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Electromagnetic Description and Unexpected Effects Metamaterials and composites: electromagnetic description and unexpected effects IEEE AP-Society Distinguished Lecture Hong Kong Chapter — 2 March 2015 City University of Hong Kong Ari Sihvola Aalto University Department of Radio Science and Engineering Finland FINLAND ESPOO Annual Meeting of the American Physical Society 29 December 1959 There’s plenty of room at the bottom! Today’s keywords • scales & levels • geometry–matter interaction • emergence/enhancement of losses • effective description and complex constitutive relations • mixing rules METAMATERIALS ? METAMATERIALS ? Wikipedia definition (changing…) 2006: In electromagnetism (covering areas like optics and photonics), a meta material (or metamaterial) is an object that gains its (electromagnetic) material properties from its structure rather than inheriting them directly from the materials it is composed of. This term is particularly used when the resulting material has properties not found in naturally-formed substances. 2014: Metamaterials gain their properties not from their composition, but from their exactingly-designed structures. Their precise shape, geometry, size, orientation and arrangement can affect the waves of light or sound in an unconventional manner, creating material properties which are unachievable with conventional materials. A. Sihvola (2003): Electromagnetic emergence in metamaterials, in Advances in Electomagnetics of Complex Media and Metamaterials, (S. Zouhdi et al., eds), NATO Science Series, 89, 1-17. A. Sihvola (2007): Metamaterials in electromagnetics. Metamaterials, 1, 2-11. Scales in (electromagnetic) material modeling İ ȝ ȟ ȗ Collection of dipole and multipole moments Physical structure (meta)material levels and phenomena • materialization • mathematization • realization • idealization • synthetization • projection • localization • homogenization • fabrication • parametrization • modularization - non-unique - wash-out of microsopic structure - several realizations - loss of details - multiple platforms - emergence, paradigm change For example, in EE context... • electromagnetics vs. circuit theory – Cellular Neural Networks, non- linear circuits – simple rules, complex behavior • Open Systems Interconnection model for the communication system – abstraction layers • Emergence: 1+ 1 > 2 • Reductionism: 1+ 1 < 2 • Inheritance of properties – children vs. parents • Multiple platforms (case: computer) – the essence of a computer is independent of the technology of the electronic circuits Emergence everywhere! • anisotropy • chirality and optical activity • artificial magnetism • structural colors from photonic band gaps • …. effects determined by the geometry of the small-scale 500 nm Blau, Physics Today Jan. 2004, p.18-20 Structural color and surface topography of emerald ash borer beetle (Agrilus planipennis) wings. (Domingue et al., PNAS, 111(39)14106-11, September 30, 2014) Example • emergence – losses from lossless building blocks – case of radially anisotropic (RA) sphere H e 3H i H eff H e H i 3H e Yu.V. Obnosov. Periodic heterogeneous structures: ”New explicit solutions and effective characteristics of refraction of an imposed field,” SIAM Journal on Applied Mathematics, 59(4):1267–1287, 1999. J. Helsing, R.C. McPhedran, and G.W. Milton: ”Spectral super-resolution in metamaterial composites,” New J. Physics, 13(11):115005, 2011. H eff / H e H e 3H i H eff H e H i 3H e Cylinder with semicircular cross section 2 1 S 2 (2 3W ) 12Li (W 2/2 (W/2) W 2 4 1) 12Li (W 2/2 (W/2) W 2 4 1) D 2 2 1 3S 2 (2 W ) / 4 D 2 (H ) D1(1/H ) H 1 W Į1Į2complex for –3 < İ < –1/3 H 1 M. Pitkonen (2010): Journal of Electromagnetic Waves and Applications, 24, 1267–1277 N. Mohammadi Estakhri and A. Alù (2013): Physics of unbounded, broadband absorption/gain e ciency in plasmonic nanoparticles, Phys. Rev. B, 87, 205418 ƥ Losses from lossless: non-dissipative damping? • chessboard & hemidisk: sharp corners • continuosly curved shapes? • RA sphere Electrostatic response of isotropic (3D) sphere p DabsE D p D abs H V E 0 H 1 Drude model: D 3 Z 2 H 2 H (Z) 1 p Z 2 Zp Singularity for Zres H 2 3 Localized Surface Plasmon, Electrostatic Resonance H (O) H(O)1 D(O) 3 H(O)2 Ag (Drude model) RA sphere: anisotropic H H rurur H t (I urur ) RU/RA sphere radially uniaxial/anisotropic sphere ൞ RA sphere from the onion structure H i H e H t pH i (1 p)H e 1 Hr p/Hi (1p)/He di p di de layer thicknesses di , de Anisotropy of RU sphere H H rurur H t (I urur ) Dipole response isotropic: D DI Electrostatic response of isotropic ------------ RA sphere H 1 H 2 H 1 8H / H D 3 D 3 r r t r H 2 H r 4 H r 1 8H t / H r T. Rimpiläinen, H. Wallén, H. Kettunen, A. Sihvola: Electrical response of systropic sphere, IEEE TAP, 60(11), 5348-5355, 2012 K. Schulgasser: Sphere assemblage model for polycrystals and symmetric materials, J. Appl. Phys., 54(3), 1380-1382, 1983 H tan hyperbolic DPS (indefinite) H rad DNG hyperbolic (indefinite) H H rad tan 8 Anomalous losses Invisibility Invisibility Invisibility Re{D} 0 Invisibility Singularity Re{D} 0 Invisibility Singularity Re{D} 0 Sign of the (real part) of D Invisibility + + + – – – – – + – + + Singularity Re{D} 0 H rad 0.1 H rad 3 H tan 5.5 H tan 1/ 3 H rad 3 H rad 5 H tan 5 / 3 H tan 3/ 5 anomalous losses invisibility anomalous singularity losses RU sphere: complex polarizability Re Im H t 1 Intact RA sphere • is an idealization • but is it a true limit of a realistic RA sphere? Undefined origin? 2b Isotropic RU layer PEC core C o 0 D oD 2a int H 2 H 18H /H r r t r H 2 H C Dint 3 D 3 r r H r 4 H r 1 8H t /H r H r 4 H r C § b · C H r 2 H r ¨ ¸ © a ¹ 1 8H t / H r Limit towards the intact sphere? § b · C H r 2 H r ¨ ¸ C o 0 © a ¹ D o D ( D ) 1 8H t / H r int RU 1 Q (1 ) 2 H t ! 1/8 § b · H 2Q 1 (real) lim ¨ ¸ 0 H r b / ao0© a ¹ t 1/8 H r D o D RU 1 8H t / H r jE jE § b · § b · § b · § b · ¨ ¸ ¨ ¸ cos¨E log ¸ jsin¨ E log ¸ limit b/a o 0 does not exist! © a ¹ © a ¹ © a ¹ © a ¹ H t § b · 1/ 8 + small Re z 0 lim ¨ ¸ 0 H ^ ` D o D RU ! r losses b / ao0© a ¹ H tan 1 -10 b/a 10 Wallén, Kettunen, and Sihvola, “Anomalous absorption, plasmonic resonances, and invisibility of radially anisotropic spheres,” Radio Science, 50(1)18-28, 2015. H tan 1 j 0.1 -10 b/a 10 Wallén, Kettunen, and Sihvola, “Anomalous absorption, plasmonic resonances, and invisibility of radially anisotropic spheres,” Radio Science, , 50(1)18-28, 2015. lim lim D z lim lim D losseso0 core sizeo0 core sizeo0 losseso0 • The original result (the intact RA sphere polarizability) is a faithful limit (for all values of the permittivity components) of the punctured RA sphere – provided that finite intrinsic losses (arbitrarily small) are present • Hence: instead of ”emergence of loss” – ”super-enhancement of loss” From quasistatics towards dynamics… • How regular is the limit ʘї0 ? • Static polarizability ї Rayleigh scattering • Dynamics: Mie scattering – Mie series obviously more complicated than in the isotropic sphere case… jQ (x) RA Qabs Wallén, Kettunen, and Sihvola, “Anomalous absorption, plasmonic resonances, and invisibility of radially anisotropic spheres,” Radio Science, 15(1)18-28, 2015. RA sphere from the onion structure H i H e H t pH i (1 p)H e 1 Hr p/Hi (1p)/He di p di de layer thicknesses di , de hyperbolic H i H 0 H e t H r ! 0 (H e ! 0) H t ! 0 H t 0 H r ! 0 H r 0 H ! 0 ”DPS” t ”DNG” H r 0 hyperbolic p Hyperbolic/indefinite onion structure • indefinite below 386 nm H r • negative definite: 369 … 469 nm • indefinite above 469 nm H i Ag H t H e 5 p 0.6 … back to simple mixtures… History of homogenization: ”a play in five acts” • Maxwell Garnett (the mixing principle) • Bruggeman (effective medium idea) • Beran, Hashin, Shtrikman, Bergman (bounding and limits) • Percolation (precursor of emergence) • Computational possibilities (brute-force) • 6th scene (?): metamaterials paradigm Chr. Brosseau, J. Phys. D: Appl. Phys. 39 (2006), 1277-94. A. Sihvola: Phot. Nanostr. Fund. Appl. 11 (2013), 364-73. Maxwell Garnett mixing formula D ! H eff E ! H i E ! f Ei (1 f )Ee H e D ! f HiEi (1 f ) HeEe 3H e Ei Ee Hi 2H e H H i e H Heff H e 3 f H e eff Hi 2He f (Hi H e ) J. C. Maxwell Garnett. Colours in metal glasses and metal films. Trans. the Royal Society (London), 203:385–420, 1904. Maxwell Garnett Heff He Hi He f Heff 2He Hi 2He Bruggeman (symmetric) H H H H (1 f ) e eff f i eff 0 H e 2H eff Hi 2H eff Maxwell Garnett Bruggeman H i 2 j10 H i 2 j100 Maxwell Garnett Bruggeman ANOTHER VIEW on 1 Maxwell Garnett H1 2b H 2 2a p DabsE H 0VD 3 (H1 1)(H2 2H1)(a/b) (2H1 1)(H2 H1) D 3 3 (H1 2)(H2 2H1)2(a/b) (H1 1)(H2 H1) H 0 1 H eff Heff 1 Deff 3 Heff 2 3 H2 H1 D D eff Heff H1 3(a/b) H1 3 H2 2H1 (a/b) (H2 H1) Exactly Maxwell Garnett two-phase mixing formula !!! Repercussions: • Bounds for the polarizability • Dispersion engineering • Cloaking applications Repercussions:A • Bounds for the polarizability • Dispersion engineering • Cloaking applications A MG and inverse MG (MG non-symmetric) Raisin pudding Swiss cheese ߳௜ ՜1߳௘ ߳௘ ՜ ߳௜ ݌ ՜ െ ݌ Hashin–Shtrikman bounds H MG d H eff d H invMG for ”Raisin pudding”; Swiss cheese: vice versa Raisin pudding : Hi ! H e Z.
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