Abstract
Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is fourfold: (1) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (2) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle; (3) to derive optimal bounds on the enhancement factors; (4) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation.
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Ammari, H.: An Introduction to Mathematics of Emerging Biomedical Imaging. Math. & Appl., Volume 62, Springer, Berlin, 2008
Ammari, H., Chow, Y.T., Liu, K., Zou, J.: Optimal shape design by partial spectral data. SIAM J. Sci. Compt., 37, B855–B883, 2015
Ammari, H., Chow, Y., Zou, J.: Super-resolution in highly contrasted media from the perspective of scattering coefficients. J. Math. Pures Appl., (arXiv:1410.1253)
Ammari H., Ciraolo G., Kang H., Lee H., Milton G.W.: Spectral theory of a Neumann–Poincaré-type operator and analysis of anomalous localized resonance II. Contemp. Math., 615, 1–14 (2014)
Ammari H., Ciraolo G., Kang H., Lee H., Yun K.: Spectral analysis of the Neumann–Poincaré operator and characterization of the stress concentration in anti-plane elasticity. Arch. Ration. Mech. Anal., 208, 275–304 (2013)
Ammari H., Deng Y., Millien P.: Surface plasmon resonance of nanoparticles and applications in imaging. Arch. Ration. Mech. Anal., 220, 109–153 (2016)
Ammari, H., Garnier, J., Jing, W., Kang, H., Lim, M., Sølna, K., Wang, H.: Mathematical and Statistical Methods for Multistatic Imaging, Lecture Notes in Mathematics, Volume 2098, Springer, Cham, 2013
Ammari H., Iakovleva E., Lesselier D., Perrusson G.: MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions. SIAM J. Sci. Comput., 29, 674–709 (2007)
Ammari H., Kang H., Lim M., Lee H.: Enhancement of near-cloaking. Part II: The Helmholtz equation. Comm. Math. Phys., 317, 485–502 (2013)
Ammari H., Kang H., Lee H., Lim M., Yu S.: Enhancement of near cloaking for the full Maxwell equations. SIAM J. Appl. Math., 73, 2055–2076 (2013)
Ammari H., Ruiz M., Yu S., Zhang H.: Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations. J. Differ. Equ., 261(3615–3669), 261 3615–3669 (2016)
Ammari H., Zhang H.: A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators. Commun. Math. Phys., 337, 379–428 (2015)
Ammari, H., Zhang, H.: Super-resolution in high contrast media. Proc. R. Soc. A, 2015(471), 20140946
Ando K., Kang H.: Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann–Poincaré operator. J. Math. Anal. Appl., 435, 162–178 (2016)
Ando K., Kang H., Liu H.: Plasmon resonance with finite frequencies: a validation of the quasi-static approximation for diametrically small inclusions. SIAM J. Appl. Math., 76, 731–749 (2016)
Arhab, S., Soriano, G., Ruan, Y., Maire, G., Talneau, A., Sentenac, D., Chaumet, P.C., Belkebir, K., Giovannini, H.: Nanometric resolution with far-field optical profilometry. Phys. Rev. Lett., 111, 053902, 2013
Baffou G., Girard C., Quidant R.: Mapping heat origin in plasmonic structures. Phys. Rev. Lett., 104, 136805 (2010)
Bao G., Li P.: Near-field imaging of infinite rough surfaces. SIAM J. Appl. Math., 73, 2162–2187 (2013)
Bao G., Lin J.: Near-field imaging of the surface displacement on an infinite ground plane. Inverse Probl. Imaging, 7, 377–396 (2013)
Bao G., Lin J., Triki F.: A multi-frequency inverse source problem. J. Diff. Equ., 249, 3443–3465 (2010)
Bonnetier E., Triki F.: On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D. Arch. Ration. Mech. Anal., 209, 541–567 (2013)
Born, M., Wolf, E.: Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge University Press, Cambridge 1999
Bonnet-Ben Dhia, A.S., Chesnel, L., Ciarlet, P.: T-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM Math. Model. Numer. Anal. 46, 1363–1387, 2012
Costabel M., Stephan E.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106, 367–413 (1985)
Grieser, D.: The plasmonic eigenvalue problem. Rev. Math. Phys. 26, 1450005, 2014
Giannini R., Hafner C.V., Löffler J.F.: Scaling behavior of individual nanoparticle plasmon resonances. J. Phys. Chem. C, 119, 6138–6147 (2015)
Jain P.K., Lee K.S., El-Sayed I.H., El-Sayed M.A.: Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biomedical imaging and biomedicine. J. Phys. Chem. B, 110, 7238–7248 (2006)
Kang H., Kim K., Lee H., Shin J.: Spectral properties of the Neumann–Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients. J. Lond. Math. Soc., 93, 519–546 (2016)
Kang, H., Lim, M., Yu, S.: Spectral resolution of the Neumann–Poincaré operator on intersecting disks and analysis of plamson resonance. arXiv:1501.02952
Kato, T.: Perturbation Theory for Linear Operators (2nd ed.), Springer-Verlag, Berlin 1980
Khavinson D., Putinar M., Shapiro H.S.: Poincaré’s variational problem in potential theory. Arch. Ration. Mech. Anal., 185, 143–184 (2007)
Kelly K.L., Coronado E., Zhao L.L., Schatz G.C.: The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment. J. Phys. Chem. B, 107, 668–677 (2003)
Link S., El-Sayed M.A.: Shape and size dependence of radiative, non-radiative and photothermal properties of gold nanocrystals. Int. Rev. Phys. Chem., 19, 409–453 (2000)
Mayergoyz I.D., Fredkin D.R., Zhang Z.: Electrostatic (plasmon) resonances in nanoparticles. Phys. Rev. B, 72, 155412 (2005)
Mayergoyz I.D., Zhang Z.: Numerical analysis of plasmon resonances in nanoparticules. IEEE Trans. Mag., 42, 759–762 (2006)
Miller, O.D., Hsu, C.W., Reid, M.T.H., Qiu, W., DeLacy, B.G., Joannopoulos, J.D., Soljacić, M., Johnson, S.G.: Fundamental limits to extinction by metallic nanoparticles. Phys. Rev. Lett., 112, 123903, 2014
Nguyen H.-M.: Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients. J. Math. Pures Appl. 106, 342–374 (2016)
Palomba S., Novotny L., Palmer R.E.: Blue-shifted plasmon resonance of individual size-selected gold nanoparticles. Opt. Commun., 281, 480–483 (2008)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic Press, New York 1970
Sarid, D., Challener, W.A.: Modern Introduction to Surface Plasmons: Theory, Mathematical Modeling, and Applications. Cambridge University Press, New York, 2010
Scaffardi L.B., Tocho J.O.: Size dependence of refractive index of gold nanoparticles. Nanotechnology 17, 1309–1315 (2006)
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Communicated by P. Rabinowitz
This work was supported by the ERC Advanced Grant Project MULTIMOD–267184. Hai Zhang was supported by Hong Kong RGC grant 26301016 and an initiation Grant IGN15SC05 from HKUST.
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Ammari, H., Millien, P., Ruiz, M. et al. Mathematical Analysis of Plasmonic Nanoparticles: The Scalar Case. Arch Rational Mech Anal 224, 597–658 (2017). https://doi.org/10.1007/s00205-017-1084-5
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DOI: https://doi.org/10.1007/s00205-017-1084-5