Abstract

An analysis of the clusters and the uniformity of distribution of states of polarization on the surface of a Poincaré sphere generated by rotating wave plates is given. The analysis of clusters of the states of polarization is based on a spherical radial distribution function. For uniform analysis of the distribution, two methods are proposed. The first method is based on calculation of the correlation coefficient; the second method is based on calculation of the angles between pairs of the states of polarization on the Poincaré sphere. For polarization scramblers consisting of eight or more rotating wave plates, nonclustered and near-uniform distribution of states of polarization is obtained.

© 2005 Optical Society of America

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  1. I. T. Lima, A. O. Lima, J. Zweck, C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. 15, 45–47 (2003).
    [CrossRef]
  2. L. S. Yan, Q. Yu, A. E. Willner, “Periodic polarization scrambling with uniformly distributed SOPs on the Poincaré sphere,” presented at the 28th European Conference on Optical Communication, Copenhagen, Denmark, 8–12 September, 2002.
  3. T. Fujiwara, A. Watanabe, H. Mori, “Polarization dependent loss in a Ti:LiNbO3 polarization scrambler/controller,” IEEE Photon. Technol. Lett. 8, 542–544 (1996).
    [CrossRef]
  4. D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, R. Noe, “Some enabling techniques for polarization mode dispersion compensation,” J. Light. Technol. 21, 1198–1210 (2003).
    [CrossRef]
  5. L. Glass, W. R. Tobler, “Uniform distribution of objects in a homogeneous field: cities on a plain,” Nature 233, 67–68 (1971).
    [CrossRef] [PubMed]
  6. S. Cheng, Z. Wang, X. Qin, X. Bian, “Effect of copper addition on the thermal contraction of indium melt clusters,” Rare Metals 22, 215–220 (2003).
  7. E. Matteoli, G. A. Mansoori, “A simple expression for radial distribution functions of pure fluids and mixtures,” J. Chem. Phys. 11, 4672–4678 (1995).
    [CrossRef]
  8. R. D. Evans, J. Bentley, K. L. More, G. L. Doll, J. T. Glass, “Radial distribution function analyses of amorphous carbon thin films containing various levels of silicon and hydrogen,” J. Appl. Phys. 96, 273–279 (2004).
    [CrossRef]
  9. I. M. Tkachenko, “Limiting properties of the radial distribution function in electronic liquids,” J. Phys. A Math. Gen. 29, 2599–2605 (1996).
    [CrossRef]
  10. R. W. Sinnott, “Virtues of the Haversine,” Sky Telescope 68, 159–161 (1984).
  11. J. A. Robinson, D. A. Liddle, C. A. Evans, D. L. Amsbury, “Astronaut-acquired orbital photographs as digital data for remote sensing: spatial resolution,” Int. J. Remote Sens. 23, 4403–4438 (2002).
    [CrossRef]
  12. R. Bakshi, C. A. Knoblock, S. Thakkar, “Exploiting online source to accurately geocode addresses,” in Proceedings of the 12th Annual ACM International Workshop on Geographic Information Systems (ACM Press, New York, 2004), pp. 194–203.
  13. R. Hernadez-Murillo, “Strategic interaction in tax polices among states,” Fed. Reserve Bank St. Louis Rev. 85, 47–56 (2003).
  14. R. W. Sinnott, “Celestial calendar—hunting for equilateral triple stars,” Sky Telescope 99, 100–101 (1999).
  15. E. B. Saff, A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intell. 19, 5–11 (1997).
    [CrossRef]
  16. N. I. Fisher, T. Lewis, B. J. J. Embleton, Statistical Analysis of Spherical Data (Cambridge U. Press, 1987).
    [CrossRef]
  17. P. J. Diggle, N. I. Fisher, A. J. Lee, “A comparision of tests of uniformity for spherical data,” Aust. J. Statist. 27, 53–59 (1985).
    [CrossRef]
  18. P. A. Karnezis, G. Durrant, G. Cantor, “Characterization of reinforcement distribution in cast Al-alloy/SiCp composites,” Mater. Charact. 40, 97–109 (1998).
    [CrossRef]

2004

R. D. Evans, J. Bentley, K. L. More, G. L. Doll, J. T. Glass, “Radial distribution function analyses of amorphous carbon thin films containing various levels of silicon and hydrogen,” J. Appl. Phys. 96, 273–279 (2004).
[CrossRef]

2003

S. Cheng, Z. Wang, X. Qin, X. Bian, “Effect of copper addition on the thermal contraction of indium melt clusters,” Rare Metals 22, 215–220 (2003).

I. T. Lima, A. O. Lima, J. Zweck, C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. 15, 45–47 (2003).
[CrossRef]

D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, R. Noe, “Some enabling techniques for polarization mode dispersion compensation,” J. Light. Technol. 21, 1198–1210 (2003).
[CrossRef]

R. Hernadez-Murillo, “Strategic interaction in tax polices among states,” Fed. Reserve Bank St. Louis Rev. 85, 47–56 (2003).

2002

J. A. Robinson, D. A. Liddle, C. A. Evans, D. L. Amsbury, “Astronaut-acquired orbital photographs as digital data for remote sensing: spatial resolution,” Int. J. Remote Sens. 23, 4403–4438 (2002).
[CrossRef]

1999

R. W. Sinnott, “Celestial calendar—hunting for equilateral triple stars,” Sky Telescope 99, 100–101 (1999).

1998

P. A. Karnezis, G. Durrant, G. Cantor, “Characterization of reinforcement distribution in cast Al-alloy/SiCp composites,” Mater. Charact. 40, 97–109 (1998).
[CrossRef]

1997

E. B. Saff, A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intell. 19, 5–11 (1997).
[CrossRef]

1996

T. Fujiwara, A. Watanabe, H. Mori, “Polarization dependent loss in a Ti:LiNbO3 polarization scrambler/controller,” IEEE Photon. Technol. Lett. 8, 542–544 (1996).
[CrossRef]

I. M. Tkachenko, “Limiting properties of the radial distribution function in electronic liquids,” J. Phys. A Math. Gen. 29, 2599–2605 (1996).
[CrossRef]

1995

E. Matteoli, G. A. Mansoori, “A simple expression for radial distribution functions of pure fluids and mixtures,” J. Chem. Phys. 11, 4672–4678 (1995).
[CrossRef]

1985

P. J. Diggle, N. I. Fisher, A. J. Lee, “A comparision of tests of uniformity for spherical data,” Aust. J. Statist. 27, 53–59 (1985).
[CrossRef]

1984

R. W. Sinnott, “Virtues of the Haversine,” Sky Telescope 68, 159–161 (1984).

1971

L. Glass, W. R. Tobler, “Uniform distribution of objects in a homogeneous field: cities on a plain,” Nature 233, 67–68 (1971).
[CrossRef] [PubMed]

Amsbury, D. L.

J. A. Robinson, D. A. Liddle, C. A. Evans, D. L. Amsbury, “Astronaut-acquired orbital photographs as digital data for remote sensing: spatial resolution,” Int. J. Remote Sens. 23, 4403–4438 (2002).
[CrossRef]

Bakshi, R.

R. Bakshi, C. A. Knoblock, S. Thakkar, “Exploiting online source to accurately geocode addresses,” in Proceedings of the 12th Annual ACM International Workshop on Geographic Information Systems (ACM Press, New York, 2004), pp. 194–203.

Bentley, J.

R. D. Evans, J. Bentley, K. L. More, G. L. Doll, J. T. Glass, “Radial distribution function analyses of amorphous carbon thin films containing various levels of silicon and hydrogen,” J. Appl. Phys. 96, 273–279 (2004).
[CrossRef]

Bhandare, S.

D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, R. Noe, “Some enabling techniques for polarization mode dispersion compensation,” J. Light. Technol. 21, 1198–1210 (2003).
[CrossRef]

Bian, X.

S. Cheng, Z. Wang, X. Qin, X. Bian, “Effect of copper addition on the thermal contraction of indium melt clusters,” Rare Metals 22, 215–220 (2003).

Cantor, G.

P. A. Karnezis, G. Durrant, G. Cantor, “Characterization of reinforcement distribution in cast Al-alloy/SiCp composites,” Mater. Charact. 40, 97–109 (1998).
[CrossRef]

Cheng, S.

S. Cheng, Z. Wang, X. Qin, X. Bian, “Effect of copper addition on the thermal contraction of indium melt clusters,” Rare Metals 22, 215–220 (2003).

Diggle, P. J.

P. J. Diggle, N. I. Fisher, A. J. Lee, “A comparision of tests of uniformity for spherical data,” Aust. J. Statist. 27, 53–59 (1985).
[CrossRef]

Doll, G. L.

R. D. Evans, J. Bentley, K. L. More, G. L. Doll, J. T. Glass, “Radial distribution function analyses of amorphous carbon thin films containing various levels of silicon and hydrogen,” J. Appl. Phys. 96, 273–279 (2004).
[CrossRef]

Durrant, G.

P. A. Karnezis, G. Durrant, G. Cantor, “Characterization of reinforcement distribution in cast Al-alloy/SiCp composites,” Mater. Charact. 40, 97–109 (1998).
[CrossRef]

Embleton, B. J. J.

N. I. Fisher, T. Lewis, B. J. J. Embleton, Statistical Analysis of Spherical Data (Cambridge U. Press, 1987).
[CrossRef]

Evans, C. A.

J. A. Robinson, D. A. Liddle, C. A. Evans, D. L. Amsbury, “Astronaut-acquired orbital photographs as digital data for remote sensing: spatial resolution,” Int. J. Remote Sens. 23, 4403–4438 (2002).
[CrossRef]

Evans, R. D.

R. D. Evans, J. Bentley, K. L. More, G. L. Doll, J. T. Glass, “Radial distribution function analyses of amorphous carbon thin films containing various levels of silicon and hydrogen,” J. Appl. Phys. 96, 273–279 (2004).
[CrossRef]

Fisher, N. I.

P. J. Diggle, N. I. Fisher, A. J. Lee, “A comparision of tests of uniformity for spherical data,” Aust. J. Statist. 27, 53–59 (1985).
[CrossRef]

N. I. Fisher, T. Lewis, B. J. J. Embleton, Statistical Analysis of Spherical Data (Cambridge U. Press, 1987).
[CrossRef]

Fujiwara, T.

T. Fujiwara, A. Watanabe, H. Mori, “Polarization dependent loss in a Ti:LiNbO3 polarization scrambler/controller,” IEEE Photon. Technol. Lett. 8, 542–544 (1996).
[CrossRef]

Glass, J. T.

R. D. Evans, J. Bentley, K. L. More, G. L. Doll, J. T. Glass, “Radial distribution function analyses of amorphous carbon thin films containing various levels of silicon and hydrogen,” J. Appl. Phys. 96, 273–279 (2004).
[CrossRef]

Glass, L.

L. Glass, W. R. Tobler, “Uniform distribution of objects in a homogeneous field: cities on a plain,” Nature 233, 67–68 (1971).
[CrossRef] [PubMed]

Hernadez-Murillo, R.

R. Hernadez-Murillo, “Strategic interaction in tax polices among states,” Fed. Reserve Bank St. Louis Rev. 85, 47–56 (2003).

Karnezis, P. A.

P. A. Karnezis, G. Durrant, G. Cantor, “Characterization of reinforcement distribution in cast Al-alloy/SiCp composites,” Mater. Charact. 40, 97–109 (1998).
[CrossRef]

Knoblock, C. A.

R. Bakshi, C. A. Knoblock, S. Thakkar, “Exploiting online source to accurately geocode addresses,” in Proceedings of the 12th Annual ACM International Workshop on Geographic Information Systems (ACM Press, New York, 2004), pp. 194–203.

Kuijlaars, A. B. J.

E. B. Saff, A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intell. 19, 5–11 (1997).
[CrossRef]

Lee, A. J.

P. J. Diggle, N. I. Fisher, A. J. Lee, “A comparision of tests of uniformity for spherical data,” Aust. J. Statist. 27, 53–59 (1985).
[CrossRef]

Lewis, T.

N. I. Fisher, T. Lewis, B. J. J. Embleton, Statistical Analysis of Spherical Data (Cambridge U. Press, 1987).
[CrossRef]

Liddle, D. A.

J. A. Robinson, D. A. Liddle, C. A. Evans, D. L. Amsbury, “Astronaut-acquired orbital photographs as digital data for remote sensing: spatial resolution,” Int. J. Remote Sens. 23, 4403–4438 (2002).
[CrossRef]

Lima, A. O.

I. T. Lima, A. O. Lima, J. Zweck, C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. 15, 45–47 (2003).
[CrossRef]

Lima, I. T.

I. T. Lima, A. O. Lima, J. Zweck, C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. 15, 45–47 (2003).
[CrossRef]

Mansoori, G. A.

E. Matteoli, G. A. Mansoori, “A simple expression for radial distribution functions of pure fluids and mixtures,” J. Chem. Phys. 11, 4672–4678 (1995).
[CrossRef]

Matteoli, E.

E. Matteoli, G. A. Mansoori, “A simple expression for radial distribution functions of pure fluids and mixtures,” J. Chem. Phys. 11, 4672–4678 (1995).
[CrossRef]

Menyuk, C. R.

I. T. Lima, A. O. Lima, J. Zweck, C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. 15, 45–47 (2003).
[CrossRef]

Mirvoda, V.

D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, R. Noe, “Some enabling techniques for polarization mode dispersion compensation,” J. Light. Technol. 21, 1198–1210 (2003).
[CrossRef]

More, K. L.

R. D. Evans, J. Bentley, K. L. More, G. L. Doll, J. T. Glass, “Radial distribution function analyses of amorphous carbon thin films containing various levels of silicon and hydrogen,” J. Appl. Phys. 96, 273–279 (2004).
[CrossRef]

Mori, H.

T. Fujiwara, A. Watanabe, H. Mori, “Polarization dependent loss in a Ti:LiNbO3 polarization scrambler/controller,” IEEE Photon. Technol. Lett. 8, 542–544 (1996).
[CrossRef]

Noe, R.

D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, R. Noe, “Some enabling techniques for polarization mode dispersion compensation,” J. Light. Technol. 21, 1198–1210 (2003).
[CrossRef]

Qin, X.

S. Cheng, Z. Wang, X. Qin, X. Bian, “Effect of copper addition on the thermal contraction of indium melt clusters,” Rare Metals 22, 215–220 (2003).

Robinson, J. A.

J. A. Robinson, D. A. Liddle, C. A. Evans, D. L. Amsbury, “Astronaut-acquired orbital photographs as digital data for remote sensing: spatial resolution,” Int. J. Remote Sens. 23, 4403–4438 (2002).
[CrossRef]

Saff, E. B.

E. B. Saff, A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intell. 19, 5–11 (1997).
[CrossRef]

Sandel, D.

D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, R. Noe, “Some enabling techniques for polarization mode dispersion compensation,” J. Light. Technol. 21, 1198–1210 (2003).
[CrossRef]

Sinnott, R. W.

R. W. Sinnott, “Celestial calendar—hunting for equilateral triple stars,” Sky Telescope 99, 100–101 (1999).

R. W. Sinnott, “Virtues of the Haversine,” Sky Telescope 68, 159–161 (1984).

Thakkar, S.

R. Bakshi, C. A. Knoblock, S. Thakkar, “Exploiting online source to accurately geocode addresses,” in Proceedings of the 12th Annual ACM International Workshop on Geographic Information Systems (ACM Press, New York, 2004), pp. 194–203.

Tkachenko, I. M.

I. M. Tkachenko, “Limiting properties of the radial distribution function in electronic liquids,” J. Phys. A Math. Gen. 29, 2599–2605 (1996).
[CrossRef]

Tobler, W. R.

L. Glass, W. R. Tobler, “Uniform distribution of objects in a homogeneous field: cities on a plain,” Nature 233, 67–68 (1971).
[CrossRef] [PubMed]

Wang, Z.

S. Cheng, Z. Wang, X. Qin, X. Bian, “Effect of copper addition on the thermal contraction of indium melt clusters,” Rare Metals 22, 215–220 (2003).

Watanabe, A.

T. Fujiwara, A. Watanabe, H. Mori, “Polarization dependent loss in a Ti:LiNbO3 polarization scrambler/controller,” IEEE Photon. Technol. Lett. 8, 542–544 (1996).
[CrossRef]

Willner, A. E.

L. S. Yan, Q. Yu, A. E. Willner, “Periodic polarization scrambling with uniformly distributed SOPs on the Poincaré sphere,” presented at the 28th European Conference on Optical Communication, Copenhagen, Denmark, 8–12 September, 2002.

Wust, F.

D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, R. Noe, “Some enabling techniques for polarization mode dispersion compensation,” J. Light. Technol. 21, 1198–1210 (2003).
[CrossRef]

Yan, L. S.

L. S. Yan, Q. Yu, A. E. Willner, “Periodic polarization scrambling with uniformly distributed SOPs on the Poincaré sphere,” presented at the 28th European Conference on Optical Communication, Copenhagen, Denmark, 8–12 September, 2002.

Yu, Q.

L. S. Yan, Q. Yu, A. E. Willner, “Periodic polarization scrambling with uniformly distributed SOPs on the Poincaré sphere,” presented at the 28th European Conference on Optical Communication, Copenhagen, Denmark, 8–12 September, 2002.

Zweck, J.

I. T. Lima, A. O. Lima, J. Zweck, C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. 15, 45–47 (2003).
[CrossRef]

Aust. J. Statist.

P. J. Diggle, N. I. Fisher, A. J. Lee, “A comparision of tests of uniformity for spherical data,” Aust. J. Statist. 27, 53–59 (1985).
[CrossRef]

Fed. Reserve Bank St. Louis Rev.

R. Hernadez-Murillo, “Strategic interaction in tax polices among states,” Fed. Reserve Bank St. Louis Rev. 85, 47–56 (2003).

IEEE Photon. Technol. Lett.

T. Fujiwara, A. Watanabe, H. Mori, “Polarization dependent loss in a Ti:LiNbO3 polarization scrambler/controller,” IEEE Photon. Technol. Lett. 8, 542–544 (1996).
[CrossRef]

I. T. Lima, A. O. Lima, J. Zweck, C. R. Menyuk, “Efficient computation of outage probabilities due to polarization effects in a WDM system using a reduced Stokes model and importance sampling,” IEEE Photon. Technol. Lett. 15, 45–47 (2003).
[CrossRef]

Int. J. Remote Sens.

J. A. Robinson, D. A. Liddle, C. A. Evans, D. L. Amsbury, “Astronaut-acquired orbital photographs as digital data for remote sensing: spatial resolution,” Int. J. Remote Sens. 23, 4403–4438 (2002).
[CrossRef]

J. Appl. Phys.

R. D. Evans, J. Bentley, K. L. More, G. L. Doll, J. T. Glass, “Radial distribution function analyses of amorphous carbon thin films containing various levels of silicon and hydrogen,” J. Appl. Phys. 96, 273–279 (2004).
[CrossRef]

J. Chem. Phys.

E. Matteoli, G. A. Mansoori, “A simple expression for radial distribution functions of pure fluids and mixtures,” J. Chem. Phys. 11, 4672–4678 (1995).
[CrossRef]

J. Light. Technol.

D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, R. Noe, “Some enabling techniques for polarization mode dispersion compensation,” J. Light. Technol. 21, 1198–1210 (2003).
[CrossRef]

J. Phys. A Math. Gen.

I. M. Tkachenko, “Limiting properties of the radial distribution function in electronic liquids,” J. Phys. A Math. Gen. 29, 2599–2605 (1996).
[CrossRef]

Mater. Charact.

P. A. Karnezis, G. Durrant, G. Cantor, “Characterization of reinforcement distribution in cast Al-alloy/SiCp composites,” Mater. Charact. 40, 97–109 (1998).
[CrossRef]

Math. Intell.

E. B. Saff, A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intell. 19, 5–11 (1997).
[CrossRef]

Nature

L. Glass, W. R. Tobler, “Uniform distribution of objects in a homogeneous field: cities on a plain,” Nature 233, 67–68 (1971).
[CrossRef] [PubMed]

Rare Metals

S. Cheng, Z. Wang, X. Qin, X. Bian, “Effect of copper addition on the thermal contraction of indium melt clusters,” Rare Metals 22, 215–220 (2003).

Sky Telescope

R. W. Sinnott, “Virtues of the Haversine,” Sky Telescope 68, 159–161 (1984).

R. W. Sinnott, “Celestial calendar—hunting for equilateral triple stars,” Sky Telescope 99, 100–101 (1999).

Other

R. Bakshi, C. A. Knoblock, S. Thakkar, “Exploiting online source to accurately geocode addresses,” in Proceedings of the 12th Annual ACM International Workshop on Geographic Information Systems (ACM Press, New York, 2004), pp. 194–203.

N. I. Fisher, T. Lewis, B. J. J. Embleton, Statistical Analysis of Spherical Data (Cambridge U. Press, 1987).
[CrossRef]

L. S. Yan, Q. Yu, A. E. Willner, “Periodic polarization scrambling with uniformly distributed SOPs on the Poincaré sphere,” presented at the 28th European Conference on Optical Communication, Copenhagen, Denmark, 8–12 September, 2002.

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Figures (6)

Fig. 1
Fig. 1

Example of the calculation of a spherical radial distribution function; see Eq. (1). The great circle distance dn between reference points n1 and n2 is given by Eq. (2). The number of points found in the area between circles cn and cn+1 is counted.

Fig. 2
Fig. 2

Theoretical random distribution. (a) Distribution of SOPs on the surface of the Poincaré sphere. (b) Spherical radial distribution function versus distance. For a random distribution the value of g(d) is close to 1.

Fig. 3
Fig. 3

Theoretical clustered distribution. (a) Distribution of SOPs on the surface of the Poincaré sphere. (b) Spherical radial distribution function versus distance. For the clustered distribution the first peak on the g(d) curve provides information about the parameters of the clusters. The location of the lower peaks on the curve indicates the mean distance between clusters.

Fig. 4
Fig. 4

Spherical radial distribution function versus distance for (a) three rotating wave plates and (b) Four rotating wave plates. For three rotating wave plates the distribution of SOPs on the Poincaré sphere is clustered distribution. For four rotating wave plates the value of g(d) oscillates about 1 (random distribution).

Fig. 5
Fig. 5

Spherical radial distribution function versus distance for (a) five rotating wave plates and (b) Six rotating wave plates. For five and six rotating wave plates the value of g(d) oscillates about (random distribution).

Fig. 6
Fig. 6

Effect of the number of rotating wave plates on (a) the value of the correlation coefficient and (b) the value of Fn. The value of the correlation coefficient increases and the value of Fn decreases if the number of elements increases from three to eight. From eight elements the correlation coefficient and Fn are constant.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

g ( d ) = K n A T A n K T ,
d n = 2 R arcsin [ sin 2 ( Θ 1 - Θ 2 2 ) + sin ( Θ 1 ) sin ( Θ 2 ) sin 2 ( ϕ 2 - ϕ 1 2 ) ] 1 / 2 ,
ρ = Y x x * Y x x Y x * x * ,             ρ ( - 1 , 1 ) ,
Y x x * = det ( i = 1 n x R , i , x k , i ) = det [ S 1 R , i S 1 k , i S 2 R , i S 1 k , i S 3 R , i S 1 k , i S 1 R , i S 2 k , i S 2 R , i S 2 k , i S 3 R , i S 2 k , i S 1 R , i S 3 k , i S 2 R , i S 3 k , i S 3 R , i S 3 k , i ] ,
Y x x = det ( i = 1 n x R , i , x R , i ) = det [ S 1 R , i S 1 R , i S 2 R , i S 1 R , i S 3 R , i S 1 R , i S 1 R , i S 2 R , i S 2 R , i S 2 R , i S 3 R , i S 2 R , i S 1 R , i S 3 R , i S 2 R , i S 3 R , i S 3 R , i S 3 R , i ] ,
Y x x = det ( i = 1 n x k , i , x k , i ) = det [ S 1 k , i S 1 k , i S 2 k , i S 1 k , i S 3 k , i S 1 k , i S 1 k , i S 2 k , i S 2 k , i S 2 k , i S 3 k , i S 2 k , i S 1 k , i S 3 k , i S 2 k , i S 3 k , i S 3 k , i S 3 k , i ] ,
Ψ i j = arccos ( S 1 i S 1 j + S 2 i S 2 j + S 3 i S 3 j ) , 1 i < j n ,
F n = 3 n 2 - ( 4 n π ) i = 1 n - 1 j = i + 1 n Ψ i j + sin ( Ψ i j ) .

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