Abstract

In this paper, we present a general analytical model for light scattering by arbitrary Vogel spiral arrays of circular apertures illuminated at normal incidence. This model suffices to unveil the fundamental mathematical structure of their complex Fraunhofer diffraction patterns and enables the engineering of optical beams carrying multiple values of orbital angular momentum (OAM). By performing analytical Fourier-Hankel decomposition of spiral arrays and far field patterns, we rigorously demonstrate the ability to encode specific numerical sequences onto the OAM values of diffracted optical beams. In particular, we show that these OAM values are determined by the rational approximations (i.e., the convergents) of the continued fraction expansions of the irrational angles utilized to generate Vogel spirals. These findings open novel and exciting opportunities for the manipulation of complex OAM spectra using dielectric and plasmonic aperiodic spiral arrays for a number of emerging engineering applications in singular optics, secure communication, optical cryptography, and optical sensing.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys.111(11), 113101 (2012).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. Dal Negro, “Multipolar second harmonic generation from planar arrays of Au nanoparticles with aperiodic order,” Opt. Express20(14), 15797–15806 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
  14. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
    [CrossRef] [PubMed]
  15. K. IIzuka, Elements of Photonics (John Wiley, 2002).
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  18. J. Havil, The Irrationals: A Story of the Numbers You Can't Count On (Princeton University Press, 2012).

2012 (6)

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev.6(2), 178–218 (2012).
[CrossRef]

J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express20(3), 3015–3033 (2012).
[CrossRef] [PubMed]

N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys.111(11), 113101 (2012).
[CrossRef]

J. Trevino, C. Forestiere, G. Di Martino, S. Yerci, F. Priolo, and L. Dal Negro, “Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells,” Opt. Express20(S3), A418–A430 (2012).
[CrossRef] [PubMed]

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. Dal Negro, “Multipolar second harmonic generation from planar arrays of Au nanoparticles with aperiodic order,” Opt. Express20(14), 15797–15806 (2012).
[CrossRef] [PubMed]

2011 (3)

J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett.11(5), 2008–2016 (2011).
[CrossRef] [PubMed]

S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic bandedge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express19(24), 23631–23642 (2011).
[CrossRef] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

2009 (1)

2008 (1)

2002 (1)

M. Naylor, “Golden, √2, and π flowers: a spiral story,” Math. Mag.75(3), 163–172 (2002).
[CrossRef]

Agrawal, A.

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Artoni, P.

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

Boriskina, S. V.

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev.6(2), 178–218 (2012).
[CrossRef]

Cao, H.

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Capretti, A.

Chen, J.

Dal Negro, L.

N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys.111(11), 113101 (2012).
[CrossRef]

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev.6(2), 178–218 (2012).
[CrossRef]

J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express20(3), 3015–3033 (2012).
[CrossRef] [PubMed]

J. Trevino, C. Forestiere, G. Di Martino, S. Yerci, F. Priolo, and L. Dal Negro, “Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells,” Opt. Express20(S3), A418–A430 (2012).
[CrossRef] [PubMed]

A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. Dal Negro, “Multipolar second harmonic generation from planar arrays of Au nanoparticles with aperiodic order,” Opt. Express20(14), 15797–15806 (2012).
[CrossRef] [PubMed]

J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett.11(5), 2008–2016 (2011).
[CrossRef] [PubMed]

Di Martino, G.

Forestiere, C.

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Genevet, P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Grattan, K. T. V.

Gregg, P.

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

Irrera, A.

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

Kats, M. A.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Kejalakshmy, N.

Lawrence, N.

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys.111(11), 113101 (2012).
[CrossRef]

Liew, S. F.

Miano, G.

Minissale, S.

Naylor, M.

M. Naylor, “Golden, √2, and π flowers: a spiral story,” Math. Mag.75(3), 163–172 (2002).
[CrossRef]

Negro, L. D.

Noh, H.

Parker, G. J.

Pecora, E. F.

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

Pollard, M. E.

Priolo, F.

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

J. Trevino, C. Forestiere, G. Di Martino, S. Yerci, F. Priolo, and L. Dal Negro, “Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells,” Opt. Express20(S3), A418–A430 (2012).
[CrossRef] [PubMed]

Rahman, B. M. A.

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Trevino, J.

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys.111(11), 113101 (2012).
[CrossRef]

A. Capretti, G. F. Walsh, S. Minissale, J. Trevino, C. Forestiere, G. Miano, and L. Dal Negro, “Multipolar second harmonic generation from planar arrays of Au nanoparticles with aperiodic order,” Opt. Express20(14), 15797–15806 (2012).
[CrossRef] [PubMed]

J. Trevino, S. F. Liew, H. Noh, H. Cao, and L. Dal Negro, “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Opt. Express20(3), 3015–3033 (2012).
[CrossRef] [PubMed]

J. Trevino, C. Forestiere, G. Di Martino, S. Yerci, F. Priolo, and L. Dal Negro, “Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells,” Opt. Express20(S3), A418–A430 (2012).
[CrossRef] [PubMed]

J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett.11(5), 2008–2016 (2011).
[CrossRef] [PubMed]

S. F. Liew, H. Noh, J. Trevino, L. D. Negro, and H. Cao, “Localized photonic bandedge modes and orbital angular momenta of light in a golden-angle spiral,” Opt. Express19(24), 23631–23642 (2011).
[CrossRef] [PubMed]

Walsh, G. F.

Yerci, S.

Yu, N.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

N. Lawrence, J. Trevino, and L. Dal Negro, “Aperiodic arrays of active nanopillars for radiation engineering,” J. Appl. Phys.111(11), 113101 (2012).
[CrossRef]

Laser Photon. Rev. (1)

L. Dal Negro and S. V. Boriskina, “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photon. Rev.6(2), 178–218 (2012).
[CrossRef]

Math. Mag. (1)

M. Naylor, “Golden, √2, and π flowers: a spiral story,” Math. Mag.75(3), 163–172 (2002).
[CrossRef]

Nano Lett. (1)

J. Trevino, H. Cao, and L. Dal Negro, “Circularly symmetric light scattering from nanoplasmonic spirals,” Nano Lett.11(5), 2008–2016 (2011).
[CrossRef] [PubMed]

Nanoscale (1)

E. F. Pecora, N. Lawrence, P. Gregg, J. Trevino, P. Artoni, A. Irrera, F. Priolo, and L. Dal Negro, “Nanopatterning of silicon nanowires for enhancing visible photoluminescence,” Nanoscale4(9), 2863–2866 (2012).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (2)

Science (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Other (6)

K. IIzuka, Elements of Photonics (John Wiley, 2002).

M. R. Schroeder, Number Theory in Science and Communication (Springer Verlag, 1985).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Oxford University Press, 2008).

J. Havil, The Irrationals: A Story of the Numbers You Can't Count On (Princeton University Press, 2012).

J. A. Adam, A Mathematical Nature Walk (Princeton University Press, 2009).

E. Macia, Aperiodic Structures in Condensed Matter: Fundamentals and Applications (CRC Press Taylor & Francis, 2009).

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Figures (6)

Fig. 1
Fig. 1

(a,b) GA and τ Vogel spirals with 2000 particles. (c,d) Analytically calculated far field radiation pattern of arrays in (a,b) respectively at a wavelength of 633nm for a structure with a0 = 14.5μm. The far field radiation pattern has been truncated with an angular aperture of 4°.

Fig. 2
Fig. 2

(a,b) µ and π Vogel spirals with 2000 particles. (c,d) Analytically calculated far field radiation pattern of arrays in (a,b) respectively at a wavelength of 633nm for a structure with a0 = 14.5μm. The far field radiation pattern has been truncated with a circular aperture.

Fig. 3
Fig. 3

Fourier-Hankel transforms of far field scattered radiation from GA (Fig. 1(c)), τ (Fig. 1(d)), µ (Fig. 2(c)) and π (Fig. 2(d)) Vogel spirals summed over the radial wavenumber kr.

Fig. 4
Fig. 4

(a-c) Far field radiation pattern of GA spirals with 500, 1000 and 4000 particles respectively. (d-f) Fourier-Hankel transform of (a-c), respectively, summed over the radial wavenumber kr at a wavelength of 633nm for a structure with a0 = 14.5μm. The far field radiation pattern has been truncated with an angular aperture of 4°.

Fig. 5
Fig. 5

(a,b) Far field radiation pattern of GA spirals truncated with different apertures, (a) 8°, (b) 4°. (c,d) Fourier-Hankel transform of (a,b), respectively, summed over the radial wavenumber kr at a wavelength of 633nm for a structure with a0 = 14.5μm and 2000 particles.

Fig. 6
Fig. 6

(a,b,c) Electric field pattern of radiation at 633nm from a GA spiral array (100 particles, a0 = 1.2μm) of circular apertures of radius 100nm at propagation distances of 0.2 λ, 100 λ and 400 λ, respectively. (d) Far field radiation pattern of the same GA spiral array shown in (a,b,c) for comparison.

Tables (2)

Tables Icon

Table 1 Listing of irrational angles as well as their corresponding rational approximations (p/q). E/M is a measure of the difficulty to approximate the irrational number with a given a set of rational approximates, where E is the absolute difference from the irrational value for a given p/q and M is the Hurwitz bound.

Tables Icon

Table 2 Listing of irrational angles and the peak positions in the Fourier-Hankel transform of the spirals in Figs. 3 and 5. The peaks correspond to the denominators of the rational approximations of the irrational aperture angles of the spirals.

Equations (44)

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

r n = n a 0
θ n =nα
ρ(r,θ)= n=1 N 1 r δ(r n a 0 ) δ(θnα)
E z=0 (r,θ)= E 0 r n=1 N δ(r n a 0 ) δ(θnα)
E ( ν r , ν θ )= m= + (j) m e jm ν θ 2π 0 r ρ m (r) J m (2π ν r r)dr
ρ m (r)= 1 2π π π E z=0 (r,θ) e jmθ dθ
H m { ρ m (r) }=2π 0 r ρ m (r) J m (2π ν r r)dr
ρ m (r)= E 0 2πr n=1 N δ(r n a 0 ) e jmnα
E ( ν r , ν θ )= m= + (j) m e jm ν θ 0 E 0 n=1 N δ(r n a 0 ) e jmnα J m (2π ν r r)dr
E ( ν r , ν θ )= E 0 n=1 N m= + (j) m J m (2π n a 0 ν r ) e jm( ν θ nα)
e jzcosφ = m= + (j) m J m (z) e jmφ
E ( ν r , ν θ )= E 0 n=1 N e j2π n a 0 ν r cos( ν θ nα)
E ( ν r , ν θ )= E 0 [ (a/ ν r ) J 1 (2π ν r a) ] n=1 N e j2π n a 0 ν r cos( ν θ nα)
f(m, k r )= 1 2π 0 0 2π rdrdθρ(r,θ) J m ( k r r) e imθ
f(m, k r )= 1 2π n=1 N 0 0 2π drdθδ(r n a 0 )δ(θnα) J m ( k r r) e imθ
f(m, k r )= 1 2π n=1 N J m ( k r a 0 n ) e inmα
ζ=[ a 0 ; a 1 , a 2 , a 3 ,...]= a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 a 4 +...
p n q n = a n p n1 + p n2 a n q n1 + q n2
E ( ν r , ν θ )= E 0 [ (a/ ν r ) J 1 (2π ν r a) ] n=1 N e j2π n a 0 ν r cos( ν θ nα)
f(m,2πr)= E 0 a 2π 0 0 2π d ν r d ν θ [ n=1 N e j2π n a 0 ν r cos( ν θ nα) ] J m (2πr ν r ) J 1 (2π ν r a) e im ν θ
= E 0 a 2π n=1 N 0 0 2π d ν r d ν θ e j2π n a 0 ν r cos( ν θ nα) J m (2πr ν r ) J 1 (2π ν r a) e im ν θ
J m ( k ρ ρ) e imφ = 1 2π 0 2π dβ e i k ρ ρcos(βφ)+imβ+imπ/2
f(m,2πr)= E 0 a 2π (i) m n=1 N e inmα 0 d ν r J m (2πr ν r ) J m (2π n a 0 ν r ) J 1 (2π ν r a)
f(m,β) n=1 N A mβ e inmα
E(x,y;d)= F 1 { H(2π ν x ,2π ν y ;d) E ^ (2π ν x ,2π ν y ;0) }
H( ν r ;d)= e j2πd 1/ λ 2 ν r 2
E ( ν r , ν θ )= E 0 n=1 N e j2π n a 0 ν r cos( ν θ nα)
E(r,θ;d)= F 1 { E 0 n=1 N e j2π n a 0 ν r cos( ν θ nα) e j2πd 1/ λ 2 ν r 2 }= F 1 [G( ν r , ν θ )]
E(r,θ;d)= m= + (j) m e jmθ 2π 0 ν r G m ( ν r ) J m (2π ν r r)d ν r
G m ( ν r )= 1 2π π π G( ν r , ν θ ) e jm ν θ d ν θ
G m ( ν r )= E 0 2π π π d ν θ n e j[ a 0 2π n ν r cos( ν θ nα)+2πd 1/ λ 2 ν r 2 m ν θ
G m ( ν r )= E 0 2π e j2πd 1/ λ 2 ν r 2 n=1 N π π d ν θ e j[2π a 0 n ν r cos( ν θ nα)m ν θ ]
J m ( k ρ ρ) e jmφ = 1 2π 0 2π dα e j k ρ ρcos(αφ)jmαjmπ/2
E(r,θ;d)= E 0 m= + (j) m e jmθ 0 ν r e j2πd 1/ λ 2 ν r 2 n=1 N π π d ν θ e j[2π a 0 n ν r cos( ν θ nα)m ν θ ] J m (2π ν r r)d ν r
E(r,θ;d)=2π E 0 m= + (j) m e jmθ e jmπ/2 n=1 N e jmnα 0 ν r e j2πd 1/ λ 2 ν r 2 J m (2π a 0 n ν r ) J m (2π ν r r)d ν r
J 0 ( k r R)= m= + J m ( k r r 1 ) J m ( k r r 2 ) e imθ
=2π E 0 n=1 N 0 ν r e j2πd 1/ λ 2 ν r 2 m= + e jm(θ+πnα) J m (2π a 0 n ν r ) J m (2π ν r r)d ν r
E(r,θ;d)=2π E 0 n=1 N 0 ν r e j2πd 1/ λ 2 ν r 2 J 0 ( k r R)d ν r
r 1 = a 0 n , r 2 =r, k r =2π ν r , R= r 1 2 + r 2 2 2 r 1 r 2 cos(β) , β=θ+πnα
E(r,θ;d)=2π E 0 a n=1 N 0 e j2πd 1/ λ 2 ν r 2 J 0 ( k r R) J 1 (2π ν r a)d ν r
J α (x)= (x/2) α Γ(α+1) F 0 1 (α+1; x 2 /4)
F p q ( a 1 ,..., a p ; b 1 ,..., b q ;z)= n=0 ( a 1 ) n ... ( a p ) n ( b 1 ) n ... ( b q ) n z n n!
(a) n =a(a+1)(a+2)...(a+n1), ( a 0 )=1
E(r,θ;d)=2πa E 0 n=1 N m=1 R 2m (m!) 2 0 e j2πd 1/ λ 2 ν r 2 (1) m [ (2π ν r ) 2 /4] m J 1 (a2π ν r )d ν r

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