Abstract

We introduce theoretically and realize experimentally a class of random, wide-sense stationary optical beams with uniform correlations which, on propagation in free space, produce a crescent-like intensity distribution with the maximum at an off-axis position. The crescent’s position of maximum intensity accelerates transversally at intermediate distances, and then exhibits a constant lateral shift further from the axis in the far zone of the source. We also show that on propagation in the isotropic turbulent atmosphere, the crescent beam shifts away from the axis as well, but slower than in free space, with rate depending on the strength of turbulence. These results are of importance for optical systems operating through long-range turbulent channels in which a beam must have a range-dependent tilt, e.g. on travelling around an obstacle.

© 2017 Optical Society of America

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References

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  1. M. V. Berry and N. L. Balázs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [PubMed]
  3. M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15(25), 16719–16728 (2007).
    [PubMed]
  4. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
    [PubMed]
  5. P. Vaveliuk, A. Lencina, J. A. Rodrigo, and O. Martinez Matos, “Symmetric Airy beams,” Opt. Lett. 39(8), 2370–2373 (2014).
    [PubMed]
  6. B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Super-Airy beam: self-accelerating beam with intensified main lobe,” Opt. Lett. 40(20), 4703–4706 (2015).
    [PubMed]
  7. Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010).
    [PubMed]
  8. X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21(2), 2154–2164 (2013).
    [PubMed]
  9. W. Nelson, J. P. Palastro, C. C. Davis, and P. Sprangle, “Propagation of Bessel and Airy beams through atmospheric turbulence,” J. Opt. Soc. Am. A 31(3), 603–609 (2014).
    [PubMed]
  10. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
    [PubMed]
  11. O. Korotkova, L. C. Andrews, and R. L. Phillips, “A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in LaserCom,” Opt. Eng. 43, 330–341 (2004).
  12. H. T. Eyyuboglu and E. Sermutlu, “Partially coherent Airy beam and its propagation in turbulent atmosphere,” Appl. Phys. B 110, 451–457 (2013).
  13. W. Wen and X. Chu, “Beam wander of partially coherent Airy beams,” J. Mod. Opt. 61, 379–384 (2014).
  14. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [PubMed]
  15. Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated random media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [PubMed]
  16. 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).
  17. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [PubMed]
  18. M. W. Hyde, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).
  19. F. Wang and O. Korotkova, “Random sources for beams with azimuthal intensity variation,” Opt. Lett. 41(3), 516–519 (2016).
    [PubMed]
  20. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).
  21. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [PubMed]
  22. F. Wang and O. Korotkova, “Convolution approach for beam propagation in random media,” Opt. Lett. 41(7), 1546–1549 (2016).
    [PubMed]
  23. F. Wang and O. Korotkova, “Random sources for beams with azimuthally varying polarization properties,” Opt. Express 24(14), 15446–15455 (2016).
    [PubMed]
  24. C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell-model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).
  25. D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
    [PubMed]
  26. M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).
  27. L. C. Andrews and R. L. Phillips, Laser beam propagation in turbulent Atmosphere, 2nd edition, SPIE press, Bellington, 2005.

2016 (4)

2015 (4)

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[PubMed]

B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Super-Airy beam: self-accelerating beam with intensified main lobe,” Opt. Lett. 40(20), 4703–4706 (2015).
[PubMed]

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

M. W. Hyde, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

2014 (5)

2013 (2)

H. T. Eyyuboglu and E. Sermutlu, “Partially coherent Airy beam and its propagation in turbulent atmosphere,” Appl. Phys. B 110, 451–457 (2013).

X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21(2), 2154–2164 (2013).
[PubMed]

2012 (1)

2011 (1)

2010 (2)

2007 (3)

2004 (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in LaserCom,” Opt. Eng. 43, 330–341 (2004).

2002 (1)

1979 (1)

M. V. Berry and N. L. Balázs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in LaserCom,” Opt. Eng. 43, 330–341 (2004).

Arie, A.

Avramov-Zamurovic, S.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell-model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).

Balázs, N. L.

M. V. Berry and N. L. Balázs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).

Bandres, M. A.

Basu, S.

M. W. Hyde, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

Berry, M. V.

M. V. Berry and N. L. Balázs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[PubMed]

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).

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[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).

Christodoulides, D. N.

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[PubMed]

Chu, X.

W. Wen and X. Chu, “Beam wander of partially coherent Airy beams,” J. Mod. Opt. 61, 379–384 (2014).

Davis, C. C.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[PubMed]

Efremidis, N. K.

Eyyuboglu, H. T.

X. Ji, H. T. Eyyuboğlu, G. Ji, and X. Jia, “Propagation of an Airy beam through the atmosphere,” Opt. Express 21(2), 2154–2164 (2013).
[PubMed]

H. T. Eyyuboglu and E. Sermutlu, “Partially coherent Airy beam and its propagation in turbulent atmosphere,” Appl. Phys. B 110, 451–457 (2013).

Gbur, G.

Gori, F.

Gu, Y.

Guth, S.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell-model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).

Gutiérrez-Vega, J. C.

Hyde, M. W.

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

M. W. Hyde, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

Ji, G.

Ji, X.

Jia, X.

Korotkova, O.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell-model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).

F. Wang and O. Korotkova, “Random sources for beams with azimuthal intensity variation,” Opt. Lett. 41(3), 516–519 (2016).
[PubMed]

F. Wang and O. Korotkova, “Convolution approach for beam propagation in random media,” Opt. Lett. 41(7), 1546–1549 (2016).
[PubMed]

F. Wang and O. Korotkova, “Random sources for beams with azimuthally varying polarization properties,” Opt. Express 24(14), 15446–15455 (2016).
[PubMed]

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[PubMed]

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).

Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated random media,” Opt. Lett. 37(15), 3240–3242 (2012).
[PubMed]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in LaserCom,” Opt. Eng. 43, 330–341 (2004).

Lajunen, H.

Lencina, A.

Liang, C.

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).

Liu, X.

Malek-Madani, R.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell-model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).

Martinez Matos, O.

Nelson, C.

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell-model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).

Nelson, W.

Palastro, J. P.

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in LaserCom,” Opt. Eng. 43, 330–341 (2004).

Remez, R.

Rodrigo, J. A.

Saastamoinen, T.

Santarsiero, M.

Sermutlu, E.

H. T. Eyyuboglu and E. Sermutlu, “Partially coherent Airy beam and its propagation in turbulent atmosphere,” Appl. Phys. B 110, 451–457 (2013).

Singh, B. K.

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[PubMed]

Sprangle, P.

Tong, Z.

Tsur, Y.

Vaveliuk, P.

Voelz, D.

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[PubMed]

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

Voelz, D. G.

M. W. Hyde, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

Wang, F.

Wen, W.

W. Wen and X. Chu, “Beam wander of partially coherent Airy beams,” J. Mod. Opt. 61, 379–384 (2014).

Wolf, E.

Xiao, X.

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

M. W. Hyde, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[PubMed]

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).

Am. J. Phys. (1)

M. V. Berry and N. L. Balázs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).

Appl. Phys. B (1)

H. T. Eyyuboglu and E. Sermutlu, “Partially coherent Airy beam and its propagation in turbulent atmosphere,” Appl. Phys. B 110, 451–457 (2013).

J. Mod. Opt. (1)

W. Wen and X. Chu, “Beam wander of partially coherent Airy beams,” J. Mod. Opt. 61, 379–384 (2014).

J. Opt. (2)

M. W. Hyde, S. Basu, X. Xiao, and D. G. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

M. W. Hyde, S. Basu, X. Xiao, and D. Voelz, “Producing any desired far-field mean irradiance pattern using a partially-coherent Schell-model source,” J. Opt. 17, 055607 (2015).

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

Opt. Commun. (1)

C. Nelson, S. Avramov-Zamurovic, O. Korotkova, S. Guth, and R. Malek-Madani, “Scintillation reduction in pseudo Multi-Gaussian Schell-model beams in the maritime environment,” Opt. Commun. 364, 145–149 (2016).

Opt. Eng. (1)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in LaserCom,” Opt. Eng. 43, 330–341 (2004).

Opt. Express (3)

Opt. Lett. (11)

C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[PubMed]

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

Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010).
[PubMed]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[PubMed]

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

Z. Tong and O. Korotkova, “Non-uniformly correlated beams in uniformly correlated random media,” Opt. Lett. 37(15), 3240–3242 (2012).
[PubMed]

P. Vaveliuk, A. Lencina, J. A. Rodrigo, and O. Martinez Matos, “Symmetric Airy beams,” Opt. Lett. 39(8), 2370–2373 (2014).
[PubMed]

D. Voelz, X. Xiao, and O. Korotkova, “Numerical modeling of Schell-model beams with arbitrary far-field patterns,” Opt. Lett. 40(3), 352–355 (2015).
[PubMed]

B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Super-Airy beam: self-accelerating beam with intensified main lobe,” Opt. Lett. 40(20), 4703–4706 (2015).
[PubMed]

F. Wang and O. Korotkova, “Random sources for beams with azimuthal intensity variation,” Opt. Lett. 41(3), 516–519 (2016).
[PubMed]

F. Wang and O. Korotkova, “Convolution approach for beam propagation in random media,” Opt. Lett. 41(7), 1546–1549 (2016).
[PubMed]

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).

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[PubMed]

Other (2)

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge U. Press, 2007).

L. C. Andrews and R. L. Phillips, Laser beam propagation in turbulent Atmosphere, 2nd edition, SPIE press, Bellington, 2005.

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Figures (9)

Fig. 1
Fig. 1

Spectral density of the random beam in free space for several propagation distances.

Fig. 2
Fig. 2

Schematic diagram of the experimental generation of the crescent beam in lab. LD: Laser Diode; NDF: Natural Density Filter; SLM: Spatial Light Modulator; LP: Linear Polarizer. The distances between LD and SLM, SLM and lens, lens and iris are 30cm, 30cm and 26cm, respectively. The CMOS camera is placed 4cm behind the iris to capture the instantaneous intensity profile of the crescent beam.

Fig. 3
Fig. 3

Normalized spectral density profiles of the crescent beam propagating through thin lens (focal length f = 30cm). The coherence length of the beam is kept uniform as δ = 2.4mm, and the index number are chosen as (a) n = 0, (b) n = 2, (c) n = 3, (d) n = 10.

Fig. 4
Fig. 4

Normalized spectral density profiles of the Crescent Beam propagating through thin lens (focal length f = 30cm). The index number is kept uniform as n = 3, and the coherence lengths are chosen as (a) δ = 4.8mm, (b) δ = 2.4mm, (c) δ = 1.2mm.

Fig. 5
Fig. 5

Spectral density of the random beam propagation through atmospheric turbulence for several propagation distances. The structure parameter of the turbulence is C n 2 =5× 10 13 m 2/3 .

Fig. 6
Fig. 6

(a) Spectral density of the random beam at the cross-line ( ρ y =0) with the propagation distance z = 5.0km for three different values of structure parameters. (b) The corresponding spectral density shown in Fig. 6(a) with ρx in the range from −1.0m to 0.2m.

Fig. 7
Fig. 7

Variation of the position of the intensity maxima of the crescent beam with the propagation distance for (a) different strength of turbulence, (b) different index number n and (c) different coherence width δ.

Fig. 8
Fig. 8

Dependence of the w(z) on the propagation distance for three different strengths of turbulence.

Fig. 9
Fig. 9

The normalized intensity maxima ρ m /z versus propagation distance in different strengths of turbulence.

Equations (18)

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

W (0) ( r 1 , r 2 )= p 1 ( v 1 ) H * ( r 1 , v 1 )H( r 2 , v 1 ) d 2 v 1 ,
H(r, v 1 )=τ(r)exp(ikr v 1 ),
τ(r)=exp( r 2 /2 σ 0 2 ),
W (0) ( r 1 , r 2 )= τ * ( r 1 )τ( r 2 )μ( r 2 r 1 ).
p 1 ( v 1 )= k 2 δ 2 (kδ v 1 ) 2n 2 n n!π exp( k 2 δ 2 v 1 2 2 ) cos 2 ( θ/2 ),
μ( r 2 r 1 )= L n 0 ( r d 2 2 δ 2 )exp( r d 2 2 δ 2 ) i 2 (n1/2)! 2n! r d δ exp( r d 2 2 δ 2 ) L n1/2 1 ( r d 2 2 δ 2 )cos φ d ,
S(ρ,z)= S f (ρ/z) p 1 (ρ/z;ω),
S f (ρ/z)=| A ˜ (ρ/z) | 2 / λ 2 z 2 ,
A(r)=τ(r)exp(ik r 2 /2z).
S(ρ,z)= 1 Q 3 exp( ρ 2 σ 0 2 Q 3 ){ ( Q 2 Q 3 ) n L n 0 ( 2 z 2 ρ 2 k 2 σ 0 4 δ 2 Q 3 Q 2 ) (n1/2)! n! 2 zρ k σ 0 2 δ Q 2 ( Q 2 Q 3 ) n+1/2 L n1/2 1 ( 2 z 2 ρ 2 k 2 σ 0 4 δ 2 Q 3 Q 2 )cosϕ },
Q 2 =1+ z 2 k 2 σ 0 4 , Q 3 = Q 2 +2 z 2 / k 2 σ 0 2 δ 2 .
Φ n (κ)=0.033 C n 2 ( κ 2 + κ 0 2 ) 11/6 exp( κ 2 / κ m 2 ),
S(ρ,z)= S f (ρ/z) p 1 (ρ/z) p 2 (ρ/z),
p 2 ( v 2 )= 3 2 π 3 Tz exp( 3 v 2 2 2 π 2 Tz ).
S(ρ,z)= 1 Q 3 T exp( ρ 2 σ 0 2 Q 3 T ){ ( Q 2 T Q 3 T ) n L n 0 ( 2 z 2 ρ 2 k 2 σ 0 4 δ 2 Q 3 T Q 2 T ) (n1/2)! n! 2 zρ k σ 0 2 δ Q 2 T ( Q 2 T Q 3 T ) n+1/2 L n1/2 1 ( 2 z 2 ρ 2 k 2 σ 0 4 δ 2 Q 2 T Q 3 T )cosϕ },
Q 2 T =1+ z 2 k 2 σ 0 4 + 2 π 2 T z 3 3 σ 0 2 , Q 3 T = Q 2 T +2 z 2 / k 2 σ 0 2 δ 2 .
S(u)= 1 Q 2 T exp( u 2 σ 0 2 Q 2 T / z 2 ),
w(z)= σ 0 z Q 2 T = σ 0 1 k 2 σ 0 4 + 1 z 2 + 2 π 2 Tz 3 σ 0 2 .

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