Wavevector-dependent optical properties from wavevector-independent proper conductivity tensor

We discuss the standard ab initio calculation of the refractive index by means of the scalar dielectric function and point out its inherent limitations. To overcome these, we start from the recently proposed fundamental, microscopic wave equation in materials in terms of the frequency- and wavevector-dependent dielectric tensor, and investigate under which conditions the standard treatment can be justified. Thereby, we address the question of neglecting the wavevector dependence of microscopic response functions. Furthermore, we analyze in how far the fundamental, microscopic wave equation is equivalent to the standard wave equation used in theoretical optics. In particular, we clarify the relation of the"effective"dielectric tensor used there to the microscopic dielectric tensor defined in ab initio physics.


Introduction
The conventional treatment of optical material properties in ab initio materials physics is based on the standard relation between the macroscopic dielectric function ε(ω) and the refractive index n(ω) given by 1 (see Refs. [2,Eq. (8.33)], [3,Eq. (2.17)], [4,Eqs. (18.26)], [5,Eq. (6.11)], [6, p. 534] or [7, Eq. (2.203)]) n 2 (ω) = ε(ω) . (1.1) The macroscopic dielectric function on its side is defined as the limit k → 0 of the microscopic (frequencyand wavevector-dependent) dielectric function (see Refs. [ Importantly, the dielectric function used in these equations is typically calculated from the density response function, and hence it corresponds to the longitudinal part of the dielectric tensor (see Refs. [4,Eq. (18.23)], [8], [7, § 2.6.4], [9,Vol. 1,§ 4]). This standard treatment delivers sensible results for a huge variety of materials (see Refs. [10][11][12][13][14][15][16] for recent examples). However, a recent, meritorious article by D. Sangalli et al. [17] has drawn attention to the fact that in principle, optical properties should be calculated from the wavevector-dependent current response tensor. Let us accordingly summarize the main conceptual problems of the standard treatment by means of the wavevector-independent dielectric function: 1. As a matter of principle, optical properties correspond to transverse (although not necessarily purely transverse) electromagnetic waves in a material, and hence they should not be deduced from a purely longitudinal response function (at least not in a naïve way).
2. The standard relation for the refractive index, Eq. (1.1), is only valid in the limit k → 0, whereas light waves definitely have a non-vanishing wavevector, i.e., k = 0.
3. In particular, for anisotropic media, the refractive index should at least depend on the direction of the wavevector k.
4. Moreover, for birefringent or optically active materials, there are polarization-dependent refractive indices. By contrast, Eq. (1.1) yields at best one refractive index, whose corresponding polarization cannot even be defined for vanishing wavevectors.
These considerations make it clear that in general, the calculation of optical material properties requires the wave-and the polarization vector to be taken into account, and this in turn requires a treatment based on the current response tensor (or equivalently, the dielectric tensor [1,18]). In fact, from the dielectric tensor-which naturally contains much more information than the scalar dielectric function-the density response function can be reconstructed by means of the Universal Response Relations [19][20][21], while the converse is not true (see Refs. [2,[22][23][24]). However, in the limit k → 0, such relations between different response functions cannot be evaluated due to their singular behavior (see, for example, Eq. (2.5) below, which relates the dielectric function to the density response function). Correspondingly, the authors of Ref. [17] have based their treatment on a full ab initio calculation of the wavevector-dependent response functions. While this is certainly the right approach in the most general case, the downside is that such wavevectordependent calculations are in general extremely demanding, in particular if they are supposed to yield dispersion relations.
In this article, we resume this problem to show that it is possible-at least in principle-to obtain wavevector-dependent optical properties without calculating wavevector-dependent response functions numerically, provided that the treatment is based on the proper conductivity tensor. Fittingly, a recent numerical study [25] has provided evidence that it may be precisely this quantity which is wavevector independent, at least for optical wavelengths. We stress, however, that this condition cannot simultaneously apply to all other response functions as well (see the discussion below). Concretely, it will turn out that in the case of a wavevector-independent proper conductivity tensor, the wavevector dependence of the microscopic wave equation can be absorbed entirely in the wavevector dependence of the refractive index. For the latter one can then formulate a pseudo-eigenvalue problem which is explicit in that very wavevector and which coincides with a standard equation used in theoretical optics.
The article is organized as follows: In § 2, we assemble some general relations between wavevector-dependent response functions. In § 3, we discuss the fundamental, microscopic wave equation in materials in terms of the proper conductivity tensor, and we investigate the conditions under which the standard treatment of optical material properties can be justified. In particular, we define the effective dielectric tensor in terms of the proper conductivity tensor, and we show that by rewriting the fundamental wave equation in terms of this effective dielectric tensor, the resulting equation agrees formally with the standard wave equation used in theoretical optics. The subsequent § 4 is dedicated to an in-depth analysis of this wave equation and its solutions under the assumption of a wavevector-independent proper conductivity tensor. Finally, in the appendix we reproduce some well-known optical properties within our general formalism by assuming special forms of the effective dielectric tensor.

Response functions and wavevector dependence
Fundamentally, the current response tensor χ is related to the conductivity tensor σ by means of a Universal Response Relation [19, § 6], which reads (see Refs. [7,Eqs. (2.177) and (2.198)], [22,Eq. (3.185)], [26], and for a gauge-independent derivation see Ref. [20, § 3 The current response tensor on its side is the spatial part of the fundamental response tensor [23, § 7.4], where j µ = (cρ, j) T denotes the (induced) electromagnetic four-current, and A ν = (ϕ/c, A) T the (external) four-potential. In the homogeneous limit, the fundamental response tensor is of the following general form (see e.g. Refs. [2,19,23,24]): In particular, the density response function χ can be calculated from the current response tensor as follows [22,Eq. (3.175)]: where v(k) = 1/(ε 0 |k| 2 ) denotes the Coulomb interaction kernel in Fourier space. We stress again that the dielectric function in this equation actually coincides with the longitudinal part of the dielectric tensor (see Ref. [7, § 2.6.4]) given by With this, Eq. (2.5) can be shown directly by applying the functional chain rule (see Refs. [19] and [27, § 5.1]). Importantly, the microscopic current response tensor-or equivalently, the microscopic conductivity tensor-already contains the complete information about all linear electromagnetic response properties (this insight can be traced back at least to Ref. [28]; the fact as such has also been stressed recently in Ref. [18]; for a systematic derivation of all linear electromagnetic response functions in terms of the conductivity tensor see also Ref. [19, § 5-6]). In particular, it is possible to reconstruct from the microscopic conductivity tensor the response with respect to strictly longitudinal perturbations. However, one has to stress that the corresponding response relations are formulated in terms of response functions at finite wavevectors |k| > 0 (see Ref. [17]). Hence, these relations can in general not be evaluated naïvely in the limit k → 0, but require precise knowledge about the response functions in the vicinity of the origin and about the way this origin actually has to be approached.
Finally, we remark that the response relations presented above hold in this form only for homogeneous materials, while they become more involved in the general (i.e., inhomogeneous) case [19,29]. Moreover, such quantities as the longitudinal or transverse dielectric function are actually meaningful only in the isotropic limit (see Ref. [27, § 5.1]).

Fundamental, microscopic wave equation
On a microscopic level, the most general, linear electromagnetic wave equation in materials-which requires only spatial homogeneity-reads as follows (see Refs. [1,21,27,30,31]): Here, ↔ ε (k, ω) denotes the ab initio dielectric tensor, which is defined by the linear approximation (see Refs. where E ≡ E tot . Although the ab initio derivation of the fundamental wave equation (3.1) is somewhat complicated [21, § 4.1], its ultimate meaning is just that the external field does not penetrate the material or, put differently, that the electromagnetic waves in the material correspond to the material's proper oscillations. Accordingly, in the case of an isotropic medium, the fundamental wave equation decouples into equations which are formulated in terms of the longitudinal and transverse dielectric functions. The resulting conditions for the existence of nontrivial solutions,

Wave equation used in theoretical optics
For the above purpose, we now reformulate the general wave equation With this, the fundamental wave equation (3.1) can be rewritten as and further using Eq. (3.8), we can recast it into the form Finally, by applying the vector identity and by defining the effective dielectric tensor as we can bring Eq. (3.10) into the equivalent form On the other hand, for the transverse parts we find 2 Furthermore, Eq. (3.13) decouples in the isotropic limit into two separate wave equations for the longitudinal and transverse electric field components: − Curiously, this is a wave equation for a transverse electric field, which is however formulated in terms of the longitudinal dielectric function. Nevertheless, this equation can also be justified from the fundamental wave equation in the isotropic case, i.e., from Eqs.
With this relation, one shows directly from Eq. (3.7) that the longitudinal and transverse dielectric functions are related as follows [21, § 4.4]: Together with the fundamental wave equation (3.4), this implies (3.21) Finally, by neglecting the k dependence of ε L we arrive at the assertion, Eq. (3.18). On the other hand, since the condition (3.19) will not always be true, and since the k dependence of ε L can not always be neglected, this corroborates again our initial statement (see § 1) that in the most general case, the deduction of optical material properties should not be based on the standard relation (1.1).

Fundamental wave equation in optical limit
The above considerations have shown that as a matter of principle, optical properties have to be deduced from the general wave equation encompassing the transverse subspace, i.e., from Eq. (3.1), or from its standard form given by Eq. (3.13), which can also be written as Unfortunately, this equation requires (via the relation (3.12)) knowledge of the full, i.e., frequency-and wavevector-dependent conductivity tensor, which is computationally very demanding. Moreover, even if the full conductivity tensor or the corresponding effective dielectric tensor is known, the dispersion relation ω = ω k has to be deduced from the implicit equation (4.1), where the frequency appears not only explicitly on the right-hand side but also implicitly as an argument of the dielectric tensor. Consequently, also the refractive index, which is defined by (see Refs.  Even in this case, though, the refractive index is given only implicitly by Eq. (4.1), because the modulus |k| does not only appear explicitly on the righthand side of this equation but also implicitly as an argument of the effective dielectric tensor, ε eff = ε eff (k, |k|, ω).
However, a pragmatic assumption to overcome these difficulties is the following: although the relations between different response functions are in general wavevector dependent (see § 2), it is not contradictory to assume that one particular response function is actually wavevector independent (at least approximately), whereby it has to be stressed that this assumption cannot be upheld simultaneously for all response functions (see again § 2, or the most general Universal Response Relations in Ref. [19, § 6]). In fact, the standard approach suggests that the assumption of wavevector independence actually applies to the proper conductivity tensor, or equivalently (see Eq. (2.1)), to the proper current response tensor. Note, however, that in this case the wavevector independence of the conductivity tensor implies that the density response function as well as the dielectric tensor are definitely wavevector dependent (see Eqs. (2.4) and (3.7)).
Thus, we assume that at optical wavelengths, the proper conductivity tensor is wavevector independent in the sense that equation which defines the so-called optical limit. In the following, we will investigate the general wave equation under this additional assumption. First, Eq. (4.4) implies that also the effective dielectric tensor defined by Eq. (3.12) depends only on the frequency. This in turn greatly simplifies the solution of the wave equation (4.1), which now becomes an explicit equation determining the modulus of the wavevector |k| as a function ofk and ω. In fact, since the transverse projection operator on its side depends only on the direction (and not on the modulus) of the wavevector, the frequency-and direction-dependent refractive indices can now be determined by an explicit equation, whose directional dependence comes into play via the transverse projection operator. To show this, we make the following ansatz for the electric field in Fourier space: where e(k, ω) is the so-called polarization vector, which we assume to be normalized. The polarization vector may, in principle, also depend on the frequency (though in praxi it is usually frequency independent). The ansatz (4.6) fulfills the constraint condition which guarantees that the electric field is real-valued in the space-time domain. Now, by putting Eq.  together imply that the condition (4.9) is actually equivalent to Eq. (4.8) and can therefore be discarded. On the other hand, by combining Eqs. (4.8) and (4.9) one sees directly that e(k, ω) = e * (−k, −ω) , (4.12) and since Eq.  ). Note, however, that one can in general not apply the standard methods of theoretical optics to this equation, because the effective dielectric tensor-stemming form a retarded response function calculated by the Kubo formula-is typically not hermitean (as it is assumed in theoretical optics) and hence not necessarily diagonalizable. In the following, we will study in detail the solutions of Eq. (4.8) without any particular assumption on the effective dielectric tensor.

Radiation modes and generalized plasmons
Although Eq. is fulfilled. As stressed above, for given directionk and frequency ω, this equation determines the refractive indices n λ (k, ω), which we label by an index λ ∈ N (later, we will see that λ ∈ {1, 2}). These solutions can be studied most conveniently by choosing for each directionk an orthonormal basis in R 3 , i.e., three (real) vectors e k1 , e k2 , e k3 with the property that e ki · e kj = δ ij . We further assume that e k1 and e k2 are perpendicular to k, i.e., in the transverse subspace, while e k3 =k is in the longitudinal subspace. In this basis, Eq. (4.15) takes the following form: where we have defined We emphasize that these components of the effective dielectric tensor refer to a basis in k-space (and hence not to a fixed basis in real space). In particular, this implies that the matrix appearing in Eq. (4.16) does in fact depend on the direction k, although the original effective dielectric tensor is purely frequency dependent. Given a refractive index n 2 λ which solves Eq. (4.16), one further obtains The corresponding (normalized) polarization vectors, which solve Eq. (4.8), can then be written as 19) and hence they are, in general, not purely transverse (and possibly complex at that).
In the remainder of this subsection, we will deduce some general properties of the solutions of Eqs. (4.16) and (4.18). In the next subsection, we will then derive explicit expressions for the refractive indices in the most general case of an anisotropic material, and in Appendix A we will investigate some special cases.
The first important observation concerning Eq. (4.16) is that in general, it leads to a polynomial equation of second order in n 2 , and hence there are (for each direction and frequency) at most two (possibly complex) refractive indices, which we denote by n λ (k, ω) with λ ∈ {1, 2}. This is in contrast to an ordinary eigenvalue problem for a (3 × 3)-matrix, which would in general have three solutions, and this is a consequence of the transverse projection operator appearing on the right-hand side of Eq. (4.8).
However, the shear fact that there at most two different refractive indices does not answer the question of how many radiation modes exist in the medium with a given frequency and direction. Here, we define a radiation mode as a solution (n 2 (k, ω), e(k, ω)) of the central Eq. (4.8) with the following properties: (i) the refractive index is non-zero, n 2 (k, ω) = 0, and (ii) the polarization vector has a non-vanishing transverse part, e T (k, ω) = 0. The first condition is necessary because by Eq. (4.3), a vanishing refractive index would imply that |k| = 0. The second condition excludes the purely longitudinal proper oscillations of the medium, which are usually referred to as plasmons (see comments below). In other words, the question is now the following: for a given frequency and direction, how many linearly independent polarization vectors exist which solve Eq. (4.8) and which are not purely longitudinal? In analogy to the vacuum case, one might assume that there are actually two such modes for each direction and frequency (see Appendix A.1). In order to prove this hypothesis, we take again recourse to theoretical optics (see e.g. Refs. [37, § 4.2] and [46, p. 300]), whereby we distinguish two cases depending on the determinant of the effective dielectric tensor.
Case 1: The effective dielectric tensor is invertible, hence det ↔ ε eff = 0. In this case, acting on Eq. (4.8) first with the inverse effective dielectric tensor and then with the transverse projector, we obtain where e T (k, ω) = P T (k)e(k, ω). In the transverse subspace, this now is an eigenvalue problem, which shows that the (transverse parts of the) polarization vectors can be characterized as eigenvectors of a suitably defined (2 × 2) matrix. The transverse part of the polarization vector being given, we can then calculate the longitudinal part by the explicit formula e L (k, ω) = n 2 (ω,k) ( which follows again from the central Eq. (4.8). Consequently, there are at most two polarization vectors which possess a transverse part and thereby qualify as radiation modes.

Case 2:
The effective dielectric tensor is not invertible, hence det ↔ ε eff = 0. In this case, one obvious solution of Eq. (4.16) is n 2 = 0, which does not qualify as a radiation mode. The other refractive index may be non-zero, and consequently, the possible polarization vectors are determined by the null space defined in Eq. (4.18) with only one possible refractive index n 2 = 0.
This null space could in principle even be three-dimensional; however, in this case the only non-vanishing components of the effective dielectric tensor would be ε 11 = ε 22 = n 2 , and hence there would again be two transverse and one longitudinal oscillation. Thus, even in the case of a singular effective dielectric tensor, there at most two radiation modes.
It remains to discuss whether the kernel of the dielectric tensor has a physical meaning. For this purpose, we consider the condition ↔ ε eff (ω) e(k, ω) = 0 . If this equation is supposed to give rise to a proper oscillation of the medium, then the central equation (4.8) has to be fulfilled as well. A comparison shows that in this case, we either have n 2 (k, ω) = 0 or e T (k, ω) = 0. The first possibility can again be discarded. By contrast, the second possibility states that the proper oscillation is purely longitudinal and hence corresponds to a so-called plasmon. Thus, Eq. (4.22) combined with the longitudinality condition constitutes the generalized plasmon equation, which generalizes the well-known condition (3.5) for isotropic media. In particular, since the effective dielectric tensor does not necessarily have a non-trivial kernel, this shows that plasmons do not necessarily exist in any material.

General formulae for refractive indices
In this final section, we study the refractive indices in the most general case of an anisotropic material, for which the off-diagonal components of the (effective) dielectric tensor do not vanish. In this case, Eq. (4.16) leads to the following equation, which is quadratic in n 2 : (4.23) In theoretical optics, this is sometimes referred to as the Fresnel equation (in honor of the legendary Augustin Jean Fresnel (1788-1827)) (see e.g. Refs. [45, Eq. (2.14)] or [46, p. 300]). Note, however, that this equation must not be confused with the Fresnel equations used for the intensity distribution for reflection at a material interface (see Ref. [31] for a recent discussion). For studying the solutions of Eq. (4.23), we distinguish again two cases depending on the value of ε 33 . The latter parameter coincides, via the relation Case a: It may happen that at the given frequency and direction, we obtain ε 33 (k, ω) = 0. For such frequencies, Eq. (4.23) reduces to 25) and this equation has precisely one solution n 2 given by Here, we have assumed that the denominator does not vanish (which will generally be the case for anisotropic materials).
Case b: ε 33 (k, ω) = 0. Now, Eq. (4.23) has generally two (possibly complex) solutions given by and n 2 2 = 0, where the latter refractive index can be discarded. (Thus, we recover Case 2 treated in the previous subsection.) To summarize, Eqs. (4.18) and (4.27) solve the problem of calculating the refractive indices and polarization vectors in the most general case of a possibly non-diagonalizable effective dielectric tensor. We emphasize again that all the above formulae refer to the effective dielectric tensor defined by Eq. (3.12). This can be calculated from the wavevector-independent proper conductivity tensor, but still allows one to deduce wavevector-dependent optical material properties from first principles.

Conclusion
We have concisely criticized the standard calculation of the refractive index from the scalar, wavevector-independent dielectric function. Consequently, we have based our treatment of the refractive index on the fundamental, microscopic wave equation in materials, Eq. (3.1), which involves in general a wavevector-dependent dielectric tensor. We have proven the equivalence of this fundamental, microscopic wave equation to the standard wave equation used in theoretical optics, Eq. (3.13), under the assumption that the latter refers to the effective dielectric tensor (3.12) rather than to the fundamental (ab initio) dielectric tensor (3.2). Thereby, we have shown that the combination of ab initio methods for calculating the proper conductivity tensor-which is a standard target quantity of any modern ab initio materials simulation code [47][48][49][50] (see also the discussion in Ref. [51,§ II])-with the Fresnel equation from theoretical optics, Eq. (4.23), solves the problem of calculating wavevector-dependent optical properties from wavevectorindependent response functions. Correspondingly, the central equation for the joint determination of frequency-and direction-dependent refractive indices and their respective polarization vectors is given by Eq. (4.8).
Besides these results, we have also clarified some more general theoretical questions such as the following: (i) Under which conditions does the standard formula (1.1) for the refractive index actually hold (see Eq. (3.19))? (ii) By which formula should it be replaced if these conditions are not fulfilled (see Eqs. (4.1) and (4.2))? (iii) To which response function does the assumption of wavevector independence actually apply in the standard treatment (see Eq. (4.4))? (vi) What is the relation between the dielectric tensors (3.2) and (3.12) used respectively in ab initio physics and theoretical optics (see Eq. (3.7))?

A. Applications
For the convenience of the reader, we reproduce in this appendix some standard results from the fundamental formulae (4.18) and (4.27) by introducing suitable approximations.

A.1. Vacuum
First, in the absence of a medium, the conductivity tensor vanishes and hence the effective dielectric tensor equals the identity matrix. Thus, in vacuo, the only solution of Eq. (4.16) is n 2 = 1 corresponding to the dispersion relation ω k = c|k|. Furthermore, for this refractive index there are two orthogonal polarization vectors e k1 and e k2 which are both purely transverse.

A.2. Isotropic material
Next, for an isotropic material, we have only two independent components of the (effective) dielectric tensor: Correspondingly, the refractive indices are determined by the condition We further distinguish between the following two cases. First, we consider ε eff,L (k, ω) = 0. This condition determines the frequency as a function of the direction. In fact, those frequecies ω = ω(k) for which the longitudinal dielectric function vanishes are precisely the plasmon frequencies in the isotropic case (see Eq. We note that this equation in terms of the effective transverse dielectric function is fully equivalent to its counterpart Eq. (3.6), which is formulated in terms of the fundamental (ab initio) transverse dielectric function. Furthermore, in this second case, any two transverse, mutually orthogonal vectors e k1 and e k2 can be regarded as polarization vectors, which share the same refractive index given by Eq. (A.4). Thus, for an isotropic medium we recover the well-known results described already in § 3.1.

A.3. Optical activity
A more general case, which includes the isotropic limit, is defined by the absence of the longitudinal-transverse cross-couplings, i.e., by In this case, the fundamental Eq. (4.16) simplifies to Now, the generalized plasmon equation (4.22) together with the condition e T (k, ω) = 0 simply implies ε 33 (k, ω) = 0, and the corresponding frequencies are precisely the plasmon frequencies. Furthermore, there exist at most two purely transverse polarization vectors corresponding to two, in general different, refractive indices. These can be characterized as the eigenvectors and eigenvalues of the (2 × 2) block matrix corresponding to the transverse subspace. Again, this block matrix does in general not have to be hermitean. It may, in special cases, be of the form ε 11 = ε 22 and ε 12 = −ε 21 , such that the general formula (4.27) implies n 2 ± = ε 11 ± iε 12 and the eigenvectors turn out to be of the form e ± = (e 1 ±ie 2 )/ √ 2 (see Ref. [43, p. 182)]). In this case, the dielectric tensor induces optical activity in the respective direction (see e.g. Ref. [43,Eq. (6.39)]). On the other hand, for ε 11 = ε 22 and ε 12 = ε 21 = 0, we recover again the results from the previous subsection.

A.4. Birefringence
As a matter of principle, the formalism presented in this article encompasses birefringence as well. To demonstrate this, we consider here for the sake of simplicity the case of a uniaxial birefringent material, whose proper conductivity tensor is of the form whereâ denotes a fixed unit vector in real space, the so-called optical axis.
In this case, all direction-dependent refractive indices can be calculated analytically from our formalism. For this purpose, let us define the reference indices In terms of these, the effective dielectric tensor can be written as where P L (â) and P T (â) denote the longitudinal and transverse projection operators in the direction ofâ. We first note that fork =â, i.e., if the wavevector is parallel to the optical axis, we recover the results derived in the isotropic case (see § A.2), with two purely transverse polarization vectors sharing the same refractive index n 2 1 (ω). The situation is more complicated in the case wherek =â. Then, we make the ansatz e or (k) =k ×â |k ×â| (A.11) for the first polarization vector, which is both orthogonal to the wavevector (i.e., purely transverse) and to the optical axis. One sees directly that this polarization vector indeed solves the central equation (4.8) with the refractive index n 2 or (ω) = n 2 1 (ω). Hence, the first refractive index is independent of the direction although the polarization vector is not. This solution corresponds to the so-called ordinary ray. In addition, one has to consider the extraordinary ray, for whose polarization vector we make the ansatz e ex (k, ω) = A(k, ω)k + B(k, ω)â , (A.12) such that e ex is perpendicular to e or . The two scalar functions A and B have to be determined together with the refractive index n ex (k, ω) by putting this ansatz into the central equation (4.8) and taking into account the normalization condition |e ex (k, ω)| = 1. Defining the angle α(k) betweenk andâ by cos[α(k)] =k ·â , (A. 13) we thus obtain after a lengthy but straightforward calculation the refractive index of the extraordinary ray, In particular, we see that the polarization vector of the extraordinary ray is not purely transverse but encloses an angle ϕ ex with the wavevector given by cos ϕ ex ≡k · e ex = A + B cos α . (A.17) Hence, the ray direction (which is-in general-defined by the Poynting vector) and the direction of the wavevector do not coincide-fact which accounts for the "extraordinary" behaviour. Finally, for α → 0, one can show that n 2 ex → n 2 1 and (cos ϕ ex ) → 0, thus the extraordinary ray becomes purely transverse and shares the same refractive index with the ordinary ray, which is consistent with the results discussed in the case wherek =â. All of this is in complete accordance with the standard results for birefringent materials (see e.g. Ref. [52, § 4.5]).