Abstract

At visible and infrared frequencies, metals show tantalizing promise for strong subwavelength resonances, but material loss typically dampens the response. We derive fundamental limits to the optical response of absorptive systems, bounding the largest enhancements possible given intrinsic material losses. Through basic conservation-of-energy principles, we derive geometry-independent limits to per-volume absorption and scattering rates, and to local-density-of-states enhancements that represent the power radiated or expended by a dipole near a material body. We provide examples of structures that approach our absorption and scattering limits at any frequency; by contrast, we find that common “antenna” structures fall far short of our radiative LDOS bounds, suggesting the possibility for significant further improvement. Underlying the limits is a simple metric, |χ|2/Im χ for a material with susceptibility χ, that enables broad technological evaluation of lossy materials across optical frequencies.

© 2016 Optical Society of America

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2015 (8)

J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotechnol. 10, 2–6 (2015).
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J.-P. Hugonin, M. Besbes, and P. Ben-Abdallah, “Fundamental limits for light absorption and scattering induced by cooperative electromagnetic interactions,” Phys. Rev. B 91, 180202 (2015).
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M. S. Eggleston, K. Messer, L. Zhang, E. Yablonovitch, and M. C. Wu, “Optical antenna enhanced spontaneous emission,” Proc. Natl. Acad. Sci. 112, 1704–1709 (2015).
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A. G. Polimeridis, M. T. H. Reid, S. G. Johnson, J. K. White, and A. W. Rodriguez, “On the computation of power in volume integral equation formulations,” IEEE Trans. Antennas Propag. 63, 611–620 (2015).
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E. Sachet, C. T. Shelton, J. S. Harris, B. E. Gaddy, D. L. Irving, S. Curtarolo, B. F. Donovan, P. E. Hopkins, P. A. Sharma, A. L. Sharma, J. Ihlefeld, S. Franzen, and J.-P. Maria, “Dysprosium-doped cadmium oxide as a gateway material for mid-infrared plasmonics,” Nat. Mater. 14, 414–420 (2015).
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S. Campione, I. Brener, and F. Marquier, “Theory of epsilon-near-zero modes in ultrathin films,” Phys. Rev. B 91, 121408 (2015).
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M. T. H. Reid and S. G. Johnson, “Efficient computation of power, force, and torque in BEM scattering calculations,” IEEE Trans. Antennas Propag. 63, 3588–3598 (2015).
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O. D. Miller, S. G. Johnson, and A. W. Rodriguez, “Shape-independent limits to near-field radiative heat transfer,” Phys. Rev. Lett. 115, 204302 (2015).
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2014 (10)

S. Esterhazy, D. Liu, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90, 023816 (2014).
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C. W. Hsu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Theoretical criteria for scattering dark states in nanostructured particles,” Nano Lett. 14, 2783–2788 (2014).
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O. D. Miller, S. G. Johnson, and A. W. Rodriguez, “Effectiveness of thin films in lieu of hyperbolic metamaterials in the near field,” Phys. Rev. Lett. 112, 157402 (2014).
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C. W. Hsu, B. Zhen, W. Qiu, O. Shapira, B. G. DeLacy, J. D. Joannopoulos, and M. Soljačić, “Transparent displays enabled by resonant nanoparticle scattering,” Nat. Commun. 5, 3152 (2014).
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A. G. Polimeridis, J. F. Villena, L. Daniel, and J. K. White, “Stable FFT-JVIE solvers for fast analysis of highly inhomogeneous dielectric objects,” J. Comput. Phys. 269, 280–296 (2014).
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Y. Wu, C. Zhang, N. M. Estakhri, Y. Zhao, J. Kim, M. Zhang, X. X. Liu, G. K. Pribil, A. Alù, C. K. Shih, and X. Li, “Intrinsic optical properties and enhanced plasmonic response of epitaxial silver,” Adv. Mater. 26, 6106–6110 (2014).
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C. David and F. J. Garcia de Abajo, “Surface plasmon dependence on the electron density profile at metal surfaces,” ACS Nano 8, 9558–9566 (2014).
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A. Welters, Y. Avniel, and S. G. Johnson, “Speed-of-light limitations in passive linear media,” Phys. Rev. A 90, 023847 (2014).
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O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Phys. Rev. Lett. 112, 123903 (2014).
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I. Liberal, Y. Ra’di, R. Gonzalo, I. Ederra, S. A. Tretyakov, and R. W. Ziolkowski, “Least upper bounds of the powers extracted and scattered by bi-anisotropic particles,” IEEE Trans. Antennas Propag. 62, 4726–4735 (2014).
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2013 (7)

A. Raman, W. Shin, and S. Fan, “Upper bound on the modal material loss rate in plasmonic and metamaterial systems,” Phys. Rev. Lett. 110, 183901 (2013).
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G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25, 3264–3294 (2013).
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S. Law, L. Yu, A. Rosenberg, and D. Wasserman, “All-semiconductor plasmonic nanoantennas for infrared sensing,” Nano Lett. 13, 4569–4574 (2013).
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S. Law, L. Yu, and D. Wasserman, “Epitaxial growth of engineered metals for mid-infrared plasmonics,” J. Vac. Sci. Technol. B 31, 03C121 (2013).
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C. Sauvan, J. P. Hugonin, I. S. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
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X. Liang and S. G. Johnson, “Formulation for scalable optimization of microcavities via the frequency-averaged local density of states,” Opt. Express 21, 30812–30841 (2013).
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A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
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2012 (13)

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
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C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
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P. Tassin, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “A comparison of graphene, superconductors and metals as conductors for metamaterials and plasmonics,” Nat. Photonics 6, 259–264 (2012).
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P. T. Kristensen, C. Van Vlack, and S. Hughes, “Generalized effective mode volume for leaky optical cavities,” Opt. Lett. 37, 1649–1651 (2012).
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C. Van Vlack and S. Hughes, “Finite-difference time-domain technique as an efficient tool for calculating the regularized Green function: applications to the local-field problem in quantum optics for inhomogeneous lossy materials,” Opt. Lett. 37, 2880–2882 (2012).
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W. Qiu, B. G. Delacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Optimization of broadband optical response of multilayer nanospheres,” Opt. Express 20, 18494–18504 (2012).
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M. Liertzer, L. Ge, A. Cerjan, A. D. Stone, H. E. Türeci, and S. Rotter, “Pump-induced exceptional points in lasers,” Phys. Rev. Lett. 108, 173901 (2012).
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M. Costabel, E. Darrigrand, and H. Sakly, “The essential spectrum of the volume integral operator in electromagnetic scattering by a homogeneous body,” Comptes Rendus Math. 350, 193–197 (2012).
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E. C. Dreaden, A. M. Alkilany, X. Huang, C. J. Murphy, and M. A. El-Sayed, “The golden age: gold nanoparticles for biomedicine,” Chem. Soc. Rev. 41, 2740–2779 (2012).
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S. Vassant, A. Archambault, F. Marquier, F. Pardo, U. Gennser, A. Cavanna, J. L. Pelouard, and J. J. Greffet, “Epsilon-near-zero mode for active optoelectronic devices,” Phys. Rev. Lett. 109, 237401 (2012).
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H. Hashemi, C.-W. Qiu, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “Diameter-bandwidth product limitation of isolated-object cloaking,” Phys. Rev. A 86, 013804 (2012).
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J. A. Scholl, A. L. Koh, and J. A. Dionne, “Quantum plasmon resonances of individual metallic nanoparticles,” Nature 483, 421–427 (2012).
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C. Ciracì, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernández-Domínguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the ultimate limits of plasmonic enhancement,” Science 337, 1072–1074 (2012).
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2011 (9)

Z. Ruan and S. Fan, “Design of subwavelength superscattering nanospheres,” Appl. Phys. Lett. 98, 043101 (2011).
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A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
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S. Basu and M. Francoeur, “Maximum near-field radiative heat transfer between thin films,” Appl. Phys. Lett. 98, 243120 (2011).
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M. I. Tribelsky, “Anomalous light absorption by small particles,” Europhys. Lett. 94, 14004 (2011).
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R. S. Ottens, V. Quetschke, S. Wise, A. A. Alemi, R. Lundock, G. Mueller, D. H. Reitze, D. B. Tanner, and B. F. Whiting, “Near-field radiative heat transfer between macroscopic planar surfaces,” Phys. Rev. Lett. 107, 014301 (2011).
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G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range [Invited],” Opt. Mater. Express 1, 1090–1099 (2011).
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A. Baron, E. Devaux, J. C. Rodier, J. P. Hugonin, E. Rousseau, C. Genet, T. W. Ebbesen, and P. Lalanne, “Compact antenna for efficient and unidirectional launching and decoupling of surface plasmons,” Nano Lett. 11, 4207–4212 (2011).
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A. Tittl, P. Mai, R. Taubert, D. Dregely, N. Liu, and H. Giessen, “Palladium-based plasmonic perfect absorber in the visible wavelength range and its application to hydrogen sensing,” Nano Lett. 11, 4366–4369 (2011).
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K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, “Broadband polarization-independent resonant light absorption using ultrathin plasmonic super absorbers,” Nat. Commun. 2, 517 (2011).
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2010 (7)

X. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104, 207403 (2010).
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A. F. Koenderink, “On the use of purcell factors for plasmon antennas,” Opt. Lett. 35, 4208–4210 (2010).
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A. Raman and S. Fan, “Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem,” Phys. Rev. Lett. 104, 087401 (2010).
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L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
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H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
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J. B. Khurgin and G. Sun, “In search of the elusive lossless metal,” Appl. Phys. Lett. 96, 181102 (2010).
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P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795–808 (2010).
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2009 (8)

D.-H. Kwon and D. M. Pozar, “Optimal characteristics of an arbitrary receive antenna,” IEEE Trans. Antennas Propag. 57, 3720–3727 (2009).
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T. S. van Zanten, A. Cambi, M. Koopman, B. Joosten, C. G. Figdor, and M. F. Garcia-Parajo, “Hotspots of gpianchored proteins and integrin nanoclusters function as nucleation sites for cell adhesion,” Proc. Natl. Acad. Sci. U. S. A. 106, 18557–18562 (2009).
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L. E. Sun and W. C. Chew, “A novel formulation of the volume integral equation for electromagnetic scattering,” Waves in Random and Complex Media 19, 162–180 (2009).
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A. Moroz, “Depolarization field of spheroidal particles,” J. Opt. Soc. Am. B 26, 517–527 (2009).
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M. D. Arnold and M. G. Blaber, “Optical performance and metallic absorption in nanoplasmonic systems,” Opt. Express 17, 3835–3847 (2009).
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E. K. Lau, A. Lakhani, R. S. Tucker, and M. C. Wu, “Enhanced modulation bandwidth of nanocavity light emitting devices,” Opt. Express 17, 7790–7799 (2009).
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R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, “Design of plasmonic thin-film solar cells with broadband absorption enhancements,” Adv. Mater. 21, 3504–3509 (2009).
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E. Boisselier and D. Astruc, “Gold nanoparticles in nanomedicine: preparations, imaging, diagnostics, therapies and toxicity,” Chem. Soc. Rev. 38, 1759–1782 (2009).
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2008 (3)

N. Landy, S. Sajuyigbe, J. Mock, D. Smith, and W. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100, 207402 (2008).
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M. Francoeur, M. P. Menguc, and R. Vaillon, “Near-field radiative heat transfer enhancement via surface phonon polaritons coupling in thin films,” Appl. Phys. Lett. 93, 043109 (2008).
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S. Franzen, “Surface plasmon polaritons and screened plasma absorption in indium tin oxide compared to silver and gold,” J. Phys. Chem. C 112, 6027–6032 (2008).
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2007 (11)

A. M. Gobin, M. H. Lee, N. J. Halas, W. D. James, R. A. Drezek, and J. L. West, “Near-infrared resonant nanoshells for combined optical imaging and photothermal cancer therapy,” Nano Lett. 7, 1929–1934 (2007).
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L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
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C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on metamaterials: restrictions on scattering and absorption over a frequency interval,” J. Phys. D. Appl. Phys. 40, 7146–7151 (2007).
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C. Sohl, M. Gustafsson, and G. Kristensson, “Physical limitations on broadband scattering by heterogeneous obstacles,” J. Phys. A Math. Theor. 40, 11165–11182 (2007).
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M. Gustafsson, C. Sohl, and G. Kristensson, “Physical limitations on antennas of arbitrary shape,” Proc. R. Soc. A Math. Phys. Eng. Sci. 463, 2589–2607 (2007).
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R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljačić, “Coupled-mode theory for general free-space resonant scattering of waves,” Phys. Rev. A 75, 053801 (2007).
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F. Capolino, D. Wilton, and W. Johnson, “Efficient computation of the 3d Greens function for the helmholtz operator for a linear array of point sources using the ewald method,” J. Comput. Phys. 223, 250–261 (2007).
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R. A. Shore and A. D. Yaghjian, “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers,” Radio Sci. 42, RS6S21 (2007).
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D. Peer, J. M. Karp, S. Hong, O. C. Farokhzad, R. Margalit, and R. Langer, “Nanocarriers as an emerging platform for cancer therapy,” Nat. Nanotechnol. 2, 751–760 (2007).
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F. Lopez-Tejeira, S. G. Rodrigo, L. Martin-Moreno, F. J. Garcia-Vidal, E. Devaux, T. W. Ebbesen, J. R. Krenn, P. Radko, S. I. Bozhevolnyi, M. U. Gonzalez, J. C. Weeber, and A. Dereux, “Efficient unidirectional nanoslit couplers for surface plasmons,” Nat. Phys. 3, 324–328 (2007).
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S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007).
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2006 (7)

P. K. Jain, K. S. Lee, I. H. El-Sayed, and M. A. El-Sayed, “Calculated absorption and scattering properties of gold nanoparticles of different size, shape, and composition: applications in biological imaging and biomedicine,” J. Phys. Chem. B 110, 7238–7248 (2006).
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W. L. Barnes, “Surface plasmon-polariton length scales: a route to sub-wavelength optics,” J. Opt. A Pure Appl. Opt. 8, S87–S93 (2006).
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F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006).
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E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
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2005 (8)

W. Nomura, M. Ohtsu, and T. Yatsui, “Nanodot coupler with a surface plasmon polariton condenser for optical far/near-field conversion,” Appl. Phys. Lett. 86, 181108 (2005).
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2004 (2)

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2003 (4)

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2003).
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2001 (3)

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2000 (2)

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1999 (3)

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1997 (3)

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1994 (1)

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1989 (2)

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1983 (1)

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1981 (2)

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1980 (2)

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1979 (1)

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1978 (1)

J. Andersen and V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
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1976 (3)

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1972 (2)

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Figures (7)

Fig. 1
Fig. 1

Scattering problem under consideration. An incident field Finc = (Einc Z0Hinc) T impinges on a lossy scatterer with a susceptibility tensor χ ¯ ¯ ( x , ω ) . The shape and topology of the scatterer are arbitrary: It may be periodic, extend to infinity, or consist of multiple particles. The limits presented in Sec. 3 hinge on the fact that absorption is a quadratic functional of the electromagnetic fields, whereas extinction, by the optical theorem, is the imaginary part of a linear functional of the fields. In Sec. 3, we present general limits for tensor susceptibilities and also simplified limits for metals.

Fig. 2
Fig. 2

A comparison of the metric |χ|2/Im χ, which limits absorption, scattering, and spontaneous emission rate enhancements, for conventional metals (Ag, Al, Au, etc.) [79] as well as alternative plasmonic materials including aluminum-doped ZnO (AZO) [29], highly doped InAs [30], SiC [79], TiN [81], ITO [82], and Dysprosium-doped cadmium oxide [83] (CdO:Dy). Silver, aluminum, and gold are the best materials at visible and near-infrared wavelengths, although at higher wavelengths the structural aspect ratios needed to achieve the limiting enhancements may not be possible. The dotted lines indicate wavelengths at which resonant nanorods would require aspect ratios greater than 30, approximating the highest feasible experimental aspect ratios [84]. Despite having lower maximum enhancements, AZO, doped InAs, and SiC should be able to approach optimal enhancements in the infrared with realistic aspect ratios.

Fig. 3
Fig. 3

Absorption (blue) and scattering (red) cross-sections per unit particle volume for nanoparticles of (a) gold [79] and (b) Si-doped InAs [30], illuminated by plane waves polarized along the particle rotation axis. The ellipsoid aspect ratios can be tuned to approach both the maximum absorption and maximum scattering cross-sections (black) of Eqs. (32a,32b). The dimensions of the nanoparticles are optimized at three representative wavelengths, in the visible for gold (constrained to have radii not less than 5nm) and at longer infrared wavelengths for doped InAs. Whereas the maximum-absorption particles are small to exhibit quasistatic behavior, represented by dashed lines in (a), the maximum-scattering particles are larger such that their scattering and absorption rates are equal.

Fig. 4
Fig. 4

Nonradiative LDOS enhancement for randomly oriented dipoles above a flat bulk metal. (a) Enhancement as a function of wavelength. For each metal except gold—which has significant losses—the nonradiative LDOS at the surface-plasmon frequency ωsp (dotted line) approaches the limit given by Eq. (35b). The emitter–metal separation distance is fixed at d = 0.1c/ωsp. The limits are equally attainable for conventional metals such as Al and Ag as for synthetic metals such as AZO [29] and highly doped InAs [30], and for SiC. (b) Enhancement as a function of metal–emitter separation distance d, with the frequency fixed at the surface-plasmon frequency ωsp for each metal. The limiting enhancements are asymptotically approached as the separation distance is decreased, because the quasistatic approximation of Eq. (42) becomes increasingly accurate.

Fig. 5
Fig. 5

Away from the surface-plasmon frequency of a given metal—taken here to be silver [79]—it is more difficult to reach the radiative and nonradiative LDOS limits, Eqs. (34b,34a). The emitter–metal separation d is fixed at d = 10nm for (a) and (b). (a) Nonradiative LDOS enhancements, ρnr0, for thin films (red) for various silver thicknesses (t), and for (type-I) hyperbolic metamaterials (HMMs, purple) for two silver fill fractions (ff). (b) Radiative LDOS enhancements, ρrad0 for cone and cylinder antennas, with dimensions optimized at wavelengths from λ = 450nm to λ = 850nm. (c) Scaling of ρnr and ρrad for optimized thin films and cone antennas, respectively, as a function of d (inset: log−log scale). The scaling of the optimal design appears to be 1/d3, with the structures falling short of their respective limits [the dashed line is (ρrad0)max] because they do not exhibit a |χ|2/Im χ enhancement.

Fig. 6
Fig. 6

A schematic comparison of absorption and scattering limits: multipole limits [21, 22] to the total cross-section can provide design guidelines at low frequencies, where it is difficult to achieve “plasmonic” resonances, but at higher frequencies our dissipation-based limits provide tighter limits to the per-volume response. The frequencies at which our bounds can be reached range from the bulk plasma frequency (ω = ωp) down to ω~ωp/ARmax, where ARmax is the maximum achievable aspect ratio. Included is the relevant range for silver ellipsoids (red text), assuming ARmax = 30. Plasmonic behavior at longer wavelengths is possible with materials such as AZO and doped InAs (cf. Fig. 2).

Fig. 7
Fig. 7

A comparison of the (a) resonant-frequency and (b) resonant-susceptibility frameworks in electromagnetism. The conventional resonant-frequency approach is depicted in (a): the operating frequency is real-valued and can in theory be approached arbitrarily closely by a resonance with small imaginary part–Im ωn (i.e. a high-Q resonance). Conversely, volume integral equations yield the resonant-susceptibility approach depicted in (b): metal losses, which correspond to Im ξ = Im(−1) > 0, inherently impose a minimum separation q to how closely a material resonance—restricted to lie on or below the real line—can approach the real system parameters. Moreover, quasistatic structures have real-valued eigenvalues, and thus have the potential to achieve the minimum eigenvalue separation and maximum optical response.

Tables (1)

Tables Icon

Table 1 Tabulation of higher-order term contributions to radiative LDOS limits.

Equations (86)

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D ( x ) = ε 0 E ( x ) + P ( x ) B ( x ) = μ 0 [ H ( x ) + M ( x ) ] .
( P 1 c M ) = ε 0 χ ¯ ¯ ( E Z 0 H ) = ε 0 χ ¯ ¯ F
ω Im χ ¯ ¯ > 0 ,
P abs = ε 0 ω 2 Im V F ( x ) χ ¯ ¯ ( x ) F ( x ) d V
P ext = ε 0 ω 2 Im V F inc ( x ) χ ¯ ¯ ( x ) F ( x ) d V
F inc = ( E inc Z 0 H inc ) .
P scat = P ext P abs = ε 0 ω 2 Im V [ F inc ( x ) F ( x ) ] χ ¯ ¯ ( x ) F ( x ) d V .
ρ tot = 1 π ω Im j s ^ j E s j ( x 0 ) ,
E s j ( x 0 ) = E inc , s j ( x 0 ) + E scat , s j ( x 0 ) .
E scat ( x 0 ) = V [ G E P ¯ ¯ ( x 0 , x ) P ( x ) + G E M ¯ ¯ ( x 0 , x ) M ( x ) ]
G i j E P ( x 0 , x ) = G j i E P ( x , x 0 )
G i j E M ( x 0 , x ) = μ 0 G j i H P ( x , x 0 )
s ^ j E s j ( x 0 ) = s ^ j E inc , s j ( x 0 ) + V F ˜ inc , s j χ ¯ ¯ F s j
F ˜ inc = ( E inc Z 0 H inc ) .
ρ tot = ρ 0 + 1 π ω Im j V F ˜ inc , s j χ ¯ ¯ F s j .
ρ tot ρ 0 = 1 + 2 π k 3 Im j V F ˜ inc , s j χ ¯ ¯ F s j
ρ nr ρ 0 = 2 π k 3 Im j V F s j χ ¯ ¯ F s j .
ρ rad ρ 0 = 1 + 2 π k 3 Im j V [ F ˜ inc , s j F s j ] χ ¯ ¯ F s j
δ P δ F = 0 ,
δ P scat δ F = ε 0 ω 2 [ χ ¯ ¯ F inc 2 i + ( Im χ ¯ ¯ ) F ] = 0.
F scat , opt ( x ) = i 2 [ Im χ ¯ ¯ ( x ) ] 1 χ ¯ ¯ ( x ) F inc ( x )
δ δ F = δ P abs δ F + δ P scat δ F = ε 0 ω 2 [ ( 1 ) ( Im χ ¯ ¯ ) F 2 i χ ¯ ¯ F inc ] = 0.
F abs , opt ( x ) = i [ Im χ ¯ ¯ ( x ) ] 1 χ ¯ ¯ ( x ) F inc ( x )
P s c a t ε 0 ω 8 V F inc χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ F inc d 3 x
P abs , P ext ε 0 ω 2 V F inc χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ F inc d 3 x
F s j , rad , opt ( x ) = i 2 ( Im χ ¯ ¯ ) 1 χ ¯ ¯ F ˜ inc , s j ( x )
F s j , nr , opt ( x ) = i ( Im χ ¯ ¯ ) 1 χ ¯ ¯ F ˜ inc , s j ( x )
ρ rad ρ 0 1 + π 2 k 3 j V F ˜ inc , s j χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ F ˜ inc , s j d 3 x
ρ nr ρ 0 , ρ tot ρ 0 2 π k 3 j V F ˜ inc , s j χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ F ˜ inc , s j d 3 x .
V F inc χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ F inc V χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ 2 F inc F inc ( χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ 2 ) max V | F inc | 2
1 2 ε 0 | F inc | 2 = 1 2 ε 0 | E inc | 2 + 1 2 μ 0 | H inc | 2 = U E , inc + U H , inc
P scat V ω ( U E , inc + U H , inc ) avg 4 ( χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ 2 ) max
P abs V , P ext V ω ( U E , inc + U H , inc ) avg ( χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ 2 ) max .
σ scat V k 2 ( χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ 2 ) avg
σ abs V , σ ext V 2 k ( χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ 2 ) avg .
χ ¯ ¯ ( Im χ ¯ ¯ ) 1 χ ¯ ¯ 2 = | χ ( ω ) | 2 Im χ ( ω ) ,
P scat V ω U E , inc 4 | χ ( ω ) | 2 Im χ ( ω )
P abs V ω U E , inc | χ ( ω ) | 2 Im χ ( ω ) .
σ scat V k 4 | χ ( ω ) | 2 Im χ ( ω )
σ abs V , σ ext V k | χ ( ω ) | 2 Im χ ( ω ) ,
G E P ¯ ¯ F 2 = k 6 8 π 2 [ 3 ( k r ) 6 + 1 ( k r ) 4 + 1 ( k r ) 2 ] ,
ρ rad ρ 0 | χ ( ω ) | 2 Im χ ( ω ) [ 1 32 ( k d ) 3 + 1 16 k d + O ( k L ) ] + 1
ρ nr ρ 0 | χ ( ω ) | 2 Im χ ( ω ) [ 1 8 ( k d ) 3 + 1 4 k d + O ( k L ) ]
ρ rad ρ 0 1 32 ( k d ) 3 | χ ( ω ) | 2 Im χ ( ω )
ρ nr ρ 0 , ρ tot ρ 0 1 8 ( k d ) 3 | χ ( ω ) | 2 Im χ ( ω ) .
P scat , opt = i 2 | χ | 2 Im χ ε 0 E inc ,
J scat , opt = 1 2 Re ρ E inc .
| χ ( ω ) | 2 Im χ ( ω ) = 1 ε 0 ω Re ρ ( ω ) ,
σ abs V = k Im ( χ ( ω ) 1 + L χ ( ω ) ) ,
P = i | χ | 2 Im χ E inc ,
[ σ abs ( ω ) V ] ellipsoid = k | χ ( ω ) | 2 Im χ ( ω ) ,
[ ρ nr ( ω sp ) ρ 0 ] planar 1 8 ( k d ) 3 | χ ( ω sp ) | 2 Im χ ( ω sp ) ,
[ ρ nr ( ω ) ρ 0 ] thin film 1 2 ( k d ) 3 ,
1 ε ( x ) × × E n = ( ω n c ) 2 E n ,
E ( x ) = E inc ( x ) V χ ( x ) G ¯ ¯ ( x , x , ω ) E ( x ) d V
V G ¯ ¯ ( x , x , ω ) E n ( x ) d V = ξ n E n ( x ) = 1 χ n E n ( x )
( + χ G ) e = e inc
G e n = ξ n e n ,
G = U Ξ U T ,
χ e = χ ( ω ) ( + χ ( ω ) G ) 1 e inc = U ( Ξ ξ ( ω ) ) 1 U T e inc ,
P ext = ε 0 ω 2 Im [ e inc U ( Ξ ξ ( ω ) ) 1 U T e inc ]
ρ tot ρ 0 = 1 + 2 π n 0 ω c Im [ e inc T U ( Ξ ξ ( ω ) ) 1 U T e inc ] .
P ext = ε 0 ω 2 Im n p n ξ n ξ ( ω )
ρ tot ρ 0 = 1 + 2 π k 3 Im n p n ξ n ξ ( ω )
n p n V | E inc | 2 d V
n ρ n = n V E inc , s j E inc , s j d V ,
ξ exp , opt = Re ξ ( ω )
n ξ n p n = n E i inc ( x ) E i n ( x ) ξ n E j n ( x ) E j inc ( x ) = n E i inc ( x ) E i n ( x ) G ¯ ¯ j k ( x , x ) E k n ( x ) E j inc ( x ) = E i inc ( x ) G ¯ ¯ j i ( x , x ) E j inc ( x )
ρ nr ρ 0 | χ | 2 Im χ [ 1 32 ( k d ) 3 + 1 16 k d ] + 1.
f = P abs P abs + P scat .
max P scat s . t . f < f max .
σ scat V f max ( 1 f max ) k | χ ( ω ) | 2 Im χ ( ω ) .
( σ abs V ) cs = f V k L 1 L 0 Im [ 2 / 3 L 0 L 0 ξ ( ω ) + L 1 2 / 3 L 1 ξ ( ω ) ]
( σ abs V ) cs 2 3 f V k 1 Im ξ ( ω ) 2 3 k | Re χ ( ω ) | Im χ ( ω )
β = ω c [ ε m ε d ε m + ε d + i ε m 2 ( ε m ) 2 ( ε m ε d ε m + ε d ) 3 / 2 ] ,
L prop = 1 2 Im β ,
β = ω c ε m ε d ε m + ε d .
β ( ω sp ) ω c ε d ε m ( 1 + i 2 ) ,
L prop λ 2 2 π ε m ε d ,
ρ ρ 0 1 k 3 k p 2 e 2 k p z ( Im S 21 ) d k p
k p 0 2 | ε | d
k p 2 e 2 k p z ( Im S 21 ) d k p π 2 k p 0 2 e 2 k p 0 z [ Im S 21 ] max Δ k p
[ Im S 21 ] max Δ k p 2 k p 0
[ ρ ρ 0 ] max 27 π 8 e 3 1 ( k d ) 3 1 2 ( k d ) 3
G E P ¯ ¯ = k 2 e i k r 4 π r [ ( 1 + i a 1 a 2 ) δ i j + ( 1 3 i a + 3 a 2 ) x i x j r 2 ]
δ i j δ i j = δ i i = 3 δ i j x i x j r 2 = x i x i r 2 = 1 x i x j r 2 x i x j r 2 = 1

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