Abstract

We describe an experimental implementation of a free-space 11-dimensional communication system using orbital angular momentum (OAM) modes. This system has a maximum measured OAM channel capacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulence on the OAM channel capacity are quantified. We find that increasing the turbulence leads to a degradation of the channel capacity. We are able to mitigate the effects of turbulence by increasing the spacing between detected OAM modes. This study has implications for high-dimensional quantum key distribution (QKD) systems. We describe the sort of QKD system that could be built using our current technology.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2012

2011

F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[CrossRef]

2010

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

2009

2008

2007

V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
[CrossRef]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

2006

S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[CrossRef]

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[CrossRef]

2005

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

2002

N. 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]

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
[CrossRef]

1999

1989

W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York 191, 363–381 (1989).
[CrossRef]

1974

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
[CrossRef]

1965

Arrizon, V.

Barbieri, C.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Beijersbergen, M. W.

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

Bennett, C.

C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf., 175–179 (Bangalore, 1984).

Berkhout, G. C. G.

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
[CrossRef] [PubMed]

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

Bjork, G.

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
[CrossRef]

Blauensteiner, B.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Bourennane, M.

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
[CrossRef]

N. 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.

Brassard, G.

C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int. Conf., 175–179 (Bangalore, 1984).

Carrada, R.

Cerf, N.

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
[CrossRef]

N. 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]

Courtial, J.

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
[CrossRef] [PubMed]

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

Dymale, R. C.

Fields, B. D.

W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York 191, 363–381 (1989).
[CrossRef]

Fried, D. L.

Fürst, M.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Gbur, G.

Gisin, N.

N. 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]

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
[CrossRef]

Gonzalez, L. A.

Groblacher, S.

S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[CrossRef]

Gruneisen, M. T.

Harding, C. M.

Hiskett, P. A.

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

Hughes, R. J.

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

Jennewein, T.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[CrossRef]

Johnston, R. A.

Karlsson, A.

N. 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]

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
[CrossRef]

Lane, R. G.

Lavery, M. P. J.

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
[CrossRef] [PubMed]

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

Lindenthal, M.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Lita, A. E.

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

Love, G. D.

Meyenburg, M.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Miller, A. J.

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

Miller, W. A.

Nam, S.

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

Nordholt, J. E.

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

Ömer, B.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Padgett, M. J.

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
[CrossRef] [PubMed]

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

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Perdigues, J.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Peterson, C. G.

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

Rarity, J.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Raymer, M.

B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[CrossRef]

Robertson, D. J.

Rosenberg, D.

P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt, “Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
[CrossRef]

Roux, F.

F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[CrossRef]

Ruiz, U.

Scheidl, T.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Schmitt-Manderbach, T.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Smith, B.

B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[CrossRef]

Sodnik, Z.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Sweiti, A. M.

Tiefenbacher, F.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Trojek, P.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Tyler, G.

Tyson, R. K.

Ursin, R.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Vaziri, A.

S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[CrossRef]

Weier, H.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Weihs, G.

S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[CrossRef]

Weinfurter, H.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

Wootters, W. K.

W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York 191, 363–381 (1989).
[CrossRef]

Young, A. T.

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
[CrossRef]

Zeilinger, A.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486 (2007).
[CrossRef]

S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[CrossRef]

Ann. Phys.-New York

W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York 191, 363–381 (1989).
[CrossRef]

Appl. Opt.

Astrophys. J.

A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
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Figures (5)

Fig. 1
Fig. 1

Proposed QKD Setup. Our source is a HeNe laser operating at 633 nm that is modulated by an acousto-optic modulator (AOM) to carve out single-photon pulses. The OAM and ANG states, as well as the turbulence phase screens are generated by the spatial light modulator (SLM) and 4f system. R1 and R2 are custom refractive elements used to sort the OAM and ANG modes, and the beam splitter (BS) acts as the passive basis selector. The modes are detected on an electron-multiplying camera (EMCCD).

Fig. 2
Fig. 2

Upper row: Holograms written onto the SLM to generate OAM and ANG modes. Lower row: CCD images of the corresponding modes generated (OAM mode number l = −1, 3, and 5, and ANG mode number n = 1, 5, and 9).

Fig. 3
Fig. 3

Sample Kolmogorov turbulence phase screens that are added onto the SLM for three different values of turbulence strength, D/r0 =(a) 5.12, (b) 10.25, and (c) 102.5. Each screen has a resolution of 512×512 pixels and the phase wraps from 0 to 2π.

Fig. 4
Fig. 4

Influence of Kolmogorov turbulence on the channel capacity of an OAM communication channel for a system dimensionality of N = 3,5,7,9 and 11. The channel capacity of a system using polarization modes is plotted as a dotted line for comparison.

Fig. 5
Fig. 5

Effects of changing the spacing between detected modes (MS) on the channel capacity of a 3-dimensional OAM communication system in the presence of atmospheric turbulence. For a mode-spacing of 4, the maximum channel capacity of a 3-dimensional system is seen to approach the theoretical maximum of log2(3) = 1.585.

Equations (5)

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Ψ OAM l , 0 = A 0 W ( r / R ) exp ( i l θ )
Ψ ANG n = 1 11 l = 5 5 Ψ OAM l , 0 exp ( i 2 π n l / 11 ) .
[ φ ( r 1 ) φ ( r 2 ) ] 2 = 6.88 | r 1 r 2 r 0 | 5 / 3
P d s = 1 π 0 1 ρ d ρ 0 2 π d θ e 3.44 ( D / r 0 ) 5 / 3 ( ρ sin θ / 2 ) 5 / 3 cos [ ( d s ) θ ]
C = max [ H ( x ) H ( x | y ) ] = max { P s } [ s P s log 2 ( P s ) + s P s d P d s log 2 ( P d s ) ] .

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