Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T08:00:14.228Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic approximations for the plasmon resonances of nearly touching spheres

Part of: Representations of solutions Spectral theory and eigenvalue problems Optics, electromagnetic theory - Basic methods

Published online by Cambridge University Press:  09 January 2019

O. SCHNITZER
O. SCHNITZER*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations.

Keywords

Surface plasmonsasymptotic analysiseigenvalue problems

MSC classification

Primary: 78M35: Asymptotic analysis
Secondary: 35C20: Asymptotic expansions 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 2 , April 2020 , pp. 246 - 276
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author acknowledges funding from EPSRC New Investigator Award EP/R041458/1.

References

Abramowitz, M. & Stegun, I. A. (1972) Handbook of Mathematical Functions, Dover, New York.Google Scholar
Agranovich, M. S., Katsenelenbaum, B. Z., Sivov, A. N. & Voitovich, N. N. (1999) Generalized Method of Eigenoscillations in Diffraction Theory, Vch Pub, Berlin, Germany.Google Scholar
Ammari, H., Millien, P., Ruiz, M. & Zhang, H. (2017) Mathematical analysis of plasmonic nanoparticles: the scalar case. Arch. Ration. Mech. Anal. 224(2), 597–658.CrossRefGoogle Scholar
Ammari, H., Ruiz, M., Yu, S. & Zhang, H. (2016) Mathematical analysis of plasmonic resonances for nanoparticles: the full maxwell equations. J. Differ. Equations 261(6), 3615–3669.CrossRefGoogle Scholar
Ando, K. & Kang, H. (2016) Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann–Poincaré operator. J. Math. Anal. Appl. 435(1), 162–178.CrossRefGoogle Scholar
Anker, J. N., Hall, W. P., Lyandres, O., Shah, N. C., Zhao, J. & Van Duyne, R. P. (2008) Biosensing with plasmonic nanosensors. Nat. Mater. 7(6), 442–453.CrossRefGoogle ScholarPubMed
Atwater, H. A. & Polman, A. (2010) Plasmonics for improved photovoltaic devices. Nat. Mater. 9(3), 205–213.CrossRefGoogle ScholarPubMed
Bender, C. M. & Orszag, S. A. (2013) Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer Science & Business Media, Berlin, Germany.Google Scholar
Bergman, D. J. (1979) Dielectric constant of a two-component granular composite: a practical scheme for calculating the pole spectrum. Phys. Rev. B 19(4), 2359.CrossRefGoogle Scholar
Bergman, D. J. & Farhi, A. (2018) Spectral method for the static electric potential of a charge density in a composite medium. Phys. Rev. A 97(4), 043855.CrossRefGoogle Scholar
Bergman, D. J. & Stockman, M. I. (2003) Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems. Phys. Rev. Lett. 90(2), 027402.CrossRefGoogle ScholarPubMed
Bergman, D. J. & Stroud, D. (1980) Theory of resonances in the electromagnetic scattering by macroscopic bodies. Phys. Rev. B 22(8), 3527.CrossRefGoogle Scholar
Chen, P. Y., Bergman, D. J. & Sivan, Y. (2017) Generalizing normal mode expansion of electromagnetic green’s tensor to lossy resonators in open systems. arXiv preprint arXiv:1711.00335.Google Scholar
Chen, H., Shao, L., Li, Q., & Wang, J. (2013) Gold nanorods and their plasmonic properties. Chem. Soc. Rev. 42(7), 2679–2724.CrossRefGoogle ScholarPubMed
Chuntonov, L. & Haran, G. (2011) Effect of symmetry breaking on the mode structure of trimeric plasmonic molecules. J. Phys. Chem. C 115(40), 19488–19495.CrossRefGoogle Scholar
Davis, T. J. & Gómez, D. E. (2017) Colloquium: an algebraic model of localized surface plasmons and their interactions. Rev. Mod. Phys. 89(1), 011003.CrossRefGoogle Scholar
Engheta, N. (2007) Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials. Science 317(5845), 1698–1702.CrossRefGoogle ScholarPubMed
Farhi, A. & Bergman, D. J. (2016) Electromagnetic eigenstates and the field of an oscillating point electric dipole in a flat-slab composite structure. Phys. Rev. A 93(6), 063844.CrossRefGoogle Scholar
Fredkin, D. R. & Mayergoyz, I. D. (2003) Resonant behavior of dielectric objects (electrostatic resonances). Phys. Rev. Lett. 91(25), 253902.CrossRefGoogle Scholar
Giannini, V., Fernández-Domínguez, A. I., Heck, S. C. & Maier, S. A. (2011) Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters. Chem. Rev. 111(6), 3888–3912.CrossRefGoogle ScholarPubMed
Grieser, D. The plasmonic eigenvalue problem. Rev. Math. Phys. 26(03), 1450005 (2014).CrossRefGoogle Scholar
Grieser, D., Uecker, H., Biehs, S., Huth, O., Rüting, F. & Holthaus, M. (2009) Perturbation theory for plasmonic eigenvalues. Phys. Rev. B 80(24), 245405.CrossRefGoogle Scholar
Gunnarsson, L., Rindzevicius, T., Prikulis, J., Kasemo, B., Käll, M., Zou, S. & Schatz, G. C. (2005) Confined plasmons in nanofabricated single silver particle pairs: experimental observations of strong interparticle interactions. J. Phys. Chem. B 109(3), 1079–1087.CrossRefGoogle ScholarPubMed
Hill, R. T., Mock, J. J., Urzhumov, Y., Sebba, D. S., Oldenburg, S. J., Chen, S.-Y., Lazarides, A. A., Chilkoti, A. & Smith, D. R. (2010) Leveraging nanoscale plasmonic modes to achieve reproducible enhancement of light. Nano Lett. 10(10), 4150–4154.CrossRefGoogle Scholar
Hinch, E. J. (1991) Perturbation Methods, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Huang, X., El-Sayed, I. H., Qian, W. & El-Sayed, M. A. (2006) Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods. J. Am. Chem. Soc. 128(6), 2115–2120.CrossRefGoogle ScholarPubMed
Jeffrey, D. J. & Van Dyke, M. (1978) The temperature field or electric potential around two almost touching spheres. IMA J. Appl. Math. 22(3), 337–351.CrossRefGoogle Scholar
Kauranen, M. & Zayats, A. V. (2012) Nonlinear plasmonics. Nat. Photonics 6(11), 737–748.CrossRefGoogle Scholar
Klimov, V. V. (2014) Nanoplasmonics, CRC Press, Boca Raton, FL.CrossRefGoogle Scholar
Klimov, V. V. & Guzatov, D. V. (2007) Optical properties of an atom in the presence of a two-nanosphere cluster. Quantum Electron. 37(3), 209.CrossRefGoogle Scholar
Klimov, V. V. & Guzatov, D. V. (2007) Strongly localized plasmon oscillations in a cluster of two metallic nanospheres and their influence on spontaneous emission of an atom. Phys. Rev. B 75(2), 024303.CrossRefGoogle Scholar
Klimov, V. V. & Guzatov, D. V. (2007) Plasmonic atoms and plasmonic molecules. Appl. Phys. A 89(2), 305–314.CrossRefGoogle Scholar
Klimov, V. V. & Lambrecht, A. (2009) Van der Waals forces between plasmonic nanoparticles. Plasmonics 4(1), 31–36.CrossRefGoogle Scholar
Lei, D. Y., Fernández-Domínguez, A. I., Sonnefraud, Y., Appavoo, K., Haglund, R. F. Jr., Pendry, J. B. & Maier, S. A. (2012) Revealing plasmonic gap modes in particle-on-film systems using dark-field spectroscopy. ACS Nano 6(2), 1380–1386.CrossRefGoogle ScholarPubMed
Lebedev, V., Vergeles, S. & Vorobev, P. (2010) Giant enhancement of electric field between two close metallic grains due to plasmonic resonance. Opt. Lett. 35(5), 640–642.CrossRefGoogle ScholarPubMed
Lebedev, V. V., Vergeles, S. S. & Vorobev, P. E. (2013) Surface modes in metal–insulator composites with strong interaction of metal particles. Appl. Phys. B 111(4), 577–588.CrossRefGoogle Scholar
Maier, S. A. (2007) Plasmonics: Fundamentals and Applications, Springer Science & Business Media, Berlin, Germany.CrossRefGoogle Scholar
Mayergoyz, I. D. (2013) Plasmon Resonances in Nanoparticles, Vol. 6, World Scientific, Singapore.CrossRefGoogle Scholar
Mayergoyz, I. D., Fredkin, D. R. & Zhang, Z. (2005) Electrostatic (plasmon) resonances in nanoparticles. Phys. Rev. B 72(15), 155412.CrossRefGoogle Scholar
Moon, P. H. & Spencer, D. E. (1961) Field Theory Handbook: Including Coordinate Systems, Differential Equations, and Their Solutions, Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
Muskens, O. L., Giannini, V., Sanchez-Gil, J. A. & Gómez Rivas, J. (2007) Optical scattering resonances of single and coupled dimer plasmonic nanoantennas. Opt. Express 15(26), 17736–17746.CrossRefGoogle ScholarPubMed
Nordlander, P., Oubre, C., Prodan, E., Li, K. & Stockman, M. I. (2004) Plasmon hybridization in nanoparticle dimers. Nano Lett. 4(5), 899–903.CrossRefGoogle Scholar
Ouyang, F. & Isaacson, M. (1989) Surface plasmon excitation of objects with arbitrary shape and dielectric constant. Philos. Mag. 60(4), 481–492.CrossRefGoogle Scholar
Pendry, J. B., Fernández-Domínguez, A. I., Luo, Y. & Zhao, R. (2013) Capturing photons with transformation optics. Nat. Phys. 9(8), 518–522.CrossRefGoogle Scholar
Romero, I., Aizpurua, J., Bryant, G. W. & García De Abajo, F. J. (2006) Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers. Opt. Express 14(21), 9988–9999.CrossRefGoogle ScholarPubMed
Ruppin, R. (1982) Surface modes of two spheres. Phys. Rev. B 26(6), 3440.CrossRefGoogle Scholar
Sandu, T. (2013) Eigenmode decomposition of the near-field enhancement in localized surface plasmon resonances of metallic nanoparticles. Plasmonics 8(2), 391–402.CrossRefGoogle Scholar
Sauvan, C., Hugonin, J.-P., Maksymov, I. S. & Lalanne, P. (2013) Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators. Phys. Rev. Lett. 110(23), 237401.CrossRefGoogle ScholarPubMed
Schnitzer, O. (2015) Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B 92(23), 235428.CrossRefGoogle Scholar
Schnitzer, O., Giannini, V., Craster, R. V. & Maier, S. A. Asymptotics of surface-plasmon redshift saturation at subnanometric separations. Phys. Rev. B 93(4), 041409 (2016).CrossRefGoogle Scholar
Schnitzer, O., Giannini, V., Maier, S. A. & Craster, R. V. (2016) Surface plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening-length limit. Proc. R. Soc. A 472(2191), 20160258.CrossRefGoogle ScholarPubMed
Schuller, J. A., Barnard, E. S., Cai, W., Jun, Y. C., White, J. S. & Brongersma, M. L. (2010) Plasmonics for extreme light concentration and manipulation. Nat. Mater. 9(3), 193–204.CrossRefGoogle ScholarPubMed
Sneddon, I. N. (1972) The Use of Integral Transforms, McGraw-Hill, New York, NY.Google Scholar
Sperling, R. A., Gil, P. R., Zhang, F., Zanella, M. & Parak, W. J. (2008) Biological applications of gold nanoparticles. Chem. Soc. Rev. 37(9), 1896–1908.CrossRefGoogle ScholarPubMed
Sukharev, M. & Seideman, T. (2007) Light trapping and guidance in plasmonic nanocrystals. J. Chem. Phys. 126(20), 204702.CrossRefGoogle ScholarPubMed
Van Dyke, M. D. (1975) Perturbation Methods in Fluid Dynamics, Parabolic Press, Stanford, CA.Google Scholar
Voicu, R. C. & Sandu, T. (2017) Analytical results regarding electrostatic resonances of surface phonon/plasmon polaritons: separation of variables with a twist. Proc. R. Soc. A 473(2199), 20160796.CrossRefGoogle ScholarPubMed
Yu, S. & Ammari, H. (2018) Plasmonic interaction between nanospheres. SIAM Rev. 60(2), 356–385.CrossRefGoogle Scholar