Abstract

The transverse structure of light is recognized as a resource that can be used to encode information onto photons and has been shown to be useful to enhance communication capacity as well as resolve point sources in superresolution imaging. The Laguerre–Gaussian (LG) modes form a complete and orthonormal basis set and are described by a radial index p and an orbital angular momentum (OAM) index ℓ. Earlier works have shown how to build a sorter for the radial index p or/and the OAM index ℓ of LG modes, but a scalable and dedicated LG mode sorter which simultaneous determinate p and ℓ is immature. Here we propose and experimentally demonstrate a scheme to accomplish complete LG mode sorting, which consists of a novel, robust radial mode sorter that can be used to couple radial modes to polarizations, an ℓ-dependent phase shifter and an OAM mode sorter. Our scheme is in principle efficient, scalable, and crosstalk-free, and therefore has potential for applications in optical communications, quantum information technology, superresolution imaging, and fiber optics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (2)

X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
[Crossref]

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: Concepts and experiments,” Phys. Rev. Lett. 120, 103601 (2018).
[Crossref] [PubMed]

2017 (3)

2016 (1)

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

2015 (3)

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
[Crossref]

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
[Crossref]

2014 (3)

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

E. Karimi, R. W. Boyd, P. D. L. Hoz, H. D. Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of laguerre-gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via hong-ou-mandel interference,” Phys. Rev. A 89, 013829 (2014).
[Crossref]

2013 (4)

. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354 (2013).
[Crossref]

M. Mirhosseini, O. S. Magana-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
[Crossref]

2012 (4)

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

P. Zhang, Y. Jiang, R. Liu, H. Gao, H. Li, and F. Li, “Implementing the deutsch’s algorithm with spin-orbital angular momentum of photon without interferometer,” Opt. Commun. 285, 838–841 (2012).
[Crossref]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012).
[Crossref]

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (1)

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

2009 (1)

E. Nagali, F. Sciarrino, F. D. Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

2007 (1)

P. Zhang, X. Ren, X. Zou, B. Liu, Y. Huang, and G. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75, 052310 (2007).
[Crossref]

2006 (2)

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

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

2004 (2)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

2002 (1)

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, 257901 (2002).
[Crossref] [PubMed]

2001 (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[Crossref]

1996 (1)

U. Leonhardt, “Discrete wigner function and quantum-state tomography,” Phys. Rev. A 53, 2998 (1996).
[Crossref] [PubMed]

1994 (1)

L. B. Almeida, “The fractional fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[Crossref]

1993 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Ahmed, N.

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

Aiello, A.

E. Karimi, R. W. Boyd, P. D. L. Hoz, H. D. Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of laguerre-gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Almeida, L. B.

L. B. Almeida, “The fractional fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[Crossref]

Bade, S.

S. Bade, B. Denolle, G. Trunet, N. Riguet, P. Jian, O. Pinel, and G. Labroille, “Fabrication and characterization of a mode-selective 45-mode spatial multiplexer based on multi-plane light conversion,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

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, 257901 (2002).
[Crossref] [PubMed]

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Beijersbergen, M. W.

G. C. G. Berkhout, M. P. J. Lavery, M. J. Padgett, and M. W. Beijersbergen, “Measuring orbital angular momentum superpositions of light by mode transformation,” Opt. Lett. 36, 1863–1865 (2011).
[Crossref] [PubMed]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Bent, N.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via hong-ou-mandel interference,” Phys. Rev. A 89, 013829 (2014).
[Crossref]

Berkhout, G. C. G.

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
[Crossref] [PubMed]

G. C. G. Berkhout, M. P. J. Lavery, M. J. Padgett, and M. W. Beijersbergen, “Measuring orbital angular momentum superpositions of light by mode transformation,” Opt. Lett. 36, 1863–1865 (2011).
[Crossref] [PubMed]

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
[Crossref]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

Bolduc, E.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via hong-ou-mandel interference,” Phys. Rev. A 89, 013829 (2014).
[Crossref]

Bouchard, F.

Boyd, R.

Boyd, R. W.

H. Larocque, J. Gagnon-Bischoff, D. Mortimer, Y. Zhang, F. Bouchard, J. Upham, V. Grillo, R. W. Boyd, and E. Karimi, “Generalized optical angular momentum sorter and its application to high-dimensional quantum cryptography,” Opt. Express 25, 19832–19843 (2017).
[Crossref] [PubMed]

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via hong-ou-mandel interference,” Phys. Rev. A 89, 013829 (2014).
[Crossref]

E. Karimi, R. W. Boyd, P. D. L. Hoz, H. D. Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of laguerre-gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

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X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
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X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488 (2012).
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M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
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J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
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M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
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Guo, G.

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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488 (2012).
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P. Zhang, X. Ren, X. Zou, B. Liu, Y. Huang, and G. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75, 052310 (2007).
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M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
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S. Bade, B. Denolle, G. Trunet, N. Riguet, P. Jian, O. Pinel, and G. Labroille, “Fabrication and characterization of a mode-selective 45-mode spatial multiplexer based on multi-plane light conversion,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Jiang, X.

X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
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P. Zhang, Y. Jiang, R. Liu, H. Gao, H. Li, and F. Li, “Implementing the deutsch’s algorithm with spin-orbital angular momentum of photon without interferometer,” Opt. Commun. 285, 838–841 (2012).
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E. Karimi, R. W. Boyd, P. D. L. Hoz, H. D. Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of laguerre-gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via hong-ou-mandel interference,” Phys. Rev. A 89, 013829 (2014).
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N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-gaussian mode sorter,” arXiv preprint arXiv:1803.04126 (2018).

Krenn, M.

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: Concepts and experiments,” Phys. Rev. Lett. 120, 103601 (2018).
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M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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S. Bade, B. Denolle, G. Trunet, N. Riguet, P. Jian, O. Pinel, and G. Labroille, “Fabrication and characterization of a mode-selective 45-mode spatial multiplexer based on multi-plane light conversion,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

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N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
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M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
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Lavery, M. P. J.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
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G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
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Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. J. Lavery, A. E. Willner, and R. W. Boyd, “High-dimensional free-space quantum key distribution using spin, azimuthal, and radial quantum numbers,” arXiv:1809.09986 (2018).

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J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
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E. Karimi, R. W. Boyd, P. D. L. Hoz, H. D. Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of laguerre-gauss modes,” Phys. Rev. A 89, 063813 (2014).
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Li, F.

P. Zhang, Y. Jiang, R. Liu, H. Gao, H. Li, and F. Li, “Implementing the deutsch’s algorithm with spin-orbital angular momentum of photon without interferometer,” Opt. Commun. 285, 838–841 (2012).
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F. Wang, P. Zeng, X. Wang, H. Gao, F. Li, and P. Zhang, “Towards practical high-speed high dimensional quantum key distribution using partial mutual unbiased basis of photon’s orbital angular momentum,” arXiv:1801.06582 (2018).

Li, G.

N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
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P. Zhang, Y. Jiang, R. Liu, H. Gao, H. Li, and F. Li, “Implementing the deutsch’s algorithm with spin-orbital angular momentum of photon without interferometer,” Opt. Commun. 285, 838–841 (2012).
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X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
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X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
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N. Zhao, X. Li, G. Li, and J. M. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photon. 9, 822–826 (2015).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
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P. Zhang, X. Ren, X. Zou, B. Liu, Y. Huang, and G. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75, 052310 (2007).
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X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
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T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
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X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
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P. Zhang, Y. Jiang, R. Liu, H. Gao, H. Li, and F. Li, “Implementing the deutsch’s algorithm with spin-orbital angular momentum of photon without interferometer,” Opt. Commun. 285, 838–841 (2012).
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Love, G. D.

Lu, C.

X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
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M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

Luo, Y.

X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
[Crossref]

Magana-Loaiza, O. S.

Magaña-Loaiza, O. S.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Malik, M.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

M. Mirhosseini, O. S. Magana-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
[Crossref]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012).
[Crossref]

Manzo, C.

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

Marquardt, C.

Marrucci, L.

E. Nagali, F. Sciarrino, F. D. Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

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

Martini, F. D.

E. Nagali, F. Sciarrino, F. D. Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Miatto, F. M.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via hong-ou-mandel interference,” Phys. Rev. A 89, 013829 (2014).
[Crossref]

Min, C.

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
[Crossref]

Mirhosseini, . M.

. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

Mirhosseini, M.

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

M. Mirhosseini, O. S. Magana-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
[Crossref]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012).
[Crossref]

Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. J. Lavery, A. E. Willner, and R. W. Boyd, “High-dimensional free-space quantum key distribution using spin, azimuthal, and radial quantum numbers,” arXiv:1809.09986 (2018).

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[Crossref]

Mortimer, D.

Nagali, E.

E. Nagali, F. Sciarrino, F. D. Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Nair, R.

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

Neilson, D. T.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-gaussian mode sorter,” arXiv preprint arXiv:1803.04126 (2018).

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354 (2013).
[Crossref]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University, 2000).

Niu, H.

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
[Crossref]

O’Brien, J. L.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

O’Sullivan, M. N.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012).
[Crossref]

Oliver, S.

Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. J. Lavery, A. E. Willner, and R. W. Boyd, “High-dimensional free-space quantum key distribution using spin, azimuthal, and radial quantum numbers,” arXiv:1809.09986 (2018).

Padgett, M. J.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via hong-ou-mandel interference,” Phys. Rev. A 89, 013829 (2014).
[Crossref]

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
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G. C. G. Berkhout, M. P. J. Lavery, M. J. Padgett, and M. W. Beijersbergen, “Measuring orbital angular momentum superpositions of light by mode transformation,” Opt. Lett. 36, 1863–1865 (2011).
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M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, “Measurement of the light orbital angular momentum spectrum using an optical geometric transformation,” J. Opt. 13, 064006 (2011).
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G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
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J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

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, 257901 (2002).
[Crossref] [PubMed]

Pan, J.

X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
[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, 163905 (2006).
[Crossref] [PubMed]

Peuntinger, C.

Piccirillo, B.

E. Nagali, F. Sciarrino, F. D. Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Pinel, O.

S. Bade, B. Denolle, G. Trunet, N. Riguet, P. Jian, O. Pinel, and G. Labroille, “Fabrication and characterization of a mode-selective 45-mode spatial multiplexer based on multi-plane light conversion,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Pryde, G. J.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Rafsanjani, S. M. H.

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. J. Lavery, A. E. Willner, and R. W. Boyd, “High-dimensional free-space quantum key distribution using spin, azimuthal, and radial quantum numbers,” arXiv:1809.09986 (2018).

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Ramelow, S.

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

Rehácek, J.

E. Karimi, R. W. Boyd, P. D. L. Hoz, H. D. Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of laguerre-gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Ren, X.

P. Zhang, X. Ren, X. Zou, B. Liu, Y. Huang, and G. Guo, “Demonstration of one-dimensional quantum random walks using orbital angular momentum of photons,” Phys. Rev. A 75, 052310 (2007).
[Crossref]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

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

Richardson, D. J.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photon. 7, 354 (2013).
[Crossref]

Riguet, N.

S. Bade, B. Denolle, G. Trunet, N. Riguet, P. Jian, O. Pinel, and G. Labroille, “Fabrication and characterization of a mode-selective 45-mode spatial multiplexer based on multi-plane light conversion,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Robertson, D. J.

Rodenburg, B.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

M. Mirhosseini, O. S. Magana-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
[Crossref]

Ryf, R.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-gaussian mode sorter,” arXiv preprint arXiv:1803.04126 (2018).

Sánchez-Soto, L. L.

E. Karimi, R. W. Boyd, P. D. L. Hoz, H. D. Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of laguerre-gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Santamato, E.

E. Nagali, F. Sciarrino, F. D. Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Sciarrino, F.

E. Nagali, F. Sciarrino, F. D. Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
[Crossref] [PubMed]

Shi, Z.

. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

Sit, A.

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

Spreeuw, R.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Su, Z.

X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
[Crossref]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2001).
[Crossref]

Trunet, G.

S. Bade, B. Denolle, G. Trunet, N. Riguet, P. Jian, O. Pinel, and G. Labroille, “Fabrication and characterization of a mode-selective 45-mode spatial multiplexer based on multi-plane light conversion,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Tsang, M.

M. Tsang, R. Nair, and X.-M. Lu, “Quantum theory of superresolution for two incoherent optical point sources,” Phys. Rev. X 6, 031033 (2016).

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

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

Upham, J.

Vaziri, A.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Wang, F.

F. Wang, P. Zeng, X. Wang, H. Gao, F. Li, and P. Zhang, “Towards practical high-speed high dimensional quantum key distribution using partial mutual unbiased basis of photon’s orbital angular momentum,” arXiv:1801.06582 (2018).

Wang, J.

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

Wang, X.

X. Wang, Y. Luo, H. Huang, M. Chen, Z. Su, C. Liu, C. Chen, W. Li, Y. Fang, X. Jiang, J. Zhang, L. Li, N. Liu, C. Lu, and J. Pan, “18-qubit entanglement with six photons’ three degrees of freedom,” Phys. Rev. Lett. 120, 260502 (2018).
[Crossref]

F. Wang, P. Zeng, X. Wang, H. Gao, F. Li, and P. Zhang, “Towards practical high-speed high dimensional quantum key distribution using partial mutual unbiased basis of photon’s orbital angular momentum,” arXiv:1801.06582 (2018).

Wang, Z.

Weihs, G.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

White, A. G.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref] [PubMed]

Willner, A.

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

Willner, A. E.

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. H. Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Y. Zhou, M. Mirhosseini, S. Oliver, J. Zhao, S. M. H. Rafsanjani, M. P. J. Lavery, A. E. Willner, and R. W. Boyd, “High-dimensional free-space quantum key distribution using spin, azimuthal, and radial quantum numbers,” arXiv:1809.09986 (2018).

Woerdman, J.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Xu, X.

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
[Crossref]

Yan, Y.

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

Yang, J. Y.

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

Yu, C.

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
[Crossref]

Yuan, X.

T. Lei, M. Zhang, Y. Li, P. Jia, G. N. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by dammann gratings,” Light. Sci Appl 4, e257 (2015).
[Crossref]

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

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

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

Zeilinger, A.

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: Concepts and experiments,” Phys. Rev. Lett. 120, 103601 (2018).
[Crossref] [PubMed]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Zeng, P.

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Figures (5)

Fig. 1
Fig. 1 (a) Implementation of the FRFT with a lens. The LG mode keeps invariant in the final plane because it is the eigenmode of FRFT. Here, a mode with p = 1, ℓ = 1 is presented as an example. (b) Schematic diagram of a FRFT module. The FRFT module consists of two sets of lenses (each set of lenses is combined a PL with a normal lens (NL)) and can perform a FRFT of order π/2 and π to right-hand (R) and left-handed (L) circular polarization respectively. The inset shows an example of a PL. The black lines denote the direction of the fast axis.

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Fig. 2
Fig. 2 Experimental setup of the LG mode sorter. The LG mode is generated by a spatial light modulator (SLM). The polarizer and the half-wave plate (HWP) set the photons to be horizontally polarized. The FRFT module is used to realize a radial mode sorter. The ℓ-dependent phase shifter is realized by a Sagnac interferometer and a Dove prism. The OAM sorter performs a coordinate transform to efficiently separate the OAM modes. SMF: single-mode fiber; L: lens; PL: polarization-dependent lens; NL: normal lens; HWP: half-wave plate; QWP: quarter-wave plate; PBS: polarization beam splitter.

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Fig. 3
Fig. 3 Experimental results for the radial mode sorter when ℓ of the input LG modes is 0. The first four columns show that a LG mode of odd (even) p is sorted to Camera2 (Camera1). The last column shows the result when the input is a coherent superposition state of p = 0 and p = 1.

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Fig. 4
Fig. 4 Experimental results for the LG mode sorter. The input state is shown on the top of the images. When ℓ is non-positive, it can be seen that p = 0 and p = 1 modes are sorted to Camera1 and Camera2 respectively, and the ℓ value determines the vertical position of sorted modes. When ℓ is positive, the LG modes with an odd (even) value of p + ℓ are directed to Camera2 (Camera1), while the vertical position of sorted mode is still determined by ℓ.

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Fig. 5
Fig. 5 The measured crosstalk matrix of the LG mode sorter.

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Equations (7)

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ℱ a [ LG p ℓ ( r 0 , θ 0 ) ] = exp [ − i 2 a ⋅ ( p + | ℓ | / 2 ) ] ⋅ LG p ℓ ( r , θ ) ,
z = π w 0 2 λ tan a 2 ,                     f = π w 0 2 λ sin a .
α ( r ) = π r 2 2 λ f 0 ,
| Φ in 〉 = E L | L 〉 + E R | R 〉 ,
| Φ out 〉 = E L e i 2 α ( r ) | R 〉 + E R e − i 2 α ( r ) | L 〉 .
LG p ℓ ⊗ ( | L 〉 + | R 〉 ) → LG p ℓ ⊗ ( | L 〉 + e i π ( p + | ℓ | 2 ) | R 〉 ) .
                                  LG p ℓ ⊗ ( | L 〉 + | R 〉 ) → FRFT LG p ℓ ⊗ ( | L 〉 + e i π ( p + | ℓ | 2 ) | R 〉 ) → shifter LG p ℓ ⊗ ( | L 〉 + e i π ( p + | ℓ | 2 + ℓ 2 ) | R 〉 )                                     = { LG p ℓ ⊗ ( | L 〉 + e i π ( p + ℓ ) | R 〉 ) , ℓ > 0 LG p ℓ ⊗ ( | L 〉 + e i π p | R 〉 ) , ℓ ≤ 0 .

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