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Introduction to Metamaterials FEATURE ARTICLE INTRODUCTION TO METAMATERIALS BY MAREK S. WARTAK, KOSMAS L. TSAKMAKIDIS AND ORTWIN HESS etamaterials (MMs) are artificial structures LEFT-HANDED MATERIALS designed to have properties not available in To describe the basic properties of metamaterials, let us nature [1]. They resemble natural crystals as recall Maxwell’s equations M they are build from periodically arranged (e.g., square) unit cells, each with a side length of a. The ∂H ∂E ∇×E =−μμand ∇×H = εε (1) unit cells are not made of physical atoms or molecules but, 00rr∂tt∂ instead, contain small metallic resonators which interact μ ε with an external electromagnetic wave that has a wave- where the r and r are relative permeability and permit- λ ⎡ ∂ ∂ ∂ ⎤ length . The manner in which the incident light wave tivity, respectively, and L =⎢ ,,⎥ . From the above ⎣⎢∂xy∂ ∂z ⎦⎥ interacts with these metallic “meta-atoms” of a metamate- equations, one obtains the wave equation rial determines the medium’s electromagnetic proper- ties – which may, hence, be made to enter highly unusual ∂2E ∇=−2E εμεμ (2) regimes, such as one where the electric permittivity and 00rr∂t 2 the magnetic permeability become simultaneously (in the ε μ same frequency region) negative. If losses are ignored and r and r are considered as real numbers, then one can observe that the wave equation is ε The response of a metamaterial to an incident electromag- unchanged when we simultaneously change signs of r and μ netic wave can be classified by ascribing to it an effective r . (averaged over the volume of a unit cell) permittivity ε ε ε μ μ μ To understand why such materials are also called left- eff = 0 r and effective permeability eff = 0 r . In order to introduce such a description, one requires that the size handed materials (LHM), let us assume a time-harmonic of the artificial resonators characterized by a be much and plane-wave variation for fields in Maxwell’s equa- smaller than the wavelength λ, i.e. a n λ. As long as this tions (1) criterion is fulfilled, one may normally assume that the E(x,y,z,t) = Eeiωt-ikAr (3) ε response of the medium is local, i.e. that the values of eff μ where we have introduced wavevector k. Similar expres- and eff averaged over a given unit cell do not depend on the wavevector nor on the corresponding values of these sion holds for H. Then, Maxwell’s equations take the form parameters at neighboring unit cells. Hence, in such a k x E = -ωμ μ H (4) medium, the effects of spatial dispersion may legitimately 0 r ωε ε be ignored. k x H = + 0 rE (5) In this short review, we shall explain the fundamental From the above equations and definition of cross product, one can immediately see that for ε > 0 and μ > 0 the vec- M.S. Wartak characteristics of (negative-refractive-index) metamateri- r r als, together with the manner in which their building tors E,H and k form a right-handed triplet of vectors, and <[email protected]>, ε μ Department of blocks (meta-atoms) can be constructed. We shall also if r < 0 and r < 0 they form a left-handed system (see Physics and concisely outline an exemplary application that is enabled Fig. 1) – from where the designation of these media as Computer Science, by such media, namely stopping of light. Finally, we con- left-handed arises. Wilfrid Laurier University, Waterloo, clude by outlining key challenges that need to be tackled Ontario N2L 3C5 before the deployment of metamaterials in such applica- tions becomes more functional and efficient. K.L. Tsakmakidis and O. Hess, Department of Physics, Imperial College London, South Kensington Campus, London SUMMARY SW7 2AZ UK The concept of man-made structures known as metamaterials is discussed along with their diverse applications, like stopping light Fig. 1 (a) Right-hand orientation of vectors E, H, k for the and optical cloaking. ε μ case when r > 0, r > 0. (b) Left-hand orientation of ε μ vectors E, H, k for the case when r < 0, r < 0. 30 C PHYSICS IN CANADA / VOL. 67, NO. 1 ( Jan.-Mar. 2011 ) ... METAMATERIALS (WARTAK ET AL.)AA An important dif- Metamaterials with negative effective permittivity in ference between the microwave regime regular dielectrics It is well-known that metals at optical frequencies are charac- and left-handed terized by an electric permittivity that varies with frequency metamaterials can according to the following, so called Drude, relation be realized when one considers prop- ⎡ ω2 ⎤ p (8) agation of a ray εω()=− ε0 ⎢1 ⎥ through the bound- ⎣⎢ ωω()+ i γ⎦⎥ ary between left- 2 2 Ne handed and right- where ω p = is the plasma frequency, i.e. the frequency mε0 handed media, as with which the collection of free electrons (plasma) oscillates shown in Fig. 2. in the presence of an external driving field, N, e and m being, Here, 1 is the inci- respectively, the electronic density, charge and mass, and γ is dent ray, 2 is the Fig. 2 Reflection and refraction at the interface of two media with n >0 the rate with which the amplitude of the plasma oscillation reflected ray, 3 is 1 and n > 0 (ray 3) or n < 0 (ray 4). decreases. One can directly infer from Eq.(8) that, e.g., when the refracted ray 2 2 γ ω ω ε = 0 and < p it is < 0, i.e. the medium is characterized by when second medi- ω a negative electric permittivity. Typical values for p are in the um is the right-handed, and 4 is the refracted ray when the sec- ultraviolet regime, while for γ a typical value (e.g., for copper) ond medium is left-handed. γ . 13 ω ω is 4 x 10 rad/s. Unfortunately, for all frequencies < p for which ε < 0, it is also ω n γ, i.e. the dominant term in Eq.(8) Light crossing the interface at non-normal incidence undergoes is the imaginary part of the plasma electric permittivity, which refraction, that is a change in its direction of propagation. The is associated with losses (light absorption). angle of refraction depends on the absolute value of the refrac- tive index of the medium and it is described by Snell’s law A method for over- coming this limita- θ θ n1sin 1 = n2sin 2 (6) tion was first pro- posed and analyzed By matching the field components at the dielectric interfaces, in detail by Pendry one may readily verify that in the case where the second medi- et al. [6], based on um is double-negative (ε < 0, μ < 0), the refraction of light r r the observation occurs on the same side of the normal as the incident beam that the plasma fre- (see Fig. 2). Thus, we see that the double-negative medium quency depends (that is medium with ε < 0 and μ < 0) behaves as a medium r r critically on the exhibiting a negative (effective) refractive index. density and mass of the collective ELEMENTARY CELL OF METAMATERIAL electronic motion. FORMED BY SRR AND THIN WIRE They considered Fig. 3 Schematic illustration of a period the structure illus- In recent years several elements of various structures have been arrangement of infinitely long thin considered as building blocks (unit cells) of metamaterials [2-4]. trated in Fig. 3, wires, used in the creation of an wherein thin metal- Provided that the dimensions of such unit cells are much small- effective plasma medium at lic wires (infinite er than the wavelength, one can determine the effective relative microwave frequencies. μ ε in the vertical magnetic permeability r , and electric permittivity r , by proper averaging techniques. In the case where both of these direction, z) of parameters are negative, the correct value that should be attrib- radius r are periodically arranged on a horizontal plane (xy). uted to the effective medium’s refractive index is given by The unit cell of the periodic structure is a square whose sides -i(ωt-kz) have length equal to a. If an electric field E = E0e z is n =− με (7) incident on the structure, then the (free) electrons inside the eff r r wires will be forced to move in the direction of the incident The negativity of the real part of a medium’s refractive index field. If the wavelength of the incident field is considerably in the case where the real parts of the permittivity and perme- larger compared to the side length of the unit cell, λ a, then ability are negative is a general result, valid for all kinds of pas- the whole structure will appear (to the incident electromagnet- sive and active media [5]. However, caution needs to be exer- ic field) as an effective medium whose electrons (confined in cised in the cases where either the permittivity or the perme- the wires) move in the +z direction. The crucial observation ability of the metamaterial is active [5]. here is that, since the electrons are confined to move only inside the thin wires, the effective electron density of the whole 2 Let us now discuss how one can create media having negative πr structure (effective medium) is Neff = N 2 , with N being the permittivity and negative permeability. a LA PHYSIQUE AU CANADA / Vol. 67, No. 1 ( jan. à mars 2011 ) C 31 ... METAMATERIALS (WARTAK ET AL.) electron density inside each wire. Thus, for sufficiently thin To this end, consid- wires the effective electron density, Neff , of the engineered er a three-dimen- medium can become much smaller compared to N, thereby sional periodic ω substantially decreasing the effective plasma frequency, p, of repetition of the the engineered medium.
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