Abstract

Gaussian spatial-polarization entanglement in a coherent vectorial paraxial light field is studied. Detection of spatial-polarization entanglement through fringe movement on rotation of a linear polarizer, with the light field passing through the polarizer, is outlined. The fringe movement is shown to be a sufficient condition for the detection of spatial-polarization entanglement in coherent paraxial vector light fields. Two Gaussian light fields with a small relative tilt but with significant spatial overlap and with orthogonal polarizations are shown to possess close to 1 ebit of spatial-polarization entanglement. Tunable Gaussian spatial-polarization entanglement is experimentally demonstrated in a folded Mach–Zehnder interferometer.

© 2020 Optical Society of America

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2019 (1)

A. Selyem, C. Rosales-Guzmán, S. Croke, A. Forbes, and S. Franke-Arnold, “Basis independent tomography of complex vectorial light fields by Stokes projections,” Phys. Rev. A 100, 063842 (2019).
[Crossref]

2018 (6)

E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes, “Entanglement beating in free space through spin-orbit coupling,” Light Sci. Appl. 7, 18009 (2018).
[Crossref]

F. D. Zela, “Optical approach to concurrence and polarization,” Opt. Lett. 43, 2603–2606 (2018).
[Crossref]

E. Otte, I. Nape, C. Rosales-Guzmán, A. Vallés, C. Denz, and A. Forbes, “Recovery of nonseparability in self-healing vector Bessel beams,” Phys. Rev. A 98, 053818 (2018).
[Crossref]

M. H. M. Passos, W. F. Balthazar, J. A. de Barros, C. E. R. Souza, A. Z. Khoury, and J. A. O. Huguenin, “Classical analog of quantum contextuality in spin-orbit laser modes,” Phys. Rev. A 98, 062116 (2018).
[Crossref]

S. Chaturvedi and N. Mukunda, “Entanglement and complete positivity: relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2018).
[Crossref]

S. Asokan and J. S. Ivan, “Radial-angular entanglement in Laguerre–Gaussian mode superpositions,” J. Opt. Soc. Am. A 35, 785–793 (2018).
[Crossref]

2017 (3)

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Light. Technol. 36, 292–301 (2017).
[Crossref]

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
[Crossref]

C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint, “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017).
[Crossref]

2016 (9)

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

P. Li, B. Wang, and X. Zhang, “High-dimensional encoding based on classical nonseparability,” Opt. Express 24, 15143–15159 (2016).
[Crossref]

B. P. da Silva, M. A. Leal, C. E. R. Souza, E. F. Galvão, and A. Z. Khoury, “Spin-orbit laser mode transfer via a classical analogue of quantum teleportation,” J. Phys. B: At. Mol. Opt. Phys. 49, 055501 (2016).
[Crossref]

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Gräfe, M. Heinrich, S. Nolte, M. Duparré, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

W. Balthazar, C. Souza, D. Caetano, E. Galvão, J. Huguenin, and A. Khoury, “Tripartite nonseparability in classical optics,” Opt. Lett. 41, 5797–5800 (2016).
[Crossref]

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
[Crossref]

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94, 030304 (2016).
[Crossref]

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
[Crossref]

2015 (9)

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

S. Prabhakar, S. G. Reddy, A. Aadhi, C. Perumangatt, G. K. Samanta, and R. P. Singh, “Violation of Bell’s inequality for phase-singular beams,” Phys. Rev. A 92, 023822 (2015).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40, 4887–4890 (2015).
[Crossref]

B. Perez-Garcia, J. Francis, M. McLaren, R. I. Hernandez-Aranda, A. Forbes, and T. Konrad, “Quantum computation with classical light: the Deutsch Algorithm,” Phys. Lett. A 379, 1675–1680 (2015).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

2014 (3)

A. F. Abouraddy, K. H. Kagalwala, and B. E. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

2013 (2)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72 (2013).
[Crossref]

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 88, 013830 (2013).
[Crossref]

2011 (1)

X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett 36, 4110–4112 (2011).
[Crossref]

2010 (1)

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

2009 (1)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009).
[Crossref]

2001 (2)

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[Crossref]

P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A 64, 042315 (2001).
[Crossref]

2000 (1)

A. Uhlmann, “Fidelity and concurrence of conjugated states,” Phys. Rev. A 62, 032307 (2000).
[Crossref]

1998 (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

1996 (1)

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824 (1996).
[Crossref]

1993 (1)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref]

1988 (1)

F. Gori, G. Guattari, C. Palma, and C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[Crossref]

Aadhi, A.

S. Prabhakar, S. G. Reddy, A. Aadhi, C. Perumangatt, G. K. Samanta, and R. P. Singh, “Violation of Bell’s inequality for phase-singular beams,” Phys. Rev. A 92, 023822 (2015).
[Crossref]

Abouraddy, A. F.

C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint, “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72 (2013).
[Crossref]

Agarwal, G. S.

P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 88, 013830 (2013).
[Crossref]

Aiello, A.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Gräfe, M. Heinrich, S. Nolte, M. Duparré, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Akhouayri, H.

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

Al Qasimi, A.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
[Crossref]

Alfano, R. R.

Ali, H.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
[Crossref]

Alonso, M. A.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
[Crossref]

Asokan, S.

Balthazar, W.

Balthazar, W. F.

M. H. M. Passos, W. F. Balthazar, J. A. de Barros, C. E. R. Souza, A. Z. Khoury, and J. A. O. Huguenin, “Classical analog of quantum contextuality in spin-orbit laser modes,” Phys. Rev. A 98, 062116 (2018).
[Crossref]

Banzer, P.

Bennett, C. H.

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824 (1996).
[Crossref]

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref]

Berg-Johansen, S.

Borges, C. V. S.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref]

Brüning, R.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Gräfe, M. Heinrich, S. Nolte, M. Duparré, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

Bužek, V.

P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A 64, 042315 (2001).
[Crossref]

Caetano, D.

Carvacho, G.

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B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Light. Technol. 36, 292–301 (2017).
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B. Perez-Garcia, J. Francis, M. McLaren, R. I. Hernandez-Aranda, A. Forbes, and T. Konrad, “Quantum computation with classical light: the Deutsch Algorithm,” Phys. Lett. A 379, 1675–1680 (2015).
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A. Selyem, C. Rosales-Guzmán, S. Croke, A. Forbes, and S. Franke-Arnold, “Basis independent tomography of complex vectorial light fields by Stokes projections,” Phys. Rev. A 100, 063842 (2019).
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Galvão, E. F.

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C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
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M. H. M. Passos, W. F. Balthazar, J. A. de Barros, C. E. R. Souza, A. Z. Khoury, and J. A. O. Huguenin, “Classical analog of quantum contextuality in spin-orbit laser modes,” Phys. Rev. A 98, 062116 (2018).
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C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
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B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
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B. Perez-Garcia, J. Francis, M. McLaren, R. I. Hernandez-Aranda, A. Forbes, and T. Konrad, “Quantum computation with classical light: the Deutsch Algorithm,” Phys. Lett. A 379, 1675–1680 (2015).
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B. P. da Silva, M. A. Leal, C. E. R. Souza, E. F. Galvão, and A. Z. Khoury, “Spin-orbit laser mode transfer via a classical analogue of quantum teleportation,” J. Phys. B: At. Mol. Opt. Phys. 49, 055501 (2016).
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A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
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F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
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Little, B.

Little, B. J.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
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P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 88, 013830 (2013).
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J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
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[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

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V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94, 030304 (2016).
[Crossref]

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N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

McLaren, M.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
[Crossref]

B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
[Crossref]

B. Perez-Garcia, J. Francis, M. McLaren, R. I. Hernandez-Aranda, A. Forbes, and T. Konrad, “Quantum computation with classical light: the Deutsch Algorithm,” Phys. Lett. A 379, 1675–1680 (2015).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

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P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A 64, 042315 (2001).
[Crossref]

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Mouane, O.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
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P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
[Crossref]

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S. Chaturvedi and N. Mukunda, “Entanglement and complete positivity: relevance and manifestations in classical scalar wave optics,” Fortschr. Phys. 66, 1700077 (2018).
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E. Otte, I. Nape, C. Rosales-Guzmán, A. Vallés, C. Denz, and A. Forbes, “Recovery of nonseparability in self-healing vector Bessel beams,” Phys. Rev. A 98, 053818 (2018).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Light. Technol. 36, 292–301 (2017).
[Crossref]

Ndagano, B.

E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes, “Entanglement beating in free space through spin-orbit coupling,” Light Sci. Appl. 7, 18009 (2018).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Light. Technol. 36, 292–301 (2017).
[Crossref]

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
[Crossref]

B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
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Okoro, C.

Ornigotti, M.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Gräfe, M. Heinrich, S. Nolte, M. Duparré, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
[Crossref]

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E. Otte, I. Nape, C. Rosales-Guzmán, A. Vallés, C. Denz, and A. Forbes, “Recovery of nonseparability in self-healing vector Bessel beams,” Phys. Rev. A 98, 053818 (2018).
[Crossref]

E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes, “Entanglement beating in free space through spin-orbit coupling,” Light Sci. Appl. 7, 18009 (2018).
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[Crossref]

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M. H. M. Passos, W. F. Balthazar, J. A. de Barros, C. E. R. Souza, A. Z. Khoury, and J. A. O. Huguenin, “Classical analog of quantum contextuality in spin-orbit laser modes,” Phys. Rev. A 98, 062116 (2018).
[Crossref]

Peres, A.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref]

Perez-Garcia, B.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
[Crossref]

B. Perez-Garcia, J. Francis, M. McLaren, R. I. Hernandez-Aranda, A. Forbes, and T. Konrad, “Quantum computation with classical light: the Deutsch Algorithm,” Phys. Lett. A 379, 1675–1680 (2015).
[Crossref]

Perumangatt, C.

S. Prabhakar, S. G. Reddy, A. Aadhi, C. Perumangatt, G. K. Samanta, and R. P. Singh, “Violation of Bell’s inequality for phase-singular beams,” Phys. Rev. A 92, 023822 (2015).
[Crossref]

Piccirillo, B.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94, 030304 (2016).
[Crossref]

Prabhakar, S.

S. Prabhakar, S. G. Reddy, A. Aadhi, C. Perumangatt, G. K. Samanta, and R. P. Singh, “Violation of Bell’s inequality for phase-singular beams,” Phys. Rev. A 92, 023822 (2015).
[Crossref]

Qian, X.-F.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
[Crossref]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett 36, 4110–4112 (2011).
[Crossref]

Reddy, S. G.

S. Prabhakar, S. G. Reddy, A. Aadhi, C. Perumangatt, G. K. Samanta, and R. P. Singh, “Violation of Bell’s inequality for phase-singular beams,” Phys. Rev. A 92, 023822 (2015).
[Crossref]

Rosales-Guzman, C.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Light. Technol. 36, 292–301 (2017).
[Crossref]

Rosales-Guzmán, C.

A. Selyem, C. Rosales-Guzmán, S. Croke, A. Forbes, and S. Franke-Arnold, “Basis independent tomography of complex vectorial light fields by Stokes projections,” Phys. Rev. A 100, 063842 (2019).
[Crossref]

E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes, “Entanglement beating in free space through spin-orbit coupling,” Light Sci. Appl. 7, 18009 (2018).
[Crossref]

E. Otte, I. Nape, C. Rosales-Guzmán, A. Vallés, C. Denz, and A. Forbes, “Recovery of nonseparability in self-healing vector Bessel beams,” Phys. Rev. A 98, 053818 (2018).
[Crossref]

B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
[Crossref]

Roux, F. S.

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
[Crossref]

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P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A 64, 042315 (2001).
[Crossref]

Saleh, B. E.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72 (2013).
[Crossref]

Samanta, G. K.

S. Prabhakar, S. G. Reddy, A. Aadhi, C. Perumangatt, G. K. Samanta, and R. P. Singh, “Violation of Bell’s inequality for phase-singular beams,” Phys. Rev. A 92, 023822 (2015).
[Crossref]

Samlan, C.

C. Samlan and N. K. Viswanathan, “Quantifying classical entanglement using polarimetry: Spatially-inhomogeneously polarized beams,” arXiv:1506.07112 (2015).

Sandeau, N.

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

Sciarrino, F.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94, 030304 (2016).
[Crossref]

Selyem, A.

A. Selyem, C. Rosales-Guzmán, S. Croke, A. Forbes, and S. Franke-Arnold, “Basis independent tomography of complex vectorial light fields by Stokes projections,” Phys. Rev. A 100, 063842 (2019).
[Crossref]

Singh, R. P.

S. Prabhakar, S. G. Reddy, A. Aadhi, C. Perumangatt, G. K. Samanta, and R. P. Singh, “Violation of Bell’s inequality for phase-singular beams,” Phys. Rev. A 92, 023822 (2015).
[Crossref]

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C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824 (1996).
[Crossref]

Souza, C.

Souza, C. E. R.

M. H. M. Passos, W. F. Balthazar, J. A. de Barros, C. E. R. Souza, A. Z. Khoury, and J. A. O. Huguenin, “Classical analog of quantum contextuality in spin-orbit laser modes,” Phys. Rev. A 98, 062116 (2018).
[Crossref]

B. P. da Silva, M. A. Leal, C. E. R. Souza, E. F. Galvão, and A. Z. Khoury, “Spin-orbit laser mode transfer via a classical analogue of quantum teleportation,” J. Phys. B: At. Mol. Opt. Phys. 49, 055501 (2016).
[Crossref]

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R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[Crossref]

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

Sroor, H.

Stiller, B.

Szameit, A.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Gräfe, M. Heinrich, S. Nolte, M. Duparré, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
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[Crossref]

S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
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F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
[Crossref]

Toussaint, K. C.

Uhlmann, A.

A. Uhlmann, “Fidelity and concurrence of conjugated states,” Phys. Rev. A 62, 032307 (2000).
[Crossref]

Vallés, A.

E. Otte, I. Nape, C. Rosales-Guzmán, A. Vallés, C. Denz, and A. Forbes, “Recovery of nonseparability in self-healing vector Bessel beams,” Phys. Rev. A 98, 053818 (2018).
[Crossref]

Vamivakas, A. N.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
[Crossref]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

Vetter, C.

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Gräfe, M. Heinrich, S. Nolte, M. Duparré, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

Viswanathan, N. K.

C. Samlan and N. K. Viswanathan, “Quantifying classical entanglement using polarimetry: Spatially-inhomogeneously polarized beams,” arXiv:1506.07112 (2015).

Vitelli, C.

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94, 030304 (2016).
[Crossref]

Wang, B.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), p. 349.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

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C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824 (1996).
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C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
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Zhang, X.

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B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
[Crossref]

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D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Gräfe, M. Heinrich, S. Nolte, M. Duparré, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
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B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Light. Technol. 36, 292–301 (2017).
[Crossref]

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J. Phys. B: At. Mol. Opt. Phys. (1)

B. P. da Silva, M. A. Leal, C. E. R. Souza, E. F. Galvão, and A. Z. Khoury, “Spin-orbit laser mode transfer via a classical analogue of quantum teleportation,” J. Phys. B: At. Mol. Opt. Phys. 49, 055501 (2016).
[Crossref]

Laser Photon. Rev. (1)

D. Guzman-Silva, R. Brüning, F. Zimmermann, C. Vetter, M. Gräfe, M. Heinrich, S. Nolte, M. Duparré, A. Aiello, M. Ornigotti, and A. Szameit, “Demonstration of local teleportation using classical entanglement,” Laser Photon. Rev. 10, 317–321 (2016).
[Crossref]

Light Sci. Appl. (1)

E. Otte, C. Rosales-Guzmán, B. Ndagano, C. Denz, and A. Forbes, “Entanglement beating in free space through spin-orbit coupling,” Light Sci. Appl. 7, 18009 (2018).
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Nat. Photonics (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72 (2013).
[Crossref]

Nat. Phys. (1)

B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397 (2017).
[Crossref]

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A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
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F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
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Opt. Lett. (5)

Optica (4)

Phys. Lett. A (1)

B. Perez-Garcia, J. Francis, M. McLaren, R. I. Hernandez-Aranda, A. Forbes, and T. Konrad, “Quantum computation with classical light: the Deutsch Algorithm,” Phys. Lett. A 379, 1675–1680 (2015).
[Crossref]

Phys. Rev. A (13)

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

E. Otte, I. Nape, C. Rosales-Guzmán, A. Vallés, C. Denz, and A. Forbes, “Recovery of nonseparability in self-healing vector Bessel beams,” Phys. Rev. A 98, 053818 (2018).
[Crossref]

M. H. M. Passos, W. F. Balthazar, J. A. de Barros, C. E. R. Souza, A. Z. Khoury, and J. A. O. Huguenin, “Classical analog of quantum contextuality in spin-orbit laser modes,” Phys. Rev. A 98, 062116 (2018).
[Crossref]

V. D’Ambrosio, G. Carvacho, F. Graffitti, C. Vitelli, B. Piccirillo, L. Marrucci, and F. Sciarrino, “Entangled vector vortex beams,” Phys. Rev. A 94, 030304 (2016).
[Crossref]

A. Selyem, C. Rosales-Guzmán, S. Croke, A. Forbes, and S. Franke-Arnold, “Basis independent tomography of complex vectorial light fields by Stokes projections,” Phys. Rev. A 100, 063842 (2019).
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P. Chowdhury, A. S. Majumdar, and G. S. Agarwal, “Nonlocal continuous-variable correlations and violation of Bell’s inequality for light beams with topological singularities,” Phys. Rev. A 88, 013830 (2013).
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S. Prabhakar, S. G. Reddy, A. Aadhi, C. Perumangatt, G. K. Samanta, and R. P. Singh, “Violation of Bell’s inequality for phase-singular beams,” Phys. Rev. A 92, 023822 (2015).
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R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
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C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
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M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824 (1996).
[Crossref]

P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A 64, 042315 (2001).
[Crossref]

A. Uhlmann, “Fidelity and concurrence of conjugated states,” Phys. Rev. A 62, 032307 (2000).
[Crossref]

Phys. Rev. Lett. (2)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895 (1993).
[Crossref]

X.-F. Qian, T. Malhotra, A. N. Vamivakas, and J. H. Eberly, “Coherence constraints and the last hidden optical coherence,” Phys. Rev. Lett. 117, 153901 (2016).
[Crossref]

Phys. Scr. (1)

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics-emerging links,” Phys. Scr. 91, 063003 (2016).
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K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and B. E. Saleh, “Optical coherency matrix tomography,” Sci. Rep. 5, 15333 (2015).
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Other (5)

C. Samlan and N. K. Viswanathan, “Quantifying classical entanglement using polarimetry: Spatially-inhomogeneously polarized beams,” arXiv:1506.07112 (2015).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2004).

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement, Vol. 35 of International Series of Monographs in Natural Philosophy (Elsevier, 2013).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), p. 349.

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Figures (4)

Fig. 1.
Fig. 1. Schematic diagram of a folded Mach–Zehnder interferometer setup using a PBS.
Fig. 2.
Fig. 2. (a) Plots the variation of Gaussian spatial-polarization entanglement $E(\tilde \Psi )$ with relative tilt ${\delta _\alpha }$ for various polarizer angles $\phi$. (b) Plots the variation of concurrence ${C^2}({\rho _p})$ (solid line) and degree of polarization ${S^2}({\rho _p})$ (dashed line) with polarizer angle $\phi$ for relative tilt ${\delta _\alpha } = 2.4 \times {10^{ - 4}}$ radians. The solid line in (c) plots the variation of Gaussian spatial-polarization entanglement $E(\tilde \Psi )$ with polarizer angle $\phi$ for the relative tilt ${\delta _\alpha } = 3 \times {10^{ - 4}}$ radians, the dashed line plots the same for ${\delta _\alpha } = 2.4 \times {10^{ - 4}}$ radians, and the dotted line for ${\delta _\alpha } = 1.5 \times {10^{ - 4}}$ radians.
Fig. 3.
Fig. 3. (a1)–(a5) plot the fringe pattern obtained on passage of the recombined light field emerging from the PBS through polarizer P3 alone (refer to Fig. 1). P3 is rotated from 70 to 110 degrees in steps of 10 degrees. (c1)–(c5) plot the fringe pattern obtained on passage of the recombined light field emerging from the PBS through polarizers P2 and P3, with polarizer P2 (at 45 degrees) inserted between P3 and the PBS. (b1)–(b5) and (d1)–(d5) plot the fringes shown in (a1)–(a5) and (c1)–(c5) reoriented through a 45 degree coordinate transformation in order to make the fringe shift or fringe stationarity more explicit. The solid line of (e1) plots the diagonal entries of (a1), the dotted line plots the diagonal entries of (a3), and the dashed line plots the diagonal entries of (a5). Similarly, the solid line of (e2) plots the diagonal entries of (c1), the dotted line plots the diagonal entries of (c3), and the dashed line plots the diagonal entries of (c5). In all the experiments, the initial polarizer P1 is fixed at 45 degrees.
Fig. 4.
Fig. 4. (a) plots the variation of estimated Stokes parameters ${S_1}$ (solid line), ${S_2}$ (dashed line), and ${S_3}$ (dotted line) with polarizer angle $\phi$ for a particular tilt as obtained from ${\rho _p}$. (b) plots the variation of ${C^2}({\rho _p})$ (solid line) and ${S^2}({\rho _p})$) (dashed line) with polarizer angle $\phi$ for a particular tilt. The $\circ$ in (c) plots the estimated spatial-polarization entanglement $E(\tilde \Psi )$ for three different tilts (shown by the solid, dashed, and dotted lines) when P2 is not inserted after the PBS for varying $\phi$. The $\Box$ in (c) plots the estimated spatial-polarization entanglement $E(\tilde \Psi )$ for a particular tilt when P2 is inserted after the PBS (see Fig. 1).

Equations (36)

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

Ψ ~ ( x , y ) = a 1 ψ 1 ( x , y ) | 0 + a 2 ψ 2 ( x , y ) | 1 ,
Ψ ~ ( x , y ) = [ a 1 ψ 1 ( x , y ) a 2 ψ 2 ( x , y ) ] ,
Ψ ~ ( x , y ) = [ a 1 a 2 ] ψ 0 ( x , y ) .
[ C θ 2 S θ C θ S θ C θ S θ 2 ] [ a 1 a 2 ] ψ 0 ( x , y ) = ( a 1 C θ + a 2 S θ ) [ C θ S θ ] ψ 0 ( x , y ) .
I ( x , y ) = ( | a 1 | 2 C θ 2 + | a 2 | 2 S θ 2 + a 1 a 2 S θ C θ + a 2 a 1 S θ C θ ) × | ψ 0 ( x , y ) | 2 = | a 1 C θ + a 2 S θ | 2 | ψ 0 ( x , y ) | 2 .
[ C θ 2 S θ C θ S θ C θ S θ 2 ] [ a 1 ψ 1 ( x , y ) a 2 ψ 2 ( x , y ) ] = ( a 1 C θ ψ 1 ( x , y ) + a 2 S θ ψ 2 ( x , y ) ) [ C θ S θ ] .
I θ ( x , y ) = | a 1 | 2 C θ 2 | ψ 1 ( x , y ) | 2 + | a 2 | 2 S θ 2 | ψ 2 ( x , y ) | 2 + a 1 a 2 S θ C θ ψ 1 ( x , y ) ψ 2 ( x , y ) + a 1 a 2 S θ C θ ψ 1 ( x , y ) ψ 2 ( x , y ) = | a 1 C θ ψ 1 ( x , y ) + a 2 S θ ψ 2 ( x , y ) | 2 .
Ψ ~ ( r , χ ) = [ ψ ( r ) e i χ ψ ( r ) e i χ ] ,
[ C θ 2 S θ C θ S θ C θ S θ 2 ] [ ψ ( r ) e i χ ψ ( r ) e i χ ] = ψ ( r ) ( C θ e i χ + S θ e i χ ) [ C θ S θ ] .
I ( r , χ , θ ) = ( 1 + S 2 θ C 2 χ ) | ψ ( r ) | 2 .
ρ ( x , y ) = Ψ ~ ( x , y ) Ψ ~ ( x , y ) = [ | a 1 | 2 | ψ 1 ( x , y ) | 2 a 1 a 2 ψ 1 ( x , y ) ψ 2 ( x , y ) a 1 a 2 ψ 1 ( x , y ) ψ 2 ( x , y ) | a 2 | 2 | ψ 2 ( x , y ) | 2 ] .
ρ p = [ ρ 00 ρ 01 ρ 10 ρ 11 ] = [ | a 1 | 2 | ψ 1 ( x , y ) | 2 d x d y a 1 a 2 ψ 1 ( x , y ) ψ 2 ( x , y ) d x d y a 1 a 2 ψ 1 ( x , y ) ψ 2 ( x , y ) d x d y | a 2 | 2 | ψ 2 ( x , y ) | 2 d x d y ] .
S 0 = ρ 00 + ρ 11 ,
S 1 = ρ 00 ρ 11 ,
S 2 = ρ 01 + ρ 10 ,
S 3 = i ( ρ 10 ρ 01 ) .
S ( ρ p ) = λ + λ .
C ( ρ p ) = 2 λ + λ .
E ( Ψ ~ ) = λ + log ( λ + ) λ log ( λ ) ,
Ψ ~ θ ( x , y ) = [ C θ 2 S θ C θ S θ C θ S θ 2 ] [ 1 0 0 i ] [ a 1 ψ 1 ( x , y ) a 2 ψ 2 ( x , y ) ] = ( a 1 C θ ψ 1 ( x , y ) + i a 2 S θ ψ 2 ( x , y ) ) [ C θ S θ ] .
I ~ θ ( x , y ) = | a 1 | 2 C θ 2 | ψ 1 ( x , y ) | 2 + | a 2 | 2 S θ 2 | ψ 2 ( x , y ) | 2 i a 1 a 2 S θ C θ ψ 1 ( x , y ) ψ 2 ( x , y ) + i a 1 a 2 S θ C θ ψ 1 ( x , y ) ψ 2 ( x , y ) = | a 1 C θ ψ 1 ( x , y ) + i a 2 S θ ψ 2 ( x , y ) | 2 .
ρ 00 = I 0 ( x , y ) d x d y
ρ 01 = ( I 45 ( x , y ) I 135 ( x , y ) ) 2 d x d y + i ( I ~ 45 ( x , y ) I ~ 135 ( x , y ) ) 2 d x d y
ρ 10 = ( I 45 ( x , y ) I 135 ( x , y ) ) 2 d x d y i ( I ~ 45 ( x , y ) I ~ 135 ( x , y ) ) 2 d x d y
ρ 11 = I 90 ( x , y ) d x d y
Ψ ~ ( x , y ) = [ C ϕ S ϕ ] ψ 0 ( x , y ) , w h e r e ψ 0 ( x , y ) = a 0 exp ( ( x 2 + y 2 ) w 0 2 ) .
ψ ( x , y ) = a 0 exp ( ( x 2 + y 2 ) w z 2 ) exp ( i κ ( x 2 + y 2 ) 2 R z ) × exp ( i κ z + i ϕ z ) , w h e r e w z = w 0 1 + ( z z R ) 2 , a n d R z = z ( 1 + ( z R z ) 2 ) .
Ψ ~ ( x , y ) = [ C ϕ ψ 1 ( x , y ) S ϕ ψ 2 ( x , y ) ] ,
ψ 2 ( x , y ) = a 0 exp ( ( ( x + z δ α ) 2 + y 2 ) w z 2 ) × exp ( i κ ( ( x + z δ α ) 2 + y 2 ) 2 R z ) × exp ( i κ ( z x δ α ) + i ϕ z ) .
ρ 00 = C ϕ 2 ,
ρ 01 = C ϕ S ϕ exp ( z 2 δ α 2 2 w 2 κ 2 δ α 2 w 2 8 ( 1 + z R ) 2 i κ z δ α 2 2 ) ,
ρ 10 = C ϕ S ϕ exp ( z 2 δ α 2 2 w 2 κ 2 δ α 2 w 2 8 ( 1 + z R ) 2 + i κ z δ α 2 2 ) ,
ρ 11 = S ϕ 2 .
λ ± = t r ( ρ p ) ± t r 2 ( ρ p ) 4 d e t ( ρ p ) 2 ,
t r ( ρ p ) = 1 ,
d e t ( ρ p ) = C ϕ 2 S ϕ 2 [ 1 exp [ z 2 δ α 2 w z 2 κ 2 δ α 2 w z 2 4 ( 1 z R z ) 2 ] ] .

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