Geometrical analysis of wave propagation in left-handed metamaterials. Part I

Progress In Electromagnetics Research C, Vol. 4, 103–119, 2008 GEOMETRICAL ANALYSIS OF WAVE PROPAGATION IN LEFT-HANDED METAMATERIALS. PART I C. Bellver-Cebreros and M. Rodriguez-Danta Departamento de Fisica Aplicada III, Escuela Superior de Ingenieros Universidad de Sevilla Camino de los Descubrimientos s/n, 41092-Seville, Spain Abstract—Geometrical analyses of basic equations of electromagnetics waves propagation in anisotropic dielectric materials with magnetic isotropy are presented in two complementary papers (Part I and Part II). In the present one, analysis arises from quadrics associated with relative dielectric tensor (˜ε) compatible with conical surfaces that represent the general plane wave equation (relation of dispersion). This study systematizes rays propagation in left-handed materials (LHMs) exhibiting dielectric anisotropy and magnetic isotropy. In particular, indefinite dielectric media where dielectric permittivities are not all the same sign, have been investigated. Graphical alternative procedures for ray tracing in these media are presented. 1. INTRODUCTION Forty years ago, the concept of left-handed metamaterials was studied in detail by Veselago [1]. Left-handed metamaterials (LHMs) are media with dielectric permittivity ε<0 and magnetic permeability µ<0, exhibiting unique electromagnetic properties, such as negative refraction. Over the last ten years, LHMs-related research has deserved a great deal of attention since the works of Pendry [2] and Smith et al. [3], who developed the first experimental LHM structure with split-ring resonators in the range of microwaves. Since then, a large amount of papers have dealt with this subject [4, 6–17]. Reflection and transmission in these materials with different dispersion relations have been well studied in [18, 19]. Anisotropic character of LHMs is frequently expected. Isotropic LHMs are difficult to prepare for experiments and they are actually anisotropic in nature. For this reason, alternative reinterpretations 104 Bellver-Cebreros and Rodriguez-Danta (that we consider original) of basic equations of wave propagation in anisotropic dielectric materials with magnetic isotropy are presented. The methodology is valid for both positive and negative values of dielectric permittivities (as well in uniaxial as in biaxial media) and magnetic permeability and allows the study of propagation of locally plane electromagnetic waves in LHMs [21]. Reinterpretation of basic equations is made in Section 2, where some features of quadrics associated to the dielectric tensor of these media are outlined. In Section 3, geometrical methodology is applied to ray tracing in RHMs and LHMs with dielectric anisotropy and magnetic isotropy. In particular, the so called indefinite media are considered [4, 5]. This term refers to a material for which the eigenvalues of the permittivity and permeability tensor are not all the same sign. In the complementary part of this paper [21], another kind of geometrical analysis of light propagation in RH and LH media, involving local properties of dielectric permittivity tensor and M¨ohr’s plane graphical construction has been carried out. 2. GEOMETRICAL INTERPRETATION OF PLANE WAVE EQUATION Let us consider linear media, with dielectric anisotropy and magnetic isotropy. An orthonormal cartesian frame (u1, u2and u3) along the main directions of relative dielectric permittivity tensor ˜εis used. It is assumed that ε1>ε 2>ε 3are the eigenvalues of the relative dielectric tensor, ˜ε, that can be positive or negative. In dealing with non-conducting media free of currents and charges (j=0;= 0), the relation of dispersion or general plane wave equation can also be written as [22]: ν−(n·ν)n=a εν(1) where unit vector νis defined as ν=E/|E|, bound vector εν= ˜εν=D/(ε0|E|). Unit vector nis defined as n=k/k, where kis the wave vector whose length is equal to ω/vp. Parameters ωand vp are the angular frequency and the phase velocity, respectively. Finally, parameter ais defined as: a=µrv2 p/c2. Then we have easily shown that when angle θbetween extraordinary ray direction and wave-vector is greater than π/2, negative refraction occurs and relative magnetic permeability must be negative (µr=−1, LHM). If θ=π, the phenomenon is known as imperfect backward-wave and is a consequence of anisotropy. Progress In Electromagnetics Research C, Vol. 4, 2008 105 Equation (1) is our starting point as the basic plane wave equation for these media. Since we restrict our study to monochromatic waves, dispersion can be disregarded and eigenvalues εiare then constants depending only on the material. When dealing with uniaxial media, one of the plane waves exhibits isotropic behaviour. Obviously, the subject of our study is the other eigen mode. In what follows, we are only concerned with the extraordinary mode: Poynting vector does not coincide with the direction of the wave vector k. From the dot product of Eq. (1) and vector εν, two important relations are obtained: σν=a|εν|2; cos θ=a|εν|(2) where θis the angle between Eand Dand σν=εν·ν. On the other hand, in [22] the authors propose an alternative expression of Fresnel’s equation of wave normals in terms of intrinsic components (τn,σ n) of bound vector εn. This equation reads: τ2 n+σ2 n−σnI−1 a+a∆ = 0 (3) where Iand ∆ are the trace and the determinant of tensor ˜ε, respectively. Eq. (3) describes the locus of the end of εnin the plane cartesian coordinate system (σ, τ). This locus is a circumference centered at point ((I−1/a)/2,0) of radius Rgiven by R=((I−1/a)/2)2−a∆. If tis an unit vector in the same direction as Poynting vector S, the dot product of tand ngives the angle between the wave vector k and the ray direction that, as expected, is also θ: t·n=a|εν|=µrv2 p c2|εν|(4) Then, values of θ>π/2 occur with µr<0. Similarly, if θis greater than π/2, the normal component σνof vector ενis also negative and one can see from (2), that this only occurs for µr<0. Let us give a new geometrical interpretation of Fresnel’s equation of wave normals, given by Eq. (3). Let n(α1,α 2,α 3) be an unit vector along any direction of propagation given by n=k/k. Intrinsic components of vector εnbound to nverify the following three equations [22]: τ2 n+σ2 n=ε2 1α2 1+ε2 2α2 2+ε2 3α2 3(5) σn=ε1α2 1+ε2α2 2+ε3α2 3(6) 1=α2 1+α2 2+α2 3(7) 106 Bellver-Cebreros and Rodriguez-Danta If Eq. (5) is multiplied by the unit, Eq. (6) by 1 a−I, and Eq. (7) by a∆ and if addition is made, Eq. (3) is reobtained, because this sum vanishes. Thus, the sum of the three right-hand sides of Eqs. (5), (6) and (7) gives: ε2 1+ε11 a−I+a∆α2 1+ε2 2+ε21 a−I+a∆α2 2 +ε2 3+ε31 a−I+a∆α2 3=0 =⇒(8) =⇒λ1α2 1+λ2α2 2+λ3α2 3= 0 (9) with λi=ε2 i+εi1 a−I+a∆. Property 1. Coefficients λifrom Eq. (9) are equal to the power of point (εi,0) belonging to σaxis with respect to the circumference given by (3). Proof: It is well known that the power P(,) of any point with respect to a circumference is obtained by substituting the point coordinates into the equation of the circumference. Substitution of coordinates (εi,0) into (3) gives: P(εi,0) = ε2 i−εiI−1 a+a∆=λi that coincides, in fact, with λi. Property 2. The locus described by the Eq. (9): λ1α2 1+λ2α2 2+λ3α2 3=0 describes a cone in the space of wave vector directions. Proof: Power of a point with respect to a circumference C1is positive when the point lies outside C1and negative when the point lies inside. The power vanishes when the point belongs to C1. In the present case, when dealing with Eq. (9), one of the (εi,0) (i=1,2,3) points lies in a different region than the others (see Fig. 1); if one of the points lies outside the circumference given by (3), the other two points lie inside. Then two of the λicoefficients in Eq. (9) are the same sign and the sign of the other coefficient is opposite to them, consequently the locus described by Fresnel’s equation of wave normals in the space of wave vector directions is a cone. In what follows, we refer to this cone as Fresnel’s cone. Property 3: The locus described by the first equation in (2): σν=µr v2 p c2|εν|2(10) Progress In Electromagnetics Research C, Vol. 4, 2008 107 ε εε 1 2 3 τ σ On n ε Figure 1. Power of points εiwith respect to the Fresnel’s circumference (locus of the ends of bound vector ε n ). Power of point ε1with respect to the circumference is positive, whereas powers of ε2 and ε3are negative. is also a cone in the space of wave vector directions. Proof: If ν(α1,α 2,α 3) is an unit vector along the direction of vector E, intrinsic components of vector εν(parallel to D) verify the following equations: τ2 ν+σ2 ν=ε2 1α2 1+ε2 2α2 2+ε2 3α2 3(11) σν=ε1α2 1+ε2α2 2+ε3α2 3(12) Multiplying Eq. (11) by the unit, Eq. (12) by −1 aand adding them, Eq. (10) is reobtained, because the sum vanishes. Therefore τ2 ν+σ2 ν−1 aσν=0 =⇒ ε1ε1−1 aα2 1+ε2ε2−1 aα2 2+ε3ε3−1 aα2 3= 0 (13) It is immediately seen that Eq. (13) also describes a cone. We call this cone, vector Dcone. A further analysis involving M¨ohr’s plane construction and other geometrical analysis of Eqs. (2) and (3) is carried out in Part II of this work. Finally, let us assume an unit vector ν(α1,α 2,α 3), along any direction in space R3(that can be the direction of propagation, direction of electric field E, and so on), the dot product of νand the bound vector ενto this direction is given by: σν=εν·ν=ε1α2 1+ε2α2 2+ε3α2 3=⇒±1=ε1X2 1+ε2X2 2+ε3X2 3(14) 108 Bellver-Cebreros and Rodriguez-Danta where Xi=αi √σν. Last equation represents Cauchy’s quadric (See Fig. 2). An important property of Cauchy’s quadric [24] is that its normal at every point is collinear with bound vector εν. This property will be used next. ε v v P O 13 2 1 12 3 3 2 (ε α ( ( ( ) ) , ,, , XXX ( ( ,, σv tangent plane Cauchy's quadric ε α ε α σvσv 132 ,, 132 ααα αα α Figure 2. Cauchy’s quadric for a medium with dielectric anisotropy, where νis an unit vector referred to space R3. In this case, eigenvalues of ˜εare all the same sign. If the eigenvalues εiare not all the same sign, we have hyperboloids (one-sheeted and two-sheeted) depending on the sign of σν. In fact, the asymptotic cone (ε1X2 1+ε2X2 2+ε3X2 3= 0), divides the space into two regions, one for σν>0 (one-sheeted hyperboloid) and the other one for σν<0 (two-sheeted hyperboloid) (see Fig. 3). It must be pointed out that the directions of vector νthat lie inside the asymptotic cone provide angles between νand ενless than π/2. The directions of νoutside the cone give angles greater than π/2. In order to state the correspondence between the proposed methods and those used by other authors, uniaxial media are considered. If methodology proposed in [23–26] is applied, one has that the section of wave vector surface for the extraordinary propagation for ε1=ε2>0, ε3<0 and µr=−1 is given by: −X2 1 |ε3|+X2 3 |ε1|=−1 (15) where Xi=(kic)/ω.Ifφand ψare the angles that nand tform with Progress In Electromagnetics Research C, Vol. 4, 2008 109 Figure 3. Cauchy’s quadric for an indefinite dielectric media, where eigenvalues of ˜εare not all the same sign. The quadric consists of a two-sheeted hyperboloid and an one-sheeted one. Asymptotic cone, that divides the space into two regions, is also shown. u1axis, respectively, it is easily shown that: tan ψ=−|ε3| ε1 tan φ(16) Our proposed alternative method attains the same result (16), by using Cauchy’s quadric section given by: ε1x2 1−|ε3|x2 3=−1 3. RAY TRACING IN RHM AND LHM INDEFINITE MEDIA WITH DIELECTRIC ANISOTROPY In [22], it was shown that in uniaxial right-handed media, vector εnis collinear with the extraordinary ray direction. In uniaxial left-handed media, εnalso lies in the same plane that ν,ενand n. Since the angle between Eand Dis the same as the angle between tand nand Cauchy’s theorem [22] states that εn·ν(= n·εν) also vanishes, vector εnis collinear with t, but may be antiparallel. Our method is applied to four media with dielectric anisotropy and magnetic isotropy [4] with the following material parameters: This kind of media are named indefinite because the eigenvalues of their permittivity tensors are not all the same sign. Figure 3 shows the plot of the two shells (double-sheeted and onesheeted hyperboloids) of Cauchy’s quadric for dielectric parameters of 110 Bellver-Cebreros and Rodriguez-Danta Medium Class ε1ε2ε3µr “1” RHM 1 1 −2 1 “2” LHM 2 −1−1−1 “3” RHM 2 −1−1 1 “4” LHM 1 1 −2−1 media “1” and “4”. The asymptotic cone that divides the space into two regions is also shown. Let us consider three cases that deserve attention in the study of propagation of plane waves in these media. 3.1. Direction of Electric Field Is Known This means that unit vector νis given. Then, allowed directions of ν can only intercept the positive (or negative) shell of Cauchy’s quadric, depending on the sign of µr. Asymptotic cone can be regarded as the boundary of allowed directions of νfor RH and LH media. In this case, the graphical procedure is as follows: 1. Positive shell (if µr>0), or negative one (if µr<0) of Cauchy’s quadric and unit vector νare drawn. 2. Let Pbe the intersection point of νdirection with Cauchy’s quadric. The gradient of Cauchy’s quadric at Pgives the direction of εν. The angle between νand ενis θ. 3. To obtain phase velocity vpor parameter a, it suffices to substitute coordinates of Pinto the equation of vector Dcone. Figure 4 shows the intersection between the one-sheeted surface of Cauchy’s quadric and the vector Dcone for a RH medium (medium “1”) and for v2 p=c2/2. Because of the symmetry of uniaxial media, a two dimensional representation can also be used. Without loss of generality, medium “4” is considered. Meridian sections of the negative shell of Cauchy’s quadric and direction of given vector νare depicted in Fig. 5. 3.2. Direction of Propagation (n)IsKnown Let us outline the procedure in a 2-D plot. 1. Meridian sections of positive and negative shells of Cauchy’s quadric are drawn. Progress In Electromagnetics Research C, Vol. 4, 2008 111 2. Let Qbe the intersection point of ndirection with Cauchy’s quadric. The gradient of Cauchy’s quadric at Qgives the direction of εn. Vector tis along εnor −εn, depending on the sign of µr,as Eq. (2) states. Figure 4. Intersection between the one-sheeted surface of Cauchy’s quadric and the vector Dcone for a RH medium (medium “1”) and for v2 p=c2/2. v v ε θ Cauchy's quadric (two sheets) Meridian section of x x 3 1 Figure 5. Plane graphical determination of angle θbetween ν (parallel to E) and ε ν (parallel to D), or between direction of propagation, nand extraordinary ray direction t, for a left-handed indefinite dielectric medium (uniaxial medium “4”), when νis given. Meridian section of Cauchy’s quadric (two-sheeted) is shown. 118 Bellver-Cebreros and Rodriguez-Danta J. A. Kong, “A study of using metamaterials as antenna substrate to enhace gain,” Progress In Electromagnetics Research, PIER 51, 295–328, 2005. 11. Chen, H., B. I. Wu, and J. A. Kong, “Review of electromagnetic theory in left-handed materials,” J. of Electromagn. Waves and Appl., Vol. 20, No. 15, 2137–2151, 2006. 12. Yang, R., Y. Xie, P. Wang, and T. Yang, “Conjugate left- and right-handed material bilayered substrates qualify the subwavelength cavity resonator microstrip antennas as sensors,” J. of Electromagn. Waves and Appl., Vol. 20, No. 15, 2113–2122, 2006. 13. Grzegorczyk, T. M. and J. A. Kong, “Review of left-handed metamaterials: Evolution from theoretical and numerical studies to potential applications,” J. of Electromagn. Waves and Appl., Vol. 20, No. 14, 2053–2064, 2006. 14. Yang, R., Y. Xie, P. Wang, and L. Li, “Microstrip antennas with left-handed materials substrates,” J. of Electromagn. Waves and Appl., Vol. 20, No. 9, 1221–1233, 2006. 15. Wang, M. Y. and J. Xu, “FDTD study on scattering of metallic column covered by double-negative metamaterial,” J. of Electromagn. Waves and Appl., Vol. 21, No. 14, 1905–1914, 2007. 16. Li, Z. and Cui, T. J., “Novel waveguide directional couplers using left-handed materials,” J. of Electromagn. Waves and Appl., Vol. 21, No. 8, 1053–1062, 2007. 17. Abdalla, M. A. and Hu, Z., “On the study of left-handed coplanar waveguide coupler on ferrite substrate,” Progress In Electromagnetics Research Letters, Vol. 1, 69–75, 2008. 18. Lindell, I. V., S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media-media with negative parameters, capable of supporting backward waves,” Microw. Opt. Tech. Lett., Vol. 31, 129–133, 2001. 19. Smith, D. R. and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett., Vol. 90, 077405, 2003. 20. Boltasseva, A. and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials, Vol. 2, 1–17, 2008. 21. Bellver-Cebreros, C. and M. Rodriguez-Danta, “Fresnel equation revisited: Application to left-handed media with dielectric anisotropy,” EOS Topical Meeting on Diffractive Optics 2007 , 133–134, European Optical Society, 2007. Progress In Electromagnetics Research C, Vol. 4, 2008 119 22. Bellver-Cebreros, C. and M. Rodriguez-Danta, “Eikonal equation, alternative expression of Fresnel’s equation and M¨ohr’s construction in optical anisotropic media,” Opt. Commun., Vol. 189, 193– 209, 2001. 23. Bellver-Cebreros, C. and M. Rodriguez-Danta, “Geometrical analysis of wave propagation in left-handed metamaterials. Part II,” Progress In Electromagnetics Research C , Vol. 4, 85–102, 2008. 24. Fung, Y. C., Foundations of Solid Mechanics, Prentice Hall, Englewood Cliffs, New Jersey, 1965. 25. Born, M. and E. Wolf, Principles of Optics, Pergamon Press, New York, 1999. 26. Scharf, T., Polarized Light in Liquid Crystals and Polymers, John Wiley and Sons, Hoboken, New Jersey, 2007. 27. Akhmanov, S. A. and S. Yu. Nikitin, Physical Optics, Clarendon Press, Oxford, 1999. 28. P´erez, J. H., Optique G´eometrique et Ondulatoire, Masson, Paris, 1984.