Abstract

The longitudinal electric field produced by focusing a radially polarized beam is applied in confocal laser scanning microscopy by introducing a higher-order transverse mode, combined with a technique of polarization conversion for signal detection. This technique improves signal detection corresponding to the longitudinally polarized field under a small confocal pinhole, enabling full utilization of the small focal spot characteristic of the longitudinal field. Detailed numerical and experimental studies demonstrate the enhanced spatial resolution in confocal imaging that detects a scattering signal using a higher-order radially polarized beam. Our method can be widely applied in various imaging techniques that detect coherent signals such as second-harmonic generation microscopy.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]
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    [Crossref]
  33. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
    [Crossref]

2020 (1)

P. Meng, H. Pham, S. F. Pereira, and H. P. Urbach, “Demonstration of lateral resolution enhancement by focusing amplitude modulated radially polarized light in a confocal imaging system,” J. Opt. 22(4), 045605 (2020).
[Crossref]

2018 (2)

2016 (1)

R. Drevinskas, J. Zhang, M. Beresna, M. Gecevičius, A. G. Kazanskii, Y. P. Svirko, and P. G. Kazansky, “Laser material processing with tightly focused cylindrical vector beams,” Appl. Phys. Lett. 108(22), 221107 (2016).
[Crossref]

2015 (2)

2014 (2)

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113(26), 263901 (2014).
[Crossref]

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63(1), 23–32 (2014).
[Crossref]

2013 (2)

2012 (2)

Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 29(11), 2439–2443 (2012).
[Crossref]

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3(1), 998 (2012).
[Crossref]

2011 (2)

2009 (1)

2008 (1)

2007 (3)

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

2004 (1)

2003 (3)

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

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[Crossref]

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Near-field scattering of longitudinal fields,” Appl. Phys. Lett. 82(25), 4596–4598 (2003).
[Crossref]

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref]

2000 (2)

1998 (1)

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45(8), 1681–1698 (1998).
[Crossref]

1997 (1)

C. J. R. Sheppard and P. Török, “An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy,” Bioimaging 5(4), 205–218 (1997).
[Crossref]

1987 (1)

1977 (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. Lond. A 253(1274), 358–379 (1959).
[Crossref]

Beresna, M.

R. Drevinskas, J. Zhang, M. Beresna, M. Gecevičius, A. G. Kazanskii, Y. P. Svirko, and P. G. Kazansky, “Laser material processing with tightly focused cylindrical vector beams,” Appl. Phys. Lett. 108(22), 221107 (2016).
[Crossref]

Beversluis, M.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[Crossref]

Beversluis, M. R.

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Near-field scattering of longitudinal fields,” Appl. Phys. Lett. 82(25), 4596–4598 (2003).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref]

Bouhelier, A.

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Near-field scattering of longitudinal fields,” Appl. Phys. Lett. 82(25), 4596–4598 (2003).
[Crossref]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[Crossref]

Boyd, R.

Brown, T.

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref]

Carlini, A. R.

Chen, Y.

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113(26), 263901 (2014).
[Crossref]

Y. Chen, D. Zhang, L. Han, G. Rui, X. Wang, P. Wang, and H. Ming, “Surface-plasmon-coupled emission microscopy with a polarization converter,” Opt. Lett. 38(5), 736–738 (2013).
[Crossref]

Choudhury, A.

Courjon, D.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[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]

Drevinskas, R.

R. Drevinskas, J. Zhang, M. Beresna, M. Gecevičius, A. G. Kazanskii, Y. P. Svirko, and P. G. Kazansky, “Laser material processing with tightly focused cylindrical vector beams,” Appl. Phys. Lett. 108(22), 221107 (2016).
[Crossref]

Ecoffey, C.

Failla, A. V.

Foreman, M. R.

M. R. Foreman and P. Török, “Computational methods in vectorial imaging,” J. Mod. Opt. 58(5-6), 339–364 (2011).
[Crossref]

Gecevicius, M.

R. Drevinskas, J. Zhang, M. Beresna, M. Gecevičius, A. G. Kazanskii, Y. P. Svirko, and P. G. Kazansky, “Laser material processing with tightly focused cylindrical vector beams,” Appl. Phys. Lett. 108(22), 221107 (2016).
[Crossref]

Gregg, P.

Grosjean, T.

Gu, M.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3(1), 998 (2012).
[Crossref]

Han, L.

Hartschuh, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[Crossref]

Hashimoto, N.

Hibi, T.

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63(1), 23–32 (2014).
[Crossref]

Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011).
[Crossref]

Higdon, P. D.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45(8), 1681–1698 (1998).
[Crossref]

Horanai, H.

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63(1), 23–32 (2014).
[Crossref]

Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011).
[Crossref]

Ipponjima, S.

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63(1), 23–32 (2014).
[Crossref]

Jackel, S.

Karimi, E.

Kazanskii, A. G.

R. Drevinskas, J. Zhang, M. Beresna, M. Gecevičius, A. G. Kazanskii, Y. P. Svirko, and P. G. Kazansky, “Laser material processing with tightly focused cylindrical vector beams,” Appl. Phys. Lett. 108(22), 221107 (2016).
[Crossref]

Kazansky, P. G.

R. Drevinskas, J. Zhang, M. Beresna, M. Gecevičius, A. G. Kazanskii, Y. P. Svirko, and P. G. Kazansky, “Laser material processing with tightly focused cylindrical vector beams,” Appl. Phys. Lett. 108(22), 221107 (2016).
[Crossref]

Kozawa, Y.

Kunz, R. E.

Kurihara, M.

Lan, T.-H.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3(1), 998 (2012).
[Crossref]

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, X.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3(1), 998 (2012).
[Crossref]

Lukosz, W.

Lumer, Y.

Machavariani, G.

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Marrucci, L.

L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, “Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy,” Optica 2(10), 900–903 (2015).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Matsunaga, D.

Meir, A.

Meixner, A. J.

Meng, P.

P. Meng, H. Pham, S. F. Pereira, and H. P. Urbach, “Demonstration of lateral resolution enhancement by focusing amplitude modulated radially polarized light in a confocal imaging system,” J. Opt. 22(4), 045605 (2020).
[Crossref]

P. Meng, S. Pereira, and P. Urbach, “Confocal microscopy with a radially polarized focused beam,” Opt. Express 26(23), 29600–29613 (2018).
[Crossref]

Ming, H.

Moshe, I.

Nemoto, T.

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63(1), 23–32 (2014).
[Crossref]

Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011).
[Crossref]

Novotny, L.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[Crossref]

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Near-field scattering of longitudinal fields,” Appl. Phys. Lett. 82(25), 4596–4598 (2003).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Pereira, S.

Pereira, S. F.

P. Meng, H. Pham, S. F. Pereira, and H. P. Urbach, “Demonstration of lateral resolution enhancement by focusing amplitude modulated radially polarized light in a confocal imaging system,” J. Opt. 22(4), 045605 (2020).
[Crossref]

Pham, H.

P. Meng, H. Pham, S. F. Pereira, and H. P. Urbach, “Demonstration of lateral resolution enhancement by focusing amplitude modulated radially polarized light in a confocal imaging system,” J. Opt. 22(4), 045605 (2020).
[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]

Ramachandran, S.

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. Lond. A 253(1274), 358–379 (1959).
[Crossref]

Rubano, A.

Rui, G.

Sato, A.

Sato, S.

Sheppard, C.

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

Sheppard, C. J. R.

Steiner, M.

Svirko, Y. P.

R. Drevinskas, J. Zhang, M. Beresna, M. Gecevičius, A. G. Kazanskii, Y. P. Svirko, and P. G. Kazansky, “Laser material processing with tightly focused cylindrical vector beams,” Appl. Phys. Lett. 108(22), 221107 (2016).
[Crossref]

Tang, W. T.

Tien, C.-H.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun. 3(1), 998 (2012).
[Crossref]

Török, P.

M. R. Foreman and P. Török, “Computational methods in vectorial imaging,” J. Mod. Opt. 58(5-6), 339–364 (2011).
[Crossref]

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45(8), 1681–1698 (1998).
[Crossref]

C. J. R. Sheppard and P. Török, “An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy,” Bioimaging 5(4), 205–218 (1997).
[Crossref]

Urbach, H. P.

P. Meng, H. Pham, S. F. Pereira, and H. P. Urbach, “Demonstration of lateral resolution enhancement by focusing amplitude modulated radially polarized light in a confocal imaging system,” J. Opt. 22(4), 045605 (2020).
[Crossref]

Urbach, P.

Wang, P.

Wang, X.

Wilson, T.

P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45(8), 1681–1698 (1998).
[Crossref]

T. Wilson, A. R. Carlini, T. Wilson, and A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12(4), 227–229 (1987).
[Crossref]

T. Wilson, A. R. Carlini, T. Wilson, and A. R. Carlini, “Size of the detector in confocal imaging systems,” Opt. Lett. 12(4), 227–229 (1987).
[Crossref]

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

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. Lond. A 253(1274), 358–379 (1959).
[Crossref]

Xie, X.

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113(26), 263901 (2014).
[Crossref]

Yan, L.

Yang, K.

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113(26), 263901 (2014).
[Crossref]

Yew, E. Y. S.

Yokoyama, H.

S. Ipponjima, T. Hibi, Y. Kozawa, H. Horanai, H. Yokoyama, S. Sato, and T. Nemoto, “Improvement of lateral resolution and extension of depth of field in two-photon microscopy by a higher-order radially polarized beam,” Microscopy 63(1), 23–32 (2014).
[Crossref]

Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011).
[Crossref]

Youngworth, K.

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001).
[Crossref]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7(2), 77–87 (2000).
[Crossref]

Zhang, D.

Zhang, J.

R. Drevinskas, J. Zhang, M. Beresna, M. Gecevičius, A. G. Kazanskii, Y. P. Svirko, and P. G. Kazansky, “Laser material processing with tightly focused cylindrical vector beams,” Appl. Phys. Lett. 108(22), 221107 (2016).
[Crossref]

Zhou, J.

X. Xie, Y. Chen, K. Yang, and J. Zhou, “Harnessing the point-spread function for high-resolution far-field optical microscopy,” Phys. Rev. Lett. 113(26), 263901 (2014).
[Crossref]

Züchner, T.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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Figures (6)

Fig. 1.
Fig. 1. Schematic diagram of confocal detection with polarization conversion. (a) Geometrical representation of the detected path of the optical system, where êx, êy, êr, and êϕ represent unit vectors along the x, y, radial, and azimuthal directions, respectively; (b) conceptual diagram of the image formation with polarization conversion for longitudinally and transversely oriented dipoles.
Fig. 2.
Fig. 2. Numerical simulations of a small scatterer image obtained by CLSM with and without polarization conversion. Conventional detection using (a) LP-Gauss and (b) RP-LG0,1 beams; detection with polarization conversion using (c) LP-Gauss, (d) RP-LG0,1, and (e) RP-LG5,1 beams. The first and second rows show the intensity distributions of the focal spot (the sum of all polarization components) and its intensity profiles along the x-axis across the center spot, respectively. The black, blue, and red solid lines of the intensity profiles in the second row are the total, transverse, and longitudinal components, respectively. The calculated images with different pinhole diameters (as indicated on the left) are shown in the third to sixth rows. The scale bar represents 1λm (= λ / n). The calculated region is 4λm × 4λm. The color scale is normalized to the maximum value of each condition. Finally, NA = 1.4 and n = 1.52.
Fig. 3.
Fig. 3. Lateral FWHM size of a small scatterer image as a function of the pinhole diameter. The size was measured in units of λm (= λ / n). NA = 1.4 and n = 1.52.
Fig. 4.
Fig. 4. Experimental setup. LCOS-SLM, liquid crystal on a silicon spatial light modulator; HWP, half-wave plate.
Fig. 5.
Fig. 5. The measured images of an isolated 100-nm gold particle acquired with different confocal pinhole sizes (indicated on the left). (a) Conventional detection using an LP-Gauss beam; detection with polarization conversion using (b) RP-LG0,1 and (c) RP-LG5,1 beams. The scale bar in (a) represents 250 nm. (d–f) Intensity profiles (red dashed lines) along the x-axis of the images corresponding to (a)–(c), acquired by 0.4 AU, and the simulation results (blue solid lines).
Fig. 6.
Fig. 6. Images of clusters of gold particles with a nominal diameter of 150 nm acquired with a confocal pinhole of 0.4 AU. (a) Conventional detection using an LP-Gauss beam; detection with polarization conversion using (b) RP-LG0,1 and (c) RP-LG5,1 beams. The scale bars in (a)–(c) represent 500 nm. (d–f) Magnified images of the region indicated by red rectangles in (a)–(c); scale bars in (d)–(f) represent 200 nm. (g) Normalized intensity profiles along the red dashed lines in (d)–(f).

Equations (12)

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I d ( x d , y d ; x s , y s ) = | e d ( x d M x s , y d M y s ) | 2 ,
I sig ( x s , y s ) = 0 2 π 0 R I d ( r p cos φ p , r p sin φ p ; x s , y s ) r p d r p d φ p ,
p ( x s , y s ) = ( α x x 0 0 0 α y y 0 0 0 α z z ) e ill ( x s , y s ) ,
( E c x E c y ) = 1 cos θ 1 ( p x 2 [ ( 1 + cos θ 1 ) ( 1 cos θ 1 ) cos 2 ϕ ] p y 2 ( 1 cos θ 1 ) sin 2 ϕ p z sin θ 1 cos ϕ p x 2 ( 1 cos θ 1 ) sin 2 ϕ + p y 2 [ ( 1 + cos θ 1 ) + ( 1 cos θ 1 ) cos 2 ϕ ] p z sin θ 1 sin ϕ ) ,
( e d , x e d , y e d , z ) = i A π 0 2 π 0 α 2 E 2 exp [ i k r d sin θ 2 cos ( ϕ ϕ d ) ] exp ( i k z d cos θ 2 ) sin θ 2 d θ 2 d ϕ ,
E 2 = 1 2 ( E c x [ 1 + cos θ 2 ( 1 cos θ 2 ) cos 2 ϕ ] E c y ( 1 cos θ 2 ) sin 2 ϕ E c x ( 1 cos θ 2 ) sin 2 ϕ + E c y [ 1 + cos θ 2 + ( 1 cos θ 2 ) cos 2 ϕ ] 2 E c x cos ϕ sin θ 2 2 E c y sin ϕ sin θ 2 ) .
( E c x E c y ) = ( E c x cos ϕ + E c y sin ϕ E c x sin ϕ E c y cos ϕ ) = 1 cos θ 1 ( p x cos θ 1 cos ϕ + p y cos θ 1 sin ϕ p z sin θ 1 p x sin ϕ p y cos ϕ ) .
E 2 = 1 cos θ 1 ( p x 4 ( A cos ϕ + B cos 3 ϕ ) + p y 4 ( C sin ϕ + B sin 3 ϕ ) p z 2 ( D E cos 2 ϕ ) p x 4 ( A sin ϕ + B sin 3 ϕ ) p y 4 ( C cos ϕ + B cos 3 ϕ ) + p z 2 E sin 2 ϕ p x 2 ( D E cos 2 ϕ ) + p y 2 E sin 2 ϕ + p z F cos ϕ ) ,
A = 1 + cos θ 2 + ( 1 + 3 cos θ 2 ) cos θ 1 , A = 1 + 3 cos θ 2 ( 1 cos θ 2 ) cos θ 1 , B = ( 1 cos θ 2 ) ( 1 cos θ 2 ) cos θ 1 , C = 1 cos θ 2 + ( 3 + cos θ 2 ) cos θ 1 , C = 3 + cos θ 2 + ( 1 cos θ 2 ) cos θ 1 , D = ( 1 + cos θ 2 ) sin θ 1 , D = ( 1 + cos θ 1 ) sin θ 2 , E = ( 1 cos θ 2 ) sin θ 1 , E = ( 1 cos θ 1 ) sin θ 2 , F = sin θ 1 sin θ 2 .
e d = [ i 2 p x K 1 I cos ϕ d + i 2 p y K 1 I I sin ϕ d i 2 ( p x cos 3 ϕ d + p y sin 3 ϕ d ) K 3 I p z K 0 I p z K 2 I cos 2 ϕ d i 2 p x K 1 I I I sin ϕ d i 2 p y K 1 I V cos ϕ d i 2 ( p x sin 3 ϕ d p y cos 3 ϕ d ) K 3 I p z K 2 I sin 2 ϕ d p x K 0 I I ( p x cos 2 ϕ d + p y sin 2 ϕ d ) K 2 I I + 2 i p z K 1 V cos ϕ ] .
K n N ( r d , z d ) = 0 α 2 cos θ 2 cos θ 1 sin θ 2 O n N J n ( k r d sin θ 2 ) exp ( i k z d cos θ 2 ) d θ 2 ,
O 0 I = ( 1 + cos θ 2 ) sin θ 1 , O 0 I I = ( 1 + cos θ 1 ) sin θ 2 , O 1 I = 1 + cos θ 2 + ( 1 + 3 cos θ 2 ) cos θ 1 , O 1 I I = 1 cos θ 2 + ( 3 + cos θ 2 ) cos θ 1 , O 1 I I I = 1 + 3 cos θ 2 ( 1 cos θ 2 ) cos θ 1 , O 1 I V = 3 + cos θ 2 + ( 1 cos θ 2 ) cos θ 1 , O 1 V = sin θ 1 sin θ 2 , O 2 I = ( 1 cos θ 2 ) sin θ 1 , O 2 I I = ( 1 cos θ 1 ) sin θ 2 , O 3 I = ( 1 cos θ 1 ) ( 1 cos θ 2 ) .

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