Abstract

The average intensity of the multiple Bessel Gaussian beams (mBGBs), which comprise the summation of the Bessel function and a Gaussian function, are investigated based on the extended Huygens-Fresnel principle and the Rytov theory. The weak turbulence just leads the mBGBs diverge and has no influence on the angular distribution of both the mean field and the average intensity. Therefore, the angular distribution of the average intensity depends on the average in the free space. When the order difference between any two sub beams of the mBGBs is the integer multiple of the minimum order difference, there are the symmetric side lobes of the average intensity distribution and its angular frequency is equal to the minimum order difference. Moreover, for the mBGBs with two sub beams, the initial phase change of the different sub beams could make the average intensity distribution rotate in opposite direction. This paper provides the theoretical basis for the investigation of the mBGBs propagation and the application of the sub beam detection and the beam multiplexing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

W. Wanjun, W. Zhensen, S. Qingchao, and B. Lu, “Propagation of Bessel Gaussian beams through non-Kolmogorov turbulence based on Rytov theory,” Opt. Express 26(17), 21712–21724 (2018).
[Crossref] [PubMed]

L. Tang, H. Wang, X. Zhang, and S. Zhu, “Propagation properties of partially coherent Lommel beams in non-Kolmogorov turbulence,” Opt. Commun. 427, 79–84 (2018).
[Crossref]

2017 (1)

2016 (3)

2015 (3)

A. A. Kovalev and V. V. Kotlyar, “Lommel modes,” Opt. Commun. 338, 117–122 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

P. Birch, I. Ituen, R. Young, and C. Chatwin, “Long-distance Bessel beam propagation through Kolmogorov turbulence,” J. Opt. Soc. Am. A 32(11), 2066–2073 (2015).
[Crossref] [PubMed]

2014 (2)

2012 (2)

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108(17), 173604 (2012).
[Crossref] [PubMed]

2009 (1)

C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009).
[Crossref]

2008 (2)

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008).
[Crossref]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

2007 (3)

H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88(2), 259–265 (2007).
[Crossref]

Y. Cai and X. Lü, “Propagation of Bessel and Bessel-Gaussian beams through an unapertured or apertured misaligned paraxial optical systems,” Opt. Commun. 274(1), 1–7 (2007).
[Crossref]

X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007).
[Crossref]

2005 (1)

L. Wang, X. Wang, and B. Lü, “Propagation properties of partially coherent modified Bessel–Gauss beams,” Optik (Stuttg.) 116(2), 65–70 (2005).
[Crossref]

2003 (1)

2001 (1)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[Crossref]

1994 (1)

1992 (1)

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
[Crossref]

1988 (1)

1987 (2)

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Andrews, L. C.

L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992).
[Crossref]

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[Crossref]

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Bao-Suan, C.

C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009).
[Crossref]

Baykal, Y.

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Belafhal, A.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

Birch, P.

Boufalah, F.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

Cai, Y.

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Y. Cai and X. Lü, “Propagation of Bessel and Bessel-Gaussian beams through an unapertured or apertured misaligned paraxial optical systems,” Opt. Commun. 274(1), 1–7 (2007).
[Crossref]

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Chatwin, C.

Chen, M.

Cheng, M.

Dalil-Essakali, L.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

Deacon, K. S.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Dholakia, K.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[Crossref]

Dogariu, A.

Durnin, J.

J. Durnin, J. H. Eberly, and J. J. Miceli, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988).
[Crossref] [PubMed]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. H. Eberly, and J. J. Miceli, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988).
[Crossref] [PubMed]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eliel, E. R.

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108(17), 173604 (2012).
[Crossref] [PubMed]

Eyyuboglu, H. T.

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008).
[Crossref]

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

H. T. Eyyuboğlu, “Propagation of higher order Bessel-Gaussian beams in turbulence,” Appl. Phys. B 88(2), 259–265 (2007).
[Crossref]

Ez-zariy, L.

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
[Crossref]

Garces-Chavez, V.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[Crossref]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Guo, L.

Hardalaç, F.

H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40(2), 343–351 (2008).
[Crossref]

Hu, B.

Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Huang, Q.

Ituen, I.

Ji, X.

X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007).
[Crossref]

Ji-Xiong, P.

C. Bao-Suan and P. Ji-Xiong, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18(3), 1033–1039 (2009).
[Crossref]

Korotkova, O.

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Kotlyar, V. V.

Kovalev, A. A.

Lavery, M. P. J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Li, J.

Li, L.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Li, Y.

Löffler, W.

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108(17), 173604 (2012).
[Crossref] [PubMed]

Lu, B.

Lü, B.

X. Ji and B. Lü, “Focal shift and focal switch of Bessel–Gaussian beams passing through a lens system with or without aperture,” Opt. Laser Technol. 39(3), 562–568 (2007).
[Crossref]

L. Wang, X. Wang, and B. Lü, “Propagation properties of partially coherent modified Bessel–Gauss beams,” Optik (Stuttg.) 116(2), 65–70 (2005).
[Crossref]

Lü, X.

Y. Cai and X. Lü, “Propagation of Bessel and Bessel-Gaussian beams through an unapertured or apertured misaligned paraxial optical systems,” Opt. Commun. 274(1), 1–7 (2007).
[Crossref]

Meyers, R. E.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Miceli, J. J.

Molisch, A. F.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Qingchao, S.

Ramachandran, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ren, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ruschin, S.

Salakhutdinov, V. D.

V. D. Salakhutdinov, E. R. Eliel, and W. Löffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108(17), 173604 (2012).
[Crossref] [PubMed]

Sermutlu, E.

H. T. Eyyuboğlu, E. Sermutlu, Y. Baykal, Y. Cai, and O. Korotkova, “Intensity fluctuations in J-Bessel-Gaussian beams of all orders propagating in turbulent atmosphere,” Appl. Phys. B 93(2–3), 605–611 (2008).
[Crossref]

Shih, Y.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Shirai, T.

Sibbett, W.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197(4-6), 239–245 (2001).
[Crossref]

Skidanov, R. V.

Soifer, V. A.

Tang, L.

L. Tang, H. Wang, X. Zhang, and S. Zhu, “Propagation properties of partially coherent Lommel beams in non-Kolmogorov turbulence,” Opt. Commun. 427, 79–84 (2018).
[Crossref]

Tunick, A. D.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100(6), 061126 (2012).
[Crossref]

Tur, M.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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L. Tang, H. Wang, X. Zhang, and S. Zhu, “Propagation properties of partially coherent Lommel beams in non-Kolmogorov turbulence,” Opt. Commun. 427, 79–84 (2018).
[Crossref]

Wang, J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Wang, L.

L. Wang, X. Wang, and B. Lü, “Propagation properties of partially coherent modified Bessel–Gauss beams,” Optik (Stuttg.) 116(2), 65–70 (2005).
[Crossref]

Wang, X.

L. Wang, X. Wang, and B. Lü, “Propagation properties of partially coherent modified Bessel–Gauss beams,” Optik (Stuttg.) 116(2), 65–70 (2005).
[Crossref]

Wanjun, W.

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Wolf, E.

Xie, G.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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L. Tang, H. Wang, X. Zhang, and S. Zhu, “Propagation properties of partially coherent Lommel beams in non-Kolmogorov turbulence,” Opt. Commun. 427, 79–84 (2018).
[Crossref]

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Zhao, Z.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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Zhu, S.

L. Tang, H. Wang, X. Zhang, and S. Zhu, “Propagation properties of partially coherent Lommel beams in non-Kolmogorov turbulence,” Opt. Commun. 427, 79–84 (2018).
[Crossref]

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Opt. Express (2)

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[Crossref]

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Optik (Stuttg.) (2)

L. Wang, X. Wang, and B. Lü, “Propagation properties of partially coherent modified Bessel–Gauss beams,” Optik (Stuttg.) 116(2), 65–70 (2005).
[Crossref]

L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “Effects of a turbulent atmosphere on an aperture Lommel-Gaussian beam,” Optik (Stuttg.) 127(23), 11534–11543 (2016).
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Figures (9)

Fig. 1
Fig. 1 Average intensity of the Bessel Gaussian beam. (a) l0 = 0.02m, (b) l0 = 0.002m.
Fig. 2
Fig. 2 RMSPE of the average intensity of the Bessel Gaussian beam calculated by different methods. (a) variation with L, (b) variation with l0.
Fig. 3
Fig. 3 Average intensity of the mBGBs (An = 1, L = 1000m, β = 200m−1) (a) RyEJ0, n = [2,5], (b) Rytov, n = [2,5], (c) HF, n = [2,5], (d) RyEJ0, n = [3,6], (e) Rytov, n = [3,6], (f) HF, n = [3,6], (g) RyEJ0, n = [0, 2, 5], (h) Rytov, n = [0, 2, 5], (i) HF, n = [0, 2, 5].
Fig. 4
Fig. 4 Average intensity of the mBGBs. (a) n = [2,5], (b) n = [3,6], (c) n = [0, 2, 5].
Fig. 5
Fig. 5 Average intensity of mBGBs with different order sub beams (RyEJ0, An = 1, L = 1000m, β = 200m−1). (a) n = [2,5], (b) n = [0, 2], (c) n = [0, 5], (d) n = [3,6], (e) n = [0, 3, 6], (f) n = [1,3,5].
Fig. 6
Fig. 6 Average intensity of the mBGBs with two different amplitude sub beams. (RyEJ0, n = [2,5], L = 1000m, β = 200m−1). (a) An = [0.01, 1], (b) An = [0.1, 1], (c) An = [1, 0.5], (d) An = [1, 0.1].
Fig. 7
Fig. 7 Average intensity of the mBGBs with three different amplitude sub beams. (RyEJ0, n = [0, 2, 5], L = 1000m, β = 200m−1). (a) An = [0.5, 1, 1], (b) An = [0.2, 1, 1], (c) An = [1, 0.6, 1], (d) An = [1, 0.2, 1].
Fig. 8
Fig. 8 Average intensity of the mBGBs with two different initial phase sub beams. (RyEJ0, n = [2,5], an = 1, L = 1000m, β = 200m−1) (a) φ0n = [0, 0], (b) φ0n = [π/4, 0], (c) φ0n = [π/2, 0], (d) φ0n = [3π/4, 0], (e) φ0n = [0, π/10], (f) φ0n = [0, π/5], (g) φ0n = [0, 3π/10].
Fig. 9
Fig. 9 Average intensity of the mBGBs. (RyEJ0, n = [2,5], An = 1, β = 500m−1) (a) L = 50m, (b) L = 100m, (c) L = 200m, (d) L = 500m.

Tables (1)

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Table 1 RMSPE of the mBGBs average intensity calculated by different methods

Equations (16)

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U(r, φ r )= n= N 1 N 2 a n exp(kα r 2 )exp[ in( φ r φ 0n ) ] J n (βr)
U(r,L)= exp(ikL) 1+2iαL exp[ i β 2 L+2α k 2 r 2 2k( 1+2iαL ) ] n= N 1 N 2 A n J n ( β r 1+2iαL )exp(in φ r )
I(r,L)=U(r,L) U * (r,L) = 1 1+4 α 2 L 2 exp[ 2α k 2 r 2 +2α β 2 L 2 k( 1+4 α 2 L 2 ) ] × n= N 1 N 2 n'= N 1 N 2 A n A n' * J n ( βr 1+2iαL ) J n' * ( βr 1+2iαL )exp[ i( nn' ) φ r ]
I(r,φ)=I(r,L)exp[2 E 1 (0,0)+ E 2 (r,r)]
E 1 (r,r)= Φ 2 (r,L)
E 2 ( r 1 , r 2 )= Φ 1 ( r 1 ,L) Φ 1 * ( r 2 ,L)
Φ 1 (r,L)=ik 0 L dz dv(K,z) exp[ iγKr i κ 2 γ 2k (Lz) ] × n= N 1 N 2 A n J n [ β(Lz) k( 1+2iαL ) | K kr Lz | ]exp( in φ Kr ) n= N 1 N 2 A n J n ( βr 1+2iαL )exp( in φ r )
Φ 2 (r,L)= k 2 0 L dz 0 z d z ' dv(K,z)dv( K ' , z ' ) ×exp[ iγ( K+ γ ' K ' )r i ( K+ γ ' K ' ) 2 γ 2k (Lz) iκ ' 2 γ ' 2k (z z ' ) ] × { n= N 1 N 2 A n J n [ β(Lz) k( 1+2iαL ) | K+ γ ' K ' kr Lz | ]exp( in φ KK'r ) } n= N 1 N 2 A n J n ( βr 1+2iαL )exp(in φ r )
exp( i φ κr )= [ κ(Lz)exp(i φ κ )rkexp(i φ r ) ] | κ(Lz)exp(i φ κ )rkexp(i φ r ) | = { κ(Lz)exp[ i( φ κ φ r ) ]rk } [ κ 2 (Lz) 2 2κrk(Lz)cos( φ κ φ r )+ r 2 k 2 ] 0.5 exp(i φ r )
E 2 (r,r)=2π k 2 0 L dη d 2 κ Φ n (κ)exp[ i(γγ*)Kr i κ 2 2k (γγ*)(Lη) ] × n= N 1 N 2 A n J n [ β(Lz) k( 1+2iαL ) | K kr Lz | ]exp( in φ Kr ) n= N 1 N 2 A n J n ( βr 1+2iαL )exp( in φ r ) × n= N 1 N 2 A n * J n * [ β(Lz) k( 1+2iαL ) | K kr Lz | ]exp( in φ Kr ) n= N 1 N 2 A n * J n * ( βr 1+2iαL )exp( in φ r )
E 1 (r,r)= Φ 2 (r,L) = E 1 (0,0)=π k 2 0 L dη d 2 κ Φ n (K)
J n [ β(Lη) k( 1+2iαL ) | K kr Lη | ]exp(in φ Kr ) J n ( βr 1+2iαL )exp(in φ r ) J 0 [ β(Lη) k( 1+2iαL ) | K kr Lη | ] J 0 ( βr 1+2iαL )
E 2 (r,r)=2π k 2 0 L dη d 2 κ Φ n (κ) ×exp[ i(γγ*)Kr i κ 2 2k (γγ*)(Lη) ] × J 0 [ β(Lη) k( 1+2iαL ) | K kr Lη | ] J 0 * [ β(Lη) k( 1+2iαL ) | K kr Lη | ] | J 0 ( βr 1+2iαL ) | 2
I(r, φ r )= b 2 π(kαib+1/ ρ 0 2 ) exp[ β 2 +4 b 2 r 2 4(kαib+1/ ρ 0 2 ) ] × n 1 = N 1 N 2 n 2 = N 1 N 2 0 0 2π d s 2 d φ s 2 A n 1 A n 2 * s 2 J n 2 (β s 2 )exp(i n 2 φ s 2 ) × [ibrexp(i φ r )+ s 2 exp(i φ s 2 )/ ρ 0 2 ] n 1 [ b 2 r 2 + s 2 2 / ρ 0 4 2ibr s 2 cos( φ r φ s 2 )/ ρ 0 2 ] n 1 /2 × J n 1 { β [ b 2 r 2 + s 2 2 / ρ 0 4 2ibr s 2 cos( φ r φ s 2 )/ ρ 0 2 ] 1/2 kαib+1/ ρ 0 2 } ×exp [ kα s 2 2 (ibkα+kα/ ρ 0 2 + b 2 ) s 2 2 2br s 2 (ikα+b)cos( φ r φ s 2 ) kαib+1/ ρ 0 2 ]
ρ 0 2 = π 2 k 2 z/3 0 dκ κ 3 Φ n (κ)
( σ I 2 ¯ ) 1/2 = ( s m ( I m ' / I m -1 ) 2 s m ) 1/2

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