Abstract

Higher order correlation beams, that is, two-photon beams obtained from the process of spontaneous parametric down-conversion pumped by Hermite-Gauss or Laguerre-Gauss beams of any order, can be used to encode information in many modes, opening the possibility of quantum communication with large alphabets. In this paper we calculate, analytically, the fourth-order correlation function for the Hermite-Gauss and Laguerre-Gauss coherent and partially coherent correlation beams propagating through a strong turbulent medium. We show that fourth-order correlation functions for correlation beams have, under certain conditions, expressions similar to those of intensities of classical beams and are degraded by turbulence in a similar way as the classical beams. Our results can be useful in establishing limits for the use of two-photon beams in quantum communications with larger alphabets under atmospheric turbulence.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Mode analysis of higher-order transverse-mode correlation beams in a turbulent atmosphere

H. Avetisyan and C. H. Monken
Opt. Lett. 42(1) 101-104 (2017)

Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence

Tomohiro Shirai, Aristide Dogariu, and Emil Wolf
J. Opt. Soc. Am. A 20(6) 1094-1102 (2003)

Partially coherent beam propagation in atmospheric turbulence [Invited]

Greg Gbur
J. Opt. Soc. Am. A 31(9) 2038-2045 (2014)

References

  • View by:
  • |
  • |
  • |

  1. J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547 (2009).
    [Crossref]
  2. Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
    [Crossref]
  3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature (London) 412, 313 (2001).
    [Crossref]
  4. D. Bruss, L. Faoro, C. Macchiavello, and G. M. Palma, “Quantum entanglement and classical communication through a depolarising channel,” J. Mod. Opt. 47, 325 (2000).
    [Crossref]
  5. C. King, “The capacity of the quantum depolarizing channel,” IEEE Trans. Inform. Theory 49, 221 (2003).
    [Crossref]
  6. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
    [Crossref] [PubMed]
  7. F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
    [Crossref]
  8. D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed turbulence,” Opt. Express 19, 24596 (2011).
    [Crossref] [PubMed]
  9. M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195 (2012).
    [Crossref] [PubMed]
  10. F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
    [Crossref]
  11. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448 (2004).
    [Crossref] [PubMed]
  12. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
    [Crossref]
  13. B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt Express 19, 6671 (2011).
    [Crossref] [PubMed]
  14. 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, 060503 (2014).
    [Crossref] [PubMed]
  15. M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” arXiv:1507.06551.
  16. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
    [Crossref]
  17. F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A: Math. Theor. 47, 195302 (2014).
    [Crossref]
  18. A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: coherence and capacity,” Science 287, 287 (2000).
    [Crossref] [PubMed]
  19. S. E. Skipetrov, “Information transfer through disordered media by diffuse waves,” Phys. Rev. E 67, 036621 (2003).
    [Crossref]
  20. D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
    [Crossref]
  21. B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
    [Crossref]
  22. A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85, 013826 (2012).
    [Crossref]
  23. B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
    [Crossref]
  24. M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
    [Crossref]
  25. N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay in atmospheric turbulence,” arXiv:1408.3324.
  26. S. E. Skipetrov, “Quantum theory of dynamic multiple light scattering in fluctuating disordered media,” Phys. Rev. A 75, 053808 (2007).
    [Crossref]
  27. S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
    [Crossref] [PubMed]
  28. J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
    [Crossref] [PubMed]
  29. C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
    [Crossref] [PubMed]
  30. W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
    [Crossref] [PubMed]
  31. H. Di Lorenzo Pires, J. Woudenberg, and M. P. van Exter, “Statistical properties of two-photon speckles,” Phys. Rev. A 85, 033807 (2012).
    [Crossref]
  32. M. Cande and S. E. Skipetrov, “Quantum versus classical effects in two-photon speckle patterns,” Phys. Rev. A 87, 013846 (2013).
    [Crossref]
  33. M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A,  88, 053836 (2013).
    [Crossref]
  34. S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
    [Crossref]
  35. A. K. Jha and R. W. Boyd, “Effects of atmospheric turbulence on the entanglement of spatial two-qubit states,” Phys. Rev. A 81, 053832 (2010).
    [Crossref]
  36. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916 (2009).
    [Crossref] [PubMed]
  37. C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599 (2010).
    [Crossref]
  38. P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
    [Crossref]
  39. K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
    [Crossref]
  40. S. Smolka, O. L. Muskens, Ad Lagendijk, and P. Lodahl, “Angle-resolved photon-coincidence measurements in a multiple-scattering medium,” Phys. Rev. A 83, 043819 (2011).
    [Crossref]
  41. M. P. van Exter, J. Woudenberg, H. Di Lorenzo Pires, and W. H. Peeters, “Bosonic, fermionic, and anyonic symmetry in two-photon random scattering,” Phys. Rev. A 85, 033823 (2012).
    [Crossref]
  42. M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
    [Crossref]
  43. S. P. Walborn, C.H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
    [Crossref]
  44. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592 (2002).
    [Crossref]
  45. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094 (2003).
    [Crossref]
  46. M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261 (2003).
    [Crossref]
  47. J. C. Owens, “Optical refractive index of air: dependence on pressure, temperature and composition,” Appl. Opt. 6, 51 (1967).
    [Crossref] [PubMed]
  48. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  49. V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (SPIE, 1993).
  50. M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).
  51. R. L. Fante, “Wave propagation in random Media: a system approach,” Progress in Optics,  22, 341398 (1985).
  52. A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. (Doki) Acad. Sci. U.S.S.R. 30, 301 (1941).
  53. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (transl. for NOAA by Israel Program for Scientific Translations, 1971).
  54. T. von Kármán, “Progress in the Statistical Theory of Turbulence,” Proc. Nat. Acad. Sci. USA 34, 530 (1948).
    [Crossref] [PubMed]
  55. S. M. Rytov, “Diffraction of light by ultrasonic waves,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya (Bulletin of the Academy of Sciences of the USSR, Physical Series) 2, 223259 (1937).
  56. A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in a turbulent media,” Proc. IEEE 63, 790 (1975).
    [Crossref]
  57. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  58. R. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652 (1971).
    [Crossref] [PubMed]
  59. Z. I. Feizulin and Yu. A. Kravtsov, “Expansion of a laser beam in a turbulent medium,” Izv. Vyssh., Uchebn. Zaved. Radiofiz. 24, 1351 (1967).
  60. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894 (1979).
    [Crossref]
  61. V. I. Tatarskii and V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Progress in Optics III, E. Wolf, ed. (Elsevier, 1980), pp. 207–256.
  62. F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
    [Crossref]
  63. C. H. Monken, P. H. Souto Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A. 57, 3123 (1998).
    [Crossref]
  64. B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
    [Crossref] [PubMed]
  65. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
    [Crossref]
  66. H. van Cittert, “Dié wahrscheinliche schwingungsverteilung in einer von einer lichtquelle direkt oder mittels einer linse beleuchteten ebene,” Physica 1, 201 (1934).
    [Crossref]
  67. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica V, 785 (1938).
    [Crossref]
  68. B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
    [Crossref]
  69. C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Clock synchronization by remote detection of correlated photon pairs,” New J. Phys. 11, 045011 (2009).
    [Crossref]
  70. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
    [Crossref] [PubMed]
  71. N. N. Lebedev, Special Functions and Their Applications (Prentice-Hall, 1965).
  72. A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter,” Opt. Comm. 181, 35 (2000).
    [Crossref]
  73. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097 (2002).
    [Crossref]
  74. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  75. A. Yang, E. Zhang, X. Ji, and B. Li, “Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence,” Opt. Express 16, 8366 (2008).
    [Crossref] [PubMed]
  76. O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
    [Crossref]
  77. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
    [Crossref]

2015 (2)

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
[Crossref]

2014 (3)

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, 060503 (2014).
[Crossref] [PubMed]

F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A: Math. Theor. 47, 195302 (2014).
[Crossref]

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

2013 (3)

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

M. Cande and S. E. Skipetrov, “Quantum versus classical effects in two-photon speckle patterns,” Phys. Rev. A 87, 013846 (2013).
[Crossref]

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A,  88, 053836 (2013).
[Crossref]

2012 (6)

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85, 013826 (2012).
[Crossref]

H. Di Lorenzo Pires, J. Woudenberg, and M. P. van Exter, “Statistical properties of two-photon speckles,” Phys. Rev. A 85, 033807 (2012).
[Crossref]

M. P. van Exter, J. Woudenberg, H. Di Lorenzo Pires, and W. H. Peeters, “Bosonic, fermionic, and anyonic symmetry in two-photon random scattering,” Phys. Rev. A 85, 033823 (2012).
[Crossref]

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[Crossref]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195 (2012).
[Crossref] [PubMed]

2011 (5)

D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed turbulence,” Opt. Express 19, 24596 (2011).
[Crossref] [PubMed]

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

S. Smolka, O. L. Muskens, Ad Lagendijk, and P. Lodahl, “Angle-resolved photon-coincidence measurements in a multiple-scattering medium,” Phys. Rev. A 83, 043819 (2011).
[Crossref]

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[Crossref]

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt Express 19, 6671 (2011).
[Crossref] [PubMed]

2010 (8)

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

A. K. Jha and R. W. Boyd, “Effects of atmospheric turbulence on the entanglement of spatial two-qubit states,” Phys. Rev. A 81, 053832 (2010).
[Crossref]

J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[Crossref] [PubMed]

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599 (2010).
[Crossref]

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
[Crossref]

S. P. Walborn, C.H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

2009 (6)

C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Clock synchronization by remote detection of correlated photon pairs,” New J. Phys. 11, 045011 (2009).
[Crossref]

J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916 (2009).
[Crossref] [PubMed]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[Crossref] [PubMed]

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547 (2009).
[Crossref]

D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
[Crossref]

2008 (1)

2007 (1)

S. E. Skipetrov, “Quantum theory of dynamic multiple light scattering in fluctuating disordered media,” Phys. Rev. A 75, 053808 (2007).
[Crossref]

2005 (2)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref] [PubMed]

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[Crossref] [PubMed]

2004 (2)

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448 (2004).
[Crossref] [PubMed]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[Crossref]

2003 (4)

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094 (2003).
[Crossref]

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261 (2003).
[Crossref]

C. King, “The capacity of the quantum depolarizing channel,” IEEE Trans. Inform. Theory 49, 221 (2003).
[Crossref]

S. E. Skipetrov, “Information transfer through disordered media by diffuse waves,” Phys. Rev. E 67, 036621 (2003).
[Crossref]

2002 (3)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592 (2002).
[Crossref]

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097 (2002).
[Crossref]

2001 (2)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

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

2000 (4)

D. Bruss, L. Faoro, C. Macchiavello, and G. M. Palma, “Quantum entanglement and classical communication through a depolarising channel,” J. Mod. Opt. 47, 325 (2000).
[Crossref]

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: coherence and capacity,” Science 287, 287 (2000).
[Crossref] [PubMed]

A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter,” Opt. Comm. 181, 35 (2000).
[Crossref]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

1998 (1)

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

1985 (1)

R. L. Fante, “Wave propagation in random Media: a system approach,” Progress in Optics,  22, 341398 (1985).

1979 (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894 (1979).
[Crossref]

1975 (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in a turbulent media,” Proc. IEEE 63, 790 (1975).
[Crossref]

1971 (1)

1967 (2)

J. C. Owens, “Optical refractive index of air: dependence on pressure, temperature and composition,” Appl. Opt. 6, 51 (1967).
[Crossref] [PubMed]

Z. I. Feizulin and Yu. A. Kravtsov, “Expansion of a laser beam in a turbulent medium,” Izv. Vyssh., Uchebn. Zaved. Radiofiz. 24, 1351 (1967).

1948 (1)

T. von Kármán, “Progress in the Statistical Theory of Turbulence,” Proc. Nat. Acad. Sci. USA 34, 530 (1948).
[Crossref] [PubMed]

1941 (1)

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. (Doki) Acad. Sci. U.S.S.R. 30, 301 (1941).

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica V, 785 (1938).
[Crossref]

1937 (1)

S. M. Rytov, “Diffraction of light by ultrasonic waves,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya (Bulletin of the Academy of Sciences of the USSR, Physical Series) 2, 223259 (1937).

1934 (1)

H. van Cittert, “Dié wahrscheinliche schwingungsverteilung in einer von einer lichtquelle direkt oder mittels einer linse beleuchteten ebene,” Physica 1, 201 (1934).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Andersen, U. L.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[Crossref] [PubMed]

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

Balents, L.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: coherence and capacity,” Science 287, 287 (2000).
[Crossref] [PubMed]

Bao, X.-H.

Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
[Crossref]

Baranger, H. U.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: coherence and capacity,” Science 287, 287 (2000).
[Crossref] [PubMed]

Barnett, S.

Bartley, T.

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
[Crossref]

Beenakker, C. W. J.

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

Bianchini, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
[Crossref]

Boyd, R. W.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195 (2012).
[Crossref] [PubMed]

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

A. K. Jha and R. W. Boyd, “Effects of atmospheric turbulence on the entanglement of spatial two-qubit states,” Phys. Rev. A 81, 053832 (2010).
[Crossref]

Bruss, D.

D. Bruss, L. Faoro, C. Macchiavello, and G. M. Palma, “Quantum entanglement and classical communication through a depolarising channel,” J. Mod. Opt. 47, 325 (2000).
[Crossref]

Buchleitner, A.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay in atmospheric turbulence,” arXiv:1408.3324.

Bunkin, F. V.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in a turbulent media,” Proc. IEEE 63, 790 (1975).
[Crossref]

Cande, M.

M. Cande and S. E. Skipetrov, “Quantum versus classical effects in two-photon speckle patterns,” Phys. Rev. A 87, 013846 (2013).
[Crossref]

Chan, K. W. C.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Cheng, J.

Courtial, J.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448 (2004).
[Crossref] [PubMed]

A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter,” Opt. Comm. 181, 35 (2000).
[Crossref]

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, 060503 (2014).
[Crossref] [PubMed]

da Cunha Pereira, M. V.

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A,  88, 053836 (2013).
[Crossref]

Dashen, R.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894 (1979).
[Crossref]

Di Lorenzo Pires, H.

H. Di Lorenzo Pires, J. Woudenberg, and M. P. van Exter, “Statistical properties of two-photon speckles,” Phys. Rev. A 85, 033807 (2012).
[Crossref]

M. P. van Exter, J. Woudenberg, H. Di Lorenzo Pires, and W. H. Peeters, “Bosonic, fermionic, and anyonic symmetry in two-photon random scattering,” Phys. Rev. A 85, 033823 (2012).
[Crossref]

Dixon, P. B.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Dogariu, A.

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094 (2003).
[Crossref]

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261 (2003).
[Crossref]

Dudley, A.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Eberly, J. H.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Eliel, E. R.

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt Express 19, 6671 (2011).
[Crossref] [PubMed]

Elser, D.

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
[Crossref]

Fante, R. L.

R. L. Fante, “Wave propagation in random Media: a system approach,” Progress in Optics,  22, 341398 (1985).

Faoro, L.

D. Bruss, L. Faoro, C. Macchiavello, and G. M. Palma, “Quantum entanglement and classical communication through a depolarising channel,” J. Mod. Opt. 47, 325 (2000).
[Crossref]

Feizulin, Z. I.

Z. I. Feizulin and Yu. A. Kravtsov, “Expansion of a laser beam in a turbulent medium,” Izv. Vyssh., Uchebn. Zaved. Radiofiz. 24, 1351 (1967).

Fickler, R.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” arXiv:1507.06551.

Filpi, L. A. P.

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A,  88, 053836 (2013).
[Crossref]

Fink, M.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” arXiv:1507.06551.

Forbes, A.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Franke-Arnold, S.

Gauthier, D. J.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

Gbur, G.

Gibson, G.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097 (2002).
[Crossref]

Giovannini, D.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

Gochelashvily, K. S.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in a turbulent media,” Proc. IEEE 63, 790 (1975).
[Crossref]

Gomonay, H. V.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85, 013826 (2012).
[Crossref]

Gong, W.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Goyal, S.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Han, S.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Handsteiner, J.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” arXiv:1507.06551.

Hardy, N. D.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Heim, B.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
[Crossref]

Ho, C.

C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Clock synchronization by remote detection of correlated photon pairs,” New J. Phys. 11, 045011 (2009).
[Crossref]

Howell, J. C.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Howland, G. A.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Huck, A.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[Crossref] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (SPIE, 1993).

Jha, A. K.

A. K. Jha and R. W. Boyd, “Effects of atmospheric turbulence on the entanglement of spatial two-qubit states,” Phys. Rev. A 81, 053832 (2010).
[Crossref]

Ji, X.

Khan, I.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

Killoran, N.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

King, C.

C. King, “The capacity of the quantum depolarizing channel,” IEEE Trans. Inform. Theory 49, 221 (2003).
[Crossref]

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. (Doki) Acad. Sci. U.S.S.R. 30, 301 (1941).

Konrad, T.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Korotkova, O.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[Crossref]

Kravtsov, Yu. A.

Z. I. Feizulin and Yu. A. Kravtsov, “Expansion of a laser beam in a turbulent medium,” Izv. Vyssh., Uchebn. Zaved. Radiofiz. 24, 1351 (1967).

Krenn, M.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” arXiv:1507.06551.

Kurtsiefer, C.

C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Clock synchronization by remote detection of correlated photon pairs,” New J. Phys. 11, 045011 (2009).
[Crossref]

Lagendijk, A.

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[Crossref] [PubMed]

Lagendijk, Ad

S. Smolka, O. L. Muskens, Ad Lagendijk, and P. Lodahl, “Angle-resolved photon-coincidence measurements in a multiple-scattering medium,” Phys. Rev. A 83, 043819 (2011).
[Crossref]

Lamas-Linares, A.

C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Clock synchronization by remote detection of correlated photon pairs,” New J. Phys. 11, 045011 (2009).
[Crossref]

Lavery, M. P. J.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195 (2012).
[Crossref] [PubMed]

Leach, J.

Lebedev, N. N.

N. N. Lebedev, Special Functions and Their Applications (Prentice-Hall, 1965).

Leonhard, N. D.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay in atmospheric turbulence,” arXiv:1408.3324.

Leuchs, G

D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
[Crossref]

Leuchs, G.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

Li, B.

Li, C.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599 (2010).
[Crossref]

Lodahl, P.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

S. Smolka, O. L. Muskens, Ad Lagendijk, and P. Lodahl, “Angle-resolved photon-coincidence measurements in a multiple-scattering medium,” Phys. Rev. A 83, 043819 (2011).
[Crossref]

J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[Crossref] [PubMed]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[Crossref] [PubMed]

Lu, C-Y.

Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
[Crossref]

Lutkenhaus, N.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Lutomirski, R.

Macchiavello, C.

D. Bruss, L. Faoro, C. Macchiavello, and G. M. Palma, “Quantum entanglement and classical communication through a depolarising channel,” J. Mod. Opt. 47, 325 (2000).
[Crossref]

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097 (2002).
[Crossref]

Mafu, M.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Magaa-Loaiza, O. S.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

Mair, A.

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

Malik, M.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195 (2012).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Mari, E.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[Crossref]

Marquardt, C.

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

Marquardt, Ch.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

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, 060503 (2014).
[Crossref] [PubMed]

McLaren, M.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Mirhosseini, M.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195 (2012).
[Crossref] [PubMed]

Moerman, J. J. D.

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

Monken, C. H.

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A,  88, 053836 (2013).
[Crossref]

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt Express 19, 6671 (2011).
[Crossref] [PubMed]

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

Monken, C.H.

S. P. Walborn, C.H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

Mortensen, N. A.

J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[Crossref] [PubMed]

Moustakas, A. L.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: coherence and capacity,” Science 287, 287 (2000).
[Crossref] [PubMed]

Muskens, O. L.

S. Smolka, O. L. Muskens, Ad Lagendijk, and P. Lodahl, “Angle-resolved photon-coincidence measurements in a multiple-scattering medium,” Phys. Rev. A 83, 043819 (2011).
[Crossref]

O’Neil, A. T.

A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter,” Opt. Comm. 181, 35 (2000).
[Crossref]

O’Sullivan, M.

O’Sullivan, M. N.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

Oesch, D. W.

OSullivan, M. N.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Ott, J. R.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[Crossref] [PubMed]

Owens, J. C.

Padget, M. J.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

Padgett, M.

Padgett, M. J.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195 (2012).
[Crossref] [PubMed]

Pádua, S.

S. P. Walborn, C.H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

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

Palma, G. M.

D. Bruss, L. Faoro, C. Macchiavello, and G. M. Palma, “Quantum entanglement and classical communication through a depolarising channel,” J. Mod. Opt. 47, 325 (2000).
[Crossref]

Pan, J.-W.

Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
[Crossref]

Pas’ko, V.

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref] [PubMed]

Peeters, W. H.

M. P. van Exter, J. Woudenberg, H. Di Lorenzo Pires, and W. H. Peeters, “Bosonic, fermionic, and anyonic symmetry in two-photon random scattering,” Phys. Rev. A 85, 033823 (2012).
[Crossref]

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

Peng, C.-Z.

Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
[Crossref]

Petruccione, F.

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

Peuntinger, C.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

Pors, B.-J.

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt Express 19, 6671 (2011).
[Crossref] [PubMed]

Prokhorov, A. M.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in a turbulent media,” Proc. IEEE 63, 790 (1975).
[Crossref]

Pu, J.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599 (2010).
[Crossref]

Rao, R.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599 (2010).
[Crossref]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

Rodenburg, B.

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195 (2012).
[Crossref] [PubMed]

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

Romanato, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[Crossref]

Roux, F. S.

F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
[Crossref]

F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A: Math. Theor. 47, 195302 (2014).
[Crossref]

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[Crossref]

Rytov, S. M.

S. M. Rytov, “Diffraction of light by ultrasonic waves,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya (Bulletin of the Academy of Sciences of the USSR, Physical Series) 2, 223259 (1937).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Sabuncu, M.

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Salem, M.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261 (2003).
[Crossref]

Sanchez, D. J.

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, 060503 (2014).
[Crossref] [PubMed]

Semenov, A. A.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85, 013826 (2012).
[Crossref]

Sengupta, A. M.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: coherence and capacity,” Science 287, 287 (2000).
[Crossref] [PubMed]

Sergienko, A. V.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Shapiro, J. H.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547 (2009).
[Crossref]

Shatokhin, V. N.

F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
[Crossref]

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay in atmospheric turbulence,” arXiv:1408.3324.

Shen, X.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Shirai, T.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261 (2003).
[Crossref]

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094 (2003).
[Crossref]

Shishov, V. I.

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in a turbulent media,” Proc. IEEE 63, 790 (1975).
[Crossref]

Simon, D. S.

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

Simon, S. H.

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: coherence and capacity,” Science 287, 287 (2000).
[Crossref] [PubMed]

Skipetrov, S. E.

M. Cande and S. E. Skipetrov, “Quantum versus classical effects in two-photon speckle patterns,” Phys. Rev. A 87, 013846 (2013).
[Crossref]

S. E. Skipetrov, “Quantum theory of dynamic multiple light scattering in fluctuating disordered media,” Phys. Rev. A 75, 053808 (2007).
[Crossref]

S. E. Skipetrov, “Information transfer through disordered media by diffuse waves,” Phys. Rev. E 67, 036621 (2003).
[Crossref]

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, 060503 (2014).
[Crossref] [PubMed]

Smolka, S.

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

S. Smolka, O. L. Muskens, Ad Lagendijk, and P. Lodahl, “Angle-resolved photon-coincidence measurements in a multiple-scattering medium,” Phys. Rev. A 83, 043819 (2011).
[Crossref]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[Crossref] [PubMed]

Souto Ribeiro, P. H.

S. P. Walborn, C.H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

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

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, 060503 (2014).
[Crossref] [PubMed]

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[Crossref]

Sych, D.

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
[Crossref]

Tamburini, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[Crossref]

Tatarskii, V. I.

V. I. Tatarskii and V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Progress in Optics III, E. Wolf, ed. (Elsevier, 1980), pp. 207–256.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (transl. for NOAA by Israel Program for Scientific Translations, 1971).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (SPIE, 1993).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

Thide, B.

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[Crossref]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

Toppel, F.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85, 013826 (2012).
[Crossref]

Vallone, G.

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, 060503 (2014).
[Crossref] [PubMed]

van Cittert, H.

H. van Cittert, “Dié wahrscheinliche schwingungsverteilung in einer von einer lichtquelle direkt oder mittels einer linse beleuchteten ebene,” Physica 1, 201 (1934).
[Crossref]

van Exter, M. P.

M. P. van Exter, J. Woudenberg, H. Di Lorenzo Pires, and W. H. Peeters, “Bosonic, fermionic, and anyonic symmetry in two-photon random scattering,” Phys. Rev. A 85, 033823 (2012).
[Crossref]

H. Di Lorenzo Pires, J. Woudenberg, and M. P. van Exter, “Statistical properties of two-photon speckles,” Phys. Rev. A 85, 033807 (2012).
[Crossref]

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

Vasnetsov, M.

Vasylyev, D. Yu.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85, 013826 (2012).
[Crossref]

Vaziri, A.

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

Venderbos, J. W. F.

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

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, 060503 (2014).
[Crossref] [PubMed]

Vogel, W.

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85, 013826 (2012).
[Crossref]

von Kármán, T.

T. von Kármán, “Progress in the Statistical Theory of Turbulence,” Proc. Nat. Acad. Sci. USA 34, 530 (1948).
[Crossref] [PubMed]

Walborn, S. P.

S. P. Walborn, C.H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

Wang, T.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599 (2010).
[Crossref]

Weihs, G.

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

Wellens, T.

F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
[Crossref]

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

Wittmann, C.

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

Wittmann, Ch.

D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
[Crossref]

Woerdman, J. P.

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt Express 19, 6671 (2011).
[Crossref] [PubMed]

Wolf, E.

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094 (2003).
[Crossref]

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261 (2003).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592 (2002).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

Woudenberg, J.

H. Di Lorenzo Pires, J. Woudenberg, and M. P. van Exter, “Statistical properties of two-photon speckles,” Phys. Rev. A 85, 033807 (2012).
[Crossref]

M. P. van Exter, J. Woudenberg, H. Di Lorenzo Pires, and W. H. Peeters, “Bosonic, fermionic, and anyonic symmetry in two-photon random scattering,” Phys. Rev. A 85, 033823 (2012).
[Crossref]

Yang, A.

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097 (2002).
[Crossref]

Yuan, Z.-S.

Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
[Crossref]

Yura, H. T.

Zavorotny, V. U.

V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (SPIE, 1993).

Zavorotnyi, V. U.

V. I. Tatarskii and V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Progress in Optics III, E. Wolf, ed. (Elsevier, 1980), pp. 207–256.

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

Zeilinger, A.

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

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” arXiv:1507.06551.

Zernike, F.

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica V, 785 (1938).
[Crossref]

Zhan, J.

Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
[Crossref]

Zhang, E.

Zhang, P.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Zhu, W.

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599 (2010).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (2)

B. Heim, D. Elser, T. Bartley, M. Sabuncu, C. Wittmann, D. Sych, C. Marquardt, and G. Leuchs, “Atmospheric channel characteristics for quantum communication with continuous polarization variables,” Appl. Phys. B 98635 (2010).
[Crossref]

C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B 99, 599 (2010).
[Crossref]

C. R. (Doki) Acad. Sci. U.S.S.R. (1)

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. (Doki) Acad. Sci. U.S.S.R. 30, 301 (1941).

IEEE J. Sel. Top. Quantum Electron. (1)

J. H. Shapiro, “The quantum theory of optical communications,” IEEE J. Sel. Top. Quantum Electron. 15, 1547 (2009).
[Crossref]

IEEE Trans. Inform. Theory (1)

C. King, “The capacity of the quantum depolarizing channel,” IEEE Trans. Inform. Theory 49, 221 (2003).
[Crossref]

Izv. Vyssh., Uchebn. Zaved. Radiofiz. (1)

Z. I. Feizulin and Yu. A. Kravtsov, “Expansion of a laser beam in a turbulent medium,” Izv. Vyssh., Uchebn. Zaved. Radiofiz. 24, 1351 (1967).

Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya (Bulletin of the Academy of Sciences of the USSR, Physical Series) (1)

S. M. Rytov, “Diffraction of light by ultrasonic waves,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya (Bulletin of the Academy of Sciences of the USSR, Physical Series) 2, 223259 (1937).

J. Math. Phys. (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894 (1979).
[Crossref]

J. Mod. Opt. (1)

D. Bruss, L. Faoro, C. Macchiavello, and G. M. Palma, “Quantum entanglement and classical communication through a depolarising channel,” J. Mod. Opt. 47, 325 (2000).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Phys. A: Math. Theor. (1)

F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A: Math. Theor. 47, 195302 (2014).
[Crossref]

Nature (London) (1)

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

New J. Phys. (5)

D. Elser, T. Bartley, B. Heim, Ch. Wittmann, D. Sych, and G Leuchs, “Feasibility of free space quantum key distribution with coherent polarization states,” New J. Phys. 11, 045014 (2009).
[Crossref]

B. Heim, C. Peuntinger, N. Killoran, I. Khan, C. Wittmann, Ch. Marquardt, and G. Leuchs, “Atmospheric continuous-variable quantum communication,” New J. Phys. 16, 113018 (2014).
[Crossref]

F. Tamburini, E. Mari, A. Sponselli, B. Thide, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012).
[Crossref]

M. Mirhosseini, O. S. Magaa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padget, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

C. Ho, A. Lamas-Linares, and C. Kurtsiefer, “Clock synchronization by remote detection of correlated photon pairs,” New J. Phys. 11, 045011 (2009).
[Crossref]

Opt Express (1)

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt Express 19, 6671 (2011).
[Crossref] [PubMed]

Opt. Comm. (1)

A. T. O’Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter,” Opt. Comm. 181, 35 (2000).
[Crossref]

Opt. Commun. (1)

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, “Long-distance propagation of partially coherent beams through atmospheric turbulence,” Opt. Commun. 216, 261 (2003).
[Crossref]

Opt. Eng. (2)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “A model for a partially coherent Gaussian beam in atmospheric turbulence with application in lasercom,” Opt. Eng. 43, 330–341 (2004).
[Crossref]

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097 (2002).
[Crossref]

Opt. Express (5)

Phys. Rep. (1)

S. P. Walborn, C.H. Monken, S. Pádua, and P. H. Souto Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

Phys. Rev. A (15)

F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
[Crossref]

B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
[Crossref]

M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lutkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
[Crossref]

A. A. Semenov, F. Toppel, D. Yu. Vasylyev, H. V. Gomonay, and W. Vogel, “Homodyne detection for atmosphere channels,” Phys. Rev. A 85, 013826 (2012).
[Crossref]

S. E. Skipetrov, “Quantum theory of dynamic multiple light scattering in fluctuating disordered media,” Phys. Rev. A 75, 053808 (2007).
[Crossref]

H. Di Lorenzo Pires, J. Woudenberg, and M. P. van Exter, “Statistical properties of two-photon speckles,” Phys. Rev. A 85, 033807 (2012).
[Crossref]

M. Cande and S. E. Skipetrov, “Quantum versus classical effects in two-photon speckle patterns,” Phys. Rev. A 87, 013846 (2013).
[Crossref]

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A,  88, 053836 (2013).
[Crossref]

S. Smolka, J. R. Ott, A. Huck, U. L. Andersen, and P. Lodahl, “Continuous-wave spatial quantum correlations of light induced by multiple scattering,” Phys. Rev. A 86, 033814 (2012).
[Crossref]

A. K. Jha and R. W. Boyd, “Effects of atmospheric turbulence on the entanglement of spatial two-qubit states,” Phys. Rev. A 81, 053832 (2010).
[Crossref]

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[Crossref]

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. OSullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A 84, 043807 (2011).
[Crossref]

S. Smolka, O. L. Muskens, Ad Lagendijk, and P. Lodahl, “Angle-resolved photon-coincidence measurements in a multiple-scattering medium,” Phys. Rev. A 83, 043819 (2011).
[Crossref]

M. P. van Exter, J. Woudenberg, H. Di Lorenzo Pires, and W. H. Peeters, “Bosonic, fermionic, and anyonic symmetry in two-photon random scattering,” Phys. Rev. A 85, 033823 (2012).
[Crossref]

Phys. Rev. A. (1)

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

Phys. Rev. E (1)

S. E. Skipetrov, “Information transfer through disordered media by diffuse waves,” Phys. Rev. E 67, 036621 (2003).
[Crossref]

Phys. Rev. Lett. (8)

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, 060503 (2014).
[Crossref] [PubMed]

S. Smolka, A. Huck, U. L. Andersen, A. Lagendijk, and P. Lodahl, “Observation of spatial quantum correlations induced by multiple scattering of nonclassical light,” Phys. Rev. Lett. 102, 193901 (2009).
[Crossref] [PubMed]

J. R. Ott, N. A. Mortensen, and P. Lodahl, “Quantum interference and entanglement induced by multiple scattering of light,” Phys. Rev. Lett. 105, 090501 (2010).
[Crossref] [PubMed]

C. W. J. Beenakker, J. W. F. Venderbos, and M. P. van Exter, “Two-photon speckle as a probe of multi-dimensional entanglement,” Phys. Rev. Lett. 102, 193601 (2009).
[Crossref] [PubMed]

W. H. Peeters, J. J. D. Moerman, and M. P. van Exter, “Observation of two-photon speckle patterns,” Phys. Rev. Lett. 104, 173601 (2010).
[Crossref] [PubMed]

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref] [PubMed]

B. E. A. Saleh, M. C. Teich, and A. V. Sergienko, “Wolf equations for two-photon light,” Phys. Rev. Lett. 94, 223601 (2005).
[Crossref] [PubMed]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87, 123602 (2001).
[Crossref] [PubMed]

Physica (2)

H. van Cittert, “Dié wahrscheinliche schwingungsverteilung in einer von einer lichtquelle direkt oder mittels einer linse beleuchteten ebene,” Physica 1, 201 (1934).
[Crossref]

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica V, 785 (1938).
[Crossref]

Physics Reports (1)

Z.-S. Yuan, X.-H. Bao, C-Y. Lu, J. Zhan, C.-Z. Peng, and J.-W. Pan, “Entangled photons and quantum communication,” Physics Reports 497, 140 (2010).
[Crossref]

Proc. IEEE (1)

A. M. Prokhorov, F. V. Bunkin, K. S. Gochelashvily, and V. I. Shishov, “Laser irradiance propagation in a turbulent media,” Proc. IEEE 63, 790 (1975).
[Crossref]

Proc. Nat. Acad. Sci. USA (1)

T. von Kármán, “Progress in the Statistical Theory of Turbulence,” Proc. Nat. Acad. Sci. USA 34, 530 (1948).
[Crossref] [PubMed]

Progress in Optics (1)

R. L. Fante, “Wave propagation in random Media: a system approach,” Progress in Optics,  22, 341398 (1985).

Rev. Mod. Phys. (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145 (2002).
[Crossref]

Science (1)

A. L. Moustakas, H. U. Baranger, L. Balents, A. M. Sengupta, and S. H. Simon, “Communication through a diffusive medium: coherence and capacity,” Science 287, 287 (2000).
[Crossref] [PubMed]

Other (13)

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” arXiv:1507.06551.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005).
[Crossref]

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay in atmospheric turbulence,” arXiv:1408.3324.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (transl. for NOAA by Israel Program for Scientific Translations, 1971).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
[Crossref]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

N. N. Lebedev, Special Functions and Their Applications (Prentice-Hall, 1965).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, Wave Propagation in Random Media (Scintillation) (SPIE, 1993).

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
[Crossref]

V. I. Tatarskii and V. U. Zavorotnyi, “Strong fluctuations in light propagation in a randomly inhomogeneous medium,” Progress in Optics III, E. Wolf, ed. (Elsevier, 1980), pp. 207–256.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the system considered. NLC represents a nonlinear crystal, hp, hs and hi represent the amplitude response functions for the pump beam (p) and for the down-converted photons (s, i). D1 and D2 represent the detectors and P2 represents the fourth order correlation function, a measurable quantity which is proportional to the two-photon joint detection probability.

Fig. 2
Fig. 2

The normalized two-photon speckle (TPS) for the HG10 pump case under turbulence conditions: strong (dashed), strong-to-moderate (dashed-dotted), moderate (dotted) and no turbulence (solid). The normalization is made by dividing Eq. (37) by its maximum value (1/πW2e) in the absence of turbulence. For simplicity, we took σ sp 2 ( z ) = 0.4 σ R 2 for the Kolmogorov spectrum.

Fig. 3
Fig. 3

Normalized two-photon speckle profile for the case when the χ(2) crystal is pumped by (a) a fully coherent (σμ → ∞) Gaussian beam and (b) a partially coherent (σμ = 2mm) Gaussian beam, both under the conditions kp = 10−7m−1, C n 2 = 10 13.6 m 2 / 3, l0 = 0.01m, σs = 5mm.

Equations (80)

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

n ( R ) 1 + 7.9 × 10 5 P ( R ) T ( R ) ,
Φ ( κ ) = 0.033 C n 2 κ 11 / 3 , 1 / L 0 κ 1 / l 0 ,
Φ ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 κ m 2 ) ,
Φ ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 / κ m 2 ) ( κ 2 + 1 / L 0 2 ) 11 / 6 ,
U ( r , L ) = k e i k L 2 π i L d 2 s U 0 ( s , 0 ) exp [ i k | s r | 2 2 L + ψ ( r , s ) ] ,
E 1 ( 0 , 0 ; 0 , 0 ) E 1 ( 0 ) = ψ 2 ( r , s ) + 1 2 ψ 1 2 ( r , s ) = 2 π 2 k 2 L 0 d κ κ Φ n ( κ ) ,
E 2 ( r 1 , r 2 ; s 1 , s 2 ) = ψ 1 ( r 1 , s 1 ) ψ 1 * ( r 2 , s 2 ) = 4 π 2 k 2 L 0 1 d ξ 0 d κ κ Φ n ( κ ) J 0 ( κ | ( 1 ξ ) p + ξ Q | ) ,
E 3 ( r 1 , r 2 ; s 1 , s 2 ) = ψ 1 ( r 1 , s 1 ) ψ 2 ( r 2 , s 2 ) = 4 π 2 k 2 L 0 1 d ξ 0 d κ κ Φ n ( κ ) J 0 ( κ | ( 1 ξ ) p + ξ Q | ) exp [ i L κ 2 k ξ ( 1 ξ ) ] ,
A ( R 1 , R 2 ) e i k p z d S E p ( S ) e i k p 2 z ( r S ) 2 .
A ( R 1 , R 2 ) E p ( r 1 + r 2 2 , z ) .
P 2 ( x 1 , x 2 ) = | A ( x 1 , x 2 ) | 2 = | 0 , 0 | E ^ 2 ( + ) ( x 2 ) E ^ 1 ( + ) ( x 1 ) | ψ | 2 ,
E ^ 1 ( + ) ( x 1 ) = d r h s ( r 1 , r ) a ^ s ( r , t 1 z 1 / c ) ,
E ^ 2 ( + ) ( x 2 ) = d r h i ( r 2 , r ) a ^ i ( r , t 2 z 2 / c ) ,
h j ( r j , r ) = k j e i k j z j i 2 π z j exp { i k j 2 z j | r j r | 2 + ψ ( j ) ( r j , r ; k j ) } ,
| ψ = d r d r E p ( r + r 2 ) δ ( r r ) a ^ s ( r ) a ^ i ( r ) | 0 , 0 = d r E p ( r ) | 1 r , 1 r ,
P 2 ( r 1 , r 2 ) = k 2 4 π 2 z 2 d r d r E p ( r ) E p * ( r ) exp { i k 2 z [ | r 1 r | 2 | r 1 r | 2 + | r 2 r | 2 | r 2 r | 2 ] } × exp [ ψ ( r , r 1 ) + ψ * ( r , r 1 ) + ψ ( r , r 2 ) + ψ * ( r , r 2 ) ] ,
exp [ ] = exp [ 4 E 1 ( 0 ) + 2 E 2 ( 0 , 0 ; r , r ) + E 2 ( r 1 , r 2 ; r , r ) + E 2 ( r 1 , r 1 ; r , r ) + 2 Re E 3 ( r 1 , r 2 ; 0 , 0 ) ] ,
P 2 ( r , r ) = k e e σ sp 2 ( z ) 4 π 2 z 2 d r d r E p ( r ) E p * ( r ) exp { i k 2 [ | r r | 2 | r r | 2 ] 2 D sp ( | r r | ) } .
D sp ( Q ) = 8 π 2 k 2 z 0 1 d ξ 0 d κ κ Φ ( κ ) [ 1 J 0 ( κ ξ Q ) ] ,
σ sp 2 ( z ) = 8 π 2 k 2 z 0 1 d ξ 0 d κ κ Φ ( κ ) [ 1 cos ( z κ 2 k ξ ( 1 ξ ) ) ] .
4 E 1 ( 0 ) + E 2 ( r 1 , r 2 ; r , r ) + E 2 ( r 2 , r 1 ; r , r ) = 1 2 [ D sp ( p , Q ) + D sp ( p , Q ) ] ,
2 E 2 ( 0 ; r , r ) + 2 Re E 3 ( r 1 , r 2 ; 0 ) = 8 π 2 k 2 z 0 1 d ξ 0 d κ κ Φ ( κ ) × [ J 0 ( κ ξ Q ) J 0 ( κ ξ p ) cos ( z k 2 k ξ ( 1 ξ ) ) ] = D sp ( p ) D sp ( Q ) + 4 π 2 z 3 0 1 d ξ 0 d κ κ 5 Φ ( κ ) J 0 ( κ ξ p ) ξ 2 ( 1 ξ ) 2 ,
D sp ( p , Q ) = 8 π 2 k 2 z 0 1 d ξ 0 d κ κ Φ ( κ ) [ 1 J 0 ( κ | 1 ξ | p + ξ Q ) ] .
P 2 ( r 1 , r 2 ) = k 2 4 π 2 z 2 exp [ ( 1.58 σ R , p 2 Λ 0 , p W 0 2 2 3 ρ p l 2 ) p 2 0.043 π 2 C n 2 z 3 p 7 / 3 ] × d S d Q E p ( S + Q / 2 ) E p * ( S Q / 2 ) exp [ i k p z ( S Q r Q ) ] × exp [ ( 1.58 σ R , p 2 Λ 0 . p W 0 2 + 2 3 ρ p l 2 ) Q 2 ] ,
P 1 ( r 1 ) = d r | E p ( r ) | 2 | h 1 ( r 1 , r ) | 2 .
P 1 ( r 1 ) = ( k 2 π z ) 2 d r | E p ( r ) | 2 exp [ ψ ( r 1 , r ; k ) + ψ * ( r 1 , r ; k ) ] .
exp [ ψ ( r 1 , r ; k ) + ψ * ( r 1 , r ; k ) ] = exp [ 2 σ r1 2 T ] ,
P 1 ( r 1 ) = ( k 2 π z ) 2 exp [ 2 σ r 2 T ] d r | E p ( r ) | 2 = ( k 2 π z ) 2 exp [ 2 σ r 2 T ] × I p | z = 0 .
E p ( r ) = U m n HG ( r x , r y , 0 ) = B m , n H m ( 2 W 0 r x ) H n ( 2 W 0 r y ) exp ( r 2 W 0 2 ) ,
P 2 ( r , r ) = k 2 e σ sp 2 ( z ) 4 π 2 z 2 | B m , n | 2 d S d Q H m [ 2 W 0 ( S x + Q x 2 ) ] H m [ 2 W 0 ( S x Q x 2 ) ] × H n [ 2 W 0 ( S y + Q y 2 ) ] H n [ 2 W 0 ( S y Q y 2 ) ] × exp [ 2 W 0 2 ( S x 2 + S y 2 ) ] exp [ 1 2 W 0 2 ( Q x 2 + Q y 2 ) ] × exp [ i k p z ( S x Q x + S y Q y ) ] exp [ i k p z ( r x Q x + r y Q y ) ] × exp [ 3.16 σ R , p 2 Λ 0 , p W 0 2 ( Q x 2 + Q y 2 ) ] ,
Q = r r , S = 1 2 ( r r ) , Q = | Q | , S = | S | .
P 2 ( r , r ) = e σ sp 2 ( z ) 2 π W LT 2 exp [ 2 r 2 W LT 2 ] k = 0 m l = 0 n ( m k ) ( n l ) [ W 2 2 W LT 2 ] k + l H 2 k [ 2 W LT r x ] H 2 l [ 2 W LT r y ] k ! l ! ,
P 2 ( r , r ) = | B m , n | 2 W 0 2 W 2 H m 2 [ 2 W r x ] H n 2 [ 2 W r y ] exp [ 2 r 2 W 2 ] ,
H m 2 ( x ) = 2 m ( m ! ) 2 k = 0 m H 2 k ( x ) 2 k ( k ! ) 2 ( m k ) ! .
P 2 ( r 1 , r 2 ) = 1 2 π W L T 1 2 exp [ ( 1.58 σ R , p 2 Λ p W 2 2 3 ρ p l 2 ) p 2 0.043 π 2 C n 2 z 3 p 7 / 3 ] e 2 r 2 W L T 1 2 × k = 0 m l = 0 n ( m k ) ( n l ) [ W 2 2 W L T 1 2 ] k + l H 2 k [ 2 W L T 1 ( r x 2 + r x 1 2 ) ] H 2 l [ 2 W L T 1 ( r y 1 + r y 2 2 ) ] k ! l ! ,
P 2 ( r 1 , r 2 ) = | B m , n | 2 W 0 2 W 2 H m 2 [ 2 W ( r x 2 + r x 1 2 ) ] H n 2 [ 2 W ( r y 1 + r y 2 2 ) ] exp [ 2 r 2 W 2 ] ,
P 2 ( 10 ) ( r , r ) = e σ sp 2 ( z ) 2 π W LT 2 exp [ 2 r 2 W LT 2 ] ( 1 + 4 W 2 W LT 4 r x 2 W 2 W LT 2 ) .
U m n LG ( x , y , z ) = k = 0 m + n i k b ( m , n , k ) U m + n k , k HG ( x , y , z ) ,
b ( m , n , k ) = ( m + n k ) ! k ! 2 m + n m ! n ! 1 k ! d n d t n [ ( 1 t ) m ( 1 + t ) n ] t = 0 ,
P 2 ( r 1 , r 2 ) = k 2 4 π 2 z 2 d r d r W ( c ) ( r , r ) × exp { i k 2 z [ | r 1 r | 2 | r 1 r | 2 + | r 2 r | 2 | r 2 r | 2 ] } × exp [ 1 2 [ D sp ( p , Q ) + D sp ( p , Q ) ] + D sp ( p ) D sp ( Q ) 0.043 π 2 C n 2 z 3 p 7 / 3 ] ,
P 2 ( r , r ) = k 2 e σ sp 2 ( z ) 4 π 2 z 2 d r d r W ( c ) ( r , r ) × exp [ i k z [ | r r | 2 | r r | 2 ] 2 D sp ( | r r | ) ] .
W ( 0 ) ( r 1 , r 2 , ω ) = S ( 0 ) ( r 1 , ω ) S ( 0 ) ( r 2 , ω ) μ ( 0 ) ( r 1 , r 2 , ω ) ,
S ( 0 ) ( r , ω ) = M exp [ | r | 2 2 σ s 2 ] , μ ( 0 ) ( r 1 , r 2 , ω ) = exp [ | r 1 r 2 | 2 2 σ μ 2 ]
P 2 ( r , r ) = k 2 e σ sp 2 ( z ) 4 π 2 z 2 d S d Q W ( c ) ( S + Q / 2 , S Q / 2 ) exp { i k p z [ S Q r Q ] 2 D sp ( Q ) } = M k 2 e σ sp 2 ( z ) 4 π 2 z 2 d S d Q exp [ S 2 2 σ s 2 Q 2 2 σ Δ 2 ] exp { i k p z [ S Q r Q ] 2 D sp ( Q ) } ,
P 2 ( r , r ) = M e σ sp 2 ( z ) Δ 2 ( z ) exp { r 2 2 σ s 2 Δ 2 ( z ) } ,
Δ 2 ( z ) = 1 + 1 ( k p σ s ) 2 ( 1 4 σ s 2 + 1 σ μ 2 ) z 2 + 1.58 C n 2 k p 1 / 6 σ s 2 z 13 / 6 .
Δ 2 ( z ) = 1 + 1 ( k p σ s ) 2 ( 1 4 σ s 2 + 1 σ μ 2 ) z 2 + 0.55 C n 2 l 0 1 / 3 σ s 2 z 3 .
P 2 N ( r , r ) = 1 + ( z / 2 k p σ s 2 ) 2 Δ 2 ( z ) exp { r 2 2 σ s 2 Δ 2 ( z ) } .
W ( 0 ) ( r 1 , r 2 , ω ) = m n β m , n ( ω ) ϕ n ( 0 ) * ( r 1 , ω ) ϕ n ( 0 ) ( r 2 , ω ) ,
β m , n ( ω ) = M ( π a + b + c ) ( b a + b + c ) m + n ,
ϕ ( 0 ) ( r , ω ) = ϕ ( 0 ) ( r x , r y , ω ) = B m , n H m [ 2 W 0 r x ] H n [ 2 W 0 r y ] exp [ r x 2 + r y 2 W 0 2 ] ,
P 2 ( pcoh ) ( r , r ) = k 2 e σ sp 2 ( z ) 4 π 2 z 2 m n β m , n | B m , n | 2 × d r d r H m [ 2 W 0 r x ] H m [ 2 W 0 r x ] H n [ 2 W 0 r y ] H n [ 2 W 0 r y ] × exp [ r x 2 + r x 2 + r y 2 + r y 2 W 0 2 ] exp { i k z [ | r r | 2 | r r | 2 ] 2 D sp ( | r r | ) } .
P 2 ( pcoh ) ( x , x ) = m = 0 n = 0 β m , n | B m , n | 2 P 2 ( coh ) ( x , x ) ,
I = 4 π 2 z 3 0 1 d ξ 0 d κ κ 5 Φ ( κ ) J 0 ( κ ξ p ) ξ 2 ( 1 ξ ) 2 .
Φ ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 κ m 2 ) ,
0 κ 4 / 3 exp [ κ 2 κ m 2 ] J 0 ( κ ξ p ) d κ = 0.033 C n 2 Γ ( 7 / 6 ) κ 7 / 3 2 F 1 1 ( 7 6 ; 1 ; ξ 2 p 2 κ m 2 4 ) ,
F 1 1 ( a ; c ; z ) ! Γ ( c ) Γ ( c a ) z a , Re ( z ) 1 ,
0 κ 4 / 3 exp [ κ 2 κ m 2 ] J 0 ( κ ξ p ) d κ 0.033 C n 2 Γ ( 7 / 6 ) 2 Γ ( 1 7 / 6 ) ( ξ 2 p 2 4 ) 7 / 6 0.016 C n 2 ξ 7 / 3 p 7 / 3 .
I = 0.046 π 2 z 3 C n 2 p 7 / 3 0 1 ξ 1 / 3 ( 1 ξ ) 2 d ξ = 0.043 π 2 z 3 C n 2 p 7 / 3 .
P 2 ( r , r ) = k 2 e σ sp 2 ( z ) 4 π 2 z 2 d S d Q H m [ 2 W 0 ( S x + Q x 2 ) ] H m [ 2 W 0 ( S x Q x 2 ) ] × H n [ 2 W 0 ( S y + Q y 2 ) ] H n [ 2 W 0 ( S y Q y 2 ) ] × exp [ 2 W 0 2 ( S x 2 + S y 2 ) ] exp [ 1 2 W 0 2 ( Q x 2 + Q y 2 ) ] × exp [ i k p z ( S x Q x + S y Q y ) ] exp [ i k p z ( r x Q x + r y Q y ) ] × exp [ 3.16 σ R , p 2 Λ 0 , p W 0 2 ( Q x 2 + Q y 2 ) ] .
I x = d S x H m [ 2 W 0 ( S x + Q x 2 ) ] H m [ 2 W 0 ( S x Q x 2 ) ] exp [ ( 2 W 0 2 S x 2 i k p Q x z S x ) ] .
2 W 0 2 S x 2 i k p Q x z S x = ( 2 W 0 S x i k p Q x W 0 2 2 z ) 2 + k p 2 Q x 2 W 0 2 8 z 2 .
2 W 0 ( S x ± Q x 2 ) = ξ + Q x 2 W 0 ( i k p W 0 2 2 z ± 1 ) .
I x = W 0 2 exp [ 1 2 ( k p Q x W 0 2 z ) 2 ] d ξ H m [ ξ + η ] H m [ ξ + ζ ] e ξ 2 ,
d ξ H m [ ξ + η ] H n [ ξ + ζ ] e ξ 2 = 2 n π m ! ζ n m L m n m ( 2 η ζ ) , [ m n ] ,
I x = W 0 2 exp [ 1 2 ( k p Q x W 0 2 z ) 2 ] 2 m π m ! L m [ ( 1 W 0 2 + ( k p W 0 2 z ) 2 ) Q x 2 ] .
I y = W 0 2 exp [ 1 2 ( k p Q y W 0 2 z ) 2 ] 2 n π n ! L n [ ( 1 W 0 2 + ( k p W 0 2 z ) 2 ) Q y 2 ] .
P 2 ( r , r ) = k 2 e σ sp 2 ( z ) 4 π 2 z 2 d Q x d Q y L m ( α Q x 2 ) L n ( α Q y 2 ) exp [ β ( Q x 2 + Q y 2 ) ] exp [ i k p z ( r x Q x + r y Q y ) ] ,
α 1 W 0 2 + ( k p W 0 2 z ) 2 , β 3.16 σ R , p 2 Λ 0 , p W 0 2 + 1 2 W 0 2 + 1 2 ( k p W 0 2 z ) 2 = 3.16 σ R , p 2 Λ 0 , p W 0 2 + α 2 .
J x = ¯ d Q x L m [ α Q x 2 ] exp [ ( β Q x 2 + i k p r x z Q x ) ] = exp [ 1 β ( k p r x 2 z ) 2 ] d Q x L m [ α Q x 2 ] exp [ ( β Q x + i k p r x 2 β z ) 2 ] .
J x = 1 β exp [ 1 β ( k p r x 2 z ) 2 ] d ξ L m ( α β ξ 2 ) e ( ξ η ) 2 .
L m ( x ) = k = 0 m ( m k ) ( 1 ) k k ! x k ,
J x = 1 β exp [ 1 β ( k p r x 2 z ) 2 ] k = 0 m ( m k ) ( 1 ) k k ! ( α β ) k d ξ e ( ξ η ) 2 ξ 2 k .
d x e ( x y ) 2 x n = ( 2 i ) n π H n ( i y ) ,
J x = 1 β exp [ 1 β ( k p r x 2 z ) 2 ] k = 0 m ( m k ) ( 1 ) k k ! ( 2 i ) 2 k ( α β ) k H 2 k [ k p r x 2 β z ] .
J y = 1 β exp [ 1 β ( k p r x 2 z ) 2 ] l = 0 m ( n l ) ( 1 ) l l ! ( 2 i ) 2 l ( α β ) l H 2 l [ k p r x 2 β z ] .
P 2 ( r , r ) = k 2 e σ sp 2 ( z ) 4 π z 2 β exp [ 1 β ( k p r 2 z ) 2 ] k = 0 m l = 0 n ( m k ) ( n l ) [ α 2 2 β 2 ] k + l H 2 k [ k p r x 2 β z ] H 2 l [ k p r x 2 β z ] k ! l ! .
W = W 0 1 + Λ 0 , p 2 = W 0 Λ 0 , p 1 + 1 Λ 0 , p 2 = 2 z k p W 0 1 + 1 Λ 0 . p 2 ,
α = 1 W 0 2 ( 1 + 1 Λ 0 2 ) = W 2 k p 2 4 z 2 , β = 3.16 σ R , p 2 Λ 0 , p W 0 2 + α 2 = 3.16 σ R , p 2 Λ p W 2 + W 2 k p 2 8 z 2 , k p 2 β z = k p 2 z 3.16 σ R , p 2 Λ p W 2 + W 2 k p 2 8 z 2 = 2 W 1 + 3.16 σ R , p 2 Λ p W 2 8 z 2 k p 2 W 2 2 W LT ,
P 2 ( r , r ) = e σ sp 2 ( z ) 2 π W LT 2 exp [ 2 r 2 W LT 2 ] k = 0 m l = 0 n ( m k ) ( n l ) ( W / W LT ) 2 k + 2 l 2 k + l k ! l ! H 2 k [ 2 W LT r x ] H 2 l [ 2 W LT r y ] .

Metrics