Abstract

Light with helical phase structures, carrying quantized orbital angular momentum (OAM), has many applications in both classical and quantum optics, such as high-capacity optical communications and quantum information processing. Frequency conversion is a basic technique to expand the frequency range of the fundamental light. The frequency conversion of OAM-carrying light gives rise to new physics and applications such as up-conversion detection of images and generation of high dimensional OAM entanglements. Quasi-phase matching (QPM) nonlinear crystals are good candidates for frequency conversion, particularly due to their high-valued effective nonlinear coefficients and no walk-off effect. Here we report the first experimental second-harmonic generation (SHG) of an OAM-carried light with a QPM crystal, where a UV light with OAM of 100 is generated. OAM conservation is verified using a specially designed interferometer. With a pump beam carrying an OAM superposition of opposite sign, we observe interesting interference phenomena in the SHG light; specifically, a photonics gear-like structure is obtained that gives direct evidence of OAM conservation, which will be very useful for ultra-sensitive angular measurements. Besides, we also develop a theory to reveal the underlying physics of the phenomena. The methods and theoretical analysis shown here are also applicable to other frequency conversion processes, such as sum frequency generation and difference-frequency generation, and may also be generalized to the quantum regime for single photons.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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2013 (5)

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

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[CrossRef] [PubMed]

Y. C. Lin, K. F. Huang, and Y. F. Chen, “The formation of quasi-nondiffracting focused beams with second-harmonic generation of flower Laguerre–Gaussian modes,” Laser Phys. 23(11), 115405 (2013).
[CrossRef]

G.-H. Shao, Z.-J. Wu, J.-H. Chen, F. Xu, and Y.-Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase matching,” Phys. Rev. A 88(6), 063827 (2013).
[CrossRef]

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]

2012 (5)

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
[CrossRef]

G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral orbital angular momentum transfer via four-wave mixing in Rb vapor,” Phys. Rev. Lett. 108(24), 243601 (2012).
[CrossRef] [PubMed]

D.-S. Ding, Z.-Y. Zhou, W. Huang, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Experimental up-conversion of images,” Phys. Rev. A 86(3), 033803 (2012).
[CrossRef]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[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. Photon. 6(7), 488–496 (2012).
[CrossRef]

2011 (4)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–244 (2011).
[CrossRef]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photon. 5(6), 335–342 (2011).
[CrossRef]

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

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high- dimensional two-photon entanglement and violations of generalized Bell inequality,” Nat. Phys. 7(9), 677–680 (2011).
[CrossRef]

2010 (2)

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

2009 (3)

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
[CrossRef] [PubMed]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[CrossRef] [PubMed]

2008 (4)

Q.-F. Chen, B.-S. Shi, Y.-S. Zhang, and G.-C. Guo, “Entanglement of orbital angular momentum states of photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78(5), 053810 (2008).
[CrossRef]

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

E. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492(3), 883–922 (2008).
[CrossRef]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).

2007 (1)

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

2006 (1)

2004 (1)

2002 (1)

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

2001 (2)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

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

1997 (1)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic-generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[CrossRef]

1996 (1)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

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]

1970 (1)

Ahmed, N.

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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Albota, M. A.

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic-generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[CrossRef]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

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]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high- dimensional two-photon entanglement and violations of generalized Bell inequality,” Nat. Phys. 7(9), 677–680 (2011).
[CrossRef]

Aolita, L.

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

Arlt, J.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Arnold, A. S.

G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral orbital angular momentum transfer via four-wave mixing in Rb vapor,” Phys. Rev. Lett. 108(24), 243601 (2012).
[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, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

Barreiro, J. T.

J. T. Barreiro, T.-C. 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.

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]

Bernet, S.

Bowman, R.

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

Boyd, R. W.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Buller, G. S.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high- dimensional two-photon entanglement and violations of generalized Bell inequality,” Nat. Phys. 7(9), 677–680 (2011).
[CrossRef]

Chen, J.-H.

G.-H. Shao, Z.-J. Wu, J.-H. Chen, F. Xu, and Y.-Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase matching,” Phys. Rev. A 88(6), 063827 (2013).
[CrossRef]

Chen, Q.-F.

Q.-F. Chen, B.-S. Shi, Y.-S. Zhang, and G.-C. Guo, “Entanglement of orbital angular momentum states of photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78(5), 053810 (2008).
[CrossRef]

Chen, Y. F.

Y. C. Lin, K. F. Huang, and Y. F. Chen, “The formation of quasi-nondiffracting focused beams with second-harmonic generation of flower Laguerre–Gaussian modes,” Laser Phys. 23(11), 115405 (2013).
[CrossRef]

Cizmar, T.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photon. 5(6), 335–342 (2011).
[CrossRef]

Collins, S. A.

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic-generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[CrossRef]

D’Ambrosio, V.

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

Dada, A. C.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high- dimensional two-photon entanglement and violations of generalized Bell inequality,” Nat. Phys. 7(9), 677–680 (2011).
[CrossRef]

De Martini, F.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[CrossRef] [PubMed]

Del Re, L.

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

Dennis, M. R.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[CrossRef]

Dholakia, K.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photon. 5(6), 335–342 (2011).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic-generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[CrossRef]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

Ding, D.-S.

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[CrossRef] [PubMed]

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
[CrossRef]

D.-S. Ding, Z.-Y. Zhou, W. Huang, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Experimental up-conversion of images,” Phys. Rev. A 86(3), 033803 (2012).
[CrossRef]

Dolinar, S.

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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Elias, E. M.

E. M. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492(3), 883–922 (2008).
[CrossRef]

Fazal, I. M.

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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Fedrizzi, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
[CrossRef] [PubMed]

Fickler, R.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[CrossRef] [PubMed]

Franke-Arnold, S.

G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral orbital angular momentum transfer via four-wave mixing in Rb vapor,” Phys. Rev. Lett. 108(24), 243601 (2012).
[CrossRef] [PubMed]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).

Fürhapter, S.

Guo, G.-C.

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[CrossRef] [PubMed]

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
[CrossRef]

D.-S. Ding, Z.-Y. Zhou, W. Huang, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Experimental up-conversion of images,” Phys. Rev. A 86(3), 033803 (2012).
[CrossRef]

Q.-F. Chen, B.-S. Shi, Y.-S. Zhang, and G.-C. Guo, “Entanglement of orbital angular momentum states of photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78(5), 053810 (2008).
[CrossRef]

Huang, H.

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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Huang, K. F.

Y. C. Lin, K. F. Huang, and Y. F. Chen, “The formation of quasi-nondiffracting focused beams with second-harmonic generation of flower Laguerre–Gaussian modes,” Laser Phys. 23(11), 115405 (2013).
[CrossRef]

Huang, W.

D.-S. Ding, Z.-Y. Zhou, W. Huang, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Experimental up-conversion of images,” Phys. Rev. A 86(3), 033803 (2012).
[CrossRef]

Ireland, D. G.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

Jack, B.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[CrossRef]

Jesacher, A.

Jha, A. K.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

Karimi, E.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[CrossRef] [PubMed]

King, R. P.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[CrossRef]

Krenn, M.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[CrossRef] [PubMed]

Kwek, L. C.

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

Kwiat, P. G.

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

Langford, N. K.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
[CrossRef] [PubMed]

Lapkiewicz, R.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[CrossRef] [PubMed]

Lavery, M. P. J.

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]

Leach, J.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high- dimensional two-photon entanglement and violations of generalized Bell inequality,” Nat. Phys. 7(9), 677–680 (2011).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

Li, Y.

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

Lin, Y. C.

Y. C. Lin, K. F. Huang, and Y. F. Chen, “The formation of quasi-nondiffracting focused beams with second-harmonic generation of flower Laguerre–Gaussian modes,” Laser Phys. 23(11), 115405 (2013).
[CrossRef]

Lu, Y.-Q.

G.-H. Shao, Z.-J. Wu, J.-H. Chen, F. Xu, and Y.-Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase matching,” Phys. Rev. A 88(6), 063827 (2013).
[CrossRef]

MacDonald, M. P.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Mair, A.

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

Marrucci, L.

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

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[CrossRef] [PubMed]

Maurer, C.

Molina-Terriza, G.

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

Nagali, E.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[CrossRef] [PubMed]

O’Holleran, K.

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[CrossRef]

Padgett, M.

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

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).

Padgett, M. J.

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]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–244 (2011).
[CrossRef]

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high- dimensional two-photon entanglement and violations of generalized Bell inequality,” Nat. Phys. 7(9), 677–680 (2011).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, “Isolated optical vortex knots,” Nat. Phys. 6(2), 118–121 (2010).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic-generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[CrossRef]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

Paterson, L.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Piccirillo, B.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[CrossRef] [PubMed]

Plick, W. N.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[CrossRef] [PubMed]

Ramelow, S.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[CrossRef] [PubMed]

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
[CrossRef] [PubMed]

Ratschbacher, L.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete tunable color entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
[CrossRef] [PubMed]

Ren, 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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Ritsch-Marte, M.

Romero, J.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

Sansoni, L.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

Santamato, E.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[CrossRef] [PubMed]

Schaeff, C.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[CrossRef] [PubMed]

Sciarrinao, F.

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

Sciarrino, F.

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103(1), 013601 (2009).
[CrossRef] [PubMed]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photon. 3(12), 720–723 (2009).
[CrossRef]

Shao, G.-H.

G.-H. Shao, Z.-J. Wu, J.-H. Chen, F. Xu, and Y.-Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase matching,” Phys. Rev. A 88(6), 063827 (2013).
[CrossRef]

Shi, B.-S.

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[CrossRef] [PubMed]

D.-S. Ding, Z.-Y. Zhou, W. Huang, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Experimental up-conversion of images,” Phys. Rev. A 86(3), 033803 (2012).
[CrossRef]

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
[CrossRef]

Q.-F. Chen, B.-S. Shi, Y.-S. Zhang, and G.-C. Guo, “Entanglement of orbital angular momentum states of photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78(5), 053810 (2008).
[CrossRef]

Sibbett, W.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Simpson, N. B.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

Spagnolo, N.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Sulssarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrinao, “Photonics 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]

Sulssarenko, S.

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

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.

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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Vaziri, A.

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

Volke-Sepulveda, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[CrossRef] [PubMed]

Walborn, S. P.

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

Walker, G.

G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral orbital angular momentum transfer via four-wave mixing in Rb vapor,” Phys. Rev. Lett. 108(24), 243601 (2012).
[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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Wei, T.-C.

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

Weihs, G.

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

Willner, A. E.

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. Photon. 6(7), 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(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Wong, F. N. C.

Wu, Z.-J.

G.-H. Shao, Z.-J. Wu, J.-H. Chen, F. Xu, and Y.-Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase matching,” Phys. Rev. A 88(6), 063827 (2013).
[CrossRef]

Xu, F.

G.-H. Shao, Z.-J. Wu, J.-H. Chen, F. Xu, and Y.-Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase matching,” Phys. Rev. A 88(6), 063827 (2013).
[CrossRef]

Yan, 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. Photon. 6(7), 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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Yao, A. M.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–244 (2011).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[CrossRef] [PubMed]

Yue, 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. Photon. 6(7), 488–496 (2012).
[CrossRef]

Zeilinger, A.

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Zhang, Y.-S.

Q.-F. Chen, B.-S. Shi, Y.-S. Zhang, and G.-C. Guo, “Entanglement of orbital angular momentum states of photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78(5), 053810 (2008).
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Zhou, Z.-Y.

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
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D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
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D.-S. Ding, Z.-Y. Zhou, W. Huang, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Experimental up-conversion of images,” Phys. Rev. A 86(3), 033803 (2012).
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Zou, X.-B.

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
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D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
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Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
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Figures (4)

Fig. 1
Fig. 1

Different blocks used in experiments. HWP (QWP): half-wave plate (quarter-wave plate); PBS: polarization beam splitter; BS: beam splitter; M1-10: mirrors; L1-2: lenses; VPP: vortex phase plate; PPKTP: periodically poled KTP; SLM: spatial light modulator; CCD: charge coupled device camera. Block-(a) is used to prepare a single OAM pump beam with proper polarization using VPPs. Block-(b) is the SHG part. Block-(c) is a specially designed balanced interferometer used to determine the OAM value of the input light. Block-(d) is a modified Sagnac interferometer for generating the superposition of OAM states of opposite sign using VPPs. Block-(e) has the same function as block-(d), but uses an SLM instead of a VPP.

Fig. 2
Fig. 2

Experimental demonstration of OAM conservation in the SHG process. Column (a) and (b) give the intensity profile and interference pattern for the pump beam; Column (c) and (d) give the corresponding image for the SHG light. The top and bottom sets of panels are for pump beams carrying OAM of 2 and 20, respectively. Corresponding theoretical images are paired below the experimental images.

Fig. 3
Fig. 3

SHG results with an input light in state |l+ e iβ | l with (a) l=2 and (b) l=20 generated via the VPP using block-(d). The first and third images of each row are the respective interference patterns for the pump light and the corresponding SHG light obtained directly from the block-(b) using CCD camera. The second and fourth images are the corresponding theoretical patterns.

Fig. 4
Fig. 4

SHG results with pump states of the form |l+ e iβ | l and | l 1 + e iβ | l 2 generated using block-(e) via an SLM. Rows (a)−-(e) present images corresponding to states |l+ e iβ | l with l=2,3,9,20,50 for the pump beam; row (f) corresponds to state | l 1 + e iβ | l 2 with l 1 =7 , l 2 =8 . The first image in each row is the phase diagram of SLM for generating a specific OAM-carrying light. The second and fourth images are the respective interference patterns for the pump light, projected onto the diagonal polarization direction, and the SHG light, directly observed after block-(b) using CCD camera. The third and fifth images are the corresponding theoretical patterns.

Equations (24)

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| Ψ in =|H|0,
| Ψ ( θ,δ ) out = 1 2 [(cos(2δ)|Hsin(2δ)|V)|l e i(4θ+π/2 ) (sin(2δ)|H+cos(2δ)|V)| l ],
E SHG sin 2 (2δ)L G 0 2l e i8θ cos 2 (2δ)L G 0 2l +Γsin(4δ) e i(4θ+ π 2 ) L G l 0 ,
L G p l (r,φ,x)= 2p! π( | l |+p )! 1 w( x ) ( 2 r w( x ) ) | l | L p | l | ( 2 r 2 w ( x ) 2 ) exp( r 2 w ( x ) 2 )exp( ik r 2 2R( x ) )exp( i[ 2p+| l |+1 ]ς( x ) )exp( ilα )
{ d E 1z dx =i K 1 E 2z E 3z e iΔkx d E 2z dx =i K 2 E 1z E 3z e iΔkx d E 3z dx =ig K 3 E 1z E 2z e iΔkx
| Ψ in =|H|0,
|H=( 1 0 ), |V=( 0 1 ).
U ^ QWP ( φ )= 1 2 ( icos(2φ) sin(2φ) sin(2φ) i+cos(2φ) ), U ^ HWP ( θ )= 1 2 ( cos(2θ) sin(2θ) sin(2θ) cos(2θ) ),
U ^ QWP ( φ ) U ^ HWP ( θ )( 1 0 )=a( θ,φ )|H+b( θ,φ )|V,
a( θ,φ )= 1 2 [icos(2θ)cos(2φ2θ)], b( θ,φ )= 1 2 [isin(2θ)+sin(2φ2θ)]
| Ψ ( θ,φ ) out =a( θ,φ )|H|l+b( θ,φ )|V| l .
| Ψ ( θ ) out = 1 2 (|H|l+ e i(4θ+π/2 ) |V| l ).
| Ψ ( θ,δ ) out = 1 2 [(cos(2δ)|Hsin(2δ)|V)|l e i(4θ+π/2 ) (sin(2δ)|H+cos(2δ)|V)| l ]
| Ψ ( θ,δ ) out V = 1 2 (sin(2δ)|l+ e i(4θ+π/2 ) cos(2δ)| l ).
E SHG sin 2 (2δ)L G 0 2l e i8θ cos 2 (2δ)L G 0 2l +Γsin(4δ) e i(4θ+ π 2 ) L G l 0
E SHG sin(4δ)L G 0 2l e i8θ sin(4δ)L G 0 2l +Γcos(4δ) e i(4θ+ π 2 ) L G l 0 .
E SHG ( r 0 , α 0 ,0){ L G 0 l ( r 0 , α 0 ,0)L G 0 l ( r 0 , α 0 ,0) case a L G 0 l ( r 0 , α 0 ,0)L G 0 l ( r 0 , α 0 ,0) case b L G 0 l ( r 0 , α 0 ,0)L G 0 l ( r 0 , α 0 ,0) case c
E( r,α,x )= i λB exp(ikx) 0 2π 0 E SHG ( r 0 , α 0 ,0) exp{ ik 2B ×[A r 0 2 2r r 0 cos(α α 0 )+D r 2 } r 0 d r 0 d α 0
E SHG (r,α,x)= 2 | l |! 1 w 0 2 ( 2 w 0 ) 2l i λB exp( ikD 2B r 2 ikx)exp( k 2 r 2 4ξ B 2 ) ×{ i 2l ξ 2l1 ( kr 2B ) 2l exp(i2lα) case a l! ξ l1 L l 0 ( k 2 r 2 4ξ B 2 ) case c
ξ= 2 w 0 2 + ikA B .
0 2π exp[in θ 1 +ikbrcos( θ 1 θ 2 )]d θ 1 =2πexp[in( π 2 θ 2 ) J n (kbr)
0 exp(a x 2 ) J v (2bx) x 2n+v+1 dx = n! 2 b v a nv1 exp( b 2 a ) L n v ( b 2 a )
( A B C D )=( 1 x 0 1 ).
E SHG (r,α,z)= i λx exp(ikx)exp( r 2 w 2 )×{ i 2l l! ( 2 r w ) 2l exp(i2lα) case a L l 0 ( r 2 w 2 ) case c { L G 0 2l (r,α,x) case a L G l 0 (r,α,x) case c

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