Abstract

Using three-dimensional (3D) coherency vector (9 × 1), we develop a new 3D polarization algebra to calculate the polarization properties of all polarization sensitive optical systems, especially when the incident optical field is partially polarized or un-polarized. The polarization properties of a high numerical aperture (NA) microscope objective (NA = 1.25 immersed in oil) are analyzed based on the proposed 3D polarization algebra. Correspondingly, the polarization simulation of this high NA optical system is performed by the commercial software VirtualLAB Fusion. By comparing the theoretical calculations with polarization simulations, a perfect matching relation is obtained, which demonstrates that this 3D polarization algebra is valid to quantify the 3D polarization properties for all polarization sensitive optical systems.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Three-dimensional polarization algebra

Colin J. R. Sheppard, Marco Castello, and Alberto Diaspro
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Three-dimensional polarization ray tracing calculus for partially polarized light

Haiyang Zhang, Yi Li, Changxiang Yan, and Junqiang Zhang
Opt. Express 25(22) 26973-26986 (2017)

References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2018 (1)

K. Xu, “Monolithically integrated Si gate-controlled light-emitting device: science and properties,” J. Opt. 20(2), 024014 (2018).
[Crossref]

2017 (2)

W. He, Y. Fu, L. Zhang, J. Wang, Y. Zheng, and Y. Li, “Three-dimensional polarization aberration functions in optical system based on three-dimensional polarization ray-tracing calculus,” Opt. Commun. 387, 128–134 (2017).
[Crossref]

H. Zhang, Y. Li, C. Yan, and J. Zhang, “Three-dimensional polarization ray tracing calculus for partially polarized light,” Opt. Express 25(22), 26973–26986 (2017).
[Crossref] [PubMed]

2016 (2)

2015 (4)

H. Di, D. Hua, L. Yan, X. Hou, and W. Xin, “Polarization analysis and corrections of different telescopes in polarization lidar,” Appl. Opt. 54(3), 389–397 (2015).
[Crossref]

K. Xu, H. Liu, and Z. Zhang, “Gate-controlled diode structure based electro-optical interfaces in standard silicon-CMOS integrated circuitry,” Appl. Opt. 54(21), 6420–6424 (2015).
[Crossref] [PubMed]

J. B. Breckinridge, W. Sze, T. Lam, and R. A. Chipman, “Polarization aberrations in astronomical telescopes: the point spread function,” Publ. Astron. Soc. Pac. 127(951), 445–468 (2015).
[Crossref]

Z. Zhang, H. Fan, H. F. 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 (2)

G. M. Mikheev, V. Vanyukov, T. N. Mogileva, A. P. Puzyr, V. S. Bondar, and Y. Svirko, “Effect of laser-radiation polarization on the nonlinear scattering of light in nanodiamond suspensions,” Tech. Phys. Lett. 40(7), 609–613 (2014).
[Crossref]

J. Atwood, W. Skidmore, G. C. Anupama, R. Manjunath, K. Reddy, and A. K. Sen, “Polarimetric analysis of the thirty meter telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).
[Crossref]

2013 (1)

2012 (1)

2011 (6)

2010 (2)

T. S. Partnership, “Optical simulation software,” Nat. Photonics 4(4), 256–257 (2010).
[Crossref]

J. J. Gil and I. S. Jose, “3D polarimetric purity,” Opt. Commun. 283(22), 4430–4434 (2010).
[Crossref]

2009 (2)

2008 (1)

2007 (1)

J. J. Gil, “Polarimetric characterization of light and media: Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40(40), 1–47 (2007).
[Crossref]

2006 (1)

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[Crossref] [PubMed]

2004 (1)

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

2002 (2)

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 016615 (2002).
[Crossref] [PubMed]

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81(9), 1576–1578 (2002).
[Crossref]

1994 (1)

1979 (1)

1962 (1)

M. Gell-Mann, “Symmetries of Baryons and Mesons, Murray Gell-Mann,” Phys. Rev. 125(3), 1067–1084 (1962).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[Crossref] [PubMed]

Anupama, G. C.

J. Atwood, W. Skidmore, G. C. Anupama, R. Manjunath, K. Reddy, and A. K. Sen, “Polarimetric analysis of the thirty meter telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).
[Crossref]

Atwood, J.

J. Atwood, W. Skidmore, G. C. Anupama, R. Manjunath, K. Reddy, and A. K. Sen, “Polarimetric analysis of the thirty meter telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).
[Crossref]

Bondar, V. S.

G. M. Mikheev, V. Vanyukov, T. N. Mogileva, A. P. Puzyr, V. S. Bondar, and Y. Svirko, “Effect of laser-radiation polarization on the nonlinear scattering of light in nanodiamond suspensions,” Tech. Phys. Lett. 40(7), 609–613 (2014).
[Crossref]

Breckinridge, J. B.

J. B. Breckinridge, W. Sze, T. Lam, and R. A. Chipman, “Polarization aberrations in astronomical telescopes: the point spread function,” Publ. Astron. Soc. Pac. 127(951), 445–468 (2015).
[Crossref]

N. Clark and J. B. Breckinridge, “Polarization compensation of Fresnel aberrations in telescopes,” Proc. SPIE 8146, 1509–1511 (2011).

Cao, Y.

Castello, M.

Chipman, R. A.

Chon, J. W. M.

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81(9), 1576–1578 (2002).
[Crossref]

Clark, N.

N. Clark and J. B. Breckinridge, “Polarization compensation of Fresnel aberrations in telescopes,” Proc. SPIE 8146, 1509–1511 (2011).

Colin, J. R.

Correas, J. M.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Crabtree, K.

Di, H.

Diaspro, A.

Fan, H.

Z. Zhang, H. Fan, H. F. 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]

Ferreira, C.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Friberg, A. T.

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34(21), 3394–3396 (2009).
[Crossref] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 016615 (2002).
[Crossref] [PubMed]

Fu, Y.

W. He, Y. Fu, L. Zhang, J. Wang, Y. Zheng, and Y. Li, “Three-dimensional polarization aberration functions in optical system based on three-dimensional polarization ray-tracing calculus,” Opt. Commun. 387, 128–134 (2017).
[Crossref]

W. He, Y. Fu, Y. Zheng, L. Zhang, J. Wang, Z. Liu, and J. Zheng, “Polarization properties of a corner-cube retroreflector with three-dimensional polarization ray-tracing calculus,” Appl. Opt. 52(19), 4527–4535 (2013).
[Crossref] [PubMed]

Gan, X.

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81(9), 1576–1578 (2002).
[Crossref]

Gell-Mann, M.

M. Gell-Mann, “Symmetries of Baryons and Mesons, Murray Gell-Mann,” Phys. Rev. 125(3), 1067–1084 (1962).
[Crossref]

Gil, J. J.

J. J. Gil and I. S. Jose, “3D polarimetric purity,” Opt. Commun. 283(22), 4430–4434 (2010).
[Crossref]

J. J. Gil, “Polarimetric characterization of light and media: Physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys. 40(40), 1–47 (2007).
[Crossref]

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Gu, M.

J. W. M. Chon, X. Gan, and M. Gu, “Splitting of the focal spot of a high numerical-aperture objective in free space,” Appl. Phys. Lett. 81(9), 1576–1578 (2002).
[Crossref]

He, W.

W. He, Y. Fu, L. Zhang, J. Wang, Y. Zheng, and Y. Li, “Three-dimensional polarization aberration functions in optical system based on three-dimensional polarization ray-tracing calculus,” Opt. Commun. 387, 128–134 (2017).
[Crossref]

W. He, Y. Fu, Y. Zheng, L. Zhang, J. Wang, Z. Liu, and J. Zheng, “Polarization properties of a corner-cube retroreflector with three-dimensional polarization ray-tracing calculus,” Appl. Opt. 52(19), 4527–4535 (2013).
[Crossref] [PubMed]

Hou, X.

Hua, D.

Huang, W.

Z. Zhang, H. Fan, H. F. 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]

Huen, T.

Jose, I. S.

J. J. Gil and I. S. Jose, “3D polarimetric purity,” Opt. Commun. 283(22), 4430–4434 (2010).
[Crossref]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 016615 (2002).
[Crossref] [PubMed]

Kuhn, M.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

Lam, T.

J. B. Breckinridge, W. Sze, T. Lam, and R. A. Chipman, “Polarization aberrations in astronomical telescopes: the point spread function,” Publ. Astron. Soc. Pac. 127(951), 445–468 (2015).
[Crossref]

Li, S.

Li, Y.

W. He, Y. Fu, L. Zhang, J. Wang, Y. Zheng, and Y. Li, “Three-dimensional polarization aberration functions in optical system based on three-dimensional polarization ray-tracing calculus,” Opt. Commun. 387, 128–134 (2017).
[Crossref]

H. Zhang, Y. Li, C. Yan, and J. Zhang, “Three-dimensional polarization ray tracing calculus for partially polarized light,” Opt. Express 25(22), 26973–26986 (2017).
[Crossref] [PubMed]

Lindfors, K.

Liu, H.

Liu, Z.

Manjunath, R.

J. Atwood, W. Skidmore, G. C. Anupama, R. Manjunath, K. Reddy, and A. K. Sen, “Polarimetric analysis of the thirty meter telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).
[Crossref]

McClain, S. C.

McGuire, J. P.

Melero, P. A.

J. J. Gil, J. M. Correas, P. A. Melero, and C. Ferreira, “Generalized polarization algebra,” Monografías del Seminario Matemático García de Galdeano 31, 161–167 (2004).

Mikheev, G. M.

G. M. Mikheev, V. Vanyukov, T. N. Mogileva, A. P. Puzyr, V. S. Bondar, and Y. Svirko, “Effect of laser-radiation polarization on the nonlinear scattering of light in nanodiamond suspensions,” Tech. Phys. Lett. 40(7), 609–613 (2014).
[Crossref]

Mogileva, T. N.

G. M. Mikheev, V. Vanyukov, T. N. Mogileva, A. P. Puzyr, V. S. Bondar, and Y. Svirko, “Effect of laser-radiation polarization on the nonlinear scattering of light in nanodiamond suspensions,” Tech. Phys. Lett. 40(7), 609–613 (2014).
[Crossref]

Partnership, T. S.

T. S. Partnership, “Optical simulation software,” Nat. Photonics 4(4), 256–257 (2010).
[Crossref]

Pu, J.

Puzyr, A. P.

G. M. Mikheev, V. Vanyukov, T. N. Mogileva, A. P. Puzyr, V. S. Bondar, and Y. Svirko, “Effect of laser-radiation polarization on the nonlinear scattering of light in nanodiamond suspensions,” Tech. Phys. Lett. 40(7), 609–613 (2014).
[Crossref]

Qu, J.

Z. Zhang, H. Fan, H. F. 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]

Reddy, K.

J. Atwood, W. Skidmore, G. C. Anupama, R. Manjunath, K. Reddy, and A. K. Sen, “Polarimetric analysis of the thirty meter telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).
[Crossref]

Sen, A. K.

J. Atwood, W. Skidmore, G. C. Anupama, R. Manjunath, K. Reddy, and A. K. Sen, “Polarimetric analysis of the thirty meter telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).
[Crossref]

Setälä, T.

T. Setälä, K. Lindfors, and A. T. Friberg, “Degree of polarization in 3D optical fields generated from a partially polarized plane wave,” Opt. Lett. 34(21), 3394–3396 (2009).
[Crossref] [PubMed]

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 016615 (2002).
[Crossref] [PubMed]

Sheppard, C. J.

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 016615 (2002).
[Crossref] [PubMed]

Skidmore, W.

J. Atwood, W. Skidmore, G. C. Anupama, R. Manjunath, K. Reddy, and A. K. Sen, “Polarimetric analysis of the thirty meter telescope (TMT) for modeling instrumental polarization characteristics,” Proc. SPIE 9150, 915013 (2014).
[Crossref]

Svirko, Y.

G. M. Mikheev, V. Vanyukov, T. N. Mogileva, A. P. Puzyr, V. S. Bondar, and Y. Svirko, “Effect of laser-radiation polarization on the nonlinear scattering of light in nanodiamond suspensions,” Tech. Phys. Lett. 40(7), 609–613 (2014).
[Crossref]

Sze, W.

J. B. Breckinridge, W. Sze, T. Lam, and R. A. Chipman, “Polarization aberrations in astronomical telescopes: the point spread function,” Publ. Astron. Soc. Pac. 127(951), 445–468 (2015).
[Crossref]

Tang, W. T.

Toussaint, K. C.

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[Crossref] [PubMed]

Tu, Y.

Vanyukov, V.

G. M. Mikheev, V. Vanyukov, T. N. Mogileva, A. P. Puzyr, V. S. Bondar, and Y. Svirko, “Effect of laser-radiation polarization on the nonlinear scattering of light in nanodiamond suspensions,” Tech. Phys. Lett. 40(7), 609–613 (2014).
[Crossref]

Wang, J.

W. He, Y. Fu, L. Zhang, J. Wang, Y. Zheng, and Y. Li, “Three-dimensional polarization aberration functions in optical system based on three-dimensional polarization ray-tracing calculus,” Opt. Commun. 387, 128–134 (2017).
[Crossref]

W. He, Y. Fu, Y. Zheng, L. Zhang, J. Wang, Z. Liu, and J. Zheng, “Polarization properties of a corner-cube retroreflector with three-dimensional polarization ray-tracing calculus,” Appl. Opt. 52(19), 4527–4535 (2013).
[Crossref] [PubMed]

Wang, X.

Wyrowski, F.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58(5–6), 449–466 (2011).
[Crossref]

Xin, W.

Xu, H. F.

Z. Zhang, H. Fan, H. F. 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]

Xu, K.

Yan, C.

Yan, L.

Yang, Y.

Yew, E. Y.

Yun, G.

Zhang, H.

Zhang, J.

Zhang, L.

W. He, Y. Fu, L. Zhang, J. Wang, Y. Zheng, and Y. Li, “Three-dimensional polarization aberration functions in optical system based on three-dimensional polarization ray-tracing calculus,” Opt. Commun. 387, 128–134 (2017).
[Crossref]

W. He, Y. Fu, Y. Zheng, L. Zhang, J. Wang, Z. Liu, and J. Zheng, “Polarization properties of a corner-cube retroreflector with three-dimensional polarization ray-tracing calculus,” Appl. Opt. 52(19), 4527–4535 (2013).
[Crossref] [PubMed]

Zhang, Z.

Zheng, J.

Zheng, Y.

W. He, Y. Fu, L. Zhang, J. Wang, Y. Zheng, and Y. Li, “Three-dimensional polarization aberration functions in optical system based on three-dimensional polarization ray-tracing calculus,” Opt. Commun. 387, 128–134 (2017).
[Crossref]

W. He, Y. Fu, Y. Zheng, L. Zhang, J. Wang, Z. Liu, and J. Zheng, “Polarization properties of a corner-cube retroreflector with three-dimensional polarization ray-tracing calculus,” Appl. Opt. 52(19), 4527–4535 (2013).
[Crossref] [PubMed]

Appl. Opt. (8)

T. Huen, “Reflectance of thinly oxidized silicon at normal incidence,” Appl. Opt. 18(12), 1927–1932 (1979).
[Crossref] [PubMed]

J. P. McGuire and R. A. Chipman, “Polarization aberrations. 1. Rotationally symmetric optical systems,” Appl. Opt. 33(22), 5080–5100 (1994).
[Crossref] [PubMed]

G. Yun, K. Crabtree, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus I: definition and diattenuation,” Appl. Opt. 50(18), 2855–2865 (2011).
[Crossref] [PubMed]

G. Yun, S. C. McClain, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus II: retardance,” Appl. Opt. 50(18), 2866–2874 (2011).
[Crossref] [PubMed]

W. He, Y. Fu, Y. Zheng, L. Zhang, J. Wang, Z. Liu, and J. Zheng, “Polarization properties of a corner-cube retroreflector with three-dimensional polarization ray-tracing calculus,” Appl. Opt. 52(19), 4527–4535 (2013).
[Crossref] [PubMed]

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Figures (11)

Fig. 1
Fig. 1 Local Cartesian coordinate system of the propagation light.
Fig. 2
Fig. 2 Rotation transformation of Cartesian coordination systems.
Fig. 3
Fig. 3 A high NA microscope objective (NA = 1.25 immersed in liquid oil).
Fig. 4
Fig. 4 Polarization properties of exiting light from the investigated high NA optical system interacting with a 3D totally polarized light in the central FoV. (a) 3D DoP distribution. (b), (c) ellipticity and azimuth distributions on exit pupil plane. (d), (e) ellipticity and azimuth distributions on sagittal plane. (f), (g) ellipticity and azimuth distributions on tangential plane.
Fig. 5
Fig. 5 Polarization properties of exiting light from the investigated high NA optical system interacting with a 3D totally polarized light in the marginal FoV. (a) 3D DoP distribution. (b), (c) ellipticity and azimuth distributions on exit pupil plane. (d), (e) ellipticity and azimuth distributions on sagittal plane. (f), (g) ellipticity and azimuth distributions on tangential plane.
Fig. 6
Fig. 6 Polarization properties of exiting light from the investigated high NA optical system interacting with a 3D partially polarized light in the central FoV. (a) 3D DoP distribution. (b), (c) ellipticity and azimuth distributions on exit pupil plane. (d), (e) ellipticity and azimuth distributions on sagittal plane. (f), (g) ellipticity and azimuth distributions on tangential plane.
Fig. 7
Fig. 7 Polarization properties of exiting light from the investigated high NA optical system interacting with a 3D partially polarized light in the marginal FoV. (a) 3D DoP distribution. (b), (c) ellipticity and azimuth distributions on exit pupil plane. (d), (e) ellipticity and azimuth distributions on sagittal plane. (f), (g) ellipticity and azimuth distributions on tangential plane.
Fig. 8
Fig. 8 Polarization simulation of exiting light from the investigated high NA optical system interacting with a 3D totally polarized light on the: (a) exit pupil plane (b) sagittal plane (c) tangential plane in the central FoV.
Fig. 9
Fig. 9 Polarization simulation of exiting light from the investigated high NA optical system objective interacting with a 3D totally polarized light on the: (a) exit pupil plane (b) sagittal plane (c) tangential plane in the marginal FoV.
Fig. 10
Fig. 10 Polarization simulation of exiting light from the investigated high NA optical system interacting with a 3D partially polarized light on the: (a) exit pupil plane (b) sagittal plane (c) tangential plane in the central FoV.
Fig. 11
Fig. 11 Polarization simulation of exiting light from the investigated high NA optical system interacting with a 3D partially polarized light on the: (a) exit pupil plane (b) sagittal plane (c) tangential plane in the marginal FoV.

Tables (1)

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Table 1 3D coherency vectors (9 × 1) of incident lights.

Equations (23)

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Φ 3×3 =E(r,t) E (r,t)=( E x (r,t) E x * (r,t) E x (r,t) E y * (r,t) E x (r,t) E z * (r,t) E y (r,t) E x * (r,t) E y (r,t) E y * (r,t) E y (r,t) E z * (r,t) E z (r,t) E x * (r,t) E z (r,t) E y * (r,t) E z (r,t) E z * (r,t) ),
σ 0 = 2 3 ( 1 0 0 0 1 0 0 0 1 ), σ 1 =( 0 1 0 1 0 0 0 0 0 ), σ 2 =( 0 i 0 i 0 0 0 0 0 ), σ 3 =( 1 0 0 0 1 0 0 0 0 ), σ 4 =( 0 0 1 0 0 0 1 0 0 ), σ 5 =( 0 0 i 0 0 0 i 0 0 ), σ 6 =( 0 0 0 0 0 1 0 1 0 ), σ 7 =( 0 0 0 0 0 i 0 i 0 ), σ 8 = 1 3 ( 1 0 0 0 1 0 0 0 2 ).
Φ 3×3 = 1 2 i=0 8 s i σ i ,
S 9×1 = ( s 0 , s 1 , s 2 , s 3 , s 4 , s 5 , s 6 , s 7 , s 8 ) T .
Φ 9×1 = ( ϕ xx , ϕ xy , ϕ xz , ϕ yx , ϕ yy , ϕ yz , ϕ zx , ϕ zy , ϕ zz ) T .
S 9×1 = Q 9×9 × Φ 9×1 ,
Φ ' 9×1,l = N 9×9,l Φ 9×1,l ,
S 4×1 ' = A 4×4 ( J 2×2 J 2×2 * ) A 4×4 -1 S 4×1 ,
S 4×1 =( < E x 2 >+< E y 2 > < E x 2 >< E y 2 > < E x E y >+< E x E y > i(< E x E y >< E x E y >) ), A 4×4 =( 1 0 0 1 1 0 0 1 0 1 1 0 0 i i 0 ),
S 9×1,l =( < E x 2 >+< E y 2 > < E x E y >+< E x E y > i(< E x E y >< E x E y >) < E x 2 >< E y 2 > 0 0 0 0 < E x 2 >+< E y 2 > ).
N 9×9,l = 1 2 ( N 11 N 12 0 N 14 N 15 0 0 0 0 N 21 N 22 0 N 24 N 25 0 0 0 0 0 0 0 0 0 0 0 0 0 N 41 N 42 0 N 44 N 45 0 0 0 0 N 51 N 52 0 N 54 N 55 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ),
( E x E y E z )= T 3×3 ( E x1 E y1 E z1 ),
T 3×3 =( cosα 0 sinα 0 1 0 sinα 0 cosα )( 1 0 0 0 cosβ sinβ 0 sinβ cosβ )=T( k in ),
α=arctan( b/c ),β=arcsin( a/ a 2 + b 2 + c 2 ).
Φ 3×3 =E E =(T( k in ) E l ) (T( k in ) E l ) , Φ 3×3,l = E l E l .
Φ 9×1 = R 9×9 Φ 9×1 ,l ,
R 9×9 =T( k in )T ( k in ) * ,
N 9×9 =(T( k out )T ( k out ) * ) N 9×9,l (T( k in )T ( k in ) * ) 1 .
N Total = q=m,1 1 N q = N m ... N q ... N 1 .
Φ 9×1 ' = N Total Φ 9×1 .
Q 9×9 =( 6 3 0 0 1 0 0 0 0 3 3 0 1 -i 0 0 0 0 0 0 0 0 0 0 1 -i 0 0 0 0 1 i 0 0 0 0 0 0 6 3 0 0 -1 0 0 0 0 3 3 0 0 0 0 0 0 1 -i 0 0 0 0 0 1 i 0 0 0 0 0 0 0 0 0 1 i 0 6 3 0 0 0 0 0 0 0 - 2 3 3 ).
N 11 = m 11 + m 21 + m 12 + m 22 , N 12 = m 13 + m 23 i( m 14 + m 24 ), N 14 = m 13 m 23 +i( m 14 + m 24 ), N 15 = m 11 + m 21 m 12 m 22 , N 21 = m 31 i m 41 + m 32 i m 42 , n 22 = m 33 i m 43 i( m 34 i m 44 ), N 24 = m 33 ++ m 44 +i( m 43 + m 34 ), N 25 = m 31 m 32 i( m 41 m 42 ), N 41 = m 31 + m 32 +i( m 41 + m 42 ), N 42 = m 33 + m 44 i( m 34 m 43 ), N 44 = m 33 m 44 i( m 43 + m 34 ), N 45 = m 31 m 32 +i( m 41 m 42 ), N 51 = m 11 m 21 + m 12 m 22 , N 52 = m 13 m 23 i( m 14 m 24 ), N 54 = m 13 + m 23 +i( m 14 m 24 )], N 55 = m 11 m 21 m 12 + m 22 , N ij =0(i=3,6,...,9,j=1,2,...,9).
M= A 4×4 ( J 2×2 J 2×2 ) A 4×4 1 .

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