Abstract

An analytical model for electromagnetic twisted Gaussian Schell-model array (EM TGSMA) beams is introduced. We derive the analytical expression for the cross-spectral density matrix (CSDM) of such beam propagating in free space and investigate the spectral density, degree of polarization (DOP) and degree of coherence (DOC) in detail. Twist effects of the evolutions of DOC and DOP during propagation are also demonstrated. It is found that the twist effects of DOC and DOP can be influenced by the beam width and coherence length. In particular, senses of rotation can also be controlled by adjusting the twist strength. Furthermore, we find that twist phase always causes DOC to rotate in an opposite direction compared with that of DOP. Our results might be beneficial for free-space communications of the partially coherent beams endowed with twist.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267(2003).
    [Crossref]
  2. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
    [Crossref] [PubMed]
  3. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007).
    [Crossref] [PubMed]
  4. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
    [Crossref]
  5. M. Guo and D. Zhao, “Polarization properties of stochastic electromagnetic beams modulated by a wavefront-folding interferometer,” Opt. Express 26, 8581–8593 (2018).
    [Crossref] [PubMed]
  6. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
    [Crossref]
  7. Z. Mei, E. Shchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21, 17512–17519 (2013).
    [Crossref] [PubMed]
  8. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
    [Crossref] [PubMed]
  9. Z. Mei and Y. Mao, “Electromagnetic sinc Schell-model beams and their statistical properties,” Opt. Express 22, 22534–22546 (2014).
    [Crossref] [PubMed]
  10. Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
    [Crossref]
  11. X. Du and D. Zhao, “Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere,” Opt. Express 17, 4257–4262 (2009).
    [Crossref] [PubMed]
  12. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A-Pure Appl. Opt. 9, 1123 (2007).
    [Crossref]
  13. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907–1916 (2004).
    [Crossref]
  14. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
    [Crossref]
  15. D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
    [Crossref]
  16. V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
    [Crossref]
  17. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
    [Crossref] [PubMed]
  18. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003).
    [Crossref]
  19. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
    [Crossref] [PubMed]
  20. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [Crossref]
  21. R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
    [Crossref]
  22. S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001).
    [Crossref]
  23. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [Crossref]
  24. L. Wan and D. Zhao, “Controllable rotating Gaussian Schell-model beams,” Opt. Lett. 44, 735–738 (2019).
    [Crossref] [PubMed]
  25. L. Wan and D. Zhao, “Twisted Gaussian Schell-model array beams,” Opt. Lett. 43, 3554–3557 (2018).
    [Crossref] [PubMed]
  26. A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
    [Crossref]
  27. F. Gori and M. Santarsiero, “Devising genuine twisted cross-spectral densities,” Opt. Lett. 43, 595–598 (2018).
    [Crossref] [PubMed]
  28. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40, 5662–5665 (2015).
    [Crossref] [PubMed]
  29. L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
    [Crossref]

2019 (1)

2018 (3)

2015 (1)

2014 (2)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Z. Mei and Y. Mao, “Electromagnetic sinc Schell-model beams and their statistical properties,” Opt. Express 22, 22534–22546 (2014).
[Crossref] [PubMed]

2013 (1)

2009 (1)

2008 (4)

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
[Crossref]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437–18442 (2008).
[Crossref] [PubMed]

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[Crossref]

2007 (2)

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15, 16909–16915 (2007).
[Crossref] [PubMed]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A-Pure Appl. Opt. 9, 1123 (2007).
[Crossref]

2006 (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

2005 (1)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[Crossref]

2004 (1)

2003 (4)

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 polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
[Crossref] [PubMed]

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003).
[Crossref]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

2001 (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001).
[Crossref]

2000 (1)

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

1997 (1)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

1996 (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

1994 (1)

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

1993 (1)

Berger, V.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Bloch, I.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003).
[Crossref]

Cai, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Chen, Y.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Chen, Z.

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
[Crossref]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A-Pure Appl. Opt. 9, 1123 (2007).
[Crossref]

Costard, E.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Denning, R. G.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

Du, X.

Friberg, A. T.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

Gauthier-Lafaye, O.

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

Gori, F.

Guo, M.

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

Korotkova, O.

Liu, L.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
[Crossref]

Mao, Y.

Mei, Z.

Mukunda, N.

Ponomarenko, S. A.

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001).
[Crossref]

Pu, J.

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49–51 (2008).
[Crossref]

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
[Crossref]

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A-Pure Appl. Opt. 9, 1123 (2007).
[Crossref]

Santarsiero, M.

Sharp, D. N.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Shchepakina, E.

Shirai, T.

Simon, R.

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[Crossref]

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

Tervonen, E.

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

Turberfield, A. J.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Turunen, J.

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

Wan, L.

Wang, F.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Wang, X.

Wolf, E.

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
[Crossref]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[Crossref]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21, 1907–1916 (2004).
[Crossref]

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

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

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
[Crossref]

Zhang, Z.

Zhao, C.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Zhao, D.

Zhu, Y.

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

J. Appl. Phys. (1)

V. Berger, O. Gauthier-Lafaye, and E. Costard, “Photonic band gaps andholography,” J. Appl. Phys. 82, 60–64 (1997).
[Crossref]

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

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagnetic beams in a turbulent atmosphere,” J. Opt. A-Pure Appl. Opt. 9, 1123 (2007).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Phys. (1)

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2003).
[Crossref]

Nature (2)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426, 421–424 (2003).
[Crossref] [PubMed]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404, 53–56 (2000).
[Crossref] [PubMed]

Opt. Commun. (3)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[Crossref]

D. Zhao and E. Wolf, “Light beams whose degree of polarization does not change on propagation,” Opt. Commun. 281, 3067–3070 (2008).
[Crossref]

A. T. Friberg, E. Tervonen, and J. Turunen, “Focusing of twisted Gaussian Schell-model beams,” Opt. Commun. 106, 127–132 (1994).
[Crossref]

Opt. Express (6)

Opt. Lett. (6)

Phys. Lett. A (2)

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

Z. Chen and J. Pu, “Stochastic electromagnetic vortex beam and its propagation,” Phys. Lett. A 372, 2734–2740 (2008).
[Crossref]

Phys. Rev. A (1)

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, and O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89, 013801 (2014).
[Crossref]

Phys. Rev. E (1)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E 64036618 (2001).
[Crossref]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5, 331 (1996).
[Crossref]

Other (1)

L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics,” (Cambridge University, 1995).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Average intensity generated by an EM TGSMA source at several propagation distances in free space and setting: uxx = −uyy (mm−2); rows 1 (a1-a4) and 2 (b1-b4): σ1xx = σ1yy = 1mm, σ2xx = σ2yy = 0.3mm. δ1xx = δ1yy = 1mm, δ2xx = δ2yy = 0.3mm; rows 3 (c1-c4) and 4 (d1-d4): σ1xx = δ1xx = 0.4mm, σ2xx = δ2xx = 0.5mm. σ1yy = δ1yy = 0.5mm,; σ2yy = δ2yy = 0.4mm.
Fig. 2
Fig. 2 DOP generated by an EM TGSMA source at several propagation distances in free space with parameters as in Fig. 1.
Fig. 3
Fig. 3 DOC generated by an EM TGSMA source at several propagation distances in free space with parameters as in Fig. 1.
Fig. 4
Fig. 4 The DOP on the axis of an EM TGSMA beam as a function of z for σ1xx = δ1xx = 0.4mm, σ2xx = δ2xx = 0.5mm. σ1yy = δ1yy = 0.5mm, σ2yy = δ2yy = 0.4mm. (a) N1 = N2 = 2, (b) and (c) uxx = −uyy = −3mm−2.
Fig. 5
Fig. 5 The DOC of EM TGSMA beam at z = 5m and yd = 0 with parameters as in Fig. 4.

Equations (14)

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W i j ( r 1 , r 2 , z , ω ) = E i * ( r 1 , z , ω ) E j ( r 2 , z , ω ) , ( i , j = x , y ) ,
W i j ( r 1 , r 2 , r 0 ) = W i j 0 ( r 1 r 0 , r 2 r 0 ) exp  ( 2 π i υ i j r 0 ) ,
W i j ( r 1 , r 2 ) = R i j ( r 1 , r 2 ) exp  ( 2 π i υ i j r m ) ,
R i j ( r 1 , r 2 ) = p ( r 0 ) W i j 0 ( r + D / 2 , r D / 2 ) exp  ( 2 π i υ i j r ) d 2 r .
W i j ( r 1 , r 2 ) = exp   [ ( x 1 x 2 ) 2 2 δ 1 i j 2 ] exp [ ( y 1 y 2 ) 2 2 δ 2 i j 2 ] × P n 1 = P cos   [ C 1 i j ( x 1 x 2 ) ] Q n 2 = Q cos [ C 2 i j ( y 1 y 2 ) ] × exp   ( i u i j x 1 y 2 ) exp   ( i u i j x 2 y 1 ) .
τ ( r ) = exp  ( x 2 4 σ 1 i j 2 ) exp  ( y 2 4 σ 2 i j 2 ) ,
W i j ( r 1 , r 2 ) = exp   ( x 1 2 + x 2 2 4 σ 1 i j 2 ) exp   ( y 1 2 + y 2 2 4 σ 2 i j 2 ) × exp   [ ( x 1 x 2 ) 2 2 δ 1 i j 2 ] exp [ ( y 1 y 2 ) 2 2 δ 2 i j 2 ] × P n 1 = P cos   [ C 1 i j ( x 1 x 2 ) ] Q n 2 = Q cos [ C 2 i j ( y 1 y 2 ) ] × exp   ( i u i j x 1 y 2 ) exp   ( i u i j x 2 y 1 ) .
W i j ( ρ 1 , ρ 2 , z ) = W i j ( r 1 , r 2 ) H z * ( ρ 1 , r 1 ) H z ( ρ 2 , r 2 ) d 2 r 1 d 2 r 2 ,
H z ( ρ , r ) = i k ( 2 π z ) 1 exp  [ i k ( ρ r ) 2 / ( 2 z ) ] .
W i j ( ρ 1 , ρ 2 , z ) = E i j n 1 = P P Q n 2 = Q ( k π σ 1 i j ) 2 A i j B i j D i j z 2 exp   [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] × exp   [ k 2 σ 1 i j 2 2 z 2 ( x 1 x 2 ) 2 ] × { exp [ 1 4 D i j ( 2 m i j 1 + + i k z ( y 1 y 2 ) ) 2 + 1 4 A i j γ i j + + 2 + 1 4 B i j β i j + 2 ]   + exp [ 1 4 D i j ( 2 m i j 1 + + i k z ( y 1 y 2 ) ) 2 + 1 4 A i j γ i j + 2 + 1 4 B i j β i j + 2 ] + exp [ 1 4 D i j ( 2 m i j 1 + i k z ( y 1 y 2 ) ) 2 + 1 4 A i j γ i j + 2 + 1 4 B i j β i j 2 ] + exp [ 1 4 D i j ( 2 m i j 1 + i k z ( y 1 y 2 ) ) 2 + 1 4 A i j γ i j 2 + 1 4 B i j β i j 2 ] } ,
E i j = 1 ( i = j ) ,    E i j = 0 ( i j ) ; γ i j ± = i k 2 z ( y 1 + y 2 ) m i j 2 ± + g i j ± 4 D i j i C 2 i j ; β i j ± = i k 2 z ( x 1 + x 2 ) + k 2 σ 1 i j 2 z 2 ( x 1 x 2 ) ± i C 1 i j ; m i j 1 ± = i u i j 4 B i j β i j ± ;   m i j 2 ± = u i j k σ 1 i j 2 z [ β i j ± ; 2 B i j ( x 1 x 2 ) ] ; A i j = 1 α 2 i j u i j 2 k 2 σ 1 i j 4 4 B i j z 2 + u i j 2 σ 1 i j 2 2 1 4 D i j ( i u i j 2 k σ 1 i j 2 2 B i j z + i k z ) 2 ; g i j ± = 2 ( i u i j 2 k σ 1 i j 2 2 B i j z + i k z ) [ 2 m i j 1 ± + i k ( y 1 y 2 ) / z ] ; B i j = 1 α 1 i j + k 2 σ 1 i j 2 2 z 2 ; D i j = 1 2 σ 2 i j 2 + u i j 2 4 B i j ; 1 α 1 i j = 1 8 σ 1 i j 2 + 1 2 δ 1 i j 2 ; 1 α 2 i j = 1 8 σ 2 i j 2 + 1 2 δ 2 i j 2 .
S ( ρ , z ) = T r W ( ρ , ρ , z ) = W x x ( ρ , ρ , z ) + W y y ( ρ , ρ , z ) ;
P ( ρ , z ) = | W x x ( ρ , ρ , z ) W y y ( ρ , ρ , z ) | W x x ( ρ , ρ , z ) + W y y ( ρ , ρ , z ) .
μ ( ρ 1 , ρ 2 , z ) = W x x ( ρ 1 , ρ 2 , z ) + W y y ( ρ 1 , ρ 2 , z ) S ( ρ 1 , z ) S ( ρ 2 , z ) .

Metrics