Surface Polaritons in Composite Media with Time Dispersion of Permittivity and Permeability

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1 Physics of the Solid State, Vol. 4, No. 11, 1, pp Translated from Fizika Tverdogo Tela, Vol. 4, No. 11, 1, pp Original Russian Text opyright 1 by espyatykh, ugaev, ikshteœn. MGNETISM N FERROELETRIITY Surface Polaritons in omposite Media with Time ispersion of Permittivity and Permeability Yu. I. espyatykh,. S. ugaev, and I. E. ikshteœn Institute of Radio Engineering and Electronics (Fryazino ranch), Russian cademy of Sciences, pl. Vvedenskogo 1, Fryazino, Moscow oblast, 1411 Russia Received March 1, 1 bstract The propagation of electromagnetic waves in a composite medium based on an array of conducting wires in a ferromagnetic nonconducting matrix is discussed. It is demonstrated that, in certain ranges of frequencies and wavelengths, the composite under investigation can possess properties inherent in a left-handed medium. The regions of the existence of bulk and surface localized electromagnetic waves are explored. onsideration is given to the dispersion of surface electromagnetic waves in thick layers of the composite. 1 MIK Nauka/Interperiodica. 1. INTROUTION The design of new composites is of considerable physical and applied interest, because composite media can possess specific properties that are not observed in ordinary materials. In recent years, important theoretical and experimental results have been obtained for composite materials based on different conducting elements arranged in a dielectric nonmagnetic matrix. Under electromagnetic radiation, a periodic array composed of conducting elements is equivalent to an effective medium with a time dispersion of the permittivity and permeability when the radiation wavelength is much longer than both the element dimension and the lattice spacing. Pendry et al. [1] proved that, in the long-wavelength limit, a three-dimensional array composed of very thin conducting straight wires behaves as an electron plasma in a metal. For a plane wave with frequency ω and wave vector k, the permittivity ε of the electron plasma with frequency ω p is represented by the relationship εω ( ) = (1) ωω ( iγ ) and does not depend on the wave vector over a wide range of wavelengths. It is of importance that the permittivity ε(ω) is negative at ω < ω p and γ =. lthough the damping parameter γ is substantially less than the plasma frequency (γ ω p ), the plasma waves in metals are observed only in the ultraviolet and visible frequency ranges. This is explained by the fact that metals exhibit rather high values of the electron concentration n, the plasma frequency ω p = (ne /4πm) 1/, and the damping parameter γ. For example, these parameters for aluminum are as follows: ω p = 15 ev and γ =.1 ev. In [1], it was demonstrated that, in artificial materials made up of thin conducting wires, the mean concentra- ω p tion of charge carriers and the effective plasma frequency can be decreased by six orders of magnitude and the effective mass of charge carriers can be increased by one order of magnitude as compared to the effective electron mass in the bulk metal. In this case, the ratio γ/ω p remains sufficiently small (γ/ω p.1 for aluminum). The plasma resonance in these materials, unlike the bulk metal, occurs in the microwave range. Periodic systems composed of different-type ring conducting elements (wire rings, split rings, helices, layers in the form of split rings, etc.) with a considerable effective permeability have been proposed and examined in recent works [ 6]. These systems are characterized by the resonant frequency of electromagnetic wave absorption. The dispersion of their effective permeability µ(ω) in the long-wavelength range is described by the formula Fω µω ( ) = , () ω ω + iωγ where ω is the resonant frequency, Γ is the dissipation factor, and F is a constant. It is worth noting that all these quantities depend on the internal structure and concentration of the conducting elements. s is the case with the transverse diagonal component of the permeability of a homogeneous magnetized isotropic ferromagnet, the effective permeability () in the vicinity of the resonance is negative in the high-frequency range. Smith et al. [6] noted that, by combining thin conducting straight wires and the aforementioned ring elements with a large inductance and capacitance, it is possible to construct composites whose effective permittivity and permeability are negative in a certain range of frequencies. This would allow experimental observations of the unusual effects, which were theoretically predicted by Pafomov [7] and Veselago [8 1] in the 195s 196s /1/411-1$1. 1 MIK Nauka/Interperiodica

2 SURFE POLRITONS IN OMPOSITE MEI WITH TIME ISPERSION 11 Earlier works [7 1] dealt primarily with the propagation of uniform plane electromagnetic waves in right-handed and left-handed media and with wave reflection from the interface between these media. However, it is important to know the specific features of the spectrum of excitations occurring in materials with a time dispersion of the general form, for example, the features of the spectrum of bulk and surface long waves in plane-parallel layers of a composite. In the present work, we examined the spectrum of surface polaritons in a layer of a composite material based on a periodic lattice of conducting wires [1] in a ferromagnetic nonconducting matrix.. THEORETIL NLYSIS Let us analyze the dispersion of surface electromagnetic waves in a sufficiently thick layer of a composite material. This layer covers the region L/ y L/ in space and is composed of a ferrodielectric matrix involving a simple cubic lattice of thin conducting or superconducting wires (similar to the structure comprising a regular array of conducting wires in a dielectric nonmagnetic matrix [1]). In the case when the lattice spacing is very small compared to the layer thickness, the electromagnetic radiation wavelength, and the penetration depth of the electromagnetic wave, the high-frequency properties of the composite can be described within the continuous approximation. The volume percentage of the conducting material is small, and the permittivity tensor εˆ coincides with that calculated in [1], that is, εˆ = εδ ik, () where the quantity ε is defined by formula (1). The effective plasma frequency ω p and the parameter γ of damping at the expense of resistance losses in the conductors are expressed through the parameters of the conducting wires as follows: ω p = πc /a ln( a/r), γ = 4a ω p /r σ, (4) where σ is the conductivity, r is the wire radius, a is the lattice spacing, and c is the velocity of light in free space. Note that the γ/ω p ratio is inversely proportional to the bulk concentration of the conducting material; hence, its value cannot be inappropriately low. Generally speaking, it is preferable to use superconductors for preparing composites with a high Q-factor in the microwave range. Hereafter, we assume that the damping effect in a metal and a ferromagnet is small and can be ignored. ccording to Pendry et al. [5], who calculated the permeability of the lattice of conducting rods arranged in a dielectric matrix, the permeability of this composite is proportional to the bulk concentration of the metal. Since the bulk concentration is assumed to be rather low, the effective permeability of the lattice of conducting elements in the ferromagnetic matrix is approximately equal to the permeability of the matrix. Therefore, for the layer in a tangential external magnetic field H n z, when the excitation amplitude varies with time t as exp(iωt), the permittivity is represented in the form and the permeability tensor is given by (5) µ iν µˆ = iν µ. (6) 1 Here, = ω/ω M, p = ω p /ω M, ω M = 4πgM, g > is the gyromagnetic ratio, M is the saturation magnetization of the magnetic matrix, µ 1 = , ν = , (7) = H /4πM ( ). The electromagnetic field inside and outside the composite satisfies the Maxwell equations: [ e] = ik b, d =, (8) [ h] = ik d, b =, where k = ω/c. The continuity conditions of the tangential components of the electric e and magnetic h fields and the normal components of the electric d = εˆe and magnetic b = µˆ h inductions are met at the layer boundary. We will restrict our consideration to the case of a transverse electric wave whose field is specified as {h x, h y, e z } and propagates along the n x axis. The solution to the system of Maxwell equations (8) with the boundary conditions at the interface will be sought under the assumption that the dependence of the electric field component e z on the x and y coordinates is described by the expression e z where ε = p, a 1 exp( ikx q y), y> L/ [ a exp( q 1 y) + a = (9) exp( q 1 y)] exp( ikx), L/ y L/ a 4 exp( ikx + q y), y< L/, q k = k, q 1 = k k εµ, µ = ( µ ν )/µ. The expression for the magnetic induction b follows immediately from the first equation of system (8), and PHYSIS OF THE SOLI STTE Vol. 4 No. 11 1

3 1 ESPYTYKH et al. the constants a 1, a, a, and a 4 are determined from the boundary conditions. We omit simple calculations and write the dispersion relationship for electromagnetic waves localized in the vicinity of the layer: (1) where µ ν. In the limiting case of short wavelengths, we have q q 1 and relationship (1) describes the dispersion of a amon Eshbach wave. The inequality (11) is the necessary and sufficient condition for the electromagnetic emission to be absent. Equation (1) is transcendental, and, hence, its solutions in the general form can be found only by numerical methods. From the standpoint of spin-wave electronics, the principal interest is in analyzing surface electromagnetic modes whose field amplitude exponentially decreases deep in the layer. In addition to inequality (11), the region of the existence of surface electromagnetic waves in the plane k is limited by the inequality (1) If this inequality is not satisfied, the electromagnetic wave is a bulk wave. For short wavelengths or thick layers, we have q 1 L 1 and the dispersion relationship (1) takes the simple form ( q 1 + µ q ) ( ν/µ ) k =. (1) Since the left-hand side of Eq. (1) is an even function of the frequency and the wave number k, the dispersion curves in the half-plane k are symmetric with respect to the ordinate axis. fter the appropriate rearrangement of the terms and squaring of the right-hand and left-hand sides of Eq. (1), we obtain the biquadratic equation for determining the dependence k(), that is, where [ q 1 + µ q ( ν/µ ) k ] sinh( q 1 L) +µ q q 1 cosh( q 1 L) =, k > k k > k εµ. k 4 1 k =, = µ [ 4 ( µ + 1/µ ) ], (14) 1 = k µ [ µ ( 1 + εµ ) ( ε + µ )( µ + 1/µ )], = k 4 µ ( ε µ ). One of the solutions to Eq. (14) is physically meaningless and results from the squaring of Eq. (1). t k k M k, we have εµ 1 and k ( k p ) , ( ) M ( + 1/) ± sgnk = ( ω M /c), 1 = ( + 1). (15) The signs ± in relationship (15) correspond to the propagation of the surface electromagnetic wave along the lower and upper boundaries of the layer. The dispersion of short surface waves depends on the normalized effective plasma frequency p. The surface waves are forward waves when p < + 1/ and backward waves when p > + 1/. This implies that the change in sign of the permittivity ε leads to a change in sign of the group velocity of the surface wave. The regions of existence of the bulk and surface localized wave solutions and the dispersion curves of the surface electromagnetic mode in the layer at different ratios of the quantities p,, 1 = ( + 1), and = + 1 are qualitatively shown in Figs In Figs. 1 4, straight lines 1 (q = ), which are described by the expression k = κ ( ) = k M, (16) separate the localized and nonlocalized wave solutions. urves and, which are represented by the relationship k = κ 1 ( ) = k M ( p )( )/( 1 ) (17) and whose points correspond to q 1 =, are the boundaries between the regions of existence of bulk and surface waves. The curve described by formula (17) is doubly connected, because the inverse function (κ 1 ) is determined from the biquadratic equation and has two positive branches. s a consequence, the region of the existence of bulk waves is singly connected, whereas the region of the existence of surface waves is doubly connected. The upper branch of the curve corresponding to relationship (17) is of no interest, because it completely lies in the region of nonlocalized solutions. For p < 1 (Fig. 1), the region of the existence of bulk waves (region I) is bounded by straight line 1 from the left, the lower κ 1 () branch [curve, relationship (17)] from below, and the straight line = 1 from above. The low-frequency region of the existence of surface waves (region ) is located to the right of straight line 1 and below the κ 1 () branch (curve ). The high-frequency region of the existence of surface waves (region I) lies to the right of straight line 1 and above the straight line = 1. In region, the dispersion curve of the surface electromagnetic wave (curve PHYSIS OF THE SOLI STTE Vol. 4 No. 11 1

4 SURFE POLRITONS IN OMPOSITE MEI WITH TIME ISPERSION 1 + 1/ I + 1/ I 1 p 1 I p 1 IV 1 I Fig. 1. Regions of the existence of (I) bulk and (, I) surface localized waves and the dispersion curves of surface electromagnetic waves (curves and ) at p < 1. Fig.. Regions of the existence of (I) bulk and (, I) surface localized waves and (IV) nonlocalized backward waves and the dispersion curves of surface electromagnetic waves (curves and ) at 1 < p < + 1/. p p + 1/ 1 I IV 1 I + 1/ 1 IV 1 I I Fig.. Regions of the existence of (I) bulk and (, I) surface localized waves and (IV) nonlocalized backward waves and the dispersion curves of surface electromagnetic waves (curves and ) at + 1/ < p <. Fig. 4. Regions of the existence of (I) bulk and (, I) surface localized waves and (IV) nonlocalized backward waves and the dispersion curves of surface electromagnetic waves (curves and ) at p >. ) terminates in the straight line = κ () = k M at point with the coordinates k = k = k M, (18) = = 1 p ( + p ) 1/ and continuously goes into the branch of bulk waves in the lower branch of the curve = κ 1 () (curve ) at point with the coordinates k = k = κ 1 ( ), =. (19) In this case, the frequency satisfies the equation =, = p, 1 = + ( 1+ ) p, = 4 p. PHYSIS OF THE SOLI STTE Vol. 4 No. 11 1

5 14 ESPYTYKH et al. In region I, the branch of surface waves (curve ) terminates in the straight line = 1 at point with the coordinates It should be emphasized that, in the vicinity of this point, the long-wave approximation becomes incorrect. In the upper region I, the surface waves are backward waves; the dispersion branch of surface waves continuk = k = k M 1 1/, = = 1. () The quantity µ has a feature at = 1. onsequently, 1 at 1, the depth q 1 of penetration of the surface wave into the composite tends to zero and the longwave approximation in the vicinity of the line = 1 turns out to be inapplicable at any parameter p. It follows from relationship (15) that, at, the wave frequency tends to a value of = + 1/, which coincides with the upper boundary of the spectrum of surface waves in a ferromagnetic layer free from conducting elements. However, at π/a, it is necessary to take into account the periodicity of the system. In regions and I, the surface waves are forward waves at any k value. In the case where 1 < p < (Figs., ), the region of bulk localized electromagnetic waves in the plane k is bounded by the straight lines = κ () k M (line 1) and = 1 from the left and below and by the lower branch of the curve = κ 1 () from above. The remaining doubly connected part of the k plane with the values lying to the right of straight line 1 (regions, I) is the region in which the surface electromagnetic waves can exist. The dispersion of surface waves in region (curve ) is positive. The terminal point of this curve is specified by coordinates (18), and the terminal point, at which the surface wave changes over to the bulk wave, is defined by coordinates (). In region I, the dispersion branch of surface waves terminates at point with coordinates (19) and asymptotically approaches the straight line = + 1/ with an increase in. t p < + 1/, the surface waves are forward waves (Fig. ). When p > + 1/ (Fig. ), the dispersion of surface waves can become negative with an increase in. For p > (Fig. 4), the regions of the existence of bulk and surface localized wave solutions are virtually identical to those described in the preceding case. The surface waves also occur in the lower region (curve ) and are forward waves. The terminal points of the dispersion curve of surface waves (curve ) are specified by the same coordinates as in the case where 1 < p <. However, as the effective plasma frequency p increases, the dispersion curve approaches line 1 represented by the expression = κ() and contracts to the intersection point of straight lines 1 (q = ) and = 1 with the coordinates k = k M 1, = 1. (1) ously goes into the branch of bulk waves at the point with the coordinates given by the formula (19). Kaganov and Shalaeva [11] described the high-frequency branch of surface polaritons. In the system under investigation, this branch is absent, because we disregard the electric polarizability of the ferromagnetic matrix. This branch appears when the permittivity of the ferromagnet ε > 1 is taken into account. For this purpose, it is necessary to calculate preliminarily the effective permittivity of the composite. It is very important that the given composite is characterized by the region of nonlocalized bulk solutions in the k plane (Figs. 4, region IV), in which the inequalities ε <, µ <, k <, and k k < k εµ are met simultaneously. It is easy to demonstrate that the sufficient condition for existence of this region is the fulfillment of the inequality p > 1. () In this range of frequencies and wave numbers, the composite behaves as a left-handed medium and the phase velocity of the bulk electromagnetic wave is oppositely directed to the group velocity.. ONLUSION Thus, the interaction of magnetization oscillations with plasma oscillations leads to the appearance of an additional branch of surface electromagnetic waves in the long-wavelength excitation spectrum of the composite layer. In the composite layer, unlike a purely ferromagnetic layer, the surface electromagnetic waves can be forward, backward, and mixed waves depending on the plasma frequency (and its related sign of the effective permittivity). It is worth noting that, in a certain frequency range, the studied composite with nonrigid requirements imposed on the parameters of conducting elements can possess properties inherent in a left-handed medium. The main advantage of this composite over the composite material proposed by Smith et al. [6] is that it can be produced using planar technology. For example, multilayer structures composed of yttrium iron garnet films alternating with two-dimensional conducting arrays that are prepared by photolithography are suitable for observing the violation of Snell s law. In this case, the main difficulties are most likely associated with the choice of the optimum parameters for conducting elements to provide appropriate attenuation of electromagnetic waves and with the necessity of preparing a sufficiently large bulk system. The results obtained in this work on the propagation of electromagnetic waves in a composite material based on a three-dimensional cubic lattice composed of conducting elements are also valid for composite media with a two-dimensional array of conducting wires aligned parallel to the z axis. PHYSIS OF THE SOLI STTE Vol. 4 No. 11 1

6 SURFE POLRITONS IN OMPOSITE MEI WITH TIME ISPERSION 15 KNOWLEGMENTS This work was supported by the Russian Foundation for asic Research (project nos , , and ), the ommittee of Scientific Research of Poland, and the International enter of Science and Technology (project no. 15). REFERENES 1. J.. Pendry,. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett. 76, 477 (1996).. M. V. Kostin and V. V. Shevchenko, Radiotekh. Élektron. (Moscow), 156 (1988).. M. V. Kostin, Radiotekh. Élektron. (Moscow) 5, 44 (199). 4. M. V. Kostin and V. V. Shevchenko, Radiotekh. Élektron. (Moscow) 7, 199 (199). 5. J.. Pendry,. J. Holden,. J. Robbins, and W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 75 (1999). 6.. R. Smith, Willie J. Padilla,.. Vier, et al., Phys. Rev. Lett. 84, 4184 (). 7. V. E. Pafomov, Zh. Éksp. Teor. Fiz. 6, 185 (1959) [Sov. Phys. JETP 9, 11 (1959)]. 8. V. G. Veselago, Fiz. Tverd. Tela (Leningrad) 8, 571 (1966) [Sov. Phys. Solid State 8, 854 (1966)]. 9. V. G. Veselago, P. V. Glushkov, and. M. Prokhorov, Radiotekh. Élektron. (Moscow) 1, 1 (1967). 1. V. G. Veselago, Usp. Fiz. Nauk 9, 517 (1967) [Sov. Phys. Usp. 1, 59 (1968)]. 11. M. I. Kaganov and T. I. Shalaeva, Zh. Éksp. Teor. Fiz. 96, 185 (1989) [Sov. Phys. JETP 69, 17 (1989)]. Translated by O. orovik-romanova PHYSIS OF THE SOLI STTE Vol. 4 No. 11 1

Extinction properties of a sphere with negative permittivity and permeability

PERGAMON Solid State Communications 116 (2000) 411 415 www.elsevier.com/locate/ssc Extinction properties of a sphere with negative permittivity and permeability R. Ruppin* Department of Physics and Applied