Abstract

We analyze the degree of polarization of random, statistically stationary electromagnetic fields in the focal region of a high-numerical-aperture imaging system. The Richards–Wolf theory for focusing is employed to compute the full 3×3 electric coherence matrix, from which the degree of polarization is obtained by using a recent definition for general three-dimensional electromagnetic waves. Significant changes in the state of partial polarization, compared with that of the incident illumination, are observed. For example, a wave consisting of two orthogonal and uncorrelated incident-electric-field components produces rings of full polarization in the focal plane. These effects are explained by considering the distribution of the spectral densities of the three electric field components as well as the correlations between them.

© 2005 Optical Society of America

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References

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  1. L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [Crossref] [PubMed]
  2. B. Sick, B. Hecht, L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
    [Crossref] [PubMed]
  3. I. Ichimura, S. Hayashi, G. S. Kino, “High-density optical recording using a solid immersion lens,” Appl. Opt. 36, 4339–4348 (1997).
    [Crossref] [PubMed]
  4. P. A. Maia Neto, H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
    [Crossref]
  5. B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [Crossref]
  6. K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [Crossref] [PubMed]
  7. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [Crossref]
  8. Q. Zhan, J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002).
    [Crossref] [PubMed]
  9. R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  10. C. J. R. Sheppard, “Focal distributions and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).
    [Crossref]
  11. J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
    [Crossref]
  12. J. J. Stamnes, ed., Electromagnetic Fields in the Focal Region, Vol. 168 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 2001).
  13. J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
    [Crossref]
  14. J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhaylan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
    [Crossref]
  15. L. A. Chernov, Wave Propagation in a Random Medium, Part III (McGraw-Hill, New York, 1960).
  16. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [Crossref]
  17. W. Wang, A. T. Friberg, E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–496 (1997).
    [Crossref]
  18. T. Setälä, A. Shevchenko, M. Kaivola, A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E 66, 016615 (2002).
    [Crossref]
  19. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  20. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).
  21. J. C. Samson, J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
    [Crossref]
  22. R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
    [Crossref]
  23. J. Ellis, A. Dogariu, E. Wolf, “The concept of polarization in near field optics,” CLEO/IQEC and PhAST Technical Digest on CD-ROM (Optical Society of America, Washington, D.C., 2004), paper IWG4.
  24. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  25. T. Setälä, K. Lindfors, M. Kaivola, J. Tervo, A. T. Friberg, “Intensity fluctuations and degree of polarization in three-dimensional thermal light fields,” Opt. Lett. 29, 2587–2589 (2004).
    [Crossref] [PubMed]
  26. The validity of the Debye approximation requires that the Fresnel number NF≫1,which may not hold for a very small NA.
  27. S. K. Rhodes, K. A. Nugent, A. Roberts, “Precision measurement of the electromagnetic fields in the focal region of a high-numerical-aperture lens using a tapered fiber probe,” J. Opt. Soc. Am. A 19, 1689–1693 (2002).
    [Crossref]

2004 (1)

2003 (3)

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

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[Crossref]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhaylan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[Crossref]

2002 (3)

2001 (1)

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

2000 (4)

B. Sick, B. Hecht, L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

P. A. Maia Neto, H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
[Crossref]

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

1999 (1)

C. J. R. Sheppard, “Focal distributions and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).
[Crossref]

1998 (1)

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[Crossref]

1997 (2)

1988 (1)

1983 (1)

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[Crossref]

1980 (1)

J. C. Samson, J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
[Crossref]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Barakat, R.

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[Crossref]

Beversluis, M. R.

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

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

Brown, T. G.

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

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

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium, Part III (McGraw-Hill, New York, 1960).

Dhaylan, V.

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhaylan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[Crossref]

Dogariu, A.

J. Ellis, A. Dogariu, E. Wolf, “The concept of polarization in near field optics,” CLEO/IQEC and PhAST Technical Digest on CD-ROM (Optical Society of America, Washington, D.C., 2004), paper IWG4.

Dorn, R.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Ellis, J.

J. Ellis, A. Dogariu, E. Wolf, “The concept of polarization in near field optics,” CLEO/IQEC and PhAST Technical Digest on CD-ROM (Optical Society of America, Washington, D.C., 2004), paper IWG4.

Friberg, A. T.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Hayashi, S.

Hecht, B.

B. Sick, B. Hecht, L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Ichimura, I.

Jain, M.

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhaylan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[Crossref]

Jiang, D.

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[Crossref]

Kaivola, M.

Kino, G. S.

Leger, J. R.

Lekner, J.

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[Crossref]

Leuchs, G.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Lindfors, K.

Lotsberg, J. K.

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhaylan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[Crossref]

Maia Neto, P. A.

P. A. Maia Neto, H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
[Crossref]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Novotny, L.

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

B. Sick, B. Hecht, L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Nugent, K. A.

Nussenzveig, H. M.

P. A. Maia Neto, H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
[Crossref]

Olson, J. V.

J. C. Samson, J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
[Crossref]

Quabis, S.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

Rhodes, S. K.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Roberts, A.

Samson, J. C.

J. C. Samson, J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
[Crossref]

Setälä, T.

Sheppard, C. J. R.

C. J. R. Sheppard, “Focal distributions and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).
[Crossref]

Shevchenko, A.

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

Sick, B.

B. Sick, B. Hecht, L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Sithambaranathan, G. S.

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhaylan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[Crossref]

Stamnes, J. J.

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhaylan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[Crossref]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[Crossref]

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Tervo, J.

Turunen, J.

Wang, W.

Wolf, E.

W. Wang, A. T. Friberg, 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, E. Wolf, “Electromagnetic diffraction in op-tical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

J. Ellis, A. Dogariu, E. Wolf, “The concept of polarization in near field optics,” CLEO/IQEC and PhAST Technical Digest on CD-ROM (Optical Society of America, Washington, D.C., 2004), paper IWG4.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Youngworth, K. S.

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

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

Zhan, Q.

Appl. Opt. (1)

Europhys. Lett. (1)

P. A. Maia Neto, H. M. Nussenzveig, “Theory of optical tweezers,” Europhys. Lett. 50, 702–708 (2000).
[Crossref]

Geophys. J. R. Astron. Soc. (1)

J. C. Samson, J. V. Olson, “Some comments on the descriptions of the polarization states of waves,” Geophys. J. R. Astron. Soc. 61, 115–129 (1980).
[Crossref]

J. Opt. A (1)

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

R. Barakat, “n-fold polarization measures and associated thermodynamic entropy of N partially coherent pencils of radiation,” Opt. Acta 30, 1171–1182 (1983).
[Crossref]

Opt. Commun. (4)

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[Crossref]

J. J. Stamnes, G. S. Sithambaranathan, M. Jain, J. K. Lotsberg, V. Dhaylan, “Focusing of electromagnetic waves into a biaxial crystal,” Opt. Commun. 226, 107–123 (2003).
[Crossref]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[Crossref]

C. J. R. Sheppard, “Focal distributions and Hertz potentials,” Opt. Commun. 160, 191–194 (1999).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. E (1)

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

Phys. Rev. Lett. (3)

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

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

B. Sick, B. Hecht, L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

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

B. Richards, E. Wolf, “Electromagnetic diffraction in op-tical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[Crossref]

Other (7)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, New York, 1998).

L. A. Chernov, Wave Propagation in a Random Medium, Part III (McGraw-Hill, New York, 1960).

J. J. Stamnes, ed., Electromagnetic Fields in the Focal Region, Vol. 168 of SPIE Milestone Series (SPIE Press, Bellingham, Wash., 2001).

The validity of the Debye approximation requires that the Fresnel number NF≫1,which may not hold for a very small NA.

J. Ellis, A. Dogariu, E. Wolf, “The concept of polarization in near field optics,” CLEO/IQEC and PhAST Technical Digest on CD-ROM (Optical Society of America, Washington, D.C., 2004), paper IWG4.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

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Figures (6)

Fig. 1
Fig. 1

Illustration of the geometry and notation used in the analysis of tightly focused electromagnetic fields.

Fig. 2
Fig. 2

Contour plots of the intensity distribution I ( r ,   ω ) in the focal plane ( u = 0 ) for (a) 2D-unpolarized light [ | μ xy 0 ( ω ) | = 0 ] and (b) partially polarized light with | μ xy 0 ( ω ) | = 0.5 and β 0 ( ω ) = 0 . The NA of the optical system is 0.9. The value of I ( r ,   ω ) has been normalized to 100 at the origin. The axes are v x = v cos ϕ and v y = v sin ϕ .

Fig. 3
Fig. 3

Polarization of the incident 2D-unpolarized light when focused through an optical system of NA = 0.9 : (a) degree of polarization P 3 ( v ,   ϕ ,   ω ) in the focal plane ( u = 0 ) , (b) P 3 ( v ,   ω ) (solid curve) and the degrees of correlation | μ xy ( v ,   ω ) | (dashed curve) and | μ xz ( v ,   ω ) | (dotted–dashed curve) along the azimuth ϕ = π / 4 in the focal plane, (c) intensities ϕ xx ( v ,   ω ) (solid curve) and ϕ zz ( v ,   ω ) (dashed curve) on the same azimuth as that in (b). The intensities have been normalized by the total intensity at the origin. The axes in (a) are defined as v x = v cos ϕ and v y = v sin ϕ .

Fig. 4
Fig. 4

Distribution of the 3D degree of polarization in the focal plane ( u = 0 ) as a function of the NA of the optical system and the coordinate v for incident 2D-unpolarized light.

Fig. 5
Fig. 5

Plots of the 3D degree of polarization for varying degrees of correlation of the incident field. The curves are along the azimuth ϕ = π / 4 in the focal plane ( u = 0 ) of an imaging system with NA = 0.9 . The solid curve is for 2D-unpolarized incident light [ | μ xy 0 ( ω ) | = 0 ] , whereas the others are for | μ xy 0 ( ω ) | = 0.4 (dashed curve), | μ xy 0 ( ω ) | = 0.8 (dotted–dashed curve), and | μ xy 0 ( ω ) | = 0.98 (dotted curve). For the cases | μ xy 0 ( ω ) | > 0 , we have chosen β 0 ( ω ) = 0 .

Fig. 6
Fig. 6

Distribution of the 3D degree of polarization in the focal plane for various partially polarized incident beams. In (a) and (c), the phase of the correlation coefficient is β 0 ( ω ) = 0 , while in (b) and (d) β 0 ( ω ) = π / 2 . In (a) and (b), the degree of correlation of the incident field is | μ xy 0 ( ω ) | = 0.5 , whereas in (c) and (d) | μ xy 0 ( ω ) | = 0.9 . The axes are defined as v x = v cos ϕ and v y = v sin ϕ .

Equations (31)

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ϕ ij ( r ,   ω ) = E i * ( r ,   ω ) E j ( r ,   ω ) , ( i ,   j ) = ( x ,   y ,   z ) .
P 3 2 ( r ,   ω ) = 3 2   tr [ Φ 3 2 ( r ,   ω ) ] tr 2 [ Φ 3 ( r ,   ω ) ] - 1 3 ,
μ ij ( r ,   ω ) ϕ ij ( r ,   ω ) [ ϕ ii ( r ,   ω ) ϕ jj ( r ,   ω ) ] 1 / 2 .
P 3 2 ( r ,   ω )
= 1 - ij 3 [ 1 - | μ ij ( r ,   ω ) | 2 ] ϕ ii ( r ,   ω ) ϕ jj ( r ,   ω ) [ ϕ xx ( r ,   ω ) + ϕ yy ( r ,   ω ) + ϕ zz ( r ,   ω ) ] 2 ,
E i 0 ( r ,   ω ) = E i 0 ( ω ) A 0 ( x ,   y ) exp ( ikz ) e i , i = ( x ,   y ) ,
E ( r ,   ω ) = E x ( r ,   ω ) e x + E y ( r ,   ω ) e y + E z ( r ,   ω ) e z ,
E x ( r ,   ω ) = - E x 0 ( ω )   ik 2   [ I 0 ( r ,   ω ) + I 2 ( r ,   ω ) cos 2 ϕ ] ,
E y ( r ,   ω ) = - E x 0 ( ω )   ik 2   I 2 ( r ,   ω ) sin 2 ϕ ,
E z ( r ,   ω ) = - E x 0 ( ω ) kI 1 ( r ,   ω ) cos ϕ .
I 0 ( r ,   ω ) = 0 α 0 A ( α ) sin α ( 1 + cos α ) J 0 v   sin α sin α 0 × exp iu   cos α sin 2   α 0 d α ,
I 1 ( r ,   ω ) = 0 α 0 A ( α ) sin 2   α J 1 v   sin α sin α 0 × exp iu   cos α sin 2   α 0 d α ,
I 2 ( r ,   ω ) = 0 α 0 A ( α ) sin α ( 1 - cos α ) J 2 v   sin α sin α 0 exp iu   cos α sin 2   α 0 d α ,
u = kz   sin 2   α 0 ,
v = k ρ sin α 0 ,
A ( α ) = A 0 ( f sin α ) cos 1 / 2   α ,
Φ 3 ( r ,   ω ) = M * ( r ,   ω ) Φ 0 ( ω ) M T ( r ,   ω ) ,
M ( r ,   ω ) = - ik 2   I 0 ( r ,   ω ) + I 2 ( r ,   ω ) cos 2 ϕ I 2 ( r ,   ω ) sin 2 ϕ I 2 ( r ,   ω ) sin 2 ϕ I 0 ( r ,   ω ) - I 2 ( r ,   ω ) cos 2 ϕ - 2 iI 1 ( r ,   ω ) cos ϕ - 2 iI 1 ( r ,   ω ) sin ϕ
ϕ ij 0 ( ω ) = E i 0 * ( ω ) E j 0 ( ω ) , ( i ,   j ) = ( x ,   y ) .
Φ 0 ( ω ) = ϕ xx 0 ( ω ) | μ xy 0 ( ω ) | [ ϕ xx 0 ( ω ) ϕ yy 0 ( ω ) ] 1 / 2 exp [ i β 0 ( ω ) ] | μ xy 0 ( ω ) | [ ϕ xx 0 ( ω ) ϕ yy 0 ( ω ) ] 1 / 2 exp [ - i β 0 ( ω ) ] ϕ yy 0 ( ω ) ,
    P 3 ( r ,   ω ) = 1 4 + 3 | I 1 ( r ,   ω ) | 2 + Re [ I 0 * ( r ,   ω ) I 2 ( r ,   ω ) ] | I 0 ( r ,   ω ) | 2 + 2 | I 1 ( r ,   ω ) | 2 + | I 2 ( r ,   ω ) | 2 2 1 / 2 ,
P 3 ( r ,   ω ) = 1
  | μ xy ( r ,   ω ) | 2 = | μ xz ( r ,   ω ) | 2
= | μ yz ( r ,   ω ) | 2 = 1 .
E i ( r ,   ω ) = a i ( r ,   ω ) E x 0 ( ω ) + b i ( r ,   ω ) E y 0 ( ω ) ,
i = ( x ,   y ,   z ) ,
| μ ij ( r ,   ω ) | 2 = 1
  | a i ( r ,   ω ) b j ( r ,   ω ) - b i ( r ,   ω ) a j ( r ,   ω ) | 2
× [ | μ xy 0 ( ω ) | 2 - 1 ] ϕ xx 0 ( ω ) ϕ yy 0 ( ω ) = 0
a i ( r ,   ω ) b j ( r ,   ω ) = b i ( r ,   ω ) a j ( r ,   ω ) for all ( ij ) = ( xy ,   xz ,   yz ) .
a x ( r ,   ω ) b x ( r ,   ω ) = a y ( r ,   ω ) b y ( r ,   ω ) = a z ( r ,   ω ) b z ( r ,   ω ) ,

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