Abstract

We theoretically study the generation of orbital angular momentum (OAM) based on the four-wave mixing process in a diamond-type homogeneously broadened $^{85}{\rm Rb}$ atomic system. We use density matrix formalism at a weak field limit to explain the origin of vortex translation between different optical fields and a generated signal. We show how the singularities, which are omnipresent in the phases of the input optical vortex beams, can be profoundly mapped to atomic coherence in the transverse plane, which holds the origin of OAM translation. This translation process works well even for a moderately intense probe and control field, which enhances medium nonlinearity. The generation and manipulation of OAM of the light beam in a nonlinear medium may have important applications in optical tweezers and quantum information processing systems.

© 2020 Optical Society of America

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2019 (6)

X. Pan, S. Yu, Y. Zhou, K. Zhang, K. Zhang, S. Lv, S. Li, W. Wang, and J. Jing, “Orbital-angular-momentum multiplexed continuous-variable entanglement from four-wave mixing in hot atomic vapor,” Phys. Rev. Lett. 123, 070506 (2019).
[Crossref]

Y. Hong, Z. Wang, D. Ding, and B. Yu, “Ultraslow vortex four-wave mixing via multiphoton quantum interference,” Opt. Express 27, 29863–29874 (2019).
[Crossref]

H. R. Hamedi, J. Ruseckas, E. Paspalakis, and G. Juzeliūnas, “Transfer of optical vortices in coherently prepared media,” Phys. Rev. A 99, 033812 (2019).
[Crossref]

Z. Amini Sabegh, M. A. Maleki, and M. Mahmoudi, “Microwave-induced orbital angular momentum transfer,” Sci. Rep. 9, 3519 (2019).
[Crossref]

H.-H. Wang, J. Wang, Z.-H. Kang, L. Wang, J.-Y. Gao, Y. Chen, and X.-J. Zhang, “Transfer of orbital angular momentum of light using electromagnetically induced transparency,” Phys. Rev. A 100, 013822 (2019).
[Crossref]

Y. Sebbag, Y. Barash, and U. Levy, “Generation of coherent mid-ir light by parametric four-wave mixing in alkali vapor,” Opt. Lett. 44, 971–974 (2019).
[Crossref]

2018 (6)

R. F. Offer, D. Stulga, E. Riis, S. Franke-Arnold, and A. S. Arnold, “Spiral bandwidth of four-wave mixing in rb vapour,” Commun. Phys. 1, 84 (2018).
[Crossref]

H. R. Hamedi, J. Ruseckas, and G. Juzeliūnas, “Exchange of optical vortices using an electromagnetically-induced-transparency–based four-wave-mixing setup,” Phys. Rev. A 98, 013840 (2018).
[Crossref]

J. D. Swaim, E. M. Knutson, O. Danaci, and R. T. Glasser, “Multimode four-wave mixing with a spatially structured pump,” Opt. Lett. 43, 2716–2719 (2018).
[Crossref]

A. Chopinaud, M. Jacquey, B. Viaris de Lesegno, and L. Pruvost, “High helicity vortex conversion in a rubidium vapor,” Phys. Rev. A 97, 063806 (2018).
[Crossref]

N. Prajapati, A. M. Akulshin, and I. Novikova, “Comparison of collimated blue-light generation in atoms via the and lines,” J. Opt. Soc. Am. B 35, 1133–1139 (2018).
[Crossref]

Z. Wang, J. Yang, Y. Sun, and Y. Zhang, “Interference patterns of vortex beams based on photonic band gap structure,” Opt. Lett. 43, 4354–4357 (2018).
[Crossref]

2017 (8)

L. J. Pereira, W. T. Buono, D. S. Tasca, K. Dechoum, and A. Z. Khoury, “Orbital-angular-momentum mixing in type-ii second-harmonic generation,” Phys. Rev. A 96, 053856 (2017).
[Crossref]

G.-H. Shao, S.-C. Yan, W. Luo, G.-W. Lu, and Y.-Q. Lu, “Orbital angular momentum (oam) conversion and multicasting using n-core supermode fiber,” Sci. Rep. 7, 1062 (2017).
[Crossref]

R. N. Lanning, Z. Xiao, M. Zhang, I. Novikova, E. E. Mikhailov, and J. P. Dowling, “Gaussian-beam-propagation theory for nonlinear optics involving an analytical treatment of orbital-angular-momentum transfer,” Phys. Rev. A 96, 013830 (2017).
[Crossref]

A. M. Akulshin, N. Rahaman, S. A. Suslov, and R. J. McLean, “Amplified spontaneous emission at in two-photon excited rubidium vapor,” J. Opt. Soc. Am. B 34, 2478–2484 (2017).
[Crossref]

A. M. Akulshin, D. Budker, and R. J. McLean, “Parametric wave mixing enhanced by velocity-insensitive two-photon excitation in rb vapor,” J. Opt. Soc. Am. B 34, 1016–1022 (2017).
[Crossref]

Y. S. Ihn, K.-K. Park, Y. Kim, Y.-T. Chough, and Y.-H. Kim, “Intensity correlation in frequency upconversion via four-wave mixing in rubidium vapor,” J. Opt. Soc. Am. B 34, 2352–2357 (2017).
[Crossref]

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Philos. Trans. R. Soc. London, Ser. A 375, 20150444 (2017).
[Crossref]

S. Franke-Arnold, “Optical angular momentum and atoms,” Philos. Trans. R. Soc. London, Ser. A 375, 20150435 (2017).
[Crossref]

2016 (1)

2015 (4)

2014 (1)

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
[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, 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, 243601 (2012).
[Crossref]

2009 (2)

J. T. Mendonça, B. Thidé, and H. Then, “Stimulated Raman and Brillouin backscattering of collimated beams carrying orbital angular momentum,” Phys. Rev. Lett. 102, 185005 (2009).
[Crossref]

A. M. Akulshin, R. J. McLean, A. I. Sidorov, and P. Hannaford, “Coherent and collimated blue light generated by four-wave mixing in rb vapour,” Opt. Express 17, 22861–22870 (2009).
[Crossref]

2007 (1)

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

2006 (1)

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97, 163903 (2006).
[Crossref]

2003 (2)

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[Crossref]

S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003).
[Crossref]

2002 (3)

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[Crossref]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B 4, S17–S19 (2002).
[Crossref]

1999 (1)

J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83, 4967–4970 (1999).
[Crossref]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[Crossref]

1992 (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. Photonics 6, 488–496 (2012).
[Crossref]

Akulshin, A. M.

Allen, L.

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B 4, S17–S19 (2002).
[Crossref]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

L. Allen, M. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics (Elsevier, 1999), Vol. 39, pp. 291–372.
[Crossref]

Amini Sabegh, Z.

Z. Amini Sabegh, M. A. Maleki, and M. Mahmoudi, “Microwave-induced orbital angular momentum transfer,” Sci. Rep. 9, 3519 (2019).
[Crossref]

Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97, 163903 (2006).
[Crossref]

Arnold, A. S.

R. F. Offer, D. Stulga, E. Riis, S. Franke-Arnold, and A. S. Arnold, “Spiral bandwidth of four-wave mixing in rb vapour,” Commun. Phys. 1, 84 (2018).
[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, 243601 (2012).
[Crossref]

Babiker, M.

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Philos. Trans. R. Soc. London, Ser. A 375, 20150444 (2017).
[Crossref]

L. Allen, M. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics (Elsevier, 1999), Vol. 39, pp. 291–372.
[Crossref]

Barash, Y.

Barbieri, C.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97, 163903 (2006).
[Crossref]

Barnett, S. M.

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Philos. Trans. R. Soc. London, Ser. A 375, 20150444 (2017).
[Crossref]

Barreiro, S.

S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003).
[Crossref]

Bianchini, A.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97, 163903 (2006).
[Crossref]

Boyd, R. W.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
[Crossref]

Budker, D.

A. M. Akulshin, D. Budker, and R. J. McLean, “Parametric wave mixing enhanced by velocity-insensitive two-photon excitation in rb vapor,” J. Opt. Soc. Am. B 34, 1016–1022 (2017).
[Crossref]

A. M. Akulshin, F. P. Bustos, N. Rahaman, and D. Budker, “Polychromatic forward-directed sub-doppler emission from sodium vapour,” ArXiv:1909.01156 (2019).

Buono, W. T.

L. J. Pereira, W. T. Buono, D. S. Tasca, K. Dechoum, and A. Z. Khoury, “Orbital-angular-momentum mixing in type-ii second-harmonic generation,” Phys. Rev. A 96, 053856 (2017).
[Crossref]

Bustos, F. P.

A. M. Akulshin, F. P. Bustos, N. Rahaman, and D. Budker, “Polychromatic forward-directed sub-doppler emission from sodium vapour,” ArXiv:1909.01156 (2019).

Chen, Y.

H.-H. Wang, J. Wang, Z.-H. Kang, L. Wang, J.-Y. Gao, Y. Chen, and X.-J. Zhang, “Transfer of orbital angular momentum of light using electromagnetically induced transparency,” Phys. Rev. A 100, 013822 (2019).
[Crossref]

Chopinaud, A.

A. Chopinaud, M. Jacquey, B. Viaris de Lesegno, and L. Pruvost, “High helicity vortex conversion in a rubidium vapor,” Phys. Rev. A 97, 063806 (2018).
[Crossref]

Chough, Y.-T.

Corkum, P. B.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
[Crossref]

Curtis, J. E.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[Crossref]

Danaci, O.

Dechoum, K.

L. J. Pereira, W. T. Buono, D. S. Tasca, K. Dechoum, and A. Z. Khoury, “Orbital-angular-momentum mixing in type-ii second-harmonic generation,” Phys. Rev. A 96, 053856 (2017).
[Crossref]

Ding, D.

Ding, D.-S.

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, 488–496 (2012).
[Crossref]

Dowling, J. P.

R. N. Lanning, Z. Xiao, M. Zhang, I. Novikova, E. E. Mikhailov, and J. P. Dowling, “Gaussian-beam-propagation theory for nonlinear optics involving an analytical treatment of orbital-angular-momentum transfer,” Phys. Rev. A 96, 013830 (2017).
[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, 488–496 (2012).
[Crossref]

Franke-Arnold, S.

R. F. Offer, D. Stulga, E. Riis, S. Franke-Arnold, and A. S. Arnold, “Spiral bandwidth of four-wave mixing in rb vapour,” Commun. Phys. 1, 84 (2018).
[Crossref]

S. Franke-Arnold, “Optical angular momentum and atoms,” Philos. Trans. R. Soc. London, Ser. A 375, 20150435 (2017).
[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, 243601 (2012).
[Crossref]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[Crossref]

Frumker, E.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
[Crossref]

Gao, J.-Y.

H.-H. Wang, J. Wang, Z.-H. Kang, L. Wang, J.-Y. Gao, Y. Chen, and X.-J. Zhang, “Transfer of orbital angular momentum of light using electromagnetically induced transparency,” Phys. Rev. A 100, 013822 (2019).
[Crossref]

Gao, W.

Z. Zhu, W. Gao, C. Mu, and H. Li, “Reversible orbital angular momentum photon–phonon conversion,” Optica 3, 212–217 (2016).
[Crossref]

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X. Pan, S. Yu, Y. Zhou, K. Zhang, K. Zhang, S. Lv, S. Li, W. Wang, and J. Jing, “Orbital-angular-momentum multiplexed continuous-variable entanglement from four-wave mixing in hot atomic vapor,” Phys. Rev. Lett. 123, 070506 (2019).
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X. Pan, S. Yu, Y. Zhou, K. Zhang, K. Zhang, S. Lv, S. Li, W. Wang, and J. Jing, “Orbital-angular-momentum multiplexed continuous-variable entanglement from four-wave mixing in hot atomic vapor,” Phys. Rev. Lett. 123, 070506 (2019).
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R. N. Lanning, Z. Xiao, M. Zhang, I. Novikova, E. E. Mikhailov, and J. P. Dowling, “Gaussian-beam-propagation theory for nonlinear optics involving an analytical treatment of orbital-angular-momentum transfer,” Phys. Rev. A 96, 013830 (2017).
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Zhang, Y.

Zhou, Y.

X. Pan, S. Yu, Y. Zhou, K. Zhang, K. Zhang, S. Lv, S. Li, W. Wang, and J. Jing, “Orbital-angular-momentum multiplexed continuous-variable entanglement from four-wave mixing in hot atomic vapor,” Phys. Rev. Lett. 123, 070506 (2019).
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Zhu, L.

L. Zhu and J. Wang, “Arbitrary manipulation of spatial amplitude and phase using phase-only spatial light modulators,” Sci. Rep. 4, 7441 (2015).
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Zhu, Z.-H.

H.-J. Wu, L.-W. Mao, Y.-J. Yang, C. Rosales-Guzmán, W. Gao, B.-S. Shi, and Z.-H. Zhu, “Radial modal transitions of Laguerre-Gauss modes during parametric upconversion: towards the full-field selection rule of spatial modes,” ArXiv e-print 1912.05585 (2019).

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Commun. Phys. (1)

R. F. Offer, D. Stulga, E. Riis, S. Franke-Arnold, and A. S. Arnold, “Spiral bandwidth of four-wave mixing in rb vapour,” Commun. Phys. 1, 84 (2018).
[Crossref]

J. Opt. B (1)

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B 4, S17–S19 (2002).
[Crossref]

J. Opt. Soc. Am. B (5)

Nat. Photonics (1)

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

Nat. Phys. (1)

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

Opt. Express (3)

Opt. Lett. (5)

Optica (1)

Philos. Trans. R. Soc. London, Ser. A (2)

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Philos. Trans. R. Soc. London, Ser. A 375, 20150444 (2017).
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S. Franke-Arnold, “Optical angular momentum and atoms,” Philos. Trans. R. Soc. London, Ser. A 375, 20150435 (2017).
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Phys. Rev. A (7)

L. J. Pereira, W. T. Buono, D. S. Tasca, K. Dechoum, and A. Z. Khoury, “Orbital-angular-momentum mixing in type-ii second-harmonic generation,” Phys. Rev. A 96, 053856 (2017).
[Crossref]

A. Chopinaud, M. Jacquey, B. Viaris de Lesegno, and L. Pruvost, “High helicity vortex conversion in a rubidium vapor,” Phys. Rev. A 97, 063806 (2018).
[Crossref]

H. R. Hamedi, J. Ruseckas, E. Paspalakis, and G. Juzeliūnas, “Transfer of optical vortices in coherently prepared media,” Phys. Rev. A 99, 033812 (2019).
[Crossref]

R. N. Lanning, Z. Xiao, M. Zhang, I. Novikova, E. E. Mikhailov, and J. P. Dowling, “Gaussian-beam-propagation theory for nonlinear optics involving an analytical treatment of orbital-angular-momentum transfer,” Phys. Rev. A 96, 013830 (2017).
[Crossref]

H.-H. Wang, J. Wang, Z.-H. Kang, L. Wang, J.-Y. Gao, Y. Chen, and X.-J. Zhang, “Transfer of orbital angular momentum of light using electromagnetically induced transparency,” Phys. Rev. A 100, 013822 (2019).
[Crossref]

H. R. Hamedi, J. Ruseckas, and G. Juzeliūnas, “Exchange of optical vortices using an electromagnetically-induced-transparency–based four-wave-mixing setup,” Phys. Rev. A 98, 013840 (2018).
[Crossref]

A. S. Zibrov, M. D. Lukin, L. Hollberg, and M. O. Scully, “Efficient frequency up-conversion in resonant coherent media,” Phys. Rev. A 65, 051801 (2002).
[Crossref]

Phys. Rev. Lett. (10)

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, 243601 (2012).
[Crossref]

S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003).
[Crossref]

J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83, 4967–4970 (1999).
[Crossref]

X. Pan, S. Yu, Y. Zhou, K. Zhang, K. Zhang, S. Lv, S. Li, W. Wang, and J. Jing, “Orbital-angular-momentum multiplexed continuous-variable entanglement from four-wave mixing in hot atomic vapor,” Phys. Rev. Lett. 123, 070506 (2019).
[Crossref]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97, 163903 (2006).
[Crossref]

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
[Crossref]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
[Crossref]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[Crossref]

J. T. Mendonça, B. Thidé, and H. Then, “Stimulated Raman and Brillouin backscattering of collimated beams carrying orbital angular momentum,” Phys. Rev. Lett. 102, 185005 (2009).
[Crossref]

Sci. Rep. (3)

L. Zhu and J. Wang, “Arbitrary manipulation of spatial amplitude and phase using phase-only spatial light modulators,” Sci. Rep. 4, 7441 (2015).
[Crossref]

G.-H. Shao, S.-C. Yan, W. Luo, G.-W. Lu, and Y.-Q. Lu, “Orbital angular momentum (oam) conversion and multicasting using n-core supermode fiber,” Sci. Rep. 7, 1062 (2017).
[Crossref]

Z. Amini Sabegh, M. A. Maleki, and M. Mahmoudi, “Microwave-induced orbital angular momentum transfer,” Sci. Rep. 9, 3519 (2019).
[Crossref]

Other (3)

A. M. Akulshin, F. P. Bustos, N. Rahaman, and D. Budker, “Polychromatic forward-directed sub-doppler emission from sodium vapour,” ArXiv:1909.01156 (2019).

L. Allen, M. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics (Elsevier, 1999), Vol. 39, pp. 291–372.
[Crossref]

H.-J. Wu, L.-W. Mao, Y.-J. Yang, C. Rosales-Guzmán, W. Gao, B.-S. Shi, and Z.-H. Zhu, “Radial modal transitions of Laguerre-Gauss modes during parametric upconversion: towards the full-field selection rule of spatial modes,” ArXiv e-print 1912.05585 (2019).

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Figures (6)

Fig. 1.
Fig. 1. (a) Simple illustration of the model system. Two strong input fields, ${\Omega _1}$ and ${\Omega _2}$, interact with the Rb atoms in a vapor cell and generate two additional phase-matched output signals, ${\Omega _3}$ and ${\Omega _g}$. (b) The perfect phase-matching configuration such that $\Delta \vec k = ({\vec k_1} + {\vec k_2}) - ({\vec k_3} + {\vec k_g})$ becomes zero. (c) Schematic representation of the four-level atomic system. The energy states of $^{85}{\rm Rb}$ are defined as $|1\rangle = 5{S_{\frac{1}{2}}}$, $|2\rangle = 5{P_{\frac{3}{2}}}$, $|3\rangle = 5{D_{\frac{5}{2}}}$, $|4\rangle = 6{P_{\frac{3}{2}}}$.
Fig. 2.
Fig. 2. (a) Amplitude (Re[${\Omega _1}$]) and (b) phase structure of Laguerre–Gaussian probe beam. (c) Absorption (Im[${\rho _{41}}$]) and (d) phase profile of the atomic coherence, ${\rho _{41}}$ in the transverse plane when ${\Omega _1}$ possesses the optical vortex. (e) Simultaneous absorption profile (Im[${\rho _{34}}$]) corresponding to atomic coherence ${\rho _{34}}$. The parameters are ${m_1} = 0$, ${l_1} = 2$, ${w_1} = 90\,\,\unicode{x00B5}{\rm m}$, $\Omega _1^0 = 1.2\gamma$, $\Omega _2^0 = 1.2\gamma$, $\Omega _3^0 = 0.005\gamma$, ${\Delta _{21}} = 0$, ${\Delta _{32}} = 0$, ${\Delta _{34}} = 0$, $\gamma = {\gamma _3}$.
Fig. 3.
Fig. 3. Solid green curve ($i = g$) depicts the evolution of the FWM-based CBL signal (${\Omega _g}$) at different propagation lengths along with its phase structure. The dotted blue curve ($i = 3$) represents the evolution of infrared signal (${\Omega _3}$). We multiply ${10^1}$ and ${10^4}$ with the infrared field intensity ($|{\Omega _3}/\gamma {|^2}$) and FWM signal intensity ($|{\Omega _g}/\gamma {|^2}$), respectively. Inset figure compares the normalized intensity profile of the output ${\Omega _g}$ and input ${\Omega _1}$. The parameters are ${m_1} = 0$, ${l_1} = 2$, ${w_1} = 90\,\,\unicode{x00B5}{\rm m}$, $\Omega _1^0 = 1.2\gamma$, $\Omega _2^0 = 1.2\gamma$, $\Omega _3^0 = 0.005\gamma$, ${\Delta _{21}} = 0$, ${\Delta _{32}} = 0$, ${\Delta _{34}} = 0$, $\gamma = {\gamma _3}$.
Fig. 4.
Fig. 4. Transfer of different OAM from probe beam (${\Omega _1}$) to FWM signal (${\Omega _g}$). First row shows phase profile of ${\Omega _1}$ due to different OAM at $z$ = 0. Second and third rows depict the phase and intensity profile of ${\Omega _g}$ at $z = 50\; {\rm mm}$. Other parameters are the same as shown in Fig. 3.
Fig. 5.
Fig. 5. Transfer of control beam’s OAM (${l_2}$) is demonstrated. (a) and (b) Input phase and intensity profile of the control beam. (c) and (d) Output phase and intensity profile of the FWM signal. (e) Comparison of normalized intensity profile of input control beam and output FWM signal. The parameters are ${m_2} = 2$, ${l_2} = 2$, and ${w_2} = 90\,\,\unicode{x00B5}{\rm m}$. Other parameters are the same as shown in Fig. 3.
Fig. 6.
Fig. 6. Simultaneous transfer of probe OAM (${l_1} = 2$) and control OAM (${l_2} = 2$) into the FWM signal such that ${l_g} = {l_1} + {l_2}$. (a) and (b) The phase structure and normalized intensity profile of the input probe and control beam. (c) and (d) Output phase structure and intensity profile of the FWM signal. The parameters are ${m_1} = 0$, ${m_2} = 0$, ${l_1} = 2$, ${l_2} = 2$, ${w_1} = 90\,\,\unicode{x00B5}{\rm m}$, ${w_2} = 90\,\,\unicode{x00B5}{\rm m}$. (e) and (f) Phase and intensity profile of FWM signal when ${l_1} = + 2$, ${l_2} = - 2$. Other parameters are the same as shown in Fig. 3.

Equations (18)

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E j ( r , t ) = e ^ j E 0 j ( r ) e i ( k j z ω j t ) + c.c. ,
H = ω 21 | 2 2 | + ( ω 21 + ω 32 ) | 3 3 | + ( ω 21 + ω 32 ω 34 ) | 4 4 | Ω 1 e i ω 1 t | 2 1 | Ω 2 e i ω 2 t | 3 2 | Ω 3 e i ω 3 t | 3 4 | + h.c. ,
Ω 1 = d 21 . e ^ 1 E 01 , Ω 2 = d 32 . e ^ 2 E 02 , Ω 3 = d 34 . e ^ 3 E 03 .
H I = U H U i U U t ,
U = e i ( ω 1 | 2 2 | + ( ω 1 + ω 2 ) | 3 3 | + ( ω 1 + ω 2 ω 3 ) | 4 4 | ) t .
H I = Δ 21 | 2 2 | ( Δ 21 + Δ 32 ) | 3 3 | ( Δ 21 + Δ 32 Δ 34 ) | 4 4 | Ω 1 | 2 1 | Ω 2 | 3 2 | Ω 3 | 3 4 | + h . c . ,
Δ 21 = ω 1 ω 21 , Δ 32 = ω 2 ω 32 , Δ 34 = ω 3 ω 34 .
ρ ˙ = i [ H I , ρ ] + L ρ ,
ρ ˙ 11 = γ 2 ρ 22 + γ 4 ρ 44 i Ω 1 ρ 12 + i Ω 1 ρ 21 , ρ ˙ 12 = [ i Δ 21 γ 1 + γ 2 2 ] ρ 12 i Ω 2 ρ 13 + i Ω 1 ( ρ 22 ρ 11 ) , ρ ˙ 13 = [ i ( Δ 21 + Δ 32 ) γ 1 + γ 3 2 ] ρ 13 + i Ω 1 ρ 23 i Ω 2 ρ 12 i Ω 3 ρ 14 , ρ ˙ 14 = [ i ( Δ 21 + Δ 32 Δ 34 ) γ 1 + γ 4 2 ] ρ 14 i Ω 3 ρ 13 + i Ω 1 ρ 24 , ρ ˙ 22 = γ 2 ρ 22 + γ 3 2 ρ 33 + i Ω 1 ρ 12 i Ω 1 ρ 21 + i Ω 2 ρ 32 i Ω 2 ρ 23 , ρ ˙ 23 = [ i Δ 32 γ 3 + γ 2 2 ] ρ 23 + i Ω 2 ( ρ 33 ρ 22 ) + i Ω 1 ρ 13 i Ω 3 ρ 24 , ρ ˙ 24 = [ i ( Δ 32 Δ 34 ) γ 4 + γ 2 2 ] ρ 24 + i Ω 1 ρ 14 + i Ω 2 ρ 34 i Ω 3 ρ 23 , ρ ˙ 33 = γ 3 ρ 33 + i Ω 2 ρ 23 + i Ω 3 ρ 43 i Ω 2 ρ 32 i Ω 3 ρ 34 , ρ ˙ 34 = [ i Δ 34 γ 4 + γ 3 2 ] ρ 34 + i Ω 3 ( ρ 44 ρ 33 ) + i Ω 2 ρ 24 , ρ ˙ 44 = γ 4 ρ 44 + γ 3 2 ρ 33 i Ω 3 ρ 43 + i Ω 3 ρ 34 .
( 2 1 c 2 2 t 2 ) E = 4 π c 2 2 P t 2 ,
P = N ( d 43 ρ 34 e i ω 3 t + d 14 ρ 41 e i ω g t + c.c. ) .
Ω 3 z = i 2 k 3 2 Ω 3 + i η 3 ρ 34 ,
Ω g z = i 2 k g 2 Ω g + i η g ρ 41 .
Ω j ( r , ϕ , z ) = Ω j 0 w j w j ( z ) ( r 2 w j ( z ) ) | l | e r 2 w j 2 ( z ) L m l [ 2 r 2 w j 2 ( z ) ] × e i l ϕ e i k j r 2 2 R ( z ) e i ( 2 m + | l | + 1 ) tan 1 ( z z 0 ) , r = x 2 + y 2 , ϕ = tan 1 ( y x ) , ψ ( m , l , z ) = ( 2 m + | l | + 1 ) tan 1 ( z / z 0 ) .
ρ ij = ρ ij ( 0 ) + Ω 3 ρ ij ( 1 ) + Ω 3 ρ ij ( 2 ) ,
ρ 34 = i | Ω 1 | 2 | Ω 2 | 2 Ω 3 ( Γ 14 + Γ 23 ) Γ 12 Γ 23 ϝ 1 Γ 14 Γ 34 ϝ 2 ;
ϝ 1 = Γ 13 + | Ω 2 | 2 Γ 12 + | Ω 1 | 2 Γ 23 , ϝ 2 = Γ 24 + | Ω 2 | 2 Γ 34 + | Ω 1 | 2 Γ 14 , ρ 41 = i Ω 1 Ω 2 Ω 3 ϝ 3 Γ 41 Γ 21 ϝ 4 ϝ 5 ; ϝ 3 = [ Γ 42 + | Ω 2 | 2 Γ 43 | Ω 1 | 2 Γ 32 ] , ϝ 4 = [ Γ 42 + | Ω 2 | 2 Γ 43 + | Ω 1 | 2 Γ 41 ] , ϝ 5 = [ Γ 31 + | Ω 2 | 2 Γ 21 + | Ω 1 | 2 Γ 32 ] ,
Γ 21 = i Δ 21 γ 2 + γ 1 2 , Γ 31 = i ( Δ 21 + Δ 32 ) γ 3 + γ 1 2 , Γ 32 = i Δ 32 γ 3 + γ 2 2 , Γ 41 = i ( Δ 21 + Δ 32 Δ 34 ) γ 4 + γ 1 2 , Γ 42 = i ( Δ 32 Δ 34 ) γ 4 + γ 2 2 , Γ 43 = i Δ 34 γ 4 + γ 3 2 .

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