Abstract

The nanofocusing performance of hourglass plasmonic waveguides is studied analytically and numerically. Nonlocal effects in the linearly tapered metal-air-metal stack that makes up the device are taken into account within a hydrodynamical approach. Using this hourglass waveguide as a model structure, we show that spatial dispersion drastically modifies the propagation of surface plasmons in metal voids, such as those generated between touching particles. Specifically, we investigate how nonlocal corrections limit the enormous field enhancements predicted by local electromagnetic treatments of geometric singularities. Finally, our results also indicate the emergence of nonlocality assisted tunnelling of plasmonic modes across hourglass contacts as thick as 0.5 nm.

© 2013 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]

2013 (6)

T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett.110, 263901 (2013).
[CrossRef] [PubMed]

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B88, 115401 (2013).
[CrossRef]

C. Ciracì, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” ChemPhysChem14, 1109–1116 (2013).
[CrossRef] [PubMed]

J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett.13, 564–569 (2013).
[CrossRef]

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

2012 (9)

A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett.12, 5946–5953 (2012).
[CrossRef] [PubMed]

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett.108, 106802 (2012).
[CrossRef] [PubMed]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun.3, 825 (2012).
[CrossRef] [PubMed]

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
[CrossRef] [PubMed]

A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal Effects in the Nanofocusing Performance of Plasmonic Tips,” Nano Lett.12, 3308–3314 (2012).
[CrossRef] [PubMed]

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012).
[CrossRef]

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature (London)483, 421–427 (2012).
[CrossRef]

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

2011 (1)

S. Raza, G. Toscano, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

2010 (1)

G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010).
[CrossRef]

2009 (1)

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett.9, 887–891 (2009).
[CrossRef] [PubMed]

2008 (3)

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008).
[CrossRef]

D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys.104, 034311 (2008).
[CrossRef]

A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics14, 1515–1521 (2008).
[CrossRef]

2007 (1)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon.1, 641–648 (2007).
[CrossRef]

2006 (1)

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett.97, 053002 (2006).
[CrossRef] [PubMed]

2005 (1)

R. Ruppin, “Non-local optics of the near field lens,” J. Phys.: Condens. Matter17, 1803–1810 (2005).
[CrossRef]

2004 (1)

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93, 137404 (2004).
[CrossRef] [PubMed]

2001 (1)

R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Commun.190, 205–209 (2001).
[CrossRef]

1997 (1)

K. Nerkararyan, “Superfocusing of a surface polariton in a wedge-like structure,” Phys. Lett.237, 103–105 (1997).
[CrossRef]

1987 (1)

R. Fuchs and F. Claro, “Multipolar response of small metallic sphere: Nonlocal theory,” Phys. Rev. B35, 3722–3727 (1987).
[CrossRef]

Aizpurua, J.

T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett.110, 263901 (2013).
[CrossRef] [PubMed]

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012).
[CrossRef]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun.3, 825 (2012).
[CrossRef] [PubMed]

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
[CrossRef] [PubMed]

Andrews, S. R.

A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics14, 1515–1521 (2008).
[CrossRef]

Ashcroft, N.

N. Ashcroft and D. Mermin, Solid State Physics (Holt, Rinehart, and Winston, 1976).

Baumberg, J. J.

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012).
[CrossRef]

Boardman, A. D.

A. D. Boardman, Electromagnetic Surface Modes (Wiley, New York, 1982).

Bokor, J.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Borisov, A. G.

T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett.110, 263901 (2013).
[CrossRef] [PubMed]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun.3, 825 (2012).
[CrossRef] [PubMed]

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012).
[CrossRef]

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
[CrossRef] [PubMed]

Bosman, M.

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B88, 115401 (2013).
[CrossRef]

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

Cabrini, S.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Chang, D. E.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett.97, 053002 (2006).
[CrossRef] [PubMed]

Chilkoti, A.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Choo, H.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Christensen, T.

S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B88, 115401 (2013).
[CrossRef]

Ciraci, C.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Ciracì, C.

C. Ciracì, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” ChemPhysChem14, 1109–1116 (2013).
[CrossRef] [PubMed]

Claro, F.

R. Fuchs and F. Claro, “Multipolar response of small metallic sphere: Nonlocal theory,” Phys. Rev. B35, 3722–3727 (1987).
[CrossRef]

Dionne, J. A.

J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett.13, 564–569 (2013).
[CrossRef]

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature (London)483, 421–427 (2012).
[CrossRef]

Duan, H.

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

Esteban, R.

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun.3, 825 (2012).
[CrossRef] [PubMed]

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012).
[CrossRef]

Fernandez-Dominguez, A. I.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Fernández-Domínguez, A. I.

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal Effects in the Nanofocusing Performance of Plasmonic Tips,” Nano Lett.12, 3308–3314 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett.12, 5946–5953 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett.108, 106802 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics14, 1515–1521 (2008).
[CrossRef]

Fuchs, R.

R. Fuchs and F. Claro, “Multipolar response of small metallic sphere: Nonlocal theory,” Phys. Rev. B35, 3722–3727 (1987).
[CrossRef]

García de Abajo, F. J.

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008).
[CrossRef]

García-Etxarri, A.

J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett.13, 564–569 (2013).
[CrossRef]

García-Vidal, F. J.

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett.108, 106802 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics14, 1515–1521 (2008).
[CrossRef]

Gramotnev, D. K.

D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys.104, 034311 (2008).
[CrossRef]

Halas, N. J.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon.1, 641–648 (2007).
[CrossRef]

Hawkeye, M. M.

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012).
[CrossRef]

Hemmer, P. R.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett.97, 053002 (2006).
[CrossRef] [PubMed]

Hill, R. T.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Horsfield, A. P.

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal Effects in the Nanofocusing Performance of Plasmonic Tips,” Nano Lett.12, 3308–3314 (2012).
[CrossRef] [PubMed]

Jauho, A.

G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010).
[CrossRef]

Jauho, A. P.

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

S. Raza, G. Toscano, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

Jeppesen, C.

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

Kazansky, A. K.

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
[CrossRef] [PubMed]

Kim, M. K.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Koh, A. L.

J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett.13, 564–569 (2013).
[CrossRef]

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature (London)483, 421–427 (2012).
[CrossRef]

Lal, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon.1, 641–648 (2007).
[CrossRef]

Link, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon.1, 641–648 (2007).
[CrossRef]

Lukin, M. D.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett.97, 053002 (2006).
[CrossRef] [PubMed]

Luo, Y.

A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett.12, 5946–5953 (2012).
[CrossRef] [PubMed]

Maier, S. A.

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal Effects in the Nanofocusing Performance of Plasmonic Tips,” Nano Lett.12, 3308–3314 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett.12, 5946–5953 (2012).
[CrossRef] [PubMed]

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett.108, 106802 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics14, 1515–1521 (2008).
[CrossRef]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Marinica, D. C.

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
[CrossRef] [PubMed]

Martín-Moreno, L.

A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics14, 1515–1521 (2008).
[CrossRef]

Mermin, D.

N. Ashcroft and D. Mermin, Solid State Physics (Holt, Rinehart, and Winston, 1976).

Mock, J. J.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Mortensen, N. A.

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B88, 115401 (2013).
[CrossRef]

S. Raza, G. Toscano, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010).
[CrossRef]

Nerkararyan, K.

K. Nerkararyan, “Superfocusing of a surface polariton in a wedge-like structure,” Phys. Lett.237, 103–105 (1997).
[CrossRef]

Nordlander, P.

T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett.110, 263901 (2013).
[CrossRef] [PubMed]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun.3, 825 (2012).
[CrossRef] [PubMed]

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
[CrossRef] [PubMed]

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett.9, 887–891 (2009).
[CrossRef] [PubMed]

Oztürk, Z. F.

G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Pendry, J. B.

C. Ciracì, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” ChemPhysChem14, 1109–1116 (2013).
[CrossRef] [PubMed]

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal Effects in the Nanofocusing Performance of Plasmonic Tips,” Nano Lett.12, 3308–3314 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett.12, 5946–5953 (2012).
[CrossRef] [PubMed]

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett.108, 106802 (2012).
[CrossRef] [PubMed]

Prodan, E.

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett.9, 887–891 (2009).
[CrossRef] [PubMed]

Raza, S.

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B88, 115401 (2013).
[CrossRef]

S. Raza, G. Toscano, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

Ruppin, R.

R. Ruppin, “Non-local optics of the near field lens,” J. Phys.: Condens. Matter17, 1803–1810 (2005).
[CrossRef]

R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Commun.190, 205–209 (2001).
[CrossRef]

Savage, K. J.

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012).
[CrossRef]

Scholl, J. A.

J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett.13, 564–569 (2013).
[CrossRef]

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature (London)483, 421–427 (2012).
[CrossRef]

Schuck, P. J.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Seok, T. J.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Smith, D. R.

C. Ciracì, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” ChemPhysChem14, 1109–1116 (2013).
[CrossRef] [PubMed]

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Sørensen, A. S.

D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett.97, 053002 (2006).
[CrossRef] [PubMed]

Staffaroni, M.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Stockman, M. I.

D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys.104, 034311 (2008).
[CrossRef]

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93, 137404 (2004).
[CrossRef] [PubMed]

Teperik, T. V.

T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Robust subnanometric plasmon ruler by rescaling of the nonlocal optical response,” Phys. Rev. Lett.110, 263901 (2013).
[CrossRef] [PubMed]

Toscano, G.

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

S. Raza, G. Toscano, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010).
[CrossRef]

Urzhumov, Y.

C. Ciraci, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernandez-Dominguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement, Science337, 1072–1074 (2012).
[CrossRef] [PubMed]

Vogel, M. W.

D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys.104, 034311 (2008).
[CrossRef]

Wiener, A.

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal Effects in the Nanofocusing Performance of Plasmonic Tips,” Nano Lett.12, 3308–3314 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett.12, 5946–5953 (2012).
[CrossRef] [PubMed]

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett.108, 106802 (2012).
[CrossRef] [PubMed]

Wu, M. C.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Wubs, M.

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B88, 115401 (2013).
[CrossRef]

S. Raza, G. Toscano, A. P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010).
[CrossRef]

Xiao, S.

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010).
[CrossRef]

Yablonovitch, E.

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Yan, M.

G. Toscano, M. Wubs, S. Xiao, M. Yan, Z. F. Oztürk, A. Jauho, and N. A. Mortensen, “Plasmonic nanostructures: local versus nonlocal response” in Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE7757(2010).
[CrossRef]

Yan, W.

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

Yang, J. K. W

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

Zuloaga, J.

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett.9, 887–891 (2009).
[CrossRef] [PubMed]

ACS Nano (1)

A. Wiener, H. Duan, M. Bosman, A. P. Horsfield, J. B. Pendry, J. K. W Yang, S. A. Maier, and A. I. Fernández-Domínguez, “Electron-energy loss study of nonlocal effects in connected plasmonic nanoprisms,” ACS Nano7, 6287–6296 (2013).
[CrossRef] [PubMed]

ChemPhysChem (1)

C. Ciracì, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” ChemPhysChem14, 1109–1116 (2013).
[CrossRef] [PubMed]

IEEE Journal of Selected Topics in Quantum Electronics (1)

A. I. Fernández-Domínguez, L. Martín-Moreno, F. J. García-Vidal, S. R. Andrews, and S. A. Maier, “Spoof surface plasmon polariton modes propagating along periodically corrugated wires,” IEEE Journal of Selected Topics in Quantum Electronics14, 1515–1521 (2008).
[CrossRef]

J. Appl. Phys. (1)

D. K. Gramotnev, M. W. Vogel, and M. I. Stockman, “Optimized nonadiabatic nanofocusing of plasmons by tapered metal rods,” J. Appl. Phys.104, 034311 (2008).
[CrossRef]

J. Phys. Chem. C (1)

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008).
[CrossRef]

J. Phys.: Condens. Matter (1)

R. Ruppin, “Non-local optics of the near field lens,” J. Phys.: Condens. Matter17, 1803–1810 (2005).
[CrossRef]

Nano Lett. (5)

A. I. Fernández-Domínguez, Y. Luo, A. Wiener, J. B. Pendry, and S. A. Maier, “Theory of three-dimensional nanocrescent light harvesters,” Nano Lett.12, 5946–5953 (2012).
[CrossRef] [PubMed]

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum description of the plasmon resonances of a nanoparticle dimer,” Nano Lett.9, 887–891 (2009).
[CrossRef] [PubMed]

D. C. Marinica, A. K. Kazansky, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer,” Nano Lett.12, 1333–1339 (2012).
[CrossRef] [PubMed]

J. A. Scholl, A. García-Etxarri, A. L. Koh, and J. A. Dionne, “Observation of quantum tunneling between two plasmonic nanoparticles,” Nano Lett.13, 564–569 (2013).
[CrossRef]

A. Wiener, A. I. Fernández-Domínguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal Effects in the Nanofocusing Performance of Plasmonic Tips,” Nano Lett.12, 3308–3314 (2012).
[CrossRef] [PubMed]

Nanophotonics (1)

G. Toscano, S. Raza, W. Yan, C. Jeppesen, S. Xiao, M. Wubs, A. P. Jauho, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in plasmonic waveguiding with extreme light confinement,” Nanophotonics2, 161–240 (2013).
[CrossRef]

Nat. Commun. (1)

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun.3, 825 (2012).
[CrossRef] [PubMed]

Nature (London) (2)

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the Quantum Regime in Tunnelling Plasmonics,” Nature (London)491, 574–577 (2012).
[CrossRef]

J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature (London)483, 421–427 (2012).
[CrossRef]

Nature Photon. (2)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon.1, 641–648 (2007).
[CrossRef]

H. Choo, M. K. Kim, M. Staffaroni, T. J. Seok, J. Bokor, S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch, “Nanofocusing in a metal–insulator–metal gap plasmon waveguide with a three-dimensional linear taper,” Nature Photon.6, 838–844 (2012).
[CrossRef]

Opt. Commun. (1)

R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Commun.190, 205–209 (2001).
[CrossRef]

Phys. Lett. (1)

K. Nerkararyan, “Superfocusing of a surface polariton in a wedge-like structure,” Phys. Lett.237, 103–105 (1997).
[CrossRef]

Phys. Rev. B (3)

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Plasmonics: Metallic Nanostructures and Their Optical Properties VIII Proc. SPIE (1)

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Figures (5)

Fig. 1
Fig. 1

(a) 3D rendering of a linearly tapered metal-insulator-metal (MIM) hourglass waveguide with translational invariance in the direction normal to the page (y-direction). The arrow indicates the direction of SPP propagation. (b) Schematic xz-plane view of the structure depicted in panel (a), with input gap thickness D, hourglass waist thickness W, arm length L′, SPP path length L, and gap (metal) permittivity ε1 and (ε2). The hourglass angle α is measured wall to wall. Note that the x and z axes are displaced for clarity. The coordinate origin is located at the center of the hourglass waist.

Fig. 2
Fig. 2

(a) Dispersion relation for the lowest SPP mode supported by a silver-air-silver MIM waveguide with gap thickness d, ranging from 1 nm to 50 nm. Black dashed (red solid) lines show local (nonlocal, β = 0.0036c) results. The inset renders the modal propagation length as a function of frequency. (b) Schematic of the infinite MIM geometry used in panel (a), fully characterised by the gap thickness d, gap permittivity ε1 and metal permittivity ε2.

Fig. 3
Fig. 3

Real part of the electric z-field component (Ez/E0) evaluated at 0.7ωSPP, plotted along the surface of a silver-air-silver plasmonic hourglass waveguide with L = 1000 nm, D = 60 nm and W = 0 nm. Dots and solid lines render FEM and analytical WKB calculations, respectively. Local (nonlocal) results are plotted in black (red).

Fig. 4
Fig. 4

(a) Frequency dependent electric z-field amplitude enhancement |Ez/E0| evaluated at the waist of the waveguide, for different hourglass waist thicknesses W, for fixed L = 1000 nm and α = 3.44°. Good agreement between WKB analytical (lines) and FEM numerical (dots) results is observed. Nonlocal and local predictions are shown in red and black, respectively (b) |Ez/E0| for ω = 0.7ωSPP, plotted as a function of the hourglass angle α. The hourglass waist is fixed to W = 0.1 nm, with all other parameters taking the same values as in panel (a).

Fig. 5
Fig. 5

(a) |Ez/E0| evaluated at 0.7ωSPP, along the surface of a silver-air-silver plasmonic hourglass waveguide with length L = 1000 nm and input gap thickness D = 60 nm. Nonlocal (local) FEM numerical results are plotted in red (black). Results for hourglass waist thicknesses of both W = 0.15 nm and W = 5 nm are rendered. (b) and (c) Nonlocal |Ez/E0| map for the W = 0.15 nm and W = 5 nm cases in panel (a). (d) Same as panel (a), but for W < 0 (g = W /(tan(α/2)). Results for separation metal thicknesses of g = 0.5 nm and g = 5 nm are rendered. (e) and (f) Nonlocal |Ez/E0| map for the g = 0.5 nm and g = 5 nm cases in panel (d).

Equations (7)

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E z ( x ) = A ( x ) e i ϕ ( x )
tanh [ q 1 ( x ) d ( x ) / 2 ] q 1 ( x ) ε 2 + q 2 ( x ) ε 1 + k ( x ) 2 ( ε 2 ε 1 ) q 3 ( x ) = 0
tanh [ q 1 ( x ) d ( x ) / 2 ] q 1 ( x ) ε 2 = q 2 ( x ) ε 1
S x ( z , x ) Re { k ( x ) ε 2 * e 2 Re { q 2 ( x ) } z } | ε 1 ε 2 e q 1 ( x ) d ( x ) / 2 + e q 1 ( x ) d ( x ) / 2 e q 2 ( x ) d ( x ) / 2 1 q 1 ( x ) | 2
S x ( z , x ) 2 Re { k ( x ) ε 1 * [ cosh ( 2 Re { q 1 ( x ) } z ) + cosh ( 2 Im { q 1 ( x ) } z ) ] } | 1 q 1 ( x ) | 2
× ( × E ) = ω 2 c 2 ε 0 ε E μ 0 J d ,
β 2 [ J d ] + ω ( ω + i γ ) J d = i ω ω P 2 ε 0 E .

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