Abstract

We introduce a new class of non-uniformly correlated beams that are called rectangular Hermite non-uniformly correlated (RHNUC) beams, which possess rectangular symmetry in their degree of coherence. It is shown that, in free space and in turbulence, these beams possess self-focusing properties and that the position of the focus can be adjusted in 3-D space by manipulating the correlation properties of the source. Furthermore, it is demonstrated that, by choosing different mode orders and correlation lengths along two transverse directions, one creates astigmatic beams that can be designed to have a high and near-constant intensity over an extended propagation range.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  2. G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus,” Opt. Lett. 28(18), 1627–1629 (2003).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  5. F. Gori, V. R. Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
    [Crossref]
  6. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [Crossref] [PubMed]
  7. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [Crossref] [PubMed]
  8. 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]
  9. L. Ma and S. A. Ponomarenko, “Free-space propagation of optical coherence lattices and periodicity reciprocity,” Opt. Express 23(2), 1848–1856 (2015).
    [Crossref] [PubMed]
  10. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  11. D. Wu, F. Wang, and Y. Cai, “High-order nonuniformly correlated beams,” Opt. Laser Technol. 99, 230–237 (2018).
    [Crossref]
  12. 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] [PubMed]
  13. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
    [Crossref]
  14. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31(9), 2038–2045 (2014).
    [Crossref]
  15. Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam wander of partially coherent array beams through non-Kolmogorov turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
    [Crossref] [PubMed]
  16. F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
    [Crossref]
  17. F. Wang, Y. Cai, H. T. Eyyuboğlu, 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] [PubMed]
  18. X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
    [Crossref] [PubMed]
  19. J. Wang, H. Wang, S. Zhu, and Z. Li, “Second-order moments of a twisted Gaussian Schell-model beam in anisotropic turbulence,” J. Opt. Soc. Am. A 35(7), 1173–1179 (2018).
    [Crossref]
  20. X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
    [Crossref] [PubMed]
  21. 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] [PubMed]
  22. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
    [Crossref] [PubMed]
  23. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
    [Crossref] [PubMed]
  24. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [Crossref]
  25. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
    [Crossref]
  26. C. Ding, M. Koivurova, J. Turunen, and L. Pan, “Self-focusing of a partially coherent beam with circular coherence,” J. Opt. Soc. Am. A 34(8), 1441–1447 (2017).
    [Crossref]

2018 (3)

2017 (1)

2016 (1)

2015 (4)

2014 (1)

2013 (5)

2012 (2)

2011 (1)

2009 (1)

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

2008 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

2007 (2)

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

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

2003 (1)

2001 (1)

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

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

Baykal, Y.

Cai, Y.

D. Wu, F. Wang, and Y. Cai, “High-order nonuniformly correlated beams,” Opt. Laser Technol. 99, 230–237 (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] [PubMed]

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

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] [PubMed]

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]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[Crossref] [PubMed]

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

F. Wang, Y. Cai, H. T. Eyyuboğlu, 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] [PubMed]

Chen, Y.

Ding, C.

Eyyuboglu, H. T.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Gao, Z.

Gbur, G.

Gori, F.

F. Gori, V. R. 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] [PubMed]

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, Y.

Koivurova, M.

Korotkova, O.

Lajunen, H.

Li, Z.

Liu, L.

Liu, X.

Ma, L.

Mandel, L.

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

Mei, Z.

Pan, L.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

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

Ponomarenko, S. A.

Saastamoinen, T.

Sahin, S.

Sánchez, V. R.

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

Santarsiero, M.

F. Gori, V. R. 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] [PubMed]

Shen, Y.

Shirai, T.

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

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Turunen, J.

Visser, T. D.

Wang, F.

Wang, H.

Wang, J.

Wolf, E.

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

Wu, D.

D. Wu, F. Wang, and Y. Cai, “High-order nonuniformly correlated beams,” Opt. Laser Technol. 99, 230–237 (2018).
[Crossref]

Yu, J.

Yuan, Y.

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

Zeng, A.

Zhang, B.

Zhu, S.

Appl. Phys. Lett. (1)

F. Wang, X. Liu, L. Liu, Y. Yuan, and Y. Cai, “Experimental study of the scintillation index of a radially polarized beam with controllable spatial coherence,” Appl. Phys. Lett. 103(9), 091102 (2013).
[Crossref]

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

F. Gori, V. R. 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)

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

D. Wu, F. Wang, and Y. Cai, “High-order nonuniformly correlated beams,” Opt. Laser Technol. 99, 230–237 (2018).
[Crossref]

Opt. Lett. (11)

X. Liu, J. Yu, Y. Cai, and S. A. Ponomarenko, “Propagation of optical coherence lattices in the turbulent atmosphere,” Opt. Lett. 41(18), 4182–4185 (2016).
[Crossref] [PubMed]

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

F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[Crossref] [PubMed]

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

Y. Huang, A. Zeng, Z. Gao, and B. Zhang, “Beam wander of partially coherent array beams through non-Kolmogorov turbulence,” Opt. Lett. 40(8), 1619–1622 (2015).
[Crossref] [PubMed]

F. Wang, Y. Cai, H. T. Eyyuboğlu, 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] [PubMed]

X. Liu, Y. Shen, L. Liu, F. Wang, and Y. Cai, “Experimental demonstration of vortex phase-induced reduction in scintillation of a partially coherent beam,” Opt. Lett. 38(24), 5323–5326 (2013).
[Crossref] [PubMed]

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

G. Gbur and T. D. Visser, “Can spatial coherence effects produce a local minimum of intensity at focus,” Opt. Lett. 28(18), 1627–1629 (2003).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

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

Phys. Rev. A (1)

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]

Proc. SPIE (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[Crossref]

Other (2)

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

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

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Figures (11)

Fig. 1
Fig. 1 Density plot of the absolute value of the DOC of RHNUC beams for different beam orders with x0=y0=0 and different shift parameters with m = n = 2 (a) in the x1x2 plane with y1 = y2 = 0 (b) in the x1y1 plane with x2 = y2 = 0.
Fig. 2
Fig. 2 The normalized intensity of RHNUC beams at different distances for different values of the beam orders and the correlation lengths with no shifting (a) m = n = 0, rcx = rcy = 3 cm (b) m = n = 2, rcx = rcy = 3 cm (c) m = n = 2, rcx = rcy = 4 cm.
Fig. 3
Fig. 3 Normalized intensity on-axis of RHNUC beams propagation in free space (a) with rcx = rcy = 3 cm for different beam orders (b) with m = n = 2 for different correlation lengths.
Fig. 4
Fig. 4 The astigmatic normalized intensity of RHNUC beams at different distances for different values of the beam orders and the correlation lengths with no shifting (a)m = 2, n = 0, rcx = rcy = 3 cm (b) m = n = 2, rcx = 5 cm, rcy = 3 cm.
Fig. 5
Fig. 5 (a) Focal distance (b) Normalized intensity maximum in x (y) direction of RHNUC beams versus the correlation lengths for different beam orders.
Fig. 6
Fig. 6 Normalized intensity of astigmatic RHNUC beams propagation in free space for different beam orders and correlation lengths. Sx and Sy are the normalized intensities in x and y directions, respectively.
Fig. 7
Fig. 7 The normalized intensity of RHNUC beams at different distances in turbulence for different beam parameters with no shifting (a) m = n = 0, rcx = rcy = 3 cm (b) m = n = 2, rcx = rcy = 3 cm (c) m = n = 2, rcx = rcy = 4 cm.
Fig. 8
Fig. 8 Normalized intensity of astigmatic RHNUC beams propagation in turbulence for different beam orders and correlation lengths. Sx and Sy are the normalized intensities in x and y directions, respectively.
Fig. 9
Fig. 9 Density plot of the absolute value of the DOC of RHNUC beams for different beam orders and correlation lengths with no shifting in free space (a) m = n = 2, rcx = rcy = 3 cm; and in turbulence (b) m = n = 2, rcx = rcy = 3 cm (c) m = n = 0, rcx = rcy = 3 cm (d) m = n = 2, rcx = rcy = 4 cm.
Fig. 10
Fig. 10 The normalized intensity of RHNUC beams in free space with m = n = 2, rcx = rcy = 3 cm (a) at several propagation distances with x0 = y0 = 2 cm; (b) with different values of the shift parameters at z = 1.2 km.
Fig. 11
Fig. 11 (a) Shift distance of the intensity maximum of RHNUC beams versus the propagation distance in free space (solid line) and in turbulence (dash line); and (b) degradation rate of shift distance of the intensity maximum of the beam in turbulence with different shift parameters.

Equations (26)

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W ( r 1 , r 2 ) = E * ( r 1 ) E ( r 2 ) ω ,
W ( r 1 , r 2 ) = S ( r 1 ) S ( r 2 ) μ ( r 1 , r 2 ) ,
W ( r 1 , r 2 ) = exp ( r 1 2 + r 2 2 w 0 2 ) μ ( r 1 , r 2 ) ,
μ ( r 1 , r 2 ) = G 0 H 2 m ( r 2 2 r 1 2 r c 2 ) exp [ ( r 2 2 r 1 2 ) r c 4 ] ,
S ( r ) = exp ( r 2 / w 0 2 ) ,
μ ( r 1 , r 2 ) = μ x ( x 1 , x 2 ) μ y ( y 1 , y 2 ) = G 0 x H 2 m [ ( x 1 x 0 ) 2 ( x 2 x 0 ) 2 r c x 2 ] exp [ ( ( x 1 x 0 ) 2 ( x 2 x 0 ) 2 ) 2 r c x 4 ] × G 0 y H 2 n [ ( y 1 y 0 ) 2 ( y 2 y 0 ) 2 r c y 2 ] exp [ ( ( y 1 y 0 ) 2 ( y 2 y 0 ) 2 ) 2 r c y 4 ] ,
W ( r 1 , r 2 ) = I ( v ) V 0 * ( r 1 , v ) V 0 ( r 2 , v ) d 2 v ,
W ( r 1 , r 2 ) = W i ( v 1 , v 2 ) V 0 * ( r 1 , v 1 ) V 0 ( r 2 , v 2 ) d 2 v 1 d 2 v 2 ,
W i ( v 1 , v 2 ) = I ( v 1 ) I ( v 2 ) δ ( v 1 , v 2 ) ,
I ( v ) = ( 4 π ) 1 ( 2 a ) 2 m + 1 ( 2 b ) 2 n + 1 v x 2 m v y 2 n exp ( v x 2 a 2 v y 2 b 2 ) ,
V 0 ( r , v ) = exp ( r 2 w 0 2 ) exp [ i k v x ( x x 0 ) 2 i k v y ( y y 0 ) 2 ] ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 W 0 ( 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 .
Φ n ( κ ) = A ( α ) C n 2 ( κ 2 + κ 0 2 ) α / 2 exp ( κ 2 / κ m 2 ) ,
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( α π / 2 ) , c ( α ) = [ 2 π A ( α ) 3 Γ ( 5 α / 2 ) ] 1 / ( α 5 ) ,
T = A ( 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 ) = I ( v ) P ( ρ 1 , ρ 2 , v , z ) d 2 v ,
P ( ρ 1 , ρ 2 , v , z ) = ( k 2 π z ) 2 V 0 * ( r 1 , v ) V 0 ( r 2 , v ) exp [ i k 2 z ( r 1 ρ 1 ) 2 + i k 2 z ( r 2 ρ 2 ) 2 ] exp { π 2 k 2 z T 3 [ ( ρ 1 ρ 2 ) 2 + ( ρ 1 ρ 2 ) ( r 1 r 2 ) + ( r 1 r 2 ) 2 ] } d 2 r 1 d 2 r 2 .
P ( ρ 1 , ρ 2 , v , z ) = exp [ i k 2 z ( ρ 1 2 ρ 2 2 ) ] exp [ ( w 0 2 k 2 8 z 2 + 1 3 π 2 k 2 z T ) ( ρ 1 ρ 2 ) 2 ] w 0 2 2 w z x w z y × exp { 1 w z x | i [ k w 0 2 4 z ( 1 2 v x z ) 1 3 π 2 k z 2 T ] ( ρ 1 x ρ 2 x ) + ( ρ 1 x + ρ 2 x 2 ) 2 x v x x 0 | 2 } × exp { 1 w z y | i [ k w 0 2 4 z ( 1 2 v y z ) 1 3 π 2 k z 2 T ] ( ρ 1 y ρ 2 y ) + ( ρ 1 y + ρ 2 y 2 ) 2 x v y x 0 | 2 } ,
w z x 2 = w 0 2 2 ( 1 2 v x z ) 2 + ( 2 z k w 0 ) 2 + 4 π 2 z 3 3 T ,
w z y 2 = w 0 2 2 ( 1 2 v y z ) 2 + ( 2 z k w 0 ) 2 + 4 π 2 z 3 3 T .
S ( ρ , z ) = W ( ρ , ρ , z ) .
μ ( ρ 1 , ρ 2 , z ) = W ( ρ 1 , ρ 2 , z ) W ( ρ 1 , ρ 1 , z ) W ( ρ 2 , ρ 2 , z ) .
S D = ρ x 2 + ρ y 2 ,
η = | S D t S D f | S D f × 100 % ,

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