Abstract

We propose a single phase-only optical element that transforms different orbital angular momentum (OAM) modes into localized spots at separated angular positions on a transverse plane. We refer to this element as an angular lens since it separates out OAM modes in a manner analogous to how a converging lens separates out transverse wave-vector modes at the focal plane. We also simulate the proposed angular lens using a spatial light modulator and experimentally demonstrate its working. Our work can have important implications for OAM-based classical and quantum communication applications.

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

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References

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    [Crossref] [PubMed]
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2017 (1)

2015 (1)

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, and et al., “Mode division multiplexing using an orbital angular momentum mode sorter and mimo-dsp over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

2014 (1)

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

2013 (1)

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

2012 (3)

2011 (2)

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83, 053829 (2011).
[Crossref]

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13, 083030 (2011).
[Crossref]

2010 (3)

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref] [PubMed]

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

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

2009 (2)

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17, 8287–8293 (2009).
[Crossref] [PubMed]

2008 (1)

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78, 043810 (2008).
[Crossref]

2007 (3)

2006 (2)

G. M. Nikolopoulos, K. S. Ranade, and G. Alber, “Error tolerance of two-basis quantum-key-distribution protocols using qudits and two-way classical communication,” Phys. Rev. A 73, 032325 (2006).
[Crossref]

E. Yao, S. Franke-Arnold, J. Courtial, S. Barnett, and M. Padgett, “Fourier relationship between angular position and optical orbital angular momentum,” Opt. Express 14, 9071–9076 (2006).
[Crossref] [PubMed]

2004 (3)

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

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

J. Cortese, “Holevo-schumacher-westmoreland channel capacity for a class of qudit unital channels,” Phys. Rev. A 69, 022302 (2004).
[Crossref]

2003 (2)

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

M. Vasnetsov, J. Torres, D. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Optics letters 28, 2285–2287 (2003).
[Crossref] [PubMed]

2002 (4)

V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, “Quantum key distribution for d-level systems with generalized bell states,” Phys. Rev. A 65, 052331 (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, 127902 (2002).
[Crossref] [PubMed]

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
[Crossref] [PubMed]

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

2001 (2)

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of laguerre–gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

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

2000 (1)

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[Crossref] [PubMed]

1998 (1)

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[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, 8185–8189 (1992).
[Crossref] [PubMed]

1974 (2)

O. Bryngdahl, “Geometrical transformations in optics,” JOSA 64, 1092–1099 (1974).
[Crossref]

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[Crossref]

Agarwal, G. S.

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83, 053829 (2011).
[Crossref]

Ahmed, N.

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, and et al., “Mode division multiplexing using an orbital angular momentum mode sorter and mimo-dsp over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Alber, G.

G. M. Nikolopoulos, K. S. Ranade, and G. Alber, “Error tolerance of two-basis quantum-key-distribution protocols using qudits and two-way classical communication,” Phys. Rev. A 73, 032325 (2006).
[Crossref]

Allen, L.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Almeida, M. P.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Arie, A.

Arlt, J.

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of laguerre–gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

Arrizón, V.

Bagherinezhad, S.

V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, “Quantum key distribution for d-level systems with generalized bell states,” Phys. Rev. A 65, 052331 (2002).
[Crossref]

Bahraminasab, A.

V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, “Quantum key distribution for d-level systems with generalized bell states,” Phys. Rev. A 65, 052331 (2002).
[Crossref]

Bao, C.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Barbieri, M.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Barnett, S.

Barnett, S. M.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17, 8287–8293 (2009).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78, 043810 (2008).
[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, 013601 (2004).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

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

Beijersbergen, M. W.

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

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Berkhout, G. C.

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, 2110–2115 (2012).
[Crossref] [PubMed]

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999), 7th ed.
[Crossref]

Bourennane, M.

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

Boyd, R. W.

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

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83, 053829 (2011).
[Crossref]

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17, 8287–8293 (2009).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78, 043810 (2008).
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun4 (2013).
[Crossref] [PubMed]

Bozinovic, N.

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

Brunner, N.

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B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

T. C. Ralph, K. J. Resch, and A. Gilchrist, “Efficient toffoli gates using qudits,” Phys. Rev. A 75, 022313 (2007).
[Crossref]

Ramachandran, S.

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

Ranade, K. S.

G. M. Nikolopoulos, K. S. Ranade, and G. Alber, “Error tolerance of two-basis quantum-key-distribution protocols using qudits and two-way classical communication,” Phys. Rev. A 73, 032325 (2006).
[Crossref]

Ren, Y.

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, and et al., “Mode division multiplexing using an orbital angular momentum mode sorter and mimo-dsp over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

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

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Resch, K. J.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

T. C. Ralph, K. J. Resch, and A. Gilchrist, “Efficient toffoli gates using qudits,” Phys. Rev. A 75, 022313 (2007).
[Crossref]

Ritsch-Marte, M.

Robertson, D. A.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

Robertson, D. J.

Romero, J.

Romero, L. A.

Ruiz, U.

Sasaki, M.

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

Shi, Z.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun4 (2013).
[Crossref] [PubMed]

Skeldon, K.

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

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Steinhoff, N.

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13, 083030 (2011).
[Crossref]

Stoltenberg, K. E.

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13, 083030 (2011).
[Crossref]

Takeoka, M.

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

Torner, L.

M. Vasnetsov, J. Torres, D. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Optics letters 28, 2285–2287 (2003).
[Crossref] [PubMed]

Torres, J.

M. Vasnetsov, J. Torres, D. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Optics letters 28, 2285–2287 (2003).
[Crossref] [PubMed]

Tur, M.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

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

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Vasnetsov, M.

Vaziri, A.

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

Vértesi, T.

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref] [PubMed]

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Weihs, G.

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

White, A. G.

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Willner, A. E.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

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

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999), 7th ed.
[Crossref]

Xie, G.

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, and et al., “Mode division multiplexing using an orbital angular momentum mode sorter and mimo-dsp over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Yan, Y.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yao, E.

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78, 043810 (2008).
[Crossref]

E. Yao, S. Franke-Arnold, J. Courtial, S. Barnett, and M. Padgett, “Fourier relationship between angular position and optical orbital angular momentum,” Opt. Express 14, 9071–9076 (2006).
[Crossref] [PubMed]

Yue, Y.

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

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Zeilinger, A.

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

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[Crossref] [PubMed]

Zhao, Z.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Zukowski, M.

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418–4421 (2000).
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J. Mod. Opt. (1)

J. Arlt, R. Kuhn, and K. Dholakia, “Spatial transformation of laguerre–gaussian laser modes,” J. Mod. Opt. 48, 783–787 (2001).

J. Opt. Soc. Am. A (2)

JOSA (1)

O. Bryngdahl, “Geometrical transformations in optics,” JOSA 64, 1092–1099 (1974).
[Crossref]

Nat. Commun. (1)

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Nat. Phys. (1)

B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
[Crossref]

Nature (1)

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

New J. Phys. (1)

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13, 083030 (2011).
[Crossref]

Opt. Commun. (1)

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[Crossref]

Opt. Express (5)

Optica (1)

Optics letters (1)

M. Vasnetsov, J. Torres, D. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Optics letters 28, 2285–2287 (2003).
[Crossref] [PubMed]

Phys. Rev. A (7)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78, 043810 (2008).
[Crossref]

J. Cortese, “Holevo-schumacher-westmoreland channel capacity for a class of qudit unital channels,” Phys. Rev. A 69, 022302 (2004).
[Crossref]

T. C. Ralph, K. J. Resch, and A. Gilchrist, “Efficient toffoli gates using qudits,” Phys. Rev. A 75, 022313 (2007).
[Crossref]

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83, 053829 (2011).
[Crossref]

V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, “Quantum key distribution for d-level systems with generalized bell states,” Phys. Rev. A 65, 052331 (2002).
[Crossref]

G. M. Nikolopoulos, K. S. Ranade, and G. Alber, “Error tolerance of two-basis quantum-key-distribution protocols using qudits and two-way classical communication,” Phys. Rev. A 73, 032325 (2006).
[Crossref]

Phys. Rev. Lett. (10)

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref] [PubMed]

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

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[Crossref] [PubMed]

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
[Crossref] [PubMed]

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref] [PubMed]

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

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

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. W. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

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

Sci. Rep. (1)

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. A. Nguyen, D. A. Nolan, M.-J. Li, and et al., “Mode division multiplexing using an orbital angular momentum mode sorter and mimo-dsp over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Science (1)

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

Other (3)

J. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1996), 2nd ed.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999), 7th ed.
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun4 (2013).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 (a) The schematic illustration of the working of a converging lens. The transverse wave-vector modes q1 and q2 with phase profiles e i q 1 ρ and e i q 2 ρ get localized at separate spatial locations. (b) The schematic illustration of the working of our proposed angular lens. The OAM modes 1 and 2 with phase profiles e i 1 ϕ and e i 2 ϕ get localized at separate angular positions.
Fig. 2
Fig. 2 (a) The thickness function of the proposed angular lens. (b) The experimental Setup. SLM is spatial light modulator, L stands for a converging lens, and M stands for a mirror.
Fig. 3
Fig. 3 Transformation of constant-intensity OAM modes by the angular lens. (a), (b), and (c) are the combined intensity patterns for various observed at z = 65 cm. (d)–(f) are the theoretical diffraction patterns as obtained by numerically evaluating the integral in Eq. (2) for the same set of parameters. The screen size in all the above plots is 6.27 mm × 6.27 mm. A constant background of 250 counts has been subtracted from the each CCD camera pixel.
Fig. 4
Fig. 4 Scaling of diffraction patterns with z. (a), (b), and (c) are the combined intensity patterns observed at three z values. (d)–(f) are the theoretical diffraction patterns as obtained by numerically evaluating the integral in Eq. (2) for the same set of parameters. The screen size in all the above plots is 6.27 mm × 6.27 mm. A constant background of 250 counts has been subtracted from the each CCD camera pixel.
Fig. 5
Fig. 5 Cross-talk analysis. (a) The theoretically defined bins for different OAM modes. (b) Experimental detection probability of the 19 input OAM modes. (c) The theoretical detection probability of the 19 input OAM modes.
Fig. 6
Fig. 6 Transformation of LG modes by the proposed angular lens. (a), (b), and (c) are the combined intensity patterns for various with p = 0 observed at z = 100 cm. (d)–(f) are the theoretical diffraction patterns as obtained by numerically evaluating the integral in Eq. (2) for the same set of parameters. The screen size in all the above plots is 5.63 mm × 5.63 mm. A constant background of 170 counts has been subtracted from the each CCD camera pixel.

Equations (4)

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

T ang ( ρ , ϕ ) = exp [ i ( α ϕ 2 β ρ ) ] .
E z = z ( ρ , ϕ ) = e i k z i λ z e i k 2 z ρ 2 0 0 2 π E z = 0 ( ρ , ϕ ) × T ang ( ρ , ϕ ) e i k 2 z ρ 2 e i k ρ ρ z cos ( ϕ ϕ ) ρ d ρ d ϕ .
E z = z 1 ( 1 ) ( ρ , ϕ ) = e i k z 1 i λ z 1 e i k ρ 2 2 z 1 0 D 1 / 2 0 2 π E z = 0 ( ϕ ) × T ang ( ρ , ϕ ) e i k 2 z 1 ρ 2 e i k ρ ρ z 1 cos ( ϕ ϕ ) ρ d ρ d ϕ .
| E z = z 2 ( 2 ) ( ρ , ϕ ) | 2 = | E z = z 1 ( 1 ) ( ρ / a , ϕ ) | 2 , if z 2 = a 2 z 1 , β 2 = β 1 a and  D 2 = D 1 a ,

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