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

where α = E ∙ D ∈ ℝ is the usual dot (or inner) product, which is symmetric, and F = E ∧ D ∈ ∧²ℝ³ 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 E ∧ D) 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 A ∥ B (i.e., bivectors corresponding to the same plane) can be regarded as directed angles turning either the same way, A ⇈ B, or the opposite way, A ⇅ B. Only when |A| = |B| and A ⇈ B do we say that A = B. There is a crucial difference between a vector a ∈ ℝ³ and a bivector F ∈ ∧² ℝ³: 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 = E ∙ D -E ∧ D = α - F. Whence

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

where F = β F̂. The outer product of a vector and a bivector produces a trivector V = a∧F = F∧a∈∧³ℝ³ 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 a┛F = d ∈ ℝ³ we have, if F = E ∧ D, d = (a ℙ E)D - (a ℙ D)E; the right contraction F ┗ a is such that F ┗ a = -a ┛ F [10]. The geometric product aF is then aF = a ┛ F + a ∧ F = d + V, i.e., the sum of a vector and a trivector. An arbitrary element u ∈ Cℓ 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 = 〈u〉 a = 〈u〉₁, F = 〈u〉₂, V = 〈u〉₃ , 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:

A k -blade of Cℓ ₃ (with k = 0,1,2,3) is an element u_k such that u_k = 〈u_k〉_k, where 〈u_k〉_k is a homogeneous multivector of grade k, i.e., u_k ∈ ∧^kℝ³ (assuming that ∧⁰∝³ = ℝ and ∧⁰ℝ³ = ℝ³). Any trivector can be written as V = β e ₁₂₃, where β ∈ ℝ and e ₁₂₃ = V̂ 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 a ∧ b = (a×b)e ₁₂₃ and a×b = -(a∧b)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

where e ₁₂ = e ₁ ∧ e ₂ = e ₁ e ₂, e ₃₁ = e ₃ ∧ e ₁ = e ₃ e ₁ and e ₂₃ = e ₂ ∧ e ₃ = e ₂ e ₃ constitute a basis for the subspace ∧²ℝ³ 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 :

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 ⊕ ∧² ℝ³. 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: ε_jk = 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| = |〈ED〉₂ = |E∧D|) 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 = s ∧ t/sin (θ) = sr, where r ² = 1, so that D = D _∥ + D _┴ with D _∥ = s ∙ D _∙ = |D|cos(θ) and D _┴ = r ∙ D _┴ = |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 E ┴ D 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

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

where r_ϕ ∈ ⊕ ∧² ℝ³ (an even multivector) is called a rotor and satisfies the relation r_ϕ r̃_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

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

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

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 = E∧D = β F̂ = ϱs∧t, where β = ϱsin(θ), ϱ = ED (with E = |E|, D = |D|) and F̂ = sr is a unit bivector as s∙r = 0 (r, s and t are unit vectors). Angle θ is such that cos(θ) = s∙t = 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.

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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 η_i = 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

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

Whence,

One should note that γ =cosh(ξ), γβ = sinh(ξ) and $\alpha =\frac{\sqrt{{\epsilon }_1}{\epsilon }_3}{{\epsilon }_2},$, thus leading to β = tanh(ξ), ξ = ln[(1 + β)/(1 - β)]/2 = ln(ε ₃/ε ₁)/4 and tan$\tan \left(\frac{\delta }{2}\right)=\sqrt{\frac{{\epsilon }_3}{{\epsilon }_1}}\tan \left(\frac{\varphi }{2}\right)$ 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

 

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 $\frac{{\tau }_1}{{\tau }_3}=\sqrt{\frac{{\epsilon }_3}{{\epsilon }_1}}\left(\frac{{\gamma }_1}{{\gamma }_3}\right).$.

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3. Eigenwaves in biaxial crystals

For electromagnetic field variation of the form exp[i(k∙r - ωt)] = exp[ik ₀(n∙r - 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

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

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

or, explicitly,

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

3.1 Refractive index surfaces Introducing two vectors

Introducing two vectors

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

Whence

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

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 a ⊗ b is such that the outer product is related to it through a ∧ b = a ⊗ b - b ⊗ a. One should bear in mind, however, that bivector a ∧ b has a direct geometric relation with vectors a and b whereas no such relation exists with dyadic a ⊗ b.

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

with κ > 1 and ζ > 1, in order to study the evolution of the refractive index surfaces $\sqrt{{\alpha }_0}$ 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 ₁ ∧ c 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 $\sqrt{{\alpha }_0}$ n _±(k̂) = 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, $\sqrt{{\alpha }_0}$ n _± (k̂), for the two eigenwaves of a lossless nonmagnetic crystal. [Media 1]

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Fig. 4. (77 kB) The two refractive index surfaces, $\sqrt{{\alpha }_0}$ n _± (k̂), on the plane c ₁ ∧ c ₂ : surface $\sqrt{{\alpha }_0}$ n _- (k̂) is in red and corresponds to the ordinary wave only in the uniaxial case; surface $\sqrt{{\alpha }_0}$ n ₊ (k̂) is in green and corresponds to the extraordinary wave only in the uniaxial case. [Media 2]

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3.2 Electromagnetic fields and energy velocity

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

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

Obviously that, when k̂ = c ₁ or k̂ = c ₂ (i.e., for waves propagating along an optic axis), we have k̂ ∧ 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

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

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

This last result shows the meaning of the two vectors b _± whenever k̂ ┴ 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

For α ₀ n ² _± = 1 we just get, from Eq. (32), θ _± = 0 as stated previously. This same result also occurs whenever k̂ ∙ 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:

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

This means that the phase velocity is the projection of the energy velocity along the direction k̂ of the wave normal. From Eq. (33) we also get, for α ₀ n ² _± = 1 or k̂ ∙ b _± = 0, v ^(±) _e = v ^(±) _p which reinforces our earlier statement that directions k̂ ∙ 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 v_e = |v _e| is such that v ₃ ≤ v_e ≤ v ₁ with v _1,3 = c/$\sqrt{{\epsilon }_{1,3}}$ (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/$\sqrt{{\epsilon }_2}$. 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 = $\sqrt{{\alpha }_0}$ = c/ $\sqrt{{\epsilon }_2}$.

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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.