Abstract

In this paper, we study the focusing of a stochastic electromagnetic beam by a bifocal lens. By taking the electromagnetic Gaussian Schell-model (EGSM) beam as an example, the changes in the spectral density, in the spectral degree of coherence, and in the spectral degree of polarization of the EGSM beam as the beam is focused by an unapertured bifocal lens are investigated. It is shown that the spectral density, the spectral degree of coherence, and the spectral degree of polarization of the focused electromagnetic EGSM beams depend upon the coherence lengths and focal lengths of the bifocal lens. The influence of the coherence lengths and the focal lengths on the focused spectral density, the spectral degree of coherence, and the spectral degree of polarization are investigated in great detail.

© 2008 Optical Society of America

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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. H. Roychowdhury and E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57-60 (2003).
    [CrossRef]
  3. O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian schell-model source,” Opt. Lett. 29, 1173-1175 (2004).
    [CrossRef] [PubMed]
  4. H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
    [CrossRef]
  5. 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, 1078-1180 (2003).
    [CrossRef] [PubMed]
  6. Y. Li, H. Lee, and E. Wolf, “Spectra-coherence and polarization in Young's interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63-72 (2006).
    [CrossRef]
  7. J. Pu, O. Korotkova, and E. Wolf, “Invariance and non-invariance of the spectrum and of the degree of polarization of stochastic electromagnetic beams on propagation,” Opt. Lett. 31, 2097-2099 (2006).
    [CrossRef] [PubMed]
  8. H. F. Schouten, T. D. Visser, and E. Wolf, “New effect in Young's interference experiment with partially coherent light,” Opt. Lett. 28, 1182-1184 (2003).
    [CrossRef] [PubMed]
  9. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
    [CrossRef]
  10. O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
    [CrossRef]
  11. H. Roychowdhury and E. Wolf, “Young's interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268-274 (2005).
    [CrossRef]
  12. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarizaton of statistically electromagnetic fields,” Opt. Commun. 248, 333-337 (2005).
    [CrossRef]
  13. 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]
  14. O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382-2385 (2004).
    [CrossRef]
  15. J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610-1-056610-6 (2007).
    [CrossRef]
  16. P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 3, 6051-6060 (2005).
    [CrossRef]
  17. S. R. Seshadri, “Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams,” Appl. Phys. (N.Y.) 87, 4084-4093 (2000).
    [CrossRef]
  18. S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321-2325 (2000).
    [CrossRef]
  19. B. Lu and X. Ji, “Focal shift in Gaussian beams focused by a spherically aberrated bifocal lens,” Opt. Commun. 189, 47-53 (2001).
    [CrossRef]
  20. S. Sanyal and A. Ghosh, “High tolerance to spherical aberrations and defects of focus with a birefringent lens,” Appl. Opt. 41, 4611-4619 (2002).
    [CrossRef] [PubMed]
  21. X. Ji and B. Lu, “Focal switch of flattened Gaussian beams focused by a bifocal lens system,” Optik (Stuttgart) 114, 7-12 (2003).
    [CrossRef]
  22. S. Sanyal, Y. Kawata, A. Ghosh, and S. Mandal, “Frequency response characteristics of a birefringent lens with off-axis aberrations,” Appl. Opt. 43, 3838-3847 (2004).
    [CrossRef] [PubMed]
  23. X. Liu, X. Cai, S. Chang, and C.Grover, “Bifocal optical system for distant object tracking,” Opt. Express 13, 136-141 (2005).
    [CrossRef] [PubMed]
  24. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713-720 (1988).
    [CrossRef]
  25. If the focal shift can be neglected, the best focus is just at the geometrical focus; see Y. J. Li, “Focal shift in small-Fresnel-number focusing systems of different relative aperture,” J. Opt. Soc. Am. A 20, 234-239 (2003).
    [CrossRef]

2007 (1)

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610-1-056610-6 (2007).
[CrossRef]

2006 (2)

Y. Li, H. Lee, and E. Wolf, “Spectra-coherence and polarization in Young's interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63-72 (2006).
[CrossRef]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and non-invariance of the spectrum and of the degree of polarization of stochastic electromagnetic beams on propagation,” Opt. Lett. 31, 2097-2099 (2006).
[CrossRef] [PubMed]

2005 (6)

X. Liu, X. Cai, S. Chang, and C.Grover, “Bifocal optical system for distant object tracking,” Opt. Express 13, 136-141 (2005).
[CrossRef] [PubMed]

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 3, 6051-6060 (2005).
[CrossRef]

H. Roychowdhury and E. Wolf, “Young's interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268-274 (2005).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarizaton of statistically electromagnetic fields,” Opt. Commun. 248, 333-337 (2005).
[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]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

2004 (5)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian schell-model source,” Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

S. Sanyal, Y. Kawata, A. Ghosh, and S. Mandal, “Frequency response characteristics of a birefringent lens with off-axis aberrations,” Appl. Opt. 43, 3838-3847 (2004).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382-2385 (2004).
[CrossRef]

2003 (6)

2002 (1)

2001 (1)

B. Lu and X. Ji, “Focal shift in Gaussian beams focused by a spherically aberrated bifocal lens,” Opt. Commun. 189, 47-53 (2001).
[CrossRef]

2000 (2)

S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. 39, 2321-2325 (2000).
[CrossRef]

S. R. Seshadri, “Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams,” Appl. Phys. (N.Y.) 87, 4084-4093 (2000).
[CrossRef]

1988 (1)

Cai, X.

Chang, S.

Dogariu, A.

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

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

Ellis, J.

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

Friberg, A. T.

Ghosh, A.

Goudail, F.

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 3, 6051-6060 (2005).
[CrossRef]

Grover, C.

Ji, X.

X. Ji and B. Lu, “Focal switch of flattened Gaussian beams focused by a bifocal lens system,” Optik (Stuttgart) 114, 7-12 (2003).
[CrossRef]

B. Lu and X. Ji, “Focal shift in Gaussian beams focused by a spherically aberrated bifocal lens,” Opt. Commun. 189, 47-53 (2001).
[CrossRef]

Kawata, Y.

Korotkova, O.

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610-1-056610-6 (2007).
[CrossRef]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and non-invariance of the spectrum and of the degree of polarization of stochastic electromagnetic beams on propagation,” Opt. Lett. 31, 2097-2099 (2006).
[CrossRef] [PubMed]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[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]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian schell-model source,” Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382-2385 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Lee, H.

Y. Li, H. Lee, and E. Wolf, “Spectra-coherence and polarization in Young's interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63-72 (2006).
[CrossRef]

Li, Y.

Y. Li, H. Lee, and E. Wolf, “Spectra-coherence and polarization in Young's interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63-72 (2006).
[CrossRef]

Li, Y. J.

Liu, X.

Lu, B.

X. Ji and B. Lu, “Focal switch of flattened Gaussian beams focused by a bifocal lens system,” Optik (Stuttgart) 114, 7-12 (2003).
[CrossRef]

B. Lu and X. Ji, “Focal shift in Gaussian beams focused by a spherically aberrated bifocal lens,” Opt. Commun. 189, 47-53 (2001).
[CrossRef]

Mandal, S.

Ponomarenko, S.

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

Pu, J.

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610-1-056610-6 (2007).
[CrossRef]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and non-invariance of the spectrum and of the degree of polarization of stochastic electromagnetic beams on propagation,” Opt. Lett. 31, 2097-2099 (2006).
[CrossRef] [PubMed]

Réfrégier, P.

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 3, 6051-6060 (2005).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and E. Wolf, “Young's interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268-274 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

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

Salem, M.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian schell-model source,” Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

Sanyal, S.

Schouten, H. F.

Seshadri, S. R.

S. R. Seshadri, “Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams,” Appl. Phys. (N.Y.) 87, 4084-4093 (2000).
[CrossRef]

Turunen, J.

Visser, T. D.

Wolf, E.

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610-1-056610-6 (2007).
[CrossRef]

Y. Li, H. Lee, and E. Wolf, “Spectra-coherence and polarization in Young's interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63-72 (2006).
[CrossRef]

J. Pu, O. Korotkova, and E. Wolf, “Invariance and non-invariance of the spectrum and of the degree of polarization of stochastic electromagnetic beams on propagation,” Opt. Lett. 31, 2097-2099 (2006).
[CrossRef] [PubMed]

H. Roychowdhury and E. Wolf, “Young's interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268-274 (2005).
[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]

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

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian schell-model source,” Opt. Lett. 29, 1173-1175 (2004).
[CrossRef] [PubMed]

O. Korotkova and E. Wolf, “Spectral degree of coherence of a random three-dimensional electromagnetic field,” J. Opt. Soc. Am. A 21, 2382-2385 (2004).
[CrossRef]

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

H. F. Schouten, T. D. Visser, and E. Wolf, “New effect in Young's interference experiment with partially coherent light,” Opt. Lett. 28, 1182-1184 (2003).
[CrossRef] [PubMed]

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, 1078-1180 (2003).
[CrossRef] [PubMed]

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

Appl. Opt. (3)

Appl. Phys. (N.Y.) (1)

S. R. Seshadri, “Polarization properties of partially coherent Gaussian Schell-model electromagnetic beams,” Appl. Phys. (N.Y.) 87, 4084-4093 (2000).
[CrossRef]

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

Opt. Commun. (8)

Y. Li, H. Lee, and E. Wolf, “Spectra-coherence and polarization in Young's interference pattern formed by stochastic electromagnetic beams,” Opt. Commun. 265, 63-72 (2006).
[CrossRef]

B. Lu and X. Ji, “Focal shift in Gaussian beams focused by a spherically aberrated bifocal lens,” Opt. Commun. 189, 47-53 (2001).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233, 225-230 (2004).
[CrossRef]

H. Roychowdhury and E. Wolf, “Young's interference experiment with light of any state of coherence and polarization,” Opt. Commun. 252, 268-274 (2005).
[CrossRef]

J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarizaton of statistically electromagnetic fields,” Opt. Commun. 248, 333-337 (2005).
[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]

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

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249, 379-385 (2005).
[CrossRef]

Opt. Express (2)

P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 3, 6051-6060 (2005).
[CrossRef]

X. Liu, X. Cai, S. Chang, and C.Grover, “Bifocal optical system for distant object tracking,” Opt. Express 13, 136-141 (2005).
[CrossRef] [PubMed]

Opt. Lett. (4)

Optik (Stuttgart) (1)

X. Ji and B. Lu, “Focal switch of flattened Gaussian beams focused by a bifocal lens system,” Optik (Stuttgart) 114, 7-12 (2003).
[CrossRef]

Phys. Lett. A (1)

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

Phys. Rev. E (1)

J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E 75, 056610-1-056610-6 (2007).
[CrossRef]

Waves Random Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Media 14, 513-523 (2004).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Spectral density distribution in several selected planes. (a) x z plane, (b) y z plane, (c) z = 10 mm plane, (d) z = 100 mm plane, (e) z = 150 mm plane, (f) z = 300 mm plane. Here, we choose ω = 3 × 10 15 s 1 , δ x x = 0.1 mm , δ y y = 0.05 mm , σ = 0.3 mm , f x = 100 mm , f y = 300 mm , B x x = 1 , B y y = 1 3 .

Fig. 2
Fig. 2

Spectral density on the z axis. The curves in (a) are associated with different values of the parameter f x . The curves in (b) are associated with different values of the parameters δ x x and δ y y . The other parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Distribution of the spectral degree of coherence μ ( 0 , 0 , x , y , z ; ω ) in several selected planes. (a) x z plane, (b) y z plane, (c) z = 10 mm plane, (d) z = 100 mm plane, (e) z = 150 mm plane, (f) z = 300 mm plane. The other parameters are the same as in Fig. 1.

Fig. 4
Fig. 4

Spectral degree of coherence along the x axis and the y axis in the focal plane. The other parameters are the same as in Fig. 1.

Fig. 5
Fig. 5

Spectral degree of coherence along the x axis and the y axis in the focal plane for different values of δ x x and δ y y . The other parameters are the same as in Fig. 1.

Fig. 6
Fig. 6

Distribution of the spectral degree of polarization in several selected planes. (a) x z plane, (b) y z plane, (c) z = 10 mm plane, (d) z = 100 mm plane, (e) z = 150 mm plane, (f) z = 300 mm plane. Here, f x = 100 mm , f y = 100 mm , The other parameters are the same as in Fig. 1.

Fig. 7
Fig. 7

Distribution of spectral degree of polarization in several selected planes. (a) x z plane, (b) y z plane, (c) z = 10 mm plane, (d) z = 100 mm plane, (e) z = 150 mm plane, (f) z = 300 mm plane. Here, f x = 100 mm , f y = 300 mm . The other parameters are the same as in Fig. 1.

Fig. 8
Fig. 8

Spectral degree of polarization along the z axis for different values of δ x x and δ y y . The other parameters are the same as in Fig. 7.

Fig. 9
Fig. 9

Spectral degree of polarization along the z axis for different values of f x . Here, f y = 300 mm , δ x x = 1 mm , δ y y = 0.5 mm . The other parameters are the same as in Fig. 7.

Equations (18)

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

W ( 0 ) ( ρ 1 , ρ 2 , ω ) = [ W x x ( 0 ) ( ρ 1 , ρ 2 , ω ) W x y ( 0 ) ( ρ 1 , ρ 2 , ω ) W y x ( 0 ) ( ρ 1 , ρ 2 , ω ) W y y ( 0 ) ( ρ 1 , ρ 2 , ω ) ] ,
W x x ( 0 ) ( ρ 1 , ρ 2 , ω ) = S ( ω ) exp [ ρ 1 2 + ρ 2 2 4 σ 2 ] exp [ ( ρ 1 ρ 2 ) 2 2 δ x x 2 ] ,
W y y ( 0 ) ( ρ 1 , ρ 2 , ω ) = S ( ω ) B exp [ ρ 1 2 + ρ 2 2 4 σ 2 ] exp [ ( ρ 1 ρ 2 ) 2 2 δ y y 2 ] .
W x y ( 0 ) ( ρ 1 , ρ 2 , ω ) = W y x ( 0 ) ( ρ 1 , ρ 2 , ω ) = 0 ,
S ( ρ , ω ) = Tr W ( ρ , ρ , ω ) ,
μ ( ρ 1 , ρ 2 , ω ) = Tr W ( ρ 1 , ρ 2 , ω ) Tr W ( ρ 1 , ρ 1 , ω ) Tr W ( ρ 2 , ρ 2 , ω ) ,
P ( ρ , ω ) = 1 4 Det W ( ρ , ρ , ω ) [ Tr W ( ρ , ρ , , ω ) ] 2 ,
P ( 0 ) = 1 B 1 + B .
W x x ( 0 ) ( x 1 , y 1 , x 2 , y 2 , 0 , ω ) = S ( ω ) exp [ x 1 2 + y 1 2 + x 2 2 + y 2 2 4 σ 2 ] exp [ x 1 2 + y 1 2 + x 2 2 + y 2 2 2 ( x 1 x 2 + y 1 y 2 ) 2 δ x x 2 ] ,
W y y ( 0 ) ( x 1 , y 1 , x 2 , y 2 , 0 , ω ) = S ( ω ) B exp [ x 1 2 + y 1 2 + x 2 2 + y 2 2 4 σ 2 ] exp [ x 1 2 + y 1 2 + x 2 2 + y 2 2 2 ( x 1 x 2 + y 1 y 2 ) 2 δ y y 2 ] ,
W x y ( 0 ) ( x 1 , y 1 , x 2 , y 2 , 0 , ω ) = W y x ( 0 ) ( x 1 , y 1 , x 2 , y 2 , 0 , ω ) = 0 .
W i j ( x 1 , y 1 , x 2 , y 2 , z , ω ) = ( ω 2 π B c ) 2 W i j ( 0 ) ( x 1 , y 1 , x 2 , y 2 , 0 , ω ) exp ( i ω 2 B c { A x x 1 2 + A y y 1 2 2 ( x 1 x 1 + y 1 y 1 ) + D ( x 1 2 + y 1 2 ) [ A x x 2 2 + A y y 2 2 2 ( x 2 x 2 + y 2 y 2 ) + D ( x 2 2 + y 2 2 ) ] } ) d x 1 d y 1 d x 2 d y 2 ( i , j = x , y ) ,
[ A j B j C j D j ] = [ 1 z f j z 1 f j 1 ] , ( j = x , y ) .
W i j ( x 1 , y 1 , x 2 , y 2 , z , ω ) = S ( ω ) B i j ( ω 2 π c z ) 2 exp [ i ω 2 z c ( x 1 2 x 2 2 ) ] 2 π 4 a x a x * ( 1 δ i j 2 ) 2 exp [ ( i ω z c ) 2 a x * x 1 2 1 δ i j 2 x 1 x 2 + a x x 2 2 4 a x a x * ( 1 δ i j 2 ) 2 ] exp [ i ω 2 z c ( y 1 2 y 2 2 ) ] 2 π 4 a y a y * ( 1 δ i j 2 ) 2 exp [ ( i ω z c ) 2 a y * y 1 2 1 δ i j 2 y 1 y 2 + a y y 2 2 4 a y a y * ( 1 δ i j 2 ) 2 ] ,
a x = 1 4 σ 2 + 1 2 δ i j 2 + i ω ( 1 z f x ) 2 z c ,
a x * = 1 4 σ 2 + 1 2 δ i j 2 i ω ( 1 z f x ) 2 z c ,
a y = 1 4 σ 2 + 1 2 δ i j 2 + i ω ( 1 z f y ) 2 z c ,
a y * = 1 4 σ 2 + 1 2 δ i j 2 i ω ( 1 z f y ) 2 z c , ( i , j = x , y )

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