problems for future research in generalized Lorenz–Mie theories and related topics, review and prospectus [Invited],” Appl. Opt.52, 897–916(2013).

2011 (3)

2010 (2)

J. Chen, J. Ng, P. Wang, and Z. Lin, “Analytical partial wave expansion of vector Bessel beam and its application to optical binding,” Opt. Lett. 35, 1674–1676 (2010).
[CrossRef]

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

2009 (1)

2008 (1)

2006 (3)

2004 (1)

2002 (1)

1997 (2)

1995 (2)

1994 (4)

1993 (2)

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

1991 (1)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

1989 (2)

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

S. A. Schaub, J. P. Barton, and 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]

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]

1982 (1)

1979 (2)

1978 (1)

1975 (1)

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

1972 (1)

1971 (1)

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres. Part 1-multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

1970 (1)

W. H. Carter, “Bandlimited angular spectrum approximation to a scalar dipole field,” Opt. Commun. 2, 142–148 (1970).
[CrossRef]

1966 (1)

1959 (1)

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

1946 (1)

G. Goertzel, “Angular correlation of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

S. A. Schaub, J. P. Barton, and 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, 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]

Ambrosio, L. A.

Arfken, G.

G. Arfken, “Orthoginality,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 594, Eq. (11.59).

G. Arfken, “Bessel functions of the first kind, Jν(x),” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 587, Eq. (11.1.16b).

Barber, P. W.

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

C. Yeh, S. Colak, and P. W. Barber, “Scattering by sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
[CrossRef]

Barbosa, L. C.

Barton, J. P.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

S. A. Schaub, J. P. Barton, and 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, 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]

Bonod, N.

B. Stout, B. Rolly, M. Fall, J. Hazart, and N. Bonod, “Laser-particle interactions in shaped beams: beam power normalization,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
[CrossRef]

Bouchal, Z.

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 1–23 (2006).
[CrossRef]

Boyde, L.

Bruning, J. H.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres. Part 1-multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Carter, W. H.

W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201(1972).
[CrossRef]

W. H. Carter, “Bandlimited angular spectrum approximation to a scalar dipole field,” Opt. Commun. 2, 142–148 (1970).
[CrossRef]

Casperson, L. W.

Cesar, C. L.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[CrossRef]

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, P. S. J. Russell, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Phys. OptarXiv:1003.2392v3.

Chalut, K. J.

Chen, J.

Cižmár, T.

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 1–23 (2006).
[CrossRef]

Clemmow, P. C.

P. C. Clemmow, “Plane wave representation,” in The Plane Wave Spectrum Representation of Electromagnetic Fields(Pergamon, 1966), pp. 11–38.

Colak, S.

Cruz, C. H. B.

Davis, L. W.

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

Dociu, A.

A. Dociu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
[CrossRef]

Doicu, A.

Euser, T. G.

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, P. S. J. Russell, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Phys. OptarXiv:1003.2392v3.

Fall, M.

B. Stout, B. Rolly, M. Fall, J. Hazart, and N. Bonod, “Laser-particle interactions in shaped beams: beam power normalization,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
[CrossRef]

Fontes, A.

Fuller, K. A.

Garbos, M. K.

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, P. S. J. Russell, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Phys. OptarXiv:1003.2392v3.

Goertzel, G.

G. Goertzel, “Angular correlation of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “The angular spectrum of plane waves,” in Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48–51.

Gouesbet, G.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrarily shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

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

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

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

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

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

G. Gouesbet and J. A. Lock, “List of problems for future research in generalized Lorenz–Mie theories and related topics, review and prospectus [Invited],” Appl. Opt.52, 897–916(2013).

G. Gouesbet and G. Gréhan, “General Lorenz–Mie theory in the strict sense, and other GLMTs,” in Generalized Lorenz–Mie Theory (Springer, 2011), pp. 37–88.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, “Combinations of Bessel functions, exponentials, and powers,” in Table of Integrals, Series, and Products (Academic, 1965), p. 718, Eq. (6.633.2).

Gréhan, G.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrarily shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

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

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

G. Gouesbet and G. Gréhan, “General Lorenz–Mie theory in the strict sense, and other GLMTs,” in Generalized Lorenz–Mie Theory (Springer, 2011), pp. 37–88.

Guck, J.

Han, Y.

Han, Y. P.

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

Hazart, J.

B. Stout, B. Rolly, M. Fall, J. Hazart, and N. Bonod, “Laser-particle interactions in shaped beams: beam power normalization,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
[CrossRef]

Hernández-Figueroa, H. E.

Hill, S. C.

E. E. M. Khaled, S. C. Hill, and 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.

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

Kogelnik, H.

Kollárová, V.

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 1–23 (2006).
[CrossRef]

Lax, M.

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

Li, T.

Lin, Z.

Lo, Y. T.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres. Part 1-multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

Lock, J. A.

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrarily shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

J. A. Lock, “Partial-wave expansions of angular spectra of plane waves,” J. Opt. Soc. Am. A 23, 2803–2809 (2006).
[CrossRef]

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz–Mie theory. 1. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532–2544 (2004).
[CrossRef]

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

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

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

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

J. A. Lock, “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]

G. Gouesbet and J. A. Lock, “List of problems for future research in generalized Lorenz–Mie theories and related topics, review and prospectus [Invited],” Appl. Opt.52, 897–916(2013).

J. A. Lock, “Beam shape coefficients of the most general focused Gaussian laser beam for light scattering applications,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
[CrossRef]

Louisell, W. H.

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

Love, G. D.

Mackowski, D. W.

Maheu, B.

McKnight, W. B.

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

Mishra, S. R.

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

Moreira, W. L.

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, P. S. J. Russell, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Phys. OptarXiv:1003.2392v3.

Neves, A. A. R.

A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
[CrossRef]

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, P. S. J. Russell, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Phys. OptarXiv:1003.2392v3.

Ng, J.

Padilha, L. A.

Richards, B.

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

Rodriguez, E.

Rohrbach, A.

Rolly, B.

B. Stout, B. Rolly, M. Fall, J. Hazart, and N. Bonod, “Laser-particle interactions in shaped beams: beam power normalization,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
[CrossRef]

Rose, M. E.

M. E. Rose, “Formulation of the problem,” in Multipole Fields (Wiley, 1955), p. 73.

M. E. Rose, “Dual fields,” in Multipole Fields (Wiley, 1955), p. 9.

Russell, P. S. J.

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, P. S. J. Russell, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Phys. OptarXiv:1003.2392v3.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, “Combinations of Bessel functions, exponentials, and powers,” in Table of Integrals, Series, and Products (Academic, 1965), p. 718, Eq. (6.633.2).

Schaub, S. A.

S. A. Schaub, J. P. Barton, and 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, 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]

Stelzer, E. H. K.

Stout, B.

B. Stout, B. Rolly, M. Fall, J. Hazart, and N. Bonod, “Laser-particle interactions in shaped beams: beam power normalization,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
[CrossRef]

Taylor, J. M.

van de Hulst, H. C.

H. C. van de Hulst, “Mie’s formal solution,” in Light Scattering by Small Particles (Dover, 1981), pp. 119–126.

H. C. van de Hulst, “The localization principle,” in Light Scattering by Small Particles (Dover, 1981), pp. 208–209.

Wang, J. J.

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

Wang, P.

Wolf, E.

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

Wreidt, T.

Wriedt, T.

A. Dociu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
[CrossRef]

Yeh, C.

Zemánek, P.

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 1–23 (2006).
[CrossRef]

Zhang, H.

Appl. Opt. (10)

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

S. Colak, C. Yeh, and L. W. Casperson, “Scattering of focused beams by tenuous particles,” Appl. Opt. 18, 294–302 (1979).
[CrossRef]

C. Yeh, S. Colak, and P. W. Barber, “Scattering by sharply focused beams by arbitrarily shaped dielectric particles: an exact solution,” Appl. Opt. 21, 4426–4433 (1982).
[CrossRef]

L. W. Casperson and C. Yeh, “Rayleigh–Debye scattering with focused laser beams,” Appl. Opt. 17, 1637–1643 (1978).
[CrossRef]

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz–Mie theory. 1. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532–2544 (2004).
[CrossRef]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
[CrossRef]

A. Rohrbach and E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002).
[CrossRef]

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

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

L. Boyde, K. J. Chalut, and J. Guck, “Exact analytical expansion of an off-axis Gaussian laser beam using the translation theorems for the vector spherical harmonics,” Appl. Opt. 50, 1023–1033 (2011).
[CrossRef]

Appl. Phys. Lett. (1)

S. A. Schaub, J. P. Barton, and 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]

Biomed. Opt. Express (1)

IEEE Trans. Antennas Propag. (2)

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres. Part 1-multipole expansion and ray-optical solutions,” IEEE Trans. Antennas Propag. AP-19, 378–390 (1971).
[CrossRef]

E. E. M. Khaled, S. C. Hill, and 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. (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]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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

J. Quant. Spectrosc. Radiat. Transfer (1)

G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrarily shaped beams: localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transfer 112, 1–27 (2011).
[CrossRef]

New J. Phys. (1)

T. Čižmár, V. Kollárová, Z. Bouchal, and P. Zemánek, “Sub-micron particle organization by self-imaging of non-diffracting beams,” New J. Phys. 8, 1–23 (2006).
[CrossRef]

Opt. Commun. (4)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

W. H. Carter, “Bandlimited angular spectrum approximation to a scalar dipole field,” Opt. Commun. 2, 142–148 (1970).
[CrossRef]

A. Dociu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136, 114–124 (1997).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. (1)

G. Goertzel, “Angular correlation of gamma-rays,” Phys. Rev. 70, 897–909 (1946).
[CrossRef]

Phys. Rev. A (2)

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 beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

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

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

Other (14)

J. A. Lock, “Beam shape coefficients of the most general focused Gaussian laser beam for light scattering applications,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
[CrossRef]

M. E. Rose, “Dual fields,” in Multipole Fields (Wiley, 1955), p. 9.

M. E. Rose, “Formulation of the problem,” in Multipole Fields (Wiley, 1955), p. 73.

G. Arfken, “Bessel functions of the first kind, Jν(x),” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 587, Eq. (11.1.16b).

W. L. Moreira, A. A. R. Neves, M. K. Garbos, T. G. Euser, P. S. J. Russell, and C. L. Cesar, “Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions,” Phys. OptarXiv:1003.2392v3.

P. C. Clemmow, “Plane wave representation,” in The Plane Wave Spectrum Representation of Electromagnetic Fields(Pergamon, 1966), pp. 11–38.

J. W. Goodman, “The angular spectrum of plane waves,” in Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 48–51.

G. Gouesbet and G. Gréhan, “General Lorenz–Mie theory in the strict sense, and other GLMTs,” in Generalized Lorenz–Mie Theory (Springer, 2011), pp. 37–88.

H. C. van de Hulst, “Mie’s formal solution,” in Light Scattering by Small Particles (Dover, 1981), pp. 119–126.

G. Gouesbet and J. A. Lock, “List of problems for future research in generalized Lorenz–Mie theories and related topics, review and prospectus [Invited],” Appl. Opt.52, 897–916(2013).

B. Stout, B. Rolly, M. Fall, J. Hazart, and N. Bonod, “Laser-particle interactions in shaped beams: beam power normalization,” J. Quant. Spectrosc. Radiat. Transfer (2013), to be published.
[CrossRef]

I. S. Gradshteyn and I. M. Ryzhik, “Combinations of Bessel functions, exponentials, and powers,” in Table of Integrals, Series, and Products (Academic, 1965), p. 718, Eq. (6.633.2).

G. Arfken, “Orthoginality,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 594, Eq. (11.59).

H. C. van de Hulst, “The localization principle,” in Light Scattering by Small Particles (Dover, 1981), pp. 208–209.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (166)

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

EFD=EFD(ρ,z)ux,
cBFD=EFD(ρ,z)uy,
EFD(ρ,z)=Dexp(ikz)exp(Dρ2),
ρ2=(x2+y2)/w02,
D=1/(1+2isz/w0).
s=1/(kw0).
ELP=ExLPux+EyLPuy+EzLPuz,
BLP=BxLPux+ByLPuy+BzLPuz,
ExLP=EFD(ρ,z)[1+s2(4D2x2/w02D3ρ4)+s4(2D24D3ρ2D4ρ4+16D4ρ2x2/w022D5ρ64D5ρ4x2/w02+D6ρ8/2)],
EyLP=EFD(ρ,z)(4xy/w02)[s2(D2)+s4(4D4ρ2D5ρ4)],
EzLP=EFD(ρ,z)(2iDsx/w0)[1+s2(2D+4D2ρ2D3ρ4)+s4(6D26D3ρ2+17D4ρ46D5ρ6+D6ρ8/2)],
cBxLP=0,
cByLP=EFD(ρ,z)[1+s2(2D2ρ2D3ρ4)+s4(2D2+4D3ρ2+5D4ρ44D5ρ6+D6ρ8/2)],
cBzLP=EFD(ρ,z)(2iDsy/w0)[1+s2(2D+2D2ρ2D3ρ4)+s4(6D2+6D3ρ2+3D4ρ44D5ρ6+D6ρ8/2)].
Edual=Exdualux+Eydualuy+Ezdualuz,
Bdual=Bxdualux+Bydualuy+Bzdualuz,
Exdual=cByLP,
Eydual=cBxLP=0,
Ezdual=cBzLP,
cBxdual=EyLP,
cBydual=ExLP,
cBzdual=EzLP,
Esymm=Exsymmux+Eysymmuy+Ezsymmuz,
Bsymm=Bxsymmux+Bysymmuy+Bzsymmuz,
Exsymm=EFD(ρ,z)[1+s2(2D2x2/w02+D2ρ2D3ρ4)+s4(2D4ρ4+8D4ρ2x2/w023D5ρ62D5ρ4x2/w02+D6ρ8/2)],
Eysymm=EFD(ρ,z)(2xy/w02)[s2(D2)+s4(4D4ρ2D5ρ4)],
Ezsymm=EFD(ρ,z)(2iDsx/w0)[1+s2(3D2ρ2D3ρ4)+s4(10D4ρ45D5ρ6+D6ρ8/2)],
cBxsymm=EFD(ρ,z)(2xy/w02)[s2(D2)+s4(4D4ρ2D5ρ4)],
cBysymm=EFD(ρ,z)[1+s2(2D2y2/w02+D2ρ2D3ρ4)+s4(2D4ρ4+8D4ρ2y2/w023D5ρ62D5ρ4y2/w02+D6ρ8/2)],
cBzsymm=EFD(ρ,z)(2iDsy/w0)[1+s2(3D2ρ2D3ρ4)+s4(10D4ρ45D5ρ6+D6ρ8/2)].
FFD(ρk)=0ρdρ02πdφEFD(ρ,z=0)exp[iρρkcos(φφk)]=πw02exp(ρk2w02/4),
kx=ρkcos(φk)=ksin(θk)cos(φk),
ky=ρksin(φk)=ksin(θk)sin(φk),
FES(ρk)=FExSux+FEySuy+FEzSuz=0ρdρ02πdφ[ExS(ρ,z=0)ux+EyS(ρ,z=0)uy+EzS(ρ,z=0)uz]exp[iρρkcos(φφk)],
FExLP=FFD(ρk)[αLP+(kx2/k2)βLP],
FEyLP=FFD(ρk)(kxky/k2)βLP,
FEzLP=FFD(ρk)(kx/k){αLP+[ρk2/(2k2)]βLP[ρk4/(8k4)]γLP},
FBxLP=0,
FByLP=FFD(ρk){αLP+[ρk2/(2k2)]βLP[ρk4/(8k4)]γLP},
FBzLP=FFD(ρk)(ky/k)αLP,
αLP=1+(kw0)2[ρk4/(16k4)+s2ρk2/k2]+(kw0)4[ρk8/(512k8)3s2ρk6/(32k6)+15s4ρk4/(16k4)],
βLP=(kw0)2(s2)+(kw0)4[s2ρk4/(16k4)s4ρk2/k2],
γLP=(kw0)4(s4).
FExdual=FByLP,
FEydual=FBxLP=0,
FEzdual=FBzLP,
FBxdual=FEyLP,
FBydual=FExLP,
FBzdual=FEzLP,
FExsymm=FFD(ρk)[αsymm+(kx2/k2)βsymm],
FEysymm=FFD(ρk)(kxky/k2)βsymm,
FEzsymm=FFD(ρk)(kx/k)αsymm,
FBxsymm=FFD(ρk)(kxky/k2)βsymm,
FBysymm=FFD(ρk)[αsymm+(ky2/k2)βsymm],
FBzsymm=FFD(ρk)(ky/k)αsymm,
αsymm=1+(kw0)2[ρk4/(16k4)+3s2ρk2/(4k2)]+(kw0)4[ρk8/(512k8)5s2ρk6/(64k6)+5s4ρk4/(8k4)],
βsymm=(kw0)2(s2/2)+(kw0)4[s2ρk4/(32k4)s4ρk2/(2k2)].
k·FE=0,
k·FB=0,
k×FE=kcFB,
k×cFB=kFE,
FExS=αS+(kx2/k2)βS,
FEyS=(kxky/k2)βS,
FEzS=(kx/kz)[αS+(ρk2/k2)βS],
FBxS=(kxky/kzk)(αS+βS),
FByS=[(k2ky2)/(kzk)]αS+[kx2/(kzk)]βS,
FBzS=(ky/k)αS,
E=E0exp(ikz)[cos(γ)ux+sin(γ)uy],
cB=E0exp(ikz)[sin(γ)ux+cos(γ)uy].
k=k[sin(θk)cos(φk)ux+sin(θk)sin(φk)uy+cos(θk)uz].
E=E0exp(ik·r){[cos(θk)cos(φk)cos(γ)sin(φk)sin(γ)]ux+[cos(θk)sin(φk)cos(γ)+cos(φk)sin(γ)]uysin(θk)cos(γ)uz}.
An,m=2E0[(n|m|)!/(n+|m|)!]exp(imφk)[cos(γ)τn|m|(θk)+isin(γ)mπn|m|(θk)],
Bn,m=2E0[(n|m|)!/(n+|m|)!]exp(imφk)[sin(γ)τn|m|(θk)+icos(γ)mπn|m|(θk)],
πn|m|(θk)=Pn|m|[cos(θk)]/sin(θk),
τn|m|(θk)=(d/dθk)Pn|m|[cos(θk)],
exp(ik·r)exp(ik·r)exp[ikz0cos(θk)]exp[ikρ0sin(θk)cos(φkφ0)],
x0=ρ0cos(φ0),
y0=ρ0sin(φ0).
FEx(θk,φk)=E0[cos(θk)cos(φk)cos(γ)sin(φk)sin(γ)],
FEy(θk,φk)=E0[cos(θk)sin(φk)cos(γ)+cos(φk)sin(γ)],
FEz(θk,φk)=E0sin(θk)cos(γ).
E0cos(γ)=FFD(ρk)cos(φk)V(θk)/cos(θk),
E0sin(γ)=FFD(ρk)sin(φk)W(θk),
VLP(θk)=1+s2(T4/16)+s4(T8/5122T6/64T4/16),
Vdual(θk)=1+s2(T4/16+2T2/4)+s4(T8/5124T6/64+5T4/16),
Vsymm(θk)=1+s2(T4/16+T2/4)+s4(T8/5123T6/64+2T4/16),
WLP(θk)=1+s2(T4/16+4T2/4)+s4(T8/5126T6/64+15T4/16),
Wdual(θk)=1+s2(T4/16+2T2/4)+s4(T8/5124T6/64+5T4/16),
Wsymm(θk)=1+s2(T4/16+3T2/4)+s4(T8/5125T6/64+10T4/16),
T=kw0sin(θk).
An,m=[1/(2π2)][(n|m|)!/(n+|m|)]0ρkdρk02πdφkexp(imφk)E0[cos(γ)τn|m|(θk)+isin(γ)mπn|m|(θk)]exp[ikz0cos(θk)]exp[ikρ0sin(θk)cos(φkφ0)],
Bn,m=[1/(2π2)][(n|m|)!/(n+|m|)!]0ρkdρk02πdφkexp(imφk)E0[sin(γ)τn|m|(θk)+icos(γ)mπn|m|(θk)]exp[ikz0cos(θk)]exp[ikρ0sin(θk)cos(φkφ0)].
An,±m=[(i)m1/(2π)][(nm)!/(n+m)!]0[ρkdρk/cos(θk)]exp[ikz0cos(θk)]FFD(ρk)×{[V(θk)τnm(θk)+cos(θk)W(θk)mπnm(θk)]exp[±i(1m)φ0]Jm1[kρ0sin(θk)][V(θk)τnm(θk)cos(θk)W(θk)mπnm(θk)]exp[±i(1m)φ0]Jm+1[kρ0sin(θk)]},
An,0=(i/π)cos(φ0)0[ρkdρk/cos(θk)]exp[ikz0cos(θk)]FFD(ρk)V(θk)×τn0(θk)J1[kρ0sin(θk)],
Bn,±m=±[(i)m/(2π)][(nm)!/(n+m)!]0[ρkdρk/cos(θk)]exp[ikz0cos(θk)]FFD(ρk)×{[cos(θk)W(θk)τnm(θk)+V(θk)mπnm(θk)]exp[±i(1m)φ0]Jm1[kρ0sin(θk)]+[cos(θk)W(θk)τnm(θk)V(θk)mπnm(θk)]exp[±i(1m)φ0]Jm+1[kρ0sin(θk)]}
Bn,0=(i/π)sin(φ0)0[ρkdρk/cos(θk)]exp[ikz0cos(θk)]FFD(ρk)cos(θk)W(θk)×τn0(θk)J1[kρ0sin(θk)].
mπnm(θk)(1/2)[(n+m)!/(nm)!](n+1/2)1m{Jm1[(n+1/2)θk]+Jm+1[(n+1/2)θk]},
πnm(θk)(1/2)[(n+m)!/(nm)!](n+1/2)1m{Jm1[(n+1/2)θk]Jm+1[(n+1/2)θk]},
exp[ikz0cos(θk)]exp(ikz0)exp(ikz0θk2/2).
An,±m[k2/(2π)][i/(n+1/2)]m1exp(ikz0)×{exp[±i(1m)φ0]Lm1+exp[±i(1m)φ0]Lm+1},
An,0(ik2/π)(n+1/2)cos(φ0)exp(ikz0)L1,
Bn,±m[k2/(2π)][im/(n+1/2)m1]exp(ikz0)×{exp[±i(1m)φ0}Lm1exp[±i(1m)φ0]Lm+1},
Bn,0(ik2/π)(n+1/2)sin(φ0)exp(ikz0)L1,
Lj=0θkdθkFFD(kθk)exp(ikz0θk2/2)Jj[(n+1/2)θk]Jj(kρ0θk).
Lj=(2s2D0)exp[s2D0(n+1/2)2]exp(ρ02D0/w02)Ij[2s2kρ0D0(n+1/2)],
D0=1/(12isz0/w0)
An,1=An,1=iBn,1=iBn,1=exp[s2(n+1/2)2].
An,1=An,1=iBn,1=iBn,1=[k2/(2π)]L0
L0=0θkdθkFEx(kθk)J0[(n+1/2)θk].
An,1=An,1=iBn,1=iBn,1=Ex[ρ=(n+1/2)/k,z=0].
ELP=(ExLPux+EyLPuy+EzLPuz)exp[ikzcos(θk)],
BLP=(BxLPux+ByLPuy+BzLPuz)exp[ikzcos(θk)],
ExLP=[1sin2(θk)/2]J0[kρsin(θk)]+[sin2(θk)/2]cos(2φ)J2[kρsin(θk)],
EyLP=[sin2(θk)/2]sin(2φ)J2[kρsin(θk)],
EzLP=isin(θk)cos(θk)cos(φ)J1[kρsin(θk)],
cBxLP=0,
cByLP=cos(θk)J0[kρsin(θk)],
cBzLP=isin(θk)sin(φ)J1[kρsin(θk)].
w2.405/[ksin(θk)],
Exdual=cByLP,
Eydual=cBxLP=0,
Ezdual=cBzLP,
cBxdual=EyLP,
cBydual=ExLP,
cBzdual=EzLP,
Exsymm=(1/2){1+cos(θk)[sin2(θk)/2]}J0[kρsin(θk)]+[sin2(θk)/4]cos(2φ)J2[kρsin(θk)],
Eysymm=[sin2(θk)/4]sin(2φ)J2[kρsin(θk)],
Ezsymm=i[sin(θk)/2][1+cos(θk)]cos(φ)J1[kρsin(θk)],
cBxsymm=[sin2(θk)/4]sin(2φ)J2[kρsin(θk)],
cBysymm=(1/2){1+cos(θk)[sin2(θk)/2]}J0[kρsin(θk)][sin2(θk)/4]cos(2φ)J2[kρsin(θk)],
cBzsymm=i[sin(θk)/2][1+cos(θk)]sin(φ)J1[kρsin(θk)].
Exaplan=(1/2)[1+cos(θk)]J0[kρsin(θk)]+(1/2)[1cos(θk)]cos(2φ)J2[kρsin(θk)],
Eyaplan=(1/2)[1cos(θk)]sin(2φ)J2[kρsin(θk)],
Ezaplan=isin(θk)cos(φ)J1[kρsin(θk)],
cBxaplan=(1/2)[1cos(θk)]sin(2φ)J2[kρsin(θk)],
cByaplan=(1/2)[1+cos(θk)]J0[kρsin(θk)](1/2)[1cos(θk)]cos(2φ)J2[kρsin(θk)],
cBzaplan=isin(θk)sin(φ)J1[kρsin(θk)].
Exgeneral=(1/2)[1+g(θk)cos(θk)]J0[kρsin(θk)]+(1/2)[1g(θk)cos(θk)]cos(2φ)J2[kρsin(θk)],
Eygeneral=(1/2)[1g(θk)cos(θk)]sin(2φ)J2[kρsin(θk)],
Ezgeneral=ig(θk)sin(θk)cos(φ)J1[kρsin(θk)].
cBxgeneral=(1/2)[1g(θk)cos(θk)]sin(2φ)J2[kρsin(θk)],
cBygeneral=(1/2)[1+g(θk)cos(θk)]J0[kρsin(θk)](1/2)[1g(θk)cos(θk)]cos(2φ)J2[kρsin(θk)],
cBzgeneral=ig(θk)sin(θk)sin(φ)J1[kρsin(θk)],
curl(E)=ikcB,
curl(cB)=ikE,
Exave={[g(θk)+1]/4}{[1+cos(θk)]J0[kρsin(θk)]+[1cos(θk)]cos(2φ)J2[kρsin(θk)]},
Eyave={[g(θk)+1]/4}[1cos(θk)]sin(2φ)J2[kρsin(θk)],
Ezave=i{[g(θk)+1]/2}sin(θk)cos(φ)J1[kρsin(θk)],
cBxave={[g(θk)+1]/4}[1cos(θk)]sin(2φ)J2[kρsin(θk)],
cByave={[g(θk)+1]/4}{[1+cos(θk)]J0[kρsin(θk)][1cos(θk)]cos(2φ)J2[kρsin(θk)]},
cBzave=i{[g(θk)+1]/2}sin(θk)sin(φ)J1[kρsin(θk)].
FEave={2π/[ksin(θk)]}δ[ρkksin(θk)](FExaveux+FEyaveuy+FEzaveuz),
FBave={2π/[ksin(θk)]}δ[ρkksin(θk)](FBxaveux+FByaveuy+FBzaveuz),
FExave={[g(θk)+1]/4}{1+cos(θk)[1cos(θk)]cos(2φk)},
FEyave={[g(θk)+1]/4}[1cos(θk)]sin(2φk),
FEzave={[g(θk)+1]/2}sin(θk)cos(φk),
FBxave={[g(θk)+1]/4}[1cos(θk)]sin(2φk),
FByave={[g(θk)+1]/4}{1+cos(θk)+[1cos(θk)]cos(2φk)},
FBzave={[g(θk)+1]/2}sin(θk)sin(φk).
cos(γ)=cos(φk),
sin(γ)=sin(φk).
An,±m=(i)m1{[g(θk)+1]/2}[(nm)!/(n+m)!]×{[τnm(θk)+mπnm(θk)]exp[±i(1m)φ0]Jm1[kρ0sin(θk)][τnm(θk)mπnm(θk)]exp[±i(1m)φ0]Jm+1[kρ0sin(θk)]},
An,0=i[g(θk)+1]cos(φ0)τn0(θk)J1[kρ0sin(θk)],
Bn,±m=±(i)m{[g(θk)+1]/2}[(nm)!/(n+m)!]×{[τnm(θk)+mπnm(θk)]exp[±i(1m)φ0]Jm1[kρ0sin(θk)]+[τnm(θk)mπnm(θk)]exp[±i(1m)φ0]Jm+1[kρ0sin(θk)]},
Bn,0=i[g(θk)+1]sin(φ0)τn0(θk)J1[kρ0sin(θk)].
An,1=An,1=iBn,1=iBn,1={[g(θk)+1]/[2n(n+1)]}[τn1(θk)+πn1(θk)].

Metrics