Abstract

The generation of hollow beams with a long focal depth from a radially polarized Bessel–Gaussian beam with a second-order vortex phase and an amplitude filter is theoretically investigated by Richards–Wolf’s integral. The null intensity on the optical axis is achieved by introducing the second-order vortex. The long focal depth is a result of the amplitude filtering based on the cosine function and Euler transformation. Numerical results indicate that the focal depth of a hollow beam is improved from 0.96λ to 2.28λ with a slight increase of the transverse size for the simplest amplitude filter design. The intensity distribution twist phenomenon of the x- and y-polarized components around the optical axis due to the introducing of the vortex phase is also discussed. It is believed that the proposed scheme can be used to achieve particle acceleration and optical trapping.

© 2014 Optical Society of America

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    [CrossRef]

2013 (12)

T. Grosjean and I. Gauthier, “Longitudinally polarized electric and magnetic optical nano-needles of ultra high lengths,” Opt. Commun. 294, 333–337 (2013).
[CrossRef]

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[CrossRef]

T. Liu, J. Tan, J. Liu, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[CrossRef]

X. Xie, H. Sun, L. Yang, S. Wang, and J. Zhou, “Effect of polarization purity of cylindrical vector beam on tightly focused spot,” J. Opt. Soc. Am. A 30, 1937–1940 (2013).
[CrossRef]

G.-Y. Chen, F. Song, and H.-T. Wang, “Sharper focal spot generated by 4π tight focusing of higher-order Laguerre–Gaussian radially polarized beam,” Opt. Lett. 38, 3937–3940 (2013).
[CrossRef]

H. Guo, X. Weng, M. Jiang, Y. Zhao, G. Sui, Q. Hu, Y. Wang, and S. Zhuang, “Tight focusing of a higher-order radially polarized beam transmitting through multi-zone binary phase pupil filters,” Opt. Express 21, 5363–5372 (2013).
[CrossRef]

S. N. Khonina, “Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions,” Opt. Eng. 52, 091711 (2013).
[CrossRef]

S. G. Reddy, A. Kumar, S. Prabhakar, and R. P. Singh, “Experimental generation of ring-shaped beams with random sources,” Opt. Lett. 38, 4441–4444 (2013).
[CrossRef]

S. N. Khonina, S. V. Alferov, and S. V. Karpeev, “Strengthening the longitudinal component of the sharply focused electric field by means of higher-order laser beams,” Opt. Lett. 38, 3223–3226 (2013).
[CrossRef]

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[CrossRef]

A. Calatayud, V. Ferrando, L. Remón, W. D. Furlan, and J. A. Monsoriu, “Twin axial vortices generated by Fibonacci lenses,” Opt. Express 21, 10234–10239 (2013).
[CrossRef]

A. Ortiz-Ambriz, S. Lopez-Aguayo, Y. V. Kartashov, V. A. Vysloukh, D. Petrov, H. Garcia-Gracia, J. C. Gutiérrez-Vega, and L. Torner, “Generation of arbitrary complex quasi-non-diffracting optical patterns,” Opt. Express 21, 22221–22231 (2013).
[CrossRef]

2012 (1)

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photon. Rev. 6, 354–392 (2012).
[CrossRef]

2011 (5)

G. H. Yuan, S. B. Wei, and X.-C. Yuan, “Nondiffracting transversally polarized beam,” Opt. Lett. 36, 3479–3481 (2011).
[CrossRef]

J. Wang, Q. Liu, Y. Liu, W. Chen, and Q. Zhan, “Discrete complex amplitude filter for ultra long optical tube,” Proc. SPIE 8097, 809722 (2011).
[CrossRef]

J. Lin, K. Yin, Y. Li, and J. Tan, “Achievement of longitudinally polarized focusing with long focal depth by amplitude modulation,” Opt. Lett. 36, 1185–1188 (2011).
[CrossRef]

X. P. Li, Y. Y. Cao, and M. Gu, “Superresolution-focal-volume induced 3.0 Tbytes/disk capacity by focusing a radially polarized beam,” Opt. Lett. 36, 2510–2512 (2011).
[CrossRef]

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760 (2011).
[CrossRef]

2010 (3)

2009 (1)

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

2008 (1)

H. Wang, L. Shi, B. Lùkyhanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (1)

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A 7, 147–153 (2005).
[CrossRef]

2004 (1)

S. F. Pereira and A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[CrossRef]

2003 (2)

D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11, 2747–2752 (2003).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

2000 (1)

1999 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Ahmad, M. A.

Alferov, S. V.

Antoniou, N.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[CrossRef]

Brown, T. G.

Calatayud, A.

Cao, Y. Y.

Capasso, F.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[CrossRef]

Chen, G.-Y.

Chen, W.

J. Wang, Q. Liu, Y. Liu, W. Chen, and Q. Zhan, “Discrete complex amplitude filter for ultra long optical tube,” Proc. SPIE 8097, 809722 (2011).
[CrossRef]

Chen, Z.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

Cheng, K.

Q. Tan, K. Cheng, Z. Zhou, and G. Jin, “Diffractive superresolution elements for radially polarized light,” J. Opt. Soc. Am. 27, 1355–1360 (2010).
[CrossRef]

Chong, C. T.

H. Wang, L. Shi, B. Lùkyhanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Dong, Y.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A 7, 147–153 (2005).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Ferrando, V.

Furlan, W. D.

Gan, F.

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[CrossRef]

Gan, X.

Ganic, D.

Garcia-Gracia, H.

Gauthier, I.

T. Grosjean and I. Gauthier, “Longitudinally polarized electric and magnetic optical nano-needles of ultra high lengths,” Opt. Commun. 294, 333–337 (2013).
[CrossRef]

Genevet, P.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[CrossRef]

Grosjean, T.

T. Grosjean and I. Gauthier, “Longitudinally polarized electric and magnetic optical nano-needles of ultra high lengths,” Opt. Commun. 294, 333–337 (2013).
[CrossRef]

Gu, M.

Guo, H.

Gutiérrez-Vega, J. C.

Hill, W. T.

Ho, S. T.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photon. Rev. 6, 354–392 (2012).
[CrossRef]

Hu, Q.

Huang, K.

Jiang, M.

Jin, G.

Q. Tan, K. Cheng, Z. Zhou, and G. Jin, “Diffractive superresolution elements for radially polarized light,” J. Opt. Soc. Am. 27, 1355–1360 (2010).
[CrossRef]

Kang, X.-L.

Karpeev, S. V.

Kartashov, Y. V.

Kats, M. A.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[CrossRef]

Kazanskiy, N. L.

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760 (2011).
[CrossRef]

Khonina, S. N.

S. N. Khonina, S. V. Alferov, and S. V. Karpeev, “Strengthening the longitudinal component of the sharply focused electric field by means of higher-order laser beams,” Opt. Lett. 38, 3223–3226 (2013).
[CrossRef]

S. N. Khonina, “Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions,” Opt. Eng. 52, 091711 (2013).
[CrossRef]

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760 (2011).
[CrossRef]

S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27, 2188–2197 (2010).
[CrossRef]

Kumar, A.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Li, X. P.

Li, Y.

Li, Y.-P.

Lin, J.

T. Liu, J. Tan, J. Liu, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[CrossRef]

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13, 4269–4274 (2013).
[CrossRef]

J. Lin, K. Yin, Y. Li, and J. Tan, “Achievement of longitudinally polarized focusing with long focal depth by amplitude modulation,” Opt. Lett. 36, 1185–1188 (2011).
[CrossRef]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076–2078 (2007).
[CrossRef]

Lin, Q.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A 7, 147–153 (2005).
[CrossRef]

Liu, J.

T. Liu, J. Tan, J. Liu, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[CrossRef]

Z. Liu, H. Zhao, J. Liu, J. Lin, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beams by spatial filtering,” Opt. Lett. 32, 2076–2078 (2007).
[CrossRef]

Liu, Q.

J. Wang, Q. Liu, Y. Liu, W. Chen, and Q. Zhan, “Discrete complex amplitude filter for ultra long optical tube,” Proc. SPIE 8097, 809722 (2011).
[CrossRef]

Liu, S.

Liu, T.

T. Liu, J. Tan, J. Liu, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[CrossRef]

Liu, Y.

J. Wang, Q. Liu, Y. Liu, W. Chen, and Q. Zhan, “Discrete complex amplitude filter for ultra long optical tube,” Proc. SPIE 8097, 809722 (2011).
[CrossRef]

Liu, Z.

Lopez-Aguayo, S.

Lùkyhanchuk, B.

H. Wang, L. Shi, B. Lùkyhanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Milam, D.

Monsoriu, J. A.

Ortiz-Ambriz, A.

Pereira, S. F.

S. F. Pereira and A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[CrossRef]

Petrov, D.

Prabhakar, S.

Pu, J.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Rao, L.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

Ravi, K.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photon. Rev. 6, 354–392 (2012).
[CrossRef]

Reddy, S. G.

Remón, L.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Sheppard, C.

H. Wang, L. Shi, B. Lùkyhanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Sheppard, C. J. R.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photon. Rev. 6, 354–392 (2012).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lùkyhanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Shi, P.

Singh, R. P.

Song, F.

Song, Y.

Sui, G.

Sun, H.

Tan, J.

T. Liu, J. Tan, J. Liu, and J. Lin, “Creation of subwavelength light needle, equidistant multi-focus, and uniform light tunnel,” J. Mod. Opt. 60, 378–381 (2013).
[CrossRef]

J. Lin, K. Yin, Y. Li, and J. Tan, “Achievement of longitudinally polarized focusing with long focal depth by amplitude modulation,” Opt. Lett. 36, 1185–1188 (2011).
[CrossRef]

Tan, Q.

Q. Tan, K. Cheng, Z. Zhou, and G. Jin, “Diffractive superresolution elements for radially polarized light,” J. Opt. Soc. Am. 27, 1355–1360 (2010).
[CrossRef]

Torner, L.

van de Nes, A. S.

S. F. Pereira and A. S. van de Nes, “Superresolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234, 119–124 (2004).
[CrossRef]

Vienne, G.

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photon. Rev. 6, 354–392 (2012).
[CrossRef]

Volotovsky, S. G.

S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” J. Mod. Opt. 58, 748–760 (2011).
[CrossRef]

S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27, 2188–2197 (2010).
[CrossRef]

Vysloukh, V. A.

Wang, H.

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[CrossRef]

H. Wang, C. J. R. Sheppard, K. Ravi, S. T. Ho, and G. Vienne, “Fighting against diffraction: apodization and near field diffraction structures,” Laser Photon. Rev. 6, 354–392 (2012).
[CrossRef]

H. Wang, L. Shi, B. Lùkyhanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Wang, H.-T.

Wang, J.

J. Wang, Q. Liu, Y. Liu, W. Chen, and Q. Zhan, “Discrete complex amplitude filter for ultra long optical tube,” Proc. SPIE 8097, 809722 (2011).
[CrossRef]

Wang, S.

Wang, Y.

Wang, Z.

Z. Wang, Y. Dong, and Q. Lin, “Atomic trapping and guiding by quasi-dark hollow beams,” J. Opt. A 7, 147–153 (2005).
[CrossRef]

Wei, J.

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[CrossRef]

Wei, S. B.

Weng, X.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Xie, X.

Yang, L.

Yei, P.

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens,” Opt. Laser Technol. 41, 241–246 (2009).
[CrossRef]

Yin, K.

Youngworth, K. S.

Yuan, G. H.

Yuan, X.-C.

Zha, Y.

Y. Zha, J. Wei, H. Wang, and F. Gan, “Creation of an ultra-long depth of focus super-resolution longitudinally polarized beam with a ternary optical element,” J. Opt. 15, 075703 (2013).
[CrossRef]

Zhan, Q.

J. Wang, Q. Liu, Y. Liu, W. Chen, and Q. Zhan, “Discrete complex amplitude filter for ultra long optical tube,” Proc. SPIE 8097, 809722 (2011).
[CrossRef]

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31, 867–869 (2006).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Focusing field intensity distribution at the xz plane for the radially polarized BG incident beam with a second-order vortex phase in the focusing system with NA=0.95. (a) Total electric energy intensity near the focal point. The dotted lines, which mark the position of the maximum intensity, are parallel to the z axis. (b)–(d) are the intensity distributions of x-, y-, and z-polarized components, respectively.

Fig. 2.
Fig. 2.

Focusing field intensity at the xz plane for a radially polarized BG beam with a second-order vortex phase in the focusing system with NA=0.95. (a) Intensity distribution at the focal plane z=0. (b) Intensity distribution along the dotted lines in Fig. 1(a). DOF of the focusing field is labeled by the dashed line.

Fig. 3.
Fig. 3.

(a) Amplitude transmittance of the complex amplitude filter for N=1, c1=1, and m1=0.5, and (b) the corresponding intensity distribution in the yz plane, where white areas represent high field intensity. (c) Longitudinal intensity profile along dashed lines, where maximum of intensity is located in (b). DOF is approximately 2.28λ, and the range of DOF is marked by the dashed lines. (d) Lateral intensity distribution at different longitudinal positions of z. AD and BC mark outer and inner diameters Do and Di, respectively.

Fig. 4.
Fig. 4.

Positions of the maximum intensity of the x- and y-polarized components on different transversal planes along the z-axis. Black and red solid curves represent the x- and y-polarized components, respectively.

Fig. 5.
Fig. 5.

Intensity distributions of the x-, y-, and z-polarized components at different transverse planes along the optical axis. (a1)–(a3) Intensity distributions at transverse plane z=1.14λ, (b1)–(b3) at z=0, and (c1)–(c3) at z=1.14λ, respectively. Intensity distributions (a1), (b1), and (c1) for the x-polarized component; (a2), (b2), and (c2) for the y- polarized component; and (a3), (b3), and (c3) for the z- polarized component. Positions of maximum intensity are marked by × for the x- and y-polarized components. There are two equal intensity peaks in the x- and y-polarized components. Field intensity of the z-polarized component is symmetrical with respect to the z-axis.

Fig. 6.
Fig. 6.

Phase distributions of the radially, azimuthally, and longitudinally polarized components at different transverse planes along the optical axis. ρmax=1.5λ. White and dark regions represent high and small phase (or intensity in the insets). Insets represent the intensity distributions of the corresponding polarized component. Total fields are the sum of the intensity of the radial, azimuthal, and longitudinal components.

Equations (8)

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E(ρ,φ,z)=iA0α02πl0(θ)V(ϕ)cos1/2θsinθ×exp[ikρsinθcos(ϕφ)]exp(ikzcosθ)P(θ,ϕ)dθdϕ,
P(θ,ϕ)=[cosϕcosθsinϕcosθsinθ].
02πV(ϕ)exp[ikρsinθcos(ϕφ)][cosϕcosθsinϕcosθsinθ]dϕ=ei2φ2[i(eiφJ3(t)eiφJ1(t))cosθ(eiφJ3(t)+eiφJ1(t))cosθ2J2(t)sinθ],
FN(θ)=p=1Ncpcos(kmpcosθ),
F1(θ)=c1cos(km1cosθ)=0.5c1[exp(ikm1cosθ)+exp(ikm1cosθ)].
Ez(ρ,φ,z)=iAc120αl0(θ)cos1/2θsin2θei2φJ2(kρsinθ)[eik(z+m1)cosθ+eik(zm1)cosθ]dθ.
Ez(ρ,φ,z)=Ez(ρ,φ,z+m1)+Ez(ρ,φ,zm1).
l0(θ)=exp[β02(sinθsinα)2]J1(2β0sinθsinα),

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