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

Progress In Electromagnetics Research C, Vol. 4, 85–102, 2008 GEOMETRICAL ANALYSIS OF WAVE PROPAGATION IN LEFT-HANDED METAMATERIALS. PART II C. Bellver-Cebreros and M. Rodriguez-Danta Departamento de F´ısica Aplicada III Escuela Superior de Ingenieros Universidad de Sevilla Camino de los Descubrimientos s/n, Seville 41092, Spain Abstract—Geometrical analysis 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 this paper, analysis arises from local properties of dielectric permittivity tensor and M¨ohr’s plane graphical construction compatible with circumferences that represent general plane wave equation. This theoretical study allows to know how rays propagate in left-handed metamaterials (LHMs) exhibiting dielectric anisotropy and magnetic isotropy. Geometrical analysis yields plane graphical procedures for ray tracing, that are extremely easy. In particular, indefinite dielectric media, where dielectric permittivities are not all the same sign, has been investigated and a joint study of materials having the same eigenvalues of ˜ε, but with opposite values of µris performed. The opposite sense of propagation of rays in “opposite media” (media with opposite values of εiand µr) has been also shown. It must be pointed out that the presented method always allows graphical plane constructions, even when dealing with biaxial media. 1. INTRODUCTION Since the theoretical study of Veselago [1] and the works of Pendry [2] and Smith et al. [3], the interest of left-handed metamaterials (LHMs) is growing [4–19]. To take into account anisotropic character of LHMs, geometrical analysis of basic equations of wave propagation in anisotropic dielectric materials with magnetic isotropy are presented in two complementary papers. The methodology is valid also for negative values of dielectric 86 Bellver-Cebreros and Rodriguez-Danta permittivities and magnetic permeability and allows the study of propagation of locally plane electromagnetic waves in LHMs. In this paper, geometrical analysis arises from local properties of dielectric permittivity tensor (˜ε) and M¨ohr’s plane graphical construction compatible with circumferences that represent general plane wave equation (relation of dispersion) (See Section 2). It must be noted that the approach presents many advantages, because it facilitates both quantitative and qualitative analysis inherent to a plane geometry. Some examples on ray tracing in RHMs and LHMs with dielectric anisotropy and magnetic isotropy are shown in Section 3, where, the so called indefinite media are considered [4–6]. 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 (Part I) of this paper [22], another kind of geometrical analysis of ray propagation in RH and LH media, involving the compatibility between quadrics associated with relative dielectric tensor (˜ε) and conical surfaces representing the general plane wave equation (relation of dispersion) has been carried out. 2. GEOMETRICAL PLANE CONSTRUCTION IN LHMS WITH DIELECTRIC ANISOTROPY 2.1. General We are only concerned with linear media, with dielectric anisotropy and magnetic isotropy (µ=µ0µr), free of currents and charges (j=0; = 0). An orthonormal cartesian frame (u1,u2and u3) along the main directions of relative dielectric permittivity tensor ˜εis used. The eigenvalues of ˜ε, are named ε1>ε 2>ε 3and they can be positive or negative. Our starting point is the corresponding relation of dispersion or general plane wave equation for these media, which can also be written as [20]: ν−(n·ν)n=aε ν (1) where unit vector νis defined as ν=E/|E|, bound vector ε ν =˜εν= D/(ε0|E|). If nis an unit vector along the direction of propagation, wave vector kis expressed as: k=kn=(ω/vp)n, where ωand vp, are the angular frequency and the phase velocity, respectively. Parameter ais equal to µrv2 p/c2. In an anisotropic medium, there are two normal eigen modes of locally plane waves for each direction of propagation. Our study only Progress In Electromagnetics Research C, Vol. 4, 2008 87 deals with the eigen mode that does not behave isotropically. This means that the directions of Poynting vector and wave vector kare non collinear. Thus, in uniaxial media, we are only interested in the extraordinary ray. In this paper, intrinsic components (τν,σ ν) of bound vectors ε ν =˜ενplay an important role. They are defined as [20]: σν=ε ν ·ν;τ2 ν=|ε ν |2−σ2 ν Since Dis parallel to ε ν (note that D=ε0|E|ε ν ), and D·n=0, the dot product of Eq. (1) and vector ε ν gives: σν=a|ε ν |2(2) that can be written in the form: τ2 ν+σ2 ν−σν a= 0 (3) Eq. (3) describes the locus of the end of the bound vector ε ν in the plane cartesian coordinate system (σ, τ). The locus is a circumference centered at point (1/(2 a),0) of radius R=1/(2|a|). Since parameter amight be negative, the radius has always a positive value and its center defines, as a function of a, the location of this circumference. The method applies for both positive and negative values of a. We call this circumference, vector Dcircumference, since Dis parallel to ε ν . It follows immediately that the angle θbetween Eand Dis given by: cos θ=a|ε ν |(4) An alternative expression of Fresnel’s equation of wave normals in terms of intrinsic components of bound vector ε n was derived in [20]: τ2 n+σ2 n−σnI−1 a+a∆ = 0 (5) where (τn,σ n) are the intrinsic componentes of ε n ,Iand ∆, trace and determinant of tensor ˜ε, respectively, and a=µrv2 p/c2. Eq. (5) describes the locus of the end of ε n in 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∆. In what follows, this circumference is named Fresnel’s circumference. If tis an unit vector along the extraordinary ray direction, defined as t=(E×H)/|E×H|, it is shown [20] that the angle between tand 88 Bellver-Cebreros and Rodriguez-Danta nis the same as the angle between Eand D(given by Eq. (4)), and then t·n=a|ε ν |. Since a=µrv2 p/c2, only media with µr=−1 give values of θ greater than π/2. This is the case of LHMs. 2.2. Graphical Procedures of Ray Tracing General graphical procedures of ray tracing based on M¨ohr’s construction, can be found in [20, 21]. In short, M¨ohr’s plane construction [23], allows to obtain the intrinsic components of bound vector ε ν from the knowledge of ν. Reciprocally, if vector ε ν is given, the direction νto which is associated, can be easily found. To perform this graphical construction only the knowledge of the eigenvalues of ˜ε is required. M¨ohr’s construction for uniaxial media is extremely easy and uses an unique circumference of diameter ε1−ε3drawn in plane (σ, τ), whose center lies on a point of σ-axis of coordinates (ε1+ε2 2,0). The equation of this circumference is: τ2+σ2−σ(ε1+ε3)+ε1ε3= 0 (6) If an unit vector nforms an angle φwith u1axis, this angle is marked off on the vertical through ε1. Let Abe the intersection point between this sloped line and M¨ohr’s circle. This point Ais the end of bound vector ε n traced from the origin (see Fig. 1). Figure 1. M¨ohr’s construction for an uniaxial indefinite medium (ε1>0 and ε3<0) from the knowledge of angle φbetween n(unit wave vector) and u1axis. The end of bound vector ε n lies on a M¨ohr’s circumference passing through ε1and ε3. In [21], a more simplified construction is also outlined, where there is no need to draw the M¨ohr’s circle. The construction is shown in Progress In Electromagnetics Research C, Vol. 4, 2008 89 Fig. 2. As an example, principal permittivities are not all the same sign (ε1=ε2>0 and ε3<0). These are the steps of the procedure: (i) Values of ε1and ε3are marked on the straight line along the direction of propagation n, that forms an angle φwith u1axis. (ii) A normal to axis u1passing through ε1and a normal to axis u3 passing through ε3are traced. The intersection of these lines is point P, that becomes the end of ε n . Figure 2. A more simplified M¨ohr’s construction in the (u1,u3) plane for an uniaxial indefinite medium (ε1>0 and ε3<0) from the knowledge of angle φbetween n(unit wave vector) and u1axis. Tracing of two straight lines is only needed. Figure 1 and Fig. 2 describe two simple graphical constructions to find ε n if vector nis known. Similarly, one can easily find nfrom ε n , as it was also outlined in [21]. Nevertheless, M¨ohr’s construction only states the tensor relation between ν(direction of E) and ε ν (parallel to D) in the (σ, τ) plane. In order to take into account the relation of dispersion given by (1), Equations (5) or (3) are needed. In fact, (3) and (5) are the loci of the end of vectors ε ν and ε n , respectively. Moreover, in both equations parameter a=µrv2 p/c2is involved. M¨ohr’s construction does not impose retrictions and can be applied to both positive and negative eigenvalues. Since Eq. (1) derives directly from Maxwell equations for plane waves, occurrence of negative values of µis also regarded. Figure 3 shows the plots of M¨ohr’s circumference, Dvector and Fresnel’s circumferences for a LH metamaterial with ε1=2;ε2=ε3= −1, and for a=−1/2. Property 1. Angle θbetween Eand Dand angle θnbetween n and ε n verify the following relation in uniaxial indefinite media: cos2θ= cos2θn 90 Bellver-Cebreros and Rodriguez-Danta Figure 3. M¨ohr’s circumference for a left-handed uniaxial indefinite medium (ε1>0, ε3<0, µr=−1). Dvector circumference (locus of the ends of bound vector ε ν ) and Fresnel’s circumference (locus of the ends of bound vector ε n ) are drawn for v2 p=c2/2. Proof: The abscissa of the point of intersection between M¨ohr’s circle given by Eq. (6) and Eq. (3) becomes: σν=ε1ε3 ε1+ε3−1 a The angle θbetween Eand Dcan be obtained from the expression: cos θ=σν |ε ν |(7) where |ε ν |2=σν/a. When dealing with the so called left-handed and right-handed indefinite media, values of acan be either positive or negative, so it is advisable to use the square of Eq. (7): cos2θ=σ2 ν |ε ν |2=aσ ν=aε 1ε3 ε1+ε3−1 a (8) Similarly, the abscissa of the point of intersection between M¨ohr’s circumference and Fresnel’s circumference, given by (5), is: σn=aε 1ε3=a∆ ε2 (9) Progress In Electromagnetics Research C, Vol. 4, 2008 91 Taking into account that σ2 n+τ2 n=|ε n |2, direct substitution of Eq. (9) into Eq. (5) leads to: |ε n |=aε 1ε3ε1+ε3−1 a The angle θnbetween nand ε n can be obtained from: cos θn=σn |ε n |(10) and, then cos2θn=σ2 n |ε n |2=σn ε1+ε3−1 a=aε 1ε3 ε1+ε3−1 a (11) Then, for indefinite media, we can write: cos2θ= cos2θn(12) Property 2. The locus of the ends of vectors ε n in plane (u1,u3) is an ellipse of semiaxes |ε1|and |ε3|. Proof: From Fig. 4, one can writes: ε n =ε1cos φu1−|ε3|sin φu3=x1u1+x3u3 and, consequently: x1 ε12 +x3 ε32 =1 (13) The locus of the ends of vectors ε n is the ellipse with semiaxes a=|ε1|and b=|ε3|. To relate vector ε n with unit vector t, we recall that in uniaxial media with positive eigenvalues of ˜ε, vector ε n is collinear with the extraordinary ray direction t, as it was shown in [20]. In uniaxial indefinite media, ε n also lies in the same plane that ν, ε ν and n. Since the angle between Eand Dis the same as the angle between tand nand Cauchy theorem [20] states that ε n ·ν(= n·ε ν ) also vanishes, vector ε n is collinear with t, but may be antiparallel. Nevertheless, from a graphical point of view, there is no uncertainty in tracing the direction of extraordinary ray. If a>0, the angle between t(the ray) and nis acute (less than π/2), and if a<0, the corresponding angle is greater than π/2, as it is depicted in Fig. 3. 92 Bellver-Cebreros and Rodriguez-Danta Figure 4. Locus of the ends of bound vector ε n in an uniaxial indefinite medium (ε1>0 and ε3<0). The locus is an ellipse of semiaxes ε1and ε3. In order to state the correspondence between the proposed methods and those used by other authors, uniaxial media are considered. In [24–27], a relation between the slopes of the direction of propagation and the extraordinary ray direction is obtained. The alternative method leads to the same relation. Property 3. There holds the following relation between the slope of the direction of propagation (tan φ) and that of the extraordinary ray direction (tan ψ), in the plane (u1,u3): tan ψ=−ε3 ε1 tan φ Proof: Let us consider the simplified construction given in Fig. 5. From the scheme, one has that: tan ψ=AB OA =|ε3|sin φ |ε1|cos φ=−ε3 ε1 tan φ 3. SOME EXAMPLES OF RAY TRACING IN RHM AND LHM UNIAXIAL INDEFINITE MEDIA In this section, the method is applied to four media with dielectric anisotropy and magnetic isotropy [4] with the following material parameters: Progress In Electromagnetics Research C, Vol. 4, 2008 93 Figure 5. Graphical construction showing the equivalence between traditional and alternative methods in ray tracing, when propagation along an uniaxial indefinite medium is considered. 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 It must be pointed out that solutions of plane and non-evanescent waves impose some restrictions on the allowed values of velocity of propagation, vpand then on possible directions of propagation. In these media allowed values of 1/a are either greater than ε1 or less than ε3. Thus, values of ain the interval ε3<1/a<ε 1are not possible. For instance, for right-handed medium “3” (µr= 1 and θ≤π/2), Eq. (3) restricts possible values of velocity of propagation vp, which must fulfill that: vp<c/ √2, because 1 a>2 to give non-evanescent waves. Consequently, possible values of directions of propagation are restricted. Let us now consider certain cases that deserve attention in the study of light propagation of plane waves in these media. It may be noted that Eq. (6) determine the bound vectors, whereas Eqs. (3) and (5) give the phase velocity vpand relative permeability µrin terms of ε ν and ε n , respectively. 100 Bellver-Cebreros and Rodriguez-Danta media, but achanges sign. From Eq. (9), one has: σn1=−σn2 where σn1and σn2are the abscissas of the points of intersection between M¨ohr circumference and Fresnel’s circumference in medium “1” and medium “2”, respectively. Figs. 12 and 13 show the property for medium “1” and medium “2”. 4. CONCLUSIONS An extension of the geometrical interpretation of the basic equations of wave propagation in terms of the intrinsic components of bound vectors ε ν =˜εν, eigenvalues εiof tensor ˜ε, relative magnetic permeability µr and phase velocity vphas been carried out in order to study left-handed metamaterials with dielectric anisotropy and magnetic isotropy. The locus described by Fresnel’s equation of wave normals in the plane of the intrinsic components of ε n is a circumference. And a circumference is the locus of another main equation describing the projection Donto E. The use of plane geometry is one of the advantages of the method, which facilitates both quantitative and qualitative analysis. The correspondence with other analogous works has been also stated. Characteristic phenomena of LHMs, like imperfect backwardwave propagation have been easily explained. A detailed discussion on wave propagation along indefinite dielectric media, where dielectric permittivities are not all the same sign, has been carried out and, as an example, four dielectric uniaxial media with given parameters have been investigated. Several properties of propagation of light along these media are demonstrated. The discussion proposes extremely easy graphical constructions to obtain the extraordinary ray direction tfrom the knowledge of n=k/k and, reciprocally, to obtain nif tis given: it suffices to trace two straight lines. Moreover, graphical procedures allow a joint study of materials having the same eigenvalues of ˜ε, but with opposite values of µr. If phase velocity vpis known, two graphical and versatile constructions are also outlined. They involve intersections between M¨ohr’s circumference and the other circumferences (loci of ε ν and ε n ). The knowledge of vpalso allows the immediate obtention of n direction: it suffices to trace a straight line. 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