Double-negative electromagnetic metamaterials due to chirality
Authors:
Habib Ammari, Wei Wu and Sanghyeon Yu
Journal:
Quart. Appl. Math. 77 (2019), 105-130
MSC (2010):
Primary 35Q60, 35C20
DOI:
https://doi.org/10.1090/qam/1516
Published electronically:
September 17, 2018
MathSciNet review:
3897921
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Abstract: The aim of this paper is to provide a mathematical theory for understanding the mechanism behind the double-negative refractive index phenomenon in chiral materials. The design of double-negative metamaterials generally requires the use of two different kinds of sub-wavelength resonators, which may limit the applicability of double-negative metamaterials. Herein, we rely on media that consist of only a single-type of dielectric resonant element, and show how the chirality of the background medium induces double-negative refractive index metamaterial, which refracts waves negatively, hence acting as a superlens. Using plasmonic dielectric particles, it is proved that both the effective electric permittivity and the magnetic permeability can be negative near some resonant frequencies. A justification of the approximation of a plasmonic particle in a chiral medium by the sum of a resonant electric dipole and a resonant magnetic dipole is provided. Moreover, the set of resonant frequencies is characterized. For an appropriate volume fraction of plasmonic particles with certain conditions on their configuration, a double-negative effective medium can be obtained when the frequency is near one of the resonant frequencies.
References
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- M. Minnaert, On musical air-bubbles and the sounds of running water, The London, Edinburgh, Dublin Philos. Mag. and J. of Sci., 16 (1933), 235–248.
- Marius Mitrea, The method of layer potentials for electromagnetic waves in chiral media, Forum Math. 13 (2001), no. 3, 423–446. MR 1831094, DOI https://doi.org/10.1515/form.2001.016
- Shin Ozawa, Point interaction potential approximation for $(-\Delta +U)^{-1}$ and eigenvalues of the Laplacian on wildly perturbed domain, Osaka J. Math. 20 (1983), no. 4, 923–937. MR 727440
- Shin Ozawa, On an elaboration of M. Kac’s theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles, Comm. Math. Phys. 91 (1983), no. 4, 473–487. MR 727196
- George C. Papanicolaou, Diffusion in random media, Surveys in applied mathematics, Vol. 1, Surveys Appl. Math., vol. 1, Plenum, New York, 1995, pp. 205–253. MR 1366209
- J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), 3966–3969.
- J. B. Pendry, A Chiral route to negative refraction, Science, 306 (2004), 1353–1355.
- V. M. Shalaev, Optical negative-index metamaterials, Nature Photonics, 1 (2007), 41–48.
- D. R. Smith, J. B. Pendry, and M. C. K. Whiltshire, Metamaterials and negative refractive index, Science, 305 (2004), 788–792.
- C. M. Soukoulis and M. Wegener, Past achievements and future challenges in the development of three-dimensional photonic materials, Nature Photonics, 5 (2011), 523–530.
- V. G. Veselago, The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$. Sov. Phys. Usp., 10 (1968), 509–514.
- V. G. Veselago and E. E. Narimanov, The left hand of brightness: past, present and future of negative index materials, Nature Materials, 5 (2006), 759–762.
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References
- Habib Ammari, Youjun Deng, and Pierre Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal. 220 (2016), no. 1, 109–153. MR 3458160, DOI https://doi.org/10.1007/s00205-015-0928-0
- H. Ammari, B. Fitzpatrick, D. Gontier, H. Lee, and H. Zhang, Minnaert resonances for acoustic waves in bubbly media, arXiv:1603.03982, 2016.
- H. Ammari, B. Fitzpatrick, H. Kang, M. Ruiz, S. Yu, and H. Zhang, Mathematical and Computational Methods in Photonics and Phononics, to appear (SAM Research Report No. 2017-05).
- H. Ammari, B. Fitzpatrick, H. Lee, S. Yu, and H. Zhang, Double-negative acoustic metamaterials, arXiv:1709.08177.
- H. Ammari, K. Hamdache, and J.-C. Nédélec, Chirality in the Maxwell equations by the dipole approximation, SIAM J. Appl. Math. 59 (1999), no. 6, 2045–2059. MR 1709796, DOI https://doi.org/10.1137/S0036139998334160
- Habib Ammari and Abdessatar Khelifi, Electromagnetic scattering by small dielectric inhomogeneities, J. Math. Pures Appl. (9) 82 (2003), no. 7, 749–842 (English, with English and French summaries). MR 2005296, DOI https://doi.org/10.1016/S0021-7824%2803%2900033-3
- H. Ammari, M. Laouadi, and J.-C. Nédélec, Low frequency behavior of solutions to electromagnetic scattering problems in chiral media, SIAM J. Appl. Math. 58 (1998), no. 3, 1022–1042. MR 1616643, DOI https://doi.org/10.1137/S0036139996310431
- Habib Ammari, Pierre Millien, Matias Ruiz, and Hai Zhang, Mathematical analysis of plasmonic nanoparticles: the scalar case, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 597–658. MR 3614756, DOI https://doi.org/10.1007/s00205-017-1084-5
- H. Ammari and J. C. Nédélec, Time-harmonic electromagnetic fields in chiral media, Modern mathematical methods in diffraction theory and its applications in engineering (Freudenstadt, 1996) Methoden Verfahren Math. Phys., vol. 42, Peter Lang, Frankfurt am Main, 1997, pp. 174–202. MR 1482562
- H. Ammari and J. C. Nédélec, Time-harmonic electromagnetic fields in thin chiral curved layers, SIAM J. Math. Anal. 29 (1998), no. 2, 395–423. MR 1616499, DOI https://doi.org/10.1137/S0036141096305504
- Habib Ammari, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations, J. Differential Equations 261 (2016), no. 6, 3615–3669. MR 3527640, DOI https://doi.org/10.1016/j.jde.2016.05.036
- Habib Ammari, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Reconstructing fine details of small objects by using plasmonic spectroscopic data, SIAM J. Imaging Sci. 11 (2018), no. 1, 1–23. MR 3742683, DOI https://doi.org/10.1137/17M1126540
- Habib Ammari, Michael S. Vogelius, and Darko Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. II. The full Maxwell equations, J. Math. Pures Appl. (9) 80 (2001), no. 8, 769–814. MR 1860816, DOI https://doi.org/10.1016/S0021-7824%2801%2901217-X
- Habib Ammari and Hai Zhang, Effective medium theory for acoustic waves in bubbly fluids near Minnaert resonant frequency, SIAM J. Math. Anal. 49 (2017), no. 4, 3252–3276. MR 3690650, DOI https://doi.org/10.1137/16M1078574
- Kazunori Ando and Hyeonbae Kang, Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator, J. Math. Anal. Appl. 435 (2016), no. 1, 162–178. MR 3423389, DOI https://doi.org/10.1016/j.jmaa.2015.10.033
- Kazunori Ando, Hyeonbae Kang, and Hongyu Liu, Plasmon resonance with finite frequencies: a validation of the quasi-static approximation for diametrically small inclusions, SIAM J. Appl. Math. 76 (2016), no. 2, 731–749. MR 3479707, DOI https://doi.org/10.1137/15M1025943
- Christodoulos Athanasiadis, George Costakis, and Ioannis G. Stratis, Electromagnetic scattering by a homogeneous chiral obstacle in a chiral environment, IMA J. Appl. Math. 64 (2000), no. 3, 245–258. MR 1764940, DOI https://doi.org/10.1093/imamat/64.3.245
- Christodoulos E. Athanasiadis, Sotiria Dimitroula, Eleftheria Kikeri, and Konstantinos I. Skourogiannis, Aspects of electromagnetic scattering in chiral media, Math. Methods Appl. Sci. 40 (2017), no. 6, 2071–2077. MR 3624081
- C. Athanasiadis, P. A. Martin, and I. G. Stratis, Electromagnetic scattering by a homogeneous chiral obstacle: boundary integral equations and low-chirality approximations, SIAM J. Appl. Math. 59 (1999), no. 5, 1745–1762. MR 1710541, DOI https://doi.org/10.1137/S003613999833633X
- R. E. Caflisch, M. J. Miksis, G. C. Papanicolaou, and L. Ting, Effective equations for wave propagation in bubbly liquids, J. Fluid Mech., 153 (1985), 259-273.
- David Colton and Rainer Kress, Inverse acoustic and electromagnetic scattering theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998. MR 1635980
- R. Figari, G. Papanicolaou, and J. Rubinstein, Remarks on the point interaction approximation, Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986) IMA Vol. Math. Appl., vol. 9, Springer, New York, 1987, pp. 45–55. MR 914984, DOI https://doi.org/10.1007/978-1-4684-6347-7_4
- Leslie L. Foldy, The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev. (2) 67 (1945), 107–119. MR 0011845
- M. Minnaert, On musical air-bubbles and the sounds of running water, The London, Edinburgh, Dublin Philos. Mag. and J. of Sci., 16 (1933), 235–248.
- Marius Mitrea, The method of layer potentials for electromagnetic waves in chiral media, Forum Math. 13 (2001), no. 3, 423–446. MR 1831094, DOI https://doi.org/10.1515/form.2001.016
- Shin Ozawa, Point interaction potential approximation for $(-\Delta +U)^{-1}$ and eigenvalues of the Laplacian on wildly perturbed domain, Osaka J. Math. 20 (1983), no. 4, 923–937. MR 727440
- Shin Ozawa, On an elaboration of M. Kac’s theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles, Comm. Math. Phys. 91 (1983), no. 4, 473–487. MR 727196
- George C. Papanicolaou, Diffusion in random media, Surveys in applied mathematics, Vol. 1, Surveys Appl. Math., vol. 1, Plenum, New York, 1995, pp. 205–253. MR 1366209
- J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), 3966–3969.
- J. B. Pendry, A Chiral route to negative refraction, Science, 306 (2004), 1353–1355.
- V. M. Shalaev, Optical negative-index metamaterials, Nature Photonics, 1 (2007), 41–48.
- D. R. Smith, J. B. Pendry, and M. C. K. Whiltshire, Metamaterials and negative refractive index, Science, 305 (2004), 788–792.
- C. M. Soukoulis and M. Wegener, Past achievements and future challenges in the development of three-dimensional photonic materials, Nature Photonics, 5 (2011), 523–530.
- V. G. Veselago, The electrodynamics of substances with simultaneously negative values of $\epsilon$ and $\mu$. Sov. Phys. Usp., 10 (1968), 509–514.
- V. G. Veselago and E. E. Narimanov, The left hand of brightness: past, present and future of negative index materials, Nature Materials, 5 (2006), 759–762.
- S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, Negative refractive index in chiral metamaterials, Phys. Rev. Lett., 102 (2009), 023901.
- J. Zhou, J. Dong, B. Wang, T. Koschny, M. Kafesaki, and C.M. Soukoulis, Negative refractive index due to chirality, Phys. Rev. B, 79 (2009), 121104(R).
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Additional Information
Habib Ammari
Affiliation:
Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
MR Author ID:
353050
Email:
habib.ammari@math.ethz.ch
Wei Wu
Affiliation:
Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
MR Author ID:
1187163
Email:
wei.wu@sam.math.ethz.ch
Sanghyeon Yu
Affiliation:
Department of Mathematics, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
MR Author ID:
959778
Email:
sanghyeon.yu@sam.math.ethz.ch
Keywords:
Plasmonic nanoparticles,
sub-wavelength resonance,
electric and magnetic resonant dipoles,
chiral materials,
effective medium theory,
double-negative metamaterials
Received by editor(s):
April 3, 2018
Published electronically:
September 17, 2018
Article copyright:
© Copyright 2018
Brown University