Abstract

Based on vectorial diffraction theory, tight focusing properties of quasi-cylindrical polarized beams (QCPBs) composed of equal fan-shaped sectors with linear polarization are investigated. We find that, for quasi-radially polarized illumination, a weak azimuthal component emerges and the circular symmetry of focus is traded in when the total number of sector N is small, but when N8 it is approaching that of a perfect radially polarized beam with a deviation smaller than 5.3% and a ratio of maximum total intensity larger than 95.5%. Meanwhile, for quasi-azimuthal polarized illumination, although weak radial and longitudinal components emerge, it is also close to that of the perfect azimuthally polarized beam when N8 with deviation smaller than 5.3% and a ratio larger than 95.0%. These results not only reveal a deep understanding of the focusing properties of QCPBs, but also provide an important contribution toward optimization of the monolithic methods for generating vector beams.

© 2014 Optical Society of America

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References

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Z. Man, C. Min, Y. Zhang, Z. Shen, and X.-C. Yuan, “Arbitrary vector beams with selective polarization states patterned by tailored polarizing films,” Laser Phys. 23, 105001 (2013).
[CrossRef]

C. Zhang, R. Wang, C. Min, S. Zhu, and X.-C. Yuan, “Experimental approach to the microscopic phase-sensitive surface plasmon resonance biosensor,” Appl. Phys. Lett. 102, 011114 (2013).
[CrossRef]

2012 (4)

2011 (5)

2009 (1)

2008 (1)

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

2007 (5)

2006 (1)

2004 (1)

2003 (1)

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

2002 (1)

2001 (1)

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

2000 (1)

1999 (1)

V. G. Niziv and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[CrossRef]

1996 (2)

1993 (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. London Ser. A 253, 358–379 (1959).
[CrossRef]

Arisaka, K.

A. Cheng, J. T. Goncalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different with spatiotemporal multiplexing,” Nat. Methods 8, 139–142 (2011).
[CrossRef]

Beresna, M.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Bomzon, Z.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

Brown, T. G.

Chen, H.

Cheng, A.

A. Cheng, J. T. Goncalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different with spatiotemporal multiplexing,” Nat. Methods 8, 139–142 (2011).
[CrossRef]

Ding, J.

Dong, X.

Dorn, R.

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

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Gahagan, K. T.

Gecevicius, M.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

Gertus, T.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

Golshani, P.

A. Cheng, J. T. Goncalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different with spatiotemporal multiplexing,” Nat. Methods 8, 139–142 (2011).
[CrossRef]

Goncalves, J. T.

A. Cheng, J. T. Goncalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different with spatiotemporal multiplexing,” Nat. Methods 8, 139–142 (2011).
[CrossRef]

Guo, C.-S.

Guo, H.

Guo, L. J.

Hao, J.

Hasman, E.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

Higuchi, T.

Hu, Q.

Imai, R.

Jackel, S.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468–1470 (2007).
[CrossRef]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Kanda, N.

Kazansky, P. G.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

Kim, G. H.

Kimura, W. D.

Kleiner, V.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

Konishi, K.

Kozawa, Y.

Kuwata-Gonokami, M.

Leuchs, G.

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

Lumer, Y.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468–1470 (2007).
[CrossRef]

Machavariani, G.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468–1470 (2007).
[CrossRef]

Man, Z.

Z. Man, C. Min, Y. Zhang, Z. Shen, and X.-C. Yuan, “Arbitrary vector beams with selective polarization states patterned by tailored polarizing films,” Laser Phys. 23, 105001 (2013).
[CrossRef]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Meir, A.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468–1470 (2007).
[CrossRef]

Min, C.

C. Zhang, R. Wang, C. Min, S. Zhu, and X.-C. Yuan, “Experimental approach to the microscopic phase-sensitive surface plasmon resonance biosensor,” Appl. Phys. Lett. 102, 011114 (2013).
[CrossRef]

Z. Man, C. Min, Y. Zhang, Z. Shen, and X.-C. Yuan, “Arbitrary vector beams with selective polarization states patterned by tailored polarizing films,” Laser Phys. 23, 105001 (2013).
[CrossRef]

Min, C. J.

Moshe, I.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468–1470 (2007).
[CrossRef]

Nesterov, A. V.

V. G. Niziv and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[CrossRef]

Ni, W.-J.

Niziv, V. G.

V. G. Niziv and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[CrossRef]

Noda, S.

K. Sakai and S. Noda, “Optical trapping of metal particles in doughnut-shaped beam emitted by photonic-crystal laser,” Electron. Lett. 43, 107–108 (2007).
[CrossRef]

Novotny, L.

Portera-Cailliau, C.

A. Cheng, J. T. Goncalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different with spatiotemporal multiplexing,” Nat. Methods 8, 139–142 (2011).
[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]

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. London Ser. A 253, 358–379 (1959).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Sakai, K.

K. Sakai and S. Noda, “Optical trapping of metal particles in doughnut-shaped beam emitted by photonic-crystal laser,” Electron. Lett. 43, 107–108 (2007).
[CrossRef]

Sato, S.

Schadt, M.

Shen, Z.

Z. Man, C. Min, Y. Zhang, Z. Shen, and X.-C. Yuan, “Arbitrary vector beams with selective polarization states patterned by tailored polarizing films,” Laser Phys. 23, 105001 (2013).
[CrossRef]

L. J. Guo, C. J. Min, G. H. Yuan, C. L. Zhang, J. G. Wang, Z. Shen, and X.-C. Yuan, “Optically stitched arbitrary fan-sectors with selective polarization states for dynamic manipulation of surface plasmon polaritons,” Opt. Express 20, 24748–24753 (2012).
[CrossRef]

Stalder, M.

Sui, G.

Swartzlander, G. A.

Tidwell, S. C.

Wang, H.-T.

Wang, J. G.

Wang, R.

C. Zhang, R. Wang, C. Min, S. Zhu, and X.-C. Yuan, “Experimental approach to the microscopic phase-sensitive surface plasmon resonance biosensor,” Appl. Phys. Lett. 102, 011114 (2013).
[CrossRef]

R. Wang, C. Zhang, Y. Yang, S. Zhu, and X.-C. Yuan, “Focused cylindrical vector beam assisted microscopic pSPR biosensor with an ultra-wide dynamic range,” Opt. Lett. 37, 2091–2093 (2012).
[CrossRef]

Wang, X.-L.

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. London Ser. A 253, 358–379 (1959).
[CrossRef]

Xu, J.

Yang, N.

Yang, Y.

Youngworth, K. S.

Yuan, G. H.

Yuan, X.-C.

C. Zhang, R. Wang, C. Min, S. Zhu, and X.-C. Yuan, “Experimental approach to the microscopic phase-sensitive surface plasmon resonance biosensor,” Appl. Phys. Lett. 102, 011114 (2013).
[CrossRef]

Z. Man, C. Min, Y. Zhang, Z. Shen, and X.-C. Yuan, “Arbitrary vector beams with selective polarization states patterned by tailored polarizing films,” Laser Phys. 23, 105001 (2013).
[CrossRef]

L. J. Guo, C. J. Min, G. H. Yuan, C. L. Zhang, J. G. Wang, Z. Shen, and X.-C. Yuan, “Optically stitched arbitrary fan-sectors with selective polarization states for dynamic manipulation of surface plasmon polaritons,” Opt. Express 20, 24748–24753 (2012).
[CrossRef]

R. Wang, C. Zhang, Y. Yang, S. Zhu, and X.-C. Yuan, “Focused cylindrical vector beam assisted microscopic pSPR biosensor with an ultra-wide dynamic range,” Opt. Lett. 37, 2091–2093 (2012).
[CrossRef]

Zhan, Q.

Zhang, B.-F.

Zhang, C.

C. Zhang, R. Wang, C. Min, S. Zhu, and X.-C. Yuan, “Experimental approach to the microscopic phase-sensitive surface plasmon resonance biosensor,” Appl. Phys. Lett. 102, 011114 (2013).
[CrossRef]

R. Wang, C. Zhang, Y. Yang, S. Zhu, and X.-C. Yuan, “Focused cylindrical vector beam assisted microscopic pSPR biosensor with an ultra-wide dynamic range,” Opt. Lett. 37, 2091–2093 (2012).
[CrossRef]

Zhang, C. L.

Zhang, Y.

Z. Man, C. Min, Y. Zhang, Z. Shen, and X.-C. Yuan, “Arbitrary vector beams with selective polarization states patterned by tailored polarizing films,” Laser Phys. 23, 105001 (2013).
[CrossRef]

Zheng, Z.

Zhu, S.

C. Zhang, R. Wang, C. Min, S. Zhu, and X.-C. Yuan, “Experimental approach to the microscopic phase-sensitive surface plasmon resonance biosensor,” Appl. Phys. Lett. 102, 011114 (2013).
[CrossRef]

R. Wang, C. Zhang, Y. Yang, S. Zhu, and X.-C. Yuan, “Focused cylindrical vector beam assisted microscopic pSPR biosensor with an ultra-wide dynamic range,” Opt. Lett. 37, 2091–2093 (2012).
[CrossRef]

Zhuang, S.

Zurita-Sánchez, J. R.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. Lett. (3)

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98, 201101 (2011).
[CrossRef]

C. Zhang, R. Wang, C. Min, S. Zhu, and X.-C. Yuan, “Experimental approach to the microscopic phase-sensitive surface plasmon resonance biosensor,” Appl. Phys. Lett. 102, 011114 (2013).
[CrossRef]

Electron. Lett. (1)

K. Sakai and S. Noda, “Optical trapping of metal particles in doughnut-shaped beam emitted by photonic-crystal laser,” Electron. Lett. 43, 107–108 (2007).
[CrossRef]

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

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

J. Phys. D (1)

V. G. Niziv and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999).
[CrossRef]

Laser Phys. (1)

Z. Man, C. Min, Y. Zhang, Z. Shen, and X.-C. Yuan, “Arbitrary vector beams with selective polarization states patterned by tailored polarizing films,” Laser Phys. 23, 105001 (2013).
[CrossRef]

Nat. Methods (1)

A. Cheng, J. T. Goncalves, P. Golshani, K. Arisaka, and C. Portera-Cailliau, “Simultaneous two-photon calcium imaging at different with spatiotemporal multiplexing,” Nat. Methods 8, 139–142 (2011).
[CrossRef]

New J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[CrossRef]

Opt. Commun. (1)

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams,” Opt. Commun. 281, 732–738 (2008).
[CrossRef]

Opt. Express (5)

Opt. Lett. (8)

Phys. Rev. Lett. (1)

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

Proc. R. Soc. London Ser. A (1)

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

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Figures (5)

Fig. 1.
Fig. 1.

(a) Schematic of the QCPB focused by a high NA lens. The origin of the coordinate system is located in the geometric focus of the lens. θ denotes the angle between the convergent ray and the optical axis, and α is the maximum incident angle. As an example, the incident beam is chosen to be QAPB with N=6. (b), (c) Intensity (denoted as background) and polarization (denoted as arrows) distribution of QRPB with (b) N=4 and (c) N=8 in the waist plane. (d), (e) Intensity and polarization distribution of QAPB with (d) N=4 and (e) N=8 in the waist plane, the location along the propagation direction where the beam radius has a minimum.

Fig. 2.
Fig. 2.

Normalized intensity of the radial, azimuthal, longitudinal, and total components of QRPB with N=4, 8, 12, and 16, respectively, at focus (XY plane, the first four rows) and through focus (YZ plane, the last row). Note that the intensity of the azimuthal component is the result after taking the logarithm since the dates vary dramatically from N=4 to N=16. The units of x, y, and z are in wavelengths.

Fig. 3.
Fig. 3.

(a) Normalized intensity profile for PRPB and QRPB along X or Y axis with N=4, 8, 12, and 16, respectively, across the focus in the plane perpendicular to the beam axis along the x or y axis. All the strengths are normalized to the maximum of the total strength of PRPB. (b) Deviation of the focusing field of QRPB from that of PRPB in the vertical axis on the left and the ratio of maximum total strength between QRPB and PRPB in the vertical axis on the right versus the total number of sectors N. (c) Ratio of the maximum strength of the longitudinal and transverse fields of QRPB versus the total number of sector N. The dotted line shows the case of PRPB illumination. (d) FWHM of total and longitudinal fields of QRPB and PRPB. (e) Normalized intensity profile for PRPB and QRPB along Z axis with N=4, 8, 12, and 16, respectively, through focus. All the strengths are normalized to the maximum of the total strength of PRPB.

Fig. 4.
Fig. 4.

Normalized intensity of the radial, azimuthal, longitudinal, and total components for QAPB with N=4, 8, 12, and 16, respectively, at focus (XY plane, the first four rows) and through focus (YZ plane, the last row). Note that the intensity of radial and longitudinal components is the result after taking the logarithm since the dates vary dramatically from N=4 to N=16. The units of x, y, and z are in wavelengths.

Fig. 5.
Fig. 5.

(a) Normalized intensity profile for PAPB and QAPB with N=4, 8, 12, and 16, respectively, across the focus in the plane perpendicular to the beam axis along the x or y axis. All the strengths are normalized to the maximum of the total strength of PAPB. (b) Deviation of focusing field of QAPB from that of PAPB in the vertical axis on the left and the ratio of maximum total strength between QAPB and PAPB in the vertical axis on the right versus the total number of sectors N.

Equations (11)

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Ex(rs,θs,φs)=iBπ0α02πcos1/2θsinθl0(θ)×[sin(Θφ)sinφ+cos(Θφ)cosθcosφ]×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ,
Ey(rs,θs,φs)=iBπ0α02πcos1/2θsinθl0(θ)×[sin(Θφ)cosφ+cos(Θφ)cosθsinφ]×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ,
Ez(rs,θs,φs)=iBπ0α02πcos1/2θsinθl0(θ)×[cos(Θφ)sinθ]×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ,
Eφ=EycosφsExsinφs,Eρ=Excosφs+Eysinφs.
Eρ(rs,θs,φs)=n=1NiBπ0α2(n1)π/N2nπ/Ncos1/2θsinθl0(θ)×{sin[(2n1)π/Nφ]sin(φsφ)+cos[(2n1)π/Nφ]cosθcos(φφs)}×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ,
Eφ(rs,θs,φs)=n=1NiBπ0α2(n1)π/N2nπ/Ncos1/2θsinθl0(θ)×{sin[(2n1)π/Nφ]cos(φsφ)+cos[(2n1)π/Nφ]cosθsin(φφs)}×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ,
Ez(rs,θs,φs)=n=1NiBπ0α2(n1)π/N2nπ/Ncos1/2θsinθl0(θ)×cos[(2n1)π/Nφ]sinθ×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ.
Eρ(rs,θs,φs)=n=1NiBπ0α2(n1)π/N2nπ/Ncos1/2θsinθl0(θ)×{sin[(2n1)π/N+π/2φ]sin(φsφ)+cos[(2n1)π/N+π/2φ]cosθcos(φφs)}×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ,
Eφ(rs,θs,φs)=n=1NiBπ0α2(n1)π/N2nπ/Ncos1/2θsinθl0(θ)×{sin[(2n1)π/N+π/2φ]cos(φsφ)+cos[(2n1)π/N+π/2φ]cosθsin(φφs)}×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ,
Ez(rs,θs,φs)=n=1NiBπ0α2(n1)π/N2nπ/Ncos1/2θsinθl0(θ)×cos[(2n1)π/N+π/2φ]sinθ×exp{ikrs[sinθssinθcos(φφs)+cosθscosθ]}dθdφ.
l0(θ)=exp[β02(sinθsinα)2]J1(2β0sinθsinα),

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