Abstract

While the motion of a classical particle bounded to a surface depends only on the local curvature, the dynamics of a quantum particle depends also on the mean surface curvature. Its influence can be experimentally observed using surface plasmon polaritons (SPPs), which are naturally surface-bounded waves. Owing to the similarity between the Schrödinger and the paraxial Helmholtz equations, this system can be used to readily examine quantum phenomena. In this work, we experimentally show a new guiding mechanism based on this quantum effect using the surface curvature of a book-cover structure, whereas in the case of an inverse book cover with the same local curvature but inverted mean curvature, the SPPs rapidly diffract. Additionally we show that, by longitudinally bending the book cover, the propagating mode width can be dynamically controlled.

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

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References

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2007 (1)

2006 (1)

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2005 (4)

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1999 (1)

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[Crossref]

1975 (1)

M. V. Berry, J. Phys. A 8, 1952 (1975).
[Crossref]

1971 (1)

H. Jensen and H. Koppe, Ann. Phys. (N.Y.) 63, 586 (1971).
[Crossref]

Arie, A.

Aubry, A.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, Nano Lett. 10, 2574 (2010).
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V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, Phys. Rev. Lett. 105, 143901 (2010).
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S. Batz and U. Peschel, Phys. Rev. A 78, 043821 (2008).
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Bekenstein, R.

A. Patsyk, M. A. Bandres, R. Bekenstein, and M. Segev, Phys. Rev. X 8, 011001 (2018).
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C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, Nat. Commun. 7, 10747 (2016).
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R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, Phys. Rev. X 4, 011038 (2014).
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Berry, M. V.

M. V. Berry, J. Phys. A 8, 1952 (1975).
[Crossref]

Boltasseva, A.

Bozhevolnyi, S. I.

A. Boltasseva, V. S. Volkov, R. B. Nielsen, E. Moreno, S. G. Rodrigo, and S. I. Bozhevolnyi, Opt. Express 16, 5252 (2008).
[Crossref]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, Phys. Rev. Lett. 100, 023901 (2008).
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R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, Adv. Opt. Mater. 6, 1800795 (2018).
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W. Liu, Z. Li, H. Cheng, C. Tang, J. Li, S. Zhang, S. Chen, and J. Tian, Adv. Mater. 30, 1706368 (2018).
[Crossref]

Cheng, H.

R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, Adv. Opt. Mater. 6, 1800795 (2018).
[Crossref]

W. Liu, Z. Li, H. Cheng, C. Tang, J. Li, S. Zhang, S. Chen, and J. Tian, Adv. Mater. 30, 1706368 (2018).
[Crossref]

Cui, T. J.

da Costa, R. C. T.

R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).
[Crossref]

Davis, C. C.

I. I. Smolyaninov, Q. Balzano, and C. C. Davis, Phys. Rev. B 72, 165412 (2005).
[Crossref]

de Sterke, C. M.

G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
[Crossref]

Della Valle, G.

G. Della Valle and S. Longhi, J. Phys. B 43, 051002 (2010).
[Crossref]

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[Crossref]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, Phys. Rev. Lett. 95, 046802 (2005).
[Crossref]

Dong, Z. G.

Dragoman, D.

D. Dragoman and M. Dragoman, Quantum-Classical Analogies, The Frontiers Collection (Springer, 2004).

Dragoman, M.

D. Dragoman and M. Dragoman, Quantum-Classical Analogies, The Frontiers Collection (Springer, 2004).

Dreisow, F.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[Crossref]

Duò, L.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[Crossref]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, Phys. Rev. Lett. 95, 046802 (2005).
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Epstein, I.

Fernández-Domínguez, A. I.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, Nano Lett. 10, 2574 (2010).
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Finazzi, M.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
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G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
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E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, Phys. Rev. Lett. 100, 023901 (2008).
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Gravesen, J.

J. Gravesen, M. Willatzen, and L. C. L. Y. Voon, Phys. Scr. 72, 105 (2005).
[Crossref]

J. Gravesen, M. Willatzen, and L. C. L. Y. Voon, J. Math. Phys. 46, 012107 (2005).
[Crossref]

Hao, Y.

Heinrich, M.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[Crossref]

Horsley, S. A. R.

Ikegami, M.

M. Ikegami, Y. Nagaoka, S. Takagi, and T. Tanzawa, Prog. Theor. Phys. Suppl. 88, 229 (1992).
[Crossref]

Jensen, H.

H. Jensen and H. Koppe, Ann. Phys. (N.Y.) 63, 586 (1971).
[Crossref]

Kaminer, I.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, Phys. Rev. X 4, 011038 (2014).
[Crossref]

Keil, R.

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[Crossref]

Khanikaev, A. B.

D. Smirnova, S. H. Mousavi, Z. Wang, Y. S. Kivshar, and A. B. Khanikaev, ACS Photon. 3, 875 (2016).
[Crossref]

Kiravittaya, S.

C. Ortix, S. Kiravittaya, O. G. Schmidt, and J. van den Brink, Phys. Rev. B 84, 045438 (2011).
[Crossref]

Kivshar, Y. S.

D. Smirnova, S. H. Mousavi, Z. Wang, Y. S. Kivshar, and A. B. Khanikaev, ACS Photon. 3, 875 (2016).
[Crossref]

Koppe, H.

H. Jensen and H. Koppe, Ann. Phys. (N.Y.) 63, 586 (1971).
[Crossref]

Laporta, P.

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[Crossref]

Lei, D. Y.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, Nano Lett. 10, 2574 (2010).
[Crossref]

Lenz, G.

G. Lenz, I. Talanina, and C. M. de Sterke, Phys. Rev. Lett. 83, 963 (1999).
[Crossref]

Li, J.

W. Liu, Z. Li, H. Cheng, C. Tang, J. Li, S. Zhang, S. Chen, and J. Tian, Adv. Mater. 30, 1706368 (2018).
[Crossref]

T. Zentgraf, J. Valentine, N. Tapia, J. Li, and X. Zhang, Adv. Mater. 22, 2561 (2010).
[Crossref]

Li, Z.

W. Liu, Z. Li, H. Cheng, C. Tang, J. Li, S. Zhang, S. Chen, and J. Tian, Adv. Mater. 30, 1706368 (2018).
[Crossref]

Lilach, Y.

Liu, H.

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, Nat. Commun. 7, 10747 (2016).
[Crossref]

Liu, W.

R. Zuo, W. Liu, H. Cheng, S. Chen, and J. Tian, Adv. Opt. Mater. 6, 1800795 (2018).
[Crossref]

W. Liu, Z. Li, H. Cheng, C. Tang, J. Li, S. Zhang, S. Chen, and J. Tian, Adv. Mater. 30, 1706368 (2018).
[Crossref]

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nat. Nanotechnol. 6, 151 (2011).
[Crossref]

Lodder, J. J.

N. G. van Kampen and J. J. Lodder, Am. J. Phys. 52, 419 (1984).
[Crossref]

Longhi, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[Crossref]

G. Della Valle and S. Longhi, J. Phys. B 43, 051002 (2010).
[Crossref]

G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[Crossref]

S. Longhi, Opt. Lett. 32, 2647 (2007).
[Crossref]

Lu, W. B.

Lucks, J. B.

V. Vitelli, J. B. Lucks, and D. R. Nelson, Proc. Natl. Acad. Sci. USA 103, 12323 (2006).
[Crossref]

Maier, S. A.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, Nano Lett. 10, 2574 (2010).
[Crossref]

Marom, E.

Martín-Moreno, L.

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, Phys. Rev. Lett. 100, 023901 (2008).
[Crossref]

McManus, T. M.

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, 2005).

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, Nat. Nanotechnol. 6, 151 (2011).
[Crossref]

Mitchell-Thomas, R. C.

Moreno, E.

A. Boltasseva, V. S. Volkov, R. B. Nielsen, E. Moreno, S. G. Rodrigo, and S. I. Bozhevolnyi, Opt. Express 16, 5252 (2008).
[Crossref]

E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, Phys. Rev. Lett. 100, 023901 (2008).
[Crossref]

Mousavi, S. H.

D. Smirnova, S. H. Mousavi, Z. Wang, Y. S. Kivshar, and A. B. Khanikaev, ACS Photon. 3, 875 (2016).
[Crossref]

Nagaoka, Y.

M. Ikegami, Y. Nagaoka, S. Takagi, and T. Tanzawa, Prog. Theor. Phys. Suppl. 88, 229 (1992).
[Crossref]

Nelson, D. R.

V. Vitelli, J. B. Lucks, and D. R. Nelson, Proc. Natl. Acad. Sci. USA 103, 12323 (2006).
[Crossref]

Nemirovsky, J.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, Phys. Rev. X 4, 011038 (2014).
[Crossref]

Ni, Z. H.

Nielsen, R. B.

Nolte, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
[Crossref]

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G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
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Nat. Commun. (1)

C. Sheng, R. Bekenstein, H. Liu, S. Zhu, and M. Segev, Nat. Commun. 7, 10747 (2016).
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Phys. Rev. Lett. (6)

A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Phys. Rev. Lett. 104, 150403 (2010).
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G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duò, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
[Crossref]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tünnermann, S. Longhi, and U. Peschel, Phys. Rev. Lett. 105, 143901 (2010).
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Phys. Rev. X (2)

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Supplementary Material (1)

NameDescription
» Supplement 1       Microscopic image of a book book-cover structure. Derivation of Eq. (4).

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Figures (4)

Fig. 1.
Fig. 1. (a) Experimental setup for the book-cover structure. Inset, curvilinear coordinates definition; (b) experimental setup for the inverse book cover. The plasmonic beam is excited from free space through a grating coupler, and an NSOM tip measures the intensity distribution.
Fig. 2.
Fig. 2. Simulation and experimental results. (a) Lumerical mode solver result for a book cover structure; (b) geometrical shape of the book-cover structure (red) and the intensity distribution of the plasmonic mode solver (blue); (c) 3D plot of the measured book-cover topography with the intensity of the SPP overlaid; (d) measured topographic profile of the book-cover structure (red) and the intensity distribution of the SPPs at beginning of the propagation (green) and after 40 μm (blue); (e) SPPs measured on a flat surface with a Gaussian grating. Dashed-dotted line defines the theoretical width of a Gaussian plasmonic beam; (f) 3D plot of the measured inverse book-cover topography with the intensity of the SPP overlaid; (g) measured topographic profile of the inverse book-cover structure (red) and the intensity distribution of the SPPs at beginning of the propagation (green) and after 50 μm (blue).
Fig. 3.
Fig. 3. Experimental results. (a) 3D plot of the measured combined book-cover topography with the intensity of the SPPs overlaid; (b) profile of the combined book-cover structure (red) and the profile of the SPPs as measured with the NSOM at the beginning of the propagation (green) and after 50 μm (blue).
Fig. 4.
Fig. 4. Longitudinal bending of the book-cover structure. (a) R2= with Cartesian and curvilinear coordinates shown; the colors in this figure visualize the bending of the structure; (b) positive R2; (c) negative R2; (d) mode width as a function of 1/R2 for L=150μm.

Equations (4)

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

iλ2πAsppZ=λ28π2nspp2Asppσ2+Veff(σ)Aspp,
Veff(σ)={λne4πR11ϵd+ϵm,|σ|R1sin(θ/2)0,|σ|>R1sin(θ/2),
n=nspp22nsppVeff.
iλ2πAsppv=λ28π2nspp2Asppσ2+Veff(σ)AsppneffR2v(q2)Aspp,

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