Abstract

With the emergence of the field of quantum communications, the appropriate choice of photonic degrees of freedom used for encoding information is of paramount importance. Highly precise techniques for measuring the polarisation, frequency, and arrival time of a photon have been developed. However, the transverse spatial degree of freedom still lacks a measurement scheme that allows the reconstruction of its full transverse structure with a simple implementation and a high level of accuracy. Here we show a method to measure the azimuthal and radial modes of Laguerre-Gaussian beams with a greater than 99 % accuracy, using a single phase screen. We compare our technique with previous commonly used methods and demonstrate the significant improvements it presents for quantum key distribution and state tomography of high-dimensional quantum states of light. Moreover, our technique can be readily extended to any arbitrary family of spatial modes, such as mutually unbiased bases, Hermite-Gauss, and Ince-Gauss. Our scheme will significantly enhance existing quantum and classical communication protocols that use the spatial structure of light, as well as enable fundamental experiments on spatial-mode entanglement to reach their full potential.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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2018 (4)

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, "Twisted photons: New quantum perspectives in high dimensions," Light Sci. Appl. 7, 17146 (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]

F. Bouchard, A. Sit, K. Heshami, R. Fickler, and E. Karimi, "Round-robin differential-phase-shift quantum key distribution with twisted photons," Phys. Rev. A 98, 010301 (2018).
[Crossref]

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, 1032–1037 (2018).
[Crossref]

2017 (5)

M. Krenn, M. Malik, M. Erhard, and A. Zeilinger, "Orbital angular momentum of photons and the entanglement of laguerre–gaussian modes," Phil. Trans. R. Soc. A 375, 20150442 (2017).
[Crossref]

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground entanglement-based quantum key distribution," Phys. Rev. Lett. 119, 200501 (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(9), 1006–1010 (2017).
[Crossref]

R. Fickler, M. Ginoya, and R. W. Boyd, "Custom-tailored spatial mode sorting by controlled random scattering," Phys. Rev. B 95, 161108 (2017).
[Crossref]

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

2016 (2)

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, "Multi-photon entanglement in high dimensions," Nat. Photonics 10, 248 (2016).
[Crossref]

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]

2015 (7)

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, "Systematic errors in current quantum state tomography tools," Phys. Rev. Lett. 114, 080403 (2015).
[Crossref] [PubMed]

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

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, "Quantum walks and wavepacket dynamics on a lattice with twisted photons," Sci. Adv. 1, e1500087 (2015).
[Crossref] [PubMed]

M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, "Significant-loophole-free test of bell’s theorem with entangled photons," Phys. Rev. Lett. 115, 250401 (2015).
[Crossref]

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[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]

W. N. Plick and M. Krenn, "Physical meaning of the radial index of laguerre-gauss beams," Phys. Rev. A 92, 063841 (2015).
[Crossref]

2014 (7)

F. S. Roux and Y. Zhang, "Projective measurements in quantum and classical optical systems," Phys. Rev. A 90, 033835 (2014).
[Crossref]

Y. Zhang, F. S. Roux, M. McLaren, and A. Forbes, "Radial modal dependence of the azimuthal spectrum after parametric down-conversion," Phys. Rev. A 89, 043820 (2014).
[Crossref]

E. Karimi, R. W. Boyd, P. De La Hoz, H. De 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]

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, "Limitations to the determination of a laguerre–gauss spectrum via projective, phase-flattening measurement," J. Opt. Soc. Am. B 31(6), A20–A23 (2014).
[Crossref]

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, "Free-space quantum key distribution by rotation-invariant twisted photons," Phys. Rev. Lett. 113, 060503 (2014).
[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. U.S.A. 111(17), 6243–6247 (2014).
[Crossref] [PubMed]

2013 (3)

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]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, "Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram," Opt. Lett. 38(18), 3546–3549 (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]

2012 (2)

V. Salakhutdinov, E. Eliel, and W. Löffler, "Full-field quantum correlations of spatially entangled photons," Phys. Rev. Lett. 108, 173604 (2012).
[Crossref] [PubMed]

E. Karimi and E. Santamato, "Radial coherent and intelligent states of paraxial wave equation," Opt. Lett. 37(13), 2484–2486 (2012).
[Crossref] [PubMed]

2010 (3)

J. F. Morizur, L. Nicholls, P. Jian, S. Armstrong, N. Treps, N. Hage, M. Hsu, W. Bowen, J. Janousek, and H.-A. Bachor, "Programmable unitary spatial mode manipulation," J. Opt. Soc. Am. A 27(11), 2524–2531 (2010).
[Crossref]

G. C. Berkhout, M. P. 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]

T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, "On mutually unbiased bases," Int. J. Quantum Inf. 8(04), 535–640 (2010).
[Crossref]

2008 (1)

A. K. Jha, M. Malik, and R. W. Boyd, "Exploring energy-time entanglement using geometric phase," Phys. Rev. Lett. 101, 180405 (2008).
[Crossref] [PubMed]

2007 (1)

S. Walborn and C. Monken, "Transverse spatial entanglement in parametric down-conversion," Phys. Rev. A 76, 062305 (2007).
[Crossref]

2006 (1)

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]

2005 (1)

A. Zeilinger, G. Weihs, T. Jennewein, and M. Aspelmeyer, "Happy centenary, photon," Nature 433, 230–238 (2005).
[Crossref] [PubMed]

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]

1994 (1)

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112(5), 321–327 (1994).
[Crossref]

1992 (3)

V. Y. Bazhenov, M. Soskin, and M. Vasnetsov, "Screw dislocations in light wavefronts," J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

N. Heckenberg, R. McDuff, C. Smith, and A. White, "Generation of optical phase singularities by computer-generated holograms," Opt. Lett. 17(3), 221–223 (1992).
[Crossref] [PubMed]

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]

1989 (1)

J. D. Franson, "Bell inequality for position and time," Phys. Rev. Lett. 62, 2205 (1989).
[Crossref] [PubMed]

1971 (1)

Abbas, A.

F. Bouchard, F. Hufnagel, D. Koutnỳ, A. Abbas, A. Sit, K. Heshami, R. Fickler, and E. Karimi, "Full characterization of a high-dimensional quantum communication channel," arXiv:1806.08018 (2018).

Abellán, C.

M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, "Significant-loophole-free test of bell’s theorem with entangled photons," Phys. Rev. Lett. 115, 250401 (2015).
[Crossref]

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

Aiello, A.

E. Karimi, R. W. Boyd, P. De La Hoz, H. De 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]

Allman, M. S.

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

Amaya, W.

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, "Significant-loophole-free test of bell’s theorem with entangled photons," Phys. Rev. Lett. 115, 250401 (2015).
[Crossref]

Armstrong, S.

Aspelmeyer, M.

A. Zeilinger, G. Weihs, T. Jennewein, and M. Aspelmeyer, "Happy centenary, photon," Nature 433, 230–238 (2005).
[Crossref] [PubMed]

Bachor, H.-A.

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, 1032–1037 (2018).
[Crossref]

Bazhenov, V. Y.

V. Y. Bazhenov, M. Soskin, and M. Vasnetsov, "Screw dislocations in light wavefronts," J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

Beijersbergen, M.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112(5), 321–327 (1994).
[Crossref]

Beijersbergen, M. W.

G. C. Berkhout, M. P. 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]

Bengtsson, I.

T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, "On mutually unbiased bases," Int. J. Quantum Inf. 8(04), 535–640 (2010).
[Crossref]

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]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, "Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram," Opt. Lett. 38(18), 3546–3549 (2013).
[Crossref] [PubMed]

Berkhout, G. C.

G. C. Berkhout, M. P. 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]

Beyer, J.

M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, "Significant-loophole-free test of bell’s theorem with entangled photons," Phys. Rev. Lett. 115, 250401 (2015).
[Crossref]

Bienfang, J. C.

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

Bierhorst, P.

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[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]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, "Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram," Opt. Lett. 38(18), 3546–3549 (2013).
[Crossref] [PubMed]

Bouchard, F.

F. Bouchard, A. Sit, K. Heshami, R. Fickler, and E. Karimi, "Round-robin differential-phase-shift quantum key distribution with twisted photons," Phys. Rev. A 98, 010301 (2018).
[Crossref]

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(9), 1006–1010 (2017).
[Crossref]

F. Bouchard, F. Hufnagel, D. Koutnỳ, A. Abbas, A. Sit, K. Heshami, R. Fickler, and E. Karimi, "Full characterization of a high-dimensional quantum communication channel," arXiv:1806.08018 (2018).

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.

R. Fickler, M. Ginoya, and R. W. Boyd, "Custom-tailored spatial mode sorting by controlled random scattering," Phys. Rev. B 95, 161108 (2017).
[Crossref]

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

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(9), 1006–1010 (2017).
[Crossref]

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, "Quantum walks and wavepacket dynamics on a lattice with twisted photons," Sci. Adv. 1, e1500087 (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]

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. De La Hoz, H. De 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]

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, "Limitations to the determination of a laguerre–gauss spectrum via projective, phase-flattening measurement," J. Opt. Soc. Am. B 31(6), A20–A23 (2014).
[Crossref]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, "Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram," Opt. Lett. 38(18), 3546–3549 (2013).
[Crossref] [PubMed]

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]

A. K. Jha, M. Malik, and R. W. Boyd, "Exploring energy-time entanglement using geometric phase," Phys. Rev. Lett. 101, 180405 (2008).
[Crossref] [PubMed]

Cai, W.-Q.

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground entanglement-based quantum key distribution," Phys. Rev. Lett. 119, 200501 (2017).
[Crossref] [PubMed]

Cao, Y.

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground entanglement-based quantum key distribution," Phys. Rev. Lett. 119, 200501 (2017).
[Crossref] [PubMed]

Cardano, F.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, "Quantum walks and wavepacket dynamics on a lattice with twisted photons," Sci. Adv. 1, e1500087 (2015).
[Crossref] [PubMed]

Carpenter, J.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, "Optical spatial mode sorter of azimuthal and radial components," arXiv:1803.04126 (2018).

Cerf, N. J.

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]

Chen, H.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, "Optical spatial mode sorter of azimuthal and radial components," arXiv:1803.04126 (2018).

Chen, Y.-A.

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground entanglement-based quantum key distribution," Phys. Rev. Lett. 119, 200501 (2017).
[Crossref] [PubMed]

Christensen, B. G.

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

Coakley, K. J.

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

Coerwinkel, R.

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112(5), 321–327 (1994).
[Crossref]

Courtial, J.

G. C. Berkhout, M. P. 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]

D’Ambrosio, V.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, "Free-space quantum key distribution by rotation-invariant twisted photons," Phys. Rev. Lett. 113, 060503 (2014).
[Crossref] [PubMed]

Dai, H.

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground entanglement-based quantum key distribution," Phys. Rev. Lett. 119, 200501 (2017).
[Crossref] [PubMed]

De La Hoz, P.

E. Karimi, R. W. Boyd, P. De La Hoz, H. De 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]

de Lisio, C.

F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, "Quantum walks and wavepacket dynamics on a lattice with twisted photons," Sci. Adv. 1, e1500087 (2015).
[Crossref] [PubMed]

Deng, L.

J. Yin, Y. Cao, Y.-H. Li, J.-G. Ren, S.-K. Liao, L. Zhang, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, M. Li, Y.-M. Huang, L. Deng, L. Li, Q. Zhang, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground entanglement-based quantum key distribution," Phys. Rev. Lett. 119, 200501 (2017).
[Crossref] [PubMed]

Dudley, 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).
[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]

Durt, T.

T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, "On mutually unbiased bases," Int. J. Quantum Inf. 8(04), 535–640 (2010).
[Crossref]

Dyer, S. D.

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

Eliel, E.

V. Salakhutdinov, E. Eliel, and W. Löffler, "Full-field quantum correlations of spatially entangled photons," Phys. Rev. Lett. 108, 173604 (2012).
[Crossref] [PubMed]

Elser, D.

Englert, B.-G.

T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, "On mutually unbiased bases," Int. J. Quantum Inf. 8(04), 535–640 (2010).
[Crossref]

Erhard, 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).
[Crossref] [PubMed]

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, "Twisted photons: New quantum perspectives in high dimensions," Light Sci. Appl. 7, 17146 (2018).
[Crossref]

M. Krenn, M. Malik, M. Erhard, and A. Zeilinger, "Orbital angular momentum of photons and the entanglement of laguerre–gaussian modes," Phil. Trans. R. Soc. A 375, 20150442 (2017).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, "Multi-photon entanglement in high dimensions," Nat. Photonics 10, 248 (2016).
[Crossref]

Erker, P.

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, 1032–1037 (2018).
[Crossref]

Farr, W. H.

L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, "Strong loophole-free test of local realism," Phys. Rev. Lett. 115, 250402 (2015).
[Crossref]

Fickler, R.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, "Twisted photons: New quantum perspectives in high dimensions," Light Sci. Appl. 7, 17146 (2018).
[Crossref]

F. Bouchard, A. Sit, K. Heshami, R. Fickler, and E. Karimi, "Round-robin differential-phase-shift quantum key distribution with twisted photons," Phys. Rev. A 98, 010301 (2018).
[Crossref]

R. Fickler, M. Ginoya, and R. W. Boyd, "Custom-tailored spatial mode sorting by controlled random scattering," Phys. Rev. B 95, 161108 (2017).
[Crossref]

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(9), 1006–1010 (2017).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, "Multi-photon entanglement in high dimensions," Nat. Photonics 10, 248 (2016).
[Crossref]

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

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Sci. Rep. (1)

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]

Other (3)

F. Bouchard, F. Hufnagel, D. Koutnỳ, A. Abbas, A. Sit, K. Heshami, R. Fickler, and E. Karimi, "Full characterization of a high-dimensional quantum communication channel," arXiv:1806.08018 (2018).

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, "Optical spatial mode sorter of azimuthal and radial components," arXiv:1803.04126 (2018).

J. M. Kahn, G. Li, X. Li, and N. Zhao, "To twist or not to twist: Capacity limits for free-space channels," In Advanced Photonics 2016 (IPR, NOMA, Sensors, Networks, SPPCom, SOF), SpM4E.1 (Optical Society of America, 2016).

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Figures (10)

Fig. 1
Fig. 1 Simplified experimental setup. A forward-propagating beam with an unknown spatial mode is made incident on a spatial light modulator (SLM). A hologram simultaneously modulating the phase and the amplitude of the incoming beam is displayed on the SLM. Subsequently, the outgoing beam is coupled to a single-mode fibre (SMF) after passing through a set of lenses and microscope objectives (not shown). The choice of lenses can be understood by considering a back-propagating beam exiting the SMF and made incident on the SLM from the back. According to the intensity-flattening technique presented here, this beam should be expanded on the SLM in order to flatten the intensity distribution of the Gaussian component.
Fig. 2
Fig. 2 Experimental details. (a) An attenuated laser diode is enlarged using a telescope with a magnification of f2/f1 = (300 mm)/(50 mm) = 6. The beam is then made incident on a first spatial light modulator (SLM-A) which reflects the incoming beam (shown in transmission here for simplicity). The output beam has the desired intensity and phase profile after passing through a 4 − f system that filters out the first order of diffraction (not shown here). (b) The beam that is to be measured is made incident on SLM-B. By considering a virtually backward-propagating beam from the 10X object, the set of lenses f3 = 200 mm and f4 = 50 mm magnifies the backward-propagating by a factor of 4, making its beam waist much larger than the beam waist of the detection mode on the SLM-B.
Fig. 3
Fig. 3 Measurement of radial modes. (a) Experimentally measured and (b) simulated cross-talk matrix of radial modes ranging from p = 0 to p = 7 in a prepare-and-measure setting. The cross-talk matrix is normalized to unity by dividing each elements by the element with maximum counts. The rows and the columns correspond to the states, |pA〉 and |pB〉, prepared and measured by Alice and Bob, respectively. A visibility of V = 98.3 % is obtained from the experimentally measured cross-talk matrix. In theory, a visibility in excess of 99 % is achieved by considering a back-propagating with a beam waist 5.4 times larger than that of the beam waist of the detection mode of the holograms. (c) The efficiency of the intensity-flattening measurement technique is shown as a function of dimensionality of radial modes. For a dimension of d, radial modes ranging from p = 0 to p = d − 1 are considered. For each dimensions, the reported efficiencies are obtained by increasing the beam waist of the back-propagating up to the point where visibilities are in excess of 99 % (dark blue), 95 % (red) and 90 % (green).
Fig. 4
Fig. 4 Performance of phase-flattening in measuring radial modes. Simulated cross-talk matrix of radial modes measurement with (a) a phase-flattening measurement scheme and (b) a phase-flattening with amplitude mask measurement scheme.
Fig. 5
Fig. 5 Measurement of a 55-dimensional space using azimuthal and radial modes. (a) Experimentally measured cross-talk matrix for a 55-dimensional space of azimuthal and radial modes. In order to show mode-dependent efficiencies, the cross-talk matrix is normalized to unity by dividing each elements by the element with maximum counts. The list of states {|, p〉} are given explicitly in the appendix-c. (b) Secret key rates obtained from lower dimensional subspaces of the full 55-dimensional space. For a given subspace, a sample of 1000 different combinations is selected. For each combination of subspaces, a secret key rate is calculated from the experimental data. The mean and the standard deviation of the secret key rates over the 1000 combinations are shown in dark blue. The maximal secret key rates obtained by searching for the optimal subspace using a genetic algorithm are shown in red. The theoretical maximal values are shown in green, given by log2(d). The shaded region corresponds to values of secret key rates inaccessible for the corresponding dimensions.
Fig. 6
Fig. 6 High-dimensional quantum state tomography. The experimentally reconstructed density matrices for a (a) 7-dimensional OAM state, (b) 5-dimensional radial state, (c) 17-dimensional OAM state and (d) 19-dimensional full-field state, are shown in the upper row along with their corresponding theory density matrices, respectively. High-dimensional states giving rise to visually interesting density matrices were chosen in order to resemble (a)-(b) a castle, (c) a sine function, and (d) a palace, where the explicit forms of the generated states are given in appendix-d. Fidelities of F = 98:7 %, 95.3 %, 95.4 %, and 93.8 %, and efficiencies of 4.7 %, 5.5 %, 8.3 %, and 6.6 % were obtained experimentally for (a)-(d), respectively.
Fig. 7
Fig. 7 Simulated tomographically reconstructed density matrices using the phase-flattening-only and the phase-flattening with amplitude masking methods. We show the tomographic reconstruction of (a) the 7-dimensional OAM castle state, (b) the 5-dimensional radial castle state, (c) the 17-dimensional OAM sine state, and (d) the 19-dimensional full-field palace state.
Fig. 8
Fig. 8 Comparison of intensity-flattening with different backward-propagating beam. Normalized intensity distribution for three different types of backward propagating beams, i.e. (a) Flat-top, (b) Gaussian and (c) Exponential. The trade-off between visibility and efficiency is shown for all three types of beams in (d) and (e), for dimension 5 and 10, respectively.
Fig. 9
Fig. 9 Comparison of intensity-flattening with different cut of the backward-propagating beam. Normalized intensity distribution for three different types of backward propagating beams with, i.e. (a) no cut (Gaussian), (b) cut at 0.5w0, (c) cut at 0.75w0 and (d) cut at w0. The trade-off between visibility and efficiency is shown for all three types of beams in (e) and (f), for dimension 5 and 10, respectively. In dimension 5, we can see in the inset that for visibilities larger than 98.5 %, a cut at 0.5w0 increasing the efficiency of the measurement, with respect to an uncut Gaussian.
Fig. 10
Fig. 10 Optimal d-dimensional subspaces. Representation of the optimal d-dimensional subspaces of the full 55-dimensional measurements. The states on the horizontal axis are explicitly given in Table 2. Yellow indicates the presence of the state in the corresponding subspace, and dark blue indicates its absence.

Tables (2)

Tables Icon

Table 1 Comparison of Intensity-Flattening and Phase-Flattening for Measuring Azimuthal and Radial Modes.

Tables Icon

Table 2 List of the States Considered in the 55-Dimensional Measurements.

Equations (8)

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0 2 π 0 LG , p * ( r , φ ) LG , p ( r , φ ) r d r d φ = δ δ p p .
0 2 π 0 r max LG , p * ( r , φ ) LG , p ( r , φ ) e r 2 / w 0 2 r d r d φ δ δ p p ,
E ft ( r , φ ) = { 1 w 0 1 π if  r w 0 0 if  r > w 0
E exp ( r , φ ) = 1 w 0 2 π exp ( r / w 0 ) .
| ψ castle ( d = 7 ) = 1 N castle ( d = 7 ) ( | = 3 + 0.5 | = 2 + 0.5 | = 1 + 0.5 | = 0 + 0.5 | = 1 + 0.5 | = 2 + | = 3 ) ,
| ψ castle ( d = 5 ) = 1 N castle ( d = 5 ) ( | p = 0 + 0.6 | p = 1 + 0.6 | p = 2 + 0.6 | p = 3 + | p = 4 ) ,
| ψ sine ( d = 17 ) = 1 N sine ( d = 17 ) = 8 8 sin ( 2 π / 17 ) | ,
| ψ palace ( d = 19 ) = 1 N palace ( d = 19 ) ( 0.5 | = 0 , p = 0 + 0.8 | = 0 , p = 1 + 0.5 | = 3 , p = 0 + 0.5 | = 3 , p = 0 + 0.5 | = 1 , p = 1 + 0.5 | = 1 , p = 1 + 0.5 | = 4 , p = 0 + 0.5 | = 4 , p = 0 + 0.5 | = 2 , p = 1 + 0.8 | = 2 , p = 1 + 0.5 | = 3 , p = 1 ) ,

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