Abstract

Along with the growing interest in using the transverse-spatial modes of light in quantum and classical optics applications, developing an accurate and efficient measurement method has gained importance. Here, we present a technique relying on a unitary mode conversion for measuring any full-field transverse-spatial mode. Our method only requires three consecutive phase modulations followed by a single mode fiber and is, in principle, error-free and lossless. We experimentally test the technique using a single spatial light modulator and achieve an average error of 4.2 % for a set of 9 different full-field Laguerre-Gauss and Hermite-Gauss modes with an efficiency of up to 70%. Moreover, as the method can also be used to measure any complex superposition state, we demonstrate its potential for quantum cryptography applications and in high-dimensional quantum state tomography.

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

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2019 (1)

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019).
[Crossref]

2018 (7)

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: Concepts and experiments,” Phys. Rev. Lett. 120(10), 103601 (2018).
[Crossref]

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7(3), 17146 (2018).
[Crossref]

M. Erhard, M. Malik, M. Krenn, and A. Zeilinger, “Experimental Greenberger-Horne-Zeilinger entanglement beyond qubits,” Nat. Photonics 12(12), 759–764 (2018).
[Crossref]

F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
[Crossref]

F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26(17), 22563–22573 (2018).
[Crossref]

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26(24), 31925–31941 (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(24), 6101–6104 (2018).
[Crossref]

2017 (5)

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” Phys. Rev. B 95(16), 161108 (2017).
[Crossref]

G. Ruffato, M. Massari, G. Parisi, and F. Romanato, “Test of mode-division multiplexing and demultiplexing in free-space with diffractive transformation optics,” Opt. Express 25(7), 7859–7868 (2017).
[Crossref]

M. J. Padgett, “Orbital angular momentum 25 years on,” Opt. Express 25(10), 11265–11274 (2017).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8(1), 15516 (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(26), 263602 (2017).
[Crossref]

2016 (2)

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. 113(48), 13642–13647 (2016).
[Crossref]

Y. Zhang, S. Prabhakar, A. H. Ibrahim, F. S. Roux, A. Forbes, and T. Konrad, “Experimentally observed decay of high-dimensional entanglement through turbulence,” Phys. Rev. A 94(3), 032310 (2016).
[Crossref]

2015 (4)

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(8), 080403 (2015).
[Crossref]

M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (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(2), e1500087 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

2014 (3)

2013 (3)

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]

C. Schulze, A. Dudley, D. Flamm, M. Duparre, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15(7), 073025 (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(6140), 1545–1548 (2013).
[Crossref]

2012 (1)

2011 (1)

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

2010 (3)

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010).
[Crossref]

J.-F. Morizur, L. Nicholls, P. Jian, S. Armstrong, N. Treps, B. 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]

T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inf. 08(04), 535–640 (2010).
[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(16), 163905 (2006).
[Crossref]

2005 (1)

2002 (2)

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

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88(12), 127902 (2002).
[Crossref]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref]

1994 (1)

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

1992 (2)

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(11), 8185–8189 (1992).
[Crossref]

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

Abbas, A.

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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(11), 8185–8189 (1992).
[Crossref]

Andrews, D. L.

D. L. Andrews and M. Babiker, The angular momentum of light (Cambridge University, 2012).

Armstrong, S.

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Babiker, M.

D. L. Andrews and M. Babiker, The angular momentum of light (Cambridge University, 2012).

Bachor, H.-A.

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Barnett, S. M.

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

Beijersbergen, M.

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010).
[Crossref]

M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

Beijersbergen, M. W.

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(11), 8185–8189 (1992).
[Crossref]

Bengtsson, I.

T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inf. 08(04), 535–640 (2010).
[Crossref]

Bent, N.

Berkhout, G.

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010).
[Crossref]

Berkhout, G. C.

Bolduc, E.

Bouchard, F.

F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
[Crossref]

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26(24), 31925–31941 (2018).
[Crossref]

F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26(17), 22563–22573 (2018).
[Crossref]

F. Brandt, M. Hiekkamäki, F. Bouchard, M. Huber, and R. Fickler, “High-dimensional quantum gates using full-field spatial modes of photons,” arXiv: 1907.13002 (2019).

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Entanglement distribution beyond qubits or: How I stopped worrying and learned to love the noise,” arXiv preprint arXiv:1904.01552 (2019).

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(12), 127902 (2002).
[Crossref]

Bowen, W.

Bowman, R.

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

Boyd, R. W.

F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
[Crossref]

F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26(17), 22563–22573 (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(24), 6101–6104 (2018).
[Crossref]

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” Phys. Rev. B 95(16), 161108 (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(26), 263602 (2017).
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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(2), e1500087 (2015).
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M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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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).
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F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26(24), 31925–31941 (2018).
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F. Brandt, M. Hiekkamäki, F. Bouchard, M. Huber, and R. Fickler, “High-dimensional quantum gates using full-field spatial modes of photons,” arXiv: 1907.13002 (2019).

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Entanglement distribution beyond qubits or: How I stopped worrying and learned to love the noise,” arXiv preprint arXiv:1904.01552 (2019).

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R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. 113(48), 13642–13647 (2016).
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R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. 113(48), 13642–13647 (2016).
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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(2), e1500087 (2015).
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N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019).
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F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8(1), 15516 (2017).
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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(12), 127902 (2002).
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N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019).
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M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
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M. P. Lavery, D. J. Robertson, G. C. 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(3), 2110–2115 (2012).
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F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
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[Crossref]

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26(24), 31925–31941 (2018).
[Crossref]

F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26(17), 22563–22573 (2018).
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R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” Phys. Rev. B 95(16), 161108 (2017).
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R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. 113(48), 13642–13647 (2016).
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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(11), 113028 (2014).
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F. Brandt, M. Hiekkamäki, F. Bouchard, M. Huber, and R. Fickler, “High-dimensional quantum gates using full-field spatial modes of photons,” arXiv: 1907.13002 (2019).

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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(11), 113028 (2014).
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Y. Zhang, S. Prabhakar, A. H. Ibrahim, F. S. Roux, A. Forbes, and T. Konrad, “Experimentally observed decay of high-dimensional entanglement through turbulence,” Phys. Rev. A 94(3), 032310 (2016).
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C. Schulze, A. Dudley, D. Flamm, M. Duparre, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15(7), 073025 (2013).
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J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

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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(26), 263602 (2017).
[Crossref]

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M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
[Crossref]

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Ginoya, M.

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” Phys. Rev. B 95(16), 161108 (2017).
[Crossref]

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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(12), 127902 (2002).
[Crossref]

Gu, X.

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: Concepts and experiments,” Phys. Rev. Lett. 120(10), 103601 (2018).
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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(8), 080403 (2015).
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S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Entanglement distribution beyond qubits or: How I stopped worrying and learned to love the noise,” arXiv preprint arXiv:1904.01552 (2019).

Hage, B.

Handsteiner, J.

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(11), 113028 (2014).
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F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
[Crossref]

F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26(17), 22563–22573 (2018).
[Crossref]

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F. Brandt, M. Hiekkamäki, F. Bouchard, M. Huber, and R. Fickler, “High-dimensional quantum gates using full-field spatial modes of photons,” arXiv: 1907.13002 (2019).

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Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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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(6140), 1545–1548 (2013).
[Crossref]

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F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26(24), 31925–31941 (2018).
[Crossref]

F. Brandt, M. Hiekkamäki, F. Bouchard, M. Huber, and R. Fickler, “High-dimensional quantum gates using full-field spatial modes of photons,” arXiv: 1907.13002 (2019).

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Entanglement distribution beyond qubits or: How I stopped worrying and learned to love the noise,” arXiv preprint arXiv:1904.01552 (2019).

Hufnagel, F.

Ibrahim, A. H.

Y. Zhang, S. Prabhakar, A. H. Ibrahim, F. S. Roux, A. Forbes, and T. Konrad, “Experimentally observed decay of high-dimensional entanglement through turbulence,” Phys. Rev. A 94(3), 032310 (2016).
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Jian, P.

Karimi, E.

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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(12), 127902 (2002).
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N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10(1), 1865 (2019).
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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(8), 080403 (2015).
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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(8), 080403 (2015).
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S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Entanglement distribution beyond qubits or: How I stopped worrying and learned to love the noise,” arXiv preprint arXiv:1904.01552 (2019).

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Konrad, T.

Y. Zhang, S. Prabhakar, A. H. Ibrahim, F. S. Roux, A. Forbes, and T. Konrad, “Experimentally observed decay of high-dimensional entanglement through turbulence,” Phys. Rev. A 94(3), 032310 (2016).
[Crossref]

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M. Erhard, M. Malik, M. Krenn, and A. Zeilinger, “Experimental Greenberger-Horne-Zeilinger entanglement beyond qubits,” Nat. Photonics 12(12), 759–764 (2018).
[Crossref]

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7(3), 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(10), 103601 (2018).
[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(11), 113028 (2014).
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M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
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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(6140), 1545–1548 (2013).
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Lam, P. K.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. 113(48), 13642–13647 (2016).
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G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010).
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M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
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F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8(1), 15516 (2017).
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M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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Nat. Commun. (2)

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Nat. Photonics (2)

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Nature (1)

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Figures (4)

Fig. 1.
Fig. 1. Sketch of the experimental setup for near-perfect spatial mode measuring. A spatial light modulator (SLM1) was used to generate a specific mode to be measured. A magnifying imaging system enlarges the beam size and thus the number of pixels illuminated in the mode transformation is increased. Three phase modulations on SLM2 transform the input mode into a Gaussian, which exclusively couples into the single mode fiber (SMF). The insets display a LG mode $l=-1, p=1$ (lower left) and the required transforming phase modulations (lower right) as an example. Note that for the mode generation we also use an amplitude masking scheme [30] and overlay all phase structures with additional gratings in the experiment (not shown).
Fig. 2.
Fig. 2. Modal cross-talk matrices for the projection measurements of the 9 lowest order (a) Laguerre-Gauss and (b) Hermite-Gauss modes. In graph (c), the SMF coupling efficiencies of the converted modes are shown with the same x-axis as used in (a) and (b). The given diagonal values show the mean and standard deviation of the normalized detection rates on the cross talk matrix diagonal.
Fig. 3.
Fig. 3. Modal cross-talk matrix of the 14 modes used in a 7-dimensional quantum cryptography protocol. The secret key rate calculated from the bit error rate is 1.98 bits per sifted photon.
Fig. 4.
Fig. 4. Real part of the density matrix reconstructed using quantum state tomography.

Equations (4)

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o t ( x , y ) = M ( x , y , t ) ¯ G ( x , y , t ) e i Φ t ( x , y ) ,
Δ Φ t ( x , y ) = arg ( o t ( x , y ) e i ϕ ) .
| ψ n { | 1 , 1 , | 1 , 0 , | 0 , 0 , | 0 , 1 , | 0 , 2 , | 1 , 0 , | 1 , 1 } { | 1 , | 2 , , | 7 } ,
| ϕ n ( k ) = 1 d m = 0 d 1 ω d ( n m + ( k 1 ) m 2 ) | ψ m ,

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