Electromagnetic Anisotropy and Bianisotropy A Field Guide Tom G Mackay University of Edinburgh, UK and Pennsylvania State University, USA Akhlesh Lakhtakia Pennsylvania State University, USA (cid:58)(cid:82)(cid:85)(cid:79)(cid:71)(cid:3)(cid:54)(cid:70)(cid:76)(cid:72)(cid:81)(cid:87)(cid:76)(cid:192)(cid:70) NEW JERSEY • LONDON (cid:127) SINGAPORE (cid:127) BEIJING (cid:127) SHANGHAI (cid:127) HONG KONG (cid:127) TAIPEI (cid:127) CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ELECTROMAGNETIC ANISOTROPY AND BIANISOTROPHY: A FIELD GUIDE Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. 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James Clerk Maxwell in his native city of Edinburgh.1 1PhotographcourtesyofGaryDoakandThe Royal Society of Edinburgh. v Acronyms and Principal Symbols Acronyms CLC cholesteric liquid crystal CSTF chiral sculptured thin film DGF dyadic Green function FCM Faraday chiral medium HBM helicoidal bianisotropic medium HCM homogenized composite medium LSPR localized surface plasmon resonance NPV negative phase velocity PPV positive phase velocity QED quantum electrodynamics SPFT strong–property–fluctuationtheory Sets C complex numbers R real numbers W angular frequencies xvii xviii Electromagnetic Anisotropy and Bianisotropy: A Field Guide Scalars c speed of light in free space 0 c˜ microscopic charge density (time domain) f volume fraction of constituent (cid:1) (cid:1) gFF local field factor √ i −1 ˜j microscopic current density (time domain) k wavenumber L correlation length cor q point charge t time v magnitude of velocity W˜ total energy density (time domain) Z impedance (cid:2) permittivity of free space 0 µ permeability of free space 0 ρ average linear particle size ρ˜ externally impressed electric charge density (time domain) e ρ˜ externally impressed magnetic charge density (time domain) m ω angular frequency 3–vectors B˜, B primitive magnetic field (time, frequency domain) D˜ , D induction electric field (time, frequency domain) E˜, E primitive electric field (time, frequency domain) H˜, H induction magnetic field (time, frequency domain) J˜ , J externally impressed electric current density e e (time, frequency domain) Acronyms and Principal Symbols xix J˜ , J externally impressed magnetic current density m m (time, frequency domain) M˜, M magnetization (time, frequency domain) P˜, P polarization (time, frequency domain) Q Beltrami field (frequency domain) k wavevector r position vector S˜, S Poynting vector (time, frequency domain) v velocity vector 6–vectors C induction electric–primitive magnetic field phasor (frequency domain) F primitive electric–induction magnetic field phasor (frequency domain) Q source current density phasor (frequency domain) N polarization–magnetizationphasor (frequency domain) 3×3 dyadics α polarizability density dyadic (frequency domain) D depolarization dyadic (frequency domain) G dyadic Green function (frequency domain) I identity dyadic T˜ Maxwell stress tensor (time domain) U shape dyadic ˜(cid:2) , (cid:2) permittivity constitutive dyadic (time, frequency domain) EB EB,EH ˜ζ , ζ magnetoelectric constitutive dyadic (time, frequency domain) EB EB,EH µ permeability constitutive dyadic (frequency domain) EH ν˜ , ν impermeability constitutive dyadic (time, frequency domain) EB EB xx Electromagnetic Anisotropy and Bianisotropy: A Field Guide ˜ξ , ξ magnetoelectric constitutive dyadic (time, frequency domain) EB EB,EH Σ[n] nth–order mass operator (frequency domain) 6×6 dyadics α polarizability density dyadic (frequency domain) D depolarization dyadic (frequency domain) G dyadic Green function (frequency domain) I identity dyadic K constitutive dyadic (frequency domain) EB,EH Σ[n] nth–order mass operator (frequency domain) Operators and functions (cid:1) (cid:2) A entry of matrix A in row m, column (cid:1) m(cid:1) AT, AT transpose of matrix or vector A −1 A inverse of matrix A † A conjugate transpose of matrix A adjA adjoint of matrix A detA determinant of matrix A C{·} conjugation transformation b∗ complex conjugate of b Im{·} imaginary part L(·) linear differential operator (6×6 dyadic) ˆp unit vector in direction of vector p P{·} spatial inversion (cid:3) P ... principal–value integration Rπ/2{·} duality transformation Re{·} real part Acronyms and Principal Symbols xxi T {·} time reversal T {·} Wigner time reversal W trA trace of matrix A τ{·} transforms K to K EH EB δ(·) Dirac delta function (cid:2)·(cid:3) ensemble average e (cid:2)·(cid:3) time average t (cid:2)(cid:2)p, q(cid:3)(cid:3) reaction of ‘p’ sources on ‘q’ fields s(·,·) switching function Preface The focus of this field guide is largely on electromagnetic fields in linear materials. Evenwiththe exclusionofnonlinearity(exceptinthe lastchap- ter), the panorama of electromagnetic properties within the ambit of this monographisvast—aviewwhichhasbeenwidelyappreciatedforthelast 10 years. Anisotropicmediumhaselectromagneticpropertieswhicharethesame inalldirections. Suchmediumsprovidethesettingforintroductorycourses onelectromagnetictheory,as encounteredathighschoolorin earlyunder- graduateclasses. Butisotropyisanabstractionwhichrequiresqualification when applied to real materials. For examples, liquids and random particu- late compositemediums may be isotropicona statisticalbasis,while cubic crystals are isotropic when viewed at macroscopic length–scales. In the frequencydomain,electromagneticallyisotropicmediumsarecharacterized simply by scalar constitutive parameters which relate the induction field phasors D and H to the primitive field phasors E and B. Often, naturally occurring materials and artificially constructed medi- ums are more accurately described as anisotropic rather than isotropic. Anisotropicmediumsexhibitdirectionallydependentelectromagneticprop- erties, such that D is not aligned with E or H is not aligned with B. Dyadics(i.e.,second–rankCartesiantensors)areneededtorelatetheprim- itive and the induction field phasors in anisotropic mediums. No wonder, these mediums exhibit a much more diverse range of phenomenons than isotropic mediums do. A visit to the mineralogicalsection of the local mu- seum should impress upon the reader the array of dazzling optical effects that may be attributed to anisotropy in crystals. Bianisotropyrepresentsthe naturalgeneralizationof anisotropy. In the electromagnetic description of a bianisotropic medium, both D and H are ix x Electromagnetic Anisotropy and Bianisotropy: A Field Guide anisotropically coupled to both E and B. Hence, in general, a linear bian- isotropicmediumischaracterizedbyfour3×3constitutivedyadics. Though seldomdescribedinstandardtextbooks,bianisotropyisinfactaubiquitous phenomenon. Supposeacertainmediumischaracterizedasanisotropicdi- electric medium by an observerin an inertial reference frame Σ. The same medium generally exhibits bianisotropic properties when viewed by an ob- server in another reference frame that translates at uniform velocity with respect to Σ. While naturally occurring materials with easily appreciable bianisotropic properties in normal environmental conditions are relatively rare(viewedfromastationaryreferenceframe),bianisotropicmediumsmay be readily conceptualizedthroughthe process ofhomogenizationof a com- posite of two or more constituent mediums. Bianisotropy looks set to play an increasingly important role in the rapidly burgeoning fields relating to complex mediums and metamaterials. Our aim in this field guide is to extend and update the standard treat- ments of crystal optics found in classical textbooks such as Born & Wolf’s Principles of Optics3 and Nye’s Physical Properties of Crystals4. We pro- videa broadoverviewofelectromagneticanisotropyandbianisotropy. The topics covered are constitutive relations (Chap. 1), examples of anisotropy and bianisotropy (Chap. 2), space–time symmetries (Chap. 3), planewave propagation (Chap. 4), dyadic Green functions including depolarization dyadics (Chap. 5), homogenization formalisms (Chap. 6), and nonlinear aspects (Chap. 7). The target audience comprises graduate students and researchersseekinganintroductorysurveyoftheelectromagnetictheoryof complex mediums. A familiarity with basic electromagnetic theory, and a commensuratelevelofmathematicalexpertise,isassumed. Anappendixis providedtoacquaintthe readerwithdyadicnotationandalgebra. SI units are adopted throughout. Tom G. Mackay, Edinburgh, Scotland Akhlesh Lakhtakia, University Park, PA July 2009 3M.Bornand E. Wolf,Principles of optics, 7th (expanded) ed, CambridgeUniversity Press,Cambridge,UK,1999. 4J.F.Nye,Physical properties of crystals,OxfordUniversityPress,Oxford,UK,1985.