Abstract

A multipole expansion, based on spherical harmonics, provides an efficient method for calculating the field in the focal region of a lens for radially polarized illumination, or other illumination polarization and phase distributions, including vortex beams. The multipole approach also has the benefit of providing a simple measure of the purity of the longitudinal field mode. The method is also convenient for calculation of fields scattered by particles and calculation of optical trapping forces.

© 2012 Optical Society of America

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  32. E. T. Whittaker, “On the partial differential equations of mathematical physics,” Math. Ann. 57, 333–355 (1903).
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  39. C. J. R. Sheppard and S. Rehman, “Highly convergent focusing of light based on rotating dipole polarization,” Appl. Opt. 50, 4463–4467 (2011).
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    [CrossRef]
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  43. S. H. Goh and C. J. R. Sheppard, “High aperture focusing through a spherical interface: application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282, 1036–1041 (2009).
    [CrossRef]
  44. K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. Chen, “The effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28, 903–911 (2011).
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2011 (2)

2010 (1)

2009 (4)

S. H. Goh and C. J. R. Sheppard, “High aperture focusing through a spherical interface: application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282, 1036–1041 (2009).
[CrossRef]

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

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]

S. H. Simpson and S. Hanna, “Optical angular momentum transfer by Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 26, 625–638 (2009).
[CrossRef]

2008 (3)

2007 (2)

2004 (2)

2003 (3)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

C. C. Sun, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (3)

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172(2001).
[CrossRef]

2000 (2)

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87(2000).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7(2000).
[CrossRef]

1999 (1)

1997 (2)

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]

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177(1997).

1996 (3)

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of polarization on forces exerted by optical tweezers,” Optik 102, 181–190 (1996).

K. F. Ren, G. Grehan, and G. Gousebet, “Prediction of reverse radiation pressure by generalized Lorentz-Mie theory,” Appl. Opt. 35, 2702–2710 (1996).
[CrossRef] [PubMed]

J. J. Stamnes and V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

1995 (1)

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

1994 (1)

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

1991 (1)

1990 (1)

R. D. Romea and W. D. Kimura, “Modeling of inverse Cherenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[CrossRef]

1981 (1)

1976 (1)

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

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]

1973 (1)

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

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. A 253, 358–379 (1959).
[CrossRef]

1919 (1)

H. Weyl, “Ausbreitung elektromagnetische Wellen über einem ebenen Leiter,” Ann. Phys. 60, 481–500 (1919).
[CrossRef]

1903 (1)

E. T. Whittaker, “On the partial differential equations of mathematical physics,” Math. Ann. 57, 333–355 (1903).
[CrossRef]

Asatryan, A. A.

Asavei, T.

T. A. Nieminen, S. Parkin, T. Asavei, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical vortex trapping and the dynamics of particle rotation,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D.L.Andrews, ed. (Elsevier, 2008), pp. 195–236.

Balla, N. K.

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

Bouchal, Z.

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Brown, T. G.

Chen, X.

Chong, C. T.

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–506(2008).
[CrossRef]

Choudhury, A.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, 1966).

Davis, L. W.

de Sterke, C. M.

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]

A. J. Devaney and G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Dhayalan, V.

J. J. Stamnes and V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7(2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7(2000).
[CrossRef]

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7(2000).
[CrossRef]

Goh, S. H.

S. H. Goh and C. J. R. Sheppard, “High aperture focusing through a spherical interface: application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282, 1036–1041 (2009).
[CrossRef]

Gousebet, G.

Grehan, G.

Hanna, S.

Hao, X.

Heckenberg, N. R.

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008).
[CrossRef] [PubMed]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

T. A. Nieminen, S. Parkin, T. Asavei, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical vortex trapping and the dynamics of particle rotation,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D.L.Andrews, ed. (Elsevier, 2008), pp. 195–236.

Helseth, L. E.

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172(2001).
[CrossRef]

Kimura, W. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Cherenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[CrossRef]

Kuang, C. F.

Lalor, E.

G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[CrossRef]

Lee, G. C. F.

Leger, J. R.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7(2000).
[CrossRef]

Lim, K. M.

Liu, X.

Loke, V. L. Y.

T. A. Nieminen, S. Parkin, T. Asavei, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical vortex trapping and the dynamics of particle rotation,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D.L.Andrews, ed. (Elsevier, 2008), pp. 195–236.

Love, G. D.

Lukyanchuk, B.

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–506(2008).
[CrossRef]

Mishchenko, M. I.

Nieminen, T. A.

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008).
[CrossRef] [PubMed]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

T. A. Nieminen, S. Parkin, T. Asavei, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical vortex trapping and the dynamics of particle rotation,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D.L.Andrews, ed. (Elsevier, 2008), pp. 195–236.

Olivik, M.

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Panofsky, W. K. H.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Wiley, 1962).

Parkin, S.

T. A. Nieminen, S. Parkin, T. Asavei, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical vortex trapping and the dynamics of particle rotation,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D.L.Andrews, ed. (Elsevier, 2008), pp. 195–236.

Patsakos, G.

Phang, J. C. H.

Phillips, M.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Wiley, 1962).

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7(2000).
[CrossRef]

Rehman, S.

C. J. R. Sheppard and S. Rehman, “Highly convergent focusing of light based on rotating dipole polarization,” Appl. Opt. 50, 4463–4467 (2011).
[CrossRef] [PubMed]

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

Ren, K. F.

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. A 253, 358–379 (1959).
[CrossRef]

Romea, R. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Cherenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[CrossRef]

Rosin, A.

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of polarization on forces exerted by optical tweezers,” Optik 102, 181–190 (1996).

Rubinsztein-Dunlop, H.

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008).
[CrossRef] [PubMed]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 1005–1017 (2003).
[CrossRef]

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

T. A. Nieminen, S. Parkin, T. Asavei, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical vortex trapping and the dynamics of particle rotation,” in Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces, D.L.Andrews, ed. (Elsevier, 2008), pp. 195–236.

Saghafi, S.

Sheppard, C.

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–506(2008).
[CrossRef]

Sheppard, C. J. R.

K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. Chen, “The effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28, 903–911 (2011).
[CrossRef]

C. J. R. Sheppard and S. Rehman, “Highly convergent focusing of light based on rotating dipole polarization,” Appl. Opt. 50, 4463–4467 (2011).
[CrossRef] [PubMed]

C. J. R. Sheppard, N. K. Balla, and S. Rehman, “Performance parameters for highly-focused electromagnetic waves,” Opt. Commun. 282, 727–734 (2009).
[CrossRef]

S. H. Goh and C. J. R. Sheppard, “High aperture focusing through a spherical interface: application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282, 1036–1041 (2009).
[CrossRef]

C. J. R. Sheppard and E. Y. S. Yew, “Performance parameters for focusing of radial polarization,” Opt. Lett. 33, 497–499(2008).
[CrossRef] [PubMed]

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G. C. Sherman, J. J. Stamnes, and E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
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C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177(1997).

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H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–506(2008).
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[CrossRef]

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

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

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C. J. R. Sheppard, “Fundamentals of superresolution,” Micron 38, 772 (2007).
[CrossRef]

Nature Photon. (1)

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nature Photon. 2, 501–506(2008).
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Optik (2)

T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of polarization on forces exerted by optical tweezers,” Optik 102, 181–190 (1996).

C. J. R. Sheppard and P. Török, “Electromagnetic field in the focal region of an electric dipole wave,” Optik 104, 175–177(1997).

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J. J. Stamnes and V. Dhayalan, “Focusing of electric dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
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Figures (8)

Fig. 1
Fig. 1

Coordinate system ( r ^ , θ , ϕ ) is the spherical coordinate of the considered position; ( k ^ , α , β ) is the coordinate of the propagation direction.

Fig. 2
Fig. 2

Strengths of the multipole orders for a lens of angular semiaperture α m for ADW (upper) and radial uniform illumination (RU) (lower).

Fig. 3
Fig. 3

Ratios | p E 1 | | p E 2 | and | p E 1 | | p M 1 | for different polarization cases. The values increase with NA. For radial illumination, ADW gives a higher value than for RU. For the generalized linear polarization cases, for | p E 1 1 | | p M 1 1 | , UTE1 has a higher value than ED, which has a higher value than MD.

Fig. 4
Fig. 4

Ratio | p E 1 | | p E 2 | for radially polarized illumination for an annular lens with obscuration angle α 0 . The value increases with obscuration. The obscuration has more effect for high NAs, and more effect on RU than ADW.

Fig. 5
Fig. 5

Strengths of the multipole orders for a lens of angular semiaperture α m , for illumination of different polarizations: MD (upper), ED (middle), and UTE1 (lower).

Fig. 6
Fig. 6

Contour plots of the electric energy density for radial polarized illumination (ADW) with l max = 10 , 20 , 30 , and the result from direct integration for comparison, for the case of α m = π / 3 .

Fig. 7
Fig. 7

Contour plots of electric energy density for ED for α m = π / 3 : three subplots are from multipole theory and correspond to l max = 10 , 15 , 20 , and the fourth plot corresponds to the result from direct integration for comparison.

Fig. 8
Fig. 8

Contour plots of electric energy density for azimuthal polarization with a phase singularity n = 1 , for a system satisfying the sine condition a ( α ) = cos α and α m = π / 3 . The plots include three subplots from the multipole theory that correspond to the three maximum multipole coefficients l max = 10 , 15 , 20 that are used for calculating, and the result from direct integration is also presented for comparison. The larger the number of the multipole coefficients used, the better is the convergence. In particular, the region far from the focal point needs a higher order of the multipole.

Equations (44)

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P l m ( cos θ ) = ( 1 ) m ( l m ) ! ( l + m ) ! P l m ( cos θ ) ,
Y l m ( θ , ϕ ) = ( 1 ) m [ 2 l + 1 4 π ( l m ) ! ( l + m ) ! ] 1 2 P l m ( cos θ ) exp ( i m ϕ ) = c l m P l m ( cos θ ) exp ( i m ϕ ) .
Y l m ( θ , ϕ ) = c l m P l m ( cos θ ) exp ( i m ϕ ) = ( 1 ) l c l m P l m ( cos α ) exp ( i m β ) = ( 1 ) l Y l m ( α , β ) .
Y l m ( α , β ) = L s Y l m ( α , β ) = i ( β ^ α Y l m ( α , β ) α ^ 1 sin α β Y l m ( α , β ) ) = i ( β ^ α Y l m ( α , β ) α ^ i m sin α Y l m ( α , β ) ) .
E ^ ( s ^ ) = l = 1 m = l l ( i ) l 1 { p E l m [ s ^ × Y l m ( α , β ) ] + p M l m [ Y l m ( α , β ) ] } ,
E ( r ¯ ) = l = 1 m = l l [ p E l m N l m ( r ¯ ) + p M l m M l m ( r ¯ ) ] ,
N l m ( r ¯ ) = × × [ r ¯ h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) ] = r ^ l ( l + 1 ) r h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) + θ ^ [ 2 r r ( r h l ( 2 ) ) θ Y l m ( θ , ϕ ) ] + ϕ ^ i m sin θ 2 r r ( r h l ( 2 ) ) Y l m ( θ , ϕ ) ,
M l m ( r ¯ ) = i k × [ r ¯ h l ( 2 ) ( k r ) Y l m ( θ , ϕ ) ] = k h l ( 2 ) ( k r ) [ θ ^ m sin θ Y l m ( θ , ϕ ) + i ϕ ^ θ Y l m ( θ , ϕ ) ] .
p E l m = i l 1 l ( l + 1 ) 0 2 π 0 π ( E ^ ( s ^ ) × s ^ ) · Y l m * ( α , β ) sin α d α d β ,
p M l m = i l 1 l ( l + 1 ) 0 2 π 0 π E ^ ( s ^ ) · Y l m * ( α , β ) sin α d α d β ,
p E l m = i l l ( l + 1 ) c l m 0 2 π 0 π ( d P l m ( cos α ) d α E α i m P l m ( cos α ) sin α E β ) e i m β sin α d α d β ,
p M l m = i l l ( l + 1 ) c l m 0 2 π 0 π ( d P l m ( cos α ) d α E β + i m P l m ( cos α ) sin α E α ) e i m β sin α d α d β .
a l m = 0 π ( d P l m ( cos α ) d α ± i m P l m ( cos α ) sin α ) ( d P l m ( cos α ) d α i m P l m ( cos α ) sin α ) sin α d α = 0 π ( d P l m ( cos α ) d α d P l m ( cos α ) d α + m 2 P l m ( cos α ) sin α P l m ( cos α ) sin α ) sin α d α = 2 l ( l + 1 ) 2 l + 1 ( l + m ) ! ( l m ) ! δ l l 0 ,
b l m = 0 π ( d P l m ( cos α ) d α ± i m P l m ( cos α ) sin α ) ( d P l m ( cos α ) d α ± i m P l m ( cos α ) sin α ) sin α d α = 0 π ( d P l m ( cos α ) d α d P l m ( cos α ) d α m 2 P l m ( cos α ) sin α P l m ( cos α ) sin α ) sin α d α = 2 l ( l + 1 ) 2 m ( 2 l + 1 ) 2 l + 1 ( l + m ) ! ( l m ) ! δ l l 0 .
E r = k π l = 1 m = l l ( 1 ) m l ( l + 1 ) ( 2 l + 1 ) 1 2 [ ( l m ) ! ( l + m ) ! ] 1 2 ( p E l m [ j l 1 ( k r ) + j l + 1 ( k r ) ] P l m ( cos θ ) ) exp ( i m ϕ ) ,
E θ = k π l = 1 m = l l ( 1 ) m l ( l + 1 ) ( 2 l + 1 ) 1 2 [ ( l m ) ! ( l + m ) ! ] 1 2 ( p E l m [ j l 1 ( k r ) l j l + 1 ( k r ) l + 1 ] d P l m ( cos θ ) d θ m 2 l + 1 l ( l + 1 ) p M l m j l ( k r ) P l m ( cos θ ) sin θ ) exp ( i m ϕ ) ,
E ϕ = i k π l = 1 m = l l ( 1 ) m l ( l + 1 ) ( 2 l + 1 ) 1 2 [ ( l m ) ! ( l + m ) ! ] 1 2 ( m p E l m [ j l 1 ( k r ) l j l + 1 ( k r ) l + 1 ] P l m ( cos θ ) sin θ 2 l + 1 l ( l + 1 ) p M l m j l ( k r ) d P l m ( cos θ ) d θ ) exp ( i m ϕ ) ,
E ^ ( s ^ ) = a ( α ) exp ( i n β ) α ^ .
p E l m = ( 1 ) n + 1 i l [ π ( 2 l + 1 ) ] 1 2 l ( l + 1 ) [ ( l n ) ! ( l + n ) ! ] 1 2 δ m n 0 0 π a ( α ) d d α P l n ( cos α ) sin α d α , p M l m = ( 1 ) n + 1 i l 1 [ π ( 2 l + 1 ) ] 1 2 n l ( l + 1 ) [ ( l n ) ! ( l + n ) ! ] 1 2 δ m n 0 0 π a ( α ) P l m ( cos α ) d α ,
E ^ ( s ^ ) = a ( α ) exp ( i n β ) β ^ ,
p E l m = ( 1 ) n + 1 i l 1 [ π ( 2 l + 1 ) ] 1 2 n l ( l + 1 ) [ ( l n ) ! ( l + n ) ! ] 1 2 δ m n 0 0 π a ( α ) P l m ( cos α ) d α , p M l m = ( 1 ) n i l [ π ( 2 l + 1 ) ] 1 2 l ( l + 1 ) [ ( l n ) ! ( l + n ) ! ] 1 2 δ m n 0 0 π a ( α ) d d α P l m ( cos α ) sin α d α .
E r = k π l = | n | ( 1 ) n l ( l + 1 ) ( 2 l + 1 ) 1 2 [ ( l n ) ! ( l + n ) ! ] 1 2 ( p E l n [ j l 1 ( k r ) + j l + 1 ( k r ) ] P l n ( cos θ ) ) exp ( i n ϕ ) , E θ = k π l = | n | ( 1 ) n l ( l + 1 ) ( 2 l + 1 ) 1 2 [ ( l n ) ! ( l + n ) ! ] 1 2 ( p E l n [ j l 1 ( k r ) l j l + 1 ( k r ) l + 1 ] d P l n ( cos θ ) d θ n 2 l + 1 l ( l + 1 ) p M l n j l ( k r ) P l n ( cos θ ) sin θ ) exp ( i n ϕ ) , E ϕ = i k π l = | n | ( 1 ) n l ( l + 1 ) ( 2 l + 1 ) 1 2 [ ( l n ) ! ( l + n ) ! ] 1 2 ( n p E l n [ j l 1 ( k r ) l j l + 1 ( k r ) l + 1 ] P l n ( cos θ ) sin θ 2 l + 1 l ( l + 1 ) p M l n j l ( k r ) d P l n ( cos θ ) d θ ) exp ( i n ϕ ) .
p E l 0 = i l [ π ( 2 l + 1 ) ] 1 2 l ( l + 1 ) 0 π a ( α ) d P l ( cos α ) d α sin α d α ,
E r = l = 1 [ 3 2 l + 1 ] 1 2 p E l 0 l ( l + 1 ) 2 [ j l 1 ( k r ) + j l + 1 ( k r ) ] P l ( cos θ ) ,
E θ = l = 1 [ 3 2 l + 1 ] 1 2 p E l 0 l ( l + 1 ) 2 [ j l 1 ( k r ) l j l + 1 ( k r ) l + 1 ] d d θ P l ( cos θ ) .
a ( α ) = sin α for     α < α m ; a ( α ) = 0 for     α > α m ,
p E 1 0 = i 4 π 3 sin 4 ( α m 2 ) ( 2 + cos α m ) , p E 2 0 = 5 π 8 sin 4 α m .
F = 2 3 π | p E 1 0 | 2 0 α m | a ( α ) | 2 sin α d α = | p E 1 0 | .
p E 1 0 = i 3 π 4 ( α m sin α m cos α m ) , p E 2 0 = 5 π 6 sin 3 α m ,
p E l 1 = i l 1 l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 π a ( α ) P l m ( cos α ) d α , p M l 1 = i l l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 π a ( α ) d d α P l m ( cos α ) sin α d α .
E ^ ( s ^ ) = a ( α ) cos 2 α 2 { ( 1 S ( α ) ) cos β α ^ ( 1 + S ( α ) ) sin β β ^ } ,
p E l 1 = i l 2 l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 α m a ( α ) cos 2 α 2 ( [ 1 S ( α ) ] d d α P l 1 ( cos α ) + [ 1 + S ( α ) ] 1 sin α P l 1 ( cos α ) ) sin α d α , p M l 1 = i l 1 2 l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 α m a ( α ) cos 2 α 2 ( [ 1 + S ( α ) ] d d α P l 1 ( cos α ) + [ 1 S ( α ) ] 1 sin α P l 1 ( cos α ) ) sin α d α .
E r = l = 1 3 l ( l + 1 ) 2 ( 2 l + 1 ) p E l 1 [ j l 1 ( k r ) + j l + 1 ( k r ) ] P l 1 ( cos θ ) cos ϕ ,
E θ = l = 1 3 l ( l + 1 ) 2 ( 2 l + 1 ) [ p E l 1 ( j l 1 ( k r ) l j l + 1 ( k r ) l + 1 ) d d θ P l 1 ( cos θ ) 2 l + 1 l ( l + 1 ) p M l 1 j l ( k r ) P l 1 ( cos θ ) sin θ ] cos ϕ ,
E ϕ = l = 1 3 l ( l + 1 ) 2 ( 2 l + 1 ) [ p E l 1 ( j l 1 ( k r ) l j l + 1 ( k r ) l + 1 ) P l 1 ( cos θ ) sin θ 2 l + 1 l ( l + 1 ) p M l 1 j l ( k r ) d d θ P l 1 ( cos θ ) ] sin ϕ .
E ^ ( s ^ ) = cos 2 α 2 ( cos β α ^ sin β β ^ ) .
p E l 1 = i p M l 1 = i l 2 l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 α m cos 2 α 2 ( d d α P l 1 ( cos α ) + P l 1 ( cos α ) sin α ) sin α d α .
p E 1 1 = i p M 1 1 = i 12 3 π 2 sin 2 α m 2 ( 7 + 4 cos α m + cos 2 α m ) , p E 2 1 = i p M 2 1 = 1 4 5 π 6 sin 2 α m cos 4 α m 2 .
E ^ ( s ^ ) = a ( α ) ( cos α cos β α ^ sin β β ^ ) .
p E l 1 = i l 2 l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 α m a ( α ) ( cos α d d α P l 1 ( cos α ) + 1 sin α P l 1 ( cos α ) ) sin α d α , p M l 1 = i l 1 2 l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 α m a ( α ) ( d d α P l 1 ( cos α ) + cot α P l 1 ( cos α ) ) sin α d α .
p E 1 1 = i 2 π 6 sin 2 α m 2 ( 4 + cos α m + cos 2 α m ) , p M 1 1 = 1 4 3 π 2 sin 2 α m , p E 2 1 = 1 8 5 π 6 sin 2 α m ( 1 + cos 2 α m ) , p M 2 1 = i 4 5 π 6 sin 2 α m cos α m .
E ^ ( s ^ ) = 2 sin β β ^ .
p E l 1 = i l l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 α m P l 1 ( cos α ) d α , p M l 1 = i l 1 l ( l + 1 ) [ π ( 2 l + 1 ) l ( l + 1 ) ] 1 2 0 α m d d α P l 1 ( cos α ) sin α d α .
p E 1 1 = i 3 π 2 sin 2 α m 2 , p M 1 1 = 1 4 3 π 2 sin 2 α m , p E 2 1 = 1 4 5 π 6 sin 2 α m , p M 2 1 = i 6 5 π 6 sin 2 α m 2 ( 2 cos 2 α m + 2 cos α m 1 ) .

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