Abstract

We introduce a new class of twisted partially coherent beams with a non-uniform correlation structure. These beams, called twisted Hermite Gaussian Schell model (THGSM) beams, have a correlation structure related to Hermite functions and a twist factor in their degree of coherence. The spectral density and total average orbital angular momentum per photon of these beams strongly depend on the distortions applied to their degree of coherence. On propagation through free space, they exhibit both self-splitting and rotation of their spectral density profile, combining the interesting effects of twisted beams and non-uniformly correlated beams. We demonstrate that we can adjust both the beam order and the twist factor of THGSM beams to improve their resistance to turbulence.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  2. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [Crossref]
  3. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
    [Crossref]
  4. J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001).
    [Crossref]
  5. X. Peng, L. Liu, F. Wang, S. Popov, and Y. Cai, “Twisted Laguerre-Gaussian Schell-model beam and its orbital angular moment,” Opt. Express 26(26), 33956–33969 (2018).
    [Crossref]
  6. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
    [Crossref]
  7. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
    [Crossref]
  8. H. Wang, X. Peng, L. Liu, F. Wang, Y. Cai, and S. A. Ponomarenko, “Generating bona fide twisted Gaussian Schell-model beams,” Opt. Lett. 44(15), 3709–3712 (2019).
    [Crossref]
  9. Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
    [Crossref]
  10. Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
    [Crossref]
  11. F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
    [Crossref]
  12. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref]
  13. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
    [Crossref]
  14. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref]
  15. J. Yu, Y. Cai, and G. Gbur, “Rectangular Hermite non-uniformly correlated beams and its propagation properties,” Opt. Express 26(21), 27894–27906 (2018).
    [Crossref]
  16. Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
    [Crossref]
  17. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref]
  18. J. Yu, F. Wang, L. Liu, Y. Cai, and G. Gbur, “Propagation properties of Hermite non-uniformly correlated beams in turbulence,” Opt. Express 26(13), 16333–16343 (2018).
    [Crossref]
  19. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
    [Crossref]
  20. J. Yu, X. Zhu, F. Wang, D. Wei, G. Gbur, and Y. Cai, “Experimental study of reducing beam wander by modulating the coherence structure of structured light beams,” Opt. Lett. 44(17), 4371–4374 (2019).
    [Crossref]
  21. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
    [Crossref]
  22. Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
    [Crossref]
  23. Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42(2), 255–258 (2017).
    [Crossref]
  24. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
  25. S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86(4), 043814 (2012).
    [Crossref]
  26. G. Gbur, “Partially coherent vortex beams,” Proc. SPIE 10549, 2 (2018).
    [Crossref]
  27. G. Gbur, Singular optics (CRC, Boca Raton2016).
  28. J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
    [Crossref]

2019 (2)

2018 (4)

2017 (1)

2015 (2)

2014 (2)

2013 (3)

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

2012 (4)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[Crossref]

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref]

S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86(4), 043814 (2012).
[Crossref]

2011 (1)

2009 (2)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
[Crossref]

2007 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

2001 (1)

1994 (1)

1993 (1)

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

Baykal, Y.

Cai, Y.

J. Yu, X. Zhu, F. Wang, D. Wei, G. Gbur, and Y. Cai, “Experimental study of reducing beam wander by modulating the coherence structure of structured light beams,” Opt. Lett. 44(17), 4371–4374 (2019).
[Crossref]

H. Wang, X. Peng, L. Liu, F. Wang, Y. Cai, and S. A. Ponomarenko, “Generating bona fide twisted Gaussian Schell-model beams,” Opt. Lett. 44(15), 3709–3712 (2019).
[Crossref]

X. Peng, L. Liu, F. Wang, S. Popov, and Y. Cai, “Twisted Laguerre-Gaussian Schell-model beam and its orbital angular moment,” Opt. Express 26(26), 33956–33969 (2018).
[Crossref]

J. Yu, F. Wang, L. Liu, Y. Cai, and G. Gbur, “Propagation properties of Hermite non-uniformly correlated beams in turbulence,” Opt. Express 26(13), 16333–16343 (2018).
[Crossref]

J. Yu, Y. Cai, and G. Gbur, “Rectangular Hermite non-uniformly correlated beams and its propagation properties,” Opt. Express 26(21), 27894–27906 (2018).
[Crossref]

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

J. Yu, Y. Chen, L. Liu, X. Liu, and Y. Cai, “Splitting and combining properties of an elegant Hermite-Gaussian correlated Schell-model beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 23(10), 13467–13481 (2015).
[Crossref]

Y. Cai, Y. Chen, and F. Wang, “Generation and propagation of partially coherent beams with nonconventional correlation functions: a review,” J. Opt. Soc. Am. A 31(9), 2083–2096 (2014).
[Crossref]

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
[Crossref]

Chen, Y.

Dolinar, S.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Eyyuboglu, H. T.

Fazal, I. M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Friberg, A. T.

Gbur, G.

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

Gu, J.

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

Gu, Y.

Huang, H.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Kim, S. M.

S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86(4), 043814 (2012).
[Crossref]

Korotkova, O.

Lajunen, H.

Lavery, M. P. J.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

Lin, Q.

Liu, L.

Liu, X.

Mei, Z.

Movilla, J. M.

Mukunda, N.

Padgett, M. J.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

Peng, X.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

Ponomarenko, S. A.

Popov, S.

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Ren, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Saastamoinen, T.

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref]

Serna, J.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Simon, R.

Speirits, F. C.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

Tervonen, E.

Tong, Z.

Tur, M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Turunen, J.

Wang, F.

Wang, H.

Wang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wei, D.

Willner, A. E.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yan, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yu, J.

Yue, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zhu, X.

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

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

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

Nat. Photonics (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

Opt. Express (5)

Opt. Lett. (10)

H. Wang, X. Peng, L. Liu, F. Wang, Y. Cai, and S. A. Ponomarenko, “Generating bona fide twisted Gaussian Schell-model beams,” Opt. Lett. 44(15), 3709–3712 (2019).
[Crossref]

J. Yu, X. Zhu, F. Wang, D. Wei, G. Gbur, and Y. Cai, “Experimental study of reducing beam wander by modulating the coherence structure of structured light beams,” Opt. Lett. 44(17), 4371–4374 (2019).
[Crossref]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref]

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Twist phase-induced reduction in scintillation of a partially coherent beam in turbulent atmosphere,” Opt. Lett. 37(2), 184–186 (2012).
[Crossref]

Z. Tong and O. Korotkova, “Beyond the classical Rayleigh limit with twisted light,” Opt. Lett. 37(13), 2595–2597 (2012).
[Crossref]

Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref]

Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[Crossref]

Z. Mei and O. Korotkova, “Random sources for rotating spectral densities,” Opt. Lett. 42(2), 255–258 (2017).
[Crossref]

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref]

Phys. Rev. A (2)

Y. Chen, J. Gu, F. Wang, and Y. Cai, “Self-splitting properties of a Hermite-Gaussian correlated Schell-model beam,” Phys. Rev. A 91(1), 013823 (2015).
[Crossref]

S. M. Kim and G. Gbur, “Angular momentum conservation in partially coherent wave fields,” Phys. Rev. A 86(4), 043814 (2012).
[Crossref]

Proc. SPIE (1)

G. Gbur, “Partially coherent vortex beams,” Proc. SPIE 10549, 2 (2018).
[Crossref]

Science (1)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref]

Other (2)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

G. Gbur, Singular optics (CRC, Boca Raton2016).

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Figures (10)

Fig. 1.
Fig. 1. Density plot of the absolute value of the DOC of THGSM beams for different values of the beam order $m$ and twist factor $u$ . The position vector ${\textbf {r}}_{2}=0$ and the DOC is plotted as a function of $(x_{1},y_{1})$ .
Fig. 2.
Fig. 2. Density plot of the normalized intensity of THGSM beams for different values of the beam order $m$ and twist factor $u$ .
Fig. 3.
Fig. 3. Density plot of the normalized intensity of THGSM beams upon propagation in free space for different beam orders $m$ and twist factors $u$ .
Fig. 4.
Fig. 4. The total average OAM per photon of THGSM beams with different beam order $m$ versus (a) the transmission distance in free space; (b) the normalized factor $u \delta ^{2}_g$ in the source plane.
Fig. 5.
Fig. 5. The normalized OAM flux density of THGSM beams with different beam order $m$ versus (a) position coordinates in the source plane; (b) propagation distance at point ( $2\mathrm {cm},2\mathrm {cm}$ ) in free space.
Fig. 6.
Fig. 6. Density plot of the normalized intensity of THGSM beams upon propagation in turbulence for different beam order $m$ and twist factor $u$ .
Fig. 7.
Fig. 7. Ratio $S(0,z)/S_{\mathrm {max}}(\boldsymbol {\rho },z)$ of the spectral intensity in the optical axis to the maximum intensity in the transverse plane of THGSM beams with different (a) twist factor and (b) beam order on propagation.
Fig. 8.
Fig. 8. The total average OAM per photon of THGSM beams with different beam order $m$ versus (a) the transmission distance; (b) the normalized factor $u \delta ^{2}_g$ at $z=1\mathrm {km}$ in turbulence.
Fig. 9.
Fig. 9. The normalized OAM flux density of THGSM beams with different beam order $m$ on propagation in turbulence at point ( $2\mathrm {cm},2\mathrm {cm}$ ); (b) Degradation rate of the normalized OAM flux density in turbulence with different beam order $m$ .
Fig. 10.
Fig. 10. The normalized OAM flux density of THGSM beams with different beam order $m$ on propagation in (a) free space, and (b) in turbulence, at a propagation distance of 3 km.

Equations (21)

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

W ( r 1 , r 2 ) = E ( r 1 ) E ( r 2 ) ω ,
W ( r 1 , r 2 ) = p ( v ) V 0 ( r 1 , v ) V 0 ( r 2 , v ) d 2 v ,
p ( v ) = ( 1 / 4 π ) ( 4 a ) 2 m + 1 ( v x v y ) 2 m exp ( a v 2 ) ,
V 0 ( r , v ) = exp ( σ r 2 ) exp { [ ( a u y + i x ) v x ( a u x i y ) v y ] } .
W ( r 1 , r 2 ) = exp ( r 1 2 + r 2 2 4 ω 0 2 ) exp [ ( r 1 r 2 ) 2 2 δ g 2 ] exp [ i u ( x 1 y 2 x 2 y 1 ) ] × H 2 m [ 1 2 a ( x 1 x 2 ) + i u a 2 ( y 1 + y 2 ) ] H 2 m [ 1 2 a ( y 1 y 2 ) i u a 2 ( x 1 + x 2 ) ] ,
μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) W ( r 1 , r 1 ) W ( r 2 , r 2 ) .
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 W ( r 1 , r 2 ) exp [ i k 2 z ( r 1 ρ 1 ) 2 + i k 2 z ( r 2 ρ 2 ) 2 ] × exp [ Ψ ( r 1 , ρ 1 ) + Ψ ( r 2 , ρ 2 ) ] d 2 r 1 d 2 r 2 ,
exp [ Ψ ( r 1 , ρ 1 ) + Ψ ( r 2 , ρ 2 ) ] = exp { ( π 2 k 2 z 3 ) [ ( ρ 1 ρ 2 ) 2 + ( ρ 1 ρ 2 ) ( r 1 r 2 ) + ( r 1 r 2 ) 2 ] 0 κ 3 Φ n ( κ ) d 2 κ } ,
T = 0 κ 3 Φ n ( κ ) d 2 κ .
T = Ω ( α ) 2 ( α 2 ) C n 2 [ β κ m 2 α exp ( κ 0 2 / κ m 2 ) Γ 1 ( 2 α / 2 , κ 0 2 / κ m 2 ) 2 κ 0 4 α ] , 3 < α < 4 ,
W ( ρ 1 , ρ 2 , z ) = C 0 exp [ i k 2 z ( ρ 2 2 ρ 1 2 ) T ( ρ 1 ρ 2 ) 2 + 1 4 A 1 ( i k z ρ 1 T ( ρ 1 ρ 2 ) ) 2 ] × exp { 1 4 A 2 [ i k z ( T A 1 ρ 1 ρ 2 ) + T ( 1 T A 1 ) ( ρ 1 ρ 2 ) ] 2 + 1 J ( G + 2 + G 2 ) } ,
A 1 = σ + i k 2 z + T , A 2 = σ i k 2 z + T T 2 4 A 1 , T = π 2 k 2 z T 3 ,
J = ( T 2 A 1 A 2 1 4 A 2 ) ( a 2 u 2 + 1 ) ( T 2 4 A 1 2 A 2 + 1 4 A 1 ) ( a 2 u 2 1 ) 1 2 A 2 a 2 u 2 + a ,
G ± = 1 4 A 1 ( k z + i T ) ( ± i a u ρ 1 x ρ 1 y ) i T 4 A 1 ( ± i a u ρ 2 x ρ 2 y ) + 1 4 A 2 [ ± i a u k z ( T 2 A 1 2 ρ 1 x ρ 2 x ) k z ( T A 1 1 ) ( T A 1 ρ 1 y ρ 2 y ) ] + 1 4 A 2 [ ± a u T ( i k A 1 z T 2 A 1 2 + 1 ) ( ρ 1 x ρ 2 x ) i T ( T A 1 1 ) 2 ( ρ 1 y ρ 2 y ) ] ,
C 0 = l = 0 m d = 0 m 4 2 m l d 1 a 2 m + 1 k 2 ( 2 m ) ! ( 2 m ) ! G + 2 m 2 l G 2 m 2 d d ! l ! z 2 A 1 A 2 ( 2 m 2 l ) ! ( 2 m 2 d ) ! J 4 m d l + 1 .
S ( ρ , z ) = W ( ρ , ρ , z ) .
O ( ρ , z ) = ε 0 k I m [ ρ 1 y ρ 2 x W ( ρ 1 , ρ 2 , z ) ρ 1 x ρ 2 y W ( ρ 1 , ρ 2 , z ) ] ρ 1 = ρ 2 = ρ ,
O n ( ρ , z ) = ω O ( ρ , z ) S p ( ρ , z ) ,
O t ( ρ , z ) = ω O ( ρ , z ) d 2 ρ S p ( ρ , z ) d 2 ρ ,
S p ( ρ , z ) = k μ 0 ω W ( ρ , ρ , z ) ,
η = O n f r e e O n t u r O n f r e e × 100 % ,

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