Constitutive Relations for Optically Active Anisotropic Media: A Review

The formal description of optical activity (OA) (circular birefringence and dichroism) has more than a 200‐year‐old history, dating back to the pioneering experiments of Arago and Biot. Despite the numerous contributions to it, including several reviews, the formalism of OA has not been treated in a self‐consistent and deductive manner, to the best of authors’ knowledge. Worse, some literature sources report different, apparently contradictory and incompatible, approaches. Willing to provide a general comprehensive review, as well as to clarify all ambiguous points, a unified, systematic, and logical approach to this topic is advanced. By applying a general formal pattern based on the energy conservation principle, the various sets of constitutive relations for optically active media are derived. The relationships allowing for conversions between different sets are presented, in view of their use in the matrix methods for computing the polarimetric responses of stratified structures. The OA tensors of the optically active crystal classes are likewise reported, and the intimate relation existing between crystal point symmetries and physical manifestations of natural OA, namely, rotatory power and longitudinal effect, is discussed. As an illustration, the theoretical developments are applied to the practically important case of planar metamaterial structures.

number of alternative forms of CRs that were advanced independently by different authors until the beginning of the second half of the 20th century, as commented in detail by Lakhtakia. [15] The formal developments were paralleled by both experimental advance and conceptual progress over the two centuries following the discovery of OA. Thus, initially thought of being typical only of enantiomorphous (or chiral) media (i.e., media exhibiting two spatially nonsuperimposable forms), OA was predicted theoretically and observed experimentally in nonenantiomorphous crystals, as discussed in detail by O'Loane. [16] Furthermore, the interest in OA measurements and characterization in the solid state (not only in natural crystals but also in polymers, molecular films, metamaterials, and other artificial structures) have exceptionally grown in the recent years. [17,18] The main purpose of the present work is to review systematically, following their historical development, the various sets of OA CRs, as well as to evidence and derive formally the relationships existing between them. A second aim, of more practical nature, is to evaluate the applicability of the different sets to the computation of the polarimetric response of OA structures, as well as to discuss their limits of validity.
The manuscript is organized as follows. Section 2 and 3 derive and present the different sets of OA CRs and their interrelations in the time and frequency domains, respectively. All relationships are formulated in the Gaussian system of units for both historical and practical reasons (OA CRs assume simpler forms in Gaussian units due to the presence of a single fundamental constant, instead of two of them in Système International (SI) units). Section 4 discusses the use of the OA CRs in Berreman's and Yeh's 4 Â 4 matrix methods, currently used for computing the polarimetric response of stratified structures. This section likewise addresses the physical constraints imposed on OA CRs and the extent of their fulfillment by the different sets. Section 5 reports the OA tensors for all optically active crystal classes and discusses their physical interpretation based on tensor decomposition. As a result, the correspondence between the point symmetries of the OA crystal classes and the two physical manifestations of OA, rotatory power (RP) and longitudinal effect (LE) are established. Section 6 converts the different sets of OA CRs and the related electromagnetic vectors from Gaussian into SI units. Finally, Section 7 illustrates the application of the OA CRs to the special but practically important case of planar metamaterial structures that can exhibit much larger OA than natural crystals and that have recently attracted considerable research interest.

Symmetric Sets of CRs in the Quasistatic Limit
For linear lightÀmatter interaction, we are concerned with here the electric induction (or displacement) D is proportional to the electric field E through the CR. [19] in which ε is the permittivity of the medium. For an anisotropic medium (a crystal, for instance) ε is a real symmetric tensor represented by a positive definite matrix. In presence of external magnetic field (Faraday effect), the tensor ε is positive definite Hermitian.
Similarly to Equation (1a), the magnetic induction (or magnetic flux density) B is related to the magnetic field H through the linear homogeneous expression [19] where μ is the permeability of the medium. Like ε, μ is a real positive definite symmetric tensor for magnetic media. If an external magnetic field is present (magnetic Faraday effect), the tensor μ is positive definite Hermitian. For nonmagnetic media, that is, natural crystals, μ ¼ I, where I is the identity. It should be noted that the CRs given by Equation (1) are strictly valid only for sufficiently slowly varying (quasistatic) fields for which the two tensors ε and μ can be assumed to still take their static values. [19] In this "quasistatic limit," the inductions are established in the material instantaneously with the respective fields applied to the latter.
Any set of electric and magnetic CRs must satisfy the continuity equation for the electromagnetic energy, expressing the principle of conservation of energy.
is the energy flux (or Pointing vector) and w is the energy density (c is the speed of light). Expanding the energy flux in accordance with the vector identity div S ¼ c 4π ðH curl E À E curl HÞ (4) and eliminating the curls with the help of the two Maxwell's equations (sourceless Ampere's law), gives Equation (6) implies that the expression between parentheses on its right-hand side must be a total derivative of time for energy to be conserved. To convince oneself that this is indeed the case for the set of CRs given by Equation (1), it is sufficient to substitute those into Equation (6), while keeping in mind that both tensors ε and μ are symmetric wherefrom the well-known result follows.
The energy conservation principle effectively couples electric and magnetic CRs and is of most general nature. [20] It constrains the specific forms of these relations not only for optically active media, as we shall see below, but likewise applies to nonreciprocal media, [21] as well as to magnetic [22] ones.
To be able to describe the phenomenon of OA (or its most common manifestations, circular birefringence and dichroism), the CRs given by Equation (1) must be extended to include additional (electric) polarization, P OA , and magnetization, M OA , inhomogeneous terms.
The physical picture behind the additional polarization term is the phenomenon of spatial dispersion of the electric permittivity consisting of the nonlocal dependence of the electric induction on the electric field. The additional polarization term P OA thus accounts for the spatial inhomogeneity of the electric field at the microscopic (atomic or molecular) level that is at the origin of OA. The spatial inhomogeneity is assumed to be weak, that is, the characteristic dimension of the microscopic structure responsible or OA is supposed to be smaller than the wavelength of the field, [19] a condition readily fulfilled in natural crystals at optical wavelengths. For a given P OA , the additional magnetization term M OA is determined from Equation (6) in order for the CRs from Equation (9) to satisfy the principle of conservation of energy expressed by Equation (2). We shall call the earlier set of CRs a "symmetric set" as it contains both polarization and magnetization terms.
As mentioned in the Introduction, the first attempt for a quantitative description of OA was done by Cauchy. [6] Based on the old elastic theory of optics, he obtained formally the correct equation of motion for a plane monochromatic wave in an optically active medium, which is equivalent to assuming the polarization P OA to be proportional to curl E (in modern notations). 40 years later, Gibbs [7] arrived at the same conclusion within the framework of the electromagnetic theory albeit without an explicit recourse to Maxwell's equations. Gibbs was the first to point out spatial dispersion as being the physical phenomenon responsible for OA. Some 10 years after Gibbs, Drude [8] reformulated Cauchy's and Gibbs' pioneering findings in modern Maxwell's electromagnetic theory framework and proposed the first set of CRs describing OA in an isotropic nonmagnetic medium.
The coefficient of proportionality f between the polarization term P OA and curl E is the OA constant of the isotropic medium.
Following the critique by Voigt [11] that CRs of the form of Equation (10) violate the conservation of energy expressed by Equation (6), Drude [9,10] later completed his initial set with a magnetization term M OA proportional to ∂E= ∂t.
Indeed, substituting Equation (11) in the right-hand side of Equation (6) and making use of Equation (5a) show that energy is conserved.
Drude provided also a qualitative physical picture for Equation (11), noticing that the electron motion induced by a spatially varying electric field must follow spiraling or helical paths similarly to current flowing in a coil and thus, possessing a magnetic moment. [10] Later, Born [23] constructed a molecular model, attributing strict physical meaning to the polarization and magnetization terms appearing in Equation (11). Pauli [24] likewise obtained Equation (11), however, in a more formal way (using the principle of conservation of energy). Some 40 years after Drude, Boys [25] built a molecular model for OA (different from that of Born) and arrived at the following set of CRs.
Boys' CR set is obviously equivalent to Drude's one (with the substitution ρ ¼ f ) if Maxwell's Equation (5a) (Faraday's law) is taken into account. Both sets can be readily generalized to anisotropic magnetic media and recast into the form where f is the OA tensor of the medium, whereas ε and μ are its permittivity and permeability tensors. Computation analogous to that of Equation (12) shows that the energy density associated with the above CR set is Equation (14) define the secondary vectors D and H as linear functions of the primary vectors E and B and of their time derivatives. We shall call Equation (14) DrudeÀBoys' CRs and the tensor f, DrudeÀBoys' OA tensor.
If the electric CR given by Equation (14a) is solved for the electric field E, one obtains where α ¼ ε À1 f is another OA tensor. It is straightforward to check that the conservation of energy is satisfied if Equation (16a) is coupled with a magnetic CR of the form Indeed, substitution of Equation (16) in the right-hand side of Equation (6) yields wherefrom the particularly simple expression for the energy density results. Notice that the energy density given by Equation (18) formally coincides with the second expression of its counterpart for optically inactive media reported in Equation (8).
It is very important to realize that, while Equation (14a) and (16a) are strictly equivalent algebraically (with the substitution α ¼ ε À1 f ), this is not so for Equation (14b) and (16b), as seen by simple inspection. Therefore, Equation (16) constitute another set of CRs, different from that of DrudeÀBoys'. This set defines the two inductions D and B as primary vectors, and the two fields E and H as secondary vectors, derived from the primary ones and their time derivatives. This second set of CRs was first reported by Voigt [14] practically at the same time as Drude's set. Unlike Drude, Voigt postulated his set of CRs to be valid not only for isotropic media but rather for all optically active crystal classes as well. However, Voigt did not accompany his set by any physical picture (as Drude did).
More than 60 years later than Voigt, Fedorov and his disciples [26,27] extended Gibb's physical picture of spatial dispersion to both permittivity and permeability, thus putting electric and magnetic vectors on an equal footing in the following symmetric set of CR.
The same set was obtained independently and almost simultaneously by Aleksandrov, [28] using slightly different reasoning. Substituting Maxwell's Equation (5) into (19) and solving for the fields transform Equation (19) into (16), thus showing their strict equivalence. Consequently, we shall refer to Equation (16) as FedorovÀVoigt's CRs with α being FedorovÀVoigt's OA tensor.
The direct relation, albeit not equivalence, between DrudeÀBoys' and FedorovÀVoigt's CRs reveals a general formal pattern for obtaining alternative sets of CRs. If an electric (magnetic) CR from a given set expressing the induction (the field) is solved for the field (the induction) instead, the OA tensor is redefined, and the equation of conservation of energy is applied, then one gets another magnetic (electric) CR. The CR thus obtained, when coupled with the redefined CR, forms an alternative set of CRs that is not equivalent to the initial set. For instance, solving Equation (14b) from DrudeÀBoys' CRs for the magnetic induction B rather than for the magnetic field H and putting f μ ¼ α 0 where α 0 is another OA tensor results in the redefined magnetic CR. and the final OA tensors coincide) or transforms a set of CRs into an alternative set (when the initial and final OA tensors are different). Therefore, we shall call the set of CRs defined by Equation (22) "dual DrudeÀBoys' CRs and the tensor f 0 dual DrudeÀBoys' OA tensor. A comparison of Equation (22) and (14) further shows that dual DrudeÀBoys' CRs and DrudeÀBoys' CRs are reciprocal to each other.
Eventually, we leave it to the reader to show that the application of the general formal pattern starting from the electric CR of dual DrudeÀBoys' set allows one to obtain Condon's set of CRs and vice versa with the OA tensors of the two sets being related through α 0 ¼ ε f 0 . Alternately, Condon's set can be obtained starting from the magnetic CR of the DrudeÀBoys' set by putting α 0 ¼ f μ (and vice versa). Figure 1 shows the interrelations between different sets of CRs that follow from the procedure of their derivation. As we shall see in Section 2.3, these interrelations, although derived in the quasistatic limit, are generally valid in the time domain. Neighboring sets in Figure 1 are derived from each other using the general pattern discussed before. Diametrically opposed sets are reciprocal to each other. The duality principle transforms DrudeÀBoys' set of CRs into its dual counterpart (as discussed), while converting the electric CRs within FedorovÀVoigt's and Condon's sets into their magnetic counterparts and vice versa. Figure 1 shows clearly that the four different sets of CRs can be derived step by step in a circular manner, starting from any set. However, it should be emphasized that the four sets thus obtained are not formally equivalent to each other: thus, if the electric (magnetic) CR of a given set is algebraically equivalent to that of its neighboring set (provided the OA tensor is redefined accordingly), then its magnetic (electric) one is not. Moreover, neither the electric, nor the magnetic CRs belonging to diametrically opposed sets are algebraically equivalent. This nonequivalence causes not only differences in the definitions of the OA tensors, but also in the values of the permittivity and permeability tensors. To establish correspondences between different sets of CRs, one must recourse either to linear differential operator theory (in the time domain) or to linear algebra (in the frequency domain). Being much simpler, we shall develop the latter approach in Section 3.

Asymmetric Sets of CRs in the Quasistatic Limit
The generic set of symmetric CRs given by Equation (9) features additional polarization, P OA , and magnetization, M OA , OArelated terms that enter the expressions on an equal footing. However, these same terms contribute differently to the related bound charge, ρ bOA , and current, J bOA , densities present in the optically active medium. [19] ρ bOA ¼ Àdiv P OA (25a) Clearly, both ρ bOA and J bOA remain invariant if the polarization and the magnetization are redefined in accordance with the relations in which Q is an arbitrary vector field. Substitution of Equation (26) into (9) yields are redefined electric induction and magnetic field, respectively. Therefore, redefining the polarization and the magnetization, while maintaining the bound charge and current densities constant formally amounts to redefine the electric induction and the magnetic field. It is straightforward to check that the two redefined vectors satisfy identically Equation (5b) (the sourceless Ampère's law). The redefinitions expressed by Equation (28) are one of the two sets of the so called "Fedorov's transformations". [27,30] Now, if the earlier Fedorov's transformation with the choice Q ¼ f T E is applied to the spatial derivative form of DrudeÀBoys' set of CR (obtained by substituting Faraday's law (5a) into Equation (14a)), one gets with the help of Equation (28) Comparing Equation (27b) and (29b) or equivalently, substituting Q ¼ f T E in Equation (26b) shows that this specific choice corresponds physically to vanishing magnetization, that is, M 0 OA ¼ 0. Consequently, we shall call the above set of CRs "asymmetric" as, unlike the generic set given by Equation (9), it features absent magnetization contribution in its magnetic CR.
Using the vector algebra identity [30] f where Equation (29) can be cast into the form In Equation (31) and (30), I is the 3 Â 3 identity; "tr" stands for trace; and ∇ is the nabla operator; thus, curl ≡ ∇Â, where "Â" denotes the vector (cross) product. The tensor g derived from DrudeÀBoys' OA tensor f through Equation (31) is called "gyration tensor"; G ¼ g∇ is the "gyration vector." Equation (32) were first established by Landau and Lifshitz [19] (in the frequency domain) and independently, by Born. [31] (Landau and Lifshitz considered the phenomenon of spatial dispersion of the dielectric permittivity, mentioned in Section 2.1, while taking advantage of vanishing magnetization due to its redefinition; Born extrapolated Drude's CRs given by Equation (11) from isotropic media to crystals, while neglecting the magnetization term contribution.) Equation (32) are called "BornÀLandau's CRs". [32] It is clear from the earlier derivation that BornÀLandau's CRs are just a variant of DrudeÀBoys' CRs with redefined electric induction and magnetic field and, strictly speaking, do not represent another set of CRs. Nevertheless, we shall treat both sets as being different for historical reasons. It should be likewise mentioned that Equation (32) were also obtained by Bokut and Serdyukov using the earlier method. [30] However, Bokut and Serdyukov started from FedorovÀVoigt's CRs rather than from DrudeÀBoys' ones and consequently, had to neglect second-order terms in the OA to be able to arrive at Equation (32).
If one applies the principle of duality expressed by Equation (24) to the set of Fedorov's transformations defined by Equation (28), one gets Fedorov's second set of transformations.
where R, like Q, is an arbitrary vector field. It is straightforward to check that Faraday's law (5a) is invariant with respect to this set of transformations. With the special choice R ¼ f 0 H, one transforms the dual DrudeÀBoys' CRs in their spatial form (obtained by substituting Ampere's law (5b) in Equation (22a)) into or, after taking into account the identity from Equation (30) We shall call the above set of asymmetric CR BokutÀSerdyukov's CRs and the gyration tensor g 0 (vector G 0 ), BokutÀSerdyukov's gyration tensor (vector). (Bokut and Serdyukov [30] obtained this set from FedorovÀVoigt's set of CRs rather than from the dual DrudeÀBoys' one, at the cost of neglecting second-order contributions in the OA.) There is an apparent symmetry between BornÀLandau's and BokutÀSerdyukov's sets of CRs. In fact, any set can be obtained from the other one by direct application of the principle of duality. However, the two sets "behave" differently from the physical point of view: BornÀLandau's set preserves the bound charge and current densities in the OA medium, as discussed in the beginning of this section, whereas BokutÀSerdyukov's set does not possess this property. Indeed, it is easy to establish from Equation (9) that the second set of Fedorov's transformations defined by Equation (33) entails the following redefinitions of the OA-related polarization and magnetization.
When substituted into Equation (25), these clearly modify the values of both the bound charge and the bound current densities. Consequently, BokutÀSerdyukov's set of CRs does not allow for a physical interpretation, unlike BornÀLandau's one (whose interpretation consists of the redefinition of the electromagnetic vectors so as to get vanishing magnetization, while preserving the bound charge and current densities).

General Time-Domain CRs
For historical reasons, as well as for manipulation convenience, we presented the OA CRs from the previous two sections in the quasistatic limit, assuming instantaneity valid only for slowly varying fields. [19] When rapidly varying electric and magnetic fields are applied to matter, the respective inductions are not established instantaneously but rather depend on the field values at all previous instants. As a result, the CRs given by Equation (1) take the most general forms of integral linear transformations, [19,33] expressed compactly through the convolution product (denoted by "*"). and (For completeness, we have written explicitly both the time (t) and the space-or position, r-dependence of the electromagnetic vectors.) The permittivity, ε, and the permeability, μ, of the medium cannot be assumed anymore to take their static values, like in the quasistatic limit, [19] but are now described by their respective time-dependent tensor kernels ε $ and μ $ .
To generalize the OA CRs from the previous two sections to the noninstantaneous action picture, that is, to extend them beyond the quasistatic limit, it is sufficient to replace the common algebraic tensorÀvector products by convolution-based ones and the (permittivity, permeability, and OA) tensors by the respective tensor kernels. Using the properties of the convolution product, it is a straightforward exercise to show that the principle of conservation of energy, expressed by Equation (2), likewise holds for the general time-dependent OA CRs. The reader interested in the mathematical foundations of the convolution-based algebra may consult the study by Yosida [34] and the references therein. Table 1 shows the four sets of symmetric time-domain OA CRs thus obtained. Notice that there can be no other sets as these exhaust all possible combinations between pairs of electric and magnetic inductions and fields. The sets are reported both in their time-derivative and spatial-derivative forms (whenever applicable). The "time-derivative form" of a CR features a polarization term linearly related (by the OA tensor) to the time derivative of a magnetic vector and a magnetization term linearly related to the time derivative of an electric vector. As shown by Aleksandrov [28] (in the special case of FedorovÀVoigt's set), the symmetric nature of this electromagnetic coupling feature is a general property imposed by the conservation of energy. The alternative equivalent "spatial-derivative form" of a CR contains curl(s) of the field(s) instead of time derivative(s) of the respective induction(s), in accordance with Maxwell's curl equations. Notice that the relation(s) with curl(s) involve vectors of the same physical nature (either electric only or magnetic only), albeit at the expense of the symmetry featured by the respective time-derivative form. Moreover, spatial-derivative forms always express the inductions as functions of the fields. The last column of Table 1 shows the OA tensor kernels associated with CRs, as well as their relations to their counterparts belonging to neighboring CRs; see Figure 1. Finally, notice that primed and unprimed symbols refer to tensor kernels belonging to reciprocal CRs. Table 2 shows BornÀLandau's and BokutÀSerdyukov's sets of asymmetric time-domain CRs, together with their respective gyration kernels and gyration vectors, as well as the pairs of electromagnetic vectors subject to the corresponding Fedorov's transformation. (The symbol " ⊗ " stands for the association of vector and convolution products.) Notice that, unlike their respective symmetric counterparts (DrudeÀBoys' and dual DrudeÀBoys' sets), BornÀLandau's and BokutÀSerdyukov's asymmetric sets are essentially of spatial-derivative form (i.e., do not possess time-derivative form counterparts). The same holds for both redefined inductions; however, both redefined fields involve time derivatives instead of spatial ones. Therefore, instead of having two, time-and spatial-derivative, forms like their symmetric counterparts, the two asymmetric sets of CRs together with their redefined fields are rather of unique mixed, spaceÀtime, form. Table 2 likewise shows that both asymmetric sets of CRs are essentially uncoupled ones, that is, within a given set, the electric (magnetic) CR involves only electric (magnetic) vectors. However, as readily seen from the last column of the table, this electromagnetic uncoupling of the CRs is achieved at the expense of the redefined magnetic (electric) field being coupled to its magnetic (electric) counterpart through the OA tensor. In other words, the electromagnetic coupling is not eliminated but is rather "shifted" from the CRs to the redefined fields. Indeed, it is fundamentally impossible to describe formally an OA medium without considering electromagnetic coupling in matter.
Before ending this section, we should note that different authors may use different names for one and the same set of OA CRs. We have named the sets in accordance with their historical discovery, in agreement with the common practice and the large majority of the literature sources. When consulting OA literature, the reader should pay particular attention to the formal structure of the given OA set rather than to its name. The same comment also holds for the frequency-domain sets of OA CRs presented in the next section.

Symmetric Sets of CRs in the Frequency Domain
If the temporal Fourier transform is applied to the permittivity tensor kernel from Equation (38a) (39) one obtains the temporal frequency-domain permittivity εðωÞ (ω is the cyclic frequency). [19,33] Most generally, εðωÞ (simply denoted ε) is frequency dependent and features both a real and an imaginary part, the imaginary part accounting for dissipation (absorption) in the material. Consequently, the permittivity ε is an indefinite complex tensor. Its real part accounts for linear birefringence, whereas its imaginary part determines the absorption and the linear dichroism; both parts respect the crystal symmetries. [19] Transparent media exhibit negligible imaginary parts of their permittivity tensors over relatively wide frequency ranges, for example, diamond from the visible to the infrared. However, causality formally imposes the coexistence of both parts of ε at all frequencies. [19,33] If the temporal Fourier transform expressed by Equation (39) is applied not only to the permittivity tensor kernel but to the time-derivative forms of the symmetric time-domain sets of CRs from Table 1 and the well-known properties of the Fourier transform of a time derivative and of a convolution product are used, then one obtains the corresponding frequencydomain CRs. All time-domain electromagnetic vectors Vðr, tÞ are replaced by their Fourier components Vðr, ωÞ (the same symbol is used for simplicity of notation), whereas all OA tensor kernels τ $ transform to their temporal frequency-domain counterparts τðωÞ, with the additional rescaling being assumed. The temporal frequency-domain CRs thus obtained appear in the form of algebraic linear homogeneous relations expressing the secondary vectors as functions of the primary ones. From a physical viewpoint, they hold for monochromatic (or time-harmonic) fields with e Àiωt time dependence. Table 3 shows the temporal frequency-domain CRs cast in matrix form. It also reports the relations between OA tensors obtained from their respective kernel counterparts from Table 1. The 6 Â 6 matrices transforming primary vectors into secondary ones are commonly called "material tensors". [33,35,36] The frequency-domain counterparts of (dual) DrudeÀBoys' and Condon's sets of CRs bear the names of (dual) Post and Tellegen, respectively. Post [33] arrived at his set through the advancedsciencenews.com www.adpr-journal.com covariant formulation of electromagnetism, whereas Tellegen [37] established his eponymous set in a physical context (electrical circuit theory) completely different from that of OA. As shown in the table, the four sets of CRs feature not only different OA tensors, but also different permittivity and permeability ones. The third column of Table 3 reports the triplets of tensors for each set. (Note that all tensors depend most generally on the frequency ω, i.e., exhibit temporal dispersion. [19,33] ) To understand the origin of the differences between permittivity and permeability tensors, reference should be made to Figure 1 from Section 2 showing the interrelations between time-domain sets of CRs. Neighboring sets with algebraically equivalent electric (magnetic) relations naturally share the same permittivity (permeability) tensor. However, the permittivity (permeability) tensors of diametrically opposed sets (i.e., sets reciprocal to each other) differ; this is because these sets do not share any algebraically equivalent relation. This entails the formal existence of two permittivity tensors (ε and ε 0 ), together with two permeability ones (μ and μ 0 ) . The situation is shown in Figure 2, in which the circular diagram from Figure 1 is transposed to the frequency domain through adding the tensor triplets from the last column of Table 3.
To establish the relations between the different tensors in Figure 2, it is sufficient to solve, with the help of Table 3, Tellegen's CRs for the secondary vectors of FedorovÀVoigt's CRs and vice versa. As these two sets are reciprocal to each other, one simply has to invert the material tensor of the first set to get that of the second set where use has been made of the algebraic identity for inverting a partitioned matrix. [30] Identification of Equation (41) with the material tensor of FedorovÀVoigt's set shows that Conversely, inverting the material tensor of FedorovÀVoigt's set and identifying the inverse thus obtained with the material tensor of Tellegen's set yields Equation (42) and (43) convert tensors from Tellegen's set of CRs into FedorovÀVoigt's one and vice versa. To establish the corresponding relations between tensors of Post's and dual Post's sets, use can be made of the last column of Table 3 connecting OA tensors of neighboring sets. Keeping in mind that "electrically" ("magnetically") related neighboring sets of CRs share the same permittivity (permeability) tensor, one thus gets from Table 3 and Figure 2 α Substituting Equation (44) into (42) and (43) allows one to establish the sought relations (after some manipulations). Table 4, 5 and 6 summarize the conversion relations for the permittivity, permeability, and OA tensors between any two of the four sets of CRs. Equating expressions from a given line in any table yields a series of valid relations.
It should be mentioned that the relations between FedorovÀVoigt's CRs and Tellegen's CRs, as well as those www.advancedsciencenews.com www.adpr-journal.com between Post's and Tellegen's ones, have been reported previously. [20,33] Konstantinova et al. [32] have applied numerical methods to establish the relationships between Tellegen's OA tensor and the BornÀLandau's gyration tensor in the special cases of certain crystal classes. If one further applies the spatial Fourier transform where V is any electromagnetic vector (electric or magnetic field or induction) to the temporal frequency-domain CRs from Table 3, then one obtains the CRs in the spatiotemporalfrequency domain. In Equation (45), k is the wave vector and m is the refraction vector. [35] It is clear that, provided identical notations are used for the electromagnetic vectors (V) and their spatial-frequency components (V $ ), the sets CRs from Table 3 formally coincide in both temporal-frequency and spatiotemporal-frequency domains. Consequently, the relations between tensors shown in Table 4-6 are valid in both domains. For this reason, all relations shown in Table 3À6 are simply referred to as belonging to the frequency domain. Notice, however, that if the temporal and the spatial Fourier transforms are applied to the spatial forms of the symmetric time-domain CRs, then a distinction between the two frequency domains has to be made; this is the case with the asymmetric time-domain CRs dealt with in the next section.
The physical meaning of the spatiotemporal-frequency domain sets of CRs consists of their applicability to the propagation of plane waves with expði k r À i ω tÞ spatiotemporal dependence. Consequently, they can also be termed "plane-wave CRs." We shall illustrate their practical use in Section 4.

Asymmetric Sets of CRs in the Frequency Domain
The application of the temporal Fourier transform expressed by Equation (39), as well as of Equation (40), to the asymmetric time-domain BornÀLandau's and BokutÀSerdyukov's sets of CRs, (together with their redefined electromagnetic vectors) shown in Table 2, converts the latter into their temporal frequency-domain counterparts. The respective pairs of permittivity and permeability tensors are those of Post's and dual Post's CRs from Table 3. The resulting expressions, gyration tensors and vectors, as well as corresponding redefined electromagnetic vectors, are shown in Table 7.
Like their symmetric counterparts from Table 3, the two asymmetric sets from Table 7 are valid for monochromatic (time-harmonic) fields. Notice that their expressions can be derived directly in the temporal frequency domain starting from Post's (and dual Post's) CRs and conducting the appropriate frequency-domain Fedorov's transformations; this has been done by Peterson [38] in the case of the pair Post's CRsÀBornÀLandau's CRs.
If one further applies the spatial Fourier transform expressed by Equation (45) not only to the electromagnetic vectors, but to the entire temporal frequency-domain asymmetric CRs from Table 7, one then obtains their spatiotemporal-frequency (or plane-wave) counterparts shown in Table 8. (The notation v Â Table 6. Conversion between OA tensors from the four sets of symmetric frequency-domain CRs.
www.advancedsciencenews.com www.adpr-journal.com used in the table stands for the antisymmetric tensor dual to the where a is an arbitrary 3D vector. [35] ) The asymmetric sets of time-domain CRs being essentially of spatial-derivative form, their spatiotemporal frequency-domain counterparts contain explicitly the refraction vector m. The latter can be written as m ¼ n u, where n is the refractive index of the medium and u is the propagation direction (i.e., the normal to the wavefront or the wave normal). [35] Therefore, the plane-wave asymmetric CRs depend (explicitly) both on the propagation direction u and on the refractive index n, which also depends (implicitly) on u in an anisotropic medium, as we shall see later.
In Table 8, this dependence is expressed symbolically by defining the generalized permittivity ε $ ðmÞ and the generalized permeability μ $ ðmÞ. The gyration vectors G and G 0 likewise depend explicitly on m, whereas the gyration tensors g and g 0 do not. Therefore, following Fedorov [39] and Jones, [40] one can clearly state that it is the gyration tensor g (g 0 ), not the tensor n g (n g 0 ), from which the gyration vector G (G 0 ) is derived through the substitution m ¼ n u, as erroneously argued by Born on page 414 of his book, [31] that is a "material constant" of the OA medium. The tensor n g (n g 0 ) depends on the direction of propagation u in an OA anisotropic medium through the (implicit) dependence of the refractive index n on u.
It should be emphasized that, unlike ε and μ, the generalized permittivity ε $ ðmÞ and the generalized permeability μ $ ðmÞ are not tensors. In fact, ε $ ðmÞ and μ $ ðmÞ are rather combinations of the permittivity and the permeability tensors ε and μ with the gyration tensors g and g 0 , respectively. Furthermore, they both depend explicitly on the refraction vector m, as already mentioned.

Use of the CRs in Calculating the Response of Stratified Structures Containing Optically Active Layers
Usually, OA media appear in the form of thick slabs (suitable for transmission measurements) or are contained in stratified structures: either as single layers on substrates or in multilayer stacks (suitable for reflection measurements). There exist two widespread 4 Â 4 matrix methods for computing the response of a stratified structure: Berreman's method and Yeh's method. While both methods are based on the propagation of the tangential components of the (electric and magnetic) fields E and H through the stratified structure in the frequency domain, they use the OA CRs in quite different ways, so we shall discuss them separately. (Note that, although using different mathematical approaches, the two methods are strictly equivalent in terms of results.)

Handling OA in Berreman's Method
In Berreman's method, [36] the two fields E and H are considered as primary vectors, whereas the two inductions D and B are secondary ones. Consequently, the description of OA in this method is based on the most general linear homogenous relation between inductions and fields (in the plane-wave response).
where a ij , i, j ¼ 1, 2 are 3 Â 3 tensors describing the optical properties of the medium. The "natural" choices a 11 ¼ ε 0 , a 12 ¼ i α 0 , a 21 ¼ Ài α 0T , and a 22 ¼ μ 0 identically transform Berreman's tensor appearing in Equation (46) into Tellegen's material tensor, as readily seen by comparing the respective entry of Table 3 with Equation (46). Tellegen's set of CRs, generally used with Berreman's formalism, is possibly the most widely used one by experimentalists involved in the characterization of optically active crystals. [41][42][43] Tellegen's set of CRs can be applied with success even to complex biological structures that are not inherently optically active but that effectively behave as such in certain measurement configurations. [43,44] If, instead of Tellegen's set of CRs, one is willing to use, for instance, Post's set, then Post's material tensor (from Table 3) should be converted to Tellegen's one with the help of Table 4 (for the permittivity) and Table 6 (for the OA tensor). (Table 5 is not used as the two sets share the same permeability). With Post's CRs, one thus obtains for Berreman's tensor from The above conversion procedure is applied invariably whatever the set of CRs, provided the corresponding group of tables (among Table 4, 5, and 6) is used. Clearly, Berreman's method is very well suited formally for implementing any set of OA CRs, Table 8. Sets of asymmetric spatiotemporal frequency-domain CRs and redefined electromagnetic vectors.

Name
Expressions Gyration tensor and gyration vector Generalized permittivity/permeability Redefined vectors with Tellegen's CRs being the "natural" choice, which explains their wide use in practice. However, this ease comes at the expense of lack of insight into how the OA CRs affect the light propagation mechanism. The latter aspect is better revealed by Yeh's method.

Handling OA in Yeh's Method
Unlike Berreman's method, Yeh's method, [45] recently extended to optically active media, [46] does not consider explicitly the two inductions D and B but rather determines the propagation eigenmodes of the electric field E through solving Fresnel's equation (or the equation of normals) [19,35] for each layer of the stratified structure. As Fresnel's equation is unique [35] one may use any set of frequency-domain OA CRs to derive it. The most straightforward procedure makes use of BornÀLandau's set from Table 8. Substituting the redefined electric induction D 0 from the plane-wave sourceless Ampère's law (for redefined vectors, Similarly, substituting the magnetic induction B from the plane-wave Faraday's law Finally, eliminating the redefined magnetic field H 0 from Equation (48) provides the following eigenequation for the electric field E.
Fresnel's equation is obtained by taking the determinant (denoted by "det") of the bracketed expression Equation (50) is a fourth-order algebraic equation for the refractive index n given the propagation direction u (together with the permittivity, permeability, and gyration tensors of the optically active medium). Equation (A1) from Appendix A reports its explicit algebraic form. For a transparent optically active medium, the four values of n are real and can be grouped into two pairs having opposite signs corresponding physically to forward and backward propagation. (If the medium is absorbing then the four values of n are complex and can likewise be grouped into two pairs whose real parts have opposite signs.) Substituting the four values of n into Equation (49) provides the respective eigenmodes of the electric field E. To obtain the eigenmodes of the magnetic field H, one makes use of the transformation equation for the redefined field H 0 from Table 8, as well as of the magnetic CR (48b).
Equation (49) and (50) show that the refractive indices and the electric field eigenmodes depend explicitly on the gyration tensor g and only implicitly on the OA tensor f from which g is derived though Equation (31). In contrast, the magnetic field eigenmodes depend explicitly on f, not on g, as shown in Equation (51).
Recall that frequency-domain BornÀLandau's CRs are the asymmetric version of Post's (symmetric) CRs. Therefore, the permittivity, ε, and the permeability, μ 0 , tensors appearing in Equation (49) and (50) are those of Post's CR from Table 3, whereas the gyration tensor g is related to Post's OA tensor f through Equation (31) (also reproduced in Table 8). To change the set of CRs, one should make use of Table 4À6 accordingly. For instance, replacing BornÀLandau's (Post's) CRs by Tellegen's CRs with the help of Table 4 and 6 transforms Fresnel's equation given by Equation (50) into and Equation (31) into together with a similar transformation for Equation (49). Eventually, Equation (51) becomes Equation (52)À(54) were established theoretically [46] (starting from Tellegen's CRs), as well as verified experimentally. [47] It is important to note that the substitution m ¼ n u for the refraction vector m we used in Equation (50) to obtain Fresnel's equation for the refractive index, as well as in Equation (49) and (51) to obtain the electric and magnetic field eigenmodes, is valid only for normal incidence (as well as, trivially, for bulk propagation), as the propagation direction u was assumed known a priori. To obtain Fresnel's equation in the case of oblique incidence, one should rather set m ¼ ½n 0 sin θ 0 n z T (instead of m ¼ n u ) into Equation (50); [45,46] the resulting equation is then solved for n z , the z-component of the refractive index (i.e., n z ¼ n u z ). [48] The same substitution applies to Equation (49) and (51) for the respective field eigenmodes once n z is determined. Here, θ is the incidence angle and n 0 is the refractive index of the ambient (n 0 ¼ 1 typically). (With this specific choice of reference frame, the z-axis is taken along the normal to the parallel interfaces of the stratified structure, whereas the y-axis is taken along the normal of the incidence plane; the x-axis is perpendicular to both.)

Choice of CRs for Optically Active Media
As already mentioned, the four sets of CRs shown in Table 1 and shown in Figure 1, although derivable from each other using a general formal pattern, are not formally equivalent to one another. This is likewise clear from Table 4 and 5, as well as from Figure 2: changing a set of CRs entails the change of either the www.advancedsciencenews.com www.adpr-journal.com permittivity or the permeability, or both of them. However, both the permittivity and the permeability are "material constants" (in Born's terms) that should not depend on the particular choice of a set of CRs. (In particular, the permittivity tensor of a nonmagnetic medium should equal the identity, be the medium optically active or not.) Formally speaking, only one set of CRs holds strictly true, whereas the remaining three are approximations thereof. As judiciously noted by Fedorov, [27] the phenomenological approach we have used to derive the different sets of CRs is incapable of pointing out the correct set. For the last purpose, it has to be supplemented with additional, physical or formal, criteria. Thus, a set of CRs definitely must 1) satisfy the principle of conservation of energy (for transparent media), 2) yield real refractive indices when used to describe light propagation in transparent media, 3) be covariant, that is, be invariant with respect to relativistic Lorentz transformations, and 4) be backed by a microscopic (molecular) physical picture.
The first criterion is automatically met by all four sets of CRs discussed as, following Voigt [11] and Fedorov, [39] we naturally implemented the principle of conservation of energy into the formal pattern used to derive the sets from each other. Nevertheless, the early version of the nonenergy-conserving Drude's set, given by Equation (10), is still sometimes used, as pointed out and criticized by Lakhtakia, [49] Konstantinova et al., [50] and Golovina et al. [51] This is certainly due to the fact that the second edition of Drude's classic book, [52] where the corrected, energy-conserving set, Equation (11), appears, was not translated into English. Another relatively common mistake that likewise amounts to violating the conservation of energy is misusing BornÀLandau's set of CRs shown in Table 2 (time-domain version) and in Table 7 (frequency-domain version). It consists of neglecting the difference between usual and transformed (redefined) magnetic field and electric induction, that is, disregarding redefined vectors in the CR expressions, as pointed out and criticized by Bokut and Serdyukov. [30] Indeed, if no distinction is made between actual and redefined vectors, the BornÀLandau's CRs take the form of the nonenergy-conserving Drude's ones in the case of an isotropic OA medium. Energy conservation is likewise violated in the optically active crystal case, as demonstrated by Silverman through numerical simulations. [53,54] Furthermore, misused BornÀLandau's CRs predict the appearance of circular dichroism upon reflection that could not be evidenced experimentally despite sufficient instrument sensitivity, as reported by Luk'yanov and Novikov. [55] The origin of this misuse most likely stems from Born's classic book, [31] where only propagation in unbound, infinite medium (bulk propagation), common to traditional OA transmission experiments, is considered. The situation is quite similar with LandauÀLifshitz's classic text, [19] where the redefinition of the magnetic field is only alluded to, despite a (generally overlooked) warning at the page bottom that the picture changes when interfaces are considered. In fact, it is easy to see that, if redefined vectors are disregarded, Equation (50) and (49), respectively, providing the refractive indices and the electric field eigenmodes remain unchanged; it is only Equation (51) for the magnetic field eigenmodes that changes. As the refractive indices and the electric field eigenmodes entirely determine bulk propagation in transmission (where the effects of the interfaces are neglected), the mistake becomes noticeable only in reflection configuration where the interfaces contribute to the response. Indeed, the redefinition of the electromagnetic vectors through Fedorov's transformations entails the modification of the classic boundary conditions-continuity of the tangential components of the fields and of the normal components of the inductions-governing the reflection and the refraction of light at the interfaces of the optically active medium, as noted by Bokut and Serdyukov. [30] Notice that the modification of the boundary conditions likewise follows from the redefinition of the Poynting vector S given by Equation (5): as S depends linearly on the redefined magnetic field, S itself must likewise be redefined to ensure the continuity of the energy flow across the interface.
To check the validity of the second criterion in the earlier list, the various forms that Berreman's material tensor, appearing in Equation (46), takes for the different sets of CRs must be inspected. Indeed, it is well known that the material tensor relating the fields to the inductions must be Hermitian for a transparent medium. However, as pointed out by Berry, [56] this is only a necessary condition, the sufficient condition being provided by the positive definiteness of the tensor in question. It is easy to see that Tellegen's material tensor, while being Hermitian, is not positive definite in general. Indeed, the quadratic form associated with it (see Table 3) is where A and B are arbitrary vectors (the superscripts "þ" and "*" stand for Hermitian conjugation and complex conjugation, respectively). While the first two terms are positive (because the permittivity and permeability tensors are positive definite), the last term is most generally complex. Consequently, Tellegen's CRs do not satisfy the second criterion. In practice, their use may result in imaginary refractive indices of transparent media for sufficiently large values of the elements of the OA tensor α 0 for certain crystal classes. FedorovÀVoigt's material tensor is not positive definite either as it is the inverse of Tellegen's one; see Equation (41) (a matrix and its inverse are either both positive definite or none of them is). Therefore, like Tellegen's CRs, FedorovÀVoigt's CRs are potentially capable of producing imaginary refractive indices when used to describe transparent optically active media exhibiting strong OA. The situation changes with Post' set of CRs and its dual counterpart. The quadratic form associated with Berreman's material tensor with implemented Post's CR given by Equation (47) is where ffiffiffiffi ffi μ 0 p is the principal square root tensor of the (positive definite) permeability tensor μ 0 . The above quadratic form is clearly positive definite. The same holds for the quadratic form associated with Berreman's tensor with implemented dual Post's CRs (with the help of Table 5 and 6).
where ffiffiffi ffi ε 0 p is the principal square root tensor of the (positive definite) permittivity tensor ε 0 . Therefore, both Post's set of CRs and its dual counterpart satisfy the second criterion from the list. In other words, using one of these sets one is always certain to obtain real refractive indices for light propagation in transparent optically active media whatever the values of the elements and the symmetries of the OA tensors f or f 0 . If Yeh's method is used instead of Berreman's one, then, instead of on Equation (46), the analysis is based on Fresnel's equation given by Equation (50), as well as on its dual obtained by applying the principle of duality (24) to it. In this case, it is easy to show (see Appendix A) that the sufficient condition for real refractive indices for propagation in a transparent medium is the positive definiteness of both permittivity and permeability tensors. Table 4 and 5 then readily show that positive definiteness can be warranted only for Post's set of CRs and is dual counterpart, that is, the same conclusion as that from the previous paragraph is arrived at.
The third criterion is well known to be satisfied by Post's set of CRs. Indeed, as shown by Post himself, [33] relativistic covariance imposes the electric field E and the magnetic induction B as primary vectors, whereas the electric induction D and the magnetic field H are secondary ones. This formal picture is, furthermore, compliant physically with the microscopic version of Maxwell's equations. [19] However, Post's set of CRs is not the unique one featuring the covariance property: dual Post's set of CR, being reciprocal to it (see Table 3), is also covariant. Indeed, the covariant description is invariant with respect to the interchange of primary and secondary vectors.
Last but not least, the fourth criterion on the list is satisfied only by DrudeÀBoys' set of CRs (and its frequency-domain counterpart, Post's set), as well as by its asymmetric variant, BornÀLandau's set. As already mentioned, both Drude [10] and Boys [25] provided OA models at the microscopic (molecular) level to justify the specific form of Equation (14) (in the isotropic case). Kuhn's classic coupled oscillators model of OA [57,58] likewise leads to DrudeÀBoys' CRs. [24,59] In modeling the phenomenon of spatial dispersion of the permittivity, Landau and Lifshitz [19] expanded linearly the permittivity tensor in the wavevector and arrived at Equation (32) (in the general anisotropic case). Recall that DrudeÀBoys' and BornÀLandau's sets of CRs are obtained from one another through mere redefinitions of the magnetic field and the electric induction leaving both charge and current densities in the optically active medium invariant. This fact further demonstrates the internal consistency between the two apparently different physical pictures (molecular modelÀspatial dispersion of permittivity). Notice that dual DrudeÀBoys' and BokutÀSerdyukov's sets of CRs are related through a similar dual redefinition; however, this redefinition violates the invariance of the charge and current densities, as discussed before. Furthermore, BokutÀSerdyukov's set implicitly presumes spatial dispersion of the permeability, which is clearly unphysical as the permeability tensor is well known to strictly equal the identity at optical frequencies. [19] The same critique applies to FedorovÀVoigt's CRs, whereby both the permittivity and the permeability are explicitly assumed to exhibit spatial dispersion. Furthermore, although having provided a qualitative physical discussion at the molecular level, Condon [29] postulated rather than derived his set of CRs. Careful analysis of Condon's physical argumentation reveals that it is similar to that given by Drude, [10] that is, it actually supports DrudeÀBoys' rather than Condon's CRs. Finally, Tellegen [37] postulated his famous set while identifying a novel electromagnetic component (the gyrator) within the framework of electrical circuit theory, that is, in a physical context completely different from that of lightÀmatter interaction described by Maxwell's electromagnetic theory we are dealing with here.
To summarize the earlier discussion, Table 9 shows the conditions OA CRs must satisfy together with their fulfillment by the different sets.
The only CR set satisfying all four conditions is the DrudeÀBoys' (Post's) set (together with its asymmetric counterpart, the BornÀLandau's set). However, if the last condition (availability of a physical picture) is relaxed, then any set of CRs can be used in practice. Indeed, for naturally OA-transparent media, the magnitude of OA is typically so weak that the second condition for real refractive indices, although potentially not fulfilled by both FedorovÀVoigt's and Condon's (Tellegen's) sets, is never violated in practice. Furthermore, as readily shown in Table 4, 5, and 6, the permittivity, permeability, and OA tensors are approximately not affected (to the first order in OA) by the change of CR set. The approximate invariance of the tensors entails the (approximate) fulfillment of the relativistic covariance condition. Consequently, all four sets satisfy the second and the third conditions from Table 9 to the first order in OA (the first condition being satisfied identically by all of them). When used to describe experiment, all four sets will yield practically identical permittivity and permeability tensors, as well as mutually compatible OA tensors, convertible from one into another using  Table 6. Nonetheless, one cannot rule out a priori the elaboration of optically active artificial media or structures (e.g., metamaterials) featuring OA that is strong enough to render the use of the DrudeÀBoys' (Post's) set of OA CRs the only possible option.

Establishing the OA Tensors for All Optically Active Crystal Classes and Isotropic Media
The OA tensors appearing in all (time-and frequency-domain) symmetric sets of CRs obey certain symmetries and exhibit certain vanishing elements depending on the symmetry properties of the crystals they describe. The symmetries and the identically vanishing elements are common to all OA tensors whatever the set of CRs used. They were first reported by Voigt [13,14] and independently reestablished, using a different approach, some 70 years later by Fedorov in the first [39] as well as tabulated in the third [60] of his series of three landmark papers. More specifically, Voigt's derivation exploits the algebraic properties of polar and axial vectors, whereas Fedorov's geometric approach is based on the systematic application of symmetry operations leaving invariant a given crystal class. Strictly speaking, the OA tensor is a pseudotensor [19] (or an axial tensor [61] ) as it behaves differently with respect to the basic symmetry operations: rotations and reflections. Under any rotation, it behaves like a true (polar) second-rank tensor, whereas under a reflection, its behavior is opposite: the change of sign of an element of the true tensor leaves this element unchanged in the pseudotensor and vice versa, as discussed by LandauÀLifshitz [19] and Nye. [61] By applying the above general rule to the group of symmetry operations proper to a given crystal class, one thus determines the specific form of its OA tensor. In particular, it is clear that a centrosymmetric crystal (or isotropic medium) cannot be optically active since, on the one hand, its properties are invariant under central point inversion, whereas, on the other hand, all its pseudotensor elements must change their signs in accordance with the above rule. Table 10 shows the general forms of the OA tensors for all optically active crystal classes (as well as noncrystalline isotropic media).
Following LandauÀLifshitz's convention, [19] the z-axis is taken either along the axis of highest symmetry (whenever present) or along the normal to the plane of symmetry (in class m). Whenever three mutually orthogonal symmetry axes exist, the coordinate axes are taken along them. This convention is at variance with the common crystallographic one where twofold symmetry axis in class 2 and the axis normal to the plane of symmetry in class m are not taken along the z-axis, but rather along the y-axis. In the crystallographic convention, the OA tensors of classes 2 and m are, respectively that is, their z-and y-nonvanishing elements are interchanged with respect to those from Table 10. Further, in the class 42m, the x-and y-axes are taken along the two orthogonal twofold axes of symmetry. If instead these are taken to lie in the two mutually orthogonal vertical planes of symmetry (i.e., in the bisectrix planes of the two twofold axes), then the OA tensor from Table 10 As shown in the table, 18 from the 21 noncentrosymmetric classes (out of the total of 32 crystal classes) are optically active, featuring 11 different OA tensors. It should be noted that, besides the 11 centrosymmetric classes where OA is forbidden, 3 noncentrosymmetric classes are likewise optically inactive; [39] thus, the absence of central symmetry in a crystal is a necessary but not a sufficient condition for the presence of OA. Table 10 likewise shows the two possible point symmetries for each OA class: enantiomorphous and polar, as well as their combination or absence: both enantiomorphous and polar or noncentrosymmetric (in the specific sense of neither enantiomorphous, nor polar). Recall that enantiomorphism (or chirality) is the existence of two forms (stereoisomers) of the same crystal or medium, which are mirror images of one another, that is, which are not superimposable through rotations only; for instance, helicity is a special case of enantiomorphism. Crystals with polar point symmetry, also known as pyroelectric crystals, exhibit permanent electric polarization. [19] Table 10 shows that there are 11 enantiomorphous and 10 polar OA crystal classes, out of which 5 are both enantiomorphous and polar. The remaining two OA classes, 4 (S 4 ) and 42m (D 2d ), are noncentrosymmetric, that is, they are neither enantiomorphous, nor polar. Therefore, the existence of enantiomorphism and/or polar structure is a sufficient but not a necessary condition of the presence of OA. In other words, all enantiomorphous and/or polar crystal classes are optically active, but the converse is not true (as demonstrated by the two noncentrosymmetric optically active classes).
Eventually, it is to be noted that, of the 11 different OA tensors shown in Table 10, the first five belong to biaxial crystals, that is, whose permittivity tensor is of the form ε ¼ diagðε 11 ε 22 ε 33 Þ, whereas the next five describe uniaxial ones, that is, having ε ¼ diagðε 11 ε 11 ε 33 Þ. (This is reported next to the crystal systems in the first column of Table 10) The last tensor, being actually a scalar, describes both isotropic cubic OA crystals as well as noncrystalline isotropic OA media (e.g., liquids), whose permittivity tensors ε are likewise scalars.

Decomposition and Physical Interpretation of the OA Tensors
Following the approaches of Voigt, [13,14] Bokut, and Fedorov, [60] as well as of Jerphagnon and Chemla, [62] we decompose identically the OA tensor τ (equal to f, α, f 0 , or α 0 depending on the OA CRs used) into isotropic, traceless symmetric and antisymmetric parts.
The three additive terms in Equation (59) correspond, respectively, to the three aforementioned tensor parts. (Notice that the symmetric part τ s ¼ 1 2 ðτ þ τ T Þ of τ is simply the sum of its isotropic and traceless symmetric parts.) Table 10 shows the OA tensor decomposition for each OA crystal class. Figure 3 shows the relation existing between the possible tensor decompositions and the point symmetries exhibited by the respective OA crystal classes.
The figure represents the three tensor parts, isotropic (I), traceless symmetric (S), and antisymmetric (A), as three ellipses and the different crystal points symmetries as different hatchings. Enantiomorphous crystals feature decompositions of types I, I þ S or I þ S þ A. Thus, enantiomorphism entails the presence of an isotropic part (I) in the OA tensor decomposition. Polar crystal OA tensors decompose into A, S þ A or I þ S þ A: therefore, polar crystals necessarily feature antisymmetric parts (A) in their tensors. Crystals that are both enantiomorphous and polar decompose as I þ S þ A, that is, they feature symmetric parts (S) in their tensors, besides isotropic and antisymmetric ones. Finally, the tensors of noncentrosymmetric (i.e., neither enantiomorphous nor polar) OA crystals consist of the traceless symmetric term (S) alone.
Jerphagnon and Chemla [62] gave a clear physical interpretation of each one of the three parts of the OA tensor decomposition. As mentioned, the most known physical manifestation of OA is the phenomenon of RP consisting of the rotation of the plane of Table 10. Optically active crystal classes in international (and in Schönflies) notations and their associated OA tensors decomposed into isotropic, traceless symmetric and antisymmetric parts. The crystal system, as well as point symmetries of each class, is likewise reported: e: enantiomorphous, p: polar, ncs: noncentrosymmetric.  (In the large majority, these directions coincide with the optic axis-or axes-of the crystal.) According to Jerphagnon and Chemla, the isotropic part (I) of the OA tensor accounts for the "molecular contribution" to RP, whereas the symmetric part (S) describes the "structural contribution" to RP. The molecular contribution is the one due to the (enantiomorphous) molecule (in a noncrystalline isotropic medium, e.g., a liquid) or to the (enantiomorphous) molecular building unit (in a crystal). The structural contribution to RP is the one resulting from the specific spatial arrangement of the molecular building unit in the crystal lattice. Note that the two contributions to RP, the molecular and the structural one, are indistinguishable experimentally (unless one could compare RP measurements in the solid crystalline state with such conducted in the liquid state; see below). Finally, the antisymmetric part (A) of the OA tensor is characteristic of the so-called LE. [63] The LE is the second physical manifestation of OA that consists of the appearance of a radial component of the electric field along the propagation direction. More specifically, it gives rise to of an out-of-phase polarization component upon reflection of linearly polarized light on a crystal with antisymmetric OA tensor, resulting in reflected light that is weakly elliptically polarized (instead of remaining linearly polarized as in the absence of OA). Therefore, this effect can be experimentally evidenced only through detecting the change in the ellipticity of polarized light upon reflection, as pointed out by Fedorov [64] in the second paper of his seminal series. The combination of the aforementioned physical interpretation of the OA tensor parts with their relations to the optically active crystal point symmetries shown in Figure 3 results in the schematic diagram shown in Figure 4.
OA in enantiomorphous nonpolar crystals manifests itself either through molecular only or through both molecular and structural contributions to RP. The first category belongs to the two isotropic cubic classes 23 and 432, whereas the biaxial class 222 and the three uniaxial classes 32, 422, and 622 pertain to the second one. Enantiomorphous nonpolar crystals featuring molecular contribution only exhibit experimentally RP in the crystalline state only; those who provide both molecular and structural contributions exhibit RP both in crystalline and in noncrystalline, for example, liquid states (if existing). [62] Polar nonenantiomorphous crystals likewise fall into two categories: those featuring either LE only or both LE and structural RP. The three uniaxial classes 3m, 4mm, and 6mm belong to the first category, while the remaining two biaxial ones, m and mm2, pertain to the second one. Crystals that are both enantiomorphous and polar (the triclinic class 1, as well as the four cyclic classes 2, 3, 4, 6) exhibit all three effects: molecular and structural RP, as well as LE. Noncentrosymmetric (i.e., both nonenantiomorphous and nonpolar) crystals, belonging to the two uniaxial classes 4 and 42m, feature only structural contribution to RP. Eventually, optically active isotropic noncrystalline media (not shown in Figure 4) exhibit only molecular RP, like the two OA cubic classes. These media are either enantiopure (i.e., consist of only one of the two enantiomorphous forms) or are homogeneous mixtures of the two enantiomorphous forms present in different amounts. [19] The experimental detection of OA is usually based on that of RP in either traditional transmission or, more recently, in reflection [47,[65][66][67] configurations. RP has been evidenced experimentally in crystals of all classes and point symmetries exhibiting it, as reviewed and discussed in detail by O'Loane. [16] The second manifestation of OA, namely, the LE alone, typical of the three nonenantiomorphous polar classes 3m, 4mm. and 6mm, has been detected experimentally by taking advantage of the presence of a resonance phenomenon strongly enhancing the effect. [65] However, the ingenious reflection experiment proposed by Fedorov et al. [68] for detecting LE far from resonance, although simulated numerically by Golovina et al., [51] has not yet been conducted experimentally, to the best of authors' knowledge.  www.advancedsciencenews.com www.adpr-journal.com So far, we have assumed implicitly that the OA tensors shown in Table 10 are real, that is, we have implicitly dealt with transparent optically active crystals exhibiting RP or LE. However, all OA tensors may as well comprise imaginary parts, of the same symmetry as the real ones, resulting in the physical manifestation of circular dichroism, that is, the differential absorption of two circularly polarized light waves of opposite handedness.

Symmetry of the Gyration Tensor
The gyration tensor g appears in (both time and frequency domain) Born Landau's CRs (see Table 2 and 7) and enters explicitly the eigenequation for the electric field E, as well as Fresnel's equation for the refractive indices; see Equation (49) and (50). (The same statement obviously applies to its dual counterpart g 0 , BokutÀSerdyukov's CRs, and the eigenequation for the magnetic field H.) The frequency-domain gyration tensor g (or g 0 ) is obtained by first converting the given OA tensor to its (dual) Post's counterpart f (or f 0 ) with the help of Table 4, 5, and 6 and then, by applying Equation (31) (or Equation (36)).
Equation (31) shows that the gyration tensor g has exactly the same symmetries and vanishing elements as the OA tensor f. These are shown in Table 10 for all OA crystal classes (as well as for isotropic OA media). The table shows that, in general, the gyration tensor does not feature definite symmetry, that is, it is either symmetric or antisymmetric depending on the crystal class. This is at variance with the statements commonly found in a number of classic texts that the composite tensor g c ¼ ε g [19,63] or, even the gyration tensor g itself, [31,61,69,70] can be considered as being symmetric. When tabulating g c ¼ ε g or g depending on the crystal class, these texts consequently report only the symmetric parts of the tensors from Table 10. As shown in Appendix B, the alleged symmetry properties of g c ¼ ε g and g are derived, respectively, from two approximations that we shall call LandauÀLifshitz's and Voigt's approximations.
As mentioned by Landau and Lifshitz, [19] as well as shown rigorously in Appendix B, the symmetry of the tensor g c ¼ ε g results from a first-order approximation in the OA of the eigenequation for the redefined electric induction D 0 . Appendix B likewise shows that the symmetry of g c entails that of g itself for OA crystal classes of sufficiently high symmetry, including all uniaxial OA crystals. Within this approximation, one can therefore use either the symmetric part g s of g (for higher-symmetry OA classes) or the product ε À1 ðε gÞ s (for lower-symmetry OA classes) instead of the gyration tensor g itself to obtain the refractive indices from Equation (50) (equivalent to Equation (A1) from Appendix A), as well as the eigenmodes of the redefined electric induction D 0 . The eigenmodes of the electric field are obtained either by inverting the electric BornÀLandau's CR from Table 8 or by solving the eigenequation (49). In either case, the gyration tensor g appearing in the respective expressions cannot be considered symmetric (in general); otherwise, the error committed would be of first order in the OA. Similarly, the eigenmodes of the magnetic field are obtained from Equation (51) involving the OA tensor f that, like g, is generally nonsymmetric. As both Berreman's and Yeh's methods for stratified structures are based on the propagation of the tangential components of the electric and magnetic field eigenmodes, the gyration and OA tensors of the optically active layered media appearing in these structures cannot be assumed symmetric but should be rather taken in their general form from Table 10. The only physical anisotropic structure for which the tensor g c ¼ ε g (or the gyration tensor g, depending on the OA crystal class) can be considered symmetric (in the LandauÀLifshitz's approximation) is the infinite (bulk) medium. This situation is approximated quite well in practice by the experimental configuration consisting of a plane-parallel optically active slab in transmission, with back-reflections from the slab facets neglected. In fact, the eigenaxes of the slab are given by the eigenmodes of the redefined electric induction, whereas the propagation in the medium is fully determined by the pair of refractive indices.
As shown in Appendix B, Voigt's approximation [11,31] consists of neglecting terms containing the product OA Â anisotropy in Fresnel's equation for the refractive indices resulting in a symmetric gyration tensor g for anisotropic nonmagnetic OA media, whatever their crystal class. However, the determination of the eigenmodes of the electric field from Equation (49) does not assume that g is symmetric. Similarly, the eigenmodes of the magnetic field, determined from Equation (51), depend on the OA tensor f that, like g, is not symmetric, in general. Furthermore, due to the use of the substitution m ¼ n u for the refraction vector, Voigt's approximation applies only to the case of normal incidence; see the comment at the end of Section 4.2. Therefore, Voigt's approximation features even larger limitations than LandauÀLifshitz's one: it is of practical interest only for determining the refractive indices, that is, the magnitude of the RP, exhibited by an optically active slab measured in normal-incidence transmission configuration (with multiple reflections neglected).
Eventually, one can apply Voigt's approximation into LandauÀLifshitz's one, thus further disregarding terms in the product OA Â anisotropy, besides those of higher orders in the OA. This reduces the approximate bicubic Fresnel's equation resulting from the eigenequation for redefined electric induction (see Appendix B) to a biquadratic one. However, the determination of the eigenmodes of both electric and magnetic fields again requires complete gyration and OA tensors (not just their symmetric parts). Moreover, the application of this combined approximation is inherently restricted to calculating only the normal incidence response of a stratified structure, as already mentioned. In an oblique incidence transmission configuration, the effect of the antisymmetric part of the gyration tensor on the optical response is nonnegligible, as shown by Konstantinova et al. [71] It should be noted that, within both LandauÀLifshitz's and Voigt's approximations, OA crystals belonging to the three nonenantiomorphous polar classes 3 m, 4 mm, and 6 mm (C 3v , C 4v , and C 6v ) appear as optically inactive as, as shown in Table 10, their OA tensor is strictly antisymmetric (recall that gyration and OA tensors feature the same symmetries). From an experimental viewpoint, slabs of these crystals will behave like optically inactive birefringent media when characterized in transmission (to the first order in the OA). However, either as substrates or as layers within stratified structures, these crystals will be effectively optically active when measured in reflection.

CRs for Optically Active Media in the SI System of Units
As mentioned in the Introduction, the sets of OA CRs take their simplest forms in the Gaussian system of units. This is due to the fact that, as far as mixed electric and magnetic relations are concerned, the Gaussian system features a single fundamental constant (the speed of light c), whereas the SI system uses two of them, namely, the permittivity and the permeability of vacuum, denoted ε 0 and μ 0 , respectively. Furthermore, the potentially advantageous rationalization of the relations in SI units (i.e., the removal of the factor 4π) is immaterial with respect to the OA CRs (except for the intermediate steps of their derivation involving the energy balance, the polarization, and the magnetization).

Time-Domain CRs in SI Units
For historical reasons essentially, we reported all OA CRs in Gaussian units. To convert formally the time-domain symmetric CRs from Gaussian units into SI ones, the following series of substitutions should be conducted.
Notice that these apply only to the inductions, as well as to the time derivatives of the fields; the fields themselves and the time derivatives of the inductions remain unchanged. In this way, Table 1 becomes Table 11.
Notice that, whatever the system of units, the time-domain OA tensor kernels have the dimension of velocity, LT À1 , whereas the one of the permittivity and the permeability tensor kernels is that of frequency, T À1 .
Still by applying Equation (60), one converts the two asymmetric time-domain sets of CRs, together with their redefined electromagnetic vectors, from Table 2 into their SI unit counterparts shown in Table 12.
Due to the electromagnetic uncoupling property of the asymmetric sets, electric (magnetic) CRs involve the electric (magnetic) permittivity of vacuum only. However, the electromagnetic coupling is still present into the redefined fields. Like their OA tensor kernel counterparts, the two time-domain gyration tensor kernels have the dimension of velocity. This follows from the fact that Equation (31) and (36), as well as their kernel analogues from Table 2, do not depend on the system of units used.

Frequency-Domain CRs in SI Units
The frequency-domain symmetric CRs in SI units follow from the time-derivative forms of their time-domain counterparts shown in Table 11. As explained in Section 3.1, if the temporal Fourier transform together with the additional substitution are conducted, then one obtains the expressions shown in Table 13. Equation (61) follows from Equation (40) and Maxwell's relation ε 0 μ 0 c 2 ¼ 1.
As discussed in Section 3.1, the temporal-frequency domain CRs formally coincide with their spatiotemporal-frequency domain (or plane-wave) counterparts, if all electromagnetic vectors are understood in terms of respective spatial Fourier components. Therefore, Table 13 also shows the plane-wave symmetric CRs.   It should be noted that, unlike their time-domain kernel counterparts, the frequency-domain OA tensors are dimensionless whatever the system of units used.
Similarly, the temporal frequency-domain asymmetric sets of CRs in SI units are obtained by applying the temporal Fourier transform to their time-domain counterparts from Table 12 and using the substitution (59). The resulting expressions are shown in Table 14.
Finally, the application of the spatial Fourier transform to the CRs from Table 14 yields the asymmetric CRs in the spatiotemporal frequency domain, reported in Table 15. These relations are valid for plane waves.
Notice that, like their time-domain counterparts, the two frequency-domain asymmetric sets in SI units feature electromagnetic uncoupling, albeit at the expense of coupled fields. The spatiotemporal frequency-domain (plane-wave) gyration tensors are dimensionless, like their OA tensor counterparts from the symmetric CRs.

Application of the CRs to Planar Metamaterials
As an example of the importance and the practical use of the OA CRs beyond natural crystals, we shall apply the theoretical developments from Section 4 and 5 to the case of planar metamaterial structures that have attracted considerable research interest over the past 20 years since the pioneering work of Smith et al. [70] Metamaterials are composite structures consisting of periodic arrangements of macroscopic "atoms" (called meta-atoms), whose size is large enough to be easily fabricated and controlled, whereas, at the same time, remains smaller than the wavelength of the incident radiation. [72] If the characteristic size of the metaatoms is comparable with the wavelength, a special treatment, going beyond the OA CR description, is required. [73] Due to their unique structure, metamaterials feature uncommon electromagnetic properties like negative permittivity and permeability values. [70] Furthermore, as a result of their subwavelength structure, metamaterials typically exhibit significant spatial dispersion that gives rise to important OA in the form of circular birefringence [74,75] and circular dichroism. [76][77][78] In contrast to metamaterials, in natural media such as optically active crystals, spatial dispersion arises at the atomic or molecular scale and consequently, appears as a weak effect only, as mentioned in Section 2.1.

Material Tensors for Planar Metamaterials
In the following paragraphs, we show how one can identify the nonvanishing elements and the symmetries of the three material tensors knowing the symmetry properties of the planar metamaterial structure under investigation. Figure 5 shows a typical metamaterial structure based on a split-ring resonator pattern. [79] The periodic array of split-ring resonators (or meta-atoms) is planar (or quasi-2D) as its dimension in the out-of-plane direction, that is, its thickness, is much smaller than either of its two in-plane dimensions. Furthermore, it is homogeneous, that is, structureless, along the out-of-plane direction. Such kinds of metamaterial structures are commonly referred to as achiral, [80] in contrast to 3D structures exhibiting chirality along the out-ofplane direction. The split-ring-based planar geometry is typical to a large number of more sophisticated planar arrangements, so, we shall take it as a model structure. Figure 6 shows the reference frame for the split-ring-type planar metamaterial in an oblique incidence illumination configuration.
The coordinate axes are clearly the principal axes of the split ring so that both permittivity and permeability tensors of the metamaterial are diagonal in the reference frame shown. To determine their specific form, consider the polarization P and the magnetization M of the metamaterial (in the frequency domain).
where χ e and χ m are the electric and magnetic susceptibilities, respectively. For a strictly planar structure, P must lie in the xy-plane whereas M must be normal to it, that is, be directed along the z-axis. Indeed, all microscopic (bound and free) charges of a planar medium can move only the xy-plane. More specifically, the two fields E and H induce in-plane electric dipoles (from the bound charges), resulting in in-plane electric moments, as well as in-plane Ampère's current loops (from the free charges) generating out-of-plane magnetic moments. Applying the two conditions P z ¼ 0 and M x ¼ M y ¼ 0 to Equation (62), while keeping in mind that χ e and χ m are diagonal, E and H are arbitrary, yielding χ e z ¼ 0 and χ m x ¼ χ m y ¼ 0. Finally, we get for the permittivity and permeability tensors Notice that if the metastructure is not strictly planar, that is, if its thickness is not much smaller than its lateral dimensions, then the unit-valued elements along the diagonals of ε and μ would be slightly different from the unit. [76] To determine the OA tensor of a planar structure, the same considerations are applied to the OA-induced polarization P OA and magnetization M OA from the general OA CRs (9) (transposed to the frequency domain). These can be expressed with the help of Post's CRs from Table 3 P As a result of the in-plane nature of P OA and the out-of-plane one of M OA , the last row of the OA tensor f must vanish, that is, In contrast, any planar structure necessarily exhibits m (or C s ) point symmetry as the xy plane is a symmetry plane of the structure; consequently, its OA tensor is of the specific form shown in Table 10 (with τ ¼ f here) with only four nonvanishing elements (f xz , f yz , f zx , and f zy ). When the m-class tensor form (see Table 10) is combined with the condition established right before, one gets that f xz and f yz are the only two nonzero elements of f. Finally, for the specific case of the split-ring resonator planar structure with x and y being its principal axes (see Figure 6), one further has f yz ¼ 0, as the yz plane is a symmetry plane of the split ring. Indeed, mirror reflection m yz ¼ diagðÀ1 1 1Þ by the yz plane of the OA tensor f must change the sign of the latter, [19] that is, m yz f m T yz ¼ Àf , which imposes f yz ¼ 0. Eventually, the OA tensor of the split-ring planar metamaterial is f ¼   www.advancedsciencenews.com www.adpr-journal.com Notice that if the ring resonator was closed instead of being split, one would have f xz ¼ 0 as then the xz-plane would be a symmetry plane too; consequently, a closed-ring metamaterial is optically inactive. Conversely, if the split ring was asymmetric with respect to the yz-plane, then both f xz and f yz would be nonvanishing; this is the general case for an achiral planar metamaterial. Note that the planar structure being essentially achiral, that is, nonenantiomorphous, its OA tensor is necessarily traceless, that is, trðf Þ ¼ 0. In accordance with the tensor decomposition given by Equation (59) and its physical interpretation from Section 5.2, achiral planar metamaterials do not exhibit molecular but rather only structural OA (together with LE).
Although derived within the framework of DrudeÀBoys' CRs, the specific form for f given by Equation (64) is likewise valid for the OA tensors (f 0 , α, and α 0 ) of the remaining three symmetric OA CRs, as readily follows from the conversion relations between OA tensors reported in the last column of Table 3. Most generally, the symmetry properties of the OA tensors do not depend on the set of (symmetric) OA CRs used, as pointed out in Section 5.1. Eventually, the gyration tensor g from the asymmetric set of BornÀLandau's CRs is given by Equation (31).
The nonvanishing elements of the three material tensors, ε, μ, and f of the planar metamaterial were determined with respect to the reference frame defined by the principal axes common to both ε and μ. Their expressions in the laboratory reference frame, necessary for the calculation of the optical response of the structure, are obtained through respective rotation transformations. Thus, if the incidence plane, that is, the plane containing the wavevector of the probing light and the z-axis, makes the angle φ with the xz-plane, as shown in Figure 6, then all three tensors must be rotated through φ about the z-axis. It is straightforward to see that the permeability tensor is invariant under such rotation (i.e., μ φ ¼ μ), whereas the permittivity, OA, and gyration tensors become, respectively ε xx cos 2 φ þ ε yy sin 2 φ ðε xx À ε yy Þ sin φ cos φ 0 ðε xx À ε yy Þ sin φ cos φ ε yy cos 2 φ þ ε xx sin 2 φ 0 0 0 1 and g φ ¼ Tensors ε φ , μ φ ¼ μ, and f φ completely determine the optical response of the planar metamaterial. They can be likewise derived using microscopic models for the metamaterial as done by Marqués et al. [81] From crystal optics viewpoint, the metamaterial behaves as an optically active biaxially anisotropic medium with magnetic structure. It is well known that the manifestation of OA in optically active anisotropic crystals depends on the propagation direction of the probing light. [16,19] In particular, to determine the measurement configurations in which the planar metamaterial appears as optically inactive in bulk propagation, that is, when neglecting the effect of both entrance and exit interfaces, one does not necessarily need to compute its optical response but one can rather make simple use of BornÀLandau's CRs in the frequency domain. [19] Indeed, the electric CR from Table 8 contains the product g m of the gyration tensor g and the refraction vector m ¼ n u. Whenever this product vanishes, there is no OA effect for light with refraction vector m propagating through the medium. For a planar metamaterial whose gyration tensor is given by Equation (66), one gets g m ¼ 0 if m ¼ ½0 n y n z T , that is, for light propagating in the yzplane. Consequently, when the incidence plane coincides with the symmetry plane (see Figure 6) of the split-ring resonator, the metamaterial appears as optically inactive (inasmuch as bulk propagation is concerned; there may be still weak OA effects due to the entrance and exit interfaces). More generally, for an arbitrary planar metamaterial whose both gyration tensor elements g zx and g zy are nonzero, the equality g m ¼ 0 holds when m ¼ ½0 0 n z T , that is, achiral planar metamaterials do not exhibit OA at normal incidence. [82]

Optical Response of Planar Metamaterials
To compute the optical response of the planar split-ring array metamaterial, use can be made of either Yeh's or Berreman's method, as explained in Section 4. Being more explicit and physically appealing, Yeh's approach will be overviewed first. Take the incidence plane as the xz plane of the laboratory reference frame so that the refraction vector of the probing light is m ¼ ½n 0 sin θ 0 n z T , where θ is the incidence angle (see Figure 6), n 0 is the refractive index of the ambient, and n z is the unknown z-component of the refractive index of the medium. Next, substitute m (and the antisymmetric tensor m Â dual to it), together with the "rotated" material tensors, from Equation (63b), (67), and (69), into Equation (50), to get Fresnel's equation for the planar metamaterial.
The values of the material tensors are ε xx ¼ 2.0, ε yy ¼ 2.2, μ zz ¼ 1.5, and α 0 xz ¼ 0.5. (In this, as well as in the following two simulations, the numerical values of the tensor elements were chosen for illustrative purposes only, i.e., they are not representative of a real metamaterial structure. A real metamaterial may be absorbing in which case the elements of the permittivity and the permeability tensors are complex rather real valued.) As the OA tensor is nonzero and real-valued, the planar metamaterial exhibits circular birefringence, as evidenced by the differential-phase plot. When the incidence plane coincides with the symmetry plane of the split-ring structure, that is, for φ ¼ 90 o , the planar metamaterial appears as optically inactive, in full agreement with the gyration tensor-based analysis at the end of the previous section. In particular, at normal incidence, achiral planar metamaterials do not exhibit OA, as already mentioned. The phase difference changes its sign when changing the illumination side with respect to the split-ring symmetry plane, a feature that has likewise been confirmed experimentally. [80,82] Its intensity increases with the incidence angle and reaches its maximum at grazing incidence when the incidence plane is normal to the split-ring symmetry plane. The magnitude difference plot exhibits the same trends; however, its intensity is much lower than that of the phase-difference one. Actually, the magnitude difference vanishes for bulk propagation through an OA medium; its weak nonzero values in the present case are due to the effect of the entrance and exit interfaces of the planar metamaterial. Figure 8 shows the polar plots of the same two quantities but for a purely imaginary OA tensor, α 0 xz ¼ i, physically corresponding to the presence of circular dichroism in the planar metamaterial. Now, the roles of the magnitude and phase differences are inversed in comparison with Figure 7: the first one exhibits a high-intensity image, whereas the second one is close to zero, with residual values due to the interface effects. Both show the same symmetry features as in Figure 7. In particular, the circular dichroism vanishes when the incidence plane coincides with the symmetry plane of the split-ring structure. These observations, confirmed experimentally on asymmetrically split-ring metastructures, [85] are in full agreement with the well-known fact that magnitude difference and the phase difference are sensitive to circular dichroism and circular birefringence, respectively. [80] Finally, when the OA tensor is complex valued, α 0 xz ¼ 0.5 þ 1.0 i, that is, when both circular birefringence and dichroism are present, like in the majority of metal-based  www.advancedsciencenews.com www.adpr-journal.com planar metamaterials, [17,76,77,80,82] both magnitude-and phasedifference polar plots exhibit nonvanishing intensities, as shown in Figure 9.
Like in the previous two cases, the magnitude and phase differences vanish along the diameter φ ¼ 90 o and change their signs upon crossing it. This behavior is characteristic of the split-ring-based planar metamaterial, as already mentioned. Most generally, planar metamaterials are essentially nonenantiomorphous so that they appear as optically inactive at normal incidence, as already mentioned. However, they exhibit structural OA, sometimes qualified as "extrinsic" OA, [82] in contrast to "intrinsic," enantiomorphism-based OA, when probed at oblique incidence. Clearly, the term "extrinsic" applies to the manifestation of OA under particular illumination conditions (oblique incidence) rather to the phenomenon itself; we have seen in Section 5 that enantiomorphism (or "intrinsic" OA) is not a prerequisite for an optically active medium or structure. In the special case of the split-ring metamaterial the "extrinsic" OA vanishes not only at normal incidence but also when the incidence plane coincides with the symmetry plane of the split ring, as discussed.

Conclusion
The formal phenomenological description of OA requires the addition of inhomogeneous, polarization and magnetization, terms to the conventional set of linear homogeneous electric and magnetic CRs describing anisotropic (transparent or absorbing) media. One finally obtains four possible symmetric sets of OA CRs relating pairs of electric and magnetic fields and inductions. If the electric or the magnetic field is redefined so at to make the inhomogeneous polarization or magnetization term vanish, one then obtains the asymmetric counterparts of two of the four symmetric sets of OA CRs.
All four symmetric sets of OA CRs, as well as the two asymmetric ones, can be used in either Berreman's or Yeh's formalisms to compute the polarimetric response of stratified structures containing optically active layers. Permittivity, permeability, and OA tensors belonging to a given set in the frequency domain are readily convertible into those of any other set. Thus, in the common case of weak OA typical of natural media, all sets can be used indifferently. However, in the specific case of strong OA potentially present in artificial structures, the DrudeÀBoys' symmetric (as well as BornÀLandau's asymmetric) time-domain sets (as well as their respective frequency-domain counterparts) are the only ones to warrant both real refractive indices for transparent optically active media and the availability of an underlying physical picture.
The application of an algebraic decomposition to the 11 OA tensors of the 18 optically active crystal classes allows one to establish a correspondence between the point symmetriesenantiomorphous, polar, both of these, or noncentrosymmetric, that is, none of these-exhibited by the crystal classes on one side and the two physical manifestations of OA, namely, the RP and the LE, on the other side. Although being by far the most common one, RP is not the unique OA effect. Moreover, the existence of enantiomorphism is not necessary for the presence of OA, as illustrated by the case of planar optically active metamaterials that is a fundamental part of modern photonics.
We believe this Review to be of particular interest to the physical optics community and especially, to researchers working not only in classical anisotropic crystal optics but also in modern metamaterial science, chemistry, and photonics and dealing with optically active structures and media.

Appendix B
The starting point for LandauÀLishitz's approximation [19] is the eigenequation for the redefined electric induction D 0 . To obtain it, the electric field E, related to D 0 through E ¼ ε $ ðmÞ À1 D 0 (see Table 8), is substituted in its own eigenequation (49).
Notice that if the two tensors ε and g commute, then ðε gÞ s ¼ ε g s ¼ g s ε, that is, only the symmetric part g s of the gyration tensor g is effective, even though g may not be symmetric. Table 10 shows that ε and g commute for most optically active crystal classes (as well as for isotropic media, obviously) except for the four lowest symmetry biaxial classes: 1 (C 1 ), 2 (C 2 ), m (C s ), and mm2 (C 2v ). Therefore, for higher-symmetry optically active crystal classes, the symmetry of ε g entails that of g itself.
To obtain Voigt's approximation, [11,31] write Fresnel's equation given by Equation (A1) for nonmagnetic (μ 0 ¼ I) optically active media as n 4 þ n 2 trðu Â ε À1 u Â Þ detðεÞ ðu ε uÞ þ detðεÞ ðu ε uÞ ¼ n 2 ðg uÞ ε ðg uÞ À n 4 ðu Â g uÞ 2 ðu ε uÞ (B5) The right-hand side of Equation (B5) expresses the OA contribution to Fresnel's equation. If one sets g ¼ 0 in it, then one obtains Fresnel's equation for the equivalent optically inactive anisotropic medium, providing the refractive indices n 1 and n 2 . If, with g 6 ¼ 0, one sets in the right-hand side n % n m and ε % n 2 m I, where n m ¼ 1 2 ðn 1 þ n 2 Þ is the refractive index of the equivalent optically inactive isotropic medium, then Equation (B5) can be cast into the compact form ðn 2 À n 2 1 Þðn 2 À n 2 2 Þ % G 2 (B6) in which the parameter G equals (To simplify the expression for G, use has been made of Lagrange's identity a 2 b 2 ¼ ða Â bÞ 2 þ ðabÞ 2 for vectors a ¼ u and b ¼ g u, as well as of u 2 ¼ 1. ) G is the so-called gyration parameter. [19,31,86,87] Being proportional to the quadratic form u g u, it depends only on the symmetric part of the gyration tensor g (as u g u ¼ u g s u, where g s is the symmetric part of g).
Clearly, Voigt's approximation amounts to neglecting terms containing the product OA Â anisotropy in nonmagnetic optically active media, as follows from the two substitutions n % n m and ε % n 2 m I into the right-hand side of Equation (B5). As a consequence, the gyration tensor g can be considered symmetric in the (approximate) Fresnel's equation for the refractive indices. Unlike the bicubic equation resulting from LandauÀLifshitz's approximation, Fresnel's equation in Voigt's approximation is biquadratic (like its exact counterpart).