An alternative model for wave propagation in anisotropic impedance-matched metamaterials

Progress In Electromagnetics Research, Vol. 141, 149–160, 2013 AN ALTERNATIVE MODEL FOR WAVE PROPAGATION IN ANISOTROPIC IMPEDANCE-MATCHED METAMATERIALS Consuelo Bellver-Cebreros*and Marcelo Rodriguez-Danta Departamento de Fisica Aplicada III, Escuela Tecnica Superior de Ingenieria, Universidad de Sevilla, Camino de los Descubrimientos s/n, Seville 41092, Spain Abstract—The propagation of light in an anisotropic impedancematched metamaterial is studied in the frame of geometrical optics. We prove that directions of fields ¯ D,¯ Band ¯v(ray velocity) are a triad of conjugate directions with respect to the inverse relative dielectric permittivity tensor and constitutes a local basis, whose reciprocal one is formed by directions of ¯ E,¯ Hfields and wave-vector ¯ k. Consequently, both dual bases are intrinsically related to the physical properties of medium. We have identified these bases with direct and reciprocal bases of a curvilinear coordinates system, showing that physics defines geometry. This identification provides a powerful tool to solve two kinds of problems (direct and inverse ones) that currently arise: In direct problems, medium properties are given and it suffices to know ˜ε= ˜µtensor at every point, to obtain the wave structure. In inverse problems, medium properties must be found for the rays to propagate along prescribed trajectories. The procedure is applied to an illustrating example. 1. INTRODUCTION When dealing with laws of propagation of monochromatic plane waves, the so called “optical transformation theory” in stating an equivalence principle between electromagnetic parameters of a physical medium expressed in Cartesian coordinates and their analogues in vacuum obtained from a curvilinear coordinate transformation, puts in correspondence the Euclidean inhomogeneous and anisotropic physical Received 5 June 2013, Accepted 6 July 2013, Scheduled 12 July 2013 * Corresponding author: Consuelo Bellver-Cebreros ([email protected]). 150 Bellver-Cebreros and Rodriguez-Danta space, with another homogeneous, isotropic and matter free space, but endowed with a locally flat metric. Thus, the ray trajectory in the physical medium is associated with a geodesic curve in virtual space. Every analogy in Physics is a guide that enable us to predict hidden relations and harmonies. Thus, if we consider that stream lines in irrotational fluid motion are the analogues of rays in geometrical optics, the invisibility problem in optics becomes similar to that of the flux of a fluid stream in the presence of an obstacle. Analogy and principle of equivalence allow, in flat problems, to recover the old theory of complex potential (conformal mapping) and, consequently, mathematical analysis is considerably simplified [1]. Nevertheless,“geometrical physics” in these media can be understood, in our opinion, not only as a direct consequence (“all occurs as if”) of the invariance of Maxwell equations in coordinate transformations [2–4], but anything else: physical properties of a medium intrinsically define the most adequate geometry. In general, at visible frequencies (430–790 THz), natural media exhibit a homogeneous and isotropic magnetic behaviour together with an inhomogeneous and anisotropic dielectric one. Certain metamaterials (to which theories developed and described in this paper can be applied) when subjected to external actions, obey linear constitutive equations: ¯ D=ε0˜ε¯ E;¯ B=µ0˜µ¯ H, with equal relative permittivity and permeability tensors, (˜ε= ˜µ). The specificity of behaviour laws enable us to endow the space with a metric biunivocally linked to the metamaterial physics. These properties can not be applied to natural media with dielectric anisotropy and magnetic isotropy [5]. In the present alternative treatment, it has been proved that in any point of these metamaterials (in absence of charges and currents) the wave front (or the ray) is getting distorted in such a way that ¯ D,¯ Bfields and ray velocity ¯v(or ¯ E,¯ Hand ¯ k) constitute a local basis of conjugated directions with respect to relative permittivity or permeability tensor evaluated at that point. 2. ALTERNATIVE ANALYTICAL MODEL In this section, from relative permittivity (or permeability) tensor and from laws of evolution in the medium, we are intended to establish vector bases consisting of conjugated directions. These bases, intrinsically related to physical properties, are put in correspondence with local vector bases that, in curvilinear coordinates, are going to define the medium geometry. Progress In Electromagnetics Research, Vol. 141, 2013 151 2.1. Conjugate Directions Let ˜ Ibe a symmetric tensor of second order, with components Iij referred to a Cartesian reference frame {¯ i1,¯ i2,¯ i3}and let ˆνand ˆµbe any two directions. The bound vector to ˆνby the tensor [6] is vector ¯ Γνdefined as ¯ Γν=˜ I·ˆν. The intrinsic component, σνof vector ¯ Γνis equal to its projection onto vector ˆνin such a way that σν=¯ Γν·ˆν. The projection of vector ¯ Γνonto ˆµis ¯ Γν·ˆµand as a consequence of Cauchy theorem [7] we arrive to: ¯ Γν·ˆµ=¯ Γµ·ˆν. Two directions ˆνand ˆµare named conjugate if [8, 9]: ¯ Γν·ˆµ=¯ Γµ·ˆν= 0 (1) Figure 1 provides a simple geometrical interpretation: A relevant property of Cauchy quadric (defined by σν=const) [7] is that its normal at every point is collinear with bound vector ¯ Γνfor any direction ˆν. The orthogonality of vectors ¯ Γνand ˆµimplies the relation of conjugation. O x P H Q π π1 2µ Γν z y ν Figure 1. Cauchy’s quadric associated with a second-order tensor ˜ I. The orthogonality of vectors ¯ Γν=˜ I·ˆνand ˆµimplies that unit vectors ˆνand ˆµare conjugate directions. Vector ˆµlies in plane π2parallel to the tangent plane of the ellipsoid at “P”. It must be noted that principal directions of a tensor are also conjugate directions. Conjugation arises as a broader concept than that of principal directions (a particular case, where ¯ Γνis collinear with ˆν), but keeping the algebraic property of diagonalisation of tensors, if expressed in covariant components. 152 Bellver-Cebreros and Rodriguez-Danta 2.2. Laws of Evolution The double Fourier transform ¯ f(ω, ¯ k) of a vector field ¯ F(t, ¯r), function of both time and position is: ¯ f¡ω, ¯ k¢=Zdtd3¯r¯ F(t, ¯r)ei(ωt−¯ k·¯r)(2) and it can be shown that [10]: ∂¯ F ∂t → −iωf(ω, ¯ k); ∇· ¯ F(t, ¯r)→i¯ k·f(ω, ¯ k) (3) ∇× ¯ F(t, ¯r)→i¯ k×f(ω, ¯ k) (4) Applying these expressions to Maxwell equations for source-free media, we have: ¯ kׯ E=ω¯ B;¯ kׯ H=−ω¯ D(5) ¯ k·¯ D= 0; ¯ k·¯ B= 0 (6) Bearing in mind that wave vector ¯ k(wave-front gradient) and ray velocity ¯v(collinear with Poynting vector), are given by [11]: ¯ k=(¯ Dׯ B)ω W; ¯v=(¯ Eׯ H) W(7) (where Wis the electromagnetic energy density), from Maxwell equations [12] or from Expression (7), the following equations in ¯vdomain arise: ¯vׯ B=−¯ E; ¯vׯ D=¯ H(8) ¯v·¯ E= 0; ¯v·¯ H= 0 (9) In order to complete the above equations, we add some specific constitutive relations of artificial media (metamaterials), where relative permittivity tensor (˜εij) and relative permeability one (˜µij) are identical when referred to the same vector basis; consequently the identity holds also for inverse tensors ˜ε−1 ij = ˜µ−1 ij = ˜τij (in what follows, ˜τ). In other words, ¯ D=ε0˜ε·¯ E;¯ E=˜τ·¯ D ε0 ;¯ B=µ0˜ε·¯ H;¯ H=˜τ·¯ B µ0 (10) It is immediate to state that in these media, electromagnetic energy density is given by W=¯ E·¯ D=¯ B·¯ H; i.e., electric energy density coincides with magnetic one, and consequently, ¯ k·¯v=ω. Under the assumptions of propagation of monochromatic (ω = constant) plane waves in the absence of current densities and charges, Equations (5), (6), (8), (9) and (10) enable us to derive some different geometrical structures and their corresponding physical laws of evolution. Progress In Electromagnetics Research, Vol. 141, 2013 153 2.2.1. Geometrical Structure of Fields a) In order to describe the laws of evolution of fields, we choose a local base (P, ¯ei/{i= 1,2,3}), consisting of vectors ¯ei, collinear with ¯ D,¯ B fields and ¯v(ray velocity), respectively. We will prove that this basis is conjugated with respect to the tensor ˜τ. The chosen vectors are: ¯e1=¯ D |¯ D|; ¯e2=¯ B |¯ B|; ¯e3=¯v |¯v|; with ¯ei=hiˆui;h1=h2=h3=1 (11) and √g= (¯e1ׯe2)·¯e3. Equations (8) and (9) prove that ¯ E(¯ H) is orthogonal to ¯ B(¯ D) and ¯v, and constitutive laws (10) prove that ¯ Eis collinear with ¯ Γu1= ˜τ·ˆu1(and ¯ His collinear with ¯ Γu2= ˜τ·ˆu2). Thus, ε0¯ E |¯ D|= ˜τ·ˆu1=¯ Γu1;µ0¯ H |¯ B|= ˜τ·ˆu2=¯ Γu2(12) From the above mentioned orthogonality relations, Equation (12) and Cauchy theorem, we can deduce that ¯ Γu1·ˆu2=¯ Γu1·ˆu3=¯ Γu2·ˆu1= ¯ Γu3·ˆu1=¯ Γu2·ˆu3=¯ Γu3·ˆu2= 0. In other words, the electric displacement field ¯ D, the magnetic induction field ¯ Band ray velocity ¯vlocally propagate along conjugated directions of the inverse relative permittivity tensor of the medium. From the above analysis, it yields the choice of the reciprocal local base (P, ¯ej/{j= 1,2,3}), that consists of vectors collinear with the bound vectors to directions ˆu1, ˆu2, ˆu3and defined as: ¯ej= ¯ Γuj hjτj =˜τ·¯ej h2 jτj−→ it yields ¯ei·¯ej=δj i(13) with τj=¯ Γuj·ˆuj(τjcoincide with the eigenvalues of the tensor, when vectors ˆuiare orthogonal). In these frames of reference (bases ¯eior ¯ei), fields have only a component (covariant or contravariant). Thus, ¯ D=D1¯e1;¯ B=B2¯e2; ¯v=v3¯e3;¯ E=E1¯e1;¯ H=H2¯e2; (14) with D1=|¯ D|/h1;B2=|¯ B|/h2;v3=|¯v|/h3;E1=|¯ D|h1τ1/ε0; H2=|¯ B|h2τ2/µ0, and the inverse relative permittivity tensor (in covariant components) is written as: ˜τ=τij ¯ei⊗¯ejwith τij =h2 iτjδj i(15) With respect to the physical meaning of τ1and τ2, we retain the expressions of electric and magnetic energy densities in a linear medium [11, 13]: We=¯ E·¯ D 2;Wm=¯ H·¯ B 2(16) 154 Bellver-Cebreros and Rodriguez-Danta Since τj=¯ Γuj·ˆuj, we can write: τ1=¯ Γu1·ˆu1=ε0¯ E·¯ D |¯ D|2=We W0 e ;τ2=¯ Γu2·ˆu2=µ0¯ H·¯ B |¯ B|2=Wm W0 m (17) Coefficient τ1is the ratio between the electric energy density in the medium, We, and the electric energy density, W0 e, carried by the wave if propagation were in vacuum and obviously, it coincides with the inverse relative permittivity associated with the ˆ u1direction. The same applies to τ2, but referred to magnetic energy density. In order to know the physical meaning of every vector of the basis that describe covariant components of fields, it remains only to investigate what vector ¯e3represents. From (7), (13), (14) and taking into account the linear character of operator ˜τ, we can write, ˜τ·(¯ Eׯ H)=E1H2˜τ¡¯e1ׯe2¢=E1H2²123˜τ·¯e3=E1H2 √gh2 3τ3²312¯e1ׯe2 =h2 1h2 2h2 3τ1τ2τ3 ε0µ0gD1B2(¯e1ׯe2)= h2 1h2 2h2 3τ1τ2τ3 ε0µ0g¡¯ Dׯ B¢(18) where components ²123 =²312 of Levi-Civita tensor are equal to 1/√g. From Equations (7), (13), (18), and taking into account that ε0µ0= 1/c2, we can write: ˜τ·ˆv=τ1τ2τ3h2 1h2 2h2 3c2 gω ¯ k=⇒¯e3=τ1τ2h2 1h2 2c2 gω|¯v|¯ k(19) We have shown that vector ¯e3has the direction of the wave-vector ¯ k, and then ¯ k=k3¯e3. To summarize, ¯ D=D1¯e1;¯ B=B2¯e2; ¯v=v3¯e3;¯ E=E1¯e1;¯ H=H2¯e2;¯ k=k3¯e3(20) Then, vectors of the reciprocal base ¯eilie along ¯ E,¯ Hand ¯ k directions. There is an intrinsic geometrical structure, associated with propagation of plane waves in these media in such a way that the Euclidean space is endowed, at every point P, with two dual bases defined from physical properties of the medium. Table 1 and Figure 2 show this geometrical structure. Table 1. Direct base vectors and reciprocal base vectors obtained from tensor ˜τ. Direct basis Reciprocal basis ¯e1collinear with ¯ D¯e1colllinear with ¯ E ¯e2collinear with ¯ B¯e2collinear with ¯ H ¯e3collinear with ¯v¯e3collinear with ¯ k Progress In Electromagnetics Research, Vol. 141, 2013 155 v k B ED H e1e1 e2 e3 e2 e3 P Figure 2. Local reciprocal bases ¯eiand ¯eiassociated with the structure of a light plane wave propagating through an anisotropic impedance-matched medium. b) Irrespective of anything given above, we may assume that the space is described by a system of curvilinear coordinates, q1,q2,q3 related to Cartesian ones, x1,x2,x3through the equations: qj=qj(x1, x2, x3)⇔xj=xj(q1, q2, q3) with j= 1,2,3,(21) that allow us to define, if the transformation is admissible [7], two reciprocal local bases at any point Pas: Direct basis: (P, ¯ej)/¯ej=∂¯r ∂qj = k=3 X k=1 ∂xk ∂qj ¯ ikwith j={1,2,3}(22) Dual basis: (P, ¯ej)/¯ej=∇qj= k=3 X k=1 ∂qj ∂xk ¯ ikwith j={1,2,3}(23) where reciprocity relation ¯ei·¯ej=δj iholds. Transformation rules between base vectors are given by ¯ei=gij ¯ej, ¯ei=gij ¯ej, where gij = ¯ei·¯ejis the metric tensor. This geometry, defined by metric tensor gij, suggests to identify local bases (11) and (13), intrinsically related to the physical properties of medium, with (22) and (23) of the just described arbitrary curvilinear coordinates system. From this correspondence, physics defines geometry. Assuming the equality of local bases, we have a powerful tool to solve two kinds of problems that currently arise: direct and inverse 156 Bellver-Cebreros and Rodriguez-Danta ones. For direct problems, medium properties are given and it suffices to know ˜ε= ˜µtensor at every point, because the method provides the wave structure. In inverse problems, we must find the medium properties for the rays to propagate along prescribed trajectories. If ray trajectory is given, i.e., geometry is known, we can determine medium properties ˜τ(See Section 3: Application). 2.2.2. Ray Evolution and Fermat Principle Unlike natural optical anisotropic media (with dielectric anisotropy and magnetic isotropy), in these media with ˜ε= ˜µ, there is an only mode of propagation and an only ray velocity ¯v, for every wave-vector ¯ k. From Equation (19) and since in this case, ¯ k·¯v=ω, we obtain, ¯v·˜τ·¯v=τ1τ2τ3h2 1h2 2h2 3c2 g(24) In accordance with the definition of ray index of refraction nr[11] as nr=c/v, and remembering that ¯vis tangent to the ray trajectory: ¯v=vd¯r/dl =v¯ t, we can write: n2 r=c2 v2=g h2 1h2 2h2 3τ1τ2τ3 (ˆu3·˜τ·ˆu3) (25) We have just found the expression of the ray index as: nr=√g h1h2h3µ¯ t·˜τ τ1τ2τ3·¯ t¶1/2 (26) with ˆu3≡¯ t=d¯r dl with |d¯r|=dl. Since nris a function both of position ¯rand of the direction of vector d¯r/dl, Equation (26) shows the complexity of light propagation in inhomogeneous and anisotropic media, even for the materials in discussion, for which there is an only mode of propagation, and we can solve for nr. Unlike the isotropic case, the distance between two infinitely near points depend on their position and also on their relative orientation. Consequently, equations of evolution can be found as geodesics in a Finsler space [14], in which the line element is given by ds =nr(¯r, d¯r/dl)dl. When values of nrare substituted into the Fermat‘s variational principle in order to use Hamiltonian and Lagrangian formalism, a problem arises because conventional Legendre transform, would obtain an identically vanishing Hamiltonian from Lagrangian, which is homogeneous of degree one in d¯r/dl. Nevertheless, Fermat principle is invariant under reparameterising, then parameter “arc length” lmay be substituted by optical path, in such a way that dl =nrdσ [15], and a non-singular Progress In Electromagnetics Research, Vol. 141, 2013 157 Lagrangian arises as L=n2 r(¯r, d¯r/dσ). Consequently, equations of evolution can be found as extremals of the functional, δZn2 rdσ = 0 (27) where distance is measured by travel time (optical path). 3. APPLICATION Let us illustrate the above theory with an application. Since direct problems have been more discussed, we deal with an inverse one: Which optical properties must the medium have for light-rays to propagate along a prescribed trajectory? For instance, we want to characterize medium properties in such a way that every plane wave that propagates along an axis, say z(wave vector ¯ kparallel to zaxis), yields a ray describing an helicoidal trajectory of axis z. For that purpose, we introduce the following coordinates system (See Figure 3). q1=z θ;q2=%;q3=z(28) expressed in cylindrical coordinates: %,θ, z. We see that coordinate plane q3= constant describes the wave-front and coordinate line q3is along the desired ray. Vectors of the direct basis are: ¯e1=∇q1=−z %θ2ˆuθ+1 θˆuz; ¯e2=∇q2= ˆu%; ¯e3=∇q3= ˆuz(29) 2 2 2 3 e line q line q q= const. q= const. 3 Figure 3. Coordinate surfaces and coordinate lines of curvilinear coordinates system (q1, q2, q3), defined as q1=z/θ;q2=%;q3=z, where cylindrical coordinates %,θand zare used. In this case, coordinate plane q3= constant describes the wave-front and coordinate line q3is along the desired ray.