Abstract

We show that an optical vortex beam, implemented classically, can be transferred to the transverse amplitude of a heralded single photon. For this purpose we have relied on the process of spontaneous parametric downconversion (SPDC) for the generation of signal and idler photon pairs, using a pump in the form of a Bessel-Gauss (BG) beam with orbital angular momentum (specifically, with topological charge l = 1 and l = 2). We have designed our source so that it operates within the short SPDC crystal regime for which, the amplitude and phase of the pump may be transferred to a heralded single photon. In order to verify the vortex nature of our heralded single photon, we have shown that the conditional angular spectrum and the transverse intensity at the single-photon level match similar measurements carried out for the pump. In addition, we have shown that when our heralded single photon is diffracted through a triangular aperture, the far-field single-photon transverse intensity exhibits the expected triangular arrangement of intensity lobes associated with the presence of orbital angular momentum.

© 2014 Optical Society of America

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  1. M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)
  2. see for example: 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] [PubMed]
  3. C. H. Monken, P. H. Souto Ribeiro, and S. Padua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
    [CrossRef]
  4. K.G. Koprulu, Y.-P. Huang, G.A. Barbosa, and P. Kumar, “Lossless single-photon shaping via heralding,” Opt. Lett. 36, 1674 (2011)
    [CrossRef] [PubMed]
  5. S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69, 023811 (2004)
    [CrossRef]
  6. R. Ramirez Alarcon, H. Cruz Ramirez, and A.B. U’Ren, “Effects of crystal length effects on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013)
    [CrossRef]
  7. J.M. Hickmann, E.J.S. Fonseca, W.C. Soares, and S. Chavez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Lights Orbital Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010)
    [CrossRef]
  8. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
    [CrossRef] [PubMed]
  9. G. Molina-Terriza, J.P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305 (2007)
    [CrossRef]
  10. 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, 662 (2010)
    [CrossRef] [PubMed]
  11. see for example: A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, and Erika Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011)
    [CrossRef]
  12. G.A. Tyler and R.W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142 (2009)
    [CrossRef] [PubMed]
  13. X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik 124, 2635 (2013)
    [CrossRef]
  14. M. McLaren, T. Mhlanga, M.J. Padgett, F.S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
    [CrossRef] [PubMed]
  15. H. Cruz Ramirez, R. Ramirez-Alarcon, Francisco J. Morelos, P. Quinto-Su, J.C. Gutierrez Vega, and A.B. U’Ren, “Observation of non-diffracting behavior at the single photon level,” Opt. Express 20, 29761 (2012)
    [CrossRef]
  16. C.L. Hernandez-Cedillo and R. Jauregui, “Spin and orbital angular momentum correlations in parametric down-conversion of Bessel beams,” J. Opt. 13, 064021 (2011)
    [CrossRef]
  17. Y. Jeronimo-Moreno and R. Jauregui, “On demand generation of propagation invariant photons with orbital angular momentum,” arXiv:1403.5474v1 (2014)
  18. S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
    [CrossRef]
  19. J. C. Gutierrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289 (2005).
    [CrossRef]
  20. A. R. Altman, K. G. Koprulu, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum Imaging of Nonlocal Spatial Correlations Induced by Orbital Angular Momentum,” Phys. Rev. Lett. 94, 123601 (2005).
    [CrossRef] [PubMed]
  21. R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Observation of a Nonlocal Optical Vortex,” Phys. Rev. Lett. 103, 033602 (2009)
    [CrossRef] [PubMed]
  22. M. McLaren, M. Agnew, J. Leach, F.S. Roux, M. J. Padgett, R.W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589 (2012)
    [CrossRef] [PubMed]
  23. A.J. Jesus-Silva, E.S. Fonseca, and J.M. Hickmann, “Measurement of the orbital angular momentum at photon level via the spatial probability distribution,” J. Mod. Opt. 59, 1194 (2012)
    [CrossRef]
  24. J.M. Hickmann, E.J.S. Fonseca, and J Jesus-Silva, “Born’s rule and the interference of photons with orbital angular momentum by a triangular slit,” EPL 96, 64006 (2011)
    [CrossRef]
  25. D.N. Klyshko, “Combine EPR and two-slit experiments: Interference of advanced waves,” Physics Letters A 132299 (1998)
    [CrossRef]
  26. D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, “Observation of two-photon ’ghost’ interference and diffraction,” Phys. Rev. Lett. 743600 (1995)
    [CrossRef] [PubMed]
  27. Y. Kang, K. Cho, Jaewoo Noh, D. L. P. Vitullo, C. Leary, and M. G. Raymer, “Remote preparation of complex spatial states of single photons and verification by two-photon coincidence experiment,” Opt. Express,  181217 (2010)
    [CrossRef] [PubMed]
  28. D. S. Tasca, R. S. Aspden, P. A. Morris, G. Anderson, R. W. Boyd, and M.J. Padgett, “The influence of non-imaging detector design on heralded ghost-imaging and ghost-diffraction examined using a triggered ICCD camera,” Opt. Express 2130460 (2013)
    [CrossRef]

2014

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

M. McLaren, T. Mhlanga, M.J. Padgett, F.S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
[CrossRef] [PubMed]

Y. Jeronimo-Moreno and R. Jauregui, “On demand generation of propagation invariant photons with orbital angular momentum,” arXiv:1403.5474v1 (2014)

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

2013

R. Ramirez Alarcon, H. Cruz Ramirez, and A.B. U’Ren, “Effects of crystal length effects on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013)
[CrossRef]

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik 124, 2635 (2013)
[CrossRef]

D. S. Tasca, R. S. Aspden, P. A. Morris, G. Anderson, R. W. Boyd, and M.J. Padgett, “The influence of non-imaging detector design on heralded ghost-imaging and ghost-diffraction examined using a triggered ICCD camera,” Opt. Express 2130460 (2013)
[CrossRef]

2012

2011

C.L. Hernandez-Cedillo and R. Jauregui, “Spin and orbital angular momentum correlations in parametric down-conversion of Bessel beams,” J. Opt. 13, 064021 (2011)
[CrossRef]

see for example: A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, and Erika Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011)
[CrossRef]

K.G. Koprulu, Y.-P. Huang, G.A. Barbosa, and P. Kumar, “Lossless single-photon shaping via heralding,” Opt. Lett. 36, 1674 (2011)
[CrossRef] [PubMed]

J.M. Hickmann, E.J.S. Fonseca, and J Jesus-Silva, “Born’s rule and the interference of photons with orbital angular momentum by a triangular slit,” EPL 96, 64006 (2011)
[CrossRef]

2010

Y. Kang, K. Cho, Jaewoo Noh, D. L. P. Vitullo, C. Leary, and M. G. Raymer, “Remote preparation of complex spatial states of single photons and verification by two-photon coincidence experiment,” Opt. Express,  181217 (2010)
[CrossRef] [PubMed]

J.M. Hickmann, E.J.S. Fonseca, W.C. Soares, and S. Chavez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Lights Orbital Angular Momentum,” Phys. Rev. Lett. 105, 053904 (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, 662 (2010)
[CrossRef] [PubMed]

2009

G.A. Tyler and R.W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142 (2009)
[CrossRef] [PubMed]

R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Observation of a Nonlocal Optical Vortex,” Phys. Rev. Lett. 103, 033602 (2009)
[CrossRef] [PubMed]

2007

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

2005

J. C. Gutierrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289 (2005).
[CrossRef]

A. R. Altman, K. G. Koprulu, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum Imaging of Nonlocal Spatial Correlations Induced by Orbital Angular Momentum,” Phys. Rev. Lett. 94, 123601 (2005).
[CrossRef] [PubMed]

2004

S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69, 023811 (2004)
[CrossRef]

2003

see for example: 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] [PubMed]

2001

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

1998

C. H. Monken, P. H. Souto Ribeiro, and S. Padua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[CrossRef]

D.N. Klyshko, “Combine EPR and two-slit experiments: Interference of advanced waves,” Physics Letters A 132299 (1998)
[CrossRef]

1995

D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, “Observation of two-photon ’ghost’ interference and diffraction,” Phys. Rev. Lett. 743600 (1995)
[CrossRef] [PubMed]

Aadhi, A

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

Agnew, M.

Altman, A. R.

A. R. Altman, K. G. Koprulu, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum Imaging of Nonlocal Spatial Correlations Induced by Orbital Angular Momentum,” Phys. Rev. Lett. 94, 123601 (2005).
[CrossRef] [PubMed]

Anderson, G.

Andersson, Erika

see for example: A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, and Erika Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011)
[CrossRef]

Aspden, R. S.

Bandres, M. A.

Barbosa, G. A.

A. R. Altman, K. G. Koprulu, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum Imaging of Nonlocal Spatial Correlations Induced by Orbital Angular Momentum,” Phys. Rev. Lett. 94, 123601 (2005).
[CrossRef] [PubMed]

Barbosa, G.A.

Barnett, S.M.

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, 662 (2010)
[CrossRef] [PubMed]

Barreiro, S.

see for example: 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] [PubMed]

Boyd, R. W.

Boyd, R.W.

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

M. McLaren, M. Agnew, J. Leach, F.S. Roux, M. J. Padgett, R.W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589 (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, 662 (2010)
[CrossRef] [PubMed]

G.A. Tyler and R.W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142 (2009)
[CrossRef] [PubMed]

Buller, G.S.

see for example: A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, and Erika Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011)
[CrossRef]

Chavez-Cerda, S.

J.M. Hickmann, E.J.S. Fonseca, W.C. Soares, and S. Chavez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Lights Orbital Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010)
[CrossRef]

Chithrabhanu, P

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

Cho, K.

Corndorf, E.

A. R. Altman, K. G. Koprulu, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum Imaging of Nonlocal Spatial Correlations Induced by Orbital Angular Momentum,” Phys. Rev. Lett. 94, 123601 (2005).
[CrossRef] [PubMed]

Cruz Ramirez, H.

R. Ramirez Alarcon, H. Cruz Ramirez, and A.B. U’Ren, “Effects of crystal length effects on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013)
[CrossRef]

H. Cruz Ramirez, R. Ramirez-Alarcon, Francisco J. Morelos, P. Quinto-Su, J.C. Gutierrez Vega, and A.B. U’Ren, “Observation of non-diffracting behavior at the single photon level,” Opt. Express 20, 29761 (2012)
[CrossRef]

Dada, A.C.

see for example: A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, and Erika Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011)
[CrossRef]

de Oliveira, A. N.

S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69, 023811 (2004)
[CrossRef]

Fonseca, E.J.S.

J.M. Hickmann, E.J.S. Fonseca, and J Jesus-Silva, “Born’s rule and the interference of photons with orbital angular momentum by a triangular slit,” EPL 96, 64006 (2011)
[CrossRef]

J.M. Hickmann, E.J.S. Fonseca, W.C. Soares, and S. Chavez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Lights Orbital Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010)
[CrossRef]

Fonseca, E.S.

A.J. Jesus-Silva, E.S. Fonseca, and J.M. Hickmann, “Measurement of the orbital angular momentum at photon level via the spatial probability distribution,” J. Mod. Opt. 59, 1194 (2012)
[CrossRef]

Forbes, A.

M. McLaren, T. Mhlanga, M.J. Padgett, F.S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
[CrossRef] [PubMed]

M. McLaren, M. Agnew, J. Leach, F.S. Roux, M. J. Padgett, R.W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589 (2012)
[CrossRef] [PubMed]

Franke-Arnold, S.

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, 662 (2010)
[CrossRef] [PubMed]

Gangi Reddy, S.

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

Gauthier, D.J.

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

Gomes, R. M.

R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Observation of a Nonlocal Optical Vortex,” Phys. Rev. Lett. 103, 033602 (2009)
[CrossRef] [PubMed]

Gutierrez Vega, J.C.

Gutierrez-Vega, J. C.

Hernandez-Cedillo, C.L.

C.L. Hernandez-Cedillo and R. Jauregui, “Spin and orbital angular momentum correlations in parametric down-conversion of Bessel beams,” J. Opt. 13, 064021 (2011)
[CrossRef]

Hickmann, J.M.

A.J. Jesus-Silva, E.S. Fonseca, and J.M. Hickmann, “Measurement of the orbital angular momentum at photon level via the spatial probability distribution,” J. Mod. Opt. 59, 1194 (2012)
[CrossRef]

J.M. Hickmann, E.J.S. Fonseca, and J Jesus-Silva, “Born’s rule and the interference of photons with orbital angular momentum by a triangular slit,” EPL 96, 64006 (2011)
[CrossRef]

J.M. Hickmann, E.J.S. Fonseca, W.C. Soares, and S. Chavez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Lights Orbital Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010)
[CrossRef]

Huang, Y.-P.

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, 662 (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, 662 (2010)
[CrossRef] [PubMed]

Jauregui, R.

Y. Jeronimo-Moreno and R. Jauregui, “On demand generation of propagation invariant photons with orbital angular momentum,” arXiv:1403.5474v1 (2014)

C.L. Hernandez-Cedillo and R. Jauregui, “Spin and orbital angular momentum correlations in parametric down-conversion of Bessel beams,” J. Opt. 13, 064021 (2011)
[CrossRef]

Jeronimo-Moreno, Y.

Y. Jeronimo-Moreno and R. Jauregui, “On demand generation of propagation invariant photons with orbital angular momentum,” arXiv:1403.5474v1 (2014)

Jesus-Silva, A.J.

A.J. Jesus-Silva, E.S. Fonseca, and J.M. Hickmann, “Measurement of the orbital angular momentum at photon level via the spatial probability distribution,” J. Mod. Opt. 59, 1194 (2012)
[CrossRef]

Jesus-Silva, J

J.M. Hickmann, E.J.S. Fonseca, and J Jesus-Silva, “Born’s rule and the interference of photons with orbital angular momentum by a triangular slit,” EPL 96, 64006 (2011)
[CrossRef]

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, 662 (2010)
[CrossRef] [PubMed]

Kang, Y.

Klyshko, D.N.

D.N. Klyshko, “Combine EPR and two-slit experiments: Interference of advanced waves,” Physics Letters A 132299 (1998)
[CrossRef]

D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, “Observation of two-photon ’ghost’ interference and diffraction,” Phys. Rev. Lett. 743600 (1995)
[CrossRef] [PubMed]

Koprulu, K. G.

A. R. Altman, K. G. Koprulu, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum Imaging of Nonlocal Spatial Correlations Induced by Orbital Angular Momentum,” Phys. Rev. Lett. 94, 123601 (2005).
[CrossRef] [PubMed]

Koprulu, K.G.

Kumar, A.

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

Kumar, P.

K.G. Koprulu, Y.-P. Huang, G.A. Barbosa, and P. Kumar, “Lossless single-photon shaping via heralding,” Opt. Lett. 36, 1674 (2011)
[CrossRef] [PubMed]

A. R. Altman, K. G. Koprulu, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum Imaging of Nonlocal Spatial Correlations Induced by Orbital Angular Momentum,” Phys. Rev. Lett. 94, 123601 (2005).
[CrossRef] [PubMed]

Leach, J.

M. McLaren, M. Agnew, J. Leach, F.S. Roux, M. J. Padgett, R.W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589 (2012)
[CrossRef] [PubMed]

see for example: A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, and Erika Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (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, 662 (2010)
[CrossRef] [PubMed]

Leary, C.

Magaña-Loaiza, O.S.

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

Mair, A.

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

Malik, M.

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

McLaren, M.

M. McLaren, T. Mhlanga, M.J. Padgett, F.S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
[CrossRef] [PubMed]

M. McLaren, M. Agnew, J. Leach, F.S. Roux, M. J. Padgett, R.W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589 (2012)
[CrossRef] [PubMed]

Mhlanga, T.

M. McLaren, T. Mhlanga, M.J. Padgett, F.S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
[CrossRef] [PubMed]

Mirhosseini, M.

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

Molina-Terriza, G.

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

Monken, C. H.

S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69, 023811 (2004)
[CrossRef]

C. H. Monken, P. H. Souto Ribeiro, and S. Padua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[CrossRef]

Morelos, Francisco J.

Morris, P. A.

Noh, Jaewoo

O’Sullivan, M.N.

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

Padgett, M. J.

Padgett, M.J.

M. McLaren, T. Mhlanga, M.J. Padgett, F.S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
[CrossRef] [PubMed]

D. S. Tasca, R. S. Aspden, P. A. Morris, G. Anderson, R. W. Boyd, and M.J. Padgett, “The influence of non-imaging detector design on heralded ghost-imaging and ghost-diffraction examined using a triggered ICCD camera,” Opt. Express 2130460 (2013)
[CrossRef]

see for example: A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, and Erika Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (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, 662 (2010)
[CrossRef] [PubMed]

Padua, S.

C. H. Monken, P. H. Souto Ribeiro, and S. Padua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[CrossRef]

Prabhakar, S.

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

Quinto-Su, P.

Ramirez Alarcon, R.

R. Ramirez Alarcon, H. Cruz Ramirez, and A.B. U’Ren, “Effects of crystal length effects on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013)
[CrossRef]

Ramirez-Alarcon, R.

Raymer, M. G.

Rodenburg, B.

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

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, 662 (2010)
[CrossRef] [PubMed]

Roux, F.S.

M. McLaren, T. Mhlanga, M.J. Padgett, F.S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
[CrossRef] [PubMed]

M. McLaren, M. Agnew, J. Leach, F.S. Roux, M. J. Padgett, R.W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589 (2012)
[CrossRef] [PubMed]

Salles, A.

R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Observation of a Nonlocal Optical Vortex,” Phys. Rev. Lett. 103, 033602 (2009)
[CrossRef] [PubMed]

Samanta, G. K.

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

Sergienko, A.V.

D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, “Observation of two-photon ’ghost’ interference and diffraction,” Phys. Rev. Lett. 743600 (1995)
[CrossRef] [PubMed]

Sheng, X.

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik 124, 2635 (2013)
[CrossRef]

Shih, Y.H.

D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, “Observation of two-photon ’ghost’ interference and diffraction,” Phys. Rev. Lett. 743600 (1995)
[CrossRef] [PubMed]

Singh, R. P.

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

Soares, W.C.

J.M. Hickmann, E.J.S. Fonseca, W.C. Soares, and S. Chavez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Lights Orbital Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010)
[CrossRef]

Souto Ribeiro, P. H.

R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Observation of a Nonlocal Optical Vortex,” Phys. Rev. Lett. 103, 033602 (2009)
[CrossRef] [PubMed]

C. H. Monken, P. H. Souto Ribeiro, and S. Padua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[CrossRef]

Strekalov, D.V.

D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, “Observation of two-photon ’ghost’ interference and diffraction,” Phys. Rev. Lett. 743600 (1995)
[CrossRef] [PubMed]

Tabosa, J. W. R.

see for example: 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] [PubMed]

Tasca, D. S.

Thebaldi, R. S.

S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69, 023811 (2004)
[CrossRef]

Torner, L.

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

Torres, J.P.

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

Toscano, F.

R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Observation of a Nonlocal Optical Vortex,” Phys. Rev. Lett. 103, 033602 (2009)
[CrossRef] [PubMed]

Tyler, G.A.

U’Ren, A.B.

R. Ramirez Alarcon, H. Cruz Ramirez, and A.B. U’Ren, “Effects of crystal length effects on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013)
[CrossRef]

H. Cruz Ramirez, R. Ramirez-Alarcon, Francisco J. Morelos, P. Quinto-Su, J.C. Gutierrez Vega, and A.B. U’Ren, “Observation of non-diffracting behavior at the single photon level,” Opt. Express 20, 29761 (2012)
[CrossRef]

Vaziri, A.

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

Vitullo, D. L. P.

Walborn, S. P.

R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Observation of a Nonlocal Optical Vortex,” Phys. Rev. Lett. 103, 033602 (2009)
[CrossRef] [PubMed]

S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69, 023811 (2004)
[CrossRef]

Weihs, G.

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

Yao, A.M.

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, 662 (2010)
[CrossRef] [PubMed]

Zeilinger, A.

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

Zhang, Y.

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik 124, 2635 (2013)
[CrossRef]

Zhu, Y.

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik 124, 2635 (2013)
[CrossRef]

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik 124, 2635 (2013)
[CrossRef]

EPL

J.M. Hickmann, E.J.S. Fonseca, and J Jesus-Silva, “Born’s rule and the interference of photons with orbital angular momentum by a triangular slit,” EPL 96, 64006 (2011)
[CrossRef]

J. Mod. Opt.

A.J. Jesus-Silva, E.S. Fonseca, and J.M. Hickmann, “Measurement of the orbital angular momentum at photon level via the spatial probability distribution,” J. Mod. Opt. 59, 1194 (2012)
[CrossRef]

J. Opt.

C.L. Hernandez-Cedillo and R. Jauregui, “Spin and orbital angular momentum correlations in parametric down-conversion of Bessel beams,” J. Opt. 13, 064021 (2011)
[CrossRef]

J. Opt. Soc. Am. A

Laser Phys.

R. Ramirez Alarcon, H. Cruz Ramirez, and A.B. U’Ren, “Effects of crystal length effects on the angular spectrum of spontaneous parametric downconversion photon pairs,” Laser Phys. 23, 055204 (2013)
[CrossRef]

Nat. Commun.

M. McLaren, T. Mhlanga, M.J. Padgett, F.S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
[CrossRef] [PubMed]

Nat. Phys.

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

see for example: A.C. Dada, J. Leach, G.S. Buller, M.J. Padgett, and Erika Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677 (2011)
[CrossRef]

Nature

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

Opt. Commun.

S. Prabhakar, S. Gangi Reddy, A Aadhi, A. Kumar, P Chithrabhanu, G. K. Samanta, and R. P. Singh, “Spatial distribution of Spontaneous Parametric Down-Converted Photons for higher order Optical Vortices,” Opt. Commun. 326, 64 (2014)
[CrossRef]

Opt. Express

Opt. Lett.

Optik

X. Sheng, Y. Zhu, Y. Zhu, and Y. Zhang, “Orbital angular momentum entangled states of vortex beam pump in non-Kolmogorov turbulence channel,” Optik 124, 2635 (2013)
[CrossRef]

Phys. Rev. A

S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69, 023811 (2004)
[CrossRef]

C. H. Monken, P. H. Souto Ribeiro, and S. Padua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[CrossRef]

Phys. Rev. Lett.

J.M. Hickmann, E.J.S. Fonseca, W.C. Soares, and S. Chavez-Cerda, “Unveiling a Truncated Optical Lattice Associated with a Triangular Aperture Using Lights Orbital Angular Momentum,” Phys. Rev. Lett. 105, 053904 (2010)
[CrossRef]

see for example: 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] [PubMed]

A. R. Altman, K. G. Koprulu, E. Corndorf, P. Kumar, and G. A. Barbosa, “Quantum Imaging of Nonlocal Spatial Correlations Induced by Orbital Angular Momentum,” Phys. Rev. Lett. 94, 123601 (2005).
[CrossRef] [PubMed]

R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro, and S. P. Walborn, “Observation of a Nonlocal Optical Vortex,” Phys. Rev. Lett. 103, 033602 (2009)
[CrossRef] [PubMed]

D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, and Y.H. Shih, “Observation of two-photon ’ghost’ interference and diffraction,” Phys. Rev. Lett. 743600 (1995)
[CrossRef] [PubMed]

Physics Letters A

D.N. Klyshko, “Combine EPR and two-slit experiments: Interference of advanced waves,” Physics Letters A 132299 (1998)
[CrossRef]

Science

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, 662 (2010)
[CrossRef] [PubMed]

Other

Y. Jeronimo-Moreno and R. Jauregui, “On demand generation of propagation invariant photons with orbital angular momentum,” arXiv:1403.5474v1 (2014)

M. Mirhosseini, O.S. Magaña-Loaiza, M.N. O’Sullivan, B. Rodenburg, M. Malik, D.J. Gauthier, and R.W. Boyd, “High-dimensional quantum cryptography with twisted light,” arXiv:1402.7113 (2014)

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Figures (6)

Fig. 1
Fig. 1

(a) Parameter space (crystal L and parameter kt), with the region where the short crystal approximation is valid shaded in yellow. While point A corresponds to the experimental conditions (see below), point B lies outside of the short-crystal regime. (b) Simulated CAS for configuration A. (c) Simulated CAS for configuration B. (d) Assumed pump angular spectrum. It may be seen that while the CAS for configuration A very nearly matches the pump angular spectrum, the CAS for configuration B differs substantially from the pump angular spectrum.

Fig. 2
Fig. 2

Back: Bessel-Gauss beam preparation. Front: Experimental setup for the transfer of optical vortices with l = 1 and l = 2 topological charge from a pump beam to a heralded single photon; note that while all measurements use a spatially-resolving detector on FP1, a second detector is either not present (for the SPDC angular spectrum), or placed on any of the planes FP1, FP2 and FP3 for measurements of the CAS, transverse intensity and far-field diffraction pattern, respectively. Inset: SPDC and pump angular spectrum on plane FP1, where we have marked the location of the idler detector and the CAS; the dotted square represents the area where the signal detector is scanned.

Fig. 3
Fig. 3

For a Bessel-Gauss optical vortex with topological charge l = 1: (a) Transverse intensity pattern, (b) angular spectrum, (c) far-field diffraction pattern through an triangular aperture

Fig. 4
Fig. 4

For a Bessel-Gauss optical vortex with topological charge l = 2: (a) Transverse intensity pattern, (b) angular spectrum, (c) far-field diffraction pattern through a triangular aperture.

Fig. 5
Fig. 5

For a pump beam with topological charge l = 1: (a) SPDC angular spectrum, (b) conditional angular spectrum of the signal-mode single photons, (c) transverse intensity distribution of signal-mode heralded single photon, (d) far-field diffraction pattern through a triangular aperture (sketched in panel c) of signal-mode heralded single photon.

Fig. 6
Fig. 6

For a pump beam with topological charge l = 2: (a) SPDC angular spectrum, (b) conditional angular spectrum of the signal-mode single photons, (c) transverse intensity distribution of signal-mode heralded single photon, (d) far-field diffraction pattern through a triangular aperture (sketched in panel c) of signal-mode heralded single photon.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

| Ψ = d k s d ω s d k i A s A i S ( k s + k i ) G ( k s , k i , ω s ) | k s , ω s s | k i , ω p ω s i
Π ^ ( k ˜ i ) = | k ˜ i , ω ˜ i i k ˜ i , ω ˜ i | i
| Ψ ( k ˜ i ) s = d k s S ( k s + k ˜ i ) G ( k s , k ˜ i , ω p ω ˜ i ) | k s , ω p ω ˜ i s .
ρ ^ s = d k ˜ i g ( k ˜ i ) | Ψ ( k ˜ i ) s Ψ ( k ˜ i ) | s .
n ^ ( k s ) = Tr ( a ^ s ( k s ) a ^ s ( k s ) ρ ^ s ) = d k ˜ i g ( k ˜ i ) | S ( k s + k ˜ i ) | 2 ,
n ^ ( k s ) = | S ( k s + k ˜ i ) | 2 .
n ˜ ( ρ s ) = Tr ( a ˜ s ( ρ s ) a ˜ s ( ρ s ) ρ ^ s ) = d k ˜ i g ( k ˜ i ) | 1 ( 2 π ) 2 d k e i k ρ s S ( k + k ˜ i ) | 2 .
n ˜ ( ρ s ) = M | S ˜ ( ρ s ) | 2 ,
S ( k ) = A exp ( w 0 2 4 | k | 2 ) I l ( k t w 0 2 | k | 2 ) exp ( i l ϕ ) ,
S ˜ ( ρ ) = A 1 μ exp { 1 μ ( i k t 2 z 2 k p + | ρ | 2 w 0 2 ) } J l ( k t | ρ | μ )

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