Abstract

The spectral degree of coherence and of polarization of some model electromagnetic beams modulated by a polarization-dependent phase-modulating device, such as a liquid-crystal spatial light modulator, acting as a random phase screen are examined on the basis of the recent theory formulated in terms of the 2×2 cross-spectral density matrix of the beam. The phase-modulating device is assumed to have strong polarization dependence that modulates only one of the orthogonal components of the electric vector, and the phase of the phase-modulating device is assumed to be a random function of position imitating a random phase screen and is assumed to obey Gaussian statistics with zero mean. The propagation of the modulated beam is also examined to show how the spectral degrees of coherence and of polarization of the beam change on propagation, even in free space. The results are illustrated by numerical examples.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 10.
  2. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 7.
  3. E. Wolf, D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
    [CrossRef]
  4. E. Wolf, T. Shirai, H. Chen, W. Wang, “Coherence filters and their uses I. Basic theory and examples,” J. Mod. Opt. 44, 1345–1353 (1997).
  5. T. Shirai, E. Wolf, H. Chen, W. Wang, “Coherence filters and their uses II. One-dimensional realization,” J. Mod. Opt. 45, 799–816 (1998).
    [CrossRef]
  6. W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
    [CrossRef]
  7. J. C. Dainty, ed., Laser Speckles and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984).
  8. A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS69 (SPIE Press, Bellingham, Wash., 1993), Sec. 6.
  9. F. Scudieri, M. Bertolotti, R. Bartolino, “Light scattered by a liquid crystal: a new quasi-thermal source,” Appl. Opt. 13, 181–185 (1974).
    [CrossRef] [PubMed]
  10. Y. Ohtsuka, “Modulation of optical coherence by ultrasonic waves,” J. Opt. Soc. Am. A 3, 1247–1257 (1986).
    [CrossRef]
  11. J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
    [CrossRef]
  12. D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
    [CrossRef]
  13. G. P. Agrawal, E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
    [CrossRef]
  14. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
    [CrossRef]
  15. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [CrossRef]
  16. E. Wolf, “Correlation-induced changes in the degree of po-larization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
    [CrossRef] [PubMed]
  17. H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
    [CrossRef]
  18. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  19. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]
  20. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
  21. This expression, which is a generalization of the transformation law for the coherence matrix, is readily derived using the analysis given in Sec. 6.4 of Ref. 20.
  22. See, for example, B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 6.1.
  23. See, for example, T. Shirai, “Liquid-crystal adaptive optics based on feedback interferometry for high-resolution retinal imaging,” Appl. Opt. 41, 4013–4023 (2002).
    [CrossRef] [PubMed]
  24. When the phase-modulating device is a reflection type, rather than a transmission type, as employed in Ref. 23, the transmission matrix must be replaced by the reflection matrix.
  25. N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
    [CrossRef]
  26. In order for the paraxial propagation equation given by Eq. (6) to be valid for analyzing the propagation of the modu-lated field produced just behind the LC SLM, the condition 1/(4σS2)+1/ση2≪2π2/λ2 must be satisfied. This condition is basically the same as the “beam condition” for scalar Gaussian Schell-model (GSM) beams (Sec. 5.6.4 of Ref. 20). For more general discussion on the beam conditions for electromagnetic GSM beams, see O. Korotkova, M. Salem, E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29, 1173–1175 (2004).
    [CrossRef] [PubMed]

2004 (1)

2003 (3)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

E. Wolf, “Correlation-induced changes in the degree of po-larization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
[CrossRef] [PubMed]

H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

2002 (1)

2001 (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

2000 (1)

1998 (3)

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

T. Shirai, E. Wolf, H. Chen, W. Wang, “Coherence filters and their uses II. One-dimensional realization,” J. Mod. Opt. 45, 799–816 (1998).
[CrossRef]

1997 (1)

E. Wolf, T. Shirai, H. Chen, W. Wang, “Coherence filters and their uses I. Basic theory and examples,” J. Mod. Opt. 44, 1345–1353 (1997).

1996 (1)

E. Wolf, D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
[CrossRef]

1994 (1)

1990 (1)

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

1986 (1)

1974 (2)

F. Scudieri, M. Bertolotti, R. Bartolino, “Light scattered by a liquid crystal: a new quasi-thermal source,” Appl. Opt. 13, 181–185 (1974).
[CrossRef] [PubMed]

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[CrossRef]

1964 (1)

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
[CrossRef]

Agrawal, G. P.

Bartolino, R.

Bertolotti, M.

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 10.

Chen, H.

T. Shirai, E. Wolf, H. Chen, W. Wang, “Coherence filters and their uses II. One-dimensional realization,” J. Mod. Opt. 45, 799–816 (1998).
[CrossRef]

E. Wolf, T. Shirai, H. Chen, W. Wang, “Coherence filters and their uses I. Basic theory and examples,” J. Mod. Opt. 44, 1345–1353 (1997).

Friberg, A. T.

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 7.

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

James, D. F. V.

E. Wolf, D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
[CrossRef]

D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

Korotkova, O.

Mandel, L.

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

Martienssen, W.

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
[CrossRef]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Ohtsuka, Y.

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

Saleh, B. E. A.

See, for example, B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 6.1.

Salem, M.

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Scudieri, F.

Shirai, T.

See, for example, T. Shirai, “Liquid-crystal adaptive optics based on feedback interferometry for high-resolution retinal imaging,” Appl. Opt. 41, 4013–4023 (2002).
[CrossRef] [PubMed]

T. Shirai, E. Wolf, H. Chen, W. Wang, “Coherence filters and their uses II. One-dimensional realization,” J. Mod. Opt. 45, 799–816 (1998).
[CrossRef]

E. Wolf, T. Shirai, H. Chen, W. Wang, “Coherence filters and their uses I. Basic theory and examples,” J. Mod. Opt. 44, 1345–1353 (1997).

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Spiller, E.

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
[CrossRef]

Takai, N.

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[CrossRef]

Teich, M. C.

See, for example, B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 6.1.

Tervonen, E.

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Turunen, J.

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Wang, W.

T. Shirai, E. Wolf, H. Chen, W. Wang, “Coherence filters and their uses II. One-dimensional realization,” J. Mod. Opt. 45, 799–816 (1998).
[CrossRef]

E. Wolf, T. Shirai, H. Chen, W. Wang, “Coherence filters and their uses I. Basic theory and examples,” J. Mod. Opt. 44, 1345–1353 (1997).

Wolf, E.

In order for the paraxial propagation equation given by Eq. (6) to be valid for analyzing the propagation of the modu-lated field produced just behind the LC SLM, the condition 1/(4σS2)+1/ση2≪2π2/λ2 must be satisfied. This condition is basically the same as the “beam condition” for scalar Gaussian Schell-model (GSM) beams (Sec. 5.6.4 of Ref. 20). For more general discussion on the beam conditions for electromagnetic GSM beams, see O. Korotkova, M. Salem, E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29, 1173–1175 (2004).
[CrossRef] [PubMed]

E. Wolf, “Correlation-induced changes in the degree of po-larization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
[CrossRef] [PubMed]

H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

G. P. Agrawal, E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17, 2019–2023 (2000).
[CrossRef]

T. Shirai, E. Wolf, H. Chen, W. Wang, “Coherence filters and their uses II. One-dimensional realization,” J. Mod. Opt. 45, 799–816 (1998).
[CrossRef]

E. Wolf, T. Shirai, H. Chen, W. Wang, “Coherence filters and their uses I. Basic theory and examples,” J. Mod. Opt. 44, 1345–1353 (1997).

E. Wolf, D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 10.

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

Am. J. Phys. (1)

W. Martienssen, E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (1)

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of optical coherence by electronically synthesized holographic gratings,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

J. Mod. Opt. (2)

E. Wolf, T. Shirai, H. Chen, W. Wang, “Coherence filters and their uses I. Basic theory and examples,” J. Mod. Opt. 44, 1345–1353 (1997).

T. Shirai, E. Wolf, H. Chen, W. Wang, “Coherence filters and their uses II. One-dimensional realization,” J. Mod. Opt. 45, 799–816 (1998).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

N. Takai, “Statistics of dynamic speckles produced by a moving diffuser under the Gaussian beam laser illumination,” Jpn. J. Appl. Phys. 13, 2025–2032 (1974).
[CrossRef]

Opt. Commun. (1)

H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

Opt. Lett. (3)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Rep. Prog. Phys. (1)

E. Wolf, D. F. V. James, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
[CrossRef]

Other (8)

When the phase-modulating device is a reflection type, rather than a transmission type, as employed in Ref. 23, the transmission matrix must be replaced by the reflection matrix.

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

This expression, which is a generalization of the transformation law for the coherence matrix, is readily derived using the analysis given in Sec. 6.4 of Ref. 20.

See, for example, B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Sec. 6.1.

J. C. Dainty, ed., Laser Speckles and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984).

A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series MS69 (SPIE Press, Bellingham, Wash., 1993), Sec. 6.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 10.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 7.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Illustration of the geometry. A random phase screen is located in the plane z=0. The superscripts (i), (t), and (z) on the symbol W denote the incident beam at z=0, the transmitted beam at z=0, and the beam propagating after a distance z, respectively.

Fig. 2
Fig. 2

Spectral degree of polarization of the field on the axis of the beam for some selected values of the polarization angle θ. The propagation distance is normalized by the Rayleigh range zR2kσS2 of the incident Gaussian beam.  

Fig. 3
Fig. 3

Spectral degree of polarization and normalized intensity (spectral density) of the field for some selected values of the normalized propagation distance z/zR and the polarization angle θ. The incident beam is a linearly polarized electromagnetic Gaussian beam.

Fig. 4
Fig. 4

Spectral degree of coherence of the field at two points located symmetrically with respect to the z axis for some selected values of the polarization angle θ at the normalized propagation distance (a) z/zR=0.071, (b) z/zR=0.25, and (c) z/zR=0.5. The incident beam is a linearly polarized electromagnetic Gaussian beam.

Tables (3)

Tables Icon

Table 1 Expressions for the Spectral Degree of Coherence and the Spectral Degree of Polarization of the Field Modulated by a Polarization-Dependent Random Phase Screen Such as an LC SLM

Tables Icon

Table 2 Expressions for the Spectral Degree of Coherence and the Spectral Degree of Polarization of the Field Modulated by a Polarization-Independent Random Phase Screen Such as a Ground Glass Plate

Tables Icon

Table 3 Expressions for the Spectral Degree of Coherence and the Spectral Degree of Polarization of the Field Modulated by the Polarization-Dependent Moving Random Phase Screen in the Limiting Case as ϕ0

Equations (59)

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

W(r1, r2, ω)
Ex*(r1, ω)Ex(r2, ω)Ex*(r1, ω)Ey(r2, ω)Ey*(r1, ω)Ex(r2, ω)Ey*(r1, ω)Ey(r2, ω),
 
S(r, ω)=tr W(r, r, ω)=|Ex(r, ω)|2+|Ey(r, ω)|2,
W(t)(ρ1, ρ2, ω)=T^(ρ1, ω)W(i)(ρ1, ρ2, ω)Tˆ(ρ2, ω)T,
η(t)(ρ1, ρ2, ω)
=tr W(t)(ρ1, ρ2, ω)tr W(t)(ρ1, ρ1, ω)tr W(t)(ρ2, ρ2, ω),
P(t)(ρ, ω)=1-4 det W(t)(ρ, ρ, ω)[tr W(t)(ρ, ρ, ω)]21/2,
W(z)(ρ1, ρ2, ω)=k2πz2z=0W(t)(ρ1, ρ2, ω)×exp-i k2z [(ρ1-ρ1)2-(ρ2-ρ2)2]d2ρ1d2ρ2,
η(z)(ρ1, ρ2, ω)
=tr W(z)(ρ1, ρ2, ω)tr W(z)(ρ1, ρ1, ω)tr W(z)(ρ2, ρ2, ω),
P(z)(ρ, ω)=1-4 det W(z)(ρ, ρ, ω)[tr W(z)(ρ, ρ, ω)]21/2,
S(i)(ρ, ω)=S0exp-|ρ|22σS2,
Wp(i)(ρ1, ρ2, ω)=S0exp-|ρ1|2+|ρ2|24σS2×cos2 θcos θ sin θsin θ cos θsin2 θ,
Wup(i)(ρ1, ρ2, ω)=12 S0exp-|ρ1|2+|ρ2|24σS2Iˆ,
T^LC(ρ, ω)=100exp[iϕ(ρ, ω)],
ϕ(ρ1, ω)ϕ(ρ2, ω)T=ϕ02exp-|ρ1-ρ2|22σϕ2,
ϕ0=|ϕ(ρ, ω)|2T,
Wp,LC(t)(ρ1, ρ2, ω)=S0exp-|ρ1|2+|ρ2|24σS2×cos2 θexp[iϕ(ρ2, ω)]Tcos θ sin θexp[-iϕ(ρ1, ω)]Tsin θ cos θexp{i[ϕ(ρ2, ω)-ϕ(ρ1, ω)]}Tsin2 θ.
exp[iϕ(ρ2, ω)]T=exp[-iϕ(ρ1, ω)]T=exp(-ϕ02/2),
exp{i[ϕ(ρ2, ω)-ϕ(ρ1, ω)]}T
=exp-ϕ021-exp-|ρ1-ρ2|22σϕ2.
Wp,LC(t)(ρ1, ρ2, ω)=S0exp-|ρ1|2+|ρ2|24σS2×cos2 θexp-ϕ022cos θ sin θexp-ϕ022sin θ cos θexp-ϕ021-exp-|ρ1-ρ2|22σϕ2sin2 θ.
ηp,LC(t)(ρ1, ρ2, ω)
=cos2 θ+exp-ϕ021-exp-|ρ1-ρ2|22σϕ2sin2 θ.
Pp,LC(t)(ρ, ω)={1-[1-exp(-ϕ02)]sin2 2θ}1/2.
ηp,LC(t)(ρ1, ρ2, ω)|θ=π/2
=exp-ϕ021-exp-|ρ1-ρ2|22σϕ2
 
Wup,LC(t)(ρ1, ρ2, ω)
=12 S0exp-|ρ1|2+|ρ2|24σS2×100exp-ϕ021-exp-|ρ1-ρ2|22σϕ2.
ηup,LC(t)(ρ1, ρ2, ω)
=121+exp-ϕ021-exp-|ρ1-ρ2|22σϕ2
T^GG(ρ, ω)=exp[iϕ(ρ, ω)]Iˆ,
Wp,GG(t)(ρ1, ρ2, ω)=S0exp-|ρ1|2+|ρ2|24σS2exp-ϕ021-exp-|ρ1-ρ2|22σϕ2×cos2 θcos θ sin θsin θ cos θsin2 θ,
ηp,GG(t)(ρ1, ρ2, ω)=exp-ϕ021-exp-|ρ1-ρ2|22σϕ2,
Wup,GG(t)(ρ1, ρ2, ω)
=12 S0exp-|ρ1|2+|ρ2|24σS2×exp-ϕ021-exp-|ρ1-ρ2|22σϕ2Iˆ,
ηup,GG(t)(ρ1, ρ2, ω)=exp-ϕ021-exp-|ρ1-ρ2|22σϕ2,
exp[iϕ(ρ2, ω)]T|ϕ0=exp[-iϕ(ρ1, ω)]T|ϕ00,
exp{i[ϕ(ρ2, ω)-ϕ(ρ1, ω)]}T|ϕ0
exp-|ρ1-ρ2|22ση2,
Wp,LC(t)(ρ1, ρ2, ω)|ϕ0
=S0exp-|ρ1|2+|ρ2|24σS2×cos2 θ00exp-|ρ1-ρ2|22ση2sin2 θ
Wup,LC(t)(ρ1, ρ2, ω)|ϕ0=12 S0exp-|ρ1|2+|ρ2|24σS2×100exp-|ρ1-ρ2|22ση2
Wp,LC(z)(ρ1, ρ2, ω)
=S0cos2 θW1(z)(ρ1, ρ2, ω)00sin2 θW2(z)(ρ1, ρ2, ω),
W1(z)(ρ1, ρ2, ω)=k2πz2z=0exp-|ρ1|2+|ρ2|24σS2×exp-i k2z [(ρ1-ρ1)2-(ρ2-ρ2)2]d2ρ1d2ρ2,
W2(z)(ρ1, ρ2, ω)=k2πz2z=0exp-|ρ1|2+|ρ2|24σS2×exp-|ρ1-ρ2|22ση2×exp-i k2z [(ρ1-ρ1)2-(ρ2-ρ2)2]d2ρ1d2ρ2.
W1(z)(ρ1, ρ2, ω)=1[ΔC(z)]2exp-|ρ1|2+|ρ2|24σS2[ΔC(z)]2×expik(|ρ2|2-|ρ1|2)2RC(z),
ΔC(z)=1+z2kσS221/2,
RC(z)=z1+2kσS2z2,
W2(z)(ρ1, ρ2, ω)=1[Δ(z)]2exp-(ρ1+ρ2)28σS2[Δ(z)]2×exp-(ρ2-ρ1)22σ2[Δ(z)]2×expik(|ρ2|2-|ρ1|2)2R(z),
Δ(z)=1+zkσSσ21/2,
R(z)=z1+kσSσz2,
1σ2=1(2σS)2+1ση2.
F(ρ1-ρ2)=exp-ϕ021-exp-|ρ1-ρ2|22σϕ2.
exp-|ρ1-ρ2|22σϕ21-|ρ1-ρ2|22σϕ2.
F(ρ1-ρ2)=exp-|ρ1-ρ2|22(σϕ/ϕ0)2,

Metrics