Abstract

Circular symmetry singular light beams (CS-SLBs) possessing spatially variant field distributions have drawn extensive attention because of their unique optical properties. However, the extraction of spatial phase and polarization distributions is always a significant but difficult issue in CS-SLB applications. Here, we propose and experimentally investigate an orthogonal polarization separation (OPS) method to retrieve the spatial phase and polarization distributions of arbitrary CS-SLBs. Theoretically, the CS-SLB, including the vortex beam (VB), cylindrical vector beam (CVB), and cylindrical vector vortex beam (CVVB), can be decomposed into two orthogonal circularly polarized sub-VBs. Therefore, once the spatial phase distributions and initial phase difference of the two components are obtained, the phase and polarization distributions of the CS-SLB can be retrieved, and its type can also be identified. Based on this analysis relationship, we first separated the CS-SLB into two circularly polarized sub-VBs and designed an astigmatic phase iterative algorithm to restore their spatial phase information. After retrieving the phases of the two components, we have experimentally obtained the spatial phase and polarization distributions of three typical CS-SLBs, including VBs, CVBs, and CVVBs. These results demonstrate that this method provides a feasible way to retrieve the variant field distributions of CS-SLBs and may have great application prospects in optical imaging, optical communication, etc.

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

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References

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2019 (2)

N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
[Crossref]

A. Volyar, M. Bretsko, Y. Akimova, and Y. Egorov, “Measurement of the vortex and orbital angular momentum spectra with a single cylindrical lens,” Appl. Opt. 58(21), 5748–5755 (2019).
[Crossref]

2018 (2)

2017 (8)

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

W. Qiao, T. Lei, Z. Wu, S. Gao, Z. Li, and X. C. Yuan, “Approach to multiplexing fiber communication with cylindrical vector beams,” Opt. Lett. 42(13), 2579–2582 (2017).
[Crossref]

C. Liang, C. Ping, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
[Crossref]

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic transforms of an optical vortex for measurement of its topological charge,” Appl. Opt. 56(14), 4095–4104 (2017).
[Crossref]

2015 (2)

2014 (4)

2013 (2)

2012 (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

2011 (5)

Z. Wang, N. Zhang, and X. C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free- space optical communication,” Opt. Express 19(2), 482–492 (2011).
[Crossref]

F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express 19(25), 25143–25150 (2011).
[Crossref]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

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

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref]

2010 (2)

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
[Crossref]

2009 (3)

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

2004 (1)

2003 (2)

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

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003).
[Crossref]

2002 (2)

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

2000 (1)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

1994 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

1989 (1)

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

1982 (1)

Agarwal, K.

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Akimova, Y.

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Barnett, S. M.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Boyd, R. W.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

Bretsko, M.

Brown, T. G.

Cai, Y.

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

C. Liang, C. Ping, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
[Crossref]

Capasso, F.

N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
[Crossref]

Chapman, H. N.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003).
[Crossref]

Checalier, P.

N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
[Crossref]

Chen, R.

Chen, S.

R. Wang, S. He, S. Chen, J. Zhang, W. Shu, H. Luo, and S. Wen, “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Opt. Lett. 43(15), 3570–3573 (2018).
[Crossref]

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
[Crossref]

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
[Crossref]

Chen, W. T.

N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
[Crossref]

Chen, X.

Chen, Y.

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

Cheng, H.

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

Coullet, P.

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

D’Aversa, G.

N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
[Crossref]

Dolinar, S.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Dong, B.

Dorn, R.

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

Du, J.

Egorov, Y.

Ersoy, O. K.

Fan, D.

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

Fatemi, F. K.

Fazal, I. M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Fienup, J. R.

Franke-Arnold, S.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

Gao, S.

Gil, G.

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Gu, B.

Han, L.

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

He, S.

He, Y.

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
[Crossref]

Henderson, C. A.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Huang, H.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ireland, D. G.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

Jack, B.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

Jha, A. K.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

Kazanskiy, N. L.

Ke, Y.

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

Khonina, S. N.

Kotlyar, V. V.

Kovalev, A. A.

Kozawa, Y.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[Crossref]

Kumar, V.

Leach, J.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

Lee, D. J.

Leger, J. R.

Lei, T.

Leuchs, G.

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

Li, P.

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

Li, S.

Li, Y.

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
[Crossref]

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
[Crossref]

Li, Z.

Liang, C.

Ling, X.

Liu, J.

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
[Crossref]

Liu, S.

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

Liu, Y.

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
[Crossref]

Liu, Z.

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

Luo, H.

R. Wang, S. He, S. Chen, J. Zhang, W. Shu, H. Luo, and S. Wen, “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Opt. Lett. 43(15), 3570–3573 (2018).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
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Mancuso, A. P.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003).
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L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
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Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
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G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011).
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G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011).
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Nugent, K. A.

C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17(14), 11905–11915 (2009).
[Crossref]

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003).
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M. Padgett, “Light’s twist,” Proc. R. Soc. London, Ser. A 470(2172), 20140633 (2014).
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M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
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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]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

Peele, A. G.

C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17(14), 11905–11915 (2009).
[Crossref]

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003).
[Crossref]

Ping, C.

Porfirev, A. P.

Qiao, W.

Qubis, S.

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

Quiney, H. M.

Ren, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

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P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
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N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
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T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
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Savelyev, D. A.

Sheppard, C. J.

Shi, Z.

N. A. Rubin, G. D’Aversa, P. Checalier, Z. Shi, W. T. Chen, and F. Capasso, “Matrix Fourier optics enables a compact full-Stokes polarization camera,” Science 365(6448), eaax1839 (2019).
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T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
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Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
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Shu, W.

R. Wang, S. He, S. Chen, J. Zhang, W. Shu, H. Luo, and S. Wen, “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Opt. Lett. 43(15), 3570–3573 (2018).
[Crossref]

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
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Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-Order Poincaré Sphere, Stokes Parameters, and the Angular Momentum of Light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
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Toussaint, K. C.

Tripathi, S.

Tur, M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Viswanathan, N. K.

Volyar, A.

Wang, F.

Wang, J.

Wang, R.

Wang, Z.

Weiner, A. M.

Wen, S.

R. Wang, S. He, S. Chen, J. Zhang, W. Shu, H. Luo, and S. Wen, “Electrically driven generation of arbitrary vector vortex beams on the hybrid-order Poincaré sphere,” Opt. Lett. 43(15), 3570–3573 (2018).
[Crossref]

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
[Crossref]

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
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Williams, G. J.

Willner, A. E.

Z. Zhao, J. Wang, S. Li, and A. E. Willner, “Metamaterials-based broadband generation of orbital angular momentum carrying vector beams,” Opt. Lett. 38(6), 932–934 (2013).
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J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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Wu, Z.

Xiang, Y.

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
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Xie, Z.

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
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J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Yang, G.

Yang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[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).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
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Ye, H.

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
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Yi, X.

Youngworth, K. S.

Yuan, X. C.

Yue, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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Zhang, J.

Zhang, N.

Zhang, X.

Y. He, Y. Li, J. Liu, X. Zhang, Y. Cai, Y. Chen, S. Chen, and D. Fan, “Switchable phase and polarization singular beams generation using dielectric metasurfaces,” Sci. Rep. 7(1), 6814 (2017).
[Crossref]

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
[Crossref]

Zhang, Y.

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

Zhang, Z.

Zhao, J.

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

Zhao, Z.

Zhou, J.

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref]

Zhou, X.

Zhuang, J.

Adv. Opt. Photonics (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
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Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
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Appl. Opt. (4)

Appl. Phys. Lett. (1)

S. Liu, L. Han, P. Li, Y. Zhang, H. Cheng, and J. Zhao, “A method for simultaneously measuring polarization and phase of arbitrarily polarized beams based on Pancharatnam-Berry phase,” Appl. Phys. Lett. 110(17), 171112 (2017).
[Crossref]

IEEE Photonics J. (1)

Y. He, H. Ye, J. Liu, Z. Xie, X. Zhang, Y. Xiang, S. Chen, Y. Li, and D. Fan, “Order-Controllable Cylindrical Vector Vortex Beam Generation by Using Spatial Light Modulator and Cascaded Metasurfaces,” IEEE Photonics J. 9(5), 6101710 (2017).
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J. Opt. Soc. Am. B (1)

Nat. Photonics (2)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Opt. Commun. (1)

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Opt. Express (12)

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
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Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
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Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18(10), 10828–10833 (2010).
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K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7(2), 77–87 (2000).
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Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
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C. A. Henderson, G. J. Williams, A. G. Peele, H. M. Quiney, and K. A. Nugent, “Astigmatic phase retrieval: an experimental demonstration,” Opt. Express 17(14), 11905–11915 (2009).
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S. Tripathi and K. C. Toussaint, “Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection,” Opt. Express 17(24), 21396–21407 (2009).
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Z. Wang, N. Zhang, and X. C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free- space optical communication,” Opt. Express 19(2), 482–492 (2011).
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C. Liang, C. Ping, F. Wang, and Y. Cai, “Radially polarized multi-Gaussian Schell-model beam and its tight focusing properties,” Opt. Express 25(26), 32475–32490 (2017).
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F. K. Fatemi, “Cylindrical vector beams for rapid polarization-dependent measurements in atomic systems,” Opt. Express 19(25), 25143–25150 (2011).
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D. J. Lee and A. M. Weiner, “Optical phase imaging using a synthetic aperture phase retrieval technique,” Opt. Express 22(8), 9380–9394 (2014).
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S. N. Khonina, D. A. Savelyev, and N. L. Kazanskiy, “Vortex phase elements as detectors of polarization state,” Opt. Express 23(14), 17845–17859 (2015).
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Opt. Lett. (6)

Photonics Res. (1)

Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017).
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Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Phys. Rev. Lett. (6)

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91(20), 203902 (2003).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

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

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[Crossref]

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[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. The schematic diagram of the API algorithm.
Fig. 2.
Fig. 2. The interference intensity distributions (first and second rows) of VBs and the spherical wave with different ratios, and the restored phase distributions (third and fourth rows), ${\varphi _0}$ is the initial phase.
Fig. 3.
Fig. 3. The experimental setup for phase and polarization retrieval of arbitrary CS-SLBs. (a) and (b) are the cross- and parallel-polarized measured images of the Q-plate. (c) and (d) are the theoretical and measured optical axis distributions of the Q-plate with q = 0.5. LP: linear polarizer; PBS: polarization beam splitter; CS-SLBGS: CS-SLBs generating system; HWP: half-wave plate; QWP: quarter-wave plate; BS: beam splitter; M: mirror. The inserted pictures are the corresponded optical intensity distributions.
Fig. 4.
Fig. 4. (a) The schematic diagram of generating VBs with Q-plates. (b) The intensity distributions of VBs (upper) and the diffraction patterns of the VBs pass through C-lens (bottom) with $l = - 3, - 2, - 1,1,2,3$.
Fig. 5.
Fig. 5. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of VBs with l = 1∼3 (from the first row to the end row). The blue ellipses represent the measured polarization states.
Fig. 6.
Fig. 6. (a) The schematic diagram of generating CVBs with Q-plates. (b)The intensity distributions of the CVBs (m = 1∼3) filtered by the LP. The white arrow indicates the transmission axis of the polarizer.
Fig. 7.
Fig. 7. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of the CVBs (m = 1∼3). The blue ellipses represent the measured polarization states.
Fig. 8.
Fig. 8. (a) The schematic diagram of generating CVVBs by cascading Q-plates. (b) The intensity distributions of CVVBs filtered by the polarizer, m= −1, l = 2 (upper); m = 2, l= −1 (lower). The white arrow indicates the transmission axis of the polarizer.
Fig. 9.
Fig. 9. (a) The interferograms (VB interferes with a spherical wave) and retrieved phases of LHCP and RHCP components. (b) The retrieved phase and polarization distributions of CVVBs with m= −1, l = 2 (upper); m = 2, l= −1 (lower). The blue ellipses represent the measured polarization states.

Equations (9)

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E v o r t e x = E 0 exp [ i ( l θ + φ 0 ) ] [ 1 0 ] = 1 2 E 0 exp [ i ( l θ + φ 0 ) ] [ 1 i ] + 1 2 E 0 exp [ i ( l θ + φ 0 ) ] [ 1 i ] ,
E v e c t o r = E 0 [ cos(m θ + φ 0 ) sin(m θ + φ 0 ) ] = 1 2 E 0 exp ( i φ 0 ) exp ( i m θ ) [ 1 i ] + 1 2 E 0 exp ( i φ 0 ) exp ( i m θ ) [ 1 i ] ,
E v e c t o r _ v o r t e x = E 0 exp ( i l θ ) [ cos(m θ + φ 0 ) sin(m θ + φ 0 ) ] = 1 2 E 0 exp ( i φ 0 ) exp [ i ( l m ) θ ] [ 1 i ] + 1 2 E 0 exp ( i φ 0 ) exp [ i ( l + m ) θ ] [ 1 i ] ,
Φ n = exp ( i k ( x cos ( θ ) y sin ( θ ) ) 2 / 2 / f ) .
ε ( p ) = 1 N j ( I j I j p ) 2 ,
S 0 = E 1 2 + E 2 2 ,
S 1 = E 1 + E 2 2 + E 1 E 2 2 ,
S 2 = ( 1 i ) E 1 + ( 1 + i ) E 2 2 + ( 1 + i ) E 1 + ( 1 i ) E 2 2 2 ( 1 + i ) E 1 + ( 1 i ) E 2 2 ,
S 3 = ( 1 + i ) E 1 2 ( ( 1 + i ) E 2 ) 2 ,

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