Abstract

We discuss vector Hankel beams with circular polarization. These beams appear as a generalization of a spherical wave with an embedded optical vortex with topological charge n. Explicit analytical relations to describe all six projections of the E- and H-field are derived. The relations are shown to satisfy Maxwell's equations. Hankel beams with clockwise and anticlockwise circular polarization are shown to have peculiar features while propagating in free space. Relations for the Poynting vector projections and the angular momentum in the far field are also obtained. It is shown that a Hankel beam with clockwise circular polarization has radial divergence (ratio between the radial and longitudinal projections of the Poynting vector) similar to that of the spherical wave, while the beam with the anticlockwise circular polarization has greater radial dependence. At n = 0, the circularly polarized Hankel beam has non-zero spin angular momentum. At n = 1, power flow of the Hankel beam with anticlockwise polarization consists of two parts: right-handed helical flow near the optical axis and left-handed helical flow in periphery. At n ≥2, power flow is directed along the right-handed helix regardless of the direction of the circular polarization. Power flow along the optical axis is the same for the Hankel beams of both circular polarizations, if they have the same topological charge.

© 2017 Optical Society of America

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References

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  1. S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85(2-3), 159–161 (1991).
    [Crossref]
  2. Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
    [Crossref]
  3. R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic fields,” Opt. Commun. 133(1-6), 315–327 (1997).
    [Crossref]
  4. R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71(3), 033411 (2005).
    [Crossref]
  5. Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. in Electr. Res. Lett. 5, 57–71 (2008).
  6. Y. Wang, W. Dou, and H. Meng, “Vector analyses of linearly and circularly polarized Bessel beams using Hertz vector potentials,” Opt. Express 22(7), 7821–7830 (2014).
    [Crossref] [PubMed]
  7. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Nonparaxial Hankel vortex beams of the first and second types,” Comput. Opt. 39, 299–304 (2015).
    [Crossref]
  8. V. V. Kotlyar and A. A. Kovalev, “Vectorial Hankel laser beams carrying orbital angular momentum,” Comput. Opt. 39(4), 449–452 (2015).
    [Crossref]
  9. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Vectorial rotating vortex Hankel laser beams,” J. Opt. 18(9), 095602 (2016).
    [Crossref]
  10. A. Cerjan and C. Cerjan, “Orbital angular momentum of Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 28(11), 2253–2260 (2011).
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    [Crossref] [PubMed]
  14. K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cedra, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
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  16. S. N. Khonina, D. A. Savelyev, and N. L. Kazanskiy, “Vortex phase elements as detectors of polarization state,” Opt. Express 23(14), 17845–17859 (2015).
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  17. S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
    [Crossref]
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  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Math. Series, 1965).

2016 (1)

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Vectorial rotating vortex Hankel laser beams,” J. Opt. 18(9), 095602 (2016).
[Crossref]

2015 (3)

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Nonparaxial Hankel vortex beams of the first and second types,” Comput. Opt. 39, 299–304 (2015).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Vectorial Hankel laser beams carrying orbital angular momentum,” Comput. Opt. 39(4), 449–452 (2015).
[Crossref]

S. N. Khonina, D. A. Savelyev, and N. L. Kazanskiy, “Vortex phase elements as detectors of polarization state,” Opt. Express 23(14), 17845–17859 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (1)

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

2011 (1)

2010 (1)

2008 (1)

Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. in Electr. Res. Lett. 5, 57–71 (2008).

2007 (1)

2006 (1)

2005 (1)

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71(3), 033411 (2005).
[Crossref]

2002 (2)

Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref] [PubMed]

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cedra, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[Crossref]

2000 (1)

1997 (1)

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic fields,” Opt. Commun. 133(1-6), 315–327 (1997).
[Crossref]

1995 (1)

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1991 (1)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85(2-3), 159–161 (1991).
[Crossref]

Alferov, S. V.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Arlt, J.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cedra, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[Crossref]

Bajer, J.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic fields,” Opt. Commun. 133(1-6), 315–327 (1997).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bouchal, Z.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic fields,” Opt. Commun. 133(1-6), 315–327 (1997).
[Crossref]

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[Crossref]

Brown, T.

Cerjan, A.

Cerjan, C.

Chávez-Cedra, S.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cedra, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[Crossref]

Dholakia, K.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cedra, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[Crossref]

Dogariu, A.

Dou, W.

Dou, W. B.

Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. in Electr. Res. Lett. 5, 57–71 (2008).

Garcés-Chávez, V.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cedra, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[Crossref]

Hacyan, S.

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71(3), 033411 (2005).
[Crossref]

Horák, R.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic fields,” Opt. Commun. 133(1-6), 315–327 (1997).
[Crossref]

Jáuregui, R.

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71(3), 033411 (2005).
[Crossref]

Karpeev, S. V.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Kazanskiy, N. L.

Khonina, S. N.

S. N. Khonina, D. A. Savelyev, and N. L. Kazanskiy, “Vortex phase elements as detectors of polarization state,” Opt. Express 23(14), 17845–17859 (2015).
[Crossref] [PubMed]

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Vectorial rotating vortex Hankel laser beams,” J. Opt. 18(9), 095602 (2016).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Vectorial Hankel laser beams carrying orbital angular momentum,” Comput. Opt. 39(4), 449–452 (2015).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Nonparaxial Hankel vortex beams of the first and second types,” Comput. Opt. 39, 299–304 (2015).
[Crossref]

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Vectorial rotating vortex Hankel laser beams,” J. Opt. 18(9), 095602 (2016).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Nonparaxial Hankel vortex beams of the first and second types,” Comput. Opt. 39, 299–304 (2015).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Vectorial Hankel laser beams carrying orbital angular momentum,” Comput. Opt. 39(4), 449–452 (2015).
[Crossref]

Laukkanen, J.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Leger, J.

Meng, H.

Mishra, S. R.

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85(2-3), 159–161 (1991).
[Crossref]

Novitsky, A. V.

Novitsky, D. V.

Olivík, M.

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[Crossref]

Savelyev, D. A.

S. N. Khonina, D. A. Savelyev, and N. L. Kazanskiy, “Vortex phase elements as detectors of polarization state,” Opt. Express 23(14), 17845–17859 (2015).
[Crossref] [PubMed]

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Soifer, V. A.

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Vectorial rotating vortex Hankel laser beams,” J. Opt. 18(9), 095602 (2016).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Nonparaxial Hankel vortex beams of the first and second types,” Comput. Opt. 39, 299–304 (2015).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Sukhov, S.

Turunen, J.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

Volke-Sepulveda, K.

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cedra, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[Crossref]

Wang, Y.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Youngworth, K.

Yu, Y. Z.

Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. in Electr. Res. Lett. 5, 57–71 (2008).

Zhan, Q.

Comput. Opt. (2)

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Nonparaxial Hankel vortex beams of the first and second types,” Comput. Opt. 39, 299–304 (2015).
[Crossref]

V. V. Kotlyar and A. A. Kovalev, “Vectorial Hankel laser beams carrying orbital angular momentum,” Comput. Opt. 39(4), 449–452 (2015).
[Crossref]

J. Mod. Opt. (1)

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[Crossref]

J. Opt. (2)

V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Vectorial rotating vortex Hankel laser beams,” J. Opt. 18(9), 095602 (2016).
[Crossref]

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

J. Opt. B Quantum Semiclassical Opt. (1)

K. Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cedra, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

S. R. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85(2-3), 159–161 (1991).
[Crossref]

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic fields,” Opt. Commun. 133(1-6), 315–327 (1997).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (2)

R. Jáuregui and S. Hacyan, “Quantum-mechanical properties of Bessel beams,” Phys. Rev. A 71(3), 033411 (2005).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Prog. in Electr. Res. Lett. (1)

Y. Z. Yu and W. B. Dou, “Vector analyses of nondiffracting Bessel beams,” Prog. in Electr. Res. Lett. 5, 57–71 (2008).

Other (2)

W. Miller, Symmetry and Separation of Variables (Addison-Wesley, 1977).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Math. Series, 1965).

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Figures (6)

Fig. 1
Fig. 1

Intensity patterns I = |Ex|2 + |Ey|2 + |Ez|2 from a Hankel beam (n = 1) with linear (а,d), clockwise circular (b,e) and anticlockwise circular (c,f) polarization at distances z = λ/4 (а-c) and z = λ/2 (d-f).

Fig. 2
Fig. 2

Higher divergence and greater longitudinal component of the electric vector of the Hankel beam with anticlockwise circular polarization compared to that with the clockwise circular polarization (k+ and k are the wavevectors).

Fig. 3
Fig. 3

Far field of the Hankel beam (area B in the blue beam) and for arbitrary paraxial beam (area A in the green beam).

Fig. 4
Fig. 4

Radial divergence of the Hankel beam with clockwise circular polarization.

Fig. 5
Fig. 5

Direction of longitudinal AM for the Hankel beam with anticlockwise polarization with unitary topological charge n = 1.

Fig. 6
Fig. 6

Longitudinal projection of the angular momentum density for Hankel beams at z = 50λ with clockwise (a, c) and anticlockwise (b, d) circular polarization with topological charge n = 0 (a, b) and n = 1 (c, d). Red color – positive values, blue color – negative values.

Equations (57)

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{ rotH=iω ε 0 E, rotE=iω μ 0 H, divE=0, divH=0,
2 P+ k 2 P=0,
E z = α x P x α y P y ,
H= i k ε 0 μ 0 rotE.
P( r,φ,z )= 1 2 i n+1 λ 3/2 ( r e iφ ) n ψ n+1/2 ( R ),
ψ ν ( R )= H ν ( 1 ) ( kR ) R ν ,
{ d 2 d ξ 2 + 1 ξ d dξ +[ 1 ( n+ 1 2 ) 2 1 ξ 2 ] } H n+1/2 ( 1 ) ( ξ )=0,
P( r,φ,z )= 1 2 i n+1 λ 3/2 r | n | e inφ ψ | n |+1/2 ( R ).
E ny + ( r,φ,z )=i E nx ( r,φ,z )=i P z = i n π λ z r | n | e inφ ψ | n |+3/2 ( R ).
E nz + ( r,φ,z )= P x i P y = e iφ ( P r + i r P φ )= = 1 2 i n1 λ 3/2 r | n |1 e i( n+1 )φ [ ( | n |n ) ψ | n |+1/2 ( R )k r 2 ψ | n |+3/2 ( R ) ],
H nx + ( r,φ,z )= i k ( E nz + y i E nx z )= = i n+1 λ 3/2 2 r | n | e inφ { ( | n |n )( n+1 ) k 1 r 2 e 2iφ ψ | n |+1/2 ( R ) + + [ | n |+2( | n |n ) e 2iφ ] ψ | n |+3/2 ( R )k( z 2 i r 2 e iφ sinφ ) ψ | n |+5/2 ( R ) },
H ny + ( r,φ,z )= i k ( E nx z E nz + x )= = i n λ 3/2 2 r | n | e inφ { ( | n |n )( n+1 ) k 1 r 2 e 2iφ ψ | n |+1/2 ( R ) [ | n |+2+( | n |n ) e 2iφ ] ψ | n |+3/2 ( R )+k( z 2 + r 2 e iφ cosφ ) ψ | n |+5/2 ( R ) },
H nz + ( r,φ,z )= i k ( i x y ) E nx = i k e iφ ( i r 1 r φ ) E nx = = i n+1 π λ 1/2 z r | n |+1 e i( n+1 )φ [ ψ | n |+5/2 ( R )( | n |n ) k 1 r 2 ψ | n |+3/2 ( R ) ].
E ny ( r,φ,z )=i E nx ( r,φ,z )=i P z = i n π λ 1/2 z r | n | e inφ ψ | n |+3/2 ( R ),
E nz ( r,φ,z )= P x +i P y = e iφ ( P r i r P φ )= = 1 2 i n1 λ 3/2 r | n |1 e i( n1 )φ [ ( | n |+n ) ψ | n |+1/2 ( R )k r 2 ψ | n |+3/2 ( R ) ],
H nx ( r,φ,z )= i k ( E nz y +i E nx z )= = i n1 λ 3/2 2 r | n | e inφ { ( | n |+n )( n1 ) k 1 r 2 e 2iφ ψ | n |+1/2 ( R ) + + [ | n |+2( | n |+n ) e 2iφ ] ψ | n |+3/2 ( R )k( z 2 +i r 2 e iφ sinφ ) ψ | n |+5/2 ( R ) },
H ny ( r,φ,z )= i k ( E nx z E nz x )= = i n λ 3/2 2 r | n | e inφ { ( | n |+n )( n1 ) k 1 r 2 e 2iφ ψ | n |+1/2 ( R ) [ | n |+2+( | n |+n ) e 2iφ ] ψ | n |+3/2 ( R )+k( z 2 + r 2 e iφ cosφ ) ψ | n |+5/2 ( R ) },
H nz ( r,φ,z )= i k ( i x y ) E nx = i k e iφ ( i r + 1 r φ ) E nx = = i n+1 π λ 1/2 z r | n |+1 e i( n1 )φ [ ( | n |+n ) k 1 r 2 ψ | n |+3/2 ( R ) ψ | n |+5/2 ( R ) ].
E 0z ± ( r,φ,z )=iπ λ 1/2 r e ±iφ ψ 3/2 ( R )=iλ R 2 ( 1+i k 1 R 1 )r e ikR±iφ ,
H 0z ± ( r,φ,z )=±iπ λ 1/2 zr e ±iφ ψ 5/2 ( R )
E 0y + ( r=0,φ,z )=i E 0x ( r=0,φ,z )= E 0y ( r=0,φ,z )=π λ 1/2 z ψ 3/2 ( z ),
H 0x + ( r=0,φ,z )= H 0x ( r=0,φ,z )= i λ 3/2 2 [ 2 ψ 3/2 ( z )k z 2 ψ 5/2 ( z ) ],
H 0y + ( r=0,φ,z )= H 0y ( r=0,φ,z )= λ 3/2 2 [ 2 ψ 3/2 ( z )k z 2 ψ 5/2 ( z ) ],
E 1z ( r=0,φ,z )= λ 3/2 ψ 3/2 ( z ),
H 1z ( r=0,φ,z )= λ 3/2 z ψ 5/2 ( z ).
I n + ( r,z )= π 2 λ r 2| n | ( 2 z 2 + r 2 ) | ψ | n |+3/2 ( R ) | 2 + + | n |n 2 λ 3 r 2| n |2 { | n |n 2 | ψ | n |+1/2 ( R ) | 2 k r 2 Re[ ψ | n |+1/2 * ( R ) ψ | n |+3/2 ( R ) ] },
I n ( r,z )= π 2 λ r 2| n | ( 2 z 2 + r 2 ) | ψ | n |+3/2 ( R ) | 2 + + | n |+n 2 λ 3 r 2| n |2 { | n |+n 2 | ψ | n |+1/2 ( R ) | 2 k r 2 Re[ ψ | n |+1/2 * ( R ) ψ | n |+3/2 ( R ) ] }.
I 1 ( r=0,z )= λ 3 | ψ 3/2 ( z ) | 2 = λ 4 π 2 z 4 [ 1+ 1 ( kz ) 2 ].
S nz ± ( r,z )= 1 2 Re{ E nx * H ny ± E ny ±* H nx ± }= = 1 2 π 2 λ r 2| n | z( 2 z 2 + r 2 )Im{ ψ | n |+3/2 * ( R ) ψ | n |+5/2 ( R ) }= = 1 2 λ 2 r 2| n | z 2 z 2 + r 2 R 2| n |+5 .
I | n | ± ( r,z )= I | n | ( r,z ).
H v (1) ( x ) 2 πx ( i ) ν+1/2 e ix ,
ψ ν ( R )= H ν (1) ( kR ) R ν ( i ) ν+1/2 λ 1/2 e ikR π R ν+1/2 .
E ny + ( r,φ,z )=i E nx ( r,φ,z )=λ r n z R n+2 e inφ+ikR ,
E nz + ( r,φ,z )=iλ r n+1 R n+2 e i( n+1 )φ+ikR ,
H nx + ( r,φ,z )=λ( i n+2 k R n+2 + i r 2 e iφ sinφ z 2 R n+3 ) r n e inφ+ikR ,
H ny + ( r,φ,z )=iλ( z 2 + r 2 e iφ cosφ R n+3 i n+2 k R n+2 ) r n e inφ+ikR ,
H nz + ( r,φ,z )=λ r n+1 z R n+3 e i( n+1 )φ+ikR .
E ny ( r,φ,z )=i E nx ( r,φ,z )=λ r n z R n+2 e inφ+ikR ,
E nz ( r,φ,z )=iλ r n+1 R n+2 ( 1 2inR k r 2 ) e i( n1 )φ+ikR ,
H nx ( r,φ,z )= λ k r n R n+3 { 2n( n1 ) e 2iφ k r 2 R 2 + + i( 2n e 2iφ n2 )R+k( z 2 +i r 2 e iφ sinφ ) } e inφ+ikR ,
H ny ( r,φ,z )= λ k r n R n+3 { i 2n( n1 ) e 2iφ k r 2 R 2 ( 2n e 2iφ +n+2 )Rik( z 2 + r 2 e iφ cosφ ) } e inφ+ikR ,
H nz ( r,φ,z )=λ r n+1 z R n+3 ( 1 2inR k r 2 ) e i( n1 )φ+ikR .
I n + ( r,z )= | E nx ( r,φ,z ) | 2 + | E ny + ( r,φ,z ) | 2 + | E nz + ( r,φ,z ) | 2 = = λ 2 r 2n R 2n+4 ( r 2 +2 z 2 ),
I n ( r,z )= | E nx ( r,φ,z ) | 2 + | E ny ( r,φ,z ) | 2 + | E nz ( r,φ,z ) | 2 = = λ 2 r 2n R 2n+4 ( r 2 +2 z 2 + 4 n 2 R 2 k 2 r 2 ).
r max + =z n3+ n 2 +2n+9 2 .
( n3 ) r 4 z 4 +2n r 2 z 2 r 6 z 6 + 4 n 2 ( n3 ) ( kz ) 2 r 2 z 2 + 4 n 2 ( n1 ) ( kz ) 2 8 n 2 ( kz ) 2 r 4 z 4 =0,
r max z n3+ n 2 +2n+9 2 .
S nx + ( r,φ,z )= 1 2 Re{ E ny +* H nz + E nz +* H ny + }= = 1 2 λ 2 r 2n+1 R 2n+5 [ ( r 2 +2 z 2 )cosφ( n+2 ) R k sinφ ],
S ny + ( r,φ,z )= 1 2 Re{ E nz +* H nx + E nx * H nz + }= = 1 2 λ 2 r 2n+1 R 2n+5 [ ( r 2 +2 z 2 )sinφ+( n+2 ) R k cosφ ],
S nx ( r,φ,z )= 1 2 Re{ E ny * H nz E nz * H ny }= = 1 2 λ 2 r 2n+1 R 2n+5 { ( 2 z 2 + r 2 )cosφ( 4n z 2 r 2 +n2 )( R k )sinφ + + 2n r 2 ( 2n+3 ) ( R k ) 2 cosφ 4 n 2 ( n1 ) r 4 ( R k ) 3 sinφ },
S ny ( r,φ,z )= E nz * ( r,φ,z ) H nx ( r,φ,z ) E nx * ( r,φ,z ) H nz ( r,φ,z )= = 1 2 λ 2 r 2n+1 R 2n+5 { ( 2 z 2 + r 2 )sinφ+( 4n z 2 r 2 +n2 )( R k )cosφ + + 2n r 2 ( 2n+3 ) ( R k ) 2 sinφ+ 4 n 2 ( n1 ) r 4 ( R k ) 3 cosφ }.
S nr + ( r,z )= S nx + cosφ+ S ny + sinφ= 1 2 λ 2 r 2n+1 R 2n+5 ( r 2 +2 z 2 ) r z S nz + , S nφ + ( r,z )= S ny + cosφ S nx + sinφ= 1 2 λ 2 ( n+2 ) r 2n+1 k R 2n+4 ,
S nr ( r,z )= 1 2 λ 2 r 2n+1 R 2n+5 [ r 2 +2 z 2 + 2n r 2 ( 2n+3 ) ( R k ) 2 ], S nφ ( r,z )= 1 2 λ 2 r 2n+1 k R 2n+4 [ 4n z 2 r 2 +n2+ 4 n 2 ( n1 ) r 4 ( R k ) 2 ].
j nz + ( r,φ,z )=r S nφ + = 1 2 λ 2 ( n+2 ) r 2n+2 k R 2n+4 ,
j nz ( r,φ,z )=r S nφ = 1 2 λ 2 r 2n+2 k R 2n+4 [ 4n z 2 r 2 +n2+ 4 n 2 ( n1 ) r 4 ( R k ) 2 ]
j 0z + ( r,φ,z )= j 0z ( r,φ,z )= λ 2 k r 2 R 4 .
j 1z ( r,φ,z )= λ 2 r 4 2k R 6 ( 4 z 2 r 2 1 ).

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