Abstract

We derive bases constructed from simple vortices and complex focus fields and show that they are useful in the description of strongly focused fields. Both scalar and electromagnetic fields are considered, and in each case two types of basis are discussed: bases that use standard polynomials but whose orthogonality condition requires a non-uniform directional weight factor, and bases that are orthogonal with uniform weight but that require new polynomials. Their performance is studied by fitting prescribed fields, where it is seen that the accuracy provided by both types of bases is comparable.

© 2017 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Elsevier, 1980).
  2. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).
    [Crossref]
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  4. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
    [Crossref] [PubMed]
  5. C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
    [Crossref]
  6. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  7. R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A.
  8. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [Crossref]
  9. C. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A 19, 1721 (2002).
    [Crossref]
  10. J. Lin, O. Rodríguez-Herrera, F. Kenny, D. Lara, and J. Dainty, “Fast vectorial calculation of the volumetric focused field distribution by using a three-dimensional fourier transform,” Opt. Express 20, 1060–1069 (2012).
    [Crossref] [PubMed]
  11. M. V. Berry, “Evanescent and real waves in quantum billiards and gaussian beams,” J. Phys. A 27, L391 (1994).
    [Crossref]
  12. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971 (1998).
    [Crossref]
  13. C. J. R. Sheppard and S. Saghafi, “Electromagnetic gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
    [Crossref]
  14. Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
    [Crossref]
  15. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electro. Lett. 7, 684–685 (1971).
    [Crossref]
  16. M. A. Alonso, R. Borghi, and M. Santarsiero, “New basis for rotationally symmetric nonparaxial fields in terms of spherical waves with complex foci,” Opt. Express 14, 6894–6905 (2006).
    [Crossref] [PubMed]
  17. N. J. Moore and M. A. Alonso, “Bases for the description of monochromatic, strongly focused, scalar fields,” J. Opt. Soc. Am. A 26, 1754–1761 (2009).
    [Crossref]
  18. N. J. Moore and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A 26, 2211–2218 (2009).
    [Crossref]
  19. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
    [Crossref]
  20. D. L. Andrews and M. Babiker, The angular momentum of light (Cambridge University, 2012).
    [Crossref]
  21. R. Gutiérrez-Cuevas and M. A. Alonso, “Polynomials of Gaussians and vortex-Gaussian beams as complete, transversely confined bases,” Opt. Lett. 42, 2205–2208 (2017).
    [Crossref] [PubMed]
  22. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [Crossref]
  23. G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1967).
  24. M. A. Alonso and N. J. T. Moore, in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, ed. (CRC, 2012), Chap. 4, pp. 97–141.
    [Crossref]
  25. K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
    [Crossref]
  26. M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
    [Crossref]
  27. M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13, 064016 (2011).
    [Crossref]
  28. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
    [Crossref]
  29. V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 3, 330–339 (2013).
    [Crossref]
  30. N. J. Moore and M. A. Alonso, “Closed form formula for mie scattering of nonparaxial analogues of gaussian beams,” Opt. Express 16, 5926–5933 (2008).
    [Crossref] [PubMed]
  31. N. J. Moore and M. A. Alonso, “Mie scattering of highly focused, scalar fields: an analytic approach,” J. Opt. Soc. Am. A 33, 1236–1243 (2016).
    [Crossref]

2017 (1)

2016 (1)

2013 (1)

V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 3, 330–339 (2013).
[Crossref]

2012 (1)

2011 (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13, 064016 (2011).
[Crossref]

2010 (1)

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

2009 (3)

2008 (1)

2007 (1)

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[Crossref]

2006 (1)

2002 (1)

1999 (1)

1998 (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971 (1998).
[Crossref]

1997 (1)

C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[Crossref]

1994 (1)

M. V. Berry, “Evanescent and real waves in quantum billiards and gaussian beams,” J. Phys. A 27, L391 (1994).
[Crossref]

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

1974 (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electro. Lett. 7, 684–685 (1971).
[Crossref]

1967 (1)

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
[Crossref]

1964 (1)

Aiello, A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

Alonso, M. A.

R. Gutiérrez-Cuevas and M. A. Alonso, “Polynomials of Gaussians and vortex-Gaussian beams as complete, transversely confined bases,” Opt. Lett. 42, 2205–2208 (2017).
[Crossref] [PubMed]

N. J. Moore and M. A. Alonso, “Mie scattering of highly focused, scalar fields: an analytic approach,” J. Opt. Soc. Am. A 33, 1236–1243 (2016).
[Crossref]

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13, 064016 (2011).
[Crossref]

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

N. J. Moore and M. A. Alonso, “Bases for the description of monochromatic, strongly focused, scalar fields,” J. Opt. Soc. Am. A 26, 1754–1761 (2009).
[Crossref]

N. J. Moore and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A 26, 2211–2218 (2009).
[Crossref]

N. J. Moore and M. A. Alonso, “Closed form formula for mie scattering of nonparaxial analogues of gaussian beams,” Opt. Express 16, 5926–5933 (2008).
[Crossref] [PubMed]

M. A. Alonso, R. Borghi, and M. Santarsiero, “New basis for rotationally symmetric nonparaxial fields in terms of spherical waves with complex foci,” Opt. Express 14, 6894–6905 (2006).
[Crossref] [PubMed]

M. A. Alonso and N. J. T. Moore, in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, ed. (CRC, 2012), Chap. 4, pp. 97–141.
[Crossref]

R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A.

Andrews, D. L.

D. L. Andrews and M. Babiker, The angular momentum of light (Cambridge University, 2012).
[Crossref]

Babiker, M.

D. L. Andrews and M. Babiker, The angular momentum of light (Cambridge University, 2012).
[Crossref]

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

M. V. Berry, “Evanescent and real waves in quantum billiards and gaussian beams,” J. Phys. A 27, L391 (1994).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Elsevier, 1980).

Courjon, D.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[Crossref]

Dainty, J.

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electro. Lett. 7, 684–685 (1971).
[Crossref]

Devaney, A. J.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

Grosjean, T.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[Crossref]

Gutiérrez-Cuevas, R.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Kenny, F.

Kotlyar, V. V.

V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 3, 330–339 (2013).
[Crossref]

Kovalev, A. A.

V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 3, 330–339 (2013).
[Crossref]

Kravtsov, Y. A.

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
[Crossref]

Lara, D.

Lin, J.

Liu, Y.

V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 3, 330–339 (2013).
[Crossref]

McCutchen, C.

McCutchen, C. W.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

Moore, N. J.

Moore, N. J. T.

M. A. Alonso and N. J. T. Moore, in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, ed. (CRC, 2012), Chap. 4, pp. 97–141.
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).
[Crossref]

O’Faolain, L.

V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 3, 330–339 (2013).
[Crossref]

Ostrovskaya, E. A.

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

Padgett, M. J.

Rodríguez-Herrera, O.

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, “Electromagnetic gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
[Crossref]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971 (1998).
[Crossref]

Santarsiero, M.

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, “Electromagnetic gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999).
[Crossref]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971 (1998).
[Crossref]

C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[Crossref]

Stafeev, S. S.

V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 3, 330–339 (2013).
[Crossref]

Szegö, G.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1967).

Török, P.

C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[Crossref]

Wolf, E.

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Elsevier, 1980).

Yao, A. M.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 3, 330–339 (2013).
[Crossref]

Electro. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electro. Lett. 7, 684–685 (1971).
[Crossref]

J. Math. Phys. (1)

A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[Crossref]

J. Mod. Opt. (1)

C. J. R. Sheppard and P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[Crossref]

J. Opt. (1)

M. A. Alonso, “The effect of orbital angular momentum and helicity in the uncertainty-type relations between focal spot size and angular spread,” J. Opt. 13, 064016 (2011).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

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

J. Phys. A (1)

M. V. Berry, “Evanescent and real waves in quantum billiards and gaussian beams,” J. Phys. A 27, L391 (1994).
[Crossref]

Opt. Commun. (1)

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (2)

K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, “Angular momenta and spin-orbit interaction of nonparaxial light in free space,” Phys. Rev. A 82, 063825 (2010).
[Crossref]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971 (1998).
[Crossref]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499 (1987).
[Crossref] [PubMed]

Radiophys. Quantum Electron. (1)

Y. A. Kravtsov, “Complex rays and complex caustics,” Radiophys. Quantum Electron. 10, 719–730 (1967).
[Crossref]

Other (7)

M. Born and E. Wolf, Principles of Optics (Elsevier, 1980).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).
[Crossref]

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

R. Borghi, M. Santarsiero, and M. A. Alonso, “Highly focused spirally polarized beams,” J. Opt. Soc. Am. A.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, 1967).

M. A. Alonso and N. J. T. Moore, in Mathematical Optics: Classical, Quantum, and Computational Methods, V. Lakshminarayanan, ed. (CRC, 2012), Chap. 4, pp. 97–141.
[Crossref]

D. L. Andrews and M. Babiker, The angular momentum of light (Cambridge University, 2012).
[Crossref]

Supplementary Material (1)

NameDescription
» Visualization 1: MOV (2170 KB)      $\theta$ dependence of the angular spectrum for $\phi=0$ (first row), intensity over the $x$-$y$ plane (second row) and the $y$-$z$ plane (third row) with $q$ ranging from 0 to 10 for four elements of the nonorthogonal scalar basis.

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Figures (10)

Fig. 1
Fig. 1 Normalized weight function for the nonorthogonal nonparaxial scalar basis, Eq. (12), for different values of m and q.
Fig. 2
Fig. 2 (a,b) θ dependence of ��n,m for ϕ = 0 for different orders and values of q. (c,d) Intensity over the x–y plane (c) and the y–z plane (d) for the orthogonal basis element ��2,1 with q = 10. The green curve on the x–y plane shows the intensity profile along the x axis.
Fig. 3
Fig. 3 Rms error as a function of q for an elliptical Gaussian beam focused by a thin lens with Δ = 0.5 and δw = 0.2 (a) for the modified basis n,m, and (b) the orthogonal basis ��n,m for truncation orders nmax = 0 (lighter curves) to nmax = 6 (darker curves).
Fig. 4
Fig. 4 Polarization vectors Vu and u × Vu for the (a) quasi-linear and (b) TE-TM bases.
Fig. 5
Fig. 5 Intensity over (left) the x–y plane, (center) x–z, and (right) the y–z plane for the orthogonal basis element 2 , 1 ( I ) with q = 10. The green curves on the x–y plane show the intensity profile along the x and y axes.
Fig. 6
Fig. 6 Normalized weight function for the nonorthogonal electromagnetic basis, Eq. (41), for different values of m and q.
Fig. 7
Fig. 7 Magnitude over the x-y plane and the x axis (green curve) for the polarized fields n , m ( + ), n , m ( II ) and n , m ( + ) with n = 0, m = 1 and q = 10.
Fig. 8
Fig. 8 Rms error as a function of q for a focused radially-polarized Laguerre-Gauss beam of order one with Δ = 0.5 (a) for the orthogonal, and (b) the nonorthogonal bases with truncation orders ranging from nmax = 0 (lighter curves) to nmax = 6 (darker curves) and m = −1, 1.
Fig. 9
Fig. 9 Rms error as a function of q for a focused linearly-polarized Gaussian beam of order one with Δ = 0.5 (a) for the orthogonal, and (b) the nonorthogonal bases with truncation orders ranging from nmax = 0 (lighter curves) to nmax = 6 (darker curves) and m = −2, 0, 2.
Fig. 10
Fig. 10 Rms error as a function of q for a focused radially-polarized Laguerre-Gauss beam of order one with Δ = 0.5 for the TE-TM (a) orthogonal, and (b) nonorthogonal bases with truncation orders ranging from lmax = 0 (lighter curves) to lmax = 6 (darker curves) and m = 0.

Equations (60)

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2 U ( r ) + k 2 U ( r ) = 0 ,
U ( r ) = 4 π A ( u ) exp ( i k u r ) d Ω ,
Λ l , m ( r ) = 4 π i l j l ( r ) Y l , m ( θ r , ϕ r ) ,
Y l , m ( θ , ϕ ) = σ m ( 2 l + 1 ) ( l m ) ! 4 π ( l + m ) ! P l ( m ) ( cos θ ) exp ( i m ϕ )
Λ l , m ( r ) = 4 π Y l , m ( θ , ϕ ) exp ( i u r ) d Ω .
𝒜 l , m ( u ; q ) = b l , m ( q ) exp ( q cos θ ) P l ( m ) [ cos Θ ( θ ; q ) ] exp ( i m ϕ ) ,
b l , m ( q ) = q ( 2 l + 1 ) ( 1 m ) ! 2 π sinh ( 2 q ) ( l + m ) ! and cos Θ ( θ ; q ) = exp ( 2 q cos θ ) cosh 2 q sinh 2 q .
n , m ( u ; q ) = b n + | m | , m ( q ) exp ( q cos θ ) { P n + | m | ( m ) [ cos Θ ( θ ; q ) ] sin | m | Θ ( θ ; q ) } sin | m | θ exp ( i m ϕ )
V ( m ) ( r ; q ) = Λ | m | , m [ r i q z ^ ] = Y | m | , m ( θ , ϕ ) exp ( q cos θ ) exp ( i r u ) d Ω
= 4 π i | m | c m j | m | [ ρ 2 + ( z i q ) 2 ] [ ρ ρ 2 ( z i q ) 2 ] | m | exp ( i m ϕ ) ,
𝒰 n , m ( r ; q ) = j = 0 n ν n , m ( j ) V ( m ) [ r ; ( 2 j + 1 ) q ] ,
W m ( θ ) = [ sin Θ ( θ ; q ) / sin θ ] | 2 m | .
𝒢 n , m ( u ; q ) = g n , m ( q ) exp ( q cos θ ) G n , m [ exp ( 2 q cos θ ) ; q ] sin | m | θ exp ( i m ϕ ) ,
exp ( 2 q ) exp ( 2 q ) G n , m ( ν ; q ) G n , m ( ν ; q ) w m ( ν ) d ν = h n , m ( q ) δ n , n , w m ( ν ) = ( 1 ln 2 ν 4 q 2 ) | m | .
𝒰 n , m ( r ; q ) = g n , m ( q ) j = 0 n γ n , m ( j ) ( q ) c m V ( m ) [ r ; ( 2 j + 1 ) q ] ,
A ˜ ( u ) = n = 0 n max m = m max m max a n , m 𝒴 n , m ( u )
a n , m = 4 π A ( u ) 𝒴 n , m * ( u ) W m ( θ ) d Ω
rms 2 = 4 π | A ( u ) A ˜ ( u ) | 2 d Ω 4 π | A ( u ) | 2 d Ω .
rms 2 = 1 n = 0 n max m = m max m max | a n , m | 2 4 π | A ( u ) | 2 d Ω .
A ( u ) = exp [ tan 2 θ 2 Δ 2 ( 1 + δ w cos 2 ϕ ) 2 ] H ( θ π / 2 ) cos θ ,
2 E ( r ) + E ( r ) = 0 , E ( r ) = 0 .
E ( r ) = 4 π A ( u ) exp ( i u r ) d Ω ,
Λ l , m ( I ) ( r ) = 4 π Z l , m ( u ) exp ( i u r ) d Ω , Λ l , m ( II ) ( r ) = 4 π Y l , m ( u ) exp ( i u r ) d Ω ,
Z l , m ( u ) = u × Y l , m ( u ) , Y l , m ( u ) = 1 l ( l + 1 ) L u Y l , m ( θ , ϕ ) ,
L u = i u × Ω = i u × ( ϕ ^ sin θ ϕ + θ ^ θ ) = i θ ^ sin θ ϕ i ϕ ^ θ ,
𝒢 n , m ( u ; q ) = V u 𝒢 n , m ( u ; q ) , 𝒵 n , m ( u ; q ) = u × V u 𝒢 n , m ( u ; q ) ,
𝒢 n , m ( u ; q ) = g n , m ( q ) exp ( q cos θ ) G n , m [ exp ( 2 q cos θ ) ; q ] sin | m | θ exp ( i m ϕ ) ,
4 π 𝒢 n , m * ( u ; q ) 𝒵 n , m ( u ; q ) d Ω = 0 ,
4 π 𝒢 n , m * ( u ; q ) 𝒢 n , m ( u ; q ) d Ω = 4 π 𝒵 n , m * ( u ; q ) 𝒵 n , m ( u ; q ) d Ω .
exp ( 2 k q ) exp ( 2 k q ) G n , m ( v ; q ) w ^ m ( v ; q ) G n , m ( v ; q ) d v = h n , m ( q ) δ n , n
w m ( v ; q ) = ( 1 + 1 2 q ln v ) 2 ( 1 1 4 q 2 ln 2 v ) | m | .
n , m ( II ) ( r ; q ) = 4 π 𝒢 n , m ( u ; q ) exp ( i u r ) d Ω = j = 0 n α n , m ( j ) ( q ) V r V ( m ) [ r ; ( 2 j + 1 ) q ] ,
V u V r = x ^ + x i y ^ × , u × V u i × V r = y ^ + y + i x ^ × .
𝒬 n , m ( u ; q ) = p n , m ( q ) exp ( q cos θ ) Q n , m [ cos Θ ( θ ; q ) ] sin | m | θ exp ( i m ϕ )
𝒬 n , m ( u ; 0 ) = p n , m ( 0 ) Q n , m ( cos θ ) sin | m | θ exp ( i m ϕ ) .
4 π 𝒬 n , m * ( u ; 0 ) V u V u 𝒬 n , m ( u ; 0 ) d Ω = δ n , n δ m , m .
1 1 Q n , m ( s ) Q n , m ( s ) ( 1 s ) | m | ( 1 + s ) | m + 2 | d s = δ n , n 2 π | p n , m ( 0 ) | 2 ,
Q n , m ( u ; q ) = V u 𝒬 n , m ( u ; q ) , 𝒳 n , m ( u ; q ) = u × V u 𝒬 n , m ( u ; q ) ,
𝒬 n , m ( u ; q ) = p n , m ( q ) exp ( q cos θ ) P n ( | m | , | m | + 2 ) [ cos Θ ( θ ; q ) ] sin | m | θ exp ( i m ϕ ) ,
p n , m ( q ) = q n ! ( 2 n + 2 | m | + 3 ) ( n + 2 | m | + 2 ) ! 2 2 | m | + 3 π sinh ( 2 q ) ( n + | m | ) ! ( n + | m | + 2 ) ! .
W m ( θ ) = [ 1 cos Θ ( θ ; q ) ] | m | [ 1 + cos Θ ( θ ; q ) ] | m | + 2 ( 1 cos θ ) | m | ( 1 + cos θ ) | m | + 2
V u ( ± ) = exp ( i π / 4 ) 2 ( V u ± i u × V u ) = exp ( i π / 4 ) ( u × ± × u ± i u × ± ) ,
V u ( ± ) = 1 + cos θ 2 exp ( i π / 4 ) exp ( ± i ϕ ) ( θ ^ ± i ϕ ^ ) ,
V r ( ± ) = exp ( i π / 4 ) [ ± + ( ± ) ± × ] .
A ˜ ( u ) = n = 0 n max m = m max m max [ c n , m ( I ) 𝒵 n , m ( u ) + c n , m ( II ) 𝒢 n , m ( u ) ]
c n , m ( I ) = 4 π 𝒵 n , m * ( u ) A ( u ) W m ( θ ) d Ω , c n , m ( II ) = 4 π 𝒢 n , m * ( u ) A ( u ) W m ( θ ) d Ω .
rms = A A ˜ A ˜ , A 2 = 4 π A * ( u ) A ( u ) d Ω .
rms 2 = 1 n = 0 n max m = m max m max [ | c n , m ( I ) | 2 + | c n , m ( II ) | 2 ] A 2 .
A ( u ) = 1 cos θ [ θ ^ T ( θ ) E ρ ( i ) ( f tan θ , ϕ , 0 ) + ϕ ^ T ( θ ) E ϕ ( i ) ( f tan θ , ϕ , 0 ) ]
T ( θ ) = 4 sin θ sin ( 3 θ / 2 ) cos ( θ / 2 ) , T ( θ ) = 4 sin θ cos ( θ / 2 ) sin ( 3 θ / 2 ) .
E ( i ) ( ρ , ϕ , 0 ) = E 0 ρ ^ ρ a exp ( ρ 2 / 2 a 2 ) ,
E ( i ) ( ρ , ϕ , 0 ) = E 0 x ^ exp ( ρ 2 / 2 a 2 ) .
exp ( 2 q ) exp ( 2 q ) G n , m ( v ) G n , m ( v ) ( 1 1 4 q 2 ln 2 v ) | m | + 1 d v = h n , m ( q ) δ n , n .
W m ( θ ) = [ 1 cos Θ ( θ ; q ) ] | m | + 1 [ 1 + cos Θ ( θ ; q ) ] | m | + 1 ( 1 cos θ ) | m | + 1 ( 1 + cos θ ) | m | + 1 ,
a b p n ( x ) p n ( x ) w ( x ) d x = h n δ n , n ,
μ j = a b w ( x ) x j d x .
p n ( x ) = | μ 0 μ 1 μ n μ 1 μ 2 μ n + 1 μ n 1 μ n μ 2 n 1 1 x x n | .
p n + 1 ( x ) = ( A n + B n x ) p n ( x ) C n p n 1 ( x )
A n = Δ n Δ n 1 ( K n + 1 Δ n K n Δ n 1 ) , B n = Δ n Δ n 1 , C n = B n 2 ,
Δ n = | μ 0 μ 1 μ n μ 1 μ 2 μ n + 1 μ n μ n + 1 μ 2 n | , K n = | μ 0 μ 1 μ n 2 μ n μ 1 μ 2 μ n 1 μ n + 1 μ n 1 μ n μ 2 n 3 μ 2 n 1 | .

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