Abstract

The capability to distinguish the handedness of circularly polarized light is a well-known intrinsic property of a chiral nanostructure. It is a long-standing controversial debate, however, whether a chiral object can also sense the vorticity, or the orbital angular momentum (OAM), of a light field. Since OAM is a spatial property, it seems rather counterintuitive that a point-like chiral object could be able to distinguish the sense of the wave-front of light carrying OAM. Here, we show that a dipolar chiral nanostructure is indeed capable of distinguishing the sign of the phase vortex of the incoming light beam. To this end, we take advantage of the conversion of the sign of OAM, carried by a linearly polarized Laguerre–Gaussian beam, into the sign of optical chirality upon tight focusing. Our study provides for a deeper insight into the discussion of chiral light–matter interactions and the respective role of OAM.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2009).
  2. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
    [Crossref]
  3. V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
    [Crossref]
  4. B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
    [Crossref]
  5. M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
    [Crossref]
  6. P. Woźniak, I. De Leon, K. Höflich, C. Haverkamp, S. Christiansen, G. Leuchs, and P. Banzer, “Chiroptical response of a single plasmonic nanohelix,” Opt. Express 26, 19275–19293 (2018).
    [Crossref]
  7. Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013).
    [Crossref]
  8. L. Hu, Y. Huang, L. Pan, and Y. Fang, “Analyzing intrinsic plasmonic chirality by tracking the interplay of electric and magnetic dipole modes,” Sci. Rep. 7, 11151 (2017).
    [Crossref]
  9. J. S. Eismann, M. Neugebauer, and P. Banzer, “Exciting a chiral dipole moment in an achiral nanostructure,” Optica 5, 954–959 (2018).
    [Crossref]
  10. S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbit-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
    [Crossref]
  11. M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. D. Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002).
    [Crossref]
  12. D. Andrews, L. Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237, 133–139 (2004).
    [Crossref]
  13. F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A 71, 055401 (2005).
    [Crossref]
  14. W. Löffler, D. Broer, and J. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A 83, 065801 (2011).
    [Crossref]
  15. F. Giammanco, A. Perona, P. Marsili, F. Conti, F. Fidecaro, S. Gozzini, and A. Lucchesini, “Influence of the photon orbital angular momentum on electric dipole transitions: negative experimental evidence,” Opt. Lett. 42, 219–222 (2017).
    [Crossref]
  16. M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
    [Crossref]
  17. K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
    [Crossref]
  18. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
    [Crossref]
  19. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
    [Crossref]
  20. K. Y. Bliokh, F. J. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796 (2015).
    [Crossref]
  21. P. Woźniak, P. Banzer, and G. Leuchs, "Selective switching of individual multipole resonances in single dielectric nanoparticles," Laser & Photon. Rev. 9, 231–240 (2015).
  22. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
    [Crossref]
  23. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
  24. P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905–10923 (2010).
    [Crossref]
  25. D. M. Lipkin, “Existence of a new conservation law in electromagnetic theory,” J. Math. Phys. 5, 696–700 (1964).
    [Crossref]
  26. G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, “Twisted-light-ion interaction: the role of longitudinal fields,” Phys. Rev. Lett. 119, 253203 (2017).
    [Crossref]
  27. K. A. Forbes and D. L. Andrews, “Optical orbital angular momentum: twisted light and chirality,” Opt. Lett. 43, 435–438 (2018).
    [Crossref]
  28. D. L. Andrews, “Quantum formulation for nanoscale optical and material chirality: symmetry issues, space and time parity, and observables,” J. Opt. 20, 033003 (2018).
    [Crossref]
  29. R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, and O. Hess, “Orbital angular momentum dichroism in nanoantennas,” Commun. Phys. 1, 87 (2018).
    [Crossref]
  30. C. Rosales-Guzmán, K. Volke-Sepulveda, and J. P. Torres, “Light with enhanced optical chirality,” Opt. Lett. 37, 3486–3488 (2012).
    [Crossref]
  31. F. Crimin, N. Mackinnon, J. B. Götte, and S. M. Barnett, “Optical helicity and chirality: conservation and sources,” Appl. Sci. 9, 828 (2019).
    [Crossref]
  32. K. Höflich, R. B. Yang, A. Berger, G. Leuchs, and S. Christiansen, “The direct writing of plasmonic gold nanostructures by electron-beam-induced deposition,” Adv. Mater. 23, 2657–2661 (2011).
    [Crossref]
  33. Z. Bomzon, V. Kleiner, and E. Hasman, “Pancharatnam-Berry phase in space-variant polarization-state manipulations with subwavelength gratings,” Opt. Lett. 26, 1 (2001).
    [Crossref]
  34. 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]

2019 (2)

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbit-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

F. Crimin, N. Mackinnon, J. B. Götte, and S. M. Barnett, “Optical helicity and chirality: conservation and sources,” Appl. Sci. 9, 828 (2019).
[Crossref]

2018 (6)

K. A. Forbes and D. L. Andrews, “Optical orbital angular momentum: twisted light and chirality,” Opt. Lett. 43, 435–438 (2018).
[Crossref]

D. L. Andrews, “Quantum formulation for nanoscale optical and material chirality: symmetry issues, space and time parity, and observables,” J. Opt. 20, 033003 (2018).
[Crossref]

R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, and O. Hess, “Orbital angular momentum dichroism in nanoantennas,” Commun. Phys. 1, 87 (2018).
[Crossref]

M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
[Crossref]

P. Woźniak, I. De Leon, K. Höflich, C. Haverkamp, S. Christiansen, G. Leuchs, and P. Banzer, “Chiroptical response of a single plasmonic nanohelix,” Opt. Express 26, 19275–19293 (2018).
[Crossref]

J. S. Eismann, M. Neugebauer, and P. Banzer, “Exciting a chiral dipole moment in an achiral nanostructure,” Optica 5, 954–959 (2018).
[Crossref]

2017 (3)

L. Hu, Y. Huang, L. Pan, and Y. Fang, “Analyzing intrinsic plasmonic chirality by tracking the interplay of electric and magnetic dipole modes,” Sci. Rep. 7, 11151 (2017).
[Crossref]

F. Giammanco, A. Perona, P. Marsili, F. Conti, F. Fidecaro, S. Gozzini, and A. Lucchesini, “Influence of the photon orbital angular momentum on electric dipole transitions: negative experimental evidence,” Opt. Lett. 42, 219–222 (2017).
[Crossref]

G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, “Twisted-light-ion interaction: the role of longitudinal fields,” Phys. Rev. Lett. 119, 253203 (2017).
[Crossref]

2015 (4)

K. Y. Bliokh, F. J. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

P. Woźniak, P. Banzer, and G. Leuchs, "Selective switching of individual multipole resonances in single dielectric nanoparticles," Laser & Photon. Rev. 9, 231–240 (2015).

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

2013 (3)

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
[Crossref]

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref]

2012 (1)

2011 (2)

W. Löffler, D. Broer, and J. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A 83, 065801 (2011).
[Crossref]

K. Höflich, R. B. Yang, A. Berger, G. Leuchs, and S. Christiansen, “The direct writing of plasmonic gold nanostructures by electron-beam-induced deposition,” Adv. Mater. 23, 2657–2661 (2011).
[Crossref]

2010 (2)

2009 (1)

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

2007 (1)

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

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]

2005 (1)

F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A 71, 055401 (2005).
[Crossref]

2004 (1)

D. Andrews, L. Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237, 133–139 (2004).
[Crossref]

2002 (1)

M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. D. Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002).
[Crossref]

2001 (1)

1964 (1)

D. M. Lipkin, “Existence of a new conservation law in electromagnetic theory,” J. Math. Phys. 5, 696–700 (1964).
[Crossref]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Andrews, D.

D. Andrews, L. Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237, 133–139 (2004).
[Crossref]

Andrews, D. L.

M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
[Crossref]

D. L. Andrews, “Quantum formulation for nanoscale optical and material chirality: symmetry issues, space and time parity, and observables,” J. Opt. 20, 033003 (2018).
[Crossref]

K. A. Forbes and D. L. Andrews, “Optical orbital angular momentum: twisted light and chirality,” Opt. Lett. 43, 435–438 (2018).
[Crossref]

M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. D. Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002).
[Crossref]

Araoka, F.

F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A 71, 055401 (2005).
[Crossref]

Babiker, M.

M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
[Crossref]

D. Andrews, L. Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237, 133–139 (2004).
[Crossref]

M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. D. Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002).
[Crossref]

Bade, K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Banzer, P.

Barnett, S. M.

F. Crimin, N. Mackinnon, J. B. Götte, and S. M. Barnett, “Optical helicity and chirality: conservation and sources,” Appl. Sci. 9, 828 (2019).
[Crossref]

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2009).

Baumberg, J. J.

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
[Crossref]

Benedetti, A.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

Bennett, C. R.

M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. D. Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002).
[Crossref]

Berger, A.

K. Höflich, R. B. Yang, A. Berger, G. Leuchs, and S. Christiansen, “The direct writing of plasmonic gold nanostructures by electron-beam-induced deposition,” Adv. Mater. 23, 2657–2661 (2011).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

Bomzon, Z.

Braun, P. V.

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

Broer, D.

W. Löffler, D. Broer, and J. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A 83, 065801 (2011).
[Crossref]

Chiu, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Christiansen, S.

P. Woźniak, I. De Leon, K. Höflich, C. Haverkamp, S. Christiansen, G. Leuchs, and P. Banzer, “Chiroptical response of a single plasmonic nanohelix,” Opt. Express 26, 19275–19293 (2018).
[Crossref]

K. Höflich, R. B. Yang, A. Berger, G. Leuchs, and S. Christiansen, “The direct writing of plasmonic gold nanostructures by electron-beam-induced deposition,” Adv. Mater. 23, 2657–2661 (2011).
[Crossref]

Clays, K.

F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A 71, 055401 (2005).
[Crossref]

Cohen, A. E.

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

Conti, F.

Crimin, F.

F. Crimin, N. Mackinnon, J. B. Götte, and S. M. Barnett, “Optical helicity and chirality: conservation and sources,” Appl. Sci. 9, 828 (2019).
[Crossref]

Cuscunà, M.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

De Giorgi, M.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

De Leon, I.

Decker, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Drezet, A.

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref]

Ebbesen, T. W.

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref]

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Eismann, J. S.

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbit-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

J. S. Eismann, M. Neugebauer, and P. Banzer, “Exciting a chiral dipole moment in an achiral nanostructure,” Optica 5, 954–959 (2018).
[Crossref]

Esposito, M.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

Fang, Y.

L. Hu, Y. Huang, L. Pan, and Y. Fang, “Analyzing intrinsic plasmonic chirality by tracking the interplay of electric and magnetic dipole modes,” Sci. Rep. 7, 11151 (2017).
[Crossref]

Fidecaro, F.

Fitzgerald, J. M.

R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, and O. Hess, “Orbital angular momentum dichroism in nanoantennas,” Commun. Phys. 1, 87 (2018).
[Crossref]

Forbes, K. A.

Frank, B.

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

Gansel, J. K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Genet, C.

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref]

Giammanco, F.

Giessen, H.

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

Gorodetski, Y.

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref]

Götte, J. B.

F. Crimin, N. Mackinnon, J. B. Götte, and S. M. Barnett, “Optical helicity and chirality: conservation and sources,” Appl. Sci. 9, 828 (2019).
[Crossref]

Gozzini, S.

Hasman, E.

Haverkamp, C.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Hein, S. M.

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

Hess, O.

R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, and O. Hess, “Orbital angular momentum dichroism in nanoantennas,” Commun. Phys. 1, 87 (2018).
[Crossref]

Höflich, K.

P. Woźniak, I. De Leon, K. Höflich, C. Haverkamp, S. Christiansen, G. Leuchs, and P. Banzer, “Chiroptical response of a single plasmonic nanohelix,” Opt. Express 26, 19275–19293 (2018).
[Crossref]

K. Höflich, R. B. Yang, A. Berger, G. Leuchs, and S. Christiansen, “The direct writing of plasmonic gold nanostructures by electron-beam-induced deposition,” Adv. Mater. 23, 2657–2661 (2011).
[Crossref]

Hu, L.

L. Hu, Y. Huang, L. Pan, and Y. Fang, “Analyzing intrinsic plasmonic chirality by tracking the interplay of electric and magnetic dipole modes,” Sci. Rep. 7, 11151 (2017).
[Crossref]

Huang, Y.

L. Hu, Y. Huang, L. Pan, and Y. Fang, “Analyzing intrinsic plasmonic chirality by tracking the interplay of electric and magnetic dipole modes,” Sci. Rep. 7, 11151 (2017).
[Crossref]

Jeffries, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Kerber, R. M.

R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, and O. Hess, “Orbital angular momentum dichroism in nanoantennas,” Commun. Phys. 1, 87 (2018).
[Crossref]

Kleiner, V.

Lembessis, V. E.

M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
[Crossref]

Leuchs, G.

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbit-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

P. Woźniak, I. De Leon, K. Höflich, C. Haverkamp, S. Christiansen, G. Leuchs, and P. Banzer, “Chiroptical response of a single plasmonic nanohelix,” Opt. Express 26, 19275–19293 (2018).
[Crossref]

P. Woźniak, P. Banzer, and G. Leuchs, "Selective switching of individual multipole resonances in single dielectric nanoparticles," Laser & Photon. Rev. 9, 231–240 (2015).

K. Höflich, R. B. Yang, A. Berger, G. Leuchs, and S. Christiansen, “The direct writing of plasmonic gold nanostructures by electron-beam-induced deposition,” Adv. Mater. 23, 2657–2661 (2011).
[Crossref]

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905–10923 (2010).
[Crossref]

Linden, S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Lipkin, D. M.

D. M. Lipkin, “Existence of a new conservation law in electromagnetic theory,” J. Math. Phys. 5, 696–700 (1964).
[Crossref]

Löffler, W.

W. Löffler, D. Broer, and J. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A 83, 065801 (2011).
[Crossref]

Lucchesini, A.

Mackinnon, N.

F. Crimin, N. Mackinnon, J. B. Götte, and S. M. Barnett, “Optical helicity and chirality: conservation and sources,” Appl. Sci. 9, 828 (2019).
[Crossref]

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]

Marrucci, L.

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]

Marsili, P.

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

Nechayev, S.

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbit-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

Neugebauer, M.

Nori, F.

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Oh, S. S.

R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, and O. Hess, “Orbital angular momentum dichroism in nanoantennas,” Commun. Phys. 1, 87 (2018).
[Crossref]

Pan, L.

L. Hu, Y. Huang, L. Pan, and Y. Fang, “Analyzing intrinsic plasmonic chirality by tracking the interplay of electric and magnetic dipole modes,” Sci. Rep. 7, 11151 (2017).
[Crossref]

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]

Passaseo, A.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

Perona, A.

Persoons, A.

F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A 71, 055401 (2005).
[Crossref]

Peschel, U.

Quabis, S.

Quinteiro, G. F.

G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, “Twisted-light-ion interaction: the role of longitudinal fields,” Phys. Rev. Lett. 119, 253203 (2017).
[Crossref]

Reiter, D. E.

R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, and O. Hess, “Orbital angular momentum dichroism in nanoantennas,” Commun. Phys. 1, 87 (2018).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Rill, M. S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Rodrguez-Fortuño, F. J.

K. Y. Bliokh, F. J. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

Romero, L.

D. Andrews, L. Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237, 133–139 (2004).
[Crossref]

Romero, L. C. D.

M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. D. Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002).
[Crossref]

Rosales-Guzmán, C.

Saile, V.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Sanvitto, D.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

Schäferling, M.

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

Schmidt-Kaler, F.

G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, “Twisted-light-ion interaction: the role of longitudinal fields,” Phys. Rev. Lett. 119, 253203 (2017).
[Crossref]

Schmiegelow, C. T.

G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, “Twisted-light-ion interaction: the role of longitudinal fields,” Phys. Rev. Lett. 119, 253203 (2017).
[Crossref]

Sibilia, C.

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
[Crossref]

Tang, Y.

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

Tarantini, I.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

Tasco, V.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

Thiel, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Todisco, F.

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

Torres, J. P.

Valev, V. K.

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
[Crossref]

Verbiest, T.

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
[Crossref]

F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A 71, 055401 (2005).
[Crossref]

Volke-Sepulveda, K.

von Freymann, G.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Wegener, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Woerdman, J.

W. Löffler, D. Broer, and J. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A 83, 065801 (2011).
[Crossref]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Wozniak, P.

P. Woźniak, I. De Leon, K. Höflich, C. Haverkamp, S. Christiansen, G. Leuchs, and P. Banzer, “Chiroptical response of a single plasmonic nanohelix,” Opt. Express 26, 19275–19293 (2018).
[Crossref]

P. Woźniak, P. Banzer, and G. Leuchs, "Selective switching of individual multipole resonances in single dielectric nanoparticles," Laser & Photon. Rev. 9, 231–240 (2015).

Yang, R. B.

K. Höflich, R. B. Yang, A. Berger, G. Leuchs, and S. Christiansen, “The direct writing of plasmonic gold nanostructures by electron-beam-induced deposition,” Adv. Mater. 23, 2657–2661 (2011).
[Crossref]

Yin, X.

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

Zayats, A. V.

K. Y. Bliokh, F. J. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

Zhao, J.

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

ACS Nano (1)

B. Frank, X. Yin, M. Schäferling, J. Zhao, S. M. Hein, P. V. Braun, and H. Giessen, “Large-area 3d chiral plasmonic structures,” ACS Nano 7, 6321–6329 (2013).
[Crossref]

ACS Photon. (1)

M. Esposito, V. Tasco, M. Cuscunà, F. Todisco, A. Benedetti, I. Tarantini, M. De Giorgi, D. Sanvitto, and A. Passaseo, “Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies,” ACS Photon. 2, 105–114 (2015).
[Crossref]

Adv. Mater. (2)

V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25, 2517–2534 (2013).
[Crossref]

K. Höflich, R. B. Yang, A. Berger, G. Leuchs, and S. Christiansen, “The direct writing of plasmonic gold nanostructures by electron-beam-induced deposition,” Adv. Mater. 23, 2657–2661 (2011).
[Crossref]

Appl. Sci. (1)

F. Crimin, N. Mackinnon, J. B. Götte, and S. M. Barnett, “Optical helicity and chirality: conservation and sources,” Appl. Sci. 9, 828 (2019).
[Crossref]

Commun. Phys. (1)

R. M. Kerber, J. M. Fitzgerald, S. S. Oh, D. E. Reiter, and O. Hess, “Orbital angular momentum dichroism in nanoantennas,” Commun. Phys. 1, 87 (2018).
[Crossref]

J. Math. Phys. (1)

D. M. Lipkin, “Existence of a new conservation law in electromagnetic theory,” J. Math. Phys. 5, 696–700 (1964).
[Crossref]

J. Opt. (2)

D. L. Andrews, “Quantum formulation for nanoscale optical and material chirality: symmetry issues, space and time parity, and observables,” J. Opt. 20, 033003 (2018).
[Crossref]

M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
[Crossref]

Laser & Photon. Rev. (1)

P. Woźniak, P. Banzer, and G. Leuchs, "Selective switching of individual multipole resonances in single dielectric nanoparticles," Laser & Photon. Rev. 9, 231–240 (2015).

Nat. Photonics (1)

K. Y. Bliokh, F. J. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

Opt. Commun. (1)

D. Andrews, L. Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237, 133–139 (2004).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Optica (1)

Phys. Rep. (1)

K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015).
[Crossref]

Phys. Rev. A (2)

F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A 71, 055401 (2005).
[Crossref]

W. Löffler, D. Broer, and J. Woerdman, “Circular dichroism of cholesteric polymers and the orbital angular momentum of light,” Phys. Rev. A 83, 065801 (2011).
[Crossref]

Phys. Rev. B (1)

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbit-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

Phys. Rev. Lett. (6)

M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. D. Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002).
[Crossref]

Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013).
[Crossref]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99, 073901 (2007).
[Crossref]

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]

G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, “Twisted-light-ion interaction: the role of longitudinal fields,” Phys. Rev. Lett. 119, 253203 (2017).
[Crossref]

Proc. R. Soc. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[Crossref]

Sci. Rep. (1)

L. Hu, Y. Huang, L. Pan, and Y. Fang, “Analyzing intrinsic plasmonic chirality by tracking the interplay of electric and magnetic dipole modes,” Sci. Rep. 7, 11151 (2017).
[Crossref]

Science (1)

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Other (2)

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University, 2009).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Supplementary Material (1)

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» Supplement 1       Supplemental Document

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Figures (2)

Fig. 1.
Fig. 1. LG0±1 beams and their field components upon tight focusing. (a) Intensity and phase distributions of the input LG0±1 beams. (b) Calculated distributions of focal electric (|E|2) and magnetic (|H|2) field intensities (and relative phases) of tightly focused (NA=0.9) beams. The distributions are normalized to the maximum value of the total energy density (ϵ02|E|2+μ02|H|2). The on-axis focal field comprises longitudinal electric and magnetic fields oscillating with a phase difference ±π/2, which corresponds to the vorticity sense of the incoming beam. (c) Linear and dephased oscillation of Ez±iHz generates optical chirality C of opposite sign for both input beams, but no zero spin density s on the optical axis. The dashed circles in first line of (b) (left column) outline the top view of the nanohelix drawn to scale.
Fig. 2.
Fig. 2. Measurement scheme and the results. (a) Simplified sketch of the experimental setup [6,24] for probing the vorticity of the incoming beam with a chiral scatterer and scanning-electron micrograph of the utilized nanostructure. (b) Dimensions of the nanohelix. The strength of its chiroptical response can be described by the chiral dipole oscillating along the z direction (helix axis). (c) Experimental and simulation spectra of the fundamental resonance of the nanohelix as a function of the vorticity of the incoming beam. The insets depict the focal optical chirality and the relative size and position of the nanohelix in the focal plane.

Equations (2)

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

E=E0rw0er2w02e±ilϕe^x,
C=ω2c2I[E*·H],