Abstract

We report a compact scheme for the generation and manipulation of photon pairs entangled in the orbital angular momentum (OAM) from the fork-poling quadratic nonlinear crystal. The χ(2)-modulation in this crystal is designed for fulfilling a tilted quasi-phase-matching geometry to ensure the efficient generation of entangled photons as well as for transferring of topological charge of the crystal to the photon pairs. Numerical results show that the OAM of photon pair is anti-correlated and the degree of OAM entanglement can be enhanced by modulating the topological charge of crystal, which indicates a feasible extension to high-dimensional OAM entanglement. These studies suggest that the fork-poling nonlinear photonic crystal a unique platform for compact generation and manipulation of high-dimensional and high-order OAM entanglement, which may have potential applications in quantum communication, quantum cryptography and quantum remote sensing.

© 2015 Optical Society of America

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    [Crossref] [PubMed]
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  4. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
    [Crossref] [PubMed]
  5. J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B: Quantum Semiclass. Opt. 7, 235–239 (2005).
    [Crossref]
  6. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
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  7. H. Di LorenzoPires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
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  8. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
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  9. V. D. Salakhutdinov, E. R. Eliel, and W. Loffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108, 173604 (2012).
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  10. R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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  11. M. Mclaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
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    [Crossref]
  13. J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A. 86, 012334 (2012).
    [Crossref]
  14. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [Crossref] [PubMed]
  15. A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83, 053829 (2011).
    [Crossref]
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    [Crossref]
  17. J. P. Meyn, C. Laue, R. Knappe, R. Wallenstein, and M. M. Fejer, “Fabrication of periodically poled lithium tantalate for UV generation with diode lasers,” Appl. Phys. B: Lasers Opt. 73, 111–114 (2001).
    [Crossref]
  18. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
    [Crossref]
  19. N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
    [Crossref] [PubMed]
  20. N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
    [Crossref] [PubMed]
  21. A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two dimensional nonlinear beam shaping,” Opt. Lett. 37, 2136–2138 (2012).
    [Crossref] [PubMed]
  22. A. Shapira, I. Juwiler, and A. Arie, “Tunable nonlinear beam shaping by non-collinear interactions,” Laser Photon. Rev. 7, L25–L29 (2013).
    [Crossref]
  23. H. Ishizuki and T. Taira, “Half-joule output optical-parametric oscillation by using 10-mm-thick periodically poled Mg-doped congruent LiNbO3,” Opt. Express 20, 20002–20010 (2012).
    [Crossref] [PubMed]
  24. T. Ellenbogen, N. V. Bloch, A. G. Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
    [Crossref]
  25. J. P. Torres, A. Alexandrescu, S. Carrasco, and L. Torner, “Quasi-phase-matching engineering for spatial control of entangled two-photon states,” Opt. Lett. 29, 376–378 (2004).
    [Crossref] [PubMed]
  26. X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
    [Crossref] [PubMed]
  27. H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
    [Crossref] [PubMed]
  28. T. Roger, J. J. F. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
    [Crossref] [PubMed]
  29. F. M. Miatto, H. Di LorenzoPires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
    [Crossref]
  30. F. M. Miatto, T. Brougham, and A. M. Yao, “Cartesian and ploar Schmidt bases for down-converted photons,” Eur. Phys. J. D 66, 183 (2012).
    [Crossref]
  31. F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011).
    [Crossref]
  32. B. Pors, F. Miatto, G. W’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
    [Crossref]
  33. M. Mclaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
    [Crossref] [PubMed]

2014 (1)

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

2013 (2)

T. Roger, J. J. F. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

A. Shapira, I. Juwiler, and A. Arie, “Tunable nonlinear beam shaping by non-collinear interactions,” Laser Photon. Rev. 7, L25–L29 (2013).
[Crossref]

2012 (9)

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two dimensional nonlinear beam shaping,” Opt. Lett. 37, 2136–2138 (2012).
[Crossref] [PubMed]

H. Ishizuki and T. Taira, “Half-joule output optical-parametric oscillation by using 10-mm-thick periodically poled Mg-doped congruent LiNbO3,” Opt. Express 20, 20002–20010 (2012).
[Crossref] [PubMed]

M. Mclaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
[Crossref] [PubMed]

F. M. Miatto, H. Di LorenzoPires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
[Crossref]

F. M. Miatto, T. Brougham, and A. M. Yao, “Cartesian and ploar Schmidt bases for down-converted photons,” Eur. Phys. J. D 66, 183 (2012).
[Crossref]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref] [PubMed]

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A. 86, 012334 (2012).
[Crossref]

V. D. Salakhutdinov, E. R. Eliel, and W. Loffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108, 173604 (2012).
[Crossref] [PubMed]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

2011 (5)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83, 053829 (2011).
[Crossref]

F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011).
[Crossref]

B. Pors, F. Miatto, G. W’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
[Crossref] [PubMed]

2010 (2)

H. Di LorenzoPires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref] [PubMed]

2009 (1)

T. Ellenbogen, N. V. Bloch, A. G. Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[Crossref]

2008 (2)

X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
[Crossref] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref] [PubMed]

2005 (1)

J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B: Quantum Semiclass. Opt. 7, 235–239 (2005).
[Crossref]

2004 (1)

2003 (1)

S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
[Crossref]

2001 (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

J. P. Meyn, C. Laue, R. Knappe, R. Wallenstein, and M. M. Fejer, “Fabrication of periodically poled lithium tantalate for UV generation with diode lasers,” Appl. Phys. B: Lasers Opt. 73, 111–114 (2001).
[Crossref]

2000 (1)

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

1998 (1)

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

1992 (2)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron 28, 2631–2654 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

1988 (1)

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[Crossref] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosencond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Agarwal, G. S.

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83, 053829 (2011).
[Crossref]

Agnew, M.

Alexandrescu, A.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Alley, C. O.

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolsky-Rosen-Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[Crossref] [PubMed]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

Arie, A.

A. Shapira, I. Juwiler, and A. Arie, “Tunable nonlinear beam shaping by non-collinear interactions,” Laser Photon. Rev. 7, L25–L29 (2013).
[Crossref]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref] [PubMed]

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two dimensional nonlinear beam shaping,” Opt. Lett. 37, 2136–2138 (2012).
[Crossref] [PubMed]

T. Ellenbogen, N. V. Bloch, A. G. Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[Crossref]

Barnett, S. M.

F. M. Miatto, H. Di LorenzoPires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
[Crossref]

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A. 86, 012334 (2012).
[Crossref]

F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Berger, V.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

Bloch, N. V.

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref] [PubMed]

T. Ellenbogen, N. V. Bloch, A. G. Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[Crossref]

Boyd, R. W.

M. Mclaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
[Crossref] [PubMed]

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83, 053829 (2011).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref] [PubMed]

Broderick, N. G. R.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Brougham, T.

F. M. Miatto, T. Brougham, and A. M. Yao, “Cartesian and ploar Schmidt bases for down-converted photons,” Eur. Phys. J. D 66, 183 (2012).
[Crossref]

Buller, G. S.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron 28, 2631–2654 (1992).
[Crossref]

Carrasco, S.

Dada, A. C.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

Di LorenzoPires, H.

F. M. Miatto, H. Di LorenzoPires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
[Crossref]

H. Di LorenzoPires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
[Crossref]

Eliel, E. R.

V. D. Salakhutdinov, E. R. Eliel, and W. Loffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108, 173604 (2012).
[Crossref] [PubMed]

B. Pors, F. Miatto, G. W’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

Ellenbogen, T.

T. Ellenbogen, N. V. Bloch, A. G. Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3, 395–398 (2009).
[Crossref]

Faccio, D.

T. Roger, J. J. F. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

Fejer, M. M.

J. P. Meyn, C. Laue, R. Knappe, R. Wallenstein, and M. M. Fejer, “Fabrication of periodically poled lithium tantalate for UV generation with diode lasers,” Appl. Phys. B: Lasers Opt. 73, 111–114 (2001).
[Crossref]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron 28, 2631–2654 (1992).
[Crossref]

Fickler, R.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

Florijn, H. C. B.

H. Di LorenzoPires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
[Crossref]

Forbes, A.

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

M. Mclaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
[Crossref] [PubMed]

Franke-Arnold, S.

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A. 86, 012334 (2012).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref] [PubMed]

Giovannini, D.

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A. 86, 012334 (2012).
[Crossref]

Gong, Y. X.

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
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T. Roger, J. J. F. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
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C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosencond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
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B. Pors, F. Miatto, G. W’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
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H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
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M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron 28, 2631–2654 (1992).
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A. Shapira, I. Juwiler, and A. Arie, “Tunable nonlinear beam shaping by non-collinear interactions,” Laser Photon. Rev. 7, L25–L29 (2013).
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N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
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A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two dimensional nonlinear beam shaping,” Opt. Lett. 37, 2136–2138 (2012).
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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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J. P. Meyn, C. Laue, R. Knappe, R. Wallenstein, and M. M. Fejer, “Fabrication of periodically poled lithium tantalate for UV generation with diode lasers,” Appl. Phys. B: Lasers Opt. 73, 111–114 (2001).
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M. Mclaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
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H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
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X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
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V. D. Salakhutdinov, E. R. Eliel, and W. Loffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108, 173604 (2012).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron 28, 2631–2654 (1992).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
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C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosencond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
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M. Mclaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
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M. Mclaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
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J. P. Meyn, C. Laue, R. Knappe, R. Wallenstein, and M. M. Fejer, “Fabrication of periodically poled lithium tantalate for UV generation with diode lasers,” Appl. Phys. B: Lasers Opt. 73, 111–114 (2001).
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M. Mclaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
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B. Pors, F. Miatto, G. W’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
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F. M. Miatto, H. Di LorenzoPires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
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F. M. Miatto, T. Brougham, and A. M. Yao, “Cartesian and ploar Schmidt bases for down-converted photons,” Eur. Phys. J. D 66, 183 (2012).
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F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011).
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X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
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J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B: Quantum Semiclass. Opt. 7, 235–239 (2005).
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S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
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Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosencond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
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M. Mclaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
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M. Mclaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
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J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A. 86, 012334 (2012).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
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S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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Pors, B.

B. Pors, F. Miatto, G. W’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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Richardson, D. J.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
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T. Roger, J. J. F. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
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J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A. 86, 012334 (2012).
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J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
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N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
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M. Mclaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014)
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M. Mclaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
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V. D. Salakhutdinov, E. R. Eliel, and W. Loffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108, 173604 (2012).
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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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A. Shapira, I. Juwiler, and A. Arie, “Tunable nonlinear beam shaping by non-collinear interactions,” Laser Photon. Rev. 7, L25–L29 (2013).
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N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
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A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two dimensional nonlinear beam shaping,” Opt. Lett. 37, 2136–2138 (2012).
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N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
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N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
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A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two dimensional nonlinear beam shaping,” Opt. Lett. 37, 2136–2138 (2012).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
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Taira, T.

Torner, L.

J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B: Quantum Semiclass. Opt. 7, 235–239 (2005).
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J. P. Torres, A. Alexandrescu, S. Carrasco, and L. Torner, “Quasi-phase-matching engineering for spatial control of entangled two-photon states,” Opt. Lett. 29, 376–378 (2004).
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Torres, J. P.

J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B: Quantum Semiclass. Opt. 7, 235–239 (2005).
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J. P. Torres, A. Alexandrescu, S. Carrasco, and L. Torner, “Quasi-phase-matching engineering for spatial control of entangled two-photon states,” Opt. Lett. 29, 376–378 (2004).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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F. M. Miatto, H. Di LorenzoPires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
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H. Di LorenzoPires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
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Walborn, S. P.

S. P. Walborn, S. Padua, and C. H. Monken, “Hyperentanglement-assisted Bell-state analysis,” Phys. Rev. A 68, 042313 (2003).
[Crossref]

Wallenstein, R.

J. P. Meyn, C. Laue, R. Knappe, R. Wallenstein, and M. M. Fejer, “Fabrication of periodically poled lithium tantalate for UV generation with diode lasers,” Appl. Phys. B: Lasers Opt. 73, 111–114 (2001).
[Crossref]

Walmsley, I. A.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref] [PubMed]

Wang, J. F.

X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. Uren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Woerdman, J. P.

B. Pors, F. Miatto, G. W’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Wright, E. M.

T. Roger, J. J. F. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

Xie, Z. D.

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
[Crossref] [PubMed]

X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
[Crossref] [PubMed]

Xu, P.

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
[Crossref] [PubMed]

X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
[Crossref] [PubMed]

Yao, A. M.

F. M. Miatto, T. Brougham, and A. M. Yao, “Cartesian and ploar Schmidt bases for down-converted photons,” Eur. Phys. J. D 66, 183 (2012).
[Crossref]

F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[Crossref] [PubMed]

Yu, X. Q.

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
[Crossref] [PubMed]

X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
[Crossref] [PubMed]

Zeilinger, A.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Zhang, C.

H. Y. Leng, X. Q. Yu, Y. X. Gong, P. Xu, Z. D. Xie, H. Jin, C. Zhang, and S. N. Zhu, “On-chip steering of entangled photons in nonlinear photonic crystals,” Nat. Commun..  2, 429 (2011).
[Crossref] [PubMed]

Zhao, J. S.

X. Q. Yu, P. Xu, Z. D. Xie, J. F. Wang, H. Y. Leng, J. S. Zhao, S. N. Zhu, and N. B. Ming, “Transforming spatial entanglement using a domain-engineering technique,” Phys. Rev. Lett. 101, 233601 (2008).
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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the fork-shaped nonlinear photonic crystal and the momentum conservation of the tilted SPDC.

Fig. 2
Fig. 2

The poling period, whose reciprocal vector is the first-order, as a function of θ. The parameter we used in the discussion is annotated. Inset graph shows the k− vector diagram.

Fig. 3
Fig. 3

Correlations between ls and li for different values of lc, lc = 2, lc = 6, lc = 12, with ps = pi = 0.

Fig. 4
Fig. 4

The Schmidt number (K) and I-concurrence (C) as a function of lc for ps = pi = 0. The smooth fit lines are to guide the eye.

Fig. 5
Fig. 5

The Schmidt number K as a function of lc for different values of the radial indices ps (pi). Inset, the correlation distribution between ls and li for two different cases: lc = 0 and lc = 20 with ps = pi = 8. The smooth fit lines are to guide the eye.

Equations (20)

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L G p , l = 2 p ! π ( | l | + p ) ! 1 ω ( z ) ( 2 ρ ω ( z ) ) | l | L p | l | ( 2 ρ 2 ω 2 ( z ) ) exp ( ρ 2 ω 2 ( z ) ) exp ( i l φ ) exp [ i k 0 r 2 z 2 ( z 2 + z R 2 ) ] exp [ i ( 2 p + | l | + 1 ) tan 1 ( z z R ) ] ,
χ ( 2 ) ( x , φ ) = d 22 sgn ( cos [ 2 π x / Λ + l c φ ] ) ,
H ^ = ε 0 d r χ ( 2 ) E ^ p ( + ) E ^ s ( ) E ^ i ( ) + h . c .
E ^ p = C 0 exp ( ( x cos θ z sin θ ) 2 + y 2 ω 0 2 ) exp [ i ( k p cos θ z + k p sin θ x ω p t ) ] ,
E j ( ) = A j d ω j d q j exp [ i ( q j · ρ + k j z z ω j t ) ] a ^ j + ( q j ) ,
H d ω s d ω i exp [ i ( ω s + ω i ω p ) t ] d q s d q i d ρ d z × exp [ i ( k p cos θ k s z k i z ) z ] exp [ i ( g n + k p sin θ ) ρ cos φ ] exp ( i l c φ ) × exp ( ( ρ cos φ cos θ z sin θ ) 2 + ( ρ sin φ ) 2 ω 0 2 ) exp [ i ( q s + q i ) ρ ] a ^ s + ( q s ) a ^ i + ( q i ) ,
k p sin θ + n 2 π Λ = 0 ,
k p cos θ k s k i = 0 ,
Ψ d q s d q i S i n c [ Δ k z 2 L ] F ( q s , q i ) a ^ s + ( q s ) a ^ i + ( q i ) | v a c > s | v a c > i ,
F ( q s , q i ) = d ρ exp ( i l c φ ) exp ( ρ 2 ω 0 2 ) exp [ i Δ k x ρ cos φ ] exp [ i Δ k y ρ sin φ ] = ρ d ρ exp ( ρ 2 ω 0 2 ) exp [ i l c ( φ + π / 2 ) ] J l c ( ρ Δ k x 2 + Δ k y 2 ) = F ( Δ k ) exp [ i l c φ ] ,
C p s p i l s l i d q s d q i S i n c [ Δ k z 2 L ] F ( Δ k ) exp [ i l c φ ] L G l s p s * ( q s ) L G l i p i * ( q i ) ,
C p s p i l s l i d ρ exp [ ρ 2 ω 0 2 ] exp [ i l c φ ] L G l s p s * ( ρ ) L G l i p i * ( ρ ) .
0 2 π d φ exp [ i ( l c l s l i ) φ ] = 2 π δ l c , l s + l i ,
C p s p i l s l i δ l c , l s + l i ( 2 3 ) ξ + 1 p s ! p i ! ( p s + | l s | ) ! ( p i + | l i | ) ! i = 0 p s j = 0 p i ( 2 3 ) i + j × ( ξ + i + j ) ! ( p s i ) ! ( | l s + i | ) ! i ! ( p i j ) ! ( | l i + j | ) ! j ! ,
k j · r j = ( k j l sin θ q j cos θ ) x + q j y + ( k j l cos θ + q j sin θ ) z ,
Δ k x = Δ k x 0 + ( q s 2 2 k s + q i 2 2 k i ) sin θ + ( q s + q i ) cos θ ,
Δ k y = q s y + q i y ,
Δ k z = Δ k z 0 + ( q s 2 2 k s + q i 2 2 k i ) cos θ ( q s + q i ) sin θ ,
Δ k x 0 = k p sin ( θ + α ) ( k s + k i ) sin θ 2 π Λ = 0
Δ k z 0 = k p cos ( θ + α ) ( k s + k i ) cos θ = 0.

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