Abstract

We report on a theoretical and numerical study of a Gaussian beam modulated by several optical vortices (OV) that carry same-sign unity topological charge (TC) and are unevenly arranged on a circle. The TC of such a multi-vortex beam equals the sum of the TCs of all OVs. If the OVs are located evenly along an arbitrary-radius circle, a simple relationship for the normalized orbital angular momentum (OAM) is derived for such a beam. It is shown that in a multi-vortex beam, OAM normalized to power cannot exceed the number of constituent vortices and decreases with increasing distance from the optical axis to the vortex centers. We show that for the OVs to appear at the infinity of such a combined beam, an infinite-energy Gaussian beam is needed. On the contrary, the total TC is independent of said distance, remaining equal to the number of constituent vortices. We show that if TC is evaluated not along the whole circle encompassing the singularity centers, but along any part of this circle, such a quantity is also invariant and conserves on propagation. Besides, a multi-spiral phase plate is studied for the first time to our knowledge, and we obtained the TC and OAM of multi-vortices generated by this plate. When propagated through a random phase screen (diffuser) the TC is unchanged, while the OAM changes by less than 10% if the random phase delay on the diffuser does not exceed half wavelength. Such multi-vortices can be used for data transmission in the turbulent atmosphere.

© 2020 Optical Society of America

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References

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2021 (1)

M. Dong, C. Zhao, Y. Cai, and Y. Yang, “Partially coherent vortex beams: fundamentals and applications,” Sci. China Phys. Mech. Astron. 64, 224201 (2021).
[Crossref]

2020 (4)

2019 (4)

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Topological stability of optical vortices diffracted by a random phase screen,” Comput. Opt. 43, 917–925 (2019).
[Crossref]

J. Zeng, X. Liu, C. Zhao, F. Wang, G. Gbur, and Y. Cai, “Spiral spectrum of a Laguerre-Gaussian beam propagating in anisotropic non-Kolmogorov turbulent atmosphere along horizontal path,” Opt. Express 27, 25342–25356 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27, 11236–11251 (2019).
[Crossref]

V. S. Vasilyev, A. I. Kapustin, R. V. Skidanov, V. V. Podlipnov, N. A. Ivliev, and S. V. Ganchevskaya, “Experimental investigation of the stability of Bessel beams in the atmosphere,” Comput. Opt. 43, 376–384 (2019).
[Crossref]

2018 (1)

2016 (4)

2014 (3)

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22, 5826–5838 (2014).
[Crossref]

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

2013 (1)

M. V. Berry, “A note on superoscillations associated with Bessel beams,” J. Opt. 15, 044006 (2013).
[Crossref]

2012 (1)

2010 (1)

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–55 (2010).
[Crossref]

2009 (1)

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[Crossref]

2008 (3)

K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16, 21315–21320 (2008).
[Crossref]

L. Burger, I. A. Litvin, and A. Forbes, “Simulation atmospheric turbulence using a phase-only spatial light modulator,” S. Afr. J. Sci. 104, 129–134 (2008).

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281, 1366–1374 (2008).
[Crossref]

2006 (2)

2005 (1)

F. Flossmann, U. T. Schwarz, and M. Maier, “Optical vortices in a Laguerre-Gaussian LG(1,0) beam,” J. Mod. Opt. 52, 1009–1017 (2005).
[Crossref]

2004 (1)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

1998 (1)

M. V. Berry, “Wave dislocation reactions in non-paraxial Gaussian beams,” J. Mod. Opt. 45, 1845–1858 (1998).
[Crossref]

1997 (2)

D. Rozas, C. T. Law, and G. A. Shwartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[Crossref]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beam carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

1993 (2)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[Crossref]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

1988 (1)

J. F. Nye, J. V. Hajnal, and J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London A 417, 7–20 (1988).
[Crossref]

1987 (2)

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

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover Publications Inc, 1979).

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Alperin, S. N.

Avramov-Zamurovic, S.

S. Avramov-Zamurovic, C. Nelson, S. Guth, O. Korotkova, and R. Malek-Madani, “Experimental study of electromagnetic Bessel-Gaussian Schell model beams propagating in a turbulent channel,” Opt. Commun. 359, 207–215 (2016).
[Crossref]

Bai, Y.

Bartal, G.

Basistiy, I. V.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Baykal, Y.

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–55 (2010).
[Crossref]

Bazhenov, V. Y.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Bekshaev, A. Y.

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281, 1366–1374 (2008).
[Crossref]

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Centrifugal transformation of the transverse structure of freely propagating paraxial light beams,” Opt. Lett. 31, 694–696 (2006).
[Crossref]

Berry, M. V.

M. V. Berry, “A note on superoscillations associated with Bessel beams,” J. Opt. 15, 044006 (2013).
[Crossref]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

M. V. Berry, “Wave dislocation reactions in non-paraxial Gaussian beams,” J. Mod. Opt. 45, 1845–1858 (1998).
[Crossref]

Boyd, R. W.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Burger, L.

L. Burger, I. A. Litvin, and A. Forbes, “Simulation atmospheric turbulence using a phase-only spatial light modulator,” S. Afr. J. Sci. 104, 129–134 (2008).

Cai, Y.

M. Dong, C. Zhao, Y. Cai, and Y. Yang, “Partially coherent vortex beams: fundamentals and applications,” Sci. China Phys. Mech. Astron. 64, 224201 (2021).
[Crossref]

J. Zeng, X. Liu, C. Zhao, F. Wang, G. Gbur, and Y. Cai, “Spiral spectrum of a Laguerre-Gaussian beam propagating in anisotropic non-Kolmogorov turbulent atmosphere along horizontal path,” Opt. Express 27, 25342–25356 (2019).
[Crossref]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22, 5826–5838 (2014).
[Crossref]

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–55 (2010).
[Crossref]

Chen, Y.

Cohen, K.

Dennis, M. R.

Dong, M.

M. Dong, C. Zhao, Y. Cai, and Y. Yang, “Partially coherent vortex beams: fundamentals and applications,” Sci. China Phys. Mech. Astron. 64, 224201 (2021).
[Crossref]

Q. Zhao, M. Dong, Y. Bai, and Y. Yang, “Measuring high orbital angular momentum of vortex beams with improved multipoint interferometer,” Photon. Res. 8, 745–749 (2020).
[Crossref]

Durnin, J.

Eyyuboglu, H. T.

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–55 (2010).
[Crossref]

Fedosejevs, R.

Fickler, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Fink, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Flossmann, F.

F. Flossmann, U. T. Schwarz, and M. Maier, “Optical vortices in a Laguerre-Gaussian LG(1,0) beam,” J. Mod. Opt. 52, 1009–1017 (2005).
[Crossref]

Forbes, A.

L. Burger, I. A. Litvin, and A. Forbes, “Simulation atmospheric turbulence using a phase-only spatial light modulator,” S. Afr. J. Sci. 104, 129–134 (2008).

Ganchevskaya, S. V.

V. S. Vasilyev, A. I. Kapustin, R. V. Skidanov, V. V. Podlipnov, N. A. Ivliev, and S. V. Ganchevskaya, “Experimental investigation of the stability of Bessel beams in the atmosphere,” Comput. Opt. 43, 376–384 (2019).
[Crossref]

Gbur, G.

Gjona, B.

Gopinath, J. T.

Gori, F.

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

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beam carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

Guattari, G.

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

Guth, S.

S. Avramov-Zamurovic, C. Nelson, S. Guth, O. Korotkova, and R. Malek-Madani, “Experimental study of electromagnetic Bessel-Gaussian Schell model beams propagating in a turbulent channel,” Opt. Commun. 359, 207–215 (2016).
[Crossref]

Hajnal, J. V.

J. F. Nye, J. V. Hajnal, and J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London A 417, 7–20 (1988).
[Crossref]

Handsteiner, J.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Hannay, J. H.

J. F. Nye, J. V. Hajnal, and J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London A 417, 7–20 (1988).
[Crossref]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beam carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[Crossref]

Ivliev, N. A.

V. S. Vasilyev, A. I. Kapustin, R. V. Skidanov, V. V. Podlipnov, N. A. Ivliev, and S. V. Ganchevskaya, “Experimental investigation of the stability of Bessel beams in the atmosphere,” Comput. Opt. 43, 376–384 (2019).
[Crossref]

Jie, Z.

Z. Jie, L. Zhi, Z. Xiaoqi, N. Xiaolong, C. Minghui, and G. Siyuan, “Simulation of partially coherent optical atmospheric turbulent transmission based on LC-SLM,” in IEEE 4th Optoelectronics Global Conference (2019).

Kapustin, A. I.

V. S. Vasilyev, A. I. Kapustin, R. V. Skidanov, V. V. Podlipnov, N. A. Ivliev, and S. V. Ganchevskaya, “Experimental investigation of the stability of Bessel beams in the atmosphere,” Comput. Opt. 43, 376–384 (2019).
[Crossref]

Karamoch, A. I.

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281, 1366–1374 (2008).
[Crossref]

Konyaev, P. A.

Korotkova, O.

S. Avramov-Zamurovic, C. Nelson, S. Guth, O. Korotkova, and R. Malek-Madani, “Experimental study of electromagnetic Bessel-Gaussian Schell model beams propagating in a turbulent channel,” Opt. Commun. 359, 207–215 (2016).
[Crossref]

Kotlyar, V.

Kotlyar, V. V.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27, 11236–11251 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Topological stability of optical vortices diffracted by a random phase screen,” Comput. Opt. 43, 917–925 (2019).
[Crossref]

Kovalev, A.

Kovalev, A. A.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27, 11236–11251 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Topological stability of optical vortices diffracted by a random phase screen,” Comput. Opt. 43, 917–925 (2019).
[Crossref]

Krenn, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Kumar, P.

P. Kumar and N. K. Nishchal, “Self-referenced interference of laterally displaced vortex beams for topological charge determination,” Opt. Commun. 459, 125000 (2020).
[Crossref]

Law, C. T.

Li, X.

Litvin, I. A.

L. Burger, I. A. Litvin, and A. Forbes, “Simulation atmospheric turbulence using a phase-only spatial light modulator,” S. Afr. J. Sci. 104, 129–134 (2008).

Liu, X.

Longman, A.

Lukin, I. P.

Lukin, V. P.

Magaña-Loaiza, O. S.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Maher, L.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier, “Optical vortices in a Laguerre-Gaussian LG(1,0) beam,” J. Mod. Opt. 52, 1009–1017 (2005).
[Crossref]

Malek-Madani, R.

S. Avramov-Zamurovic, C. Nelson, S. Guth, O. Korotkova, and R. Malek-Madani, “Experimental study of electromagnetic Bessel-Gaussian Schell model beams propagating in a turbulent channel,” Opt. Commun. 359, 207–215 (2016).
[Crossref]

Malik, M.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beam carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

Minghui, C.

Z. Jie, L. Zhi, Z. Xiaoqi, N. Xiaolong, C. Minghui, and G. Siyuan, “Simulation of partially coherent optical atmospheric turbulent transmission based on LC-SLM,” in IEEE 4th Optoelectronics Global Conference (2019).

Mirhosseini, M.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Nelson, C.

S. Avramov-Zamurovic, C. Nelson, S. Guth, O. Korotkova, and R. Malek-Madani, “Experimental study of electromagnetic Bessel-Gaussian Schell model beams propagating in a turbulent channel,” Opt. Commun. 359, 207–215 (2016).
[Crossref]

Niederriter, R. D.

Nishchal, N. K.

P. Kumar and N. K. Nishchal, “Self-referenced interference of laterally displaced vortex beams for topological charge determination,” Opt. Commun. 459, 125000 (2020).
[Crossref]

Nye, J. F.

J. F. Nye, J. V. Hajnal, and J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London A 417, 7–20 (1988).
[Crossref]

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).

Ostrovsky, E.

Padovani, C.

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

Podlipnov, V. V.

V. S. Vasilyev, A. I. Kapustin, R. V. Skidanov, V. V. Podlipnov, N. A. Ivliev, and S. V. Ganchevskaya, “Experimental investigation of the stability of Bessel beams in the atmosphere,” Comput. Opt. 43, 376–384 (2019).
[Crossref]

Porfirev, A. P.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Calculation of fractional orbital angular momentum of superpositions of optical vortices by intensity moments,” Opt. Express 27, 11236–11251 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Topological stability of optical vortices diffracted by a random phase screen,” Comput. Opt. 43, 917–925 (2019).
[Crossref]

Rodenburg, B.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Rozas, D.

Scheidl, T.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier, “Optical vortices in a Laguerre-Gaussian LG(1,0) beam,” J. Mod. Opt. 52, 1009–1017 (2005).
[Crossref]

Sennikov, V. A.

Shwartzlander, G. A.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Siemens, M. E.

Siyuan, G.

Z. Jie, L. Zhi, Z. Xiaoqi, N. Xiaolong, C. Minghui, and G. Siyuan, “Simulation of partially coherent optical atmospheric turbulent transmission based on LC-SLM,” in IEEE 4th Optoelectronics Global Conference (2019).

Skidanov, R. V.

V. S. Vasilyev, A. I. Kapustin, R. V. Skidanov, V. V. Podlipnov, N. A. Ivliev, and S. V. Ganchevskaya, “Experimental investigation of the stability of Bessel beams in the atmosphere,” Comput. Opt. 43, 376–384 (2019).
[Crossref]

Soskin, M. S.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Centrifugal transformation of the transverse structure of freely propagating paraxial light beams,” Opt. Lett. 31, 694–696 (2006).
[Crossref]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beam carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover Publications Inc, 1979).

Steinhoff, N. K.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Tang, H.

Tsesses, S.

Tyler, G. A.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Ursin, R.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Vasilyev, V. S.

V. S. Vasilyev, A. I. Kapustin, R. V. Skidanov, V. V. Podlipnov, N. A. Ivliev, and S. V. Ganchevskaya, “Experimental investigation of the stability of Bessel beams in the atmosphere,” Comput. Opt. 43, 376–384 (2019).
[Crossref]

Vasnetsov, M. V.

A. Y. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Centrifugal transformation of the transverse structure of freely propagating paraxial light beams,” Opt. Lett. 31, 694–696 (2006).
[Crossref]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beam carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Wang, F.

Wang, L. G.

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Xiaolong, N.

Z. Jie, L. Zhi, Z. Xiaoqi, N. Xiaolong, C. Minghui, and G. Siyuan, “Simulation of partially coherent optical atmospheric turbulent transmission based on LC-SLM,” in IEEE 4th Optoelectronics Global Conference (2019).

Xiaoqi, Z.

Z. Jie, L. Zhi, Z. Xiaoqi, N. Xiaolong, C. Minghui, and G. Siyuan, “Simulation of partially coherent optical atmospheric turbulent transmission based on LC-SLM,” in IEEE 4th Optoelectronics Global Conference (2019).

Yanakas, M.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Yang, Y.

M. Dong, C. Zhao, Y. Cai, and Y. Yang, “Partially coherent vortex beams: fundamentals and applications,” Sci. China Phys. Mech. Astron. 64, 224201 (2021).
[Crossref]

Q. Zhao, M. Dong, Y. Bai, and Y. Yang, “Measuring high orbital angular momentum of vortex beams with improved multipoint interferometer,” Photon. Res. 8, 745–749 (2020).
[Crossref]

Zeilinger, A.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

Zeng, J.

Zhao, C.

Zhao, Q.

Zheng, W. W.

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[Crossref]

Zheng, X.

Zhi, L.

Z. Jie, L. Zhi, Z. Xiaoqi, N. Xiaolong, C. Minghui, and G. Siyuan, “Simulation of partially coherent optical atmospheric turbulent transmission based on LC-SLM,” in IEEE 4th Optoelectronics Global Conference (2019).

Zhou, G.

Zhu, K.

Appl. Opt. (2)

Comput. Opt. (2)

V. S. Vasilyev, A. I. Kapustin, R. V. Skidanov, V. V. Podlipnov, N. A. Ivliev, and S. V. Ganchevskaya, “Experimental investigation of the stability of Bessel beams in the atmosphere,” Comput. Opt. 43, 376–384 (2019).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Topological stability of optical vortices diffracted by a random phase screen,” Comput. Opt. 43, 917–925 (2019).
[Crossref]

J. Mod. Opt. (3)

M. V. Berry, “Wave dislocation reactions in non-paraxial Gaussian beams,” J. Mod. Opt. 45, 1845–1858 (1998).
[Crossref]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[Crossref]

F. Flossmann, U. T. Schwarz, and M. Maier, “Optical vortices in a Laguerre-Gaussian LG(1,0) beam,” J. Mod. Opt. 52, 1009–1017 (2005).
[Crossref]

J. Opt. (1)

M. V. Berry, “A note on superoscillations associated with Bessel beams,” J. Opt. 15, 044006 (2013).
[Crossref]

J. Opt. A (2)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

L. G. Wang and W. W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam arrays,” J. Opt. A 11, 065703 (2009).
[Crossref]

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

J. Opt. Soc. Am. B (1)

New J. Phys. (2)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16, 113028 (2014).
[Crossref]

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Opt. Commun. (5)

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281, 1366–1374 (2008).
[Crossref]

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

S. Avramov-Zamurovic, C. Nelson, S. Guth, O. Korotkova, and R. Malek-Madani, “Experimental study of electromagnetic Bessel-Gaussian Schell model beams propagating in a turbulent channel,” Opt. Commun. 359, 207–215 (2016).
[Crossref]

P. Kumar and N. K. Nishchal, “Self-referenced interference of laterally displaced vortex beams for topological charge determination,” Opt. Commun. 459, 125000 (2020).
[Crossref]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Optica (2)

Photon. Res. (1)

Phys. Rev. A (2)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beam carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Proc. R. Soc. London A (1)

J. F. Nye, J. V. Hajnal, and J. H. Hannay, “Phase saddles and dislocations in two-dimensional waves such as the tides,” Proc. R. Soc. London A 417, 7–20 (1988).
[Crossref]

Prog. Electromagn. Res. (1)

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. 103, 33–55 (2010).
[Crossref]

S. Afr. J. Sci. (1)

L. Burger, I. A. Litvin, and A. Forbes, “Simulation atmospheric turbulence using a phase-only spatial light modulator,” S. Afr. J. Sci. 104, 129–134 (2008).

Sci. China Phys. Mech. Astron. (1)

M. Dong, C. Zhao, Y. Cai, and Y. Yang, “Partially coherent vortex beams: fundamentals and applications,” Sci. China Phys. Mech. Astron. 64, 224201 (2021).
[Crossref]

Other (4)

Z. Jie, L. Zhi, Z. Xiaoqi, N. Xiaolong, C. Minghui, and G. Siyuan, “Simulation of partially coherent optical atmospheric turbulent transmission based on LC-SLM,” in IEEE 4th Optoelectronics Global Conference (2019).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover Publications Inc, 1979).

J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).

A. E. Siegman, Lasers (University Science, 1986).

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Figures (4)

Fig. 1.
Fig. 1. (a),(b),(e),(f),(i),(j) Computed and (c),(d),(g),(h),(k),(l) experimental intensity distributions without (a),(c),(e),(g),(i),(k) the reference beam and (b),(d),(f),(h),(j),(l) with the reference beam at a distance 50 mm from the SLM, generated by the multi-singularity SPP (a)–(d) with three singularity centers inside the 4 mm radius circular diaphragm, (e)–(h) with three singularity centers, one of which is on the edge of the 4 mm radius diaphragm, and (i)–(l) with three singularity centers, one of which is outside the diaphragm. Each frame is ${10} \times 10\;{\rm mm}$.
Fig. 2.
Fig. 2. Patterns of (a) intensity and (b) phase for the beam in Eq. (3) at $m = 3$ in the source plane; (c),(f),(i) phase patterns after passing through a random phase diffuser, and patterns of (d),(g),(j) intensity and (e),(h),(k) phase upon free-space propagation ($z = 5\;{\rm m}$). Random phase variations are on the intervals(c)–(e) $[- \pi /{2},\pi /{2}]$, (f)–(h) $[- \pi ,\pi]$, and (i)–(k) $[- 3\pi /{2},{3}\pi /{2}]$. (e),(h),(k) The red circles show the area where the TC and OAM were computed. The scale bar in each figure is 5 mm.
Fig. 3.
Fig. 3. Intensity patterns for the beam in Eq. (3) at distance $z = 2\;{\rm m}$ for different numbers of constituent OVs in the beam and differently varying random phases over the diffuser: (a),(b) ${m} = {2}$, (c),(d) ${m} = {3}$, and (e),(f) ${m} = {4}$; (a),(c),(e) $|\Delta \psi |\le \pi$ and (b),(d),(f) $|\Delta \psi |\le {3}\pi {/2}$. The scale bar in each figure is 5 mm.
Fig. 4.
Fig. 4. (a),(c),(e) Intensity and (b),(d),(f) phase distributions of the beam in Eq. (3) at the distances (a),(b) $z = 0.5\;{\rm m}$, (c),(d) $z = 5\;{\rm m}$, and (e),(f) $z = 20\;{\rm m}$. The red dotted sector denotes the angles over which the asymptotic phase invariant [Eq. (26)] is evaluated.

Tables (1)

Tables Icon

Table 1. Normalized OAM, TC, and Phase Invariants μ 1 and μ 2 of the Beam in Eq. (3) upon Free-Space Propagation after Passing a Random Phase Diffuser at Different Numbers of Constituent Beams ( m ) and Different Phase Variations Δ ψ on the Diffuser (with r.m.s. Error Given in Brackets), Calculated Theoretically Using Eqs. (9) and (13) and Numerically Using Eqs. (5) and (11)

Equations (26)

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

E ( r , φ , z ) = 1 σ ( 2 w 0 ) m exp ( r 2 σ w 0 2 ) p = 0 m 1 ( r e i φ σ r p e i φ p ) ,
x = a cos φ p , y = a sin φ p ,
E ( r , φ , z ) = 1 σ ( 2 w 0 ) m exp ( r 2 σ w 0 2 ) ( r m e i m φ σ m a m ) .
W = 0 0 2 π | E | 2 r d r d φ ,
J z = i 0 0 2 π E E φ r d r d φ ,
W = π w 0 2 2 [ m ! + ( 2 a 2 w 0 2 ) m ] .
J z = 2 π m 2 m w 0 2 m 0 exp ( 2 r 2 w 0 2 ) r 2 m r d r .
J z = 2 π m 2 m w 0 2 m m ! 2 ( 2 / w 0 2 ) m + 1 = m π w 0 2 2 m ! .
J z W = m m ! m ! + ( 2 a 2 / w 0 2 ) m .
T C = lim r 1 2 π 0 2 π φ [ arg E ( r , φ , z ) ] d φ .
T C = 1 2 π lim r Im { 0 2 π 1 E E φ d φ } .
T C = 1 2 π lim r Im { 0 2 π i m σ m r m e i m φ σ m r m e i m φ a m d φ } .
T C = m .
E m ( r , φ , z = 0 ) = A ( r ) p = 1 m ( r e i φ r p e i φ p ) m p .
T C = 1 2 π lim r Im { 0 2 π i r e i φ p = 1 m m p r e i φ r p e i φ p d φ } = p = 1 m m p .
E m , n ( r , φ , z = 0 ) = A ( r ) p = 1 m ( r e i φ r p e i φ p ) m p × q = 1 n ( r e i φ r q e i φ q ) n q .
T C = p = 1 m m p q = 1 n n q .
μ g = 1 2 π lim r { 0 2 π g ( φ ) φ [ arg E ( r , φ , z ) ] d φ } ,
μ 1 = 1 2 π lim r { π / 4 π / 4 φ [ arg E ( r , φ , z ) ] d φ } ,
μ 2 = 1 2 π lim r { 0 2 π exp ( φ 2 φ 0 2 ) φ [ arg E ( r , φ , z ) ] d φ } .
E ( r , φ ) = circl ( r R ) exp ( i Ψ ( r , φ ) ) ,
Ψ ( r , φ ) = arg [ p = 1 m ( r e i φ r p e i φ p ) m p ] = p = 1 M n p arctan ( r sin φ r p sin φ p r cos φ r p cos φ p ) ,
T C = p = 1 M n p , r p < R , p = 1 , 2 , M .
J z W = p = 1 M n p r p R | A ( r ) | 2 r d r ( 0 R | A ( r ) | 2 r d r ) 1 .
exp ( i ψ ( r ) i ψ ( r ) = exp ( | r r | 2 δ 2 ) ,
μ 3 = 1 2 π lim r { π / 6 π / 6 φ [ arg E ( r , φ , z ) ] d φ } ,

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