Abstract

In this paper, we have introduced a new class of partially coherent vector vortex beams, named radially polarized multi-Gaussian Schell-model (MGSM) vortex beam, carrying the vortex phase with tunable topological charges (i.e., both integral and fractional values) as a natural extension of the radially polarized MGSM beam. The tight focusing properties of the radially polarized MGSM vortex beam passing through a high numerical aperture (NA) objective lens are investigated numerically based on the vectorial diffraction theory. Numerical results show that the focal intensity distributions of the radially polarized MGSM vortex beam can be shaped by regulating the structure of the correlation functions and the topological charge of vortex phase. In contrast with the integral vortex beam, the most intriguing property of the fractional vortex beam is that the focal intensity distribution at the focal plane can be nonuniformity and asymmetry, while such unique characteristics will vanish when the spatial coherence length is sufficiently small. Furthermore, some focal fields with novel structure, such as a focal spot with nonuniform asymmetric or an anomalous asymmetric hollow focal field, can be formed by choosing suitable fractional values of topological charge and spatial coherence length. Our results will be useful for optical trapping, especially for trapping of irregular particles or manipulation of absorbing particles.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

2018 (4)

2017 (4)

2016 (2)

L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016).
[Crossref]

Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. 41(9), 1929–1932 (2016).
[Crossref]

2015 (3)

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

2014 (1)

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

2012 (4)

2011 (3)

S. Yan, B. Yao, and R. Rupp, “Shifting the spherical focus of a 4Pi focusing system,” Opt. Express 19(2), 673–678 (2011).
[Crossref]

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

2010 (2)

2009 (3)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

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(3), 241–246 (2009).
[Crossref]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

2008 (2)

T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008).
[Crossref]

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

2006 (1)

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[Crossref]

2005 (1)

2004 (1)

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

2002 (1)

1995 (1)

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

Basistiy, I.

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

Bo, F.

Cai, Y.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

J. Zeng, X. Liu, F. Wang, C. Zhao, and Y. Cai, “Partially coherent fractional vortex beam,” Opt. Express 26(21), 26830–26844 (2018).
[Crossref]

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Chen, B.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

Chen, H.

Chen, L.

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

Chen, W.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[Crossref]

Chen, Z.

Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. 41(9), 1929–1932 (2016).
[Crossref]

Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express 23(14), 17701–17710 (2015).
[Crossref]

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

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(3), 241–246 (2009).
[Crossref]

Chong, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Ding, J.

Dong, Y.

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

Y. Dong, F. Wang, C. Zhao, and Y. Cai, “Effect of spatial coherence on propagation, tight focusing, and radiation forces of an azimuthally polarized beam,” Phys. Rev. A 86(1), 013840 (2012).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

Fan, H.

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Fang, Y.

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

Feng, B.

Gong, L.

Gori, F.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Gu, B.

Guo, H.

Guo, L.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Hong, M.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Hua, L.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

Huang, K.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[Crossref]

Huang, L.

Huang, W.

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Jabbour, T. G.

Jiao, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Kang, X. L.

Korotkova, O.

Kozawa, Y.

Kuebler, S. M.

Leger, J.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

Li, J.

Li, X.

Li, Y. P.

Li, Z.

Liang, C.

C. Ping, C. Liang, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Lin, J.

Ling, L.

Liu, L.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Liu, X.

Lu, Q.

Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

Lu, X.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Luo, X.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Mao, Y.

Mei, Z.

Niu, H.

Peng, X.

Ping, C.

Pu, J.

L. Hua, B. Chen, Z. Chen, and J. Pu, “Tight focusing of partially coherent, partially polarized vortex beams,” J. Opt. 13(7), 075702 (2011).
[Crossref]

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(3), 241–246 (2009).
[Crossref]

Qian, B.

Qin, F.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Qiu, C.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Qu, J.

H. Xu, Y. Zhou, H. Wu, H. Chen, Z. Sheng, and J. Qu, “Focus shaping of the radially polarized Laguerre- Gaussian-correlated Schell-model vortex beams,” Opt. Express 26(16), 20076–20088 (2018).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 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(3), 241–246 (2009).
[Crossref]

Rupp, R.

Sahin, S.

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Santarsiero, M.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Sato, S.

Shchepakina, E.

Sheng, Z.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Shi, P.

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Soskin, M.

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

Tan, Z.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Tang, Z.

L. Guo, Z. Tang, C. Liang, and Z. Tan, “Intensity and spatial correlation properties of tightly focused partially coherent radially polarized vortex beams,” Opt. Laser Technol. 43(4), 895–898 (2011).
[Crossref]

Tao, S.

Urbach, H.

L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016).
[Crossref]

Vasnetsov, M.

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119(5-6), 604–612 (1995).
[Crossref]

Wang, F.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008).
[Crossref]

Wang, L.

Wang, S.

Wang, X.

Wang, Y.

Wei, L.

L. Wei and H. Urbach, “Shaping the focal field of radially/azimuthally polarized phase vortex with Zernike polynomials,” J. Opt. 18(6), 065608 (2016).
[Crossref]

Wen, J.

Wolf, E.

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[Crossref]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University).

Wu, H.

Wu, J.

F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a Subwavelength Needle with Ultra-long Focal Length by Focusing Azimuthally Polarized Light,” Sci. Rep. 5(1), 9977 (2015).
[Crossref]

Xu, H.

H. Xu, Y. Zhou, H. Wu, H. Chen, Z. Sheng, and J. Qu, “Focus shaping of the radially polarized Laguerre- Gaussian-correlated Schell-model vortex beams,” Opt. Express 26(16), 20076–20088 (2018).
[Crossref]

Z. Zhang, H. Fan, H. Xu, J. Qu, and W. Huang, “Three-dimensional focus shaping of partially coherent circularly polarized vortex beams using a binary optic,” J. Opt. 17(6), 065611 (2015).
[Crossref]

Yan, S.

Yang, X.

Yang, Y.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

Yao, B.

Yei, P.

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Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
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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(3), 241–246 (2009).
[Crossref]

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[Crossref]

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Y. Fang, Q. Lu, X. Wang, W. Zhang, and L. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

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[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Schematic diagram of tight focusing of a light beam focused by a high NA objective lens. $Q({r,\;\varphi,\;z} )$ is an observation point in the focal plane.
Fig. 2.
Fig. 2. Normalized focal intensity distribution of the total component ${I_{total}}$, transverse component ${I_{tra}}$, longitudinal component ${I_z}$ and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM beam with ${\delta _0} = 0.5\textrm{mm}$ and $l = 0$ at the focal plane for different values of beam index M.
Fig. 3.
Fig. 3. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM IV beam with integral topological charge $l = 3$ at the focal plane for different values of beam index M and spatial coherence length ${\delta _0}$.
Fig. 4.
Fig. 4. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM FV beam with fractional topological charge $l = 3.5$ at the focal plane for different values of beam index M and spatial coherence length ${\delta _0}$.
Fig. 5.
Fig. 5. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM IV beam with $M = 5$ at the focal plane for different values of integral topological charge l and spatial coherence length ${\delta _0}$.
Fig. 6.
Fig. 6. Normalized focal total intensity distribution and its corresponding cross line at ${\rho _x}({{\rho_y}} )= 0$ of a tightly focused radially polarized MGSM FV beam with $M = 5$ at the focal plane for different values of fractional topological charge l and spatial coherence length ${\delta _0}$.

Equations (20)

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E ( r ; ω ) = E x ( r ; ω ) e x + E y ( r ; ω ) e y = x w 0 exp ( r 2 w 0 2 ) e x + y w 0 exp ( r 2 w 0 2 ) e y ,
W α β ( 0 ) ( r 1 , r 2 , ω ) = E α ( r 1 ; ω ) E β ( r 2 ; ω ) , ( α , β = x , y )
W α β ( 0 ) ( r 1 , r 2 ) = p α β ( v ) H α ( r 1 , v ) H β ( r 2 , v ) d 2 v ,
H α ( r , v ) = F α ( r ) exp ( 2 π i r v ) ,
F α ( r ) = α w 0 ( 2 r w 0 ) l exp ( r 2 w 0 2 ) exp ( i l ϕ ) ,
p α β ( v ) = B α β δ α β 2 C 0 × { 1 [ 1 exp ( δ α β 2 v 2 / 2 ) ] M } .
B x y ( y x ) = B x x ( y y ) = 1 , δ x y ( y x ) = δ x x ( y y ) = δ 0 .
W α β ( r 1 , r 2 , 0 ) = α 1 β 2 w 0 2 ( 2 r 1 r 2 w 0 2 ) l exp ( r 1 2 + r 2 2 w 0 2 ) exp [ i l ( ϕ 1 ϕ 2 ) ] μ α β ( r 1 r 2 ) ,
μ α β ( r 1 r 2 ) = μ 0 ( r 1 r 2 ) = 1 C 0 m = 1 M ( M m ) ( 1 ) m 1 m exp [ ( r 1 r 2 ) 2 2 m δ 0 2 ] ,
E f ( r , φ , z ) = [ E f x E f y E f z ] = i k 1 f 2 π 0 θ max 0 2 π [ l x ( θ , ϕ ) ( θ , ϕ ) + l y ( θ , ϕ ) Ω ( θ , ϕ ) l x ( θ , ϕ ) Ω ( θ , ϕ ) + l y ( θ , ϕ ) ( θ , ϕ ) l x ( θ , ϕ ) sin θ cos ϕ l y ( θ , ϕ ) sin θ sin ϕ ] × cos θ sin θ exp [ i k 1 ( z cos θ + r sin θ cos ( ϕ φ ) ) ] d ϕ d θ
( θ , ϕ ) = cos θ + si n 2 ϕ ( 1 cos θ ) , Ω ( θ , ϕ ) = cos ϕ sin ϕ ( cos θ 1 ) ,
W f α β ( r 1 , φ 1 , r 2 , φ 2 , z ) = E f α ( r 1 , φ 1 , z ) E f β ( r 2 , φ 2 , z ) , ( α , β = x , y , z )
W f x x ( r 1 , φ 1 , r 2 , φ 2 , z ) = f 2 n 1 λ 2 0 θ m a x 0 θ m a x 0 2 π 0 2 π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) × exp [ i k 1 ( 2 1 ) ] × ( cos θ 1 cos θ 2 ) 3 / 2 ( sin θ 1 sin θ 2 ) 2 cos ϕ 1 cos ϕ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 ,
W f y y ( r 1 , φ 1 , r 2 , φ 2 , z ) = f 2 n 1 λ 2 0 θ m a x 0 θ m a x 0 2 π 0 2 π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) × exp [ i k 1 ( 2 1 ) ] × ( cos θ 1 cos θ 2 ) 3 / 2 ( sin θ 1 sin θ 2 ) 2 sin ϕ 1 sin ϕ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 ,
W f z z ( r 1 , φ 1 , r 2 , φ 2 , z ) = f 2 n 1 λ 2 0 θ m a x 0 θ m a x 0 2 π 0 2 π W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) × exp [ i k 1 ( 2 1 ) ] × ( cos θ 1 cos θ 2 ) 1 / 2 ( sin θ 1 sin θ 2 ) 3 d θ 1 d θ 2 d ϕ 1 d ϕ 2
i = z cos θ i + r i sin θ i cos ( ϕ i φ i ) , ( i = 1 , 2 ) ,
W 0 ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = 1 C 0 f 2 w 0 2 ( 2 f 2 sin θ 1 sin θ 2 cos ( ϕ 1 ϕ 2 ) w 0 2 ) l exp [ f 2 w 0 2 ( si n 2 θ 1 + si n 2 θ 2 ) + i l ( ϕ 1 ϕ 2 ) ] × m = 1 M ( M m ) ( 1 ) m 1 m exp [ f 2 2 m δ 0 2 ( si n 2 θ 1 + si n 2 θ 2 2 sin θ 1 sin θ 2 cos ( ϕ 1 ϕ 2 ) ) ] .
I t r a ( r , φ , z ) = W f x x ( r , φ , r , φ , z ) + W f y y ( r , φ , r , φ , z ) ,
I z ( r , φ , z ) = W f z z ( r , φ , r , φ , z ) ,
I t o t a l ( r , φ , z ) = I t r a ( r , φ , z ) + I z ( r , φ , z ) .

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