Abstract

The effects of polarization gratings on partially coherent beams are investigated by studying a Gaussian Schell-model beam impinging on a linear polarizer whose transmission axis varies periodically along one transverse direction. Analytical expressions for the beam polarization-coherence matrix after the grating are obtained. In particular, the evolution of the degree of polarization upon propagation is analyzed. Different behaviors of the output beam, depending on the beam parameters and on the period of the grating, are exhibited. In particular, it is shown that, by suitably choosing the latter quantities, it is possible to obtain not only any desirable value of the degree of polarization of the output beam but also particular distributions of such parameters across the transverse sections of the beam.

© 2001 Optical Society of America

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References

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  1. F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
    [CrossRef]
  2. C. G. Someda, “Far field of polarization gratings,” Opt. Lett. 24, 1657–1659 (1999).
    [CrossRef]
  3. P. Rochon, V. Drnoyan, A. Natansohn, “Polarization holographic gratings in azopolymers for detecting and producing circularly polarized light,” in 1998 International Conference on Applications of Photonic Technology III: Closing the Gap Between Theory, Developments, and Applications, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 306–309 (2000).
    [CrossRef]
  4. J. Tervo, J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
    [CrossRef]
  5. F. Gori, M. Santarsiero, R. Borghi, G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
    [CrossRef]
  6. S. C. Tidwell, D. H. Ford, W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234–2239 (1990).
    [CrossRef] [PubMed]
  7. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [CrossRef]
  8. P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel-Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
    [CrossRef]
  9. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
    [CrossRef]
  10. J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
    [CrossRef]
  11. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  12. D. F. V. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
    [CrossRef]
  13. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  14. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  15. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]
  16. F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
    [CrossRef]
  17. A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
    [CrossRef]
  18. D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
    [CrossRef]
  19. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  20. F. Gori, “Mode propagation of the field generated by Collett–Wolf sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]

2000 (2)

1999 (4)

1998 (5)

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

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

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre–Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
[CrossRef]

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

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel-Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

1994 (2)

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

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

1990 (1)

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett–Wolf sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

1973 (1)

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

Agrawal, G. P.

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

Anderson, E.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

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

Drnoyan, V.

P. Rochon, V. Drnoyan, A. Natansohn, “Polarization holographic gratings in azopolymers for detecting and producing circularly polarized light,” in 1998 International Conference on Applications of Photonic Technology III: Closing the Gap Between Theory, Developments, and Applications, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 306–309 (2000).
[CrossRef]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Ford, D. H.

Goodman, J. W.

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

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, “Measuring Stokes parameters by means of a polarization grating,” Opt. Lett. 24, 584–586 (1999).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

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,” J. Eur. Opt. Soc. A Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

Greene, P. L.

Guattari, G.

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

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

Hall, D. G.

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel-Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Jaiswal, A. K.

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

James, D. F. V.

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

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

Kimura, W. D.

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Mandel, L.

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

Marti´nez-Herrero, R.

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Mehta, C. L.

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

Meji´as, P. M.

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Movilla, J. M.

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Natansohn, A.

P. Rochon, V. Drnoyan, A. Natansohn, “Polarization holographic gratings in azopolymers for detecting and producing circularly polarized light,” in 1998 International Conference on Applications of Photonic Technology III: Closing the Gap Between Theory, Developments, and Applications, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 306–309 (2000).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, R. Borghi, G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Rochon, P.

P. Rochon, V. Drnoyan, A. Natansohn, “Polarization holographic gratings in azopolymers for detecting and producing circularly polarized light,” in 1998 International Conference on Applications of Photonic Technology III: Closing the Gap Between Theory, Developments, and Applications, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 306–309 (2000).
[CrossRef]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, G. Piquero, “Use of the van Cittert–Zernike theorem for partially polarized sources,” Opt. Lett. 25, 1291–1293 (2000).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

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

Seshadri, S. R.

Someda, C. G.

Tervo, J.

Tidwell, S. C.

Tovar, A. A.

Turunen, J.

Vicalvi, S.

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

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Wolf, E.

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating surface emitting AlGaAs/GaAs quantum well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

J. Eur. Opt. Soc. A Pure Appl. Opt. (1)

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

J. Opt. Soc. Am. A (4)

Nuovo Cimento B (1)

A. K. Jaiswal, G. P. Agrawal, C. L. Mehta, “Coherence functions in the far field diffraction plane,” Nuovo Cimento B 15, 295–307 (1973).
[CrossRef]

Opt. Commun. (4)

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Opt. Lett. (5)

Other (3)

P. Rochon, V. Drnoyan, A. Natansohn, “Polarization holographic gratings in azopolymers for detecting and producing circularly polarized light,” in 1998 International Conference on Applications of Photonic Technology III: Closing the Gap Between Theory, Developments, and Applications, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 306–309 (2000).
[CrossRef]

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

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

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Figures (8)

Fig. 1
Fig. 1

Degree of polarization P for an incident fully coherent beam on axis versus z for different values of L/σI0: (a) 2, (b) 3, (c) 4, (d) 5, (e) 7, (f) 10, and (g) 30. The value of the wavelength is λ=2π×10-4 mm.

Fig. 2
Fig. 2

Degree of polarization P for an incident fully coherent beam versus z for different values of x/xL: (a) 0, (b) 0.2, (c) 0.3, (d) 0.5, (e) 0.7, (f) 1, and (g) 3. L/σI0=1 and λ=2π×10-4 mm.

Fig. 3
Fig. 3

Effect of the polarization grating on circularly polarized incident beams.

Fig. 4
Fig. 4

Superposition scheme of two polarization patterns giving rise to a (a) maximum value and (b) a minimum value of P.

Fig. 5
Fig. 5

Degree of polarization P versus x/xL for different values of z: (a) 0, (b) 0.3 m, (c) 3.2 m, (d) 6.4 m, (e) 12 m, and (f) 30 m. L/σI0=1 and λ=2π×10-4 mm.

Fig. 6
Fig. 6

Degree of polarization P at x/xL=0 versus z for different values of the degree of coherence σμ0: (a) 0.1 mm, (b) 1 mm, (c) 2 mm, (d) 3 mm, and (e) ∞. L/σI0=1 and λ=2π×10-4 mm.

Fig. 7
Fig. 7

Degree of polarization P at x/xL=0.5 versus z for different values of σμ0: (a) 0.1 mm, (b) 1 mm, (c) 2 mm, (d) 3 mm, (e) 10 mm, and (f) ∞. L/σI0=1 and λ=2π×10-4 mm.

Fig. 8
Fig. 8

Degree of polarization P for a partially coherent incident beam versus x/xL at z=10 m for different values of σμ0: (a) 0.1 mm, (b) 0.25 mm, (c) 0.5 mm, (d) 0.7 mm, (e) 1 mm, (f) 2 mm, and (g) ∞. L/σI0=1 and λ=2π×10-4 mm.

Equations (36)

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Jˆ(x1, x2, z)=Jxx(x1, x2, z)Jxy(x1, x2, z)Jyx(x1, x2, z)Jyy(x1, x2, z),
Jpq(x1, x2, z)=Ep*(x1, z; t)Eq(x2, z; t)(p, q=x, y).
Jxy(x1, x2, z)=Jyx*(x2, x1, z),
|Jpq(x1, x2, z)|2Jpp(x1, x1, z)Jqq(x2, x2, z)(p, q=x, y).
P(x, z)=[Jxx(x, x, z)-Jyy(x, x, z)]2+4|Jxy(x, x, z)|2[Jxx(x, x, z)+Jyy(x, x, z)]21/2.
Jeq(x1, x2, z)=Jxx(x1, x2, z)+Jyy(x1, x2, z).
I(x, z)=Jeq(x, x, z).
Jˆs(x1, x2)=Jsc(x1, x2, 0)1001,
Jsc(x1, x2, 0)=I0 exp-x12+x224σI02exp-(x1-x2)22σμ02,
Jˆ(x1, x2, 0)=Jsc(x1, x2, 0)C12C1S1C1S1S12×1001C22C2S2C2S2S22,
Cj=cos(γxj),Sj=sin(γxj)(j=1, 2),
Jxx(x1, x2, 0)=12Jsc(x1, x2, 0)cos[γ(x1-x2)]×{cos[γ(x1-x2)]+cos[γ(x1+x2)]},
Jyy(x1, x2, 0)=12Jsc(x1, x2, 0)cos[γ(x1-x2)]×{cos[γ(x1-x2)]-cos[γ(x1+x2)]},
Jxy(x1, x2, 0)=12Jsc(x1, x2, 0)cos[γ(x1-x2)]×{sin[γ(x1+x2)]-sin[γ(x1-x2)]}.
Jpq(x1, x2, z)=Jpq(ξ1, ξ2, 0)K*(x1, ξ1, z)×K(x2, ξ2, z)dξ1dξ2(p, q=x, y),
K(x, ξ, z)=-iλz expi 2πzλexpiπλz(x-ξ)2,
Jˆ (s, t, z)=Jsc(s, t, z)hxx(s, t, z)hxy(s, t, z)hxy*(s, t, z)hyy(s, t, z),
Jsc(s, t, z)=Jsc(x1, x2, z),
Jsc(s, t, z)=I0 exp(-iδzst)Fz expi δzstFz2exp-(αs2+βt2)Fz2
Fz2=1+(λz/π)24σI02 14σI02+1σμ02,
α=12σI02,
β=18σI02+12σμ02,
δz=2πλz.
hxx(s, t, z)=14+18 exp-4αγ2δz2Fz2G+4αγsδzFz2, 2γtFz2+exp-(α+4β)γ2δz2Fz2×exp2iδzγs-4βγtδzFz2×G+2αγsδzFz2, (δzt+2γ)γδzFz2+exp-2iγδzs+4βγtδzFz2×G+2αγsδzFz2, (δzt-2γ)γδzFz2,
hyy(s, t, z)=14+18 exp-4 αγ2δz2Fz2G+4αγsδzFz2, 2γtFz2-exp-(α+4β)γ2δz2Fz2×exp2iγδzs-4βγtδzFz2×G+2αγsδzFz2, (δzt+2γ)γδzFz2+exp-2iγδzs+4βγtδzFz2×G+2αγsδzFz2, (δzt-2γ)γδzFz2,
hxy(s, t, z)=18i exp-(α+4β)γ2δz2Fz2×exp2iγδzs-4βγtδzFz2×G+2αγsδzFz2, (δzt+2γ)γδzFz2-exp-2iγδzs+4βγtδzFz2×G+2αγsδzFz2, (δzt-2γ)γδzFz2-exp-4αγ2δz2Fz2G-4αγsδzFz2, 2γtFz2,
G±(x, y)=exp(-x)exp(iy)±exp(x)exp(-iy).
Jeq(s, t, z)=Jxx(s, t, z)+Jyy(s, t, z)=Jsc(s, t, z)12+14 exp-4αγ2δz2Fz2×G+4αγsδzFz2, 2γtFz2,
Ieq(x, z)=Jeq(x, 0, z)=I02Fz exp-αx2Fz21+exp-4αγ2δz2Fz2×cosh4αγxδzFz2.
P(x, z)=[hxx(x, 0, z)-hyy(x, 0, z)]2+4|hxy(x, 0, z)|2[hxx(x, 0, z)+hyy(x, 0, z)]21/2.
hxx()(θs, θt)=14+12 exp-(α+4β)γ24αβ×coshγπθsβλcosh2γπθtαλ+14 exp-γ2βcosh2γπθsβλ
hyy()(θs, θt)=14-12 exp-(α+4β)γ24αβ×coshγπθsβλcosh2γπθtαλ+14 exp-γ2βcosh2γπθsβλ
hxy()(θs, θt)=14i exp-γ2βsinh2γπθsβλ-2 exp-(α+4β)γ24αβ×sinh2γπθtαλcoshγπθsβλ.
P(θ)=4 expγ2(3α-4β)2αβcosh2γπθβλ+sinh22γπθβλ1/2expγ2β+cosh2γπθβλ.
Jˆ(s, t, z)=2Jscinc(s, t, z)×1+exp-2π2σI02L2001-exp-2π2σI02L2,
P=exp-2π2σI02L2.

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