Abstract

Orbital angular momentum (OAM) is an important resource in high-dimensional quantum information processing, as its quantum number can be infinite. Dove prism (DP) is a most common tool to manipulate OAM light. However, the Dove prism changes the polarization of the photon states and decreases the sorting fidelity of the interferometer. In this work, we analyze the polarization-dependent effect of the DP on OAM light manipulation in the normal single-path Sagnac interferometers (SPSIs) with beam splitter (BS) and polarizing beam splitter (PBS). The results demonstrate that the BS SPSI is more sensitive to the input polarization and the specific parameters of the DP. We have also proposed and realized a modified BS SPSI, of which the sorting fidelity can be 100% in principle and is independent on the input polarization and the transmission matrix of the DP. The experiments demonstrate that the fidelity of the modified BS SPSI is about 5%~10% higher than that of the normal one. The modified BS SPSI is easy to implement (only two more half-wave plates are required) and is stable for free running at the scale of several hours. These merits make the structure suitable for applications in critical quantum information processing tasks, such as quantum cryptography.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  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, (11)8185–8189 (1992).
    [Crossref] [PubMed]
  2. 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] [PubMed]
  3. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
    [Crossref]
  4. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
    [Crossref] [PubMed]
  5. V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
  6. M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
    [Crossref]
  7. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001).
    [Crossref]
  8. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
    [Crossref]
  9. M. Bourennane, A. Karlsson, and G. Bjork, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
    [Crossref]
  10. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
    [Crossref]
  11. J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
    [Crossref]
  12. I. B. Djordjevic, “Multidimensional QKD Based on Combined Orbital and Spin Angular Momenta of Photon,” IEEE Photonics J. 5(6), 7600112 (2013).
    [Crossref]
  13. D. S. Simon and A. V. Sergienko, “High-capacity quantum key distribution via hyperentangled degrees of freedom,” New J. Phys. 16(6), 063052 (2014).
    [Crossref]
  14. M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
    [Crossref]
  15. A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Gunthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-Dimensional Intra-City Quantum Cryptography with Structured Photons,” Optica 4(9), 1006–1010 (2017).
    [Crossref]
  16. I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).
  17. M. Erhard, M. Malik, and A. Zeilinger, “A quantum router for high-dimensional entanglement,” Quantum Sci. Technol. 2(1), 014001 (2017).
    [Crossref]
  18. Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
    [Crossref] [PubMed]
  19. M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
    [Crossref]
  20. V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
    [Crossref]
  21. 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] [PubMed]
  22. 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(15), 153601 (2010).
    [Crossref]
  23. M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20(22), 24444–24449 (2012).
    [Crossref]
  24. 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]
  25. M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
    [Crossref] [PubMed]
  26. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
    [Crossref] [PubMed]
  27. M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
    [Crossref] [PubMed]
  28. H. Zhou, D. Fu, J. Dong, P. Zhang, and X. Zhang, “Theoretical analysis and experimental verification on optical rotational Doppler effect,” Opt. Express 24(9), 10050–10056 (2016).
    [Crossref] [PubMed]
  29. H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
    [Crossref]
  30. G. F. Walsh, “Pancharatnam-Berry optical element sorter of full angular momentum eigenstate,” Opt. Express 24(6), 6689–6704 (2016).
    [Crossref] [PubMed]
  31. H. Ren, X. Li, Q. Zhang, and M. Gu, “On-chip noninterference angular momentum multiplexing of broadband light,” Science 352(6287), 805–809 (2016).
    [Crossref] [PubMed]
  32. 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] [PubMed]
  33. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
    [Crossref] [PubMed]
  34. W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112(15), 153601 (2014).
    [Crossref] [PubMed]
  35. P. H. Jones, M. Rashid, M. Makita, and O. M. Marago, “Sagnac interferometer method for synthesis of fractional polarization vortices,” Opt. Lett. 34(17), 2560–2562 (2009).
    [Crossref] [PubMed]
  36. S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, “The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express 18(26), 27205–27216 (2010).
    [Crossref]
  37. V. D’Ambrosio, E. Nagali, C. H. Monken, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Deterministic qubit transfer between orbital and spin angular momentum of single photons,” Opt. Lett. 37(2), 172–174 (2012).
    [Crossref]
  38. Q. Zeng, T. Li, X. Song, and X. Zhang, “Realization of optimized quantum controlled-logic gate based on the orbital angular momentum of light,” Opt. Express 24(8), 8186–8193 (2016).
    [Crossref] [PubMed]
  39. M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46(2), 175–179 (1999).
    [Crossref]
  40. I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4), 257–268 (2003).
    [Crossref]
  41. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999), Chap. 1.
    [Crossref]
  42. P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85(2), 441–444 (2000).
    [Crossref] [PubMed]
  43. H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94(23), 230504 (2005).
    [Crossref] [PubMed]
  44. T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24(6), 6249–6264 (2016).
    [Crossref] [PubMed]

2017 (3)

M. Erhard, M. Malik, and A. Zeilinger, “A quantum router for high-dimensional entanglement,” Quantum Sci. Technol. 2(1), 014001 (2017).
[Crossref]

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Gunthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-Dimensional Intra-City Quantum Cryptography with Structured Photons,” Optica 4(9), 1006–1010 (2017).
[Crossref]

2016 (10)

T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24(6), 6249–6264 (2016).
[Crossref] [PubMed]

G. F. Walsh, “Pancharatnam-Berry optical element sorter of full angular momentum eigenstate,” Opt. Express 24(6), 6689–6704 (2016).
[Crossref] [PubMed]

Q. Zeng, T. Li, X. Song, and X. Zhang, “Realization of optimized quantum controlled-logic gate based on the orbital angular momentum of light,” Opt. Express 24(8), 8186–8193 (2016).
[Crossref] [PubMed]

H. Zhou, D. Fu, J. Dong, P. Zhang, and X. Zhang, “Theoretical analysis and experimental verification on optical rotational Doppler effect,” Opt. Express 24(9), 10050–10056 (2016).
[Crossref] [PubMed]

H. Ren, X. Li, Q. Zhang, and M. Gu, “On-chip noninterference angular momentum multiplexing of broadband light,” Science 352(6287), 805–809 (2016).
[Crossref] [PubMed]

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
[Crossref]

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] [PubMed]

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

2015 (3)

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] [PubMed]

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

2014 (3)

D. S. Simon and A. V. Sergienko, “High-capacity quantum key distribution via hyperentangled degrees of freedom,” New J. Phys. 16(6), 063052 (2014).
[Crossref]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref] [PubMed]

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

2013 (4)

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

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

I. B. Djordjevic, “Multidimensional QKD Based on Combined Orbital and Spin Angular Momenta of Photon,” IEEE Photonics J. 5(6), 7600112 (2013).
[Crossref]

2012 (3)

2011 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

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

2010 (2)

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(15), 153601 (2010).
[Crossref]

S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, “The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express 18(26), 27205–27216 (2010).
[Crossref]

2009 (1)

2008 (1)

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

2005 (1)

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94(23), 230504 (2005).
[Crossref] [PubMed]

2004 (1)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

2003 (1)

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4), 257–268 (2003).
[Crossref]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

2001 (2)

M. Bourennane, A. Karlsson, and G. Bjork, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001).
[Crossref]

2000 (1)

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85(2), 441–444 (2000).
[Crossref] [PubMed]

1999 (1)

M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46(2), 175–179 (1999).
[Crossref]

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, (11)8185–8189 (1992).
[Crossref] [PubMed]

Ahmed, N.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

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] [PubMed]

Aolita, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Arbabi, A.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Arbabi, E.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Arnold, A. S.

Ashrafi, S.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Barreiro, J. T.

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[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(15), 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, (11)8185–8189 (1992).
[Crossref] [PubMed]

Berkhout, G. C. G.

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(15), 153601 (2010).
[Crossref]

Bjork, G.

M. Bourennane, A. Karlsson, and G. Bjork, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999), Chap. 1.
[Crossref]

Bouchard, F.

Bourennane, M.

M. Bourennane, A. Karlsson, and G. Bjork, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

Bowman, R.

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

Boyd, R. W.

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Gunthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-Dimensional Intra-City Quantum Cryptography with Structured Photons,” Optica 4(9), 1006–1010 (2017).
[Crossref]

M. Mirhosseini, O. S. Magana-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(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] [PubMed]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[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]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20(22), 24444–24449 (2012).
[Crossref]

Buchler, B.

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] [PubMed]

Cai, X.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Cai, X.-L.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

Campbell, G.

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] [PubMed]

Cao, Y.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[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(2), e1500087 (2015).
[Crossref] [PubMed]

Carvacho, G.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
[Crossref]

Chen, D.-X.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

Chen, K.

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94(23), 230504 (2005).
[Crossref] [PubMed]

Chen, L.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

Clark, T. W.

Courtial, J.

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(15), 153601 (2010).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

D’Ambrosio, V.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
[Crossref]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

V. D’Ambrosio, E. Nagali, C. H. Monken, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Deterministic qubit transfer between orbital and spin angular momentum of single photons,” Opt. Lett. 37(2), 172–174 (2012).
[Crossref]

S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, “The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express 18(26), 27205–27216 (2010).
[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(2), e1500087 (2015).
[Crossref] [PubMed]

Del Re, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Djordjevic, I. B.

I. B. Djordjevic, “Multidimensional QKD Based on Combined Orbital and Spin Angular Momenta of Photon,” IEEE Photonics J. 5(6), 7600112 (2013).
[Crossref]

Dong, J.

Dong, J.-J.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

Elser, D.

Erhard, M.

M. Erhard, M. Malik, and A. Zeilinger, “A quantum router for high-dimensional entanglement,” Quantum Sci. Technol. 2(1), 014001 (2017).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

Faraon, A.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Fickler, R.

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Gunthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-Dimensional Intra-City 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(4), 248–252 (2016).
[Crossref]

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] [PubMed]

Forbes, A.

I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).

Franke-Arnold, S.

T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24(6), 6249–6264 (2016).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Fu, D.

Fu, D.-Z.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

Gagnon-Bischoff, J.

Galajda, P.

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001).
[Crossref]

Gao, P.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Gauthier, D. J.

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

Graffitti, F.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
[Crossref]

Gu, M.

H. Ren, X. Li, Q. Zhang, and M. Gu, “On-chip noninterference angular momentum multiplexing of broadband light,” Science 352(6287), 805–809 (2016).
[Crossref] [PubMed]

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Gunthner, K.

Heim, B.

Hernandez-Aranda, R. I.

I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).

Heshami, K.

Hong, M.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Hu, C.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Huang, C.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Huber, M.

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

Johnson-Morris, B.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Jones, P. H.

Kamali, S. M.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Karimi, E.

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Gunthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-Dimensional Intra-City 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(2), e1500087 (2015).
[Crossref] [PubMed]

Karlsson, A.

M. Bourennane, A. Karlsson, and G. Bjork, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

Konrad, T.

I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).

Krenn, M.

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

Kwek, L. C.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Kwiat, P. G.

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

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).
[Crossref] [PubMed]

Larocque, H.

Lavery, M. P. J.

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref] [PubMed]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
[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(15), 153601 (2010).
[Crossref]

Leach, J.

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Lesso, J. P.

M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46(2), 175–179 (1999).
[Crossref]

Leuchs, G.

Li, F.-L.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

Li, L.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Li, T.

Li, X.

H. Ren, X. Li, Q. Zhang, and M. Gu, “On-chip noninterference angular momentum multiplexing of broadband light,” Science 352(6287), 805–809 (2016).
[Crossref] [PubMed]

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Li, Y.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Liu, C.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

Lo, H.-K.

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94(23), 230504 (2005).
[Crossref] [PubMed]

Luo, X.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Ma, X.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94(23), 230504 (2005).
[Crossref] [PubMed]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

Magana-Loaiza, O. S.

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

Makita, M.

Malik, M.

M. Erhard, M. Malik, and A. Zeilinger, “A quantum router for high-dimensional entanglement,” Quantum Sci. Technol. 2(1), 014001 (2017).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[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]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20(22), 24444–24449 (2012).
[Crossref]

Marago, O. M.

Marquardt, C.

Marrucci, L.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
[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] [PubMed]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

V. D’Ambrosio, E. Nagali, C. H. Monken, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Deterministic qubit transfer between orbital and spin angular momentum of single photons,” Opt. Lett. 37(2), 172–174 (2012).
[Crossref]

S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, “The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express 18(26), 27205–27216 (2010).
[Crossref]

Massa, 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(2), e1500087 (2015).
[Crossref] [PubMed]

Mirhosseini, M.

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[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]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20(22), 24444–24449 (2012).
[Crossref]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Monken, C. H.

Moreno, I.

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4), 257–268 (2003).
[Crossref]

Nagali, E.

Nape, I.

I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).

Ndagano, B.

I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).

O’Brien, J. L.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

O’Sullivan, M. N.

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20(22), 24444–24449 (2012).
[Crossref]

Offer, R. F.

Ormos, P.

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001).
[Crossref]

Padgett, M.

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

Padgett, M. J.

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref] [PubMed]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
[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(15), 153601 (2010).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46(2), 175–179 (1999).
[Crossref]

Paez, G.

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4), 257–268 (2003).
[Crossref]

Paparo, D.

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] [PubMed]

Perez-Garcia, B.

I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).

Peuntinger, C.

Piccirillo, B.

Preskill, J.

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85(2), 441–444 (2000).
[Crossref] [PubMed]

Pu, M.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Qassim, H.

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] [PubMed]

Qi, Q.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

Qin, F.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Radwell, N.

Rashid, M.

Ren, H.

H. Ren, X. Li, Q. Zhang, and M. Gu, “On-chip noninterference angular momentum multiplexing of broadband light,” Science 352(6287), 805–809 (2016).
[Crossref] [PubMed]

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Ren, Y.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Rodenburg, B.

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

Santamato, E.

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] [PubMed]

S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, “The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express 18(26), 27205–27216 (2010).
[Crossref]

Scholes, S.

I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).

Sciarrino, F.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
[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] [PubMed]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

V. D’Ambrosio, E. Nagali, C. H. Monken, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Deterministic qubit transfer between orbital and spin angular momentum of single photons,” Opt. Lett. 37(2), 172–174 (2012).
[Crossref]

Sergienko, A. V.

D. S. Simon and A. V. Sergienko, “High-capacity quantum key distribution via hyperentangled degrees of freedom,” New J. Phys. 16(6), 063052 (2014).
[Crossref]

Shi, Z.

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

Shor, P. W.

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85(2), 441–444 (2000).
[Crossref] [PubMed]

Simon, D. S.

D. S. Simon and A. V. Sergienko, “High-capacity quantum key distribution via hyperentangled degrees of freedom,” New J. Phys. 16(6), 063052 (2014).
[Crossref]

Sit, A.

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

Slussarenko, S.

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] [PubMed]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

V. D’Ambrosio, E. Nagali, C. H. Monken, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Deterministic qubit transfer between orbital and spin angular momentum of single photons,” Opt. Lett. 37(2), 172–174 (2012).
[Crossref]

S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, “The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express 18(26), 27205–27216 (2010).
[Crossref]

Song, X.

Sorel, M.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Spagnolo, N.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Speirits, F. C.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

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, (11)8185–8189 (1992).
[Crossref] [PubMed]

Strain, M. J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Strojnik, M.

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4), 257–268 (2003).
[Crossref]

Thompson, M. G.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Tischler, N.

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Tur, M.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Vitelli, C.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
[Crossref]

Walborn, S. P.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Walsh, G. F.

Wang, C.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Wang, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Wang, Y.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Wang, Z.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Wei, T.

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

Willner, A. E.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Willner, A. J.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

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, (11)8185–8189 (1992).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999), Chap. 1.
[Crossref]

Xie, G.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Yan, Y.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Yang, J.

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Yu, S.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Zeilinger, A.

M. Erhard, M. Malik, and A. Zeilinger, “A quantum router for high-dimensional entanglement,” Quantum Sci. Technol. 2(1), 014001 (2017).
[Crossref]

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

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] [PubMed]

Zeng, Q.

Zhang, P.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

H. Zhou, D. Fu, J. Dong, P. Zhang, and X. Zhang, “Theoretical analysis and experimental verification on optical rotational Doppler effect,” Opt. Express 24(9), 10050–10056 (2016).
[Crossref] [PubMed]

Zhang, Q.

H. Ren, X. Li, Q. Zhang, and M. Gu, “On-chip noninterference angular momentum multiplexing of broadband light,” Science 352(6287), 805–809 (2016).
[Crossref] [PubMed]

Zhang, W.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

Zhang, X.

Zhang, X.-L.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

Zhao, Z.

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Zhou, H.

Zhou, H.-L.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

Zhou, J.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

Zhu, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001).
[Crossref]

IEEE Photonics J. (1)

I. B. Djordjevic, “Multidimensional QKD Based on Combined Orbital and Spin Angular Momenta of Photon,” IEEE Photonics J. 5(6), 7600112 (2013).
[Crossref]

J. Mod. Opt. (1)

M. J. Padgett and J. P. Lesso, “Dove prisms and polarized light,” J. Mod. Opt. 46(2), 175–179 (1999).
[Crossref]

Light Sci. Appl. (1)

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017).
[Crossref]

Nat. Commun. (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]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref] [PubMed]

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).

Nat. Photonics (3)

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

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10(4), 248–252 (2016).
[Crossref]

Nat. Phys. (2)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

J. T. Barreiro, T. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

New J. Phys. (3)

D. S. Simon and A. V. Sergienko, “High-capacity quantum key distribution via hyperentangled degrees of freedom,” New J. Phys. 16(6), 063052 (2014).
[Crossref]

M. Mirhosseini, O. S. Magana-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(3), 033033 (2015).
[Crossref]

M. Krenn, N. Tischler, and A. Zeilinger, “On small beams with large topological charge,” New J. Phys. 18(3), 033012 (2016).
[Crossref]

Opt. Commun. (1)

I. Moreno, G. Paez, and M. Strojnik, “Polarization transforming properties of Dove prisms,” Opt. Commun. 220(4), 257–268 (2003).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

Optica (1)

Phys. Rev. A (3)

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94(3), 030304 (2016).
[Crossref]

M. Bourennane, A. Karlsson, and G. Bjork, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[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, (11)8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (6)

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(15), 153601 (2010).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the Orbital Angular Momentum of a Single Photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday Rotation to Sort the Orbital Angular Momentum of Light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85(2), 441–444 (2000).
[Crossref] [PubMed]

H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. 94(23), 230504 (2005).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. (1)

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] [PubMed]

Quantum Sci. Technol. (1)

M. Erhard, M. Malik, and A. Zeilinger, “A quantum router for high-dimensional entanglement,” Quantum Sci. Technol. 2(1), 014001 (2017).
[Crossref]

Sci. Adv. (2)

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] [PubMed]

M. Pu, X. Li, X. Ma, Y. Wang, Z. Zhao, C. Wang, C. Hu, P. Gao, C. Huang, H. Ren, X. Li, F. Qin, J. Yang, M. Gu, M. Hong, and X. Luo, “Catenary optics for achromatic generation of perfect optical angular momentum,” Sci. Adv. 1(9), e1500396 (2015).
[Crossref] [PubMed]

Sci. Rep. (1)

Y. Ren, L. Li, Z. Wang, S. M. Kamali, E. Arbabi, A. Arbabi, Z. Zhao, G. Xie, Y. Cao, N. Ahmed, Y. Yan, C. Liu, A. J. Willner, S. Ashrafi, M. Tur, A. Faraon, and A. E. Willner, “Orbital Angular Momentum-based Space Division Multiplexing for High-capacity Underwater Optical Communications,” Sci. Rep. 6, 33306 (2016).
[Crossref] [PubMed]

Science (3)

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a Spinning Object Using Light’s Orbital Angular Momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

H. Ren, X. Li, Q. Zhang, and M. Gu, “On-chip noninterference angular momentum multiplexing of broadband light,” Science 352(6287), 805–809 (2016).
[Crossref] [PubMed]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science 338(6105), 363–366 (2012).
[Crossref] [PubMed]

Other (2)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, 1999), Chap. 1.
[Crossref]

I. Nape, B. Ndagano, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “High-bit-rate quantum key distribution with entangled internal degrees of freedom of photons,” arXiv: 1612.09261v1 (2016).

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Figures (8)

Fig. 1
Fig. 1 The schematic diagram of a DP. There are two refractive and one total reflective interfaces for a light beam propagating through the DP. θ1 is the incident angle at the input interface. n DP is the normal of the base. There is a relative rotation angle α between the lab (with red color) and the DP (with black color) coordinate systems. The rotation of the DP is along Z axis.
Fig. 2
Fig. 2 (a) The schematic diagram of a DP with two OAM states incident from opposite directions. (b) The coordinate systems of the DP for light incident from directions a (nDP,a and n D P , a ) and b (nDP,b and n D P , b ). The minus sign before n D P , b is to satisfy the right-hand rule of the coordinate system. (c) The schematic setup for a normal BS SPSI. BS: beam splitter.
Fig. 3
Fig. 3 The sorting fidelities of the BS SPSI with the rotation angle α (the blue line) and the relative phase Δφ(the black dashed line) added by the DP, respectively. The input polarization is |H〉. Δφ = 0.159π for the blue line. The rotation angle is α = π/4 for the black dashed line.
Fig. 4
Fig. 4 The sorting fidelity (the blue line) and the output probability Pc of the state at port c (the green dashed line) of the PBS SPSI with the rotation angle α. The input polarization is |H〉 + |V〉. The DP-dependent parameters are t// = 0.9877, t = 0.9475, Δφ = 0.159π.
Fig. 5
Fig. 5 The device for measuring (a) the transmissivity coefficients t / / , t the relative phase shift Δφ of the DP. PBS: polarizing beam splitter.
Fig. 6
Fig. 6 (a) The unitary transformations to rotate the polarization. (b) nDP, a ( b ) and n D P , a ( b ) consist the coordinate system of the DP for direction a (b). H W P i a ( b ) is the fast axis of the HWP in the BS SPSI for direction a (b). (c) The proof-of-principle experimental setup of the modified BS SPSI. SLM: spatial light modulator; QWP: quarter-wave plate.
Fig. 7
Fig. 7 The sorting fidelities of the normal (blue bars) and modified (red bars) BS SPSIs for eleven OAM modes and different polarizations.(a)-(f) correspond to the sorting fidelities with |+〉, |−〉, |H〉, |V〉, |L〉 and |R〉 polarizations, respectively. The black dashed line corresponds to 0.939, the upper bound of the normal BS SPSI. All error bars are set as ±3σ, where σ is the standard deviation.
Fig. 8
Fig. 8 The real time sorting fidelity of the modified BS SPSI for free running. The input state is |+〉 s |l = 1〉 o .

Equations (21)

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J D P = R s T o u t R i n n e r T i n R s
{ J D P | / / = t / / | / / J D P | = t e i Δ φ | ,
J D P = ( t / / 0 0 t e i Δ φ ) .
J D P ( α ) = R s ( α ) ( t / / 0 0 t e i Δ φ ) R s ( α ) , R S ( α ) = ( c o s α s i n α s i n α c o s α ) .
| ψ o u t = ( U p B S I s I o ) ( | b p b | J D P ( π α ) P o ( α ) ) ( | a p a | J D P ( α ) P o ( α ) ) ( U p B S I s I o ) | ψ i n ,
U p B S = ( T 1 T 1 T T ) ,
| ψ o u t i d e a l = e 2 i l α 2 [ ( 1 e 4 i l α ) | c p + ( 1 + e 4 i l α ) | d p ] | s s | l o .
| ψ o u t = e 2 i l α 2 t / / + t { ( t / / t e i Δ φ ) s i n 2 α ( β | H s + γ | V s ) | c p + [ 2 γ ( t / / c o s 2 α + t e i Δ φ s i n 2 α ) | H s + 2 β ( t / / s i n 2 α + t e i Δ φ c o s 2 α ) | V s ] | d p } | l o ,
= I m a x I m a x + I m i n ,
= m a x { | p c | ψ | 2 , | p d | ψ | 2 } ,
| H D P , a ( t / / c o s 2 α + t s i n 2 α e i Δ φ ) | H + ( t / / t e i Δ φ ) s i n α c o s α | V , | V D P , b ( t / / s i n 2 α + t c o s 2 α e i Δ φ ) | V ( t / / t e i Δ φ ) s i n α c o s α | H .
| ψ o u t = e 2 i l α 2 ( t / / + t ) { [ ( t / / c o s 2 α + t s i n 2 α e i Δ φ ) | H ± ( t / / s i n 2 α + t c o s 2 α e i Δ φ ) | V ] | c p + ( t / / t e i Δ φ ) s i n α c o s α ( | V | H ) | d p } | l o .
P o v e r l a p = | ψ ( l 1 ) o u t , c | ψ ( l 2 ) o u t , c | 2 = 1 N 2 | [ ( t / / c o s 2 α + t s i n 2 α e i Δ φ ) H | ( t / / s i n 2 α + t c o s 2 α e i Δ φ ) V | ] [ ( t / / c o s 2 α + t s i n 2 α e i Δ φ ) | H + ( t / / s i n 2 α + t c o s 2 α e i Δ φ ) | V ] | 2 = 1 N 2 ( t / / t ) 2 c o s 2 ( 2 α ) ,
= 1 P o v e r l a p ,
| ψ o u t = 1 N J D P ( π / 4 ) | H = 1 2 N [ ( t / / + t e i Δ φ ) | H + ( t / / t e i Δ φ ) | V ] ,
U 1 , a ( γ | H + β | V ) = ± γ | / / a + β | a , U 2 , b ( γ | H + β | V ) = ± γ | / / b + β | b , U 2 , a J D P ( α ) U 1 , a = U 1 , b J D P ( π α ) U 2 , b ,
U 1 , a = U 2 , a , U 1 , b = U 2 , b , U 2 , b = U 2 , b ( α ) , U 1 , a = U 1 , a ( α ) U 2 , b ( α ) = U 1 , a ( α ) , U 1 , a ( α ) = ( c o s ( α ) s i n ( α ) s i n ( α ) c o s ( α ) ) ,
| ψ o u t = e 2 i l α N [ ( t / / t e i Δ φ e i δ p e 4 i l α ) | c p + ( t / / t e i Δ φ e i δ p e 4 i l α ) | d p ] | + s | l o .
+ = 1 2 ( t / / + t ) [ t / / + t + 2 t / / t c o s ( Δ φ δ p 4 l δ α ) ] .
= 1 2 ( t / / + t ) [ t / / + t + 2 t / / t c o s ( Δ φ δ p 4 l δ α ) ] .
H , V , L , R = 1 2 ( t / / + t ) [ t / / + t + 2 t / / t c o s ( Δ φ ) c o s ( δ p + 4 l δ α ) ] .

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