Abstract

Many approaches for producing optical vortices have been developed both for fundamental interests of science and for engineering applications. In particular, the approach with direct excitation of several emitters has a potential to control the topological charges with a control of the source conditions without any modifications of structures of the system. In this paper, we investigate the propagation properties of the optical vortices emitted from a collectively polarized electric dipole array as a simple model of the several emitters. Using an analytical approach based on the Jacobi-Anger expansion, we derive a relationship between the topological charge of the optical vortices and the source conditions of the emitter, and clarify and report our new finding; there exists an intrinsic split of the singular points in the electric field due to the spin-orbit interaction of the dipole fields.

© 2015 Optical Society of America

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References

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2015 (1)

2014 (2)

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

2013 (2)

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21, 15755–15764 (2013).
[Crossref] [PubMed]

2012 (3)

F. Ricci, W. Löffler, and M. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20, 22961–22975 (2012).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (1)

2008 (1)

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

2006 (2)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

A. A. Savchenkov, A. B. Matsko, I. Grudinin, E. A. Savchenkova, D. Strekalov, and L. Maleki, “Optical vortices with large orbital momentum: generation and interference,” Opt. Express 14, 2888–2897 (2006).
[Crossref] [PubMed]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

1993 (1)

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

1992 (2)

N. Heckenberg, R. McDuff, C. Smith, and A. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref] [PubMed]

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

1987 (1)

K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987).
[Crossref]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Ando, T.

Andrews, D. L.

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

Aoki, N.

Barnett, S. M.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Basistiy, I.

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Bazhenov, V. Y.

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Boyd, R. W.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Bradshaw, D. S.

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

Cai, X.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Chen, X.

Cheng, C.

Chujo, K.

Coles, M. M.

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

Cui, J.

De Leon, I.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fukuchi, N.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Grudinin, I.

Hara, T.

Heckenberg, N.

Huang, C.

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Inoue, T.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Johnson-Morris, B.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Karimi, E.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Lavery, M. P.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Li, X.

Li, Z.

Liang, G.

Löffler, W.

Luo, X.

Ma, X.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Maleki, L.

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Matsko, A. B.

Matsumoto, N.

McDuff, R.

Miyamoto, K.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Morita, R.

Nakamura, K.

Nicholls, K.

K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987).
[Crossref]

Nye, J.

K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987).
[Crossref]

O’Brien, J. L.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Ohtake, Y.

Okida, M.

Omatsu, T.

Padgett, M. J.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

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

Pan, W.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Pu, M.

Qassim, H.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Ricci, F.

Savchenkov, A. A.

Savchenkova, E. A.

Schulz, S. A.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Smith, C.

Sorel, M.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Soskin, M.

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Speirits, F. C.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Spreeuw, R.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Strain, M. J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Strekalov, D.

Thompson, M. G.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Upham, J.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

van Exter, M.

Vasnetsov, M. V.

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

White, A.

Williams, M. D.

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Woerdman, J.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yao, A. M.

Yu, S.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Zhang, M.

Zhao, B.

Zhu, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Adv. Opt. Photon. (1)

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

J. Phys. A: Math. Gen. (1)

K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987).
[Crossref]

Light Sci. Appl. (1)

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Nat. Photonics (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Nature (2)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Opt. Commun. (1)

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Figures (5)

Fig. 1
Fig. 1

Optical vortices of x component of electric field generated by an electric dipole array which is collectively polarized to x direction at z = 100λ. (a) An illustration of our system.(b),(d) and (f) Phase distribution of electric field in the case of (N.q) = (3, 1), (5, 2) and (7, 3) respectively. (c), (e) and (f) Phase distribution of magnetic field in the case of (N.q) = (3, 1), (5, 2) and (7, 3) respectively.

Fig. 2
Fig. 2

The topological charge of the optical vortex with respect to the source conditions with the collective polarization of (sx = 1, sy = 0). (a) The relationship between Qmin and (N, q). The phase profiles of the electric and the magnetic fields are shown in (b) ∼ (d) with the source conditions of (N, q) = (9, 5), (N, q) = (9, 4) and (N, q) = (9, 1) respectively.

Fig. 3
Fig. 3

The correspondence between the coordinates of the singular points (right) and the collective polarizations on the Poincare sphere (left). The dots of same color in both of the left and the right fiure shows the same angle of β and γ. The colered lines in the right figure show the trajectory in the same Qmin. The characters in the left figure indicate the collective polarization states. H: horizontal polarization, V: vertical polarization, D: diagonal polarization, A: anti-diagonal polarization, L: left circular polarization and R: right circular polarization.

Fig. 4
Fig. 4

The results of the numerical calculations (a) the topological charges around the coordinate of ζ1(0, 0) for each z. (b) the topological charges around the coordinate of ζ1(β, γ) at z = 100λ for each β and γ.

Fig. 5
Fig. 5

Amplitude profiles of the optical vortex in each topological charge.

Equations (45)

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E ( r ) = j = 1 N e i k | ξ j | 4 π ε 0 { k 2 ( ξ j | ξ j | × p ) × ξ j | ξ j | 2 + [ 3 ξ j | ξ j | ( ξ j | ξ j | p ) p ] ( 1 | ξ j | 3 i k | ξ j | 2 ) }
H ( r ) = j = 1 N c k 2 e i k | ξ j | 4 π ( ξ j | ξ j | × p ) 1 | ξ j | ( 1 1 i k | ξ j | )
e i k | ξ j | exp [ i k z ( 1 + ρ 2 + a 2 2 z 2 ) ] exp [ i k ρ a z cos ( ϕ ϕ j π ) ] = e i k ψ ( z ) m ( i ) m J m ( k ρ a z ) e i m ( ϕ ϕ j )
( ξ j | ξ j | × p ) × ξ j | ξ j | 2 p e i q φ j z 3 ( s x z 2 3 2 s x ξ x 2 1 2 s x ξ y 2 s y ξ x ξ y s y z 2 3 2 s y ξ y 2 1 2 s y ξ x 2 s x ξ x ξ y z ( s x ξ x + s y ξ y ) ) Λ j z e i q φ j
( ξ j | ξ j | × p ) 1 | ξ j | p e i q φ j z 3 ( s y z 2 + s y ( ξ x 2 + ξ y 2 ) s x z 2 s x ( ξ x 2 + ξ y 2 ) s x ξ y z + s y ξ x z ) Π j z e i q φ j
E ( r ) k 2 4 π ε 0 z e i k ψ ( z ) Q ( s x F Q 0 ( r ) + F Q SOI ( r ) s y F Q 0 ( r ) i F Q SOI ( r ) G Q SOI , + ( r ) )
H ( r ) c k 4 π z e i k ψ ( z ) Q ( s y F Q 0 ( r ) s x F Q 0 ( r ) i G Q SOI , ( r ) )
F Q 0 ( r ) = [ ( 1 ρ 2 + a 2 z 2 ) f Q + ρ a z 2 ( f Q + 1 + f Q 1 ) ) ] e i Q ϕ ,
F Q SOI ( r ) = s [ ρ a 2 z 2 f Q + 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q + 2 ] e i ( Q + 2 ) ϕ + s + [ ρ a 2 z 2 f Q 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q 2 ] e i ( Q 2 ) ϕ ,
G Q SOI , ± ( r ) = s ( ρ 2 z f Q a 2 z f Q + 1 ) e i ( Q + 1 ) ϕ ± s + ( ρ 2 z f Q a 2 z f Q 1 ) e i ( Q 1 ) ϕ ,
J Q ( k ρ a z ) = ( 1 ) sgn ( Q ) 1 2 | Q | t = 0 ( 1 ) t t ! ( t + | Q | ) ! ( k ρ a 2 z ) 2 t + | Q | ( 1 ) sgn ( Q ) 1 2 | Q | 1 | Q | ! ( k ρ a 2 z ) | Q |
Q F Q 0 ( r ) ( i ) | Q min | | Q min | ! ( 1 ) sgn ( Q min 1 ) 2 | Q min | ( k ρ a 2 z ) Q min e i Q min ϕ
Q F Q SOI ( r ) { 0 ( Q min = ± 1 ) s + ( i ) Q min 2 ( Q min 2 ) ! ( k ρ a 2 z ) Q min 2 a 2 4 z 2 e i ( Q min 2 ) ϕ ( Q min 2 ) ( 1 ) | Q min + 2 | s i | Q min + 2 | | Q min + 2 | ! ( k ρ a 2 z ) | Q min + 2 | a 2 4 z 2 e i ( Q min + 2 ) ϕ ( Q min 2 )
Q G Q SOI , ± ( r ) { s + ( i ) Q min 1 ( Q min 1 ) ! ( k ρ a 2 z ) Q min 1 a 2 z e i ( Q min 1 ) ϕ ( Q min 1 ) ( 1 ) | Q min + 1 | s i | Q min + 1 | | Q min + 1 | ! ( k ρ a 2 z ) | Q min + 1 | a 2 z e i ( Q min + 1 ) ϕ ( Q min 1 )
E x ( r ) k 2 4 π ε 0 z ( i k ρ ) Q min 2 ( Q min 2 ) ! ( a 2 z ) Q min e i ( Q min 2 ) ϕ [ s x Q min ( Q min 1 ) ( k ρ ) 2 e i 2 ϕ + s + ]
E y ( r ) k 2 4 π ε 0 z ( i k ρ ) Q min 2 ( Q min 2 ) ! ( a 2 z ) Q min e i ( Q min 2 ) ϕ [ s y Q min ( Q min 1 ) ( k ρ ) 2 e i 2 ϕ + i s + ] .
ζ 1 ( β , γ ) = { ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( 0 γ π 2 , 3 π 2 γ 2 π ) ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( π 2 γ 2 π )
ζ 2 ( β , γ ) = { ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( 0 γ π 2 , 3 π 2 γ 2 π ) ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( π 2 γ 2 π )
ζ ± ( β , γ ) ± λ Q min ( Q min 1 ) 2 π 2 1 sin β sin γ cos β 2 ± ( tan β 2 sin γ 1 )
D = λ Q min ( Q min 1 ) 2 π .
Q N 1 2 π c d l Arg [ E x ( r ) ] = 1 2 π x π d ϕ 1 Arg [ E x ( R N cos ϕ 1 , R N sin ϕ 1 , z ) ]
E ( r ) k 2 4 π ε 0 z m = ( i ) m J m ( k ρ a z ) e i m ϕ j = 1 N Λ j e i ( q m ) φ j
H ( r ) c k 4 π z m = ( i ) m J m ( k ρ a z ) e i m ϕ j = 1 N Π j e i ( q m ) φ j ,
[ Λ j ] x = s x ( 1 3 ρ 2 2 z 2 cos 2 ϕ 3 a 2 2 z 2 cos 2 φ j + 3 ρ a z 2 cos ϕ cos φ j ρ 2 2 z 2 sin 2 ϕ a 2 2 z 2 sin 2 φ j + ρ a z 2 sin ϕ sin φ j ) s y { ρ 2 z 2 cos ϕ sin ϕ + a 2 z 2 cos φ j sin φ j ρ a z 2 ( cos φ sin φ j + cos φ j sin φ ) } = s x ( 1 ρ 2 + a 2 z 2 ρ 2 2 z 2 cos 2 ϕ a 2 2 z 2 cos 2 φ j s + 3 ρ a z 2 cos ϕ cos φ j + ρ a z 2 sin ϕ sin φ j ) s y ( ρ 2 2 z 2 sin 2 ϕ + a 2 2 z 2 sin 2 φ j ρ a 2 z 2 sin ( ϕ + φ j ) )
j = 1 N [ Λ j ] x e i ( q m ) φ j = n = s x [ ( 1 ρ 2 + a 2 z 2 ) δ q m , n N + ρ 2 4 z 2 ( e i 2 ϕ + e i 2 ϕ ) δ q m , n N a 2 4 z 2 ( δ q m + 2 , n N + δ q m 2 , n N ) + 3 ρ a 4 z 2 ( e i ϕ + e i ϕ ) ( δ q m + 1 , n N + δ q m 1 , n N ) ρ a 4 z 2 ( e i ϕ e i ϕ ) ( δ q m + 1 , n N δ q m 1 , n N ) ] n = s y [ ρ 2 i 4 z 2 ( e i 2 ϕ e i 2 ϕ ) δ q m , n N + a 2 i 4 z 2 ( δ q m + 2 , n N δ q m 2 , n N ) ρ a i 2 z 2 ( e i ϕ δ q m + 1 , n N e i ϕ δ q m 1 , n N ) ]
j = 1 N [ Λ j ] y e i ( q m ) φ j = n = s y [ ( 1 ρ 2 + a 2 z 2 ) δ q m , n N + ρ 2 4 z 2 ( e i 2 ϕ + e i 2 ϕ ) δ q m , n N + a 2 4 z 2 ( δ q m + 2 , n N + δ q m 2 , n N ) 3 ρ a 4 z 2 ( e i ϕ e i ϕ ) ( δ q m + 1 , n N δ q m 1 , n N ) + ρ a 4 z 2 ( e i ϕ + e i ϕ ) ( δ q m + 1 , n N + δ q m 1 , n N ) ] n = s x [ ρ 2 i 4 z 2 ( e i 2 ϕ e i 2 ϕ ) δ q m , N + a 2 i 4 z 2 ( δ q m + 2 , n N δ q m 2 , n N ) ρ a i 2 z 2 ( e i ϕ δ q m + 1 , n N e i ϕ δ q m 1 , n N ) ]
j = 1 N [ Λ j ] z e i ( q m ) φ j = n = [ s x { ρ 2 z ( e i ϕ + e i ϕ ) δ q m , n N a 2 z ( δ q m + 1 , n N + δ q m 1 , n N ) } + s y { ρ i 2 z ( e i ϕ e i ϕ ) δ q m , n N a i 2 z ( δ q m + 1 , n N δ q m 1 , n N ) } ]
E x ( r ) k 2 4 π ε 0 z Q s x ( f Q ρ 2 + a 2 z 2 f Q + ρ a z 2 ( ( f Q + 1 + f Q 1 ) ) ) e i Q ϕ + Q s [ ρ a 2 z 2 f Q + 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q + 2 ] e i ( Q + 2 ) ϕ + Q s + [ ρ a 2 z 2 f Q 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q 2 ] e i ( Q 2 ) ϕ ,
E y ( r ) k 2 4 π ε 0 z Q s y ( f Q ρ 2 + a 2 z 2 f Q + ρ a z 2 ( ( f Q + 1 + f Q 1 ) ) ) e i Q ϕ i Q s [ ρ a 2 z 2 f Q + 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q + 2 ] e i ( Q + 2 ) ϕ i Q s + [ ρ a 2 z 2 f Q 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q 2 ] e i ( Q 2 ) ϕ ,
E z ( r ) k 2 4 π ε 0 z Q [ s ( ρ 2 z f Q a 2 z f Q + 1 ) e i ( Q + 1 ) ϕ + s + ( ρ 2 z f Q a 2 z f Q 1 ) e i ( Q 1 ) ϕ ] ,
Π j = 1 z 2 ( s y z 2 + s y ( ξ x 2 + ξ y 2 ) s x z 2 s x ( ξ x 2 + ξ y 2 ) s x ξ y z + s y ξ x z ) ,
j = 1 N [ Π j ] x e i ( q m ) φ j = s y n [ ( 1 ρ 2 + a 2 z 2 ) δ q m , n N + ρ a z 2 ( e i ϕ δ q m 1 , n N + e i ϕ δ q m + 1 , n N ) ] ,
j = 1 N [ Π j ] y e i ( q m ) φ j = s x n [ ( 1 ρ 2 + a 2 z 2 ) δ q m , n N + ρ a z 2 ( e i ϕ δ q m 1 , n N + e i ϕ δ q m + 1 , n N ) ] ,
j = 1 N [ Π j ] z e i ( q m ) φ j = i s n ( ρ 2 z e i ϕ δ q m , n N a 2 z δ q m + 1 , n N ) i s + n ( ρ 2 z e i ϕ δ q m , n N a 2 z δ q m 1 , n N ) ,
H x ( r ) = c k 4 π z s y e i Q ϕ n [ ( 1 ρ 2 + a 2 z 2 ) f Q + ρ a z 2 ( f Q 1 + f Q + 1 ) ] ,
H y ( r ) = c k 4 π z s x e i Q ϕ n [ ( 1 ρ 2 + a 2 z 2 ) f Q + ρ a z 2 ( f Q 1 + f Q + 1 ) ] ,
H z ( r ) = i c k 4 π z [ s e i ( Q + 1 ) ϕ n ( ρ 2 z f Q f Q + 1 ) s + e i ( Q 1 ) ϕ n ( ρ 2 z f Q a 2 z f Q 1 ) ] .
A s x ρ 2 e i 2 ϕ + s + = [ A cos β 2 ( x 2 y 2 + i 2 x y ) + cos β 2 + i sin β 2 cos γ sin β 2 sin γ ] = [ A cos β 2 ( x 2 y 2 ) + cos β 2 sin β 2 sin γ + i ( 2 A x y cos β 2 + sin β 2 sin γ ) ]
ϕ = tan 1 [ Im [ G ] Re [ G ] ] = tan 1 [ 2 A x y cos β 2 + sin β 2 cos γ A cos β 2 ( x 2 y 2 ) + cos β 2 sin β 2 sin γ ] = tan 1 [ 2 x y + 1 A tan β 2 cos γ ( x 2 y 2 ) + 1 A 1 A tan β 2 sin γ ] .
tan 1 [ y ζ y x ζ x + y + ζ y x + ζ x 1 y ζ y x ζ x y + ζ y x + ζ x ] = tan 1 y + ζ y x + ζ x + tan 1 y ζ y x ζ x
ζ x ζ y = 1 2 A tan β 2 cos γ ,
ζ y 2 ζ x 2 = 1 A 1 A tan β 2 sin γ .
ζ 1 ( β , γ ) = { ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( 0 γ π 2 , 3 π 2 γ 2 π ) ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( π 2 γ 2 π )
ζ 2 ( β , γ ) = { ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( 0 < γ π 2 , 3 π 2 γ 2 π ) ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( π 2 γ 2 π )
ζ ± ( β , γ ) ± λ Q min ( Q min 1 ) 2 π 2 1 sin β sin γ cos β 2 ± ( tan β 2 sin γ 1 )

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