Abstract

Here we employ both dynamic and geometric phase control of light to produce radially modulated vector-vortex modes, the natural modes of optical fibers. We then measure these modes using a vector modal decomposition set-up as well as a tomography measurement, the latter providing a degree of the non-separability of the vector states, akin to an entanglement measure for quantum states. We demonstrate the versatility of the approach by creating the natural modes of a step-index fiber, which are known to exhibit strong mode coupling, and measure the modal cross-talk and non-separability decay during propagation. Our approach will be useful in mode division multiplexing schemes for transport of classical and quantum states.

© 2015 Optical Society of America

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References

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2015 (3)

2014 (4)

2013 (7)

S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2, 455–474 (2013).
[Crossref]

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

D. Flamm, C. Schulze, D. Naidoo, S. Schröter, A. Forbes, and M. Duparré, “All-digital holographic tool for mode excitation and analysis in optical fibers,” J. Lightwave Technol. 31, 1023–1032 (2013).
[Crossref]

A. Dudley, Y. Li, T. Mhlanga, M. Escuti, and A. Forbes, “Generating and measuring nondiffracting vector bessel beams,” Opt. Lett. 38, 3429–3432 (2013).
[Crossref] [PubMed]

J. Carpenter, C. Xiong, M. J. Collins, J. Li, T. F. Krauss, B. J. Eggleton, A. S. Clark, and J. Schröder, “Mode multiplexed single-photon and classical channels in a few-mode fiber,” Opt. Express 21, 28794–28800 (2013).
[Crossref]

2012 (3)

2010 (1)

2009 (4)

2007 (2)

V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
[Crossref]

C. Souza, J. Huguenin, P. Milman, and A. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

2001 (1)

W. K. Wootters, “Entanglement of formation and concurrence,” Quantum Inf. Comput. 1, 27–44 (2001).

Ahmed, N.

A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

N. Ahmed, M. P. Lavery, H. Huang, G. Xie, Y. Ren, Y. Yan, and A. E. Willner, “Experimental demonstration of obstruction-tolerant free-space transmission of two 50-Gbaud QPSK data channels using Bessel beams carrying orbital angular momentum” In European Conference on Optical Communication (IEEE, 2014), pp. 1–3.

Alfano, R. R.

Arrizón, V.

Ashrafi, N.

Ashrafi, S.

Bai, N.

Bao, C.

Barnett, S.

B. Jack, J. Leach, H. Ritsch, S. Barnett, and M. Padgett, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 811, 103024 (2009).
[Crossref]

Bolle, C.

Borchardt, J.

Boyd, R.

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Brunet, C.

Burrows, E. C.

Cao, Y.

Carpenter, J.

Carrada, R.

Clark, A. S.

Collins, M. J.

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Dudley, A.

Duparré, M.

Eggleton, B. J.

Escuti, M.

Escuti, M. J.

Esmaeelpour, M.

Es-siambre, R.-J.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fini, J.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

Flamm, D.

Forbes, A.

Gnauck, A. H.

González, L. A.

Gregg, P.

Huang, H.

A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).

G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “4 20 Gbit/s mode division multiplexing over free space using vector modes and a q -plate mode (de) multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

N. Ahmed, M. P. Lavery, H. Huang, G. Xie, Y. Ren, Y. Yan, and A. E. Willner, “Experimental demonstration of obstruction-tolerant free-space transmission of two 50-Gbaud QPSK data channels using Bessel beams carrying orbital angular momentum” In European Conference on Optical Communication (IEEE, 2014), pp. 1–3.

Huguenin, J.

C. Souza, J. Huguenin, P. Milman, and A. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

Ibrahim, A. H.

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

Ismail, Y.

Jack, B.

B. Jack, J. Leach, H. Ritsch, S. Barnett, and M. Padgett, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 811, 103024 (2009).
[Crossref]

Kahn, J.

Kaiser, T.

Karimi, E.

Khoury, A.

C. Souza, J. Huguenin, P. Milman, and A. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

Kim, J.

Konrad, T.

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the non-separability of classically entangled vector vortex beams,” arXiv http://arxiv.org/abs/1502.02153 (2015).

Krauss, T. F.

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2, 455–474 (2013).
[Crossref]

S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34, 2525–2527 (2009).
[Crossref] [PubMed]

LaRochelle, S.

Lavery, M.

Lavery, M. P.

N. Ahmed, M. P. Lavery, H. Huang, G. Xie, Y. Ren, Y. Yan, and A. E. Willner, “Experimental demonstration of obstruction-tolerant free-space transmission of two 50-Gbaud QPSK data channels using Bessel beams carrying orbital angular momentum” In European Conference on Optical Communication (IEEE, 2014), pp. 1–3.

Lavery, M. P. J.

Leach, J.

B. Jack, J. Leach, H. Ritsch, S. Barnett, and M. Padgett, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 811, 103024 (2009).
[Crossref]

Li, G.

Li, J.

Li, L.

Li, Y.

Lingle, R.

Love, J.

A. W. Snyder and J. Love, “Optical waveguide theory,” 190 (Springer Science & Business Media, 1983).

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Mao, W.

Marrucci, L.

McCurdy, A.

McLaren, M.

A. Trichili, T. Mhlanga, Y. Ismail, F. S. Roux, M. McLaren, M. Zghal, and A. Forbes, “Detection of bessel beams with digital axicons,” Opt. Express 22, 17553–17560 (2014).
[Crossref] [PubMed]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the non-separability of classically entangled vector vortex beams,” arXiv http://arxiv.org/abs/1502.02153 (2015).

Messaddeq, Y.

Mhlanga, T.

Milione, G.

Milman, P.

C. Souza, J. Huguenin, P. Milman, and A. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

Mirhosseini, M.

Molisch, A.

Mumtaz, S.

Naidoo, D.

Nelson, L.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

Nguyen, T. A.

Nolan, D. A.

Padgett, M.

B. Jack, J. Leach, H. Ritsch, S. Barnett, and M. Padgett, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 811, 103024 (2009).
[Crossref]

Panicker, R.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Peckham, D.

Ramachandran, S.

Randel, S.

Ren, Y.

A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).

G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “4 20 Gbit/s mode division multiplexing over free space using vector modes and a q -plate mode (de) multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

N. Ahmed, M. P. Lavery, H. Huang, G. Xie, Y. Ren, Y. Yan, and A. E. Willner, “Experimental demonstration of obstruction-tolerant free-space transmission of two 50-Gbaud QPSK data channels using Bessel beams carrying orbital angular momentum” In European Conference on Optical Communication (IEEE, 2014), pp. 1–3.

Richardson, D.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

Ritsch, H.

B. Jack, J. Leach, H. Ritsch, S. Barnett, and M. Padgett, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 811, 103024 (2009).
[Crossref]

Roux, F. S.

A. Trichili, T. Mhlanga, Y. Ismail, F. S. Roux, M. McLaren, M. Zghal, and A. Forbes, “Detection of bessel beams with digital axicons,” Opt. Express 22, 17553–17560 (2014).
[Crossref] [PubMed]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

Rubano, A.

Ruiz, U.

Rusch, L. A.

Ryf, R.

Schmidt, O. A.

Schröder, J.

Schröter, S.

Schulze, C.

Shemirani, M.

Sierra, A.

Snyder, A. W.

A. W. Snyder and J. Love, “Optical waveguide theory,” 190 (Springer Science & Business Media, 1983).

Souza, C.

C. Souza, J. Huguenin, P. Milman, and A. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[Crossref] [PubMed]

Trichili, A.

Tur, M.

A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Vaity, P.

Wang, J.

A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Willner, A.

A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Willner, A. E.

G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “4 20 Gbit/s mode division multiplexing over free space using vector modes and a q -plate mode (de) multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

N. Ahmed, M. P. Lavery, H. Huang, G. Xie, Y. Ren, Y. Yan, and A. E. Willner, “Experimental demonstration of obstruction-tolerant free-space transmission of two 50-Gbaud QPSK data channels using Bessel beams carrying orbital angular momentum” In European Conference on Optical Communication (IEEE, 2014), pp. 1–3.

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[Crossref]

N. Ahmed, M. P. Lavery, H. Huang, G. Xie, Y. Ren, Y. Yan, and A. E. Willner, “Experimental demonstration of obstruction-tolerant free-space transmission of two 50-Gbaud QPSK data channels using Bessel beams carrying orbital angular momentum” In European Conference on Optical Communication (IEEE, 2014), pp. 1–3.

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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

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[Crossref]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Other (3)

N. Ahmed, M. P. Lavery, H. Huang, G. Xie, Y. Ren, Y. Yan, and A. E. Willner, “Experimental demonstration of obstruction-tolerant free-space transmission of two 50-Gbaud QPSK data channels using Bessel beams carrying orbital angular momentum” In European Conference on Optical Communication (IEEE, 2014), pp. 1–3.

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Supplementary Material (2)

NameDescription
» Visualization 1: AVI (1317 KB)     
» Visualization 2: AVI (1988 KB)     

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Figures (4)

Fig. 1
Fig. 1 Schematic of the step-index fiber and its modes; (a) structure of the fiber composed of a protective coating, cladding and core where the light is guided; schematic of the mode patterns and spectrum for (b) the scalar approximation and (c) the natural vector modes with partialy resolved degeneracy of the first higher order mode group.
Fig. 2
Fig. 2 Experimental setup used to generate and detect vector beams after propagating through a fiber by two separate measurement techniques: vector mode detection and tomography. Insets depict a hologram used on SLM 1 and beam profiles at different planes. SLM - spatial light modulator; HWP - half wave plate; QP - q-plate; MO - microscopic objective; L - lens; NPBS - non-polarizing beam splitter; LP - linear polarizer; QWP - quarter wave plate.
Fig. 3
Fig. 3 Example and results of the vector mode decomposition; (a) depicts the working principle by illumination with a TE01 mode. The bright spot in the TE channel on the optical axis, indicated by the green crosses, compared to zero intensity in the other channel denotes the high mode purity. Illustration of the normalized intensity of the modal transmission matrix through (b) free space, indicating the high quality of the vector beam generation, and (c) after fiber transmission, showing the appearance of mode coupling during propagation through the fiber.
Fig. 4
Fig. 4 Experimental results obtained from the full state tomography of the TE01 mode, for qualitative analysis by rotating polarizer ( Visualization 1). (a) Tomography matrix obtained by projection of the mode into six polarization and the corresponding six OAM states, measurement procedure ( Visualization 2). (b) The qubit vector density matrix showing the high degree of non-separability of the investigated mode.

Tables (1)

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Table 1 Effects of stress on degree of non-separability

Equations (6)

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{ t 2 + n 2 k 2 } e t + t { e t t [ ln ( n 2 ) ] } = β 2 e t ,
E P ( r ) = { J | | ( u P r a ) / J | | ( u P ) for r < a K | | ( w P a ) / K | | ( w P ) for r a
| , L | + Q , R ,
| , R | Q , L .
| ψ = E 11 ( r ) ( exp ( i ϕ ) | R exp ( i ϕ ) | L ) ,
C ( ρ ) = max { 0 , λ 1 λ 2 λ 3 λ 4 } ,

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