Abstract

We introduce a new metric to characterise spatial variations occurring in a time varying vector field, and we derive a rotationally invariant formula that quantifies temporal fluctuations within a consistent framework. So as to highlight the physics behind these metrics, both are derived from a well–known experiment in polarimetry. The derivation yields a set of expressions in a two–dimensional space, which is subsequently expanded to n–dimensions for special cases. The resulting expressions of the temporal and spatial metrics are incorporated into the electromagnetic theory of coherence and polarisation. Examples are given in the context of single molecule detection when measuring asymmetrically and radially polarised beams.

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  1. J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95(20), 203905 (2004).
    [CrossRef]
  2. A. F. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
    [CrossRef] [PubMed]
  3. K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot. 2(5), 311–314 (2008).
    [CrossRef]
  4. T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett. 77(19), 3979–3982 (1996).
    [CrossRef] [PubMed]
  5. E. Wolf, “Optics in terms of observable quantities,” Il Nuovo Cimento 12(6), 884–888 (1954).
    [CrossRef]
  6. P. Munro and P. Török, “Properties of high-numerical-aperture Müller-matrix polarimeters.” Opt. Lett. 33, 2428–2430 (2008).
    [CrossRef] [PubMed]
  7. T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
    [CrossRef]
  8. H. C. Jacks and O. Korotkova, “Intensity–intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A 28(6), 1139–1144 (2011).
    [CrossRef]
  9. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento 13(6), 1165–1181 (1959).
    [CrossRef]
  10. T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 16615 (2002).
    [CrossRef]
  11. U. Fano, “A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics,” Phys. Rev. 93(1), 121–123 (1954).
    [CrossRef]
  12. M. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
    [CrossRef]
  13. J. Gil, “Polarimetric characterization of light and media,” Europ. Phys. J. App. Phys. 40(1), 1–47 (2007).
    [CrossRef]
  14. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4–6), 333–337 (2005).
    [CrossRef]
  15. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  16. J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149(4–6), 230–234 (1998).
    [CrossRef]
  17. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004).
    [CrossRef] [PubMed]
  18. H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
    [CrossRef]
  19. P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Comm. 148(4–6), 300–315 (1998).
    [CrossRef]

2011 (1)

2008 (3)

P. Munro and P. Török, “Properties of high-numerical-aperture Müller-matrix polarimeters.” Opt. Lett. 33, 2428–2430 (2008).
[CrossRef] [PubMed]

K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot. 2(5), 311–314 (2008).
[CrossRef]

H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
[CrossRef]

2007 (2)

J. Gil, “Polarimetric characterization of light and media,” Europ. Phys. J. App. Phys. 40(1), 1–47 (2007).
[CrossRef]

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (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]

2005 (1)

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4–6), 333–337 (2005).
[CrossRef]

2004 (2)

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004).
[CrossRef] [PubMed]

J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95(20), 203905 (2004).
[CrossRef]

2002 (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 16615 (2002).
[CrossRef]

2000 (1)

M. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[CrossRef]

1998 (2)

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Comm. 148(4–6), 300–315 (1998).
[CrossRef]

J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149(4–6), 230–234 (1998).
[CrossRef]

1996 (1)

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett. 77(19), 3979–3982 (1996).
[CrossRef] [PubMed]

1959 (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento 13(6), 1165–1181 (1959).
[CrossRef]

1954 (2)

E. Wolf, “Optics in terms of observable quantities,” Il Nuovo Cimento 12(6), 884–888 (1954).
[CrossRef]

U. Fano, “A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics,” Phys. Rev. 93(1), 121–123 (1954).
[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]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Chemla, D. S.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett. 77(19), 3979–3982 (1996).
[CrossRef] [PubMed]

Dogariu, A.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4–6), 333–337 (2005).
[CrossRef]

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004).
[CrossRef] [PubMed]

J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95(20), 203905 (2004).
[CrossRef]

Ellis, J.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4–6), 333–337 (2005).
[CrossRef]

J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95(20), 203905 (2004).
[CrossRef]

J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29(6), 536–538 (2004).
[CrossRef] [PubMed]

Enderle, T.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett. 77(19), 3979–3982 (1996).
[CrossRef] [PubMed]

Fano, U.

U. Fano, “A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics,” Phys. Rev. 93(1), 121–123 (1954).
[CrossRef]

Friberg, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 16615 (2002).
[CrossRef]

Friberg, A. T.

H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
[CrossRef]

Gil, J.

J. Gil, “Polarimetric characterization of light and media,” Europ. Phys. J. App. Phys. 40(1), 1–47 (2007).
[CrossRef]

Ha, T.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett. 77(19), 3979–3982 (1996).
[CrossRef] [PubMed]

Higdon, P.

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Comm. 148(4–6), 300–315 (1998).
[CrossRef]

Jacks, H. C.

Juskaitis, R.

M. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[CrossRef]

Kaivola, M.

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 16615 (2002).
[CrossRef]

Korotkova, O.

Landgrave, J.

H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
[CrossRef]

Martínez-Herrero, R.

J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149(4–6), 230–234 (1998).
[CrossRef]

Martinez-Niconoff, A.

H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
[CrossRef]

Mejías, P.

J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149(4–6), 230–234 (1998).
[CrossRef]

Movilla, J.

J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149(4–6), 230–234 (1998).
[CrossRef]

Moya-Cessa, H.

H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
[CrossRef]

Moya-Cessa, J. R.

H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
[CrossRef]

Munro, P.

Neil, M.

M. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[CrossRef]

Perez-Leija, G.

H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
[CrossRef]

Piquero, G.

J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149(4–6), 230–234 (1998).
[CrossRef]

Ponomarenko, S.

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4–6), 333–337 (2005).
[CrossRef]

Ramsay, E.

K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot. 2(5), 311–314 (2008).
[CrossRef]

Reid, D.

K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot. 2(5), 311–314 (2008).
[CrossRef]

Selvin, P. R.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett. 77(19), 3979–3982 (1996).
[CrossRef] [PubMed]

Serrels, K.

K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot. 2(5), 311–314 (2008).
[CrossRef]

Setälä, T.

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 16615 (2002).
[CrossRef]

Shevchenko, A.

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 16615 (2002).
[CrossRef]

Shirai, T.

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[CrossRef]

Török, P.

P. Munro and P. Török, “Properties of high-numerical-aperture Müller-matrix polarimeters.” Opt. Lett. 33, 2428–2430 (2008).
[CrossRef] [PubMed]

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Comm. 148(4–6), 300–315 (1998).
[CrossRef]

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]

Warburton, R.

K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot. 2(5), 311–314 (2008).
[CrossRef]

Weiss, S.

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett. 77(19), 3979–3982 (1996).
[CrossRef] [PubMed]

Wilson, T.

M. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[CrossRef]

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Comm. 148(4–6), 300–315 (1998).
[CrossRef]

Wolf, E.

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4–6), 333–337 (2005).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento 13(6), 1165–1181 (1959).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Il Nuovo Cimento 12(6), 884–888 (1954).
[CrossRef]

Europ. Phys. J. App. Phys. (1)

J. Gil, “Polarimetric characterization of light and media,” Europ. Phys. J. App. Phys. 40(1), 1–47 (2007).
[CrossRef]

Il Nuovo Cimento (2)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento 13(6), 1165–1181 (1959).
[CrossRef]

E. Wolf, “Optics in terms of observable quantities,” Il Nuovo Cimento 12(6), 884–888 (1954).
[CrossRef]

J. Europ. Opt. Soc. Rap. Public. (1)

H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public. 3(5), 08014 (2008).
[CrossRef]

J. Microscopy (1)

M. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[CrossRef]

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

Nature Phot. (1)

K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot. 2(5), 311–314 (2008).
[CrossRef]

Opt. Comm. (1)

P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Comm. 148(4–6), 300–315 (1998).
[CrossRef]

Opt. Commun. (3)

T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun. 272(2), 289–292 (2007).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun. 248(4–6), 333–337 (2005).
[CrossRef]

J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149(4–6), 230–234 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. (1)

U. Fano, “A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics,” Phys. Rev. 93(1), 121–123 (1954).
[CrossRef]

Phys. Rev. E (1)

T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66(1), 16615 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett. 77(19), 3979–3982 (1996).
[CrossRef] [PubMed]

J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett. 95(20), 203905 (2004).
[CrossRef]

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

Other (1)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

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Figures (5)

Fig. 1
Fig. 1

Null-polarimeter experiment.

Fig. 2
Fig. 2

(a) Polarised part of the field distribution produced by a single molecule whose dipole moment is ( 1 , 0 , 1 ) / 3 and exhibits circular dichroism. The irradiance is represented by the grayscale in positive, and the ellipses depict the polarised part of the field at the given points. Figures (b) and (c) show the distribution of the azimuthal angle and the ellipticity of the polarisation, respectively.

Fig. 3
Fig. 3

Degree of spatial polarisation of the field produced by a single molecule at the back focal plane of a high NA lens. The figure consists of subfigures that correspond to the coordinates r i = (xi, yi, zi = 0) centred on the coordinates r j = (xj, yj, zj = 0).

Fig. 4
Fig. 4

Effect of angular perturbations on the dotp when averaging over finite areas. The examples are given column–wise for different amounts of angular perturbation increasing from left to right (left column being the static case). Top row depicts the dipole moment of the molecule at specific times. The black dot represents the initial dipole moment and the red ones depict the subsequent ones. The middle row is the dotp measuring with a CCD formed of infinitesimally small pixels. Bottom row is the dotp measured using a CCD whose pixel size is 1/8th of the total area of the beam.

Fig. 5
Fig. 5

Degree of spatial polarisation of the field produced by a radially polarised beam. The figure consists of subfigures that correspond to the coordinates r i = (xi, yi, zi = 0) centred on the coordinates r j = (xj, yj, zj = 0).

Equations (38)

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

Γ j = Γ j p + Γ j u ,
Tr [ G ˜ i Γ j G ˜ i ] = Tr [ G ˜ i Γ j p G ˜ i ] + Tr [ G ˜ i Γ j u G ˜ i ] ,
Γ j p = X ˜ j Λ j p X ˜ j ,
G ˜ i = X ˜ i Φ ˜ i X ˜ i ,
Φ ˜ i = λ ˜ i p I Λ ˜ j p ,
I p ( r i , r j ) = I ( i , j ) p = Tr [ G ˜ i Γ j p G ˜ i ] ,
I ( i , j ) p = Tr [ Γ ˜ i p ] Tr [ Γ j p ] Tr [ Γ j p Γ ˜ i p ]
Tr [ Γ j ] λ j = α j ( Tr [ I ] 1 ) ,
Tr [ Λ ˜ i Λ j ] = λ ˜ i λ j + α ˜ i α j ( Tr [ I ] 1 ) ,
I ( i , j ) p = Tr [ Λ ˜ i Λ j ] Tr [ Γ ˜ i Γ j ] .
P ( i , j ) s 1 Tr [ Λ i Λ j ] Tr [ Γ i Γ j ] Tr [ Γ i ] Tr [ Γ j ] .
P ( i , j ) s = | Tr [ Γ ( i , j ) ] | 2 Tr [ Γ i ] Tr [ Γ j ] ,
P s ( r i , r j , 0 ) = | Tr [ Γ ( r i , r j , 0 ) ] | 2 Tr [ Γ ( r i , r i , 0 ) ] Tr [ Γ ( r j , r j , 0 ) ] .
P s ( r i , r j , 0 ) = E ( r i , t ) E ( r j , t ) 2 E ( r i , t ) 2 E ( r j , t ) 2 = cos 2 θ ( i , j ) ,
Tr [ G ˜ j Γ j G ˜ j ] = Tr [ G ˜ j Γ j u G ˜ j ]
Tr [ G ˜ j Γ j G ˜ j ] = α j λ ˜ j p ( Tr [ I ] 1 ) = λ ˜ j p Tr [ Γ j u ] q ,
q = Tr [ I ] Tr [ I ] 1 .
Tr [ Γ j p ] = Tr [ Γ j ] q λ ˜ j p Tr [ G ˜ j Γ j G ˜ j ] .
Tr [ Γ j p ] = Tr [ Γ j ] q λ ˜ j p Tr [ Λ j ( λ ˜ j p I Λ ˜ j p ) ] .
Tr [ Λ j ( λ ˜ j p I Λ ˜ j p ) ] = λ ˜ j p ( Tr [ Λ j ] λ j ) ,
P n t ( r j ) = 1 q ( Tr [ Γ j ] λ j ) Tr [ Γ j ] ,
P 2 t ( r j ) = λ j p λ j p + 2 α j = 1 4 Det Γ j ( Tr [ Γ j ] ) 2 ,
P 3 t ( r j ) = λ j p λ j p + 3 α j = 3 Tr [ Γ j 2 ] 2 ( Tr [ Γ j ] ) 2 1 2 ,
I ( j , j ) p = Tr [ Φ ˜ j Λ j ] Tr [ J ˜ j Γ j ] ,
μ ( r i , r j , τ ) = Tr [ Γ ( r i , r j , τ ) ] Tr [ Γ ( r i , r i , 0 ) ] Tr [ Γ ( r j , r j , 0 ) ] ,
P s ( r i , r j , 0 ) = μ 2 ( r i , r j , 0 ) .
E ( r , t ) = L ( r ) [ 1 | r | 3 r × ( r × p ( t ) ) ] ,
Tr [ Γ j ] λ j = α j ( Tr [ I ] 1 )
Tr [ Λ j p + α j I ] λ j p α j = α j ( Tr [ I ] 1 ) ,
Tr [ Λ j p ] + α j Tr [ I ] λ j p α j = α j Tr [ I ] α j ,
λ j p = λ j p
Tr [ Λ ˜ i Λ j ] λ ˜ i λ j = α ˜ i α j ( Tr [ I ] 1 )
Tr [ ( Λ ˜ i p + α ˜ i I ) ( Λ j p + α j I ) ] = ( α ˜ i + λ ˜ i p ) ( α j + λ j p ) + α ˜ i α j ( Tr [ I ] 1 ) ,
Tr [ Λ ˜ i p Λ j p ] + α ˜ i Tr [ Λ j p ] = α j Tr [ Λ ˜ i p ] + λ ˜ i p λ j p + α j λ ˜ i p + α ˜ i λ j p
λ ˜ i p λ j p = λ ˜ i p λ j p
Tr [ Λ j ( λ ˜ j p I Λ ˜ j p ) ] = λ ˜ j p ( Tr [ Λ j ] λ j )
λ ˜ j p Tr [ Λ j ] Tr [ Λ j Λ ˜ j p ] = λ ˜ j p ( λ j p + 2 α j ( λ j p α j ) ) ,
λ ˜ j p ( λ j p + 2 α j ) λ ˜ j p ( λ j p + α j ) = λ ˜ j p α j λ ˜ j p α j = λ ˜ j p α j .

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