Abstract

Axisymmetric radiating and scattering structures whose rotational invariance is broken by non-axisymmetric excitations present an important class of problems in electromagnetics. For such problems, a cylindrical wave decomposition formalism can be used to efficiently obtain numerical solutions to the full-wave frequency-domain problem. Often, the far-field, or Fraunhofer region is of particular interest in scattering cross-section and radiation pattern calculations; yet, it is usually impractical to compute full-wave solutions for this region. Here, we propose a generalization of the Stratton-Chu far-field integral adapted for 2.5D formalism. The integration over a closed, axially symmetric surface is analytically reduced to a line integral on a meridional plane. We benchmark this computational technique by comparing it with analytical Mie solutions for a plasmonic nanoparticle, and apply it to the design of a three-dimensional polarization-insensitive cloak.

© 2013 OSA

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References

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  1. G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Opt. Express20, 4176–4188 (2012).
    [CrossRef] [PubMed]
  2. J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters9, 887–891 (2009).
    [CrossRef] [PubMed]
  3. A. D. Greenwood and J.-M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag.47, 1260–1266 (1999).
    [CrossRef]
  4. R. K. Gordon and R. Mittra, “Finite element analysis of axisymmetric radomes,” IEEE Trans. Antennas Propag.41, 975–981 (1993).
    [CrossRef]
  5. Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys.111, 053105 (2012).
    [CrossRef]
  6. C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
    [CrossRef] [PubMed]
  7. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955).
    [CrossRef]
  8. J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56, 99–107 (1939).
    [CrossRef]
  9. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons Ltd, New York, 1957).
  10. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  11. U. Leonhardt, “Optical conformal mapping,” Science312, 1777–1780 (2006).
    [CrossRef] [PubMed]
  12. Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
    [CrossRef]
  13. N. Landy and D. R. Smith, “A full-parameter unidirectional metamaterial cloak for microwaves,” Nat. Materials12, 25–28 (2013).
    [CrossRef]
  14. Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt.13, 024002 (2011).
    [CrossRef]
  15. Y. Urzhumov and D. R. Smith, “Low-loss directional cloaks without superluminal velocity or magnetic,” Opt. Lett.37, 4471 (2012).
    [CrossRef] [PubMed]

2013 (1)

N. Landy and D. R. Smith, “A full-parameter unidirectional metamaterial cloak for microwaves,” Nat. Materials12, 25–28 (2013).
[CrossRef]

2012 (4)

G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response,” Opt. Express20, 4176–4188 (2012).
[CrossRef] [PubMed]

Y. Urzhumov and D. R. Smith, “Low-loss directional cloaks without superluminal velocity or magnetic,” Opt. Lett.37, 4471 (2012).
[CrossRef] [PubMed]

Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys.111, 053105 (2012).
[CrossRef]

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

2011 (1)

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt.13, 024002 (2011).
[CrossRef]

2009 (2)

Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
[CrossRef]

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters9, 887–891 (2009).
[CrossRef] [PubMed]

2006 (2)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312, 1777–1780 (2006).
[CrossRef] [PubMed]

1999 (1)

A. D. Greenwood and J.-M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag.47, 1260–1266 (1999).
[CrossRef]

1993 (1)

R. K. Gordon and R. Mittra, “Finite element analysis of axisymmetric radomes,” IEEE Trans. Antennas Propag.41, 975–981 (1993).
[CrossRef]

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955).
[CrossRef]

1939 (1)

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56, 99–107 (1939).
[CrossRef]

Chen, H.

Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
[CrossRef]

Chilkoti, A.

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Chu, L.

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56, 99–107 (1939).
[CrossRef]

Ciracì, C.

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Fernández-Domínguez, A. I.

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Gordon, R. K.

R. K. Gordon and R. Mittra, “Finite element analysis of axisymmetric radomes,” IEEE Trans. Antennas Propag.41, 975–981 (1993).
[CrossRef]

Greenwood, A. D.

A. D. Greenwood and J.-M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag.47, 1260–1266 (1999).
[CrossRef]

Hill, R.

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Jauho, A.-P.

Jin, J.-M.

A. D. Greenwood and J.-M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag.47, 1260–1266 (1999).
[CrossRef]

Kong, J. A.

Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
[CrossRef]

Kundtz, N. B.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt.13, 024002 (2011).
[CrossRef]

Landy, N.

N. Landy and D. R. Smith, “A full-parameter unidirectional metamaterial cloak for microwaves,” Nat. Materials12, 25–28 (2013).
[CrossRef]

Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys.111, 053105 (2012).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science312, 1777–1780 (2006).
[CrossRef] [PubMed]

Luo, Y.

Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
[CrossRef]

Maier, S. A.

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Mittra, R.

R. K. Gordon and R. Mittra, “Finite element analysis of axisymmetric radomes,” IEEE Trans. Antennas Propag.41, 975–981 (1993).
[CrossRef]

Mock, J. J.

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Mortensen, N. A.

Nordlander, P.

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters9, 887–891 (2009).
[CrossRef] [PubMed]

Pendry, J. B.

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt.13, 024002 (2011).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Prodan, E.

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters9, 887–891 (2009).
[CrossRef] [PubMed]

Ran, L.

Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
[CrossRef]

Raza, S.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Smith, D. R.

N. Landy and D. R. Smith, “A full-parameter unidirectional metamaterial cloak for microwaves,” Nat. Materials12, 25–28 (2013).
[CrossRef]

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Y. Urzhumov and D. R. Smith, “Low-loss directional cloaks without superluminal velocity or magnetic,” Opt. Lett.37, 4471 (2012).
[CrossRef] [PubMed]

Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys.111, 053105 (2012).
[CrossRef]

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt.13, 024002 (2011).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

Stratton, J.

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56, 99–107 (1939).
[CrossRef]

Toscano, G.

Urzhumov, Y.

Y. Urzhumov and D. R. Smith, “Low-loss directional cloaks without superluminal velocity or magnetic,” Opt. Lett.37, 4471 (2012).
[CrossRef] [PubMed]

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Urzhumov, Y. A.

Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys.111, 053105 (2012).
[CrossRef]

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt.13, 024002 (2011).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons Ltd, New York, 1957).

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955).
[CrossRef]

Wu, B.-I.

Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
[CrossRef]

Wubs, M.

Zhang, J.

Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
[CrossRef]

Zuloaga, J.

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters9, 887–891 (2009).
[CrossRef] [PubMed]

Can. J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

A. D. Greenwood and J.-M. Jin, “Finite-element analysis of complex axisymmetric radiating structures,” IEEE Trans. Antennas Propag.47, 1260–1266 (1999).
[CrossRef]

R. K. Gordon and R. Mittra, “Finite element analysis of axisymmetric radomes,” IEEE Trans. Antennas Propag.41, 975–981 (1993).
[CrossRef]

Y. Luo, J. Zhang, H. Chen, L. Ran, B.-I. Wu, and J. A. Kong, “A rigorous analysis of plane-transformed invisibility cloaks,” IEEE Trans. Antennas Propag.57, 3926–3933 (2009).
[CrossRef]

J. Appl. Phys. (1)

Y. A. Urzhumov, N. Landy, and D. R. Smith, “Isotropic-medium three-dimensional cloaks for acoustic and electromagnetic waves,” J. Appl. Phys.111, 053105 (2012).
[CrossRef]

J. Opt. (1)

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt.13, 024002 (2011).
[CrossRef]

Nano Letters (1)

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Letters9, 887–891 (2009).
[CrossRef] [PubMed]

Nat. Materials (1)

N. Landy and D. R. Smith, “A full-parameter unidirectional metamaterial cloak for microwaves,” Nat. Materials12, 25–28 (2013).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev.56, 99–107 (1939).
[CrossRef]

Science (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312, 1777–1780 (2006).
[CrossRef] [PubMed]

C. Ciracì, R. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Other (1)

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons Ltd, New York, 1957).

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Figures (3)

Fig. 1
Fig. 1

Scattering properties of a gold nanosphere of radius R = 30 nm are calculated in the far-field implementing Eq. (8). (b) Comparison between analytical calculations based on Mie solutions (solid lines) and COMSOL 2.5D simulations (markers). (c) Differential scattering cross-section for a gold nanosphere on the E-plane (blue) and H-plane (green) as defined in (a) for θi = 0. The wavelength of the incident radiation is λ0 = 520 nm.

Fig. 2
Fig. 2

Directional cloaking device obtained by revolution around the z-axis of the transformation (15). (a) Scheme of the structure. (b) Electric field component Ey = Eϕ cosϕ + Eρ sinϕ at λ0 = 3a/20.

Fig. 3
Fig. 3

Scattering properties of the cloaking device of Fig. 2. Normalized differential scattering cross-section of the uncloaked (left) and cloaked (right) object. The top and bottom plots refer to the scattering measured at the H-plane and E-plane respectively.

Equations (33)

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E ρ , ϕ , z ( ρ , ϕ , z ) = m E ρ , ϕ , z ( m ) ( ρ , z ) e i m ϕ , H ρ , ϕ , z ( ρ , ϕ , z ) = m H ρ , ϕ , z ( m ) ( ρ , z ) e i m ϕ .
( m ) × 1 μ r ( m ) × E ( m ) ε r k 2 E ( m ) = 0 ,
E = E 0 ( x ^ cos θ i + z ^ sin θ i ) e i k 0 x sin θ i e i k 0 z cos θ i .
E z = E 0 sin θ i e i k 0 z cos θ i m i m J m ( k 0 ρ sin θ i ) e i m ϕ .
E ϕ = E 0 cos θ i k 0 ρ sin θ i e i k 0 z cos θ i m m i m J m ( k 0 ρ sin θ i ) e i m ϕ .
E ρ = i E 0 cos θ i e i k 0 z cos θ i m i m J m ( k 0 ρ sin θ i ) e i m ϕ .
E far ( r ^ ) = i k 4 π r ^ × S [ n ^ × E η r ^ × ( n ^ × H ) ] e i k r r ^ d S .
( E ϕ far E θ far ) = i k 2 m i m e i m ϕ s [ K e ( m ) E ( m ) ( ρ , z ) + η K h ( m ) H ( m ) ( ρ , z ) ] e i k z cos θ ρ d s
K e ( m ) = ( n z m J m cos θ k ρ sin θ i J m n z cos θ J m n ρ sin θ n ρ m J m cos θ k ρ sin θ i J m n z n z m J m k ρ sin θ i J m n ρ ) ,
K h ( m ) = ( i J m n z n z m J m k ρ sin θ i J m n ρ n z m J m cos θ k ρ sin θ i J m n z cos θ J m n ρ sin θ n ρ m J m cos θ k ρ sin θ ) .
σ ext = 4 π k Im { E θ far ( 0 , π ) } .
W abs = 1 2 Ω Re { J * E } d V ,
W abs = 1 2 Re { m , n 0 2 π e i ( m n ) ϕ d ϕ Ω J * ( n ) E ( m ) ρ d ρ d z } ,
W abs = π m Ω Re { J ( m ) * E ( m ) } ρ d ρ d z .
I : ρ = ρ γ z b , ϕ = ϕ , z = z , II : ρ = χ ( ρ b ) , ϕ = ϕ , z = z ,
I : ε ρ ρ = 1 + γ 2 , ε ϕ ϕ = 1 , ε ρ z = ε z ρ = γ , ε z z = 1 , II : ε ρ ρ = 1 / χ ε ϕ ϕ = χ , ε z z = χ ,
E far ( r ^ ) = i k 4 π [ r ^ × S n ^ × E e i k 0 r r ^ d S η r ^ × r ^ × S n ^ × H e i k r r ^ d S ] ,
V = S V e i k r r ^ d S ,
V = m V ( m ) = m S M l ( ϕ ) V l ( m ) ( ρ , z ) e i m ϕ e i k r r ^ d S ,
M l ( ϕ ) = ( cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ) ,
r r ^ = ( ρ cos ϕ ρ sin ϕ z ) ( sin θ cos ϕ sin θ sin ϕ cos θ ) = ρ sin θ cos ( ϕ ϕ ) + z cos θ .
M l ( ϕ ) V l ( m ) = ( 0 0 V z ( m ) ) + 1 2 ( V ρ ( m ) + i V ϕ ( m ) V ϕ ( m ) i V ρ ( m ) 0 ) e i ϕ + 1 2 ( V ρ ( m ) i V ϕ ( m ) V ϕ ( m ) + i V ρ ( m ) 0 ) e i ϕ ,
E far ( ϕ , θ ) = m E far ( m ) ( θ ) e i m ϕ .
V ( m ) = S [ v 1 ( m ) ( ρ , z ) e i m ϕ + v 2 ( m ) ( ρ , z ) e i ( m + 1 ) ϕ + + v 3 ( m ) ( ρ , z ) e i ( m + 1 ) ϕ ] e i k z cos θ e i ξ cos ϕ d S ,
v 1 ( m ) = ( 0 0 V z ( m ) ) , v 2 ( m ) = 1 2 ( V ρ ( m ) + i V ϕ ( m ) V ϕ ( m ) i V ρ ( m ) 0 ) , v 3 ( m ) = 1 2 ( V ρ ( m ) i V ϕ ( m ) V ϕ ( m ) + i V ρ ( m ) 0 ) .
0 2 π e i ξ cos φ e i m φ d φ = 2 π i m J m ( ξ ) ,
V ( m ) = 2 π i m s [ v 1 ( m ) J m + i v 2 ( m ) J m + 1 i v 3 ( m ) J m 1 ] e i k z cos θ ρ d s ,
1 2 ( V ρ ( m ) i ( J m + 1 J m 1 ) V ϕ ( m ) ( J m + 1 + J m 1 ) V ρ ( m ) ( J m + 1 + J m 1 ) + i V ϕ ( m ) ( J m + 1 J m 1 ) 2 V z ( m ) J m ) = ( V ρ ( m ) i J m V ϕ ( m ) m J m / ξ V ρ ( m ) m J m / ξ i V ϕ ( m ) J m V z ( m ) J m ) ,
E far ( m ) ( θ ) = i k 4 π M S 1 ( 0 , θ ) r ^ × ( n ^ × E ( m ) η r ^ × n ^ × H ( m ) ) ,
M S ( ϕ , θ ) = ( cos ϕ sin θ sin ϕ 0 cos ϕ cos θ sin ϕ sin θ cos ϕ sin ϕ cos θ cos θ 0 sin θ ) .
E far ( m ) ( θ ) = ( E ϕ , far ( m ) E θ , far ( m ) ) = k 2 i m + 1 s [ K e ( m ) E ( m ) ( ρ , z ) + η K h ( m ) H ( m ) ( ρ , z ) ] e i k z cos θ ρ d s
K e ( m ) = ( n z m J m cos θ k ρ sin θ i J m n z cos θ J m n ρ sin θ n ρ m J m cos θ k ρ sin θ i J m n z n z m J m k ρ sin θ i J m n ρ ) ,
K h ( m ) = ( i J m n z n z m J m k ρ sin θ i J m n ρ n z m J m cos θ k ρ sin θ i J m n z cos θ J m n ρ sin θ n ρ m J m cos θ k ρ sin θ ) .

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