Abstract

Optical vortex light can be up-converted into a second harmonic output in an isotropic medium, in which such conversion is normally forbidden, through six-wave mixing. The involvement of orbital angular momentum is tackled for the case of a Laguerre-Gaussian pump comprising l = 1 photons. By calculating quantum amplitudes for the emergent radiation states, utilizing a state-sequence method, the analysis identifies the characteristics of the emission and an entangled distribution of conserved orbital momentum. A distinctive conical spread affords a potential means of resolving the individual angular momentum content.

© 2013 OSA

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Corrections

Matt M. Coles, Mathew D. Williams, and David L. Andrews, "Second harmonic generation in isotropic media: six-wave mixing of optical vortices: erratum," Opt. Express 22, 17478-17478 (2014)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-14-17478

References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  17. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  26. J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
    [CrossRef]
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2013 (1)

D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE8637, 863717 (2013).
[CrossRef]

2012 (7)

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE8542, 85421R, (2012).
[CrossRef]

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

M. T. Cao, L. Han, R. F. Liu, H. Liu, D. Wei, P. Zhang, Y. Zhou, W. G. Guo, S. G. Zhang, H. Gao, and F. L. Li, “Deutsch’s algorithm with topological charges of optical vortices via non-degenerate four-wave mixing,” Opt. Express20(22), 24263–24271 (2012).
[CrossRef] [PubMed]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express20(22), 24444–24449 (2012).
[CrossRef] [PubMed]

D. Shwa, E. Shtranvasser, Y. Shalibo, and N. Katz, “Controllable motion of optical vortex arrays using electromagnetically induced transparency,” Opt. Express20(22), 24835–24842 (2012).
[CrossRef] [PubMed]

N. Olivier, D. DéBarre, P. Mahou, and E. Beaurepaire, “Third-harmonic generation microscopy with Bessel beams: a numerical study,” Opt. Express20(22), 24886–24902 (2012).
[CrossRef] [PubMed]

2009 (1)

2004 (2)

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys.6, 103 (2004).
[CrossRef]

2002 (4)

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A65(3), 033823 (2002).
[CrossRef]

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett.88(15), 153901 (2002).
[CrossRef] [PubMed]

R. D. Jenkins, D. L. Andrews, and L. C. Dávila Romero, “A new diagrammatic methodology for non-relativistic quantum electrodynamics,” J. Phys. At. Mol. Opt. Phys.35(3), 445–468 (2002).
[CrossRef]

L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B Quantum Semiclassical Opt.4(2), S66–S72 (2002).
[CrossRef]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
[CrossRef] [PubMed]

2000 (1)

H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett.85(2), 286–289 (2000).
[CrossRef] [PubMed]

1998 (1)

I. D. Hands, S. J. Lin, S. R. Meech, and D. L. Andrews, “A quantum electrodynamical treatment of second harmonic generation through phase conjugate six-wave mixing: Polarization analysis,” J. Chem. Phys.109(24), 10580–10586 (1998).
[CrossRef]

1997 (1)

P. Allcock and D. L. Andrews, “Six-wave mixing: secular resonances in a higher-order mechanism for second-harmonic generation,” J. Phys. At. Mol. Opt. Phys.30(16), 3731–3742 (1997).
[CrossRef]

1996 (2)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

S. Franke and S. M. Barnett, “Angular momentum in spontaneous emission,” J. Phys. At. Mol. Opt. Phys.29(10), 2141–2150 (1996).
[CrossRef]

1980 (1)

D. L. Andrews, “Harmonic-generation in free molecules,” J. Phys. At. Mol. Opt. Phys.13(20), 4091–4099 (1980).
[CrossRef]

1977 (1)

D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys.67(11), 5026–5033 (1977).
[CrossRef]

Allcock, P.

P. Allcock and D. L. Andrews, “Six-wave mixing: secular resonances in a higher-order mechanism for second-harmonic generation,” J. Phys. At. Mol. Opt. Phys.30(16), 3731–3742 (1997).
[CrossRef]

Allen, L.

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A65(3), 033823 (2002).
[CrossRef]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

Andrews, D. L.

R. D. Jenkins, D. L. Andrews, and L. C. Dávila Romero, “A new diagrammatic methodology for non-relativistic quantum electrodynamics,” J. Phys. At. Mol. Opt. Phys.35(3), 445–468 (2002).
[CrossRef]

L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B Quantum Semiclassical Opt.4(2), S66–S72 (2002).
[CrossRef]

I. D. Hands, S. J. Lin, S. R. Meech, and D. L. Andrews, “A quantum electrodynamical treatment of second harmonic generation through phase conjugate six-wave mixing: Polarization analysis,” J. Chem. Phys.109(24), 10580–10586 (1998).
[CrossRef]

P. Allcock and D. L. Andrews, “Six-wave mixing: secular resonances in a higher-order mechanism for second-harmonic generation,” J. Phys. At. Mol. Opt. Phys.30(16), 3731–3742 (1997).
[CrossRef]

D. L. Andrews, “Harmonic-generation in free molecules,” J. Phys. At. Mol. Opt. Phys.13(20), 4091–4099 (1980).
[CrossRef]

D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys.67(11), 5026–5033 (1977).
[CrossRef]

Arnaut, H. H.

H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett.85(2), 286–289 (2000).
[CrossRef] [PubMed]

Babiker, M.

L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B Quantum Semiclassical Opt.4(2), S66–S72 (2002).
[CrossRef]

Barbosa, G. A.

H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett.85(2), 286–289 (2000).
[CrossRef] [PubMed]

Barnett, S.

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

Barnett, S. M.

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys.6, 103 (2004).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A65(3), 033823 (2002).
[CrossRef]

S. Franke and S. M. Barnett, “Angular momentum in spontaneous emission,” J. Phys. At. Mol. Opt. Phys.29(10), 2141–2150 (1996).
[CrossRef]

Beaurepaire, E.

Boyd, R. W.

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express20(22), 24444–24449 (2012).
[CrossRef] [PubMed]

M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE8542, 85421R, (2012).
[CrossRef]

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett.88(15), 153901 (2002).
[CrossRef] [PubMed]

Cao, M. T.

Courtial, J.

M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE8542, 85421R, (2012).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys.6, 103 (2004).
[CrossRef]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

Dávila Romero, L. C.

R. D. Jenkins, D. L. Andrews, and L. C. Dávila Romero, “A new diagrammatic methodology for non-relativistic quantum electrodynamics,” J. Phys. At. Mol. Opt. Phys.35(3), 445–468 (2002).
[CrossRef]

L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B Quantum Semiclassical Opt.4(2), S66–S72 (2002).
[CrossRef]

DéBarre, D.

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. A54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

Duparré, M.

D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE8637, 863717 (2013).
[CrossRef]

T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express17(11), 9347–9356 (2009).
[CrossRef] [PubMed]

Flamm, D.

D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE8637, 863717 (2013).
[CrossRef]

T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express17(11), 9347–9356 (2009).
[CrossRef] [PubMed]

Forbes, A.

D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE8637, 863717 (2013).
[CrossRef]

Franke, S.

S. Franke and S. M. Barnett, “Angular momentum in spontaneous emission,” J. Phys. At. Mol. Opt. Phys.29(10), 2141–2150 (1996).
[CrossRef]

Franke-Arnold, S.

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys.6, 103 (2004).
[CrossRef]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A65(3), 033823 (2002).
[CrossRef]

S. Franke-Arnold, “Orbital angular momentum of photons, atoms, and electrons,” Proc. SPIE 8637, (in press).

Gaeta, A. L.

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett.88(15), 153901 (2002).
[CrossRef] [PubMed]

Gao, H.

Gibson, G.

Giovannini, D.

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

Guo, W. G.

Han, L.

Hands, I. D.

I. D. Hands, S. J. Lin, S. R. Meech, and D. L. Andrews, “A quantum electrodynamical treatment of second harmonic generation through phase conjugate six-wave mixing: Polarization analysis,” J. Chem. Phys.109(24), 10580–10586 (1998).
[CrossRef]

Homoelle, D.

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett.88(15), 153901 (2002).
[CrossRef] [PubMed]

Jenkins, R. D.

R. D. Jenkins, D. L. Andrews, and L. C. Dávila Romero, “A new diagrammatic methodology for non-relativistic quantum electrodynamics,” J. Phys. At. Mol. Opt. Phys.35(3), 445–468 (2002).
[CrossRef]

Kaiser, T.

Katz, N.

Lavery, M. P. J.

M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE8542, 85421R, (2012).
[CrossRef]

Leach, J.

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys.6, 103 (2004).
[CrossRef]

Li, F. L.

Lin, S. J.

I. D. Hands, S. J. Lin, S. R. Meech, and D. L. Andrews, “A quantum electrodynamical treatment of second harmonic generation through phase conjugate six-wave mixing: Polarization analysis,” J. Chem. Phys.109(24), 10580–10586 (1998).
[CrossRef]

Liu, H.

Liu, R. F.

Mahou, P.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Malik, M.

M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE8542, 85421R, (2012).
[CrossRef]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express20(22), 24444–24449 (2012).
[CrossRef] [PubMed]

Meech, S. R.

I. D. Hands, S. J. Lin, S. R. Meech, and D. L. Andrews, “A quantum electrodynamical treatment of second harmonic generation through phase conjugate six-wave mixing: Polarization analysis,” J. Chem. Phys.109(24), 10580–10586 (1998).
[CrossRef]

Mirhosseini, M.

Moll, K. D.

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical Harmonic Generation in Isotropic Materials,” Phys. Rev. Lett.88(15), 153901 (2002).
[CrossRef] [PubMed]

Naidoo, D.

D. Flamm, C. Schulze, D. Naidoo, A. Forbes, and M. Duparré, “Mode analysis using the correlation filter method,” Proc. SPIE8637, 863717 (2013).
[CrossRef]

O’Sullivan, M. N.

Olivier, N.

Padgett, M.

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys.6, 103 (2004).
[CrossRef]

Padgett, M. J.

M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE8542, 85421R, (2012).
[CrossRef]

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004).
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A65(3), 033823 (2002).
[CrossRef]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A54(5), R3742–R3745 (1996).
[CrossRef] [PubMed]

Pas’ko, V.

Robertson, D.

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J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
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New J. Phys. (1)

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S. Franke-Arnold, “Orbital angular momentum of photons, atoms, and electrons,” Proc. SPIE 8637, (in press).

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J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
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S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A65(3), 033823 (2002).
[CrossRef]

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86(1), 012334 (2012).
[CrossRef]

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Proc. SPIE (2)

M. P. J. Lavery, D. Robertson, M. Malik, B. Rodenburg, J. Courtial, R. W. Boyd, and M. J. Padgett, “The efficient sorting of light's orbital angular momentum for optical communications,” Proc. SPIE8542, 85421R, (2012).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

In the state sequence diagram (left), each row denotes a state of consecutive photon occupying number, n. Columns denote successive system states. Each vacant column and row in the tabular diagram represents an interaction, signifying a node of the corresponding Feynman diagram (right), for a given permutation. In the middle set of catawampus cells the wave-vector label for the emitted photon provides for a photon in either of the two output modes; both modes are populated in the upper set. Absorption ▬▬▬ (inclination) Emission ▬▬▬ (declination).

Fig. 2
Fig. 2

Left: Schematic depiction of conical SWM with LG input. Middle: Intensity plots (arbitrary scale) of the emission in the two output beams shown on the left, for OAM combinations (l1, l2), at a distance of 100 wavelengths from the conversion material. Right: Cross-sectional intensity distribution around the (2,2) output delivered by l0 = 1, p = 0 input (centered on the input beam axis: distance as in the middle diagram). Red indicates high intensity; blue and black indicate low and zero intensities, respectively.

Tables (1)

Tables Icon

Table 1 Relative Magnitudes of the Intensities for OAM Output Photon Pairs

Equations (9)

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M FI ( 6 ) = R,S,T,U,VI,F F| H int |V V| H int |U U| H int |T T| H int |S S| H int |R R| H int |I ( E I E R )( E I E S )( E I E T )( E I E U )( E I E V ) ,
M FI ( r )= 3 ( ck 2 ε 0 Ω ) 3 e ¯ i e ¯ j e k e l e m e n χ ( ij )( klmn ) (5) e i( 4k k k ).r .
χ ( ij )( klmn ) (5) = 1 48 r,s,t,u,v k,η,l,p [ { μ i 0v μ j vu μ k ut μ l ts μ m sr μ n r0 [ E r0 ck ][ E s0 2ck ][ E t0 3ck ][ E u0 4ck ][ E v0 2ck ] +... } +{ perm( i,j )perm( k,l,m,n ) } ],
4k= k + k ,
Γ~| ( e ¯ e ¯ ) ( ee ) 2 { 23 χ ( λλ )( μμνν ) (5) 20 χ ( λμ )( λμνν ) (5) }+ ( e ¯ e )( e ¯ e )( ee ){ 20 χ ( λλ )( μμνν ) (5) +32 χ ( λμ )( λμνν ) (5) } | 2 ,
Γ~ | { 3 χ ( λλ )( μμνν ) (5) +12 χ ( λμ )( λμνν ) (5) } | 2 ,
Γ~ | { 23 χ ( λλ )( μμνν ) (5) 20 χ ( λμ )( λμνν ) (5) } | 2 ,
d (r)=i k,η,l,p ( ck ε 0 2Ω ) 1 2 { e l,p (η) (k) a (η) (k) f l,p (r) e ikzilφ h.c. },
[ f l 0 , p 0 (r) ] 4 f ¯ l 1 , p 1 ( r ) f ¯ l 2 , p 2 ( r ) e i( 4 l 0 φ l 1 φ l 2 φ ) ,

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