Abstract

We discuss how the topological charge of an OAM-carrying plasmon (Plasmonic Vortex) can be probed by monitoring the near-field response of plasmonic nanostructures suitably arranged inside a Plasmonic Vortex Lens. The turning “on” or “off” of four gold nanorods, detected by a Scanning Near field Optical Microscope (SNOM), acts as a fingerprint of the OAM state of the PV at the nanoscale. Different configurations are studied numerically, the integrated structure is fabricated and near field characterization is performed for a particularly meaningful case.

© 2014 Optical Society of America

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References

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    [Crossref]

2012 (5)

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

S.-W. Cho, J. Park, S.-Y. Lee, H. Kim, and B. Lee, “Coupling of spin and angular momentum of light in plasmonic vortex,” Opt. Express 20(9), 10083–10094 (2012).
[Crossref] [PubMed]

P. Zilio, E. Mari, G. Parisi, F. Tamburini, and F. Romanato, “Angular momentum properties of electromagnetic field transmitted through holey plasmonic vortex lenses,” Opt. Lett. 37(15), 3234–3236 (2012).
[Crossref] [PubMed]

2010 (2)

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[Crossref] [PubMed]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10(2), 529–536 (2010).
[Crossref] [PubMed]

2009 (4)

2008 (2)

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008).
[Crossref] [PubMed]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: Unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (1)

J. Aizpurua, G. W. Bryant, L. J. Richter, F. J. García de Abajo, B. K. Kelley, and T. Mallouk, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1984 (1)

M. V. Berry, “Quantum phase factors accompanying adiabatic changes,” Proc. R. Soc. London A 392(1802), 45–57 (1984).
[Crossref]

1956 (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indiana Acad. Sci. A44, 247 (1956).

Adam, A. J. L.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[Crossref] [PubMed]

Aieta, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

Aizpurua, J.

J. Aizpurua, G. W. Bryant, L. J. Richter, F. J. García de Abajo, B. K. Kelley, and T. Mallouk, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Beermann, J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Quantum phase factors accompanying adiabatic changes,” Proc. R. Soc. London A 392(1802), 45–57 (1984).
[Crossref]

Bharadwaj, P.

Blanchard, R.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: Unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

Bretner, I.

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[Crossref] [PubMed]

Brok, J. M.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[Crossref] [PubMed]

Bryant, G. W.

J. Aizpurua, G. W. Bryant, L. J. Richter, F. J. García de Abajo, B. K. Kelley, and T. Mallouk, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005).
[Crossref]

Capasso, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

E. J. Smythe, E. Cubukcu, and F. Capasso, “Optical properties of surface plasmon resonances of coupled metallic nanorods,” Opt. Express 15(12), 7439–7447 (2007).
[Crossref] [PubMed]

Chen, W.

Cho, S.-W.

S.-W. Cho, J. Park, S.-Y. Lee, H. Kim, and B. Lee, “Coupling of spin and angular momentum of light in plasmonic vortex,” Opt. Express 20(9), 10083–10094 (2012).
[Crossref] [PubMed]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10(2), 529–536 (2010).
[Crossref] [PubMed]

Cubukcu, E.

Deutsch, B.

Gaburro, Z.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

García de Abajo, F. J.

J. Aizpurua, G. W. Bryant, L. J. Richter, F. J. García de Abajo, B. K. Kelley, and T. Mallouk, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005).
[Crossref]

Genevet, P.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

Gorodetski, Y.

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[Crossref] [PubMed]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: Unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008).
[Crossref] [PubMed]

Hasman, E.

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[Crossref] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008).
[Crossref] [PubMed]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: Unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

Kang, M.

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10(2), 529–536 (2010).
[Crossref] [PubMed]

Kats, M. A.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Kelley, B. K.

J. Aizpurua, G. W. Bryant, L. J. Richter, F. J. García de Abajo, B. K. Kelley, and T. Mallouk, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005).
[Crossref]

Kim, H.

S.-W. Cho, J. Park, S.-Y. Lee, H. Kim, and B. Lee, “Coupling of spin and angular momentum of light in plasmonic vortex,” Opt. Express 20(9), 10083–10094 (2012).
[Crossref] [PubMed]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10(2), 529–536 (2010).
[Crossref] [PubMed]

Kleiner, V.

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[Crossref] [PubMed]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: Unified geometric phase and spin-hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008).
[Crossref] [PubMed]

Lee, B.

S.-W. Cho, J. Park, S.-Y. Lee, H. Kim, and B. Lee, “Coupling of spin and angular momentum of light in plasmonic vortex,” Opt. Express 20(9), 10083–10094 (2012).
[Crossref] [PubMed]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10(2), 529–536 (2010).
[Crossref] [PubMed]

Lee, S.-Y.

S.-W. Cho, J. Park, S.-Y. Lee, H. Kim, and B. Lee, “Coupling of spin and angular momentum of light in plasmonic vortex,” Opt. Express 20(9), 10083–10094 (2012).
[Crossref] [PubMed]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10(2), 529–536 (2010).
[Crossref] [PubMed]

Leosson, K.

Lin, J.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Mallouk, T.

J. Aizpurua, G. W. Bryant, L. J. Richter, F. J. García de Abajo, B. K. Kelley, and T. Mallouk, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005).
[Crossref]

Mari, E.

Nelson, R. L.

Niv, A.

Y. Gorodetski, A. Niv, V. Kleiner, and E. Hasman, “Observation of the spin-based plasmonic effect in nanoscale structures,” Phys. Rev. Lett. 101(4), 043903 (2008).
[Crossref] [PubMed]

Novikov, S. M.

Novotny, L.

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indiana Acad. Sci. A44, 247 (1956).

Parisi, G.

Park, J.

S.-W. Cho, J. Park, S.-Y. Lee, H. Kim, and B. Lee, “Coupling of spin and angular momentum of light in plasmonic vortex,” Opt. Express 20(9), 10083–10094 (2012).
[Crossref] [PubMed]

H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10(2), 529–536 (2010).
[Crossref] [PubMed]

Planken, P. C. M.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[Crossref] [PubMed]

Richter, L. J.

J. Aizpurua, G. W. Bryant, L. J. Richter, F. J. García de Abajo, B. K. Kelley, and T. Mallouk, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71(23), 235420 (2005).
[Crossref]

Romanato, F.

Scully, M. O.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Shitrit, N.

Y. Gorodetski, N. Shitrit, I. Bretner, V. Kleiner, and E. Hasman, “Observation of optical spin symmetry breaking in nanoapertures,” Nano Lett. 9(8), 3016–3019 (2009).
[Crossref] [PubMed]

Smythe, E. J.

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Tamburini, F.

Urbach, H. P.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[Crossref] [PubMed]

Vuong, L. T.

L. T. Vuong, A. J. L. Adam, J. M. Brok, P. C. M. Planken, and H. P. Urbach, “Electromagnetic spin-orbit interactions via scattering of subwavelength apertures,” Phys. Rev. Lett. 104(8), 083903 (2010).
[Crossref] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Yang, S.

Yu, N.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Geometry of our structure, consisting in a Plasmonic Vortex Lens with m = 1, with four nanorods placed in a hole at the center of the spiral (see inset). The dashed cylinder highlights the FEM simulation domain, where the analytical plasmonic vortex expression was set as the background field.

Fig. 2
Fig. 2

Power flow as a function of the hole radius for different values of j at λ = 780 nm. For each value of the topological charge we can identify optimal values for the hole radius.

Fig. 3
Fig. 3

Calculated transmittance of an 800 nm wide hole on 200 nm thick gold film illuminated by a circularly polarized plane wave, without (blue) or with (red) four nanorods inside. Insets: Electric field norm distribution on the empty hole at 780 nm (blue frame) or on the hole with embedded nanorods at 780 nm or 1000 nm (red frame).

Fig. 4
Fig. 4

Electric field norm inside the FEM simulation domain, consisting in a hole at the center of a PVL where four nanorods have been placed. (a) and (b) are xz-cross sections of the structure, while (c) and (d) are xy-cross sections of the hole; the arrows indicate the direction of the electric field in absence of the rods. The simulations refer to the two cases j = 0 (shown in (a) and (c)) and j = 2 (shown in (b) and (d)). For a better visualization the color scales were saturated to proper values. The minima and maxima of the field for the different plots are reported respectively at the left and at the right of the color bars.

Fig. 5
Fig. 5

Scheme of the fabrication process. An EBL overlay exposure was needed. The first layer – consisting in the PVL and the marks used for alignment – was fabricated by EBL on a negative resist, followed by electrolytic growth. The sample was then realigned and the second layer – nanorods inside the hole – was fabricated.

Fig. 6
Fig. 6

SEM micrographs of a fabricated sample. (a) The sample consists in an m = 1 PVL with four nanorods placed inside the central hole. (b) Zoom in the central part of the device, highlighting the almost perfect alignment of the two exposures. (c) Tilted view highlighting the different thickness of the spiral and of the nanorods.

Fig. 7
Fig. 7

SNOM scans obtained upon illumination with (a) Case j = 0: light is focused to the center but the rods are not excited. (b) Case j = 2: all the four rods are illuminated. Inset: plot of light intensity as a function of the azimuthal angle, taken at a fixed radius of 200 nm, corresponding to the position of the rods. The observed modulation highlights the presence of the rods.

Equations (3)

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r m ( φ )= r 0 +m d 2π φ
E SPP,m ( R,φ ) e ijφ J i ( k SPP R )
j=l+s+m

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