Abstract

The aim of this paper is to provide a formal framework for designing highly focused fields with specific transversal features when the incoming beam is partially polarized. More specifically, we develop a field with a transversal component that remains unpolarized in the focal area. Special attention is paid to the design of the input beam and the development of the experiment. The implementation of such fields is possible by using an interferometric setup combined with the use of digital holography techniques. Experimental results are compared with those obtained numerically.

© 2014 Optical Society of America

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References

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  14. I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Jones matrix treatment for optical fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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2014 (2)

2013 (5)

2012 (2)

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

F. Kenny, D. Lara, O. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-na microscopy,” Opt. Express 20, 14015–14029 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (7)

2009 (1)

2008 (3)

2007 (4)

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]

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24, 1793–1798 (2007).
[Crossref]

X.L. Wang, J. Ding, W.J. Ni, C.S. Guo, and H.T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007),
[Crossref] [PubMed]

2006 (2)

2005 (1)

2004 (2)

2003 (2)

1997 (1)

1979 (1)

G. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

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]

Arrizón, V.

Auñón, J. M.

Bao, J.

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

Bao, K.

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

Barends, P.

G. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

Bekshaev, A. Y.

A. Y. Bekshaev, “A simple analytical model of the angular momentum transformation in strongly focused light beams,” Cent. Eur. J. Phys. 8, 947–960 (2010).
[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]

Blom, P.

G. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

Bokor, N.

K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector slepian harmonics and vector slepian multipole fields,” Opt. Commun. 288, 13–16 (2013).
[Crossref]

N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

Brakenhoff, G.

G. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

Brenner, P.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[Crossref] [PubMed]

Campos, J.

Capasso, F.

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

Carnicer, A.

Chen, J.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, 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]

Choudhury, A.

Dainty, C.

Davidson, N.

de Boer, J. F.

Ding, J.

Dorn, R.

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

Ergin, T.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[Crossref] [PubMed]

Fan, J. A.

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

Foreman, M. R.

Friberg, A. T.

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]

Guo, C.S.

Halas, N. J.

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

Hao, X.

Hasan, T.

Iemmi, C.

Jahn, K.

K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector slepian harmonics and vector slepian multipole fields,” Opt. Commun. 288, 13–16 (2013).
[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]

Juvells, I.

Kaivola, M.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
[Crossref]

Kenny, F.

Khonina, S. N.

Kim, K. H.

Kitamura, K.

Kozawa, Y.

Kuang, C.

Lara, D.

Lasser, T.

Lassiter, J. B.

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

Lee, B.

Leitgeb, R. A.

Leppänen, L.-P.

Lerman, G. 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] [PubMed]

Leutenegger, M.

Levy, U.

Li, J.

Li, Y.

Lindfors, K.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

K. Lindfors, T. Setälä, M. Kaivola, and A. T. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561–568 (2005).
[Crossref]

Liu, X.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, 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]

Maluenda, D.

Martinez-Herrero, R.

R. Martınez-Herrero, I. Juvells, and A. Carnicer, “Design of highly focused fields that remain unpolarized on axis,” Opt. Lett.39 (2014).
[Crossref]

Martínez-Herrero, R.

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]

Mejías, P.

Moreno, I.

Munro, P. R.

Ni, W.J.

Nieto-Vesperinas, M.

Noda, S.

Nordlander, P.

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

Park, B. H.

Pendry, J. B.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[Crossref] [PubMed]

Priimagi, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[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] [PubMed]

Rao, R.

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]

Rodríguez-Herrera, O.

Sakai, K.

Sato, S.

Setälä, T.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, 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.

Sherif, S. S.

Shevchenko, A.

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, 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]

Stenger, N.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[Crossref] [PubMed]

Török, P.

Tu, Y.

Volotovsky, S. G.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, 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, T.

Wang, W.

Wang, X.L.

Wegener, M.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[Crossref] [PubMed]

Wolf, E.

W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–496 (1997)
[Crossref]

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]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
[Crossref]

Yzuel, M. J.

Zhan, Q.

Appl. Opt. (1)

Cent. Eur. J. Phys. (1)

A. Y. Bekshaev, “A simple analytical model of the angular momentum transformation in strongly focused light beams,” Cent. Eur. J. Phys. 8, 947–960 (2010).
[Crossref]

J. Microsc. (1)

G. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light microscopy with high aperture immersion lenses,” J. Microsc. 117, 219–232 (1979).
[Crossref]

J. Opt. Soc. Am. A (6)

Nano Lett. (1)

J. A. Fan, K. Bao, J. B. Lassiter, J. Bao, N. J. Halas, P. Nordlander, and F. Capasso, “Near-normal incidence dark-field microscopy: applications to nanoplasmonic spectroscopy,” Nano Lett. 12, 2817–2821 (2012).
[Crossref] [PubMed]

Nat. Photonics (2)

K. Lindfors, A. Priimagi, T. Setälä, A. Shevchenko, A. T. Friberg, and M. Kaivola, “Local polarization of tightly focused unpolarized light,” Nat. Photonics 1, 228–231 (2007).
[Crossref]

H. Wang, L. Shi, B. Lukyanchuk, 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]

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)

K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector slepian harmonics and vector slepian multipole fields,” Opt. Commun. 288, 13–16 (2013).
[Crossref]

Opt. Express (11)

M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006).
[Crossref] [PubMed]

D. Maluenda, I. Juvells, R. Martínez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21, 5432–5439 (2013).
[Crossref] [PubMed]

K. H. Kim, B. H. Park, Y. Tu, T. Hasan, B. Lee, J. Li, and J. F. de Boer, “Polarization-sensitive optical frequency domain imaging based on unpolarized light,” Opt. Express 19, 552–561 (2011).
[Crossref] [PubMed]

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Jones matrix treatment for optical fourier processors with structured polarization,” Opt. Express 19, 4583–4594 (2011).
[Crossref] [PubMed]

F. Kenny, D. Lara, O. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-na microscopy,” Opt. Express 20, 14015–14029 (2012).
[Crossref] [PubMed]

K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18, 4518–4525 (2010).
[Crossref] [PubMed]

R. Martínez-Herrero and P. Mejías, “Angular momentum decomposition of nonparaxial light beams,” Opt. Express 18, 7965–7971 (2010).
[Crossref] [PubMed]

H.T. Wang, X.L. Wang, Y. Li, J. Chen, C.S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[Crossref] [PubMed]

G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (2008).
[Crossref] [PubMed]

M. R. Foreman, S. S. Sherif, P. R. Munro, and P. Török, “Inversion of the debye-wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express 16, 4901–4917 (2008).
[Crossref] [PubMed]

D. Maluenda, R. Martínez-Herrero, I. Juvells, and A. Carnicer, “Synthesis of highly focused fields with circular polarization at any transverse plane,” Opt. Express 22, 6859–6867 (2014).
[Crossref] [PubMed]

Opt. Lett. (8)

X.L. Wang, J. Ding, W.J. Ni, C.S. Guo, and H.T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007),
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[Crossref] [PubMed]

R. Martínez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38, 2065–2067 (2013).
[Crossref] [PubMed]

J. M. Auñón and M. Nieto-Vesperinas, “Partially coherent fluctuating sources that produce the same optical force as a laser beam,” Opt. Lett. 38, 2869–2872 (2013).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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Phys. Rev. Lett. (1)

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

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[Crossref]

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T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010).
[Crossref] [PubMed]

Other (2)

R. Martınez-Herrero, I. Juvells, and A. Carnicer, “Design of highly focused fields that remain unpolarized on axis,” Opt. Lett.39 (2014).
[Crossref]

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[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Geometry, bases and variables involved.
Fig. 2
Fig. 2 Numerical simulation of Eq. (10) for the beam proposed in Eq. (24), NA=0.85. First row: false color representation of the intensity at z = 0 and z = 2λ; second row: profiles of the irradiance and the transversal and longitudinal components.
Fig. 3
Fig. 3 Experimental setup
Fig. 4
Fig. 4 Experimental C0 distribution at the focal area z = 2λ.
Fig. 5
Fig. 5 Measure of the transversal Stokes parameters. First row: laser source; second and third rows: focal area of the microscope lens.

Tables (1)

Tables Icon

Table 1 Averaged normalized Stokes parameters.

Equations (32)

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E ( r , φ , z ) = A 0 θ 0 0 2 π P ( θ ) E 0 ( θ , ϕ ) e i k sin θ r cos ( φ ϕ ) e i k cos θ z sin θ d θ d ϕ ,
E 0 ( θ , ϕ ) = g 1 ( θ , ϕ ) v 1 ( θ , ϕ ) + g 2 ( θ , ϕ ) v 2 ( θ , ϕ ) ,
v 1 ( θ , ϕ ) = e i ϕ 2 ( e 2 + i e 1 )
v 2 ( θ , ϕ ) = e i ϕ 2 ( e 2 i e 1 ) ,
e 1 ( ϕ ) = ( sin ϕ , cos ϕ , 0 )
e 2 ( θ , ϕ ) = ( cos θ cos ϕ , cos θ sin ϕ , sin θ ) .
E = E c ( 1 2 i 2 0 1 2 i 2 0 0 0 1 ) .
E c ( r , ϕ , z ) = A 0 θ 0 0 2 π g ( θ , ϕ ) A ^ ( θ , ϕ ) P ( θ ) e i k sin θ r cos ( ϕ φ ) e i k cos θ z sin θ d θ d ϕ ,
A ^ ( θ , ϕ ) = ( cos 2 ( θ / 2 ) sin 2 ( θ / 2 ) e 2 i ϕ 1 2 sin θ e i ϕ sin 2 ( θ / 2 ) e 2 i ϕ cos 2 ( θ / 2 ) 1 2 sin θ e i ϕ ) .
G ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = g ( θ 1 , ϕ 1 ) g ( θ 2 , ϕ 2 ) ,
W c ^ ( r 1 , φ 1 , r 2 , φ 2 , z ) = E c ( r 1 , φ 1 , z ) E c ( r 2 , φ 2 , z ) .
W c ^ ( r 1 , φ 1 , r 2 , φ 2 , z ) = | A | 2 0 θ 0 0 θ 0 0 2 π 0 2 π A ^ ( θ 1 , ϕ 1 ) G ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) A ^ ( θ 2 , ϕ 2 ) P ( θ 1 ) P ( θ 2 ) exp ( i k r 1 s 1 ) exp ( i k 1 r 2 s 2 ) sin θ 1 sin θ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 .
W c T ^ ( r 1 , φ 1 , r 2 , φ 2 , z ) = | A | 2 0 θ 0 0 θ 0 0 2 π 0 2 π A T ^ ( θ 1 , ϕ 1 ) G ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) A T ^ ( θ 2 , ϕ 2 ) P ( θ 1 ) P ( θ 2 ) exp ( i k r 1 s 1 ) exp ( i k r 2 s 2 ) sin θ 1 sin θ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 .
A T ^ ( θ , ϕ ) = ( cos 2 ( θ / 2 ) sin 2 ( θ / 2 ) e 2 i ϕ sin 2 ( θ / 2 ) e 2 i ϕ cos 2 ( θ / 2 ) ) .
G T ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = A T ^ ( θ 1 , ϕ 1 ) G ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) A T ^ ( θ 2 , ϕ 2 )
W c T ^ ( r 1 , φ 1 , r 2 , φ 2 , z ) = | A | 2 0 θ 0 0 θ 0 0 2 π 0 2 π G T ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) P ( θ 1 ) P ( θ 2 ) exp ( i k r 1 s 1 ) exp ( i k r 2 s 2 ) sin θ 1 sin θ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2 .
C n ( r , z ) = | A | 2 0 θ 0 0 θ 0 0 2 π 0 2 π C n T ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) P ( θ 1 ) P ( θ 2 ) exp ( i k r 1 s 1 ) exp ( i k r 2 s 2 ) sin θ 1 sin θ 2 d θ 1 d θ 2 d ϕ 1 d ϕ 2
C n T ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = tr ( G T ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) σ n ) .
C 0 T ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = 4 h ( θ 1 , θ 2 ) cos ( m ( ϕ 1 ϕ 2 ) )
C 1 T ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = 4 i h ( θ 1 , θ 2 ) sin ( m ( ϕ 1 ϕ 2 ) )
C 2 T ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = C 3 T ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = 0
G T ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = h ( θ 1 , θ 2 ) U ^ ( ϕ 1 ) U ^ ( ϕ 2 )
U ^ ( ϕ ) = ( i e i m ϕ i e i m ϕ e i m ϕ e i m ϕ ) .
C 0 ( r , z ) = 16 π 2 | A | 2 0 θ 0 0 θ 0 P ( θ 1 ) P ( θ 2 ) h ( θ 1 , θ 2 ) J m ( k r sin θ 1 ) J m ( k r sin θ 2 ) exp ( i k r z cos θ 2 ) exp ( i k z cos θ 1 ) sin θ 1 sin θ 2 d θ 1 d θ 2
C 1 ( r , z ) = C 2 ( r , z ) = C 3 ( r , z ) = 0
G ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = h ( θ 1 , θ 2 ) ( A T ^ ( θ 2 , ϕ 2 ) ) 1 U ^ ( ϕ 1 ) U ^ ( ϕ 2 ) A T ^ ( θ 2 , ϕ 2 ) 1 .
W c ^ ( r 1 , ϕ 1 , r 2 , ϕ 2 , z ) z z = 8 π 2 | A | 2 cos ( ( m 1 ) ( ϕ 1 ϕ 2 ) ) 0 θ 0 0 θ 0 P ( θ 1 ) P ( θ 2 ) h ( θ 1 , θ 2 ) tan θ 1 tan θ 2 exp ( i k r z cos θ 2 ) exp ( i k z cos θ 1 ) J m 1 ( k r 1 sin θ 1 ) J m 1 ( k r 2 sin θ 2 ) sin θ 1 sin θ 2 d θ 1 d θ 2
I ( 0 , z ) = tr ( W c ^ ( 0 , φ , 0 , φ , z ) ) = 8 π 2 | A | 2 0 θ 0 0 θ 0 P ( θ 1 ) P ( θ 2 ) h ( θ 1 , θ 2 ) tan θ 1 tan θ 2 exp ( i k r z cos θ 2 ) exp ( i k z cos θ 1 ) sin θ 1 sin θ 2 d θ 1 d θ 2
G ^ ( θ 1 , ϕ 1 , θ 2 , ϕ 2 ) = h ( θ 1 , θ 2 ) ( a e i ( ϕ 1 ϕ 2 ) b e i ( ϕ 1 + ϕ 2 ) b e i ( ϕ 1 + ϕ 2 ) a e i ( ϕ 1 ϕ 2 ) )
a = 1 + cos θ 1 cos θ 2 b = 1 cos θ 1 cos θ 2
E s x = i sin ϕ cos θ + cos ϕ
E s y = i cos ϕ cos θ + sin ϕ

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