Abstract

The dimension of the state space for information encoding offered by the transverse structure of light is usually limited by the finite size of apertures. The widely used orbital angular momentum (OAM) number of Laguerre-Gaussian (LG) modes in free-space communications cannot achieve the theoretical maximum transmission capacity unless the radial degree of freedom is multiplexed into the protocol. While the methodology to sort the radial quantum number has been developed, the application of radial modes in quantum communications requires an additional ability to efficiently measure the superposition of LG modes in the mutually unbiased basis. Here we develop and implement a generic mode sorter that is capable of sorting the superposition of LG modes through the use of a mode converter. As a consequence, we demonstrate an 8-dimensional quantum key distribution experiment involving all three transverse degrees of freedom: spin, azimuthal, and radial quantum numbers of photons. Our protocol presents an important step towards the goal of reaching the capacity limit of a free-space link and can be useful to other applications that involve spatial modes of photons.

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

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2018 (6)

2017 (7)

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep.  7, 13882 (2017).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Yan, Y. Ren, P. Liao, Z. Zhao, N. Ahmed, Z. Wang, C. Bao, A. J. Willner, S. Ashrafi, M. Tur, and A. E. Willner, “Power loss mitigation of orbital-angular-momentum-multiplexed free-space optical links using nonzero radial index laguerre–gaussian beams,” J. Opt. Soc. Am. B 34, 1–6 (2017).
[Crossref]

J. Liñares, X. Prieto-Blanco, C. Montero-Orille, and V. Moreno, “Spatial mode multiplexing/demultiplexing by gouy phase interferometry,” Opt. Lett. 42, 93–96 (2017).
[Crossref] [PubMed]

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]

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006–1010 (2017).
[Crossref]

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. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref] [PubMed]

2016 (4)

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” Proc. Natl. Acad. Sci. U.S.A. 113, 13648–13653 (2016).
[Crossref] [PubMed]

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep.  6, 27674 (2016).
[Crossref] [PubMed]

R. Ionicioiu, “Sorting quantum systems efficiently,” Sci. Rep.  6, 25356 (2016).
[Crossref] [PubMed]

J. Wong-Campos, K. Johnson, B. Neyenhuis, J. Mizrahi, and C. Monroe, “High-resolution adaptive imaging of a single atom,” Nat. Photonics 10, 606 (2016).
[Crossref]

2015 (4)

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

X. Wang, X. Cai, Z. Su, M. Chen, D. Wu, L. Li, N. Liu, C. Lu, and J. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516 (2015).
[Crossref] [PubMed]

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]

S. Yu, “Potentials and challenges of using orbital angular momentum communications in optical interconnects,” Opt. Express 23, 3075–3087 (2015).
[Crossref] [PubMed]

2014 (4)

G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J. Morizur, “Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion,” Opt. Express 22, 15599–15607 (2014).
[Crossref] [PubMed]

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

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light. Sci. Appl.  3, e167 (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 (5)

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
[Crossref]

M. Mirhosseini, O. S. Magaña-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]

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]

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[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]

2012 (3)

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

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, 488–496 (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 (1)

2010 (3)

J. Morizur, L. Nicholls, P. Jian, S. Armstrong, N. Treps, B. Hage, M. Hsu, W. Bowen, J. Janousek, and H. Bachor, “Programmable unitary spatial mode manipulation,” J. Opt. Soc. Am. A 27, 2524–2531 (2010).
[Crossref]

E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010).
[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]

2002 (1)

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett.  88, 127902 (2002).
[Crossref] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

1992 (1)

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

Ahmed, N.

L. Li, G. Xie, Y. Yan, Y. Ren, P. Liao, Z. Zhao, N. Ahmed, Z. Wang, C. Bao, A. J. Willner, S. Ashrafi, M. Tur, and A. E. Willner, “Power loss mitigation of orbital-angular-momentum-multiplexed free-space optical links using nonzero radial index laguerre–gaussian beams,” J. Opt. Soc. Am. B 34, 1–6 (2017).
[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, 488–496 (2012).
[Crossref]

Allen, L.

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

Armstrong, S.

Ashrafi, S.

Bachor, H.

Banzer, P.

M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref] [PubMed]

Bao, C.

Barnett, S. M.

Bavaresco, J.

J. Bavaresco, N. H. Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys.  14, 1032 (2018).
[Crossref]

Beijersbergen, M. W.

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

Bennett, C. H.

C. H. Bennett and G. Brassard, in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (IEEE, New York, 1984), Vol. 175, p. 8.

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, 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.

Bourennane, M.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett.  88, 127902 (2002).
[Crossref] [PubMed]

Bowen, W.

Boyd, R. W.

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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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J. Bavaresco, N. H. Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys.  14, 1032 (2018).
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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).
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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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E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light. Sci. Appl.  3, e167 (2014).
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N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett.  88, 127902 (2002).
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J. Bavaresco, N. H. Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys.  14, 1032 (2018).
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B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep.  7, 13882 (2017).
[Crossref] [PubMed]

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

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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, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” Proc. Natl. Acad. Sci. U.S.A. 113, 13648–13653 (2016).
[Crossref] [PubMed]

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

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

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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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Lapkiewicz, R.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref] [PubMed]

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[Crossref]

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
[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]

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]

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E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light. Sci. Appl.  3, e167 (2014).
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M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
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Li, 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).

Li, G.

N. Zhao, X. Li, G. Li, and J. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photonics 9, 822–826 (2015).
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X. Wang, X. Cai, Z. Su, M. Chen, D. Wu, L. Li, N. Liu, C. Lu, and J. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516 (2015).
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X. Wang, X. Cai, Z. Su, M. Chen, D. Wu, L. Li, N. Liu, C. Lu, and J. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516 (2015).
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M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
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M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
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Magaña-Loaiza, O. S.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Malik, M.

J. Bavaresco, N. H. Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys.  14, 1032 (2018).
[Crossref]

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” Proc. Natl. Acad. Sci. U.S.A. 113, 13648–13653 (2016).
[Crossref] [PubMed]

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

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. Magaña-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]

Marquardt, C.

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006–1010 (2017).
[Crossref]

M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref] [PubMed]

Marrucci, L.

E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010).
[Crossref]

McLaren, M.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

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]

Mirhosseini, M.

D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable laguerre–gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26, 33057–33065 (2018).
[Crossref]

S. Choudhary, R. Sampson, Y. Miyamoto, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, M. Mirhosseini, and R. W. Boyd, “Measurement of the radial mode spectrum of photons through a phase-retrieval method,” Opt. Lett. 43, 6101–6104 (2018).
[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]

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. Magaña-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]

R. W. Boyd, B. Rodenburg, M. Mirhosseini, and S. M. Barnett, “Influence of atmospheric turbulence on the propagation of quantum states of light using plane-wave encoding,” Opt. Express 19, 18310–18317 (2011).
[Crossref] [PubMed]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (2018).

Miyamoto, Y.

Mizrahi, J.

J. Wong-Campos, K. Johnson, B. Neyenhuis, J. Mizrahi, and C. Monroe, “High-resolution adaptive imaging of a single atom,” Nat. Photonics 10, 606 (2016).
[Crossref]

Monroe, C.

J. Wong-Campos, K. Johnson, B. Neyenhuis, J. Mizrahi, and C. Monroe, “High-resolution adaptive imaging of a single atom,” Nat. Photonics 10, 606 (2016).
[Crossref]

Montero-Orille, C.

Moreno, V.

Morizur, J.

Mortimer, D.

Nagali, E.

E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010).
[Crossref]

Nape, I.

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep.  7, 13882 (2017).
[Crossref] [PubMed]

Ndagano, B.

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep.  7, 13882 (2017).
[Crossref] [PubMed]

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep.  6, 27674 (2016).
[Crossref] [PubMed]

Neyenhuis, B.

J. Wong-Campos, K. Johnson, B. Neyenhuis, J. Mizrahi, and C. Monroe, “High-resolution adaptive imaging of a single atom,” Nat. Photonics 10, 606 (2016).
[Crossref]

Nicholls, L.

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]

Oliver, S.

Padgett, M. J.

M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref] [PubMed]

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, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
[Crossref]

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[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]

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]

Pan, J.

X. Wang, X. Cai, Z. Su, M. Chen, D. Wu, L. Li, N. Liu, C. Lu, and J. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516 (2015).
[Crossref] [PubMed]

Pang, K.

Perez-Garcia, B.

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep.  7, 13882 (2017).
[Crossref] [PubMed]

Petruccione, F.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Peuntinger, C.

M. P. J. Lavery, C. Peuntinger, K. Günthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, “Free-space propagation of high-dimensional structured optical fields in an urban environment,” Sci. Adv. 3, e1700552 (2017).
[Crossref] [PubMed]

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006–1010 (2017).
[Crossref]

Pivoluska, M.

J. Bavaresco, N. H. Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys.  14, 1032 (2018).
[Crossref]

Plick, W. N.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

Prieto-Blanco, X.

Qassim, H.

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light. Sci. Appl.  3, e167 (2014).
[Crossref]

Qi, R.

Rafsanjani, S. M. H.

S. Choudhary, R. Sampson, Y. Miyamoto, O. S. Magaña-Loaiza, S. M. H. Rafsanjani, M. Mirhosseini, and R. W. Boyd, “Measurement of the radial mode spectrum of photons through a phase-retrieval method,” Opt. Lett. 43, 6101–6104 (2018).
[Crossref] [PubMed]

D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable laguerre–gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26, 33057–33065 (2018).
[Crossref]

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]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (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.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

Ren, Y.

K. Pang, C. Liu, G. Xie, Y. Ren, Z. Zhao, R. Zhang, Y. Cao, J. Zhao, H. Song, H. Song, L. Li, A. N. Willner, M. Tur, R. W. Boyd, and A. E. Willner, “Demonstration of a 10 mbit/s quantum communication link by encoding data on two laguerre–gaussian modes with different radial indices,” Opt. Lett. 43, 5639–5642 (2018).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Yan, Y. Ren, P. Liao, Z. Zhao, N. Ahmed, Z. Wang, C. Bao, A. J. Willner, S. Ashrafi, M. Tur, and A. E. Willner, “Power loss mitigation of orbital-angular-momentum-multiplexed free-space optical links using nonzero radial index laguerre–gaussian beams,” J. Opt. Soc. Am. B 34, 1–6 (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]

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, 488–496 (2012).
[Crossref]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (2018).

Robertson, D. J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
[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]

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. Magaña-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]

R. W. Boyd, B. Rodenburg, M. Mirhosseini, and S. M. Barnett, “Influence of atmospheric turbulence on the propagation of quantum states of light using plane-wave encoding,” Opt. Express 19, 18310–18317 (2011).
[Crossref] [PubMed]

B. Rodenburg, “Communicating with transverse modes of light,” Ph.D. thesis, University of Rochester (2015).

Rosales-Guzmán, C.

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep.  6, 27674 (2016).
[Crossref] [PubMed]

Salem, A. B.

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep.  6, 27674 (2016).
[Crossref] [PubMed]

Sampson, R.

Sansoni, L.

E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010).
[Crossref]

Santamato, E.

E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010).
[Crossref]

Schaeff, C.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

Scheidl, T.

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

Scholes, S.

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep.  7, 13882 (2017).
[Crossref] [PubMed]

Schulz, S. A.

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light. Sci. Appl.  3, e167 (2014).
[Crossref]

Sciarrino, F.

E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010).
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Shi, Z.

Sit, A.

Song, H.

Sponselli, A.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
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Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
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Steinhoff, N. K.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
[Crossref]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (2018).

Su, Z.

X. Wang, X. Cai, Z. Su, M. Chen, D. Wu, L. Li, N. Liu, C. Lu, and J. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516 (2015).
[Crossref] [PubMed]

Treps, N.

Trichili, A.

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep.  6, 27674 (2016).
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Tsang, M.

M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: Semiclassical treatment,” Phys. Rev. A 97, 023830 (2018).
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Tur, M.

Tyler, G. A.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
[Crossref]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (2018).

Upham, J.

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]

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light. Sci. Appl.  3, e167 (2014).
[Crossref]

Ursin, R.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” Proc. Natl. Acad. Sci. U.S.A. 113, 13648–13653 (2016).
[Crossref] [PubMed]

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

Valencia, N. H.

J. Bavaresco, N. H. Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys.  14, 1032 (2018).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (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. 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, 488–496 (2012).
[Crossref]

Wang, X.

X. Wang, X. Cai, Z. Su, M. Chen, D. Wu, L. Li, N. Liu, C. Lu, and J. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516 (2015).
[Crossref] [PubMed]

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, Y.

Wang, Z.

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Willner, A. E.

K. Pang, C. Liu, G. Xie, Y. Ren, Z. Zhao, R. Zhang, Y. Cao, J. Zhao, H. Song, H. Song, L. Li, A. N. Willner, M. Tur, R. W. Boyd, and A. E. Willner, “Demonstration of a 10 mbit/s quantum communication link by encoding data on two laguerre–gaussian modes with different radial indices,” Opt. Lett. 43, 5639–5642 (2018).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Yan, Y. Ren, P. Liao, Z. Zhao, N. Ahmed, Z. Wang, C. Bao, A. J. Willner, S. Ashrafi, M. Tur, and A. E. Willner, “Power loss mitigation of orbital-angular-momentum-multiplexed free-space optical links using nonzero radial index laguerre–gaussian beams,” J. Opt. Soc. Am. B 34, 1–6 (2017).
[Crossref]

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]

M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
[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, 488–496 (2012).
[Crossref]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (2018).

Willner, A. J.

Willner, A. N.

Woerdman, J. P.

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

Wong-Campos, J.

J. Wong-Campos, K. Johnson, B. Neyenhuis, J. Mizrahi, and C. Monroe, “High-resolution adaptive imaging of a single atom,” Nat. Photonics 10, 606 (2016).
[Crossref]

Wu, D.

X. Wang, X. Cai, Z. Su, M. Chen, D. Wu, L. Li, N. Liu, C. Lu, and J. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516 (2015).
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Xie, G.

Yan, Y.

L. Li, G. Xie, Y. Yan, Y. Ren, P. Liao, Z. Zhao, N. Ahmed, Z. Wang, C. Bao, A. J. Willner, S. Ashrafi, M. Tur, and A. E. Willner, “Power loss mitigation of orbital-angular-momentum-multiplexed free-space optical links using nonzero radial index laguerre–gaussian beams,” J. Opt. Soc. Am. B 34, 1–6 (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, 488–496 (2012).
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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, 488–496 (2012).
[Crossref]

Yu, S.

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. 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, 488–496 (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, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” Proc. Natl. Acad. Sci. U.S.A. 113, 13648–13653 (2016).
[Crossref] [PubMed]

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

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
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Zeng, P.

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).

Zghal, M.

A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep.  6, 27674 (2016).
[Crossref] [PubMed]

Zhang, P.

D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable laguerre–gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26, 33057–33065 (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).

Zhang, R.

Zhang, Y.

Zhao, J.

D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable laguerre–gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26, 33057–33065 (2018).
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K. Pang, C. Liu, G. Xie, Y. Ren, Z. Zhao, R. Zhang, Y. Cao, J. Zhao, H. Song, H. Song, L. Li, A. N. Willner, M. Tur, R. W. Boyd, and A. E. Willner, “Demonstration of a 10 mbit/s quantum communication link by encoding data on two laguerre–gaussian modes with different radial indices,” Opt. Lett. 43, 5639–5642 (2018).
[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]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (2018).

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N. Zhao, X. Li, G. Li, and J. Kahn, “Capacity limits of spatially multiplexed free-space communication,” Nat. Photonics 9, 822–826 (2015).
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D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable laguerre–gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26, 33057–33065 (2018).
[Crossref]

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]

J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (2018).

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

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

Light. Sci. Appl (1)

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light. Sci. Appl.  3, e167 (2014).
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Nat. Commun (1)

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).
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Nat. Photonics (3)

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, 488–496 (2012).
[Crossref]

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

J. Wong-Campos, K. Johnson, B. Neyenhuis, J. Mizrahi, and C. Monroe, “High-resolution adaptive imaging of a single atom,” Nat. Photonics 10, 606 (2016).
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Nat. Phys (1)

J. Bavaresco, N. H. Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys.  14, 1032 (2018).
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Nature (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

X. Wang, X. Cai, Z. Su, M. Chen, D. Wu, L. Li, N. Liu, C. Lu, and J. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516 (2015).
[Crossref] [PubMed]

New J. Phys (3)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across vienna,” New J. Phys.  16, 113028 (2014).
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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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M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys.  15, 013024 (2013).
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Opt. Express (7)

S. Yu, “Potentials and challenges of using orbital angular momentum communications in optical interconnects,” Opt. Express 23, 3075–3087 (2015).
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G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J. Morizur, “Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion,” Opt. Express 22, 15599–15607 (2014).
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M. Mirhosseini, O. S. Magaña-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).
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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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R. W. Boyd, B. Rodenburg, M. Mirhosseini, and S. M. Barnett, “Influence of atmospheric turbulence on the propagation of quantum states of light using plane-wave encoding,” Opt. Express 19, 18310–18317 (2011).
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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).
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D. Fu, Y. Zhou, R. Qi, S. Oliver, Y. Wang, S. M. H. Rafsanjani, J. Zhao, M. Mirhosseini, Z. Shi, P. Zhang, and R. W. Boyd, “Realization of a scalable laguerre–gaussian mode sorter based on a robust radial mode sorter,” Opt. Express 26, 33057–33065 (2018).
[Crossref]

Opt. Lett. (3)

Optica (1)

Phys. Rev. A (5)

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

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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E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010).
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M. Tsang, “Subdiffraction incoherent optical imaging via spatial-mode demultiplexing: Semiclassical treatment,” Phys. Rev. A 97, 023830 (2018).
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Phys. Rev. Lett (4)

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]

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).
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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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Proc. Natl. Acad. Sci. U.S.A. (1)

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 km,” Proc. Natl. Acad. Sci. U.S.A. 113, 13648–13653 (2016).
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Sci. Adv. (1)

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A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep.  6, 27674 (2016).
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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).
[Crossref] [PubMed]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. H. Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, “Performance analysis of d-dimensional quantum cryptography under state-dependent diffraction”, arXiv:1812.07070 (2018).

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).

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Figures (6)

Fig. 1
Fig. 1 Spatial representations of elements in the main and conjugate bases. The normalized intensity distribution of each spatial mode is presented. The upper right image is the corresponding experimental (Exp.) record of the simulated (Sim.) image for comparison. Each mode can have the two different polarizations as shown at the upper left corner. Scale bar, 1 mm.
Fig. 2
Fig. 2 Schematic of the radial superposition mode sorter. (a) Conceptual schematic of a d-dimensional radial superposition mode sorter. (b) Two-dimensional (d = 2) realization to sort |pL〉 and |pR〉. The unitary transformation U ^ 1 0 is realized by SLMs. The modes |pL〉 and |pR〉 are directed to different output ports as indicated at the last beamsplitter.
Fig. 3
Fig. 3 (a) Common-path radial mode sorter. (b) Radial mode converter which transforms |1p, 2〉 to |0p, 2〉. (c) The phase distribution as a function of radius r on the two SLMs in the radial mode converter. (d) The input and output spatial mode intensity distribution of the radial mode converter. The upper right image is the corresponding experimental (Exp.) record of the simulated (Sim.) image for comparison.
Fig. 4
Fig. 4 (a) Experimental realization of a common-path radial superposition mode sorter. By removing the radial mode converter in the dashed box it becomes a common-path radial mode sorter. (b) The measured cross-talk of the common-path radial mode sorter. (c) The measured cross-talk of the common-path radial superposition mode sorter.
Fig. 5
Fig. 5 Schematic of the QKD protocol involving three degrees of freedom. Relay lenses are omitted for simplicity.
Fig. 6
Fig. 6 Conditional probability matrix measured with µ = 0.1 pulses. Each element in this matrix represents the probability that Bob detects the corresponding symbol conditioned on the symbol sent by Alice.

Equations (9)

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

| D = | H + | V 2 , | A = | H | V 2 ,
| D = | 2 + | 2 2 , | A = | 2 | 2 2 ,
| p L = | 0 p + i | 1 p 2 , | p R = | 0 p i | 1 p 2 ,
| n ¯ p = 1 d m = 0 d 1 exp ( i 2 π n ¯ d ) | m p ,
| n ¯ p | 0 = 1 d m = 0 d 1 exp ( i 2 π m n ¯ d ) | m p | 0 1 d m = 0 d 1 exp ( i 2 π m n ¯ d ) | m p | m .
1 d m = 0 d 1 exp ( i 2 π m n ¯ d ) | m p | m 1 d m = 0 d 1 exp ( i 2 π m n ¯ d ) | 0 p | m .
F ^ [ 1 d m = 0 d 1 exp ( i 2 π m n ¯ d ) | 0 p | m ] = 1 d m = 0 d 1 exp ( i 2 π m n ¯ d ) F ^ [ | 0 p | m ] = 1 d j = 0 d 1 m = 0 d 1 exp ( i 2 π m ( j n ¯ ) d ) | 0 p | j = j = 0 d 1 δ ( j n ¯ ) | 0 p | j = | 0 p | n ¯ ,
F ^ [ | m p | k ] = 1 d j = 0 d 1 exp ( i 2 π j k d ) | m p | j .
I AB = log 2 d + F log 2 F + ( 1 F ) log 2 ( 1 F d 1 ) ,