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Optic axes of general anisotropic media: a perspective Sérgio A. Matos†, Carlos R. Paiva† and Afonso M. Barbosa† †Instituto de Telecomunicações, and Department of Electrical and Computer Engineering, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Phone: +351-218418457, Fax: +351-218418472, e-mail: [email protected]

Abstract— For simple electric (magnetic) anisotropy, a single the authors are aware, the constitutive function that mapping between the permittivity (permeability) value and the characterizes general anisotropy. We have shown that direction of space describes the medium properties. The well- general anisotropy can be understood through the new known classification of anisotropic crystals (uniaxial or biaxial) function ξ that plays a central role: when ξ is uniaxial depends only on a single constitutive function. The generalization of this taxonomy for simultaneously electric and (biaxial) the medium is uniaxial (biaxial). We should stress, magnetic anisotropy is presented using the formalism of that several papers neglect the role of the interrelation geometric algebra. For this case, the permittivity and between the electric and magnetic constitutive functions on permeability functions lose their direct interpretation on the the global type of anisotropy. In references [2-4] a general classification scheme. We found a new constitutive function that uniaxial crystal is presented as a medium described by describes the type of general anisotropy. The mathematical and uniaxial permittivity and permeability functions. However, as geometric background, provided by the geometric algebra formalism, is the keystone to derive these results in its most we show, the medium is uniaxial only if ξ is uniaxial. In general form. The binding between geometric algebra and fact, ξ can be uniaxial for biaxial electric and magnetic linear functions is proved to be a more elegant and insightful linear functions. More than an elegant way of resuming way of working with anisotropy than the usual or even anisotropy, geometric algebra brings a new geometric insight differential forms methods. into the problem that was not obvious to obtain with the I. INTRODUCTION conventional formalisms. A part from the new in this paper, we obtained new physical results as well, that Anisotropy is an old topic in as a wide are independent of the chosen formalism. However, with the range of materials with electric or magnetic anisotropy are new geometrical perspective of linear available. However, only recently did materials with both transformations these results were easier to grasp. electric and magnetic anisotropy become available through This communication is organized as follows: in section II, the development of metamaterials technology. These media some fundamental concepts to understand geometric algebra have singular properties that can be used for new exciting are introduced. In section III, a brief overview on how new applications such as invisibility cloaking devices [1]. geometric meaning, as well as new algebraic techniques, are Nevertheless, in this new area, a general theory that can fully obtained from the binding of the formalism of geometric describe the physical nature of this type of anisotropy is still algebra with linear functions. In section IV, we characterize lacking. In fact, few studies address this type of materials in a aligned anisotropy using the new constitutive function ξ . general way [2-5]. Plain tensors and dyadics, as suggested in Finally, the conclusions are presented in section V. [5], give a purely mathematical solution for this problem, thereby obnubilating its physical interpretation. By using the II. GEOMETRIC ALGEBRA CA3 new mathematical language of geometric algebra [6-11], a new framework that enables a fresh physical insight into this 3 Geometric algebra of \ , named CA3 , is a graded algebra, problem will be presented herein. As shown in [11], this i.e. a general element of A , which we call a multivector, is formalism can shed new light into the classical problem of C 3 wave propagation in a biaxial crystal. In this a sum between: a (grade 0 blade), a vector (grade 1 communication we generalize that study for general blade), a bivector (grade 2 blade) and a trivector (grade 3 anisotropy, i.e., for media with both electric and magnetic blade). In CA3 we work with two new geometric objects: anisotropy. The only restriction, apart from reciprocity, is bivectors and trivectors (Fig.1). A fundamental definition of that we consider that the eigenvectors of the permeability and geometric algebra is the geometric product (or Clifford permittivity functions should be parallel, in order to product) between two vectors: guarantee the existence of optic axes. Through this new ab= a⋅+∧ b a b (1) perspective on anisotropy we revisit the old concepts of where the scalar component of this product is the usual inner biaxial and uniaxial media by establishing broader definitions product ab⋅ , and the bivector component is given by the −1 for these concepts. The function ε (the inverse of the ab∧ . The sum of two blades gives a blade permittivity) characterizes the electrical anisotropy [11]. A with the same grade, whereas the inner product decreases the uniaxial (biaxial) function ε−1 implies a uniaxial (biaxial) grade and the outer product increases the grade. The unit medium, and, when written in the biaxial form, ε−1 defines trivector, also called the unit pseudoscalar, corresponds to the directions of the optic axes. For magnetic anisotropy it is eeee123= 1∧∧ 2 3 , where (eee123,,) are the orthonormal the μ−1 function (the inverse of the permeability) that defines 3 vectors of \ . The geometric product ue123 , where u is a anisotropy. In this paper, we obtain for the first time, as far as general multivector, corresponds to the Clifford dual According to (4) and to the property, it is possible to . The dual of a vector is a bivector, the dual of a transform any element of CA3 . The transformation of the bivector is a vector and the dual of a trivector is a scalar. The : cab=×=−∧( abe) is given by: Gibbs cross product ()× gains a new geometric 123 ff(ece) () abe∧ interpretation, shown in Fig 2, according to f −1 ()c =−123 123 =− 123 (5) detff det ab×=−∧() a be123 ⇔∧= a b () abe × 123 (2) The outer product has several advantages over the cross where abba⋅=⋅ff( ) ( ) is the adjoint function that reflects product: i) the outer product is associative which is not true the symmetry of the transformation: i) ffaa= the for the cross product; ii) the outer product does not depend ( ) ( ) on the metric, whereas the cross product needs the definition transformation is symmetric, ff()aa=− ( ) the of the normal direction and therefore is metric dependent; iii) transformation is antisymmetric. This and other new the outer product can be defined for any , whereas algebraic techniques brought by geometric algebra and linear the cross product is only defined for \3 . To remove the functions allow handling the Maxwell equations and the cross product from electromagnetism is the fist step to be constitutive relations with a new fresh analytical simplicity, able to obtain a true coordinate-free approach for complex and with the advantage of having a geometric interpretation media. of the obtained expressions. Further details about linear

functions and geometric algebra ^A3 can be found in [11] abc∧∧ and references therein. c b b IV. GENERAL ANISOTROPY ab∧ In geometric algebra CA , the constitutive relations of a 3 ab∧ general anisotropic media are given by a DEBH==εμεμ, (6) a) 00( )() b) Fig. 1. Geometric picture of a bivector a) and a trivector b). where ε (E) and μ(H) are the dielectric and magnetic linear functions, respectively. We say that the constitutive linear function f characterizes anisotropy when: i) the cab=×=− Fe123 mathematical form f (aaacc)()= α+β⋅00 implies that c is the optic axis of the medium; ii) the mathematical form

b f (aa) =α00 +β⎣⎡( accacc ⋅ab)() + ⋅ ba⎦⎤ implies that ca and

cb are the optic axes of the medium. For simple electric Fabce=∧= 123 (magnetic) anisotropy it is ε−1 ()a (μ−1 ()a ) linear function a that characterizes anisotropy, as it was shown in [11]. A Fig 2. Correspondence between the Gibbs cross product and fundamental question needs to be answered: what the outer product constitutive function characterizes general uniaxial or biaxial media for simultaneously electric and magnetic anisotropy? III. LINEAR FUNCTION IN CA 3 Based on the new algebraic techniques brought up to linear Linear function is the fundamental concept that binds the and multilinear functions, we were able to state this problem and consequential formalism with geometric in a single geometric equation: ˆ algebra. An equivalence between bivectors and kE∧ W ()⊥ = 0 (7) antisymmetric dyadic exists, according to where ()bae∧↔×⋅123 ( aIb) (3) ˆ 2 W ()EkEE⊥ = κee()n ζ ()⊥⊥− (8) where aI× is an antisymmetric dyadic with The scalar function κ kkkˆˆˆ=⋅μμdet , vector ax=++aaax yzy z and Ixxyyzz=++ is the identity e () () dyadic. However, symmetric tensors do not correspond to EEEkkkk=−⎡ ⋅μμˆˆ ⋅ ˆˆ⎤ and the linear function any object of geometric algebra. In fact, they represent not an ⊥ ⎣ ( ) ( )⎦ object, but a linear mapping between elements of CA . In −1 3 ζεμe (aa) = ⎣⎡ ( )⎦⎤ were introduced. A magnetic formulation geometric algebra, a linear function is then a linear mapping of equation (7) can be obtained by interchanging ε ↔ μ and between , and not only between vectors. The grade of a multivector is preserved according to the EH↔ , resulting in ˆ outermorphism kH∧ W ()⊥ = 0 (9) fff()()()ab∧= a ∧ b (4) where ˆ 2 ξ aa=+αβ⎡⎤ caccac ⋅ +⋅ (18) W ()HkHH⊥⊥⊥=−κmm()n ζ () (10) ( )()()00⎣⎦abba −1 ˆˆˆ After some algebra we obtain that with ζζme= , κm ()kkk=⋅εε ()det and ξζ(aa) =−εε21( ε 3)()em + μμ 2 ( 1 − μ 3 ) ζ( a) (19) HHHkkkk=−⎡⎤ ⋅εεˆˆ ⋅ ˆˆ. For general anisotropy the ⊥ ⎣⎦() () where εi (μi ) are the eigenvalues of the electric (magnetic) optic axes may not exist. Therefore, to maintain the linear function, for i = {1, 2, 3} . Eq. (19) shows clearly what classification of uniaxial and biaxial media we impose the are the roles of ζ and ζ for aligned anisotropy. When condition of aligned anisotropy: i.e. the proper vectors of the e m permittivity and permeability are the same. For this case, we ε13=ε (μ13=μ ) it is ζm ()a ()ζe ()a that characterizes can write the following mathematical biaxial forms anisotropy. The general case implies a linear combination ⎧ ⎪⎣ζeee()aa=+αβ⎡⎤ ( daddad1221 ⋅ ) +⋅ ( ) ⎦between these two functions to obtain the directions of the ⎨ (11) optic axes. In Fig 3, we show the refractive index polar plots ζ aa=+αβ⎡⎤ faffaf ⋅+⋅ ⎩⎪ mmm()⎣⎦ (1221 ) ( ) for a general biaxial medium. We also plot the directions of with αζe = 2 , βζζe =−()312 , and the mathematical biaxial axes ()dd12, and (ff12, ) . We should stress, that a general biaxial medium is only defined ⎧dee13311=+γ γ ⎨ (12) through the function ξ . The mathematical form of the ⎩dee23311=−γ γ functions ε−1 and μ−1 do not reveal the number of optic where e are the eigenvectors of ζ , and i e axes of the medium. In this example, we show that from a ⎧ ⎪γζζζζ12131=−()() − uniaxial electric function and a biaxial magnetic function the ⎨ (13) resulting medium is biaxial. In fact, we could even have both ⎪γζζζζ=− − −1 −1 ⎩ 33231()() ε and μ uniaxial resulting in a biaxial medium. The for ζζζ123<<. The eigenvalues of ζe are ζi with complete discussion about classification of general anisotropic media will be published elsewhere. i = {}1, 2, 3 . The other magnetic constants ()αβmm,,,ff12 can be obtained by interchanging ε↔μ. Both the ζ and ζ e1 e m ε23μ ε()a : ε=ε=120.25, ε= 3 0.5 functions are strong candidates to be the linear functions that μ()a : μ=1230.25, μ= 0.5, μ= 2 n+ characterize anisotropy. However, after solving the Eqs. (7) ζe ()a : ζ=121, ζ= 2, , ζ= 3 4 and (9), we verify that neither one of these functions ε32μ characterizes anisotropy. In fact, a uniaxial (biaxial) ζa c a ε13→ε ⇒θ→0 implies a uniaxial (biaxial) medium, with aem= { , } , μ13→ μ ⇒δ→0 however the optic axes directions are in general different n− f1 from ()dd12, or ()ff12, . θ The eigenvalue solution of Eq. (7) is given by d1 2 su⋅+⋅ rv⎛⎞ su ⋅−⋅ rv δ λ± =±⎜⎟ +⋅⋅()()ru sv (14) 22⎝⎠ − ε μ εμ 21 − εμ12 12 εμ21 where e ⎛⎞ 3 11⎜⎟ λ± =−αe (15) β ⎜⎟n2κ kˆ e ⎝⎠± e () and d ˆ 2 μ()kd∧ 1 ukder=−ˆ ∧, =− e ()1 123ˆˆ 123 μ()kk⋅ f2 (16) μ kdˆ ∧ − ε μ c ˆ () 2 32 b vkdes=−() ∧2 123, =− e 123 μ()kkˆˆ⋅

Imposing that nn+−= we obtain the optic axes directions −ε23μ Fig 3. Eigenwaves of a general biaxial medium. μ311γμγee+−+ 133 μγμγ 311 ee 133 ccab==, (17) 22 22 μγ31++ μγ 13 μγ 31 μγ 13 For a uniaxial medium, we have that

Therefore, the anisotropic function that truly characterizes ccddff12= ==== 1 212 and therefore ζe or ζm can anisotropy must be given by describe general uniaxial media. [8] S. A. Matos, J. R. Canto, C. R. Paiva, and A. M. V. CONCLUSIONS Barbosa, “Complex aberration effect in moving dispersive DNG media: a spacetime algebra approach,” Few studies address media with both electric and magnetic PIERS Online, vol. 4, No. 6, pp. 611 – 614, 2008. anisotropy. However, the advent of metamaterials has [9] C. R. Paiva and M. A. Ribeiro, “Doppler shift from a prompted a fresh look into the old problem of anisotropic composition of boosts with Thomas : A media in electromagnetics. In this communication, we have spacetime algebra approach,” J. of Electromagn. Waves presented a new general approach to solve this problem by and Appl., vol. 20, pp. 941-953, 2006. using geometric algebra. In fact, with this novel approach, [10] M. A. Ribeiro and C. R. Paiva, “Transformation and we have shown how geometric algebra can provide a Moving Media: A Unified Approach Using Geometric mathematical framework for general anisotropy that is far Algebra,” – Chapter in: Metamaterials and better than plain tensors and dyadics: through the direct Plasmonics: Fundamentals, Modelling, Applications manipulation of coordinate-free objects such as vectors, (NATO Science Series), Saïd Zouhdi, Ari Shivola, and bivectors and trivectors, geometric algebra reveals itself as Alexey P. Vinogradov, Editors. Dordercht The the most natural setting to study anisotropy by providing a Netherlands: Springer 2009. deeper physical and geometrical insight while avoiding [11] S. A. Matos, M. A. Ribeiro and C. R. Paiva, cumbersome calculations. We have shown that general “Anisotropy without tensors: a novel approach using anisotropy can be understood through a new function ξ that geometric algebra,” Optics Express, vol. 15, no. 23, pp. plays a central role: when ξ is uniaxial (biaxial) the medium 15175 -15186, 2007. is uniaxial (biaxial). Furthermore, the optic axes are obtained by writing ξ in the biaxial form. In summary: we have shown that general anisotropy does not depend on the mathematical form that each constitutive function (either ε or μ ) separately assumes but rather on a new single constitutive function ξ . Materials with magnetic and electric anisotropies impose nowadays a very important class of metamaterials with exciting new properties. Invisibility cloaking devices are a paradigmatic example. For the fabrication of these devices an heterogeneous anisotropic medium with a permittivity identical to the permeability function needs to be implemented.

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