Abstract

Using the transverse Hertz vector potentials, vector analyses of linearly and circularly polarized Bessel beams of arbitrary orders are presented in this paper. Expressions for the electric and magnetic fields of vector Bessel beams in free space that are rigorous solutions to the vector Helmholtz equation are derived. Their respective time averaged energy density and Poynting vector are also obtained, in order to exhibit their non-diffracting properties. Polarization patterns and magnitude profiles with different parameters are displayed. Particular emphasis is placed on the cases where the ratio of wave number over its transverse component k/kt approximately equals to one and largely exceeds it, which corresponding to the nonparaxial and paraxial condition, respectively. These results allow us to recognize that the vector Bessel beams exhibit new and important features, compared with the scalar fields.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004).
    [CrossRef] [PubMed]
  9. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013 (2)

2009 (1)

2008 (2)

A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33(14), 1563–1565 (2008).
[CrossRef] [PubMed]

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

2006 (2)

K. Volke-Sepulvea and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8(10), 867–877 (2006).
[CrossRef]

A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. 31(11), 1732–1734 (2006).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222(1-6), 75–80 (2003).
[CrossRef]

2001 (1)

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001).
[CrossRef]

2000 (2)

1999 (1)

1998 (1)

1997 (1)

1996 (2)

1995 (2)

Z. P. Jiang, Q. S. Lu, and Z. J. Liu, “Propagation of apertured Bessel beams,” Appl. Opt. 34(31), 7183–7185 (1995).
[CrossRef] [PubMed]

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

1994 (1)

1992 (2)

A. J. Cox and J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17(4), 232–234 (1992).
[CrossRef] [PubMed]

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31(7), 1527–1531 (1992).
[CrossRef]

1991 (1)

1990 (1)

R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D 42(5), 1807–1818 (1990).
[CrossRef]

1988 (1)

1987 (1)

1977 (1)

E. A. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. 45(11), 1099–1101 (1977).
[CrossRef]

Aiello, A.

April, A.

Arlt, J.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
[CrossRef]

Bandres, M. A.

Bouchal, Z.

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

Chávez-Cerda, S.

Cox, A. J.

D’Anna, J.

Dartora, C. A.

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222(1-6), 75–80 (2003).
[CrossRef]

Dholakia, K.

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
[CrossRef]

Dibble, D. C.

Dou, W. B.

Y. Z. Yu and W. B. Dou, “Generation of pseudo-Bessel beams at THz frequencies by use of binary axicons,” Opt. Express 17(2), 888–893 (2009).
[CrossRef] [PubMed]

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

Durnin, J.

Eberly, J. H.

Essex, E. A.

E. A. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. 45(11), 1099–1101 (1977).
[CrossRef]

Flores-Pérez, A.

Ford, D. H.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31(7), 1527–1531 (1992).
[CrossRef]

Friberg, A. T.

Greene, P. L.

Grogan, M. D. W.

Gutiérrez-Vega, J. C.

Hall, D. G.

Hernández-Figueroa, H. E.

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222(1-6), 75–80 (2003).
[CrossRef]

Hernández-Hernández, J.

Iturbe-Castillo, M. D.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
[CrossRef] [PubMed]

Jáuregui, R.

Jiang, Z. P.

Jordan, R. H.

Kärtner, F. X.

Kettunen, V.

Kimura, W. D.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31(7), 1527–1531 (1992).
[CrossRef]

R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D 42(5), 1807–1818 (1990).
[CrossRef]

Kuittinen, M.

Ley-Koo, E.

K. Volke-Sepulvea and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8(10), 867–877 (2006).
[CrossRef]

Liu, Z. J.

Lu, Q. S.

Miceli, J. J.

New, G. H. C.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001).
[CrossRef]

Nóbrega, K. Z.

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222(1-6), 75–80 (2003).
[CrossRef]

Olivík, M.

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

Ornigotti, M.

Putnam, W. P.

Ramachandran, S.

Ramírez, G. A.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001).
[CrossRef]

Recami, E.

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222(1-6), 75–80 (2003).
[CrossRef]

Rodríguez-Dagnino, R. M.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001).
[CrossRef]

Romea, R. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D 42(5), 1807–1818 (1990).
[CrossRef]

Saghafi, S.

Schimpf, D. N.

Sheppard, C. J. R.

Tepichín, E.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001).
[CrossRef]

Tidwell, S. C.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31(7), 1527–1531 (1992).
[CrossRef]

Turunen, J.

Vahimaa, P.

Volke-Sepulvea, K.

K. Volke-Sepulvea and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8(10), 867–877 (2006).
[CrossRef]

Volke-Sepúlveda, K.

Yu, Y. Z.

Y. Z. Yu and W. B. Dou, “Generation of pseudo-Bessel beams at THz frequencies by use of binary axicons,” Opt. Express 17(2), 888–893 (2009).
[CrossRef] [PubMed]

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

Zamboni-Rached, M.

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222(1-6), 75–80 (2003).
[CrossRef]

Am. J. Phys. (1)

E. A. Essex, “Hertz vector potentials of electromagnetic theory,” Am. J. Phys. 45(11), 1099–1101 (1977).
[CrossRef]

Appl. Opt. (2)

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. A, Pure Appl. Opt. (1)

K. Volke-Sepulvea and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A, Pure Appl. Opt. 8(10), 867–877 (2006).
[CrossRef]

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

Opt. Commun. (3)

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195(1-4), 35–40 (2001).
[CrossRef]

C. A. Dartora, M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “General formulation for the analysis of scalar diffraction-free beams using angular modulation: Mathieu and Bessel beams,” Opt. Commun. 222(1-6), 75–80 (2003).
[CrossRef]

Opt. Eng. (1)

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31(7), 1527–1531 (1992).
[CrossRef]

Opt. Express (3)

Opt. Lett. (9)

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004).
[CrossRef] [PubMed]

A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. 31(11), 1732–1734 (2006).
[CrossRef] [PubMed]

A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33(14), 1563–1565 (2008).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988).
[CrossRef] [PubMed]

A. J. Cox and J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17(4), 232–234 (1992).
[CrossRef] [PubMed]

R. H. Jordan and D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,” Opt. Lett. 19(7), 427–429 (1994).
[CrossRef] [PubMed]

C. J. R. Sheppard and S. Saghafi, “Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,” Opt. Lett. 24(22), 1543–1545 (1999).
[CrossRef] [PubMed]

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21(1), 9–11 (1996).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
[CrossRef] [PubMed]

Phy. Rev. D (1)

R. D. Romea and W. D. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phy. Rev. D 42(5), 1807–1818 (1990).
[CrossRef]

Prog. Electromagn. Res. Lett. (1)

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

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Figures (9)

Fig. 1
Fig. 1

Magnitude profiles and vector diagrams of the transverse components of the electric fields associated with linearly polarized n-order Bessel beams at three different instants: t = 0T, t = 0.25T, and t = 0.5T, respectively. (a)-(c) n = 1. (d)-(f) n = 2. Length scales in units of λ.

Fig. 2
Fig. 2

Magnitude profiles and vector diagrams of the transverse components of the electric fields associated with circularly polarized n-order Bessel beams at three different instants: t = 0T, t = 0.125T, and t = 0.25T, respectively. (a)-(c) n = 0. (d)-(f) n = 1. Length scales in units of λ.

Fig. 3
Fig. 3

Amplitude of the transverse and longitudinal components of the electric and magnetic fields, respectively, associated with linearly polarized n-order Bessel beam with parameters n = 0 and k/kt ≈7.84, corresponding to the paraxial condition. Length scales in units of λ.

Fig. 4
Fig. 4

Amplitude of the transverse and longitudinal components of the electric and magnetic fields, respectively, associated with linearly polarized n-order Bessel beam with parameters n = 0 and k/kt ≈1.02, corresponding to the nonparaxial condition. Length scales in units of λ.

Fig. 5
Fig. 5

Amplitude of the transverse and longitudinal components of the electric and magnetic fields, respectively, associated with circularly polarized n-order Bessel beam with parameters n = 2 and k/kt ≈6.17, corresponding to the paraxial condition. Length scales in units of λ.

Fig. 6
Fig. 6

Amplitude of the transverse and longitudinal components of the electric and magnetic fields, respectively, associated with circularly polarized n-order Bessel beam with parameters n = 2 and k/kt ≈1.01, corresponding to the nonparaxial condition. Length scales in units of λ.

Fig. 7
Fig. 7

Amplitude of the transverse, longitudinal and total components of the time averaged Poynting vector associated with linearly polarized n-order Bessel beam with parameters n = 2 and (a)-(c) k/kt ≈6.17, (d)-(f) k/kt ≈1.01. Length scales in units of λ.

Fig. 8
Fig. 8

Amplitude of the transverse, longitudinal and total components of the time averaged Poynting vector associated with circularly polarized n-order Bessel beam with parameters n = 1 and (a)-(c) k/kt ≈10.24, (d)-(f) k/kt ≈1.02. Length scales in units of λ.

Fig. 9
Fig. 9

Amplitude of the time averaged energy density associated with the n-order Bessel beam with parameters (a) linearly polarized, n = 2, k/kt ≈6.17, (b) linearly polarized, n = 2, k/kt ≈1.01, (c) circularly polarized, n = 1, k/kt ≈10.24, (d) circularly polarized, n = 1, k/kt ≈1.02. Length scales in units of λ.

Equations (30)

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

E e =( e )+ k 2 e ,
H e =jωε× e ,
E m =jωμ× m ,
H m =( m )+ k 2 m ,
( 2 + k 2 ) e =0,
( 2 + k 2 ) m =0.
e = J n ( k t ρ) e jnφ e j k z z ,
m = J n ( k t ρ) e jnφ e j k z z ,
m = m y ^ = J n ( k t ρ) e jnφ e j k z z y ^ ,
E xm =ωμ k z J n ( k t ρ) e jnφ e j k z z ,
E ym =0,
E zm =ωμ[j x ρ k t J n+1 ( k t ρ) jx+y ρ 2 n J n ( k t ρ)] e jnφ e j k z z ,
H xm ={[ 2xy+j x 2 j y 2 ρ 4 ( n 2 n) xy ρ 2 k t 2 ] J n ( k t ρ) + jn y 2 jn x 2 +2xy ρ 3 k t J n+1 ( k t ρ)} e jnφ e j k z z ,
H ym ={[ y 2 x 2 +j2xy ρ 4 ( n 2 n)+ k 2 x 2 + k z 2 y 2 ρ 2 ] J n ( k t ρ) + y 2 x 2 j2nxy ρ 3 k t J n+1 ( k t ρ)} e jnφ e j k z z ,
H zm = k z [n xjy ρ 2 J n ( k t ρ)+j k t y ρ J n+1 ( k t ρ)] e jnφ e j k z z .
< w m >= 1 4 μ 0 {[ k 2 k z 2 + ( k 2 + k z 2 ) n 2 + k 4 x 2 + k z 4 y 2 ρ 2 +2 n 2 n+ k z 2 y 2 k 2 x 2 ρ 4 ( n 2 n)] J n 2 + n 2 +1+ k 2 x 2 + k z 2 y 2 ρ 2 k t 2 J n+1 2 +2 (1n) k z 2 y 2 (1+n) k 2 x 2 n (n1) 2 ρ 3 k t J n J n+1 },
< S xm >=ωμ[yn ( n 2 n+ k 2 x 2 + k z 2 y 2 ) ρ 4 J n 2 2 x 2 y k t (2 n 2 n) ρ 5 J n J n+1 + 2 x 2 yn k t 2 ρ 4 J n+1 2 ],
< S ym >=ωμ[xn k t 2 y 2 n 2 +n ρ 4 J n 2 x( x 2 y 2 )n k t 2 ρ 4 J n+1 2 k z 2 nx ρ 2 J n 2 +x (2 n 2 n) x 2 (2 n 2 +n) y 2 ρ 5 k t J n J n+1 ],
< S zm >=ωμ k z [ y 2 x 2 ρ 4 ( n 2 n) J n 2 + k 2 x 2 + k z 2 y 2 ρ 2 J n 2 + y 2 x 2 ρ 3 k t J n J n+1 ],
m = m ( x ^ +j y ^ )= J n ( k t ρ) e jnφ e j k z z ( x ^ +j y ^ ).
E xm =jωμ k z J n ( k t ρ) e jnφ e j k z z ,
E ym =ωμ k z J n ( k t ρ) e jnφ e j k z z ,
E zm =ωμ k t J n+1 ( k t ρ) x+jy ρ e jnφ e j k z z ,
H xm =[ k z 2 x 2 + k 2 y 2 j k t 2 xy ρ 2 J n ( k t ρ) +(n+1) x 2 y 2 +j2xy ρ 3 k t J n+1 ( k t ρ)] e jnφ e j k z z ,
H ym =[ j k 2 x 2 +j k z 2 y 2 k t 2 xy ρ 2 J n ( k t ρ) +(n+1) j y 2 j x 2 +2xy ρ 3 k t J n+1 ( k t ρ)] e jnφ e j k z z ,
H zm = k z k t jxy ρ J n+1 ( k t ρ) e jnφ e j k z z .
< w m >= 1 4 μ 0 [ ( k 2 + k z 2 ) 2 J n 2 + k t 2 ( k 2 + k z 2 ) J n+1 2 +2 k t 2 (n+1) 2 ρ 2 J n+1 2 2 k t 3 (n+1) ρ J n J n+1 ],
< S xm >=ωμ k t [ 2y ρ k z 2 J n J n+1 + y k t ρ 2 (n+1) J n+1 2 ],
< S ym >=ωμ k t [ 2x ρ k z 2 J n J n+1 + x k t ρ 2 (n+1) J n+1 2 ],
< S zm >=ωμ k z ( k 2 + k z 2 ) J n 2 .

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