Abstract

A new kind of partially coherent vector vortex beam, namely, the partially coherent radially polarized (PCRP) beam with multiple off-axis vortices, is introduced, and the average intensity distributions of such vortex beam focused by a thin lens are investigated theoretically. It is novelty that the off-axis vortices will induce the focal intensity redistribution and reconstruction, while this remarkable characteristic will be vanished in the case of a very low coherence. In view of this distinctive feature, a new method has been put forward to shape or modulate the focal intensity distribution by elaborately tailoring the multiple off-axis vortices as well as the coherence length. More importantly, some peculiar focal fields with novel structures, such as bar-shaped, triangle-shaped, square-shaped, and pentagon-shaped hollow profiles or flat-top foci, are obtained. Our results indicate that modulating the multiple off-axis vortices provides an additional degree of freedom for focus shaping.

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

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2020 (3)

2019 (6)

H. Xu, R. Zhang, Z. Sheng, and J. Qu, “Focus shaping of partially coherent radially polarized vortex beam with tunable topological charge,” Opt. Express 27(17), 23959–23969 (2019).
[Crossref]

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

D. Shen and D. Zhao, “Measuring the topological charge of optical vortices with a twisting phase,” Opt. Lett. 44(9), 2334–2337 (2019).
[Crossref]

X. Liu, J. Zeng, and Y. Cai, “Review on vortex beams with low spatial coherence,” Adv. Phys.: X 4(1), 1626766 (2019).
[Crossref]

J. Zeng, R. Lin, X. Liu, C. Zhao, and Y. Cai, “Review on partially coherent vortex beams,” Front. Guided Wave Opt. Optoelectron. 12(3), 229–248 (2019).
[Crossref]

Y. Dong, S. Xi, B. Zhu, X. Wang, Q. Mu, S. Wang, and Z. Zhu, “The directional excitation of surface plasmon polaritons by radially polarized beam with multiple off-axis vortices,” Opt. Commun. 443, 197–201 (2019).
[Crossref]

2017 (6)

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

Z. Mei, “Modeling for Partially Spatially Coherent Vortex Beams,” IEEE Photonics J. 9(5), 1–6 (2017).
[Crossref]

K. Liu, Y. Cheng, Y. Gao, X. Li, Y. Qin, and H. Wang, “Super-resolution radar imaging based on experimental OAM beams,” Appl. Phys. Lett. 110(16), 164102 (2017).
[Crossref]

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[Crossref]

X. Zhao, J. Zhang, X. Pang, and G. Wan, “Properties of a strongly focused Gaussian beam with an off-axis vortex,” Opt. Commun. 389, 275–282 (2017).
[Crossref]

2016 (2)

2015 (5)

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random electromagnetic Gaussian-shell model vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Y. Zhang, Y. Cui, F. Wang, and Y. Cai, “Correlation singularities in a partially coherent electromagnetic beam with initially radial polarization,” Opt. Express 23(9), 11483–11492 (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]

O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
[Crossref]

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

2014 (4)

2013 (2)

2012 (1)

2011 (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

2010 (1)

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref]

2009 (3)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

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

T. V. Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the far field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009).
[Crossref]

2008 (1)

2007 (1)

2004 (1)

D. Palacios, I. Maleev, A. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[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]

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]

Bo, F.

Cai, Y.

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[Crossref]

J. Zeng, C. Liang, H. Wang, F. Wang, C. Zhao, G. Gbur, and Y. Cai, “Partially coherent radially polarized fractional vortex beam,” Opt. Express 28(8), 11493–11513 (2020).
[Crossref]

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

J. Zeng, R. Lin, X. Liu, C. Zhao, and Y. Cai, “Review on partially coherent vortex beams,” Front. Guided Wave Opt. Optoelectron. 12(3), 229–248 (2019).
[Crossref]

X. Liu, J. Zeng, and Y. Cai, “Review on vortex beams with low spatial coherence,” Adv. Phys.: X 4(1), 1626766 (2019).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[Crossref]

L. Guo, Y. Chen, X. Liu, L. Liu, and Y. Cai, “Vortex phase-induced changes of the statistical properties of a partially coherent radially polarized beam,” Opt. Express 24(13), 13714–13728 (2016).
[Crossref]

Y. Zhang, Y. Cui, F. Wang, and Y. Cai, “Correlation singularities in a partially coherent electromagnetic beam with initially radial polarization,” Opt. Express 23(9), 11483–11492 (2015).
[Crossref]

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

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]

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]

Chan, C.

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref]

Chen, Y.

L. Guo, Y. Chen, X. Liu, L. Liu, and Y. Cai, “Vortex phase-induced changes of the statistical properties of a partially coherent radially polarized beam,” Opt. Express 24(13), 13714–13728 (2016).
[Crossref]

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

Chen, Z.

Cheng, Y.

K. Liu, Y. Cheng, Y. Gao, X. Li, Y. Qin, and H. Wang, “Super-resolution radar imaging based on experimental OAM beams,” Appl. Phys. Lett. 110(16), 164102 (2017).
[Crossref]

Cui, Y.

Dai, W.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random electromagnetic Gaussian-shell model vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

de Sande, J. C. G.

Dijk, T. V.

T. V. Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the far field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009).
[Crossref]

Ding, J.

Dong, Y.

Y. Dong, S. Xi, B. Zhu, X. Wang, Q. Mu, S. Wang, and Z. Zhu, “The directional excitation of surface plasmon polaritons by radially polarized beam with multiple off-axis vortices,” Opt. Commun. 443, 197–201 (2019).
[Crossref]

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

Fan, X.

Gao, Y.

K. Liu, Y. Cheng, Y. Gao, X. Li, Y. Qin, and H. Wang, “Super-resolution radar imaging based on experimental OAM beams,” Appl. Phys. Lett. 110(16), 164102 (2017).
[Crossref]

Gbur, G.

Gori, F.

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]

Guo, H.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random electromagnetic Gaussian-shell model vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Guo, L.

Hu, J.

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[Crossref]

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]

Jabbour, T. G.

Konijnenberg, S.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

Korotkova, O.

Kuebler, S. M.

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]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

Li, H.

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[Crossref]

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, P.

Li, X.

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[Crossref]

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

K. Liu, Y. Cheng, Y. Gao, X. Li, Y. Qin, and H. Wang, “Super-resolution radar imaging based on experimental OAM beams,” Appl. Phys. Lett. 110(16), 164102 (2017).
[Crossref]

Liang, C.

Lin, R.

J. Zeng, R. Lin, X. Liu, C. Zhao, and Y. Cai, “Review on partially coherent vortex beams,” Front. Guided Wave Opt. Optoelectron. 12(3), 229–248 (2019).
[Crossref]

Lin, Z.

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref]

Liu, K.

K. Liu, Y. Cheng, Y. Gao, X. Li, Y. Qin, and H. Wang, “Super-resolution radar imaging based on experimental OAM beams,” Appl. Phys. Lett. 110(16), 164102 (2017).
[Crossref]

Liu, L.

Liu, S.

Liu, X.

X. Liu, J. Zeng, and Y. Cai, “Review on vortex beams with low spatial coherence,” Adv. Phys.: X 4(1), 1626766 (2019).
[Crossref]

J. Zeng, R. Lin, X. Liu, C. Zhao, and Y. Cai, “Review on partially coherent vortex beams,” Front. Guided Wave Opt. Optoelectron. 12(3), 229–248 (2019).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

L. Guo, Y. Chen, X. Liu, L. Liu, and Y. Cai, “Vortex phase-induced changes of the statistical properties of a partially coherent radially polarized beam,” Opt. Express 24(13), 13714–13728 (2016).
[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]

Long, Z.

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[Crossref]

Lu, X.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

Maleev, I.

D. Palacios, I. Maleev, A. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref]

Mandel, L.

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

Mao, Y.

Marathay, A.

D. Palacios, I. Maleev, A. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref]

Mei, Z.

Mi, C.

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]

Mu, Q.

Y. Dong, S. Xi, B. Zhu, X. Wang, Q. Mu, S. Wang, and Z. Zhu, “The directional excitation of surface plasmon polaritons by radially polarized beam with multiple off-axis vortices,” Opt. Commun. 443, 197–201 (2019).
[Crossref]

Ng, J.

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref]

Padgett, M. J.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Palacios, D.

D. Palacios, I. Maleev, A. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref]

Pang, X.

X. Zhao, J. Zhang, X. Pang, and G. Wan, “Properties of a strongly focused Gaussian beam with an off-axis vortex,” Opt. Commun. 389, 275–282 (2017).
[Crossref]

Peng, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

Piquero, G.

Ponomarenko, S. A.

Popov, S.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

Qi, S.

Qin, Y.

K. Liu, Y. Cheng, Y. Gao, X. Li, Y. Qin, and H. Wang, “Super-resolution radar imaging based on experimental OAM beams,” Appl. Phys. Lett. 110(16), 164102 (2017).
[Crossref]

Qu, J.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

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]

Ramírez-Sánchez, V.

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

Ren, Y.

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]

Sahin, S.

Santarsiero, M.

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB, (SPIE Press, Bellingham, WA2010).

Schouten, H. F.

T. V. Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the far field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009).
[Crossref]

Shao, Y.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

Shen, D.

Shen, Y.

Sheng, Z.

Shirai, T.

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

Swartzlander, G. A.

D. Palacios, I. Maleev, A. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref]

Tai, Y.

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[Crossref]

Tang, H.

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random electromagnetic Gaussian-shell model vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Tang, M.

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[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]

Urbach, H. P.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

Visser, T. D.

T. V. Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the far field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009).
[Crossref]

Wan, G.

X. Zhao, J. Zhang, X. Pang, and G. Wan, “Properties of a strongly focused Gaussian beam with an off-axis vortex,” Opt. Commun. 389, 275–282 (2017).
[Crossref]

Wang, F.

Wang, H.

J. Zeng, C. Liang, H. Wang, F. Wang, C. Zhao, G. Gbur, and Y. Cai, “Partially coherent radially polarized fractional vortex beam,” Opt. Express 28(8), 11493–11513 (2020).
[Crossref]

K. Liu, Y. Cheng, Y. Gao, X. Li, Y. Qin, and H. Wang, “Super-resolution radar imaging based on experimental OAM beams,” Appl. Phys. Lett. 110(16), 164102 (2017).
[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, S.

Wang, X.

Y. Dong, S. Xi, B. Zhu, X. Wang, Q. Mu, S. Wang, and Z. Zhu, “The directional excitation of surface plasmon polaritons by radially polarized beam with multiple off-axis vortices,” Opt. Commun. 443, 197–201 (2019).
[Crossref]

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

Wei, B.

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.

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

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

Wu, G.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random electromagnetic Gaussian-shell model vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

Xi, S.

Y. Dong, S. Xi, B. Zhu, X. Wang, Q. Mu, S. Wang, and Z. Zhu, “The directional excitation of surface plasmon polaritons by radially polarized beam with multiple off-axis vortices,” Opt. Commun. 443, 197–201 (2019).
[Crossref]

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]

Xu, H.

Yan, 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]

Yao, A. M.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

Zeng, J.

J. Zeng, C. Liang, H. Wang, F. Wang, C. Zhao, G. Gbur, and Y. Cai, “Partially coherent radially polarized fractional vortex beam,” Opt. Express 28(8), 11493–11513 (2020).
[Crossref]

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

X. Liu, J. Zeng, and Y. Cai, “Review on vortex beams with low spatial coherence,” Adv. Phys.: X 4(1), 1626766 (2019).
[Crossref]

J. Zeng, R. Lin, X. Liu, C. Zhao, and Y. Cai, “Review on partially coherent vortex beams,” Front. Guided Wave Opt. Optoelectron. 12(3), 229–248 (2019).
[Crossref]

Zeng, T.

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Zhang, J.

X. Zhao, J. Zhang, X. Pang, and G. Wan, “Properties of a strongly focused Gaussian beam with an off-axis vortex,” Opt. Commun. 389, 275–282 (2017).
[Crossref]

Zhang, R.

Zhang, X.

Zhang, Y.

Zhao, C.

J. Zeng, C. Liang, H. Wang, F. Wang, C. Zhao, G. Gbur, and Y. Cai, “Partially coherent radially polarized fractional vortex beam,” Opt. Express 28(8), 11493–11513 (2020).
[Crossref]

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

J. Zeng, R. Lin, X. Liu, C. Zhao, and Y. Cai, “Review on partially coherent vortex beams,” Front. Guided Wave Opt. Optoelectron. 12(3), 229–248 (2019).
[Crossref]

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
[Crossref]

Zhao, D.

Zhao, J.

Zhao, X.

X. Zhao, J. Zhang, X. Pang, and G. Wan, “Properties of a strongly focused Gaussian beam with an off-axis vortex,” Opt. Commun. 389, 275–282 (2017).
[Crossref]

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]

Zhu, B.

Y. Dong, S. Xi, B. Zhu, X. Wang, Q. Mu, S. Wang, and Z. Zhu, “The directional excitation of surface plasmon polaritons by radially polarized beam with multiple off-axis vortices,” Opt. Commun. 443, 197–201 (2019).
[Crossref]

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

Zhu, L.

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[Crossref]

Zhu, X.

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

Zhu, Z.

Y. Dong, S. Xi, B. Zhu, X. Wang, Q. Mu, S. Wang, and Z. Zhu, “The directional excitation of surface plasmon polaritons by radially polarized beam with multiple off-axis vortices,” Opt. Commun. 443, 197–201 (2019).
[Crossref]

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

Adv. Opt. Photonics (3)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[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]

Adv. Phys.: X (1)

X. Liu, J. Zeng, and Y. Cai, “Review on vortex beams with low spatial coherence,” Adv. Phys.: X 4(1), 1626766 (2019).
[Crossref]

Appl. Phys. Lett. (4)

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110(18), 181104 (2017).
[Crossref]

K. Liu, Y. Cheng, Y. Gao, X. Li, Y. Qin, and H. Wang, “Super-resolution radar imaging based on experimental OAM beams,” Appl. Phys. Lett. 110(16), 164102 (2017).
[Crossref]

J. Hu, Y. Tai, L. Zhu, Z. Long, M. Tang, H. Li, X. Li, and Y. Cai, “Optical vortex with multi-fractional orders,” Appl. Phys. Lett. 116(20), 201107 (2020).
[Crossref]

X. Lu, C. Zhao, Y. Shao, J. Zeng, S. Konijnenberg, X. Zhu, S. Popov, H. P. Urbach, and Y. Cai, “Phase detection of coherence singularities and determination of the topological charge of a partially coherent vortex beam,” Appl. Phys. Lett. 114(20), 201106 (2019).
[Crossref]

Front. Guided Wave Opt. Optoelectron. (1)

J. Zeng, R. Lin, X. Liu, C. Zhao, and Y. Cai, “Review on partially coherent vortex beams,” Front. Guided Wave Opt. Optoelectron. 12(3), 229–248 (2019).
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IEEE Photonics J. (1)

Z. Mei, “Modeling for Partially Spatially Coherent Vortex Beams,” IEEE Photonics J. 9(5), 1–6 (2017).
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F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A: Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Opt. Commun. (3)

Y. Dong, S. Xi, B. Zhu, X. Wang, Q. Mu, S. Wang, and Z. Zhu, “The directional excitation of surface plasmon polaritons by radially polarized beam with multiple off-axis vortices,” Opt. Commun. 443, 197–201 (2019).
[Crossref]

G. Wu, W. Dai, H. Tang, and H. Guo, “Beam wander of random electromagnetic Gaussian-shell model vortex beams propagating through a Kolmogorov turbulence,” Opt. Commun. 336, 55–58 (2015).
[Crossref]

X. Zhao, J. Zhang, X. Pang, and G. Wan, “Properties of a strongly focused Gaussian beam with an off-axis vortex,” Opt. Commun. 389, 275–282 (2017).
[Crossref]

Opt. Express (8)

L. Guo, Y. Chen, X. Liu, L. Liu, and Y. Cai, “Vortex phase-induced changes of the statistical properties of a partially coherent radially polarized beam,” Opt. Express 24(13), 13714–13728 (2016).
[Crossref]

X. Zhang, P. Li, S. Liu, B. Wei, S. Qi, X. Fan, S. Wang, Y. Zhang, and J. Zhao, “Autofocusing of ring Airy beams embedded with off-axial vortex singularities,” Opt. Express 28(6), 7953–7960 (2020).
[Crossref]

X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express 25(22), 26844–26852 (2017).
[Crossref]

J. Zeng, C. Liang, H. Wang, F. Wang, C. Zhao, G. Gbur, and Y. Cai, “Partially coherent radially polarized fractional vortex beam,” Opt. Express 28(8), 11493–11513 (2020).
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H. Xu, R. Zhang, Z. Sheng, and J. Qu, “Focus shaping of partially coherent radially polarized vortex beam with tunable topological charge,” Opt. Express 27(17), 23959–23969 (2019).
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Y. Zhang, Y. Cui, F. Wang, and Y. Cai, “Correlation singularities in a partially coherent electromagnetic beam with initially radial polarization,” Opt. Express 23(9), 11483–11492 (2015).
[Crossref]

C. Liang, C. Mi, F. Wang, C. Zhao, Y. Cai, and S. A. Ponomarenko, “Vector optical coherence lattices generating controllable far-field beam profiles,” Opt. Express 25(9), 9872–9885 (2017).
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T. G. Jabbour and S. M. Kuebler, “Vectorial beam shaping,” Opt. Express 16(10), 7203–7213 (2008).
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Opt. Lett. (11)

Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. 41(9), 1929–1932 (2016).
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Z. Mei, O. Korotkova, and Y. Mao, “Products of Schell-model cross-spectral densities,” Opt. Lett. 39(24), 6879–6882 (2014).
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O. Korotkova and Z. Mei, “Convolution of degrees of coherence,” Opt. Lett. 40(13), 3073–3076 (2015).
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S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
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Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
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Z. Mei, “Light sources generating self-focusing beams of variable focal length,” Opt. Lett. 39(2), 347–350 (2014).
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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).
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D. Shen and D. Zhao, “Measuring the topological charge of optical vortices with a twisting phase,” Opt. Lett. 44(9), 2334–2337 (2019).
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Phys. Rev. A (2)

T. V. Dijk, H. F. Schouten, and T. D. Visser, “Coherence singularities in the far field generated by partially coherent sources,” Phys. Rev. A 79(3), 033805 (2009).
[Crossref]

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

Phys. Rev. Lett. (3)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref]

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010).
[Crossref]

D. Palacios, I. Maleev, A. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004).
[Crossref]

Other (3)

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

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

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB, (SPIE Press, Bellingham, WA2010).

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Figures (6)

Fig. 1.
Fig. 1. (a) An illustration of the position vector relationship between the j-th off-axis vortices core and the observation point in the initial plane. (b) Arrangement of three off-axis vortices (red circular) and (c) intensity distribution of a PCRP beam carrying three off-axis vortices in the source plane, respectively.
Fig. 2.
Fig. 2. Average intensity distribution $I$ and its component ${I_x}$ and ${I_y}$ of a focused PCRP beam carrying three off-axis vortices with $l = 1,{s_0} = 0.5{w_0}$ and ${\sigma _g} = 3\textrm{mm}$ in the transverse plane at several propagation distances $z$.
Fig. 3.
Fig. 3. The same as Fig. 2 but for ${\sigma _g} = 0.5\textrm{mm}$.
Fig. 4.
Fig. 4. Average intensity distribution I of a focused PCRP beam carrying various numbers of off-axis vortices with $l = 1,{s_0} = 0.5{w_0}$ in the focal plane for different values of coherence length ${\sigma _g}$. The first column shows the location and arrangement of off-axis vortex arrays.
Fig. 5.
Fig. 5. Average intensity distribution I of a focused PCRP beam carrying various numbers of off-axis vortices with $l = 1,{\sigma _g}\textrm{ = }3\,\textrm{mm}$ in the focal plane for different values of off-axis distance ${s_0}$. The location and arrangement of off-axis vortex arrays are the same as Fig. 4 in the first column.
Fig. 6.
Fig. 6. The generation of equilateral-polygon-like flat-top focus by using a PCRP beam carrying multiple off-axis vortices with an identical topological charge of $l\textrm{ = }3$ and off-axis distance of ${s_0} = 0.5{w_0}$. The number ${N_0}$ of off-axis vortices required for generation of bar-like, triangle-like, square-like, and pentagon-like flat-top focus are in sequence 2, 3, 4, and 5, meanwhile, the location and arrangement of off-axis vortex arrays are the same as Fig. 4 in the first column, but different values of coherence length ${\sigma _g}$ are also required, respectively.

Equations (18)

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W α β ( r 1 , r 2 , ω ) = E α ( r 1 ; ω ) E β ( r 2 ; ω ) , ( α , β = x , y ) ,
W α β ( r 1 , r 2 ) = p α β ( v ) H α ( r 1 , v ) H β ( r 2 , v ) d 2 v
H α ( r , v )  =  F α ( r ) f ( r ) exp ( i k r v ) ,
H α ( r , v ) = α w 0 exp ( r 2 w 0 2 ) exp ( i l φ 10 ) exp ( i l φ 20 ) × exp ( i l φ N 0 0 ) exp ( i k r v ) = α w 0 exp ( r 2 w 0 2 ) j = 1 N 0 exp ( i l φ j 0 ) exp ( i k r v ) , ( α  =  x , y ) .
p α β ( v )  =  p ( v )  =  k 2 σ g 2 2 π exp ( k 2 σ g 2 v 2 / 2 ) ,
W α β ( r 1 , r 2 , 0 ) = α 1 β 2 w 0 2 exp ( r 1 2 + r 2 2 w 0 2 ) j = 1 N 0 exp [ i l ( φ j 10 φ j 20 ) ] exp [ ( r 1 r 2 ) 2 2 σ g 2 ] .
W α β ( ρ 1 , ρ 2 , z ) = k 2 4 π 2 B 2 W α β ( r 1 , r 2 ) exp [ i k A 2 B r 1 2 + i k B r 1 ρ 1 i k D 2 B ρ 1 2 ] × exp [ i k A 2 B r 2 2 i k B r 2 ρ 2 + i k D 2 B ρ 2 2 ] d 2 r 1 d 2 r 2 ,
r s  =  r 1  +  r 2 2 , r d = r 1 r 2 ,   ρ s = ρ 1 + ρ 2 2 ,   ρ d = ρ 1 ρ 2 ,  
W α β ( ρ 1 , ρ 2 , z ) = k 2 4 π 2 B 2 exp ( i k D B ρ s ρ d ) A α ( r s r d / 2 ) A β ( r s + r d / 2 ) p α β ( v ) × exp [ i k B ( r s ρ d + r d ρ s ) ] exp ( i k r d v ) d 2 r s d 2 r d d 2 v ,   ( α , β = x , y ) .
A x ( r )  =  x w 0 exp ( r 2 w 0 2 ) j = 1 N 0 exp ( i l φ j 0 ) exp ( i k A r 2 / 2 B ) ,
A y ( r )  =  y w 0 exp ( r 2 w 0 2 ) j = 1 N 0 exp ( i l φ j 0 ) exp ( i k A r 2 / 2 B ) .
A α ( r s r d / 2 )  =  ( k 2 π ) 2 A ~ α ( u 1 ) exp [ i k u 1 ( r s r d / 2 ) ] d 2 u 1 ,
A β ( r s + r d / 2 )  =  ( k 2 π ) 2 A ~ β ( u 2 ) exp [ i k u 2 ( r s  +  r d / 2 ) ] d 2 u 2 .
W α β ( ρ 1 , ρ 2 , z ) = k 2 4 π 2 B 2 exp ( i k D B ρ s ρ d ) A ~ α ( v  +  ( 2 ρ s ρ d ) / 2 B ) × A ~ β ( v  +  ( 2 ρ s + ρ d ) / 2 B ) p α β ( v ) d 2 v ,
W x x ( ρ , ρ , z ) = I x x ( ρ , z ) = λ 2 k 2 4 π 2 B 2 | A ~ x ( v λ  +  ρ λ B ) | 2 p ( v λ ) d 2 v λ  =    S f x ( ρ λ B ) p ( ρ λ B ) ,
W y y ( ρ , ρ , z ) = I y y ( ρ , z ) = λ 2 k 2 4 π 2 B 2 | A ~ y ( v λ  +  ρ λ B ) | 2 p ( v λ ) d 2 v λ  =    S f y ( ρ λ B ) p ( ρ λ B ) .
I ( ρ , z )  =  I x x ( ρ , z )  +  I y y ( ρ , z ) .
( A B C D ) = ( 1 z 0 1 ) ( 1 0 1 / f 1 ) = ( 1 z / f z 1 / f 1 ) .

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