An Introduction to Spatial Dispersion: Revisiting the Basic Concepts

Total Page:16

File Type:pdf, Size:1020Kb

An Introduction to Spatial Dispersion: Revisiting the Basic Concepts [email protected] SPATYPE An introduction to spatial dispersion http://www.kinsler.org/physics/ An introduction to spatial dispersion: revisiting the basic concepts Paul Kinsler∗ Cockcroft Institute, Keckwick Lane, Daresbury WA4 4AD, United Kingdom Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom and Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom (Dated: Monday 15th February, 2021) I describe three ways that the spatial properties of a wave propagation medium can cause dispersion, and pro- pose that they should form the basics for correctly understanding and naming phenomena described as “spatial dispersion”. In particular, I emphasise the specific spatial properties which generate the resulting dispersive – i.e. spatially dispersive – behaviour. The properties are geometry, structure, and dynamics. I. INTRODUCTION considered topic of temporal dispersion. The first two spa- tial dispersion mechanisms depend on inhomogeneous spatial properties, whereas the third derives from propagation-based Spatial dispersion is the non-local dependence of material properties. properties on both direction and wavelength, and is of particu- The categorization here shows that instances of spatial lar interest in electromagnetic systems and other fields where dispersion can be caused by the geometric, structural, or dy- artificial functional materials (AFMs) are finding technolog- namic properties of a system. It is not based on existing (and ical uses, such as in acoustic and elastic metamaterials. It often somewhat ad hoc) naming conventions or justifications has been long observable in crystals [1], but is also of partic- for spatial dispersion, but is instead grounded in the following ular importance in AFMs where the wavelength of radiation assertion: becomes comparable to the lattice parameters (see e.g. [2]). Further, it is also significant in the region of the material res- Spatial dispersion refers to any dispersive behaviour that onances utilized in metamaterial unit cells, where constitutive occurs (solely) as a result of spatial properties of the system. parameters such as the permittivity and permeability – and likewise their acoustic or elastic counterparts – are maximised This viewpoint is distinct from a widely used one (see e.g. or have near-zero local values [3] or where the response is [5]) where spatial dispersion is characterised as being due non-reciprocal [4]. In some materials its presence is revealed to (non-trivial) wavevector-dependence in the dispersion re- by the shape of the equi-frequency surfaces of the dispersion lations, and is then categorized into two cases ‘weak’ spatial relations – e.g. when they are non-symmetric, not elliptical or dispersion, where the spatial effects have been approximated hyperbolic, or having multiple modes with the same direction as a local effect; and ‘strong’ spatial dispersion, where non- and polarisation. local effects remain important [6, 7]. It is to be noted that Spatial dispersion usually appears as a either spatially non- so-called weak spatial dispersion is not necessarily small in local effect that produces a wavevector dependence of the ma- any practical sense, despite the implications of ‘weak’; and terial parameters, or as a non-trival wavevector dependence likewise so-called strong spatial dispersion need not be par- for the dispersion relations. Since representing spatial proper- ticularly dominant1. ties in a fourier space (here, that of wavevector k ) necessar- ily involves an integral over spatially separated points, spatial In particular, in the interest of both brevity and not repeating dispersion is intrinsically non-local. existing discussions in the literature, I do not: Although spatial dispersion can be modelled in an ad hoc 1. discuss how homogenization schemes might reduce the manner to suit intuition or an empirical fit to data, it is prefer- description of a complicated medium – such as those able to be more systematic. However, the term “spatial disper- with spatial varying temporal responses – into a simpler sion” is often used rather loosely, and so sometimes the origin one, (e.g.) perhaps one based on a series expansion in of the specific phenomenon being discussed is unclear. In an terms of the wavevector k; attempt to rectify this, here I describe three physically distinct ways that spatial dispersion can arise, briefly describing exam- 2. consider dispersion relations that have been artificially ples of each. Each of these mechanisms has (a) no time depen- constructed without reference to some specific physi- dence beyond that required to support a wave, and (b) a spec- cal system with spatial properties; e.g. putative disper- ification of time-independent material properties. The lack of sia based on suggested or “what-if” polynomials in fre- arXiv:1904.11957v4 [physics.optics] 12 Feb 2021 any (non-trivial) time dependence ensures that the phenom- quency w and wavevector k, some other specification ena treated here are entirely unrelated to the more commonly 1 In my view, it is preferrable for terms such as ‘weak’ or ‘strong’ to be only ∗https://orcid.org/0000-0001-5744-8146; used to to indicate the strength or significance of a phenomenon, not its [email protected] origin or type, or whether it might be effectively local or non-local. 1 [email protected] SPATYPE An introduction to spatial dispersion http://www.kinsler.org/physics/ of an w, k interdependence, or a non-local convolution over spatial properties. The main focus here is on spatial dispersion in electromag- w netic systems or artifical functional media such as metamateri- als, but the distinctions apply equally well to waves in acoustic or elastic materials, or indeed potentially any material system which supports wave propagation of some kind. In Section II, I will define what I mean by dispersion (i.e. specifically non- trivial dispersion) by considering the relationships between the frequency w of a wave and the wavevector k associated with that frequency. Following that the next three sections address each of geometric (Section III), structural (IV), and dynamic (V) spatial dispersions in turn. Note that the idea of k spatial dispersion as being grounded in spatial properties as opposed to (spectral) wavevector ones is mirrored in the var- FIG. 1: Dispersionless case, where the dispersion relation is linear, ious treatments of temporal dispersion: although sometimes passes through and includes the origin, and is continuous. The light seen as a consequence of a dynamic time-domain process [8] dotted line indicates that the depicted dispersion behaviour continues particularly when implemented in FDTD algorithms [9], tem- in a similar manner for higher wavevectors and frequencies. poral dispersion is more often discussed in solely terms of a (spectral) frequency response [10, 11]. Of course, in practical terms the categorization of dispersion types is really applied If all these criteria hold, then the wave is dispersionless, to our models or representations of the physical system, and and the phase and group velocities are always the same. If not some perfectly accurate microscopic view of the actual any of the criteria do not hold, then the wave is dispersive, physical system, and this is discussed in Section VI. and an initial pulse shape will change over time as its differ- In what follows, I do not describe the example systems in ent spectral components evolve at different rates. In general, a detailed manner. Only the minimum features necessary to this behaviour might be due to either the time-response of the make the point are presented, as such models are often worked medium (giving rise to temporal dispersion), or to the spatial through both exhaustively and frequently in other sources. properties of the medium (giving rise to spatial dispersion), or Here their role is simply to provide examples that typify their possibly both. role in creating a dispersive response as a result of their par- Here I focus exclusively on dispersion that results solely ticular spatial features or structure. Nonlinear effects are not from spatial properties. However, it is worth keeping in mind considered. that typically the structural elements relied on in a physical device to create those spatial properties may in reality also be temporally dispersive – notably, reflecting walls will not be II. DISPERSION perfectly reflective at all frequencies, and different refractive indices exist primarily because of temporal dispersion; some The definition of dispersion used here is based on three cri- of these issues are discussed in section VI. In what follows teria relating to how the frequency ( f or w) of a wave is related I ignore such complications because I wish to focus solely to its wavelength l or wavevector k. Note that in discrete sys- on spatial properties and their effects; but this is not to say tems, analytic solutions typically depend on an integer index that such time-domain responses may always be ignored in m (and possibly more indices as well). Since these indices practical situations. usually indicate the number or spatial frequency of the os- A dispersionless wave travelling along the z–axis with cillation in a given solution, they play the same role as the propagation speed c follows the wave equation wavevector k, and contain essentially the same physical con- 2 2 tent. In simple cases, the wavevector k can be a straightfor- ¶t E − ¶zc ¶zE = 0; (1) ward multiple of m; in any case here the wavevector associ- where ¶t ≡ d=dt, and ¶z ≡ d=dz. This equation can be Fourier ated with an index m is km, and likewise the related angular transformed in both time and space, and the field strength frequency is wm. The criteria are: terms cancelled, giving the “dispersionless” dispersion rela- tion Linear – the relationship between w and k (or between w and w2 − c2k2 = 0: (2) m or km) is linear, i.e. it lies along a single straight line. A depiction of this relationship between w and k can be seen Origin – if w = 0 then k = 0 (or m = 0), and vice versa.
Recommended publications
  • Chapter 3 Constitutive Relations
    Chapter 3 Constitutive Relations Maxwell’s equations define the fields that are generated by currents and charges. However, they do not describe how these currents and charges are generated. Thus, to find a self-consistent solution for the electromagnetic field, Maxwell’s equations must be supplemented by relations that describe the behavior of matter under the influence of fields. These material equations are known as constitutive relations. The constitutive relations express the secondary sources P and M in terms of the fields E and H, that is P = f[E] and M = f[H].1 According to Eq. (1.19) this is equivalent to D = f[E] and B = f[H]. If we expand these relations into power series ∂D 1 ∂2D D = D + E + E2 + .. (3.1) 0 ∂E 2 ∂E2 E=0 E=0 we find that the lowest-order term depends linearly on E. In most practical situ- ations the interaction of radiation with matter is weak and it suffices to truncate the power series after the linear term. The nonlinear terms come into play when the fields acting on matter become comparable to the atomic Coulomb potential. This is the territory of strong field physics. Here we will entirely focus on the linear properties of matter. 1In some exotic cases we can have P = f[E, H] and M = f[H, E], which are so-called bi- isotropic or bi-anisotropic materials. These are special cases and won’t be discussed here. 33 34 CHAPTER 3. CONSTITUTIVE RELATIONS 3.1 Linear Materials The most general linear relationship between D and E can be written as2 ∞ ∞ D(r, t) = ε ε˜(r r′, t t′) E(r′, t′) d3r′ dt′ , (3.2) 0 − − −∞Z−∞Z which states that the response D at the location r and at time t not only depends on the excitation E at r and t, but also on the excitation E in all other locations r′ and all other times t′.
    [Show full text]
  • Double Negative Dispersion Relations from Coated Plasmonic Rods∗
    MULTISCALE MODEL. SIMUL. c 2013 Society for Industrial and Applied Mathematics Vol. 11, No. 1, pp. 192–212 DOUBLE NEGATIVE DISPERSION RELATIONS FROM COATED PLASMONIC RODS∗ YUE CHEN† AND ROBERT LIPTON‡ Abstract. A metamaterial with frequency dependent double negative effective properties is constructed from a subwavelength periodic array of coated rods. Explicit power series are developed for the dispersion relation and associated Bloch wave solutions. The expansion parameter is the ratio of the length scale of the periodic lattice to the wavelength. Direct numerical simulations for finite size period cells show that the leading order term in the power series for the dispersion relation is a good predictor of the dispersive behavior of the metamaterial. Key words. metamaterials, dispersion relations, Bloch waves, simulations AMS subject classifications. 35Q60, 68U20, 78A48, 78M40 DOI. 10.1137/120864702 1. Introduction. Metamaterials are artificial materials designed to have elec- tromagnetic properties not generally found in nature. One contemporary area of research explores novel subwavelength constructions that deliver metamaterials with both a negative bulk dielectric constant and bulk magnetic permeability across cer- tain frequency intervals. These double negative materials are promising materials for the creation of negative index superlenses that overcome the small diffraction limit and have great potential in applications such as biomedical imaging, optical lithography, and data storage. The early work of Veselago [39] identified novel ef- fects associated with hypothetical materials for which both the dielectric constant and magnetic permeability are simultaneously negative. Such double negative media support electromagnetic wave propagation in which the phase velocity is antiparallel to the direction of energy flow and other unusual electromagnetic effects, such as the reversal of the Doppler effect and Cerenkov radiation.
    [Show full text]
  • Modeling Optical Metamaterials with Strong Spatial Dispersion
    Fakultät für Physik Institut für theoretische Festkörperphysik Modeling Optical Metamaterials with Strong Spatial Dispersion M. Sc. Karim Mnasri von der KIT-Fakultät für Physik des Karlsruher Instituts für Technologie (KIT) genehmigte Dissertation zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN (Dr. rer. nat.) Tag der mündlichen Prüfung: 29. November 2019 Referent: Prof. Dr. Carsten Rockstuhl (Institut für theoretische Festkörperphysik) Korreferent: Prof. Dr. Michael Plum (Institut für Analysis) KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft Erklärung zur Selbstständigkeit Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe, die wörtlich oder inhaltlich über- nommenen Stellen als solche kenntlich gemacht und die Satzung des KIT zur Sicherung guter wissenschaftlicher Praxis in der gültigen Fassung vom 24. Mai 2018 beachtet habe. Karlsruhe, den 21. Oktober 2019, Karim Mnasri Als Prüfungsexemplar genehmigt von Karlsruhe, den 28. Oktober 2019, Prof. Dr. Carsten Rockstuhl iv To Ouiem and Adam Thesis abstract Optical metamaterials are artificial media made from subwavelength inclusions with un- conventional properties at optical frequencies. While a response to the magnetic field of light in natural material is absent, metamaterials prompt to lift this limitation and to exhibit a response to both electric and magnetic fields at optical frequencies. Due tothe interplay of both the actual shape of the inclusions and the material from which they are made, but also from the specific details of their arrangement, the response canbe driven to one or multiple resonances within a desired frequency band. With such a high number of degrees of freedom, tedious trial-and-error simulations and costly experimen- tal essays are inefficient when considering optical metamaterials in the design of specific applications.
    [Show full text]
  • Plasma Waves
    Plasma Waves S.M.Lea January 2007 1 General considerations To consider the different possible normal modes of a plasma, we will usually begin by assuming that there is an equilibrium in which the plasma parameters such as density and magnetic field are uniform and constant in time. We will then look at small perturbations away from this equilibrium, and investigate the time and space dependence of those perturbations. The usual notation is to label the equilibrium quantities with a subscript 0, e.g. n0, and the pertrubed quantities with a subscript 1, eg n1. Then the assumption of small perturbations is n /n 1. When the perturbations are small, we can generally ignore j 1 0j ¿ squares and higher powers of these quantities, thus obtaining a set of linear equations for the unknowns. These linear equations may be Fourier transformed in both space and time, thus reducing the differential equations to a set of algebraic equations. Equivalently, we may assume that each perturbed quantity has the mathematical form n = n exp i~k ~x iωt (1) 1 ¢ ¡ where the real part is implicitly assumed. Th³is form descri´bes a wave. The amplitude n is in ~ general complex, allowing for a non•zero phase constant φ0. The vector k, called the wave vector, gives both the direction of propagation of the wave and the wavelength: k = 2π/λ; ω is the angular frequency. There is a relation between ω and ~k that is determined by the physical properties of the system. The function ω ~k is called the dispersion relation for the wave.
    [Show full text]
  • Dispersion Relation & Index Ellipsoids
    8/27/2020 Advanced Electromagnetics: 21st Century Electromagnetics Dispersion Relation & Index Ellipsoids Lecture Outline • Dispersion relation • Dispersion surfaces • Index ellipsoids Slide 2 1 8/27/2020 Dispersion Relation Slide 3 The Wave Vector The wave vector (wave momentum) is a vector quantity that conveys two pieces of information: 1. Wavelength and Refractive Index –The magnitude of the wave vector conveys the spatial period (i.e. wavelength) of the wave inside the material. When the frequency is known, the magnitude conveys the material’s refractive index n (more to be said later). 22 n k 0 free space wavelength 0 2. Direction –The direction of the wave is perpendicular to the wave fronts (more to be said later). ˆ kkakbkcabcˆˆ Slide 4 2 8/27/2020 The Dispersion Relation The dispersion relation for a material relates the wave vector to frequency . Essentially, it sets a rule for the values of as a function of direction and frequency. For an ordinary linear, homogeneous and isotropic (LHI) material, the dispersion relation is: 2 222n kkkabc c0 222 kkkabc 2 This can also be written as: 2 kk000 nc0 Slide 5 How to Derive the Dispersion Relation (1 of 2) The wave equation in a linear homogeneous anisotropic material is: 2 Assume no magnetic response i. e. 1 . Ek 00 r E 0 The solution to this equation is still a plane wave, but the allowed values for (modes) are more complicated. jk r ˆ E Ee00 E Eaabcˆˆ Eb Ec Substituting this solution into the wave equation leads to the following relation: 2 kk E000r0 kE k E 0 This equation has the form: abcˆˆ ˆ 0 Each (•••) term has the form: EEEabc 0 Each vector component must be set to zero independently.
    [Show full text]
  • 7 Plasmonics
    7 Plasmonics Highlights of this chapter: In this chapter we introduce the concept of surface plasmon polaritons (SPP). We discuss various types of SPP and explain excitation methods. Finally, di®erent recent research topics and applications related to SPP are introduced. 7.1 Introduction Long before scientists have started to investigate the optical properties of metal nanostructures, they have been used by artists to generate brilliant colors in glass artefacts and artwork, where the inclusion of gold nanoparticles of di®erent size into the glass creates a multitude of colors. Famous examples are the Lycurgus cup (Roman empire, 4th century AD), which has a green color when observing in reflecting light, while it shines in red in transmitting light conditions, and church window glasses. Figure 172: Left: Lycurgus cup, right: color windows made by Marc Chagall, St. Stephans Church in Mainz Today, the electromagnetic properties of metal{dielectric interfaces undergo a steadily increasing interest in science, dating back in the works of Gustav Mie (1908) and Rufus Ritchie (1957) on small metal particles and flat surfaces. This is further moti- vated by the development of improved nano-fabrication techniques, such as electron beam lithographie or ion beam milling, and by modern characterization techniques, such as near ¯eld microscopy. Todays applications of surface plasmonics include the utilization of metal nanostructures used as nano-antennas for optical probes in biology and chemistry, the implementation of sub-wavelength waveguides, or the development of e±cient solar cells. 208 7.2 Electro-magnetics in metals and on metal surfaces 7.2.1 Basics The interaction of metals with electro-magnetic ¯elds can be completely described within the frame of classical Maxwell equations: r ¢ D = ½ (316) r ¢ B = 0 (317) r £ E = ¡@B=@t (318) r £ H = J + @D=@t; (319) which connects the macroscopic ¯elds (dielectric displacement D, electric ¯eld E, magnetic ¯eld H and magnetic induction B) with an external charge density ½ and current density J.
    [Show full text]
  • Transparent Metamaterials with a Negative Refractive Index Determined by Spatial Dispersion
    Transparent metamaterials with a negative refractive index determined by spatial dispersion V.V. Slabko Siberian Federal University, Krasnoyarsk 660041, Russia Institute of Physics of the Russian Academy of Sciences, Krasnoyarsk 660036, Russia Abstract: The paper considers an opportunity for the creation of an artificial two-component metamaterial with a negative refractive index within the radio and optical frequency band, which possesses a spatial dispersion. It is shown that there exists a spectral region where, under certain ratio of volume fraction unit occupied with components, the metamaterial appears transparent without attenuating the electromagnetic wave passing through it and possessing a phase and group velocity of the opposite sign. 1. Introduction J.B. Pendry in co-authorship papers suggested the idea of possible creation of artificial metamaterials with a negative refractive index in the gigahertz frequency band [1,2]. The idea proceeded from the work by V.G. Veselago where the processes of electromagnetic wave propagation in isotropic materials with simultaneously negative permittivity ( ( ) 0) and magnetic permeability ( ( ) 0) was theoretically considered [3]. The latter leads to negative refraction index ( n 0 ), and phase propagation (wave vector k ) and energy-flow (Poynting vector S ) appear counter-directed, with the left-handed triplet of the vectors of magnetic field, electrical field, and the wave vector. Those were called optical negative-index materials (NIM). Interest to such metamaterials is aroused both by possible observation of extremely unusual properties, e.g. a negative refraction at the materials boundary with positive and negative refractive index, Doppler’s and Vavilov-Cherenkov’s effect, nonlinear wave interactions; and by possibilities in the solution of a number of practical tasks, which can overcome the resolution threshold diffracting limit in optical devices and creation of perfect shielding or “cloaking” [4-7].
    [Show full text]
  • Spatial Dispersion Consideration in Dipoles
    Basics of averaging of the Maxwell equations A. Chipouline, C. Simovski, S. Tretyakov Abstract Volume or statistical averaging of the microscopic Maxwell equations (MEs), i.e. transition from microscopic MEs to their macroscopic counterparts, is one of the main steps in electrodynamics of materials. In spite of the fundamental importance of the averaging procedure, it is quite rarely properly discussed in university courses and respective books; up to now there is no established consensus about how the averaging procedure has to be performed. In this paper we show that there are some basic principles for the averaging procedure (irrespective to what type of material is studied) which have to be satisfied. Any homogenization model has to be consistent with the basic principles. In case of absence of this correlation of a particular model with the basic principles the model could not be accepted as a credible one. Another goal of this paper is to establish the averaging procedure for metamaterials, which is rather close to the case of compound materials but should include magnetic response of the inclusions and their clusters. We start from the consideration of bulk materials, which means in the vast majority of cases that we consider propagation of an electromagnetic wave far from the interfaces, where the eigenwave in the medium has been already formed and stabilized. A basic structure for discussion about boundary conditions and layered metamaterials is a subject of separate publication and will be done elsewhere. 1. Material equations
    [Show full text]
  • Ch 6-Propagation in Periodic Media
    Bloch waves and Bandgaps Chapter 6 Physics 208, Electro-optics Peter Beyersdorf Document info Ch 6, 1 Bloch Waves There are various classes of boundary conditions for which solutions to the wave equation are not plane waves Planar conductor results in standing waves E(z) = 2E0 sin (kzz) cos (ωt) Waveguide and cavities results in modal structure ikz E(x, y, z) = Enm(x, y)e− Periodic materials result in Bloch waves 2π/Λ iK" "r E! (!r) = E(K! , !r)e− · dK! !0 Ch 6, 2 Class Outline Types of periodic media dispersion relation in layered materials Bragg reflection Coupled mode theory Surface waves Ch 6, 3 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 4 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 5 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 6 Periodic Media Many useful materials and devices may have an inhmogenous index of refraction profile that is periodic Dielectric stack optical coatings Diffraction gratings Holograms Acousto-optic devices Photonic bandgap crystals Ch 6, 7 Periodic
    [Show full text]
  • Electromagnetism - Lecture 13
    Electromagnetism - Lecture 13 Waves in Insulators Refractive Index & Wave Impedance • Dispersion • Absorption • Models of Dispersion & Absorption • The Ionosphere • Example of Water • 1 Maxwell's Equations in Insulators Maxwell's equations are modified by r and µr - Either put r in front of 0 and µr in front of µ0 - Or remember D = r0E and B = µrµ0H Solutions are wave equations: @2E µ @2E 2E = µ µ = r r r r 0 r 0 @t2 c2 @t2 The effect of r and µr is to change the wave velocity: 1 c v = = pr0µrµ0 prµr 2 Refractive Index & Wave Impedance For non-magnetic materials with µr = 1: c c v = = n = pr pr n The refractive index n is usually slightly larger than 1 Electromagnetic waves travel slower in dielectrics The wave impedance is the ratio of the field amplitudes: Z = E=H in units of Ω = V=A In vacuo the impedance is a constant: Z0 = µ0c = 377Ω In an insulator the impedance is: µrµ0c µr Z = µrµ0v = = Z0 prµr r r For non-magnetic materials with µr = 1: Z = Z0=n 3 Notes: Diagrams: 4 Energy Propagation in Insulators The Poynting vector N = E H measures the energy flux × Energy flux is energy flow per unit time through surface normal to direction of propagation of wave: @U −2 @t = A N:dS Units of N are Wm In vacuo the amplitudeR of the Poynting vector is: 1 1 E2 N = E2 0 = 0 0 2 0 µ 2 Z r 0 0 In an insulator this becomes: 1 E2 N = N r = 0 0 µ 2 Z r r The energy flux is proportional to the square of the amplitude, and inversely proportional to the wave impedance 5 Dispersion Dispersion occurs because the dielectric constant r and refractive index
    [Show full text]
  • Operator Approach to Effective Medium Theory to Overcome a Breakdown of Maxwell Garnett Approximation
    Downloaded from orbit.dtu.dk on: Oct 05, 2021 Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation Popov, Vladislav; Lavrinenko, Andrei; Novitsky, Andrey Published in: Physical Review B Link to article, DOI: 10.1103/PhysRevB.94.085428 Publication date: 2016 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Popov, V., Lavrinenko, A., & Novitsky, A. (2016). Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation. Physical Review B, 94(8), [085428]. https://doi.org/10.1103/PhysRevB.94.085428 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. PHYSICAL REVIEW B 94, 085428 (2016) Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation Vladislav Popov* Department of Theoretical Physics and Astrophysics, Belarusian State University, Nezavisimosti avenue 4, 220030, Minsk, Republic of Belarus Andrei V.
    [Show full text]
  • Vector Circuit Theory for Spatially Dispersive Uniaxial Magneto-Dielectric Slabs 3
    Machine Copy for Proofreading, JEWA (Ikonen et al.) VECTOR CIRCUIT THEORY FOR SPATIALLY DISPERSIVE UNIAXIAL MAGNETO-DIELECTRIC SLABS Pekka Ikonen, Mikhail Lapine, Igor Nefedov, and Sergei Tretyakov Radio Laboratory/SMARAD, Helsinki University of Technology P.O. Box 3000, FI-02015 TKK, Finland. Abstract—We present a general dyadic vector circuit formalism, applicable for uniaxial magneto-dielectric slabs, with strong spatial dispersion explicitly taken into account. This formalism extends the vector circuit theory, previously introduced only for isotropic and chiral slabs. Here we assume that the problem geometry imposes strong spatial dispersion only in the plane, parallel to the slab interfaces. The difference arising from taking into account spatial dispersion along the normal to the interface is briefly discussed. We derive general dyadic impedance and admittance matrices, and calculate corresponding transmission and reflection coefficients for arbitrary plane wave incidence. As a practical example, we consider a metamaterial slab built of conducting wires and split-ring resonators, and show that neglecting spatial dispersion and uniaxial nature in this structure leads to dramatic errors in calculation of transmission characteristics. 1 Introduction 2 Transmission matrix 3 Impedance and admittance matrices 4 Reflection and transmission dyadics 5 Practically realizable slabs 6 Specific examples 7 Conclusions Acknowledgment References 1. INTRODUCTION arXiv:physics/0605243v1 [physics.class-ph] 29 May 2006 It is well known that many problems dealing with reflections from multilayered media can be solved using the transmission line analogy when the eigen-polarizations are studied separately (see e.g. [1]). In this case, the amplitudes of tangential electric and magnetic fields are treated as equivalent scalar voltages and currents in the equivalent transmission line section.
    [Show full text]