Abstract

Light with orbital angular momentum (OAM), or twisted light, is widely investigated in the fields of optical communications, quantum information science, and nonlinear optics by harnessing its unbounded dimension. For nonlinear frequency conversion of twisted light, the efficiency usually decreases exponentially with increasing topological charge, which severely degrades the quality of the output modes. We here conceive and develop a method of eliminating the dependence of the conversion efficiency on the topological charge in second-harmonic generation using a single lens imaging technique. We first give the theoretical model and simulate the evolution of light beams near the image point of the lens, then the independence of the SHG conversion efficiency from topological charge is verified for different integer values of the topological charge; this independence is confirmed for varying pump power. The method can be generalized to other light–matter interactions and is a promising method of achieving higher efficiency and fidelity in the interaction of twisted light with matter.

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

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References

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2018 (2)

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

L. Zhang, X. Qiu, F. Li, H. Liu, X. Chen, and L. Chen, “Second harmonic generation with full Poincaré beams,” Opt. Express 26(9), 11678 (2018).
[Crossref]

2017 (4)

2016 (5)

F. Steinlechner, N. Hermosa, V. Pruneri, and J. P. Torres, “Frequency conversion of structured light,” Sci. Rep. 6(1), 21390 (2016).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

N. A. Chaitanya, M. V. Jabir, and G. K. Samanta, “Efficient nonlinear generation of high power, higher order, ultrafast “perfect” vortices in green,” Opt. Lett. 41(7), 1348 (2016).
[Crossref]

2015 (4)

2014 (5)

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref]

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298 (2014).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098 (2014).
[Crossref]

2013 (2)

S.-M. Li, L.-J. Kong, Z.-C. Ren, Y. Li, C. Tu, and H.-T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (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]

2012 (2)

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(7), 488–496 (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]

2011 (2)

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]

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]

Aadhi, A.

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. Photonics 6(7), 488–496 (2012).
[Crossref]

Allen, L.

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]

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]

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]

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]

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]

Belmonte, A.

Bouchard, F.

Bowman, R.

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

Boyd, R. W.

Chaitanya, N. A.

Chen, L.

Chen, L.-X.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

Chen, X.

D’Ambrosio, V.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref]

Dholakia, K.

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]

Ding, D.-S.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

Y. Li, Z.-Y. Zhou, D.-S. Ding, and B.-S. Shi, “Sum frequency generation with two orbital angular momentum carrying laser beams,” J. Opt. Soc. Am. B 32(3), 407–411 (2015).
[Crossref]

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298 (2014).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098 (2014).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

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. Photonics 6(7), 488–496 (2012).
[Crossref]

Dong, M.-X.

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

Elser, D.

Erhard, M.

M. Erhard, R. Flicker, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light: Sci. Appl. 7(3), 17146 (2018).
[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. Photonics 6(7), 488–496 (2012).
[Crossref]

Fickler, R.

Flicker, R.

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

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]

Gagnon-Bischoff, J.

Gao, W.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

Giacobino, E.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

Giner, L.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

Günthner, K.

Guo, G.-C.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

Heim, B.

Hermosa, N.

F. Steinlechner, N. Hermosa, V. Pruneri, and J. P. Torres, “Frequency conversion of structured light,” Sci. Rep. 6(1), 21390 (2016).
[Crossref]

Heshami, K.

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. Photonics 6(7), 488–496 (2012).
[Crossref]

Jabir, M. V.

Jiang, Y.-K.

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298 (2014).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

Karimi, E.

Kong, L.-J.

S.-M. Li, L.-J. Kong, Z.-C. Ren, Y. Li, C. Tu, and H.-T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).
[Crossref]

Krenn, M.

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

Larocque, H.

Laurat, J.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

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]

Leuchs, G.

Li, F.

Li, S.-M.

S.-M. Li, L.-J. Kong, Z.-C. Ren, Y. Li, C. Tu, and H.-T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).
[Crossref]

Li, X.

Li, Y.

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

Y. Li, Z.-Y. Zhou, D.-S. Ding, and B.-S. Shi, “Sum frequency generation with two orbital angular momentum carrying laser beams,” J. Opt. Soc. Am. B 32(3), 407–411 (2015).
[Crossref]

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298 (2014).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098 (2014).
[Crossref]

S.-M. Li, L.-J. Kong, Z.-C. Ren, Y. Li, C. Tu, and H.-T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).
[Crossref]

Li, Y.-H.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

S.-L. Liu, S.-K. Liu, Y.-H. Li, S. Shi, Z.-Y. Zhou, and B.-S. Shi, “Coherent frequency bridge between visible and telecommunications band for vortex light,” Opt. Express 25(20), 24290–24298 (2017).
[Crossref]

Liu, H.

Liu, S.-K.

S.-L. Liu, S.-K. Liu, Y.-H. Li, S. Shi, Z.-Y. Zhou, and B.-S. Shi, “Coherent frequency bridge between visible and telecommunications band for vortex light,” Opt. Express 25(20), 24290–24298 (2017).
[Crossref]

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

Liu, S.-L.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

S.-L. Liu, S.-K. Liu, Y.-H. Li, S. Shi, Z.-Y. Zhou, and B.-S. Shi, “Coherent frequency bridge between visible and telecommunications band for vortex light,” Opt. Express 25(20), 24290–24298 (2017).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

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Marrucci, L.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref]

Maxein, D.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

Nicolas, A.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
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M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
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M. J. Padgett, “Orbital angular momentum 25 years on,” Opt. Express 25(10), 11265 (2017).
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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).
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A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3(2), 161–244 (2011).
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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).
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Pan, H.

Pan, J.-S.

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
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Peuntinger, C.

Pruneri, V.

F. Steinlechner, N. Hermosa, V. Pruneri, and J. P. Torres, “Frequency conversion of structured light,” Sci. Rep. 6(1), 21390 (2016).
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Qiu, X.

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. Photonics 6(7), 488–496 (2012).
[Crossref]

Ren, Z.-C.

S.-M. Li, L.-J. Kong, Z.-C. Ren, Y. Li, C. Tu, and H.-T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).
[Crossref]

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Samanta, G. K.

Sciarrino, F.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref]

Shi, B.-S.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

S.-L. Liu, S.-K. Liu, Y.-H. Li, S. Shi, Z.-Y. Zhou, and B.-S. Shi, “Coherent frequency bridge between visible and telecommunications band for vortex light,” Opt. Express 25(20), 24290–24298 (2017).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

Y. Li, Z.-Y. Zhou, D.-S. Ding, and B.-S. Shi, “Sum frequency generation with two orbital angular momentum carrying laser beams,” J. Opt. Soc. Am. B 32(3), 407–411 (2015).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098 (2014).
[Crossref]

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298 (2014).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

Shi, S.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

S.-L. Liu, S.-K. Liu, Y.-H. Li, S. Shi, Z.-Y. Zhou, and B.-S. Shi, “Coherent frequency bridge between visible and telecommunications band for vortex light,” Opt. Express 25(20), 24290–24298 (2017).
[Crossref]

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298 (2014).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098 (2014).
[Crossref]

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

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]

Sit, A.

Slussarenko, S.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref]

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).
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Sponselli, A.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
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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).
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F. Steinlechner, N. Hermosa, V. Pruneri, and J. P. Torres, “Frequency conversion of structured light,” Sci. Rep. 6(1), 21390 (2016).
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Tang, R.

Torres, J. P.

F. Steinlechner, N. Hermosa, V. Pruneri, and J. P. Torres, “Frequency conversion of structured light,” Sci. Rep. 6(1), 21390 (2016).
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A. Belmonte, C. Rosales-Guzmán, and J. P. Torres, “Measurement of flow vorticity with helical beams of light,” Optica 2(11), 1002–1005 (2015).
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S.-M. Li, L.-J. Kong, Z.-C. Ren, Y. Li, C. Tu, and H.-T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).
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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. Photonics 6(7), 488–496 (2012).
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G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref]

Veissier, L.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

Villoresi, P.

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
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S.-M. Li, L.-J. Kong, Z.-C. Ren, Y. Li, C. Tu, and H.-T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).
[Crossref]

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(7), 488–496 (2012).
[Crossref]

Wang, K.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

Wang, X.-S.

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298 (2014).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
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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(7), 488–496 (2012).
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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).
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Wu, W.

Xiang, G.-Y.

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
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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(7), 488–496 (2012).
[Crossref]

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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(7), 488–496 (2012).
[Crossref]

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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. Photonics 6(7), 488–496 (2012).
[Crossref]

Zeilinger, A.

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

Zeng, H.

Zhang, L.

Zhang, W.

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098 (2014).
[Crossref]

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

Zhou, Z.-Y.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

S.-L. Liu, S.-K. Liu, Y.-H. Li, S. Shi, Z.-Y. Zhou, and B.-S. Shi, “Coherent frequency bridge between visible and telecommunications band for vortex light,” Opt. Express 25(20), 24290–24298 (2017).
[Crossref]

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

Y. Li, Z.-Y. Zhou, D.-S. Ding, and B.-S. Shi, “Sum frequency generation with two orbital angular momentum carrying laser beams,” J. Opt. Soc. Am. B 32(3), 407–411 (2015).
[Crossref]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098 (2014).
[Crossref]

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298 (2014).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

Zhu, Z.-H.

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, Y.-H. Li, S. Shi, D.-S. Ding, L.-X. Chen, W. Gao, G.-C. Guo, and B.-S. Shi, “Quantum twisted double-slits experiments: confirming wavefunctions’ physical reality,” Sci. Bull. 62(17), 1185–1192 (2017).
[Crossref]

Z.-Y. Zhou, Z.-H. Zhu, S.-L. Liu, S.-K. Liu, K. Wang, S. Shi, W. Zhang, D.-S. Ding, and B.-S. Shi, “Generation and reverse transformation of twisted light by spatial light modulator,” arXiv:1612.04482 [physics.optics].

Adv. Opt. Photon. (1)

J. Opt. Soc. Am. B (1)

Light: Sci. Appl. (3)

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular photonic quantum interface,” Light: Sci. Appl. 5(1), e16019 (2016).
[Crossref]

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

D.-S. Ding, W. Zhang, S. Shi, Z.-Y. Zhou, Y. Li, B.-S. Shi, and G.-C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. 5(10), e16157 (2016).
[Crossref]

Nat. Photonics (3)

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

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(7), 488–496 (2012).
[Crossref]

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

Opt. Express (5)

Opt. Lett. (3)

Optica (2)

Phys. Rev. (1)

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, J.-S. Pan, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Toward high-dimensional-state quantum memory in a cold atomic ensemble,” Phys. Rev. 90(4), 042301 (2014).
[Crossref]

Phys. Rev. A (3)

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]

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]

S.-M. Li, L.-J. Kong, Z.-C. Ren, Y. Li, C. Tu, and H.-T. Wang, “Managing orbital angular momentum in second-harmonic generation,” Phys. Rev. A 88(3), 035801 (2013).
[Crossref]

Phys. Rev. Lett. (3)

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Figures (4)

Fig. 1.
Fig. 1. Simulation of evolution of light field near the image point for different topological charges. (X1)-(X8) (X = a, b, c) are for l = 1, 2, 3, respectively. Different topological charges has the same propagation distances, which is from −2.5 mm to 0.5 mm. From (X1) to (X8), the propagation distances are (−2.5, −0.5, −0.1, −0.01, 0, 0.01, 0.1, 0.5) mm. For (X1), the simulation transverse area is 0. 4 mm × 0.4 mm, for other images [(X2)-(X8)], the area are 0.2 mm × 0.2 mm. The parameters for the simulation are λ=1.56 μm, u = 1050 mm, f = 50 mm, w0=1 mm, A = 1/20.
Fig. 2.
Fig. 2. Experimental setup. HWP1, HWP2: half-wave plates; PBS: polarized beam splitter; VPP: vortex phase plate; L1, L2: lenses; DM: dichroic mirror; CL1, CL2: cylindrical lenses. The inserted figure shows the spatial distances between different optical elements, where F is the focus plane of lens L1, C is the center pane of PPKTP crystal, which is the zero propagation point for theoretical simulations.
Fig. 3.
Fig. 3. SHG conversion efficiency as a function of pump power for topological charge ranging from 0 to 3. The small deviation of higher topological charge from the Gaussian beam arises from slight beam distortion for higher spatial modes because of the aberration of single-lens imaging.
Fig. 4.
Fig. 4. Images of the SHG beam and the fringes transformed with a pair of cylindrical lenses for a pump beam with integer topological charge of 1, 2, and 3.

Equations (8)

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E 2 ω ( r , α , 0 ) E ω ( r , α , 0 ) 2 .
P = n 2 c μ 0 0 + 0 2 π | E ( r , α , 0 ) | 2 r d r d α ,
P 2 ω = n 2 c μ 0 0 + 0 2 π | E 2 ω ( r , α , 0 ) | 2 r d r d α .
P 2 ω 0 + 0 2 π | E ω ( r , α , 0 ) | 4 r d r d α .
( f 2 f 1 0 0 f 1 f 2 )  4f imaging;  ( f u f 0 1 f 1 u f )  single f imaging ,
E i ( r i , α i , z ) = 1 A exp (  -  i k z ) exp [  -  i k C A r i 2 ] E o ( r i A , α i , 0 ) ,
E 0 ( r 0 , α o , 0 ) = 2 / π exp [ r 0 2 / w 0 2 ] exp [ i α 0 ] ,
E C ( r c , α C , z ) = N k z 1 exp [  -  i ( k z 1 + l α C + k r C 2 2 z 1 ] F ( 1 + l / 2 , l + 1 , b 2 a ) ,

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