Abstract

We numerically study the effect of the quantum spill-out (QSO) on the plasmon mode indices of an ultra-thin metallic slab, using the Fourier modal method (FMM). To improve the convergence of the FMM results, a novel nonlinear coordinate transformation is suggested and employed. Furthermore, we present a perturbative approach for incorporating the effects of QSO on the plasmon mode indices, which agrees very well with the full numerical results. The perturbative approach also provides additional physical insight, and is used to derive analytical expressions for the mode indices using a simple model for the dielectric function. The analytical expressions reproduce the results obtained from the numerically-challenging spill-out problem with much less effort and may be used for understanding the effects of QSO on other plasmonic structures.

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

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2019 (1)

E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in nanometer-thin gold slabs: Effect on the plasmon mode index and the plasmonic absorption,” Phys. Rev. B 99(15), 155427 (2019).
[Crossref]

2018 (2)

E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in few-nanometer metal gaps: Effect on gap plasmons and reflectance from ultrasharp groove arrays,” Phys. Rev. B 97(11), 115429 (2018).
[Crossref]

C. Chen and S. Li, “Valence electron density-dependent pseudopermittivity for nonlocal effects in optical properties of metallic nanoparticles,” ACS Photonics 5(6), 2295–2304 (2018).
[Crossref]

2017 (1)

A. Taghizadeh and I.-S. Chung, “Dynamical dispersion engineering in coupled vertical cavities employing a high-contrast grating,” Sci. Rep. 7(1), 2123 (2017).
[Crossref]

2016 (5)

A. Taghizadeh, J. Mork, and I.-S. Chung, “Numerical investigation of vertical cavity lasers with high-contrast gratings using the Fourier modal method,” J. Lightwave Technol. 34(18), 4240–4251 (2016).
[Crossref]

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

G. C. Park, A. Taghizadeh, and I.-S. Chung, “Hybrid grating reflectors: Origin of ultrabroad stopband,” Appl. Phys. Lett. 108(14), 141108 (2016).
[Crossref]

S. Learkthanakhachon, A. Taghizadeh, G. C. Park, K. Yvind, and I.-S. Chung, “Hybrid III–V/SOI resonant cavity enhanced photodetector,” Opt. Express 24(15), 16512–16519 (2016).
[Crossref]

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena—part II: High/low-contrast grating resonators,” IEEE J. Quantum Electron. 52(12), 1–8 (2016).
[Crossref]

2015 (3)

2014 (2)

C. David and F. J. G. de Abajo, “Surface plasmon dependence on the electron density profile at metal surfaces,” ACS Nano 8(9), 9558–9566 (2014).
[Crossref]

F. Bigourdan, J.-P. Hugonin, and P. Lalanne, “Aperiodic-Fourier modal method for analysis of body-of-revolution photonic structures,” J. Opt. Soc. Am. A 31(6), 1303–1311 (2014).
[Crossref]

2013 (1)

J. Ctyroky, P. Kwiecien, and I. Richter, “Analysis of hybrid dielectric-plasmonic slot waveguide structures with 3D Fourier modal methods,” J. Eur. Opt. Soc. Rapid Publ. 8, 13024 (2013).
[Crossref]

2012 (2)

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3(1), 825 (2012).
[Crossref]

J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and applications,” J. Phys. D: Appl. Phys. 45(11), 113001 (2012).
[Crossref]

2010 (5)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum plasmonics: Optical properties and tunability of metallic nanorods,” ACS Nano 4(9), 5269–5276 (2010).
[Crossref]

J. Ctyroky, P. Kwiecien, and I. Richter, “Fourier series-based bidirectional propagation algorithm with adaptive spatial resolution,” J. Lightwave Technol. 28(20), 2969–2976 (2010).
[Crossref]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27(11), 2423–2431 (2010).
[Crossref]

2009 (1)

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photonics 1(3), 484–588 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (1)

2005 (2)

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37(1-3), 107–119 (2005).
[Crossref]

H. Dötsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22(1), 240–253 (2005).
[Crossref]

2004 (2)

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

2002 (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

2001 (1)

2000 (1)

1997 (1)

1996 (2)

1995 (1)

1994 (1)

C. Themistos, B. Rahman, and K. Grattan, “Finite element analysis for lossy optical waveguides by using perturbation techniques,” IEEE Photonics Technol. Lett. 6(4), 537–539 (1994).
[Crossref]

1981 (1)

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
[Crossref]

1976 (1)

M. Hashimoto, “A perturbation method for the analysis of wave propagation in inhomogeneous dielectric waveguides with perturbed media,” IEEE Trans. Microwave Theory Tech. 24(9), 559–566 (1976).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

1970 (1)

N. D. Lang and W. Kohn, “Theory of metal surfaces: Charge density and surface energy,” Phys. Rev. B 1(12), 4555–4568 (1970).
[Crossref]

1965 (1)

G. Gabriel and M. Brodwin, “The solution of guided waves in inhomogeneous anisotropic media by perturbation and variational methods,” IEEE Trans. Microwave Theory Tech. 13(3), 364–370 (1965).
[Crossref]

Aizpurua, J.

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3(1), 825 (2012).
[Crossref]

Armaroli, A.

Bahlmann, N.

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Baumberg, J. J.

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

Bellanca, G.

Benech, P.

Berini, P.

Berolo, E.

Bienstman, P.

Bigourdan, F.

Boltasseva, A.

Borisov, A. G.

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3(1), 825 (2012).
[Crossref]

Bozhevolnyi, S. I.

Brodwin, M.

G. Gabriel and M. Brodwin, “The solution of guided waves in inhomogeneous anisotropic media by perturbation and variational methods,” IEEE Trans. Microwave Theory Tech. 13(3), 364–370 (1965).
[Crossref]

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

R. Zia, M. D. Selker, P. B. Catrysse, and M. L. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21(12), 2442–2446 (2004).
[Crossref]

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Cao, Q.

Catrysse, P. B.

Charbonneau, R.

Chen, C.

C. Chen and S. Li, “Valence electron density-dependent pseudopermittivity for nonlocal effects in optical properties of metallic nanoparticles,” ACS Photonics 5(6), 2295–2304 (2018).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Chung, I.-S.

Crozier, K. B.

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

Ctyroky, J.

J. Ctyroky, P. Kwiecien, and I. Richter, “Analysis of hybrid dielectric-plasmonic slot waveguide structures with 3D Fourier modal methods,” J. Eur. Opt. Soc. Rapid Publ. 8, 13024 (2013).
[Crossref]

J. Ctyroky, P. Kwiecien, and I. Richter, “Fourier series-based bidirectional propagation algorithm with adaptive spatial resolution,” J. Lightwave Technol. 28(20), 2969–2976 (2010).
[Crossref]

David, C.

C. David and F. J. G. de Abajo, “Surface plasmon dependence on the electron density profile at metal surfaces,” ACS Nano 8(9), 9558–9566 (2014).
[Crossref]

de Abajo, F. J. G.

C. David and F. J. G. de Abajo, “Surface plasmon dependence on the electron density profile at metal surfaces,” ACS Nano 8(9), 9558–9566 (2014).
[Crossref]

Debernardi, P.

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena—part II: High/low-contrast grating resonators,” IEEE J. Quantum Electron. 52(12), 1–8 (2016).
[Crossref]

Denecker, B.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Dötsch, H.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Esteban, R.

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3(1), 825 (2012).
[Crossref]

Evers, F.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
[Crossref]

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

Gabriel, G.

G. Gabriel and M. Brodwin, “The solution of guided waves in inhomogeneous anisotropic media by perturbation and variational methods,” IEEE Trans. Microwave Theory Tech. 13(3), 364–370 (1965).
[Crossref]

Gaylord, T. K.

Gerhardt, R.

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

Granet, G.

Grann, E. B.

Grattan, K.

C. Themistos, B. Rahman, and K. Grattan, “Finite element analysis for lossy optical waveguides by using perturbation techniques,” IEEE Photonics Technol. Lett. 6(4), 537–539 (1994).
[Crossref]

Gregersen, N.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC, 2018), chap. 6.

Griffiths, D.

D. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005), chap. 6, Pearson international edition.

Guizal, B.

Hammer, M.

Hashimoto, M.

M. Hashimoto, “A perturbation method for the analysis of wave propagation in inhomogeneous dielectric waveguides with perturbed media,” IEEE Trans. Microwave Theory Tech. 24(9), 559–566 (1976).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012), p. 375, 2nd ed.

Hertel, P.

Hugonin, J. P.

G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express 15(18), 11042–11060 (2007).
[Crossref]

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37(1-3), 107–119 (2005).
[Crossref]

Hugonin, J.-P.

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Johnson, S. G.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
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Kobayashi, T.

Kohn, W.

N. D. Lang and W. Kohn, “Theory of metal surfaces: Charge density and surface energy,” Phys. Rev. B 1(12), 4555–4568 (1970).
[Crossref]

Kwiatkowski, A.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
[Crossref]

Kwiecien, P.

J. Ctyroky, P. Kwiecien, and I. Richter, “Analysis of hybrid dielectric-plasmonic slot waveguide structures with 3D Fourier modal methods,” J. Eur. Opt. Soc. Rapid Publ. 8, 13024 (2013).
[Crossref]

J. Ctyroky, P. Kwiecien, and I. Richter, “Fourier series-based bidirectional propagation algorithm with adaptive spatial resolution,” J. Lightwave Technol. 28(20), 2969–2976 (2010).
[Crossref]

Lægsgaard, J.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC, 2018), chap. 6.

Lalanne, P.

Lang, N. D.

N. D. Lang and W. Kohn, “Theory of metal surfaces: Charge density and surface energy,” Phys. Rev. B 1(12), 4555–4568 (1970).
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A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC, 2018), chap. 6.

Learkthanakhachon, S.

Lecamp, G.

Lee, K. J.

Leosson, K.

Lezec, H. J.

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

Li, L.

Li, S.

C. Chen and S. Li, “Valence electron density-dependent pseudopermittivity for nonlocal effects in optical properties of metallic nanoparticles,” ACS Photonics 5(6), 2295–2304 (2018).
[Crossref]

Lisicka-Shrzek, E.

Magnusson, R.

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, US, 2007), p. 21.

Matias, I. R.

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37(1-3), 107–119 (2005).
[Crossref]

Mattheij, R.

Maubach, J.

Moharam, M. G.

Morand, A.

Morimoto, A.

Mork, J.

Mørk, J.

Mortensen, N. A.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
[Crossref]

Nikolajsen, T.

Nordlander, P.

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3(1), 825 (2012).
[Crossref]

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum plasmonics: Optical properties and tunability of metallic nanorods,” ACS Nano 4(9), 5269–5276 (2010).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012), p. 375, 2nd ed.

Olyslager, F.

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E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in nanometer-thin gold slabs: Effect on the plasmon mode index and the plasmonic absorption,” Phys. Rev. B 99(15), 155427 (2019).
[Crossref]

E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in few-nanometer metal gaps: Effect on gap plasmons and reflectance from ultrasharp groove arrays,” Phys. Rev. B 97(11), 115429 (2018).
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Pissoort, D.

Pommet, D. A.

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J. Ctyroky, P. Kwiecien, and I. Richter, “Analysis of hybrid dielectric-plasmonic slot waveguide structures with 3D Fourier modal methods,” J. Eur. Opt. Soc. Rapid Publ. 8, 13024 (2013).
[Crossref]

J. Ctyroky, P. Kwiecien, and I. Richter, “Fourier series-based bidirectional propagation algorithm with adaptive spatial resolution,” J. Lightwave Technol. 28(20), 2969–2976 (2010).
[Crossref]

Rockstuhl, C.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
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D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
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A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC, 2018), chap. 6.

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J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
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Setija, I.

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E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in nanometer-thin gold slabs: Effect on the plasmon mode index and the plasmonic absorption,” Phys. Rev. B 99(15), 155427 (2019).
[Crossref]

E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in few-nanometer metal gaps: Effect on gap plasmons and reflectance from ultrasharp groove arrays,” Phys. Rev. B 97(11), 115429 (2018).
[Crossref]

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

Søndergaard, T.

E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in nanometer-thin gold slabs: Effect on the plasmon mode index and the plasmonic absorption,” Phys. Rev. B 99(15), 155427 (2019).
[Crossref]

E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in few-nanometer metal gaps: Effect on gap plasmons and reflectance from ultrasharp groove arrays,” Phys. Rev. B 97(11), 115429 (2018).
[Crossref]

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC, 2018), chap. 6.

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G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
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Takahara, J.

Taki, H.

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R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena—part II: High/low-contrast grating resonators,” IEEE J. Quantum Electron. 52(12), 1–8 (2016).
[Crossref]

Toscano, G.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
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Villar, I. D.

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37(1-3), 107–119 (2005).
[Crossref]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Wilkens, L.

Wubs, M.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
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Xu, H.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
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J. Zhang, L. Zhang, and W. Xu, “Surface plasmon polaritons: physics and applications,” J. Phys. D: Appl. Phys. 45(11), 113001 (2012).
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W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
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Zia, R.

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J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum plasmonics: Optical properties and tunability of metallic nanorods,” ACS Nano 4(9), 5269–5276 (2010).
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ACS Nano (2)

J. Zuloaga, E. Prodan, and P. Nordlander, “Quantum plasmonics: Optical properties and tunability of metallic nanorods,” ACS Nano 4(9), 5269–5276 (2010).
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C. David and F. J. G. de Abajo, “Surface plasmon dependence on the electron density profile at metal surfaces,” ACS Nano 8(9), 9558–9566 (2014).
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C. Chen and S. Li, “Valence electron density-dependent pseudopermittivity for nonlocal effects in optical properties of metallic nanoparticles,” ACS Photonics 5(6), 2295–2304 (2018).
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G. C. Park, A. Taghizadeh, and I.-S. Chung, “Hybrid grating reflectors: Origin of ultrabroad stopband,” Appl. Phys. Lett. 108(14), 141108 (2016).
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IEEE J. Quantum Electron. (1)

R. Orta, A. Tibaldi, and P. Debernardi, “Bimodal resonance phenomena—part II: High/low-contrast grating resonators,” IEEE J. Quantum Electron. 52(12), 1–8 (2016).
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IEEE Photonics Technol. Lett. (1)

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Nat. Commun. (3)

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nat. Commun. 3(1), 825 (2012).
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G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nat. Commun. 6(1), 7132 (2015).
[Crossref]

W. Zhu, R. Esteban, A. G. Borisov, J. J. Baumberg, P. Nordlander, H. J. Lezec, J. Aizpurua, and K. B. Crozier, “Quantum mechanical effects in plasmonic structures with subnanometre gaps,” Nat. Commun. 7(1), 11495 (2016).
[Crossref]

Nat. Mater. (1)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
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D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
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Opt. Express (5)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quantum Electron. 37(1-3), 107–119 (2005).
[Crossref]

Phys. Rev. B (4)

E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in nanometer-thin gold slabs: Effect on the plasmon mode index and the plasmonic absorption,” Phys. Rev. B 99(15), 155427 (2019).
[Crossref]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

N. D. Lang and W. Kohn, “Theory of metal surfaces: Charge density and surface energy,” Phys. Rev. B 1(12), 4555–4568 (1970).
[Crossref]

E. J. H. Skjølstrup, T. Søndergaard, and T. G. Pedersen, “Quantum spill-out in few-nanometer metal gaps: Effect on gap plasmons and reflectance from ultrasharp groove arrays,” Phys. Rev. B 97(11), 115429 (2018).
[Crossref]

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

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D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
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A. Taghizadeh and I.-S. Chung, “Dynamical dispersion engineering in coupled vertical cavities employing a high-contrast grating,” Sci. Rep. 7(1), 2123 (2017).
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Other (4)

D. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall, 2005), chap. 6, Pearson international edition.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012), p. 375, 2nd ed.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, US, 2007), p. 21.

A. V. Lavrinenko, J. Lægsgaard, N. Gregersen, F. Schmidt, and T. Søndergaard, Numerical Methods in Photonics (CRC, 2018), chap. 6.

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Figures (4)

Fig. 1.
Fig. 1. (a) Schematic of a slab waveguide with a thickness of $d$, with two mapped regions used for coordinate transformation $x=F(x')$, see the text. We choose the mapped regions to be far from the waveguide to avoid overlap with the regions affected by the QSO, e.g. $d_1=d+20a$. (b) Real (solid) and imaginary (dashed) parts of the dielectric function $\epsilon (x)$ across the metal slab including the QSO ($d=2$ nm). The blue curve is a typical profile obtained from a DFT calculation (using a jellium model), whereas the red curve shows a fitted profile using Eq. (7) with $a=0.09$ nm.
Fig. 2.
Fig. 2. Real (a,c) and imaginary (b,d) parts of the long-range (a,b) and short-range (c,d) SPP mode indices $\beta$ with QSO, normalized to the corresponding values without QSO, $\beta _0$, versus the spill-out parameter $a$. The calculations are performed for three waveguide widths: $d=1$ nm (blue), $d=2$ nm (red), $d=5$ nm (green). The results are obtained for a gold slab ($\epsilon _m=-21.995+ 1.363i$[24] and $\epsilon _b=8.778 + 0.056i$) surrounded by glass ($\epsilon _1=2.25$) at the wavelength of 775 nm, modeled by the dielectric function in Eq. (7). The solid-lines are obtained from the FMM by solving Eq. (3) numerically, while the dashed-lines are the perturbation results using Eq. (5). The stars show the analytical results from Eqs. (8) and (9). For small values of $a$, the effect of QSO vanishes and hence, all curves converge toward one as expected, whereas for large values of $a$ the perturbative results deviate from the numerical solutions.
Fig. 3.
Fig. 3. Real (red) and imaginary (blue) parts of the short-range (a) and long-range (b) SPPs with QSO, $\beta$, normalized to the corresponding values without QSO, $\beta _0$, versus the waveguide width $d$. Three different methods are used: numerical method (FMM) with the DFT-based dielectric profile, perturbation approach with the DFT-based dielectric profile (Pert.) and analytical solution (Ana.) with the toy dielectric profile ($a=0.09$ nm for all $d$’s). The gold slab is surrounded by glass ($\epsilon _1=2.25$) and the results are obtained at 775 nm wavelength, see caption of Fig. 2.
Fig. 4.
Fig. 4. Normalized amplitude of (a) $H_y(x)$, (b) $E_z(x)$, (c) $E_x(x)$ profiles obtained for the short-range SPP of a 2 nm-thick gold slab, when QSO is included (red) or neglected (blue). QSO mainly modifies $E_x$ in a narrow region outside the slab and close to the interfaces where the field amplitude becomes very large. (d) The relative error of mode indices for the short-range (blue) and long-range (red) SPPs for the slab waveguide with classical dielectric response versus the number of Fourier functions, $N_t$, for two coordinate transformations: TR1 the transformation suggested in [22], and TR2 the transformation in Eq. (11). The relative error is defined as $|(\beta _0^{\textrm {FMM}}-\beta _0)/\beta _0|$, where $\beta _0^{\textrm {FMM}}$ is obtained by using the FMM for the classical dielectric function, $\epsilon ^{(0)}(x)$.

Equations (12)

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ϵ x x ( x ) d d x [ 1 ϵ z z ( x ) d H y ( x ) d x ] + k 0 2 ϵ x x ( x ) H y ( x ) = β 2 H y ( x ) .
H y ( 0 ) ( x ) = { A 1 e κ 1 x x < d / 2 B 1 cos ( q x ) + B 2 sin ( q x ) | x | < d / 2 A 2 e κ 2 x x > d / 2 ,
A 1 ( k 0 2 I F K E 1 F K ) h = β 2 h
P β d x | H y | 2 ϵ ( x ) = d x H y β { d d x [ 1 ϵ ( x ) d H y d x ] + k 0 2 H y ( x ) } β 0 2 β d x | H y ( 0 ) | 2 ϵ ( 0 ) ( x ) β 0 β P ( 0 ) .
β 2 β 0 2 d x | H y ( 0 ) ( x ) | 2 [ ϵ ( 0 ) ( x ) ] 1 d x | H y ( 0 ) ( x ) | 2 ϵ 1 ( x ) = β 0 2 t 0 t 0 + Δ t .
ϵ ( x ) = 1 ω p 2 ( x ) ω ( ω + i γ ) + ϵ b θ ( d / 2 | x | ) + ( ϵ 1 1 ) θ ( | x | d / 2 ) ,
ϵ t ( x ) = 1 + ϵ D 1 4 ( 1 e 2 d / a ) [ tanh ( x + d / 2 a ) + 1 ] [ tanh ( x + d / 2 a ) + 1 ] + ϵ b θ ( d / 2 | x | ) + ( ϵ 1 1 ) θ ( | x | d / 2 ) ,
t 0 = 1 ϵ 1 κ r + 1 ϵ m sinh ( q i d ) / q i ± sin ( q r d ) / q r cosh ( q i d ) ± cos ( q r d )
Δ t 2 | H y ( 0 ) ( d / 2 ) | 2 d / 2 d x { 2 ϵ p + 2 ϵ 1 + ϵ p tanh [ ( d / 2 x ) / a ] 1 ϵ 1 } + 2 | H y ( 0 ) ( d / 2 ) | 2 d / 2 d x { 2 ϵ m + ϵ b + 1 + ϵ p tanh [ ( d / 2 x ) / a ] 1 ϵ m } = a ϵ p [ log ( 2 ϵ p / ϵ m ) ϵ m ( ϵ m ϵ p ) log ( 2 + ϵ p / ϵ 1 ) ϵ 1 ( ϵ 1 + ϵ p ) ] ,
( β ) ( β 0 ) | d 1 + a k 0 ϵ 1 2 ϵ m r [ log ( 2 + ϵ p / ϵ 1 ) ] ,
( β ) ( β 0 ) | d ( β ) ( β 0 ) | d + ( 2 ϵ m r 2 ϵ 1 ϵ m i ) a k 0 ϵ 1 2 ϵ m r [ log ( 2 + ϵ p / ϵ 1 ) ] .
x = F ( x ) = { x | x | d 1 / 2 x | x | d 1 2 e 2 | x | / d 1 1 d 1 / 2 < | x | d 2 / 2 x | x | e 2 d 2 / d 1 1 [ d 1 2 + Λ d 2 2 tan ( π | x | d 2 / 2 Λ d 2 ) ] d 2 / 2 < | x | Λ / 2 .

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