## Abstract

The most widespread approach to anisotropic media is dyadic analysis. However, to get a geometrical picture of a dielectric tensor, one has to resort to a coordinate system for a matrix form in order to obtain, for example, the index-ellipsoid, thereby obnubilating the deeper coordinate-free meaning of anisotropy itself. To overcome these shortcomings we present a novel approach to anisotropy: using geometric algebra we introduce a direct geometrical interpretation without the intervention of any coordinate system. By applying this new approach to biaxial crystals we show the effectiveness and insight that geometric algebra can bring to the optics of anisotropic media.

## 1. Introduction

In Art. 794 of his celebrated treatise, originally published in 1873, James Clerk Maxwell states that “in certain media the specific capacity for electrostatic induction is different in different directions, or in other words, the electric displacement, instead of being in the same direction as the electromotive intensity, and proportional to it, is related to it by a system of linear equations” [1]. This is the physical definition of an (electrically) anisotropic medium. In electromagnetism, especially in optics and photonics, anisotropic media have always played a central role [2, 3]. There is a significant research activity in subwavelength anisotropic structures for a large variety of applications (e.g., polarization control) [4, 5]. Even new exciting potential applications such as invisibility cloaking using metamaterials do require a study of anisotropic media [6]. Apart from some attempts to use differential forms [7], the preferred coordinate-free approach to anisotropic (and bianisotropic) media is based on plain tensor (or dyadic) methods [8]. In this article we intend to show that anisotropy (as well as bianisotropy) can be more easily handled through the new mathematical approach to linear algebra provided by Clifford’s geometric algebra [9–12]. Although based on the mathematical ideas of Grassmann, Hamilton and Clifford, only recently did this mathematical approach began to reach a wider acceptability, namely through the works of David Hestenes - the most pre-eminent forerunner of this universal geometric algebra and calculus [13]. Usually, it is believed that geometric algebra is particularly useful when applied to special relativity or to relativistic quantum mechanics and general relativity [11]. Accordingly, spacetime algebra – the geometric algebra of Minkowski spacetime – has been used to solve several problems of relativistic electromagnetism [14, 15]. However, the insight and new algebraic techniques brought up to linear and multilinear functions [16] makes geometric algebra a particularly useful tool and, to the authors’ opinion, a far better framework to understand (and work on) anisotropic media than tensor methods.

Although explaining how this new approach to anisotropy can be generalized to study bianisotropic media, we focus in this article on the way geometric algebra Cℓ 3 (the geometric algebra of Euclidean space ℝ3) can handle electrical anisotropy without tensors (or dyadics). An alternative treatment to the dyadic analysis of biaxial crystals [17] is fully developed herein, which is an extension of a preliminary work presented in [18]. Namely, the two eigenwaves (±) in a biaxial crystal are fully analyzed: after obtaining the two refractive indices n ±, the fields (E ±, D ±, B ±, H ±), the angles θ ±=∢(E ±, D ±) and the energy velocities v(±)e are derived and new expressions that provide a better insight into the optical behavior of biaxial crystals are presented.

Hopefully we intend to show how this new approach, which avoids the clumsiness of tensors and dyadics in coordinate-free analyses, can facilitate the research work on anisotropy which, with the ongoing progress on metamaterials technology, is becoming increasingly more important to the optical community. One should stress that our approach should be particularly relevant whenever a coordinate-free analysis is needed either to provide insight into new anisotropic (or bianisotropic) media or leading to solutions in their most general analytical form. We do not claim, however, that this approach is the most appropriate in all circumstances: it is doubtful, for example, that our coordinate-free approach will be the most adequate to handle specific problems in guided-wave optics or in periodic layered media, where a matrix approach is, probably, still preferable.

## 2. Anisotropy in geometric algebra

Anisotropy means that the magnitude of a property can only be defined along a given direction [19]. Let us be more specific: if a medium is electrically anisotropic, an angle between vectors E (the electric field) and D (the electric displacement) exists and depends on the direction of the (Euclidean) space ℝ3 along which E is applied. In other words, anisotropy means that it is not possible to write D = ε 0 ε E , where ε 0 is the permittivity of vacuum and ε is a scalar called the (relative) dielectric permittivity of the medium. Then, the usual solution consists in introducing a permittivity (or dielectric) tensor that, in a given coordinate system, may be written as a 3×3 matrix. In fact, according to tensor algebra, a coordinate system where that 3×3 matrix can be conveyed is always implicit, although that matrix is only the specific form that the dielectric tensor takes in that particular coordinate system. In geometric algebra Cℓ 3, on the other hand, we simply state anisotropy by writing D = ε 0ε(E) , where ε(E) is a linear function ε : ℝ3 → ℝ3 that maps vectors to vectors. We call ε the dielectric function and it fully characterizes the aforementioned property of the medium that can only be defined along a given direction. Throughout we use sans-serif symbols for linear functions.

#### 2.1 Geometric algebra of the Euclidean three-dimensional space

In Cℓ 3 we introduce the geometric product u = ED, which is associative and an invertible product for vectors, as the graded sum

$u=ED=E∙D+E∧D=α+F,$

where α = ED ∈ ℝ is the usual dot (or inner) product, which is symmetric, and F = ED ∈ ∧23 is the outer (or exterior) product, which is antisymmetric. We further impose the contraction on the geometric product: aa = a 2 = |a|2 if a ∈ ℝ3. One should not confuse the outer product (which produces bivector ED) with the Gibbsian cross product (which produces vector E × D): the outer product is associative whereas the cross product is not (it satisfies the Jacobi identity and generates a Lie algebra). Basically a vector is a directed line segment whereas a bivector is a directed plane segment. The area or norm of a bivector is denoted by |F|. Parallel bivectors AB (i.e., bivectors corresponding to the same plane) can be regarded as directed angles turning either the same way, AB, or the opposite way, AB. Only when |A| = |B| and AB do we say that A = B. There is a crucial difference between a vector a ∈ ℝ3 and a bivector F ∈ ∧23: a2 = |a| = a2 ≥ 0 whereas F2 = -|F|2 = -β 2 ≤ 0. Then, from Eq. (1), we get the reverse of u, ũ, such that ũ = DE = ED -ED = α - F. Whence

$α=E∙D=(u+u˜)2,F=E∧D=(u+u˜)2,$
$∣u∣2=uu˜=EDDE=E2D2=(α+F)(α−F)=α2−F2=α2+β2=ϱ2≥0.$

One can easily show that, as the unit bivector is such that 2 = -1, then

$u=α+βF̂=ϱexp(θF̂)=ϱcos(θ)+F̂ϱsin(θ),ϱ=∣E∣∣D∣=α2+β2,$

where F = β . The outer product of a vector and a bivector produces a trivector V = aF = Fa∈∧33 which is an oriented volume element. A vector and a bivector can also be multiplied so that the result is a vector: using the left contraction aF = d ∈ ℝ3 we have, if F = ED, d = (aE)D - (aD)E; the right contraction Fa is such that Fa = -aF [10]. The geometric product aF is then aF = aF + aF = d + V, i.e., the sum of a vector and a trivector. An arbitrary element uCℓ 3, which we call a multivector, is a (graded) sum of a scalar, a vector, a bivector and a trivector: u = α + a + F + V, α = 〈ua = 〈u1, F = 〈u2, V = 〈u3 , denoting the operation of projecting onto the terms of a chosen grade k by 〈 〉k. The multivector structure of Cℓ 3 can be expressed through the direct sum of linear subspaces of homogeneous grades (or degrees) 0, 1, 2, 3, from the Grassmann (or exterior) algebra ∧ℝ3 as follows:

$Cℓ3=ℝ⊕ℝ3⊕∧2ℝ3⊕∧3ℝ3.$

A k -blade of Cℓ 3 (with k = 0,1,2,3) is an element uk such that uk = 〈ukk, where 〈ukk is a homogeneous multivector of grade k, i.e., uk ∈ ∧k3 (assuming that ∧03 = ℝ and ∧03 = ℝ3). Any trivector can be written as V = β e 123, where β ∈ ℝ and e 123 = is the unit trivector such that e 2 123 = -1. We can easily show that any multivector can be uniquely decomposed into the sum u = α + a + be 123 + β e 123. In fact any bivector is the (Clifford) dual of a vector b ∈ ℝ3, i.e., one has F = be 123 = e 123 b. For example, the relations between the outer and the cross products between two vectors a,b ∈ ℝ3 are ab = (a×b)e 123 and a×b = -(ab)e 123. Geometric algebra Cℓ 3 is a linear space of dimension 1 + 3 + 3 + 1 = 23 = 8: adopting {e 1, e 2, e 3} as an orthonormal basis for vector space ℝ3, a suitable basis for the corresponding linear space C∓ 3 is

${1︸schalars,e1,e2,e3︸vectors,e12,e31,e23︸bivectors,e123︸trivectors}$

where e 12 = e 1e 2 = e 1 e 2, e 31 = e 3e 1 = e 3 e 1 and e 23 = e 2e 3 = e 2 e 3 constitute a basis for the subspace ∧23 of bivectors (i.e., 2-blades). The subalgebra of scalars and trivectors is the center of the algebra, i.e., it consists of those elements of Cℓ 3 which commute with every element in Cℓ 3 :

$Cen(Cℓ3)=ℝ⊕∧3ℝ3.$

This subalgebra is isomorphic to the complex field ℂ. Even multivectors, which result from the geometric product of an even number of vectors, form the so-called even subalgebra Cℝ + 3 = ℝ ⊕ ∧23. This even subalgebra is isomorphic to the division ring of quaternions ℍ [10]. Although we will never use the dielectric tensor in our approach, it is easy to show how it can be obtained as soon as a coordinate system is adopted: εjk = e j ∙ε(e k).

#### 2.2 Defining anisotropy in geometric algebra

A medium is said (electrically) anisotropic if the angle θ between the electric field E and the electric displacement D is different for different directions of E. Hence, according to Eq. (4), β = β(θ)= ϱ sin(θ) (i.e., β = |F| = |〈ED2 = |ED|) depends on the direction along which E is applied. A principal dielectric axis of the anisotropic medium is a direction θ = θ 0 such that β(θ 0) = 0. Let us write E = |E|s and D = |D|t, where s 2 = t 2 = 1 (i.e., s and t are unit vectors). Then F̂ = st/sin (θ) = sr, where r 2 = 1, so that D = D + D with D = sD = |D|cos(θ) and D = rD = |D|sin(θ) as shown in Fig. 1. Moreover, a (relative) permittivity along s , ε s, can be defined as ε s = s ∙ ε(s); one has ε s =0 if ED and ε s < 0 if θ > π/2 (e.g., these two cases are possible in metamaterials). For a lossless nonmagnetic crystal, the eigenvalue equation ε(a) = λ a gives three positive eigenvalues ε 1, ε 2 and ε 3 corresponding to three unit eigenvectors e 1, e 2 and e 3 (respectively), which form an orthonormal basis for ℝ3. These eigenvectors correspond to the three principal dielectric axes of the crystal. For a biaxial crystal, one has ε 3 > ε 2 > ε 1 thereby allowing the definition of two unit vectors d 1 and d 2, such that

$d1=γ1e1+γ3e3,d2=−γ1e1+γ3e3,γ1=ε2−ε1ε3−ε1,γ3=ε3−ε2ε3−ε1,$

with γ 1 = sin(ϕ/2) and γ 3 = cos(ϕ/2). The relation between d 2 and d 1 is very neat in Cℓ 3

$d2=rϕd1r˜ϕ,rϕ=exp(ϕe312)=cos(ϕ2)+e31sin(ϕ2),$

where rϕ ∈ ⊕ ∧23 (an even multivector) is called a rotor and satisfies the relation rϕ ϕ =1. This result should be compared with the Gibbsian formula d 2 =cos(ϕ)d 1 + sin(ϕ)(d 1×e 2). One can easily show that the corresponding dielectric function is then

$ε(E)=ε2E+[(ε3−ε1)2][(E∙d1)d2+(E∙d2)d1].$

When ε 1 = ε 2 = ε and ε 3 = ε the crystal is just a uniaxial medium with d 1 = d 2 = c and

$ε(E)=ε┴E+(ε∥−ε┴)(E∙c)c.$

The isotropic case corresponds, obviously, to the limit ε = ε . A bianisotropic medium, on the other hand, is the general linear medium characterized by the constitutive relations

$D=ε0ε(E)+ξ(H),B=μ0μ(H)+ζ(E)$

where μ is the (relative) permeability and ξ and ζ are some linear functions expressing the magnetoelectric coupling.

Fig. 1. (Color online) The electrical anisotropy of a medium is characterized by the bivector F = ED = β F̂ = ϱst, where β = ϱsin(θ), ϱ = ED (with E = |E|, D = |D|) and F̂ = sr is a unit bivector as sr = 0 (r, s and t are unit vectors). Angle θ is such that cos(θ) = st = ε 0 ε s E/D, where ε S = s ∙ ε(s) is the permittivity along s and ε(s) is the dielectric function. One has D = D + D with D = D s and D = D r , where D = Dcos(θ) and D = Dsin(θ). For an isotropic medium β = 0, i.e., θ = 0 for all possible directions S.

The inverse of the dielectric function ε is the impermeability function η = ε-1 such that E = η(D)/ε 0 . If E = E 1 e 1 + E 2 e 2 + E 3 e 3 and e 1, e 2 and e 3 are the principal dielectric axes corresponding to the eigenvalues ε 1, ε 2 and ε 3 (respectively), with ε 3 > ε 2 > ε 1, then D = ε 1 E 1 e 1 + ε 2 E 2 e 2 + ε 3 E 3 e 3 and E = η 1 D 1 e 1 + η 2 e 2 + η 3 D 3 e 3 where ηi = ε -1 i (with i = 1,2,3). One can readily show that, is d 1 and d 2 are the two unit vectors that characterize ε, then c 1 and c 2 are the two unit vectors that characterize η, where

$c1=τ1e1+τ3e3,c1=−τ1e1+τ3e3,τ1=ε3ε2γ1,τ3=ε1ε2γ3,$

with τ 1 = sin(δ/2) and τ 3 = cos(δ/2). Similarly to Eq. (9), we can now write

$c2=rδc1r˜δ,rδ=exp(δe312)=cos(δ2)+e31sin(δ2).$

Whence,

$c1c2=αγ−γβ−γβγd1d2,β=ε3−ε1ε3+ε1,γ=11−β2.$

One should note that γ =cosh(ξ), γβ = sinh(ξ) and $α=ε1ε3ε2,$, thus leading to β = tanh(ξ), ξ = ln[(1 + β)/(1 - β)]/2 = ln(ε 3/ε 1)/4 and tan$tan(δ2)=ε3ε1tan(ϕ2)$ tan(ϕ/2). In Fig. 2 we show the relation between unit vectors d 1 and d 2 and unit vectors c 1 and c 2. Accordingly, in comparison with (10), one has

$η(D)=η2D+[(η3−η1)2][(D∙c1)c2+(D∙c2)c1].$

Fig. 2. (Color online) In a biaxial crystal the two unit vectors (d 1,d 2) characterize the dielectric function ε, whereas the two unit vectors (c 1,c 2) characterize the impermeability function η. One should stress that (c 1,c 2) are the optic axes of the crystal – not (d 1,d 2) . Orthonormal basis {e 1,e 2,e 3} contains the three principal dielectric axes of the crystal. One has $τ1τ3=ε3ε1(γ1γ3).$.

## 3. Eigenwaves in biaxial crystals

For electromagnetic field variation of the form exp[i(kr - ωt)] = exp[ik 0(nr - ct)], with k = k 0 n and k 0 = ω/c , Maxwell equations in Cℓ 3 may be simply written, for source-free regions, as

$n∧E=cBe123,n∧H=−cDe123.$

In this section we are going to derive the two eigenwaves (or isonormal waves) that can propagate in a lossless nonmagnetic biaxial crystal, i.e., in a medium characterized by D = ε 0ε(E) and B = μ 0 H , where the dielectric function is given by Eq. (10). From Eqs. (17) and writing n = n k̂ , we get

$n2E┴=ε(E),E┴=E−E∥=E−(E∙k̂)k̂.$

Accordingly, in terms of the impermeability function of Eq. (16) we may also write

$k̂∧w=0,w=n2η(E┴)−E┴,$

or, explicitly,

$(n2α0−1)(k̂∧E┴)+n2β0[(c1∙E┴)(k̂∧c2)+(c2∙E┴)(k̂∧c1)]=0$

where α 0 = η 2 and β 0 = (η 3 - η 1)/2

#### 3.1 Refractive index surfaces Introducing two vectors

Introducing two vectors

$u=−(k̂∧c1)e123,v=−(k̂∧c2)e123$

and applying the left contraction of bivectors c 1 and c 2 to Eq. (20) we obtain

$c1∙E┴=n2β0u21−n2(α0+β0u∙v)(c2∙E┴),c2∙E┴=n2β0v21−n2(α0+β0u∙v)(c1∙E┴).$

Whence

${1−n4β02u2v2[1−n2(α0+β0u∙v)]2}(c1∙E┴)=0.$

Therefore, the eigenwaves corresponding to the direction of propagation (the wave normal) are characterized by two distinct refractive indices (birefringence) n + and n -, such that

$1n±2=α0+β0(u∙v±u2v2).$

This can be readily shown to be in accordance with the results obtained using dyadic methods [17]. Nevertheless, the approach using geometric algebra is far less cumbersome than the one using dyadics. By the way, if a, b ∈ ℝ3, then the tensor (or dyadic) product ab is such that the outer product is related to it through ab = ab - ba. One should bear in mind, however, that bivector ab has a direct geometric relation with vectors a and b whereas no such relation exists with dyadic ab.

There is a very important conclusion to be drawn from Eq. (24): for waves propagating along c 1 or c 2 we have, according to Eq. (21), u = 0 or v = 0 (respectively) and hence n 2 ± = 1/α 0 = ε 2, i.e., the two refractive indices are equal. But then, according to the definition of optic axis, we conclude that the two unit vectors c 1 and c 2 that characterize the impermeability function are, in fact, the two optic axes of the biaxial crystal. We introduce two dimensionless parameters

$ζ=ε2ε1,κ=ε3ε2,$

with κ > 1 and ζ > 1, in order to study the evolution of the refractive index surfaces $α0$ n ±(k̂) of an anisotropic medium as shown in Fig. 3 and Fig. 4: the starting point is an isotropic medium (ζ = κ =1); then, only parameter κ is increased to obtain a uniaxial medium (parameter ζ is kept at ζ = 1); finally, with the increase of ζ (while keeping κ = 2.1), a biaxial medium is obtained. Only for the uniaxial case do ordinary and extraordinary waves exist, corresponding to a sphere and an ellipsoid as is well-known. For the biaxial case this clear distinction between the two eigenwaves cannot be maintained. This becomes clearer when looking at a single plane. In Fig 4 we consider the c 1c 2 plane (i.e, the X 1 X 3 plane of Fig. 2): a circumference of radius 1 is always present; however, for the biaxial case, this circumference does not correspond to a single eigenwave as it is only completed through the contribution of both eigenwaves, depending on the direction under consideration. Then we may say that, along those directions for which $α0$ n ±() = 1, an ordinary-like behavior is found for the corresponding eigenwave – although it is not possible to distinguish anymore between ordinary and extraordinary waves as for the uniaxial case.

Fig. 3. (496 kB) The 3D refractive index surfaces, $α0$ n ± (), for the two eigenwaves of a lossless nonmagnetic crystal. [Media 1]

Fig. 4. (77 kB) The two refractive index surfaces, $α0$ n ± (), on the plane c 1c 2 : surface $α0$ n - () is in red and corresponds to the ordinary wave only in the uniaxial case; surface $α0$ n + () is in green and corresponds to the extraordinary wave only in the uniaxial case. [Media 2]

#### 3.2 Electromagnetic fields and energy velocity

From Eqs. (22) one has |v| (c 1E ) = ±|u| (c 2E ) for the two eigenwaves. Accordingly, it is possible to define two important vectors

$b±=∣k̂∧c2∣c2±∣k̂∧c1∣c2$

from which the electromagnetic fields for each linearly polarized eigenwave can be derived:

$D±=ε0n±2E0[k̂┛(k̂∧b±)],E±=α0ε0D±+(1−α0n±2)E0b±,$
$B±=μ0H±=−n±cE0(k̂∧b±)e123.$

Obviously that, when = c 1 or = c 2 (i.e., for waves propagating along an optic axis), we have b ± = 0 and these expressions fail. In fact, as is well-known, conical refraction takes place in these particular cases [2] which should be analyzed by an alternative method and will not be discussed herein. One should note that

$k̂∙b±=∣k̂∧c2∣(k̂∙c1)±∣k̂∧c1∣(k̂∙c2)$

and hence, if is perpendicular do the plane c 1c 2, then b ± = 0. According to Eq. (27) we obtain, for α 0 n 2 ± = 1,

$D±=ε0α0E±,E±=E0[k̂┛(k̂∧b±)]=E0[b±−(k̂∙b±)k̂].$

From Eq. (27) and for b ± = 0 we also get

$D±=ε0n±2E±,E±=E0b±.$

This last result shows the meaning of the two vectors b ± whenever b ± we have an ordinary-like behavior with both D ± and E ± parallel to b ± (respectively). In the general biaxial case, the angles θ ± between E ± and D ± (one for each eigenwave) as introduced in Eq. (4) can be explicitly obtained from

$cos2(θ±)=b±2−(k̂∙b±)2b±2−α0n±2(2−α0n±2)(k̂∙b±)2.$

For α 0 n 2 ± = 1 we just get, from Eq. (32), θ ± = 0 as stated previously. This same result also occurs whenever b ± =0. After calculating the energy density and the Poynting vector, using Eqs. (27) and (28), we derive the following expressions for the energy velocity of the two eigenwaves:

$ve(±)=[b±2−α0n±2(k̂∙b±)2]vp(±)−(k̂∙b±)(1−α0n±2)vp(±)b±b±2−(k̂∙b±)2$

The phase velocity is given by v (±) p = v (±) p where v (±) p = c/n ±. From Eq. (33) we get

$k̂∙ve(±)=vp(±).$

This means that the phase velocity is the projection of the energy velocity along the direction of the wave normal. From Eq. (33) we also get, for α 0 n 2 ± = 1 or b ± = 0, v (±) e = v (±) p which reinforces our earlier statement that directions b ± = 0 present an ordinary-like behavior. From Fig. 5 we can confirm that phase and energy velocities coincide when α 0 n 2 ± = 1; for the other directions, the energy velocity is always greater than the phase velocity. Energy velocity ve = |v e| is such that v 3vev 1 with v 1,3 = c/$ε1,3$ (the maximum v 1 occurs on the X 1 -axis and the minimum v 3 on the X 3 -axis); for α 0 n 2 ± = 1 the energy velocity is v 2 = c/$ε2$. To obtain a causal medium the principal values of the dielectric function must be above unit (ε 3 > ε 2 > ε 1 > 1); otherwise the energy velocity would be greater than the speed of light in vacuum. To study media with ε 1 < 1, a causal dispersive model should be taken into account and losses included (according to the Kramers-Kronig relations); but then the two eigenwaves will not be linearly polarized anymore (the two eigenwaves will be elliptically polarized). One should stress that Eq. (33) is only valid for a lossless medium.

Fig. 5. (Color online) Normalized energy velocities v (±) e/v 2 and phase velocities v (±) p/v 2 for the two eigenwaves of a biaxial crystal. One has v 2 = $α0$ = c/ $ε2$.

## 4. Conclusion

The standard approach to anisotropic media has been the coordinate method where the problem is usually solved through the principal coordinate system of the dielectric tensor as it takes, in this specific system, its simplest diagonal form. However, a coordinate-free approach is preferable as it provides solutions in their greater generality, thereby rendering the whole physical problem easier to grasp. Dyadic analysis has been the only coordinate-free method available to date – apart from differential forms which, from our perspective, do not offer any special improvements when compared to the usual dyadic approach. Nevertheless, the dyadic approach lacks a direct physical and geometrical interpretation: whenever such an interpretation is needed one usually reduces the general expressions to a specific coordinate system.

With the novel approach herein presented we have shown how geometric algebra can provide a better mathematical framework for anisotropy than tensors and dyadics. Through the direct manipulation of coordinate-free objects such as vectors, bivectors and trivectors, geometric algebra is the most natural setting to study anisotropy, providing a deeper insight and simpler calculations, without loosing its direct geometrical interpretation. We have applied our method to the problem of the electromagnetic wave propagation in biaxial crystals: the whole treatment puts in evidence the superiority of this novel approach in the determination of the two eigenwaves of these anisotropic media. In fact, as a by-product of our analysis, we have presented several new expressions that provide a better insight to the optical behavior of biaxial crystals.

Finally, with the present work, we hope to have contributed to establish a new trend in the analysis of the optics of anisotropic media by showing how Clifford’s geometric algebra may shed light on this ancient topic which, by the ongoing research on metamaterials, has regained a new interest for optical science.

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1. J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol. 2, p. 443.
2. M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University Press, Cambridge,., 1999) pp. 790–852.
3. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley Classics Library, Hoboken, 2003).
4. I. Richter, P. C. Sun, F. Xu, and Y. Fainman, “Design considerations of form birefringence microstructures,” Appl. Opt. 34, 2421–2429 (1995). http://www.opticsinfobase.org/abstract.cfm?URI=ao-34-14-2421
[Crossref] [PubMed]
5. U. Levy, C. H. Tsai, L. Pang, and Y. Fainman, “Engineering space-time variant inhomogeneous media for polarization control,” Opt. Lett. 29, 1718–1720 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=ol-29-15-1718
[Crossref] [PubMed]
6. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9804 (2006).
[Crossref] [PubMed]
7. I. V. Lindell, Differential Forms in Electromagnetics, (IEEE Press, Piscataway, 2004) pp. 123–161.
8. I. V. Lindell, Methods for Electromagnetic Field Analysis, (IEEE Press, Piscataway, 2nd ed., 1995) pp. 17–52.
9. D. Hestenes, New Foundations for Classical Mechanics (Kluwer Academic Publishers, Dordrecht, 2nd ed., 1999).
10. P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, Cambridge, 2nd ed., 2001).
[Crossref]
11. C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).
12. L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science - An Object-oriented Approach to Geometry (Elsevier - Morgan Kaufmann Publishers, San Francisco, 2007).
13. D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf.
[Crossref]
14. P. Puska, “Covariant isotropic constitutive relations in Clifford’s geometric algebra,” Progress in Electromagnetics Research - PIER 32, 413–428 (2001). http://ceta.mit.edu/PIER/pier32/16.00080116.puska.pdf.
[Crossref]
15. C. R. Paiva and M. A. Ribeiro, “Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach,” J. Electromagn. Waves Appl. 20, 941–953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[Crossref]
16. D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, (Kluwer Academic Publishers, Dordrecht, 1984) pp. 63–136.
17. H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach, (McGraw-Hill, Singapore, 1985) pp. 215–216.
18. M. A. Ribeiro, S. A. Matos, and C. R. Paiva, “A geometric algebra approach to anisotropic media,” in Proc. 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, USA, (2007), pp. 4032–4035.
19. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, (Oxford University Press, Oxford, 1985) pp. 24–25.

#### 2006 (2)

C. R. Paiva and M. A. Ribeiro, “Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach,” J. Electromagn. Waves Appl. 20, 941–953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[Crossref]

#### 2003 (1)

D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf.
[Crossref]

#### 2001 (1)

P. Puska, “Covariant isotropic constitutive relations in Clifford’s geometric algebra,” Progress in Electromagnetics Research - PIER 32, 413–428 (2001). http://ceta.mit.edu/PIER/pier32/16.00080116.puska.pdf.
[Crossref]

#### Born, M.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University Press, Cambridge,., 1999) pp. 790–852.

#### Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach, (McGraw-Hill, Singapore, 1985) pp. 215–216.

#### Doran, C.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).

#### Dorst, L.

L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science - An Object-oriented Approach to Geometry (Elsevier - Morgan Kaufmann Publishers, San Francisco, 2007).

#### Fontijne, D.

L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science - An Object-oriented Approach to Geometry (Elsevier - Morgan Kaufmann Publishers, San Francisco, 2007).

#### Hestenes, D.

D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf.
[Crossref]

D. Hestenes, New Foundations for Classical Mechanics (Kluwer Academic Publishers, Dordrecht, 2nd ed., 1999).

D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, (Kluwer Academic Publishers, Dordrecht, 1984) pp. 63–136.

#### Lasenby, A.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).

#### Lindell, I. V.

I. V. Lindell, Differential Forms in Electromagnetics, (IEEE Press, Piscataway, 2004) pp. 123–161.

I. V. Lindell, Methods for Electromagnetic Field Analysis, (IEEE Press, Piscataway, 2nd ed., 1995) pp. 17–52.

#### Lounesto, P.

P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, Cambridge, 2nd ed., 2001).
[Crossref]

#### Mann, S.

L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science - An Object-oriented Approach to Geometry (Elsevier - Morgan Kaufmann Publishers, San Francisco, 2007).

#### Matos, S. A.

M. A. Ribeiro, S. A. Matos, and C. R. Paiva, “A geometric algebra approach to anisotropic media,” in Proc. 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, USA, (2007), pp. 4032–4035.

#### Maxwell, J. C.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol. 2, p. 443.

#### Nye, J. F.

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, (Oxford University Press, Oxford, 1985) pp. 24–25.

#### Paiva, C. R.

C. R. Paiva and M. A. Ribeiro, “Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach,” J. Electromagn. Waves Appl. 20, 941–953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[Crossref]

M. A. Ribeiro, S. A. Matos, and C. R. Paiva, “A geometric algebra approach to anisotropic media,” in Proc. 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, USA, (2007), pp. 4032–4035.

#### Puska, P.

P. Puska, “Covariant isotropic constitutive relations in Clifford’s geometric algebra,” Progress in Electromagnetics Research - PIER 32, 413–428 (2001). http://ceta.mit.edu/PIER/pier32/16.00080116.puska.pdf.
[Crossref]

#### Ribeiro, M. A.

C. R. Paiva and M. A. Ribeiro, “Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach,” J. Electromagn. Waves Appl. 20, 941–953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[Crossref]

M. A. Ribeiro, S. A. Matos, and C. R. Paiva, “A geometric algebra approach to anisotropic media,” in Proc. 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, USA, (2007), pp. 4032–4035.

#### Sobczyk, G.

D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, (Kluwer Academic Publishers, Dordrecht, 1984) pp. 63–136.

#### Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University Press, Cambridge,., 1999) pp. 790–852.

#### Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley Classics Library, Hoboken, 2003).

#### Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley Classics Library, Hoboken, 2003).

#### Am. J. Phys. (1)

D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003). http://modelingnts.la.asu.edu/pdf/OerstedMedalLecture.pdf.
[Crossref]

#### J. Electromagn. Waves Appl. (1)

C. R. Paiva and M. A. Ribeiro, “Doppler shift from a composition of boosts with Thomas rotation: A spacetime algebra approach,” J. Electromagn. Waves Appl. 20, 941–953 (2006). http://dx.doi.org/10.1163/156939306776149806.
[Crossref]

#### Progress in Electromagnetics Research - PIER (1)

P. Puska, “Covariant isotropic constitutive relations in Clifford’s geometric algebra,” Progress in Electromagnetics Research - PIER 32, 413–428 (2001). http://ceta.mit.edu/PIER/pier32/16.00080116.puska.pdf.
[Crossref]

#### Other (13)

D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, (Kluwer Academic Publishers, Dordrecht, 1984) pp. 63–136.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach, (McGraw-Hill, Singapore, 1985) pp. 215–216.

M. A. Ribeiro, S. A. Matos, and C. R. Paiva, “A geometric algebra approach to anisotropic media,” in Proc. 2007 IEEE Antennas and Propagation Society International Symposium, Honolulu, Hawaii, USA, (2007), pp. 4032–4035.

J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, (Oxford University Press, Oxford, 1985) pp. 24–25.

J. C. Maxwell, A Treatise on Electricity and Magnetism (Dover, New York, 1954) Vol. 2, p. 443.

M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University Press, Cambridge,., 1999) pp. 790–852.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley Classics Library, Hoboken, 2003).

I. V. Lindell, Differential Forms in Electromagnetics, (IEEE Press, Piscataway, 2004) pp. 123–161.

I. V. Lindell, Methods for Electromagnetic Field Analysis, (IEEE Press, Piscataway, 2nd ed., 1995) pp. 17–52.

D. Hestenes, New Foundations for Classical Mechanics (Kluwer Academic Publishers, Dordrecht, 2nd ed., 1999).

P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, Cambridge, 2nd ed., 2001).
[Crossref]

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).

L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science - An Object-oriented Approach to Geometry (Elsevier - Morgan Kaufmann Publishers, San Francisco, 2007).

### Supplementary Material (2)

 » Media 1: MOV (495 KB) » Media 2: MOV (77 KB)

### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (5)

Fig. 1. (Color online) The electrical anisotropy of a medium is characterized by the bivector F = ED = β F̂ = ϱst, where β = ϱsin(θ), ϱ = ED (with E = |E|, D = |D|) and F̂ = sr is a unit bivector as sr = 0 (r, s and t are unit vectors). Angle θ is such that cos(θ) = st = ε 0 ε s E/D, where ε S = s ∙ ε(s) is the permittivity along s and ε(s) is the dielectric function. One has D = D + D with D = D s and D = D r , where D = Dcos(θ) and D = Dsin(θ). For an isotropic medium β = 0, i.e., θ = 0 for all possible directions S.
Fig. 2. (Color online) In a biaxial crystal the two unit vectors (d 1,d 2) characterize the dielectric function ε, whereas the two unit vectors (c 1,c 2) characterize the impermeability function η. One should stress that (c 1,c 2) are the optic axes of the crystal – not (d 1,d 2) . Orthonormal basis {e 1,e 2,e 3} contains the three principal dielectric axes of the crystal. One has $τ 1 τ 3 = ε 3 ε 1 ( γ 1 γ 3 ) .$ .
Fig. 3. (496 kB) The 3D refractive index surfaces, $α 0$ n ± (), for the two eigenwaves of a lossless nonmagnetic crystal. [Media 1]
Fig. 4. (77 kB) The two refractive index surfaces, $α 0$ n ± (), on the plane c 1c 2 : surface $α 0$ n - () is in red and corresponds to the ordinary wave only in the uniaxial case; surface $α 0$ n + () is in green and corresponds to the extraordinary wave only in the uniaxial case. [Media 2]
Fig. 5. (Color online) Normalized energy velocities v (±) e /v 2 and phase velocities v (±) p /v 2 for the two eigenwaves of a biaxial crystal. One has v 2 = $α 0$ = c/ $ε 2$ .

### Equations (34)

$u = ED = E ∙ D + E ∧ D = α + F ,$
$α = E ∙ D = ( u + u ˜ ) 2 , F = E ∧ D = ( u + u ˜ ) 2 ,$
$∣ u ∣ 2 = u u ˜ = EDDE = E 2 D 2 = ( α + F ) ( α − F ) = α 2 − F 2 = α 2 + β 2 = ϱ 2 ≥ 0 .$
$u = α + β F ̂ = ϱ exp ( θ F ̂ ) = ϱ cos ( θ ) + F ̂ ϱ sin ( θ ) , ϱ = ∣ E ∣ ∣ D ∣ = α 2 + β 2 ,$
$Cℓ 3 = ℝ ⊕ ℝ 3 ⊕ ∧ 2 ℝ 3 ⊕ ∧ 3 ℝ 3 .$
${ 1 ︸ schalars , e 1 , e 2 , e 3 ︸ vectors , e 12 , e 31 , e 23 ︸ bivectors , e 123 ︸ trivectors }$
$Cen ( Cℓ 3 ) = ℝ ⊕ ∧ 3 ℝ 3 .$
$d 1 = γ 1 e 1 + γ 3 e 3 , d 2 = − γ 1 e 1 + γ 3 e 3 , γ 1 = ε 2 − ε 1 ε 3 − ε 1 , γ 3 = ε 3 − ε 2 ε 3 − ε 1 ,$
$d 2 = r ϕ d 1 r ˜ ϕ , r ϕ = exp ( ϕ e 31 2 ) = cos ( ϕ 2 ) + e 31 sin ( ϕ 2 ) ,$
$ε ( E ) = ε 2 E + [ ( ε 3 − ε 1 ) 2 ] [ ( E ∙ d 1 ) d 2 + ( E ∙ d 2 ) d 1 ] .$
$ε ( E ) = ε ┴ E + ( ε ∥ − ε ┴ ) ( E ∙ c ) c .$
$D = ε 0 ε ( E ) + ξ ( H ) , B = μ 0 μ ( H ) + ζ ( E )$
$c 1 = τ 1 e 1 + τ 3 e 3 , c 1 = − τ 1 e 1 + τ 3 e 3 , τ 1 = ε 3 ε 2 γ 1 , τ 3 = ε 1 ε 2 γ 3 ,$
$c 2 = r δ c 1 r ˜ δ , r δ = exp ( δ e 31 2 ) = cos ( δ 2 ) + e 31 sin ( δ 2 ) .$
$c 1 c 2 = α γ − γβ − γβ γ d 1 d 2 , β = ε 3 − ε 1 ε 3 + ε 1 , γ = 1 1 − β 2 .$
$η ( D ) = η 2 D + [ ( η 3 − η 1 ) 2 ] [ ( D ∙ c 1 ) c 2 + ( D ∙ c 2 ) c 1 ] .$
$n ∧ E = c Be 123 , n ∧ H = − c De 123 .$
$n 2 E ┴ = ε ( E ) , E ┴ = E − E ∥ = E − ( E ∙ k ̂ ) k ̂ .$
$k ̂ ∧ w = 0 , w = n 2 η ( E ┴ ) − E ┴ ,$
$( n 2 α 0 − 1 ) ( k ̂ ∧ E ┴ ) + n 2 β 0 [ ( c 1 ∙ E ┴ ) ( k ̂ ∧ c 2 ) + ( c 2 ∙ E ┴ ) ( k ̂ ∧ c 1 ) ] = 0$
$u = − ( k ̂ ∧ c 1 ) e 123 , v = − ( k ̂ ∧ c 2 ) e 123$
$c 1 ∙ E ┴ = n 2 β 0 u 2 1 − n 2 ( α 0 + β 0 u ∙ v ) ( c 2 ∙ E ┴ ) , c 2 ∙ E ┴ = n 2 β 0 v 2 1 − n 2 ( α 0 + β 0 u ∙ v ) ( c 1 ∙ E ┴ ) .$
${ 1 − n 4 β 0 2 u 2 v 2 [ 1 − n 2 ( α 0 + β 0 u ∙ v ) ] 2 } ( c 1 ∙ E ┴ ) = 0 .$
$1 n ± 2 = α 0 + β 0 ( u ∙ v ± u 2 v 2 ) .$
$ζ = ε 2 ε 1 , κ = ε 3 ε 2 ,$
$b ± = ∣ k ̂ ∧ c 2 ∣ c 2 ± ∣ k ̂ ∧ c 1 ∣ c 2$
$D ± = ε 0 n ± 2 E 0 [ k ̂ ┛ ( k ̂ ∧ b ± ) ] , E ± = α 0 ε 0 D ± + ( 1 − α 0 n ± 2 ) E 0 b ± ,$
$B ± = μ 0 H ± = − n ± c E 0 ( k ̂ ∧ b ± ) e 123 .$
$k ̂ ∙ b ± = ∣ k ̂ ∧ c 2 ∣ ( k ̂ ∙ c 1 ) ± ∣ k ̂ ∧ c 1 ∣ ( k ̂ ∙ c 2 )$
$D ± = ε 0 α 0 E ± , E ± = E 0 [ k ̂ ┛ ( k ̂ ∧ b ± ) ] = E 0 [ b ± − ( k ̂ ∙ b ± ) k ̂ ] .$
$D ± = ε 0 n ± 2 E ± , E ± = E 0 b ± .$
$cos 2 ( θ ± ) = b ± 2 − ( k ̂ ∙ b ± ) 2 b ± 2 − α 0 n ± 2 ( 2 − α 0 n ± 2 ) ( k ̂ ∙ b ± ) 2 .$
$v e (±) = [ b ± 2 − α 0 n ± 2 ( k ̂ ∙ b ± ) 2 ] v p (±) − ( k ̂ ∙ b ± ) ( 1 − α 0 n ± 2 ) v p (±) b ± b ± 2 − ( k ̂ ∙ b ± ) 2$
$k ̂ ∙ v e ( ± ) = v p ( ± ) .$