Gyrotropy and permittivity sensing driven by toroidal response

Using a quasi-static model of a toroidal metamaterial we demonstrate analytically that simultaneous excitation of the magnetic and toroidal dipoles in an array of subwavelength toroidal solenoids results in gyrotropic behaviour resembling conventional optical activity. We derive the polarization eigenstates of this uniaxial chiral toroidal metamaterial and show that such a medium is reciprocal, while the eigenstates are represented by two counter-rotating ellipses, one of which can be used for probing changes of the host permittivity in a manner exclusive to the toroidal metamaterial. We also show that the mechanism of permittivity sensing involving resonant toroidal response is fundamentally different from that, which has been exploited so far under the term ‘toroidal’.


Introduction
Anapoles have been introduced by Zel'dovich in 1957 [1]. He showed that parity symmetry violation implies the existence of a third vector characteristic of spin-1/ 2 particles, distinctly different from their electric and magnetic dipole moments. Zel'dovich pointed out that the anapole term is not present in the standard multipole expansion and cannot be presented in the form of a superposition of known multipole terms. He named this new dipole characteristic anapole, ('zero-pole' or 'without poles') and identified the qualitative similarities between the properties of an anapole and poloidal currents supported by a toroidal solenoid. The first experimental observation of a nuclear anapole was reported in 1997 [2]. It should be noted, however, that the anapole and the toroidal dipole moment are different objects according to the accepted present-day terminology [3].
The electromagnetic properties of toroidal current configurations have been studied extensively in the past and a number of intriguing results have been obtained theoretically. In particular, the standard multipole expansion has been generalized to include the contributions of the missing toroidal terms [3][4][5][6][7]. It has been shown that toroidal solenoids could generate static electric fields in the absence of charge separation [8]. In addition, non-reciprocal electromagnetic behaviour has been studied theoretically [9] and the possibility of constructing classical non-radiating sources (electrodynamic anapoles) has been identified [10,11]. The existence of a Lorentz-like force acting on photons has been predicted [12]. Ab initio simulations have suggested the presence of toroidal moments in certain chemical compounds [13]. Until recently, there has been little experimental work on toroidal multipoles and their properties outside the field of nuclear and particle physics [14]. This is because the effects related to macroscopic toroidization (toroidal response) in naturally occurring materials are weak compared to those resulting from magnetization and polarization (electric and magnetic dipole response). In this situation the presence of bulk toroidization is difficult to detect experimentally [15][16][17].
Metamaterials, however, offer a solution to this problem. Their elementary structural elements, metamolecules, can be designed in a way that favours the excitation of toroidal moments and (at the same time) suppresses the generation of unwanted, lower-order standard multipoles. This idea was conceived by the authors of [18] and, thus, the concept of a toroidal metamaterial was introduced for the first time. A practical way of identifying the presence of specific multipole excitations was described in [19]. The first successful experimental observation of a toroidal dipole interacting with an electromagnetic radiation was reported in 2010 for a metamolecule featuring a circular arrangement of split-ring resonators [20]. Since then, toroidal metamaterials and their properties have become an active research field [21][22][23][24][25][26][27][28][29][30]. An interesting hypothesis suggesting that dark matter is an ensemble of anapoles has been formulated [31]. At the same time an experimental investigation of an electromagnetic anapole metamaterial, a composite structure that does not interact with and is, therefore, transparent to electromagnetic waves, has been reported independently [32]. Remarkably, the anapole mode has been observed in dielectric nanoparticles as well [33].
Detecting the excitation of toroidal multipoles in a metamaterial relies on the computation of the current distribution induced in its metamolecules. This distribution is in turn used to compute the frequency dependence of the microscopic scattering powers of the lowest-order multipoles [19], including the toroidal contribution, according to the formulae derived earlier [5][6][7]. Multipole expansion technique has been applied to calculate the transmission and reflection coefficients in a metamaterial in which toroidal response plays a significant role [29]. The property of poloidal currents to generate fields which are only weakly coupled to radiation has been recently employed to create a metamaterial-based laser [27]. Important steps towards development of applications based on toroidal moments have been taken recently [34][35][36][37]. These rely upon the properties of toroidal metasurfaces [34,35] and electromagnetically induced transparency assisted by toroidal resonances [36,37].
The motivation for this work is two-fold. On the one hand, we want to address the question of reciprocity of chiral toroidal media. Earlier work [9] suggests that the validity of the reciprocity lemma may not be as universal as previously thought [12]. Predicts non-reciprocal behaviour in a toroidal medium consisting of static toroidal moments in the form of an effective Lorentz force acting on photons.
In the present manuscript the reciprocity question has been approached from a different angle. We consider a toroidal chiral medium in the form of an array of coupled dynamic toroidal dipole and magnetic dipole moments. The reciprocity of this metamaterial is determined by the mathematical properties of its eigenpolarization matrix and the latter can be obtained by following a standard, textbook procedure [38]. Indeed, conventional chiral metamolecules in the form of coupled electric and/or magnetic dipole moments are known to be reciprocal, but their toroidal counterparts (introduced in [19]), need to be analysed separately.
The meaning of the term 'reciprocity' in this case is easy to understand in the context of polarization rotation that can be observed in chiral media (polymers, sugar solutions, bio-substances) and magnetised plasmas, ferrites (Faraday rotation). Both types of media are capable of rotating the polarization plane of an incident linearly polarised wave. There is an important difference between them, however.
Consider a slab of chiral medium that rotates the polarization of an incident wave by a certain angle. If a mirror is placed at the exit face of the slab the reflected wave will traverse the slab in the opposite direction and leave at the input face. The polarization state of the reflected wave leaving the slab at the input face will be identical to that of the incident wave as reversing the propagation direction in chiral media reverses the sense of polarization rotation and zero net polarization rotation will result from the round trip. This is known as reciprocal behaviour. If the chiral media is replaced with a material exhibiting Faraday rotation the polarization state of the reflected wave will be rotated to twice the rotation angle resulting from a single pass. This is a non-reciprocal behaviour as reversing the propagation direction does not reverse the sense of polarization rotation.
The first step in solving the problem is to derive the eigenpolarization matrix of the medium. This is achieved by transforming the Maxwell's equations to the so-called kDB system, where k is the wavevector and D and B are the displacement vector and the magnetic field vector in the Maxwell's equations (see below). The eigenvalues and the eigenvectors of the matrix are determined next as they describe the wavenumbers and the polarization eigenstates of the metamaterial, respectively. The reciprocity properties of the medium are then determined by the off-diagonal components of the eigenpolarization matrix: if those components change their sign upon reversal of the propagation direction then the medium is non-reciprocal. As the off-diagonal components of the eigenpolarization matrix corresponding to the medium considered here do not change their signs under the described transformation we conclude that this metamaterial is reciprocal.
The second aim of our work is related to the problem of sensing with toroidal metamaterials. Indeed, this is not a new concept and the possibility of using a toroidal dipole moment as a permittivity sensor was identified some time ago [11]. However, the definition of 'toroidal metamaterial sensing' adopted here and the underlying physics are different from what has been considered so far. We illustrate this with both our analytical and simulation results.
The permittivity sensing mechanism we propose does not involve permittivity variation in the capacitive gap of the metamolecule whereas the mechanisms proposed by others, e.g. [29,39] is based on precisely that. This has some important consequences. A sensing mechanism that relies on placing the analyte in a capacitive gap can be implemented with e.g. split-ring resonator arrays (magnetic dipoles) and has little to do with toroidal response. Such a mechanism amounts to changing the capacitance in a LC-resonator circuit. In the latter case it is straightforward to show from the Thomson formula that the sensitivity of this method is directly proportional to the resonant frequency itself. Therefore, in a metamaterial that supports several resonant modes (e.g. [29]) one does not need the excitation of a toroidal dipole moment to obtain the highest sensitivity. Instead, the highest frequency resonance is needed.
In our sensing scheme the analyte is placed in the centre of the metamolecule where there is a strong local electric field while the permittivity in the capacitive gaps remains unaffected by the permittivity perturbation. We show that in this case the highest sensitivity is obtained with the resonance that corresponds to the excitation of a toroidal dipole and not from the highest-frequency resonance that corresponds to a magnetic dipole excitation. We also relate this feature to the explicit presence of the ambient permittivity in the denominator of the dispersion equation of the eigenstate that interacts with the toroidal medium. The contrast between these two sensing mechanisms-permittivity variation in a capacitive gap and permittivity variation in the centre of a toroidal dipole moment-has not been described in the literature so far. Understanding the difference between the two mechanisms is important for applications such as biosensors. It is outside the scope of this work, however, to determine which of the two mechanisms (or a combination of both) is the optimal one and instead we leave this question to biophysics and medical physics experts.

Toroidal solenoid in an external field
Toroidal solenoids exist in two enantiomeric forms, as shown in figure 1. The current I flowing along the wire coil has two components: the azimuthal I φ directed along the parallels of the torus and the poloidal component I P along the meridians of the toroidal surface. These current components are responsible for the excitation of the magnetic and toroidal dipole modes in the structure, respectively.
Depending on how the coil is wound, the two dipole moments point in either the same (figure 1(a)) or opposite (figure 1(b)) directions. In what follows below the two solenoids will be referred to as Type I or Type II solenoids, respectively.
Assuming that the current strength does not vary along the wire (quasi-static regime) the expressions for the magnetic and toroidal dipole moments are and respectively. In equations (1) and (2) r is the radius-vector of a particular point along the wire and the integration is performed along the closed wire loop. Performing the integrations yields and where N > 1 is the number of windings in the coil and n is vector of unit length pointing along the axis of the torus (figure 1). Note that the magnitude of the magnetic dipole moment does not depend on the number of windings and this property is the reason why toroidal shapes are preferred in applications such as inductors and transformers. Indeed, high magnetic flux in the core (∝ N) and high value of the inductance ( ∝ N 2 ) can be both achieved by increasing N. At the same time the leading-order contributor to the stray field existing outside the solenoid (and to the radiation losses, at higher frequencies) is the magnetic dipole moment. As equation (3) shows, its magnitude remains unaffected by the increase of N and so does the stray magnetic field.
Consider now a toroidal solenoid in an external electromagnetic field (see figure 2). The interaction energy W between the solenoid and the field has the form:  where B is the flux density of the external field. Note that under the quasi-static approximation assumed here the current strength does not vary along the wire and, hence, there is no charge accumulation. As a consequence, the electric multipoles are all zero. In addition, the magnetic quadrupole moment of a toroidal solenoid with homogeneous current is identically zero, as shown earlier [18]. The expressions for the interaction energy of a charge-current distribution with an external field in both the general and static cases, and in the presence of toroidal moments have been derived by Dubovik and Cheshkov some time ago [4].
The interaction energy of a wire coil carrying homogeneous current with an external magnetic field is where Φ is the flux through the coil. Equations (1)-(6) allow the flux to be written as In equation (7) and τ = 0.5πR 2 Nd. In the subsequent derivation the upper and lower signs in the expressions are assumed to apply to Type I and Type II solenoids, respectively. It has been further assumed that a gap in the wire coil (or an external capacitor C) is present in each solenoid (see figure 2). The only role of this capacitor is to compensate the relatively large inductance of the coil by creating a resonance.
The current I in turn can be obtained from Kirchhoff 's law where Q is the charge on the capacitor and the result is where , ω 2 0 = (LC) −1 and γ = ℜ/ L.

Toroidal metamaterial
Consider now a toroidal metamaterial-a periodic array of the above-described oriented solenoids, which can be realised experimentally using an approach similar to the one reported in [19]. The currents flowing in the windings of the solenoids give rise to a macroscopic magnetization M, such that The vectors M T and M m are the toroidal and the magnetic dipole components of the macroscopic magnetization, respectively. As shown earlier, the toroidal component M T can be obtained from the macroscopic toroidization vector Θ = κT = ±κτ In, where κ is the number density of solenoids. The relationship between M T and Θ is and it is assumed that the interaction between adjacent metamolecules, i.e. solenoids, is negligible (low-density approximation). Assuming that the electromagnetic field is time-harmonic, exp (−iωt + ik · r), and using equations (9) and (11) allow M T to be obtained in the form In equation (12) k is the wave vector, the quantity l = τ /S m is the effective magneto-toroidal length of the structure and f (ω) = f 1 (ω) c 2 /ω 2 T . The effective toroidal frequency ω 2 T = Lc 2 ( µ 0 κτ 2 ) −1 has been introduced earlier [18] and it is a measure of the strength of the response of the toroidal metamaterial to the external electromagnetic field. Note that ω T does not depend on the number of windings in each solenoid N and that for a fixed value of the unit cell size, 2(R + d) (assuming adjacent solenoids are in contact with each other), a fundamental lower limit ω T0 exists such that ω T ⩾ ω T0 . Since ω T0 d/c ∼ 3 the validity of the quasi-static assumption implies ω ≪ ω T0 [18].
The magnetic dipole component of the magnetization is simply M m = κS m In and after some manipulation can be written as Equations (10), (12) and (13) finally give Maxwell's equations can be used to express curl H through the displacement current in a straightforward way and the result is As equation (15) shows, the response of a toroidal metamaterial to an external electromagnetic field is magneto-electric in nature as the macroscopic magnetization is driven by both the electric and the magnetic field components that are parallel to the optic axis n. This feature is somewhat similar to the magnetoelectric effect in condensed matter physics [40]. The magnetization vector itself has two components: one is parallel and the other is perpendicular to the optic axis. In addition, the toroidal component of the magnetization depends on the permittivity of the host material ε. In other words, the magnetic response of the toroidal metamaterial is effectively controlled by its background permittivity. As will be shown below, this results in an intriguing dependence of the dispersion equations of the polarization eigenmodes on ε, a feature unique to toroidal metamaterials, and suggests a possible application for the metamaterial as a permittivity sensor.
It is straightforward to show that in the presence of a microscopic toroidization Faraday's law can be written as and the role of the electric field is effectively played by the quantity Σ = E + ∂Θ ∂t [3]. A direct consequence of this result is that Σ || (the component of Σ parallel to the interface) is continuous across the interface between two materials, and not E || as is the case in ordinary dielectrics and magnetics.

Polarization eigenstates
In what follows below, the polarization eigenstates of a bulk toroidal metamaterial will be looked at into somewhat more detail. Following [38] the kDB co-ordinate system is first introduced. The unit vectors of this system are where θ is the angle between the vectors k and n. As can be verified, the triplet {e 1 , e 2 , e 3 } is a right-handed one and e i . e j = δ ij , where δ ij is the Kronecker delta symbol. The kDB co-ordinate system can be understood as follows. As equation (16) show the third unit vector e 3 is directed along the wavevector k. The latter is perpendicular to both the displacement vector D and the magnetic field vector B in accordance with the Maxwell's div-equations. Therefore, the wavevector is perpendicular to the DB-plane itself and, hence, the other two unit vectors e 1 and e 2 must be in the DB-plane, as they are in turn perpendicular to e 3 . It should be noted, however, that D and B are not necessarily perpendicular to each other in the general case.
The magnetization vector M given by equation (14) is then transformed to the kDB system and the result is where H i = H.e i are the magnetic field components and the vector quantity A (ω, θ) is In the kDB system Maxwell's equation div B = 0 reduces to B 3 = 0, or, equivalently, H 3 + M 3 = 0. The latter allows one to obtain H 3 from H 1 and H 2 Substituting equation (19) into equation (17) yields The wave equation written in its standard form is Note that since H 3 + M 3 = 0 the third component of equation (21) is satisfied automatically in the kDB system. Substituting equation (20) into the first and second component of equation (21) yields It should be noted that the off-diagonal terms of the matrix of equation (22) do not change sign when θ changes from π/ 2 to −π/ 2, which corresponds to reversing the propagation direction. The same is true for the eigenpolarization matrix of the so-called chiral (optically active) media: polymers, sugar solutions, biological macromolecules and proteins. In contrast, the off-diagonal elements of the polarization matrix of non-reciprocal magnetized plasmas and ferrites do change sign under reversing the wave propagation direction. Thus, the toroidal medium defined through its constitutive relationship (14) is reciprocal, as expected in the absence of magnetic field [38,41]. The obtained result, however, does in no way support or disprove earlier conclusions regarding the reciprocity properties of toroidal media (see e.g. [9] and [12]) [9] exploits conditions, which are clearly outside the validity of the quasi-static regime, while [12] considers a medium with permanent toroidization, which is not the case here. Formulating the general conditions under which a toroidal medium is reciprocal/non-reciprocal lies outside the scope of this work.
The dispersion equation of the polarization eigenstates propagating in the medium can be obtained by setting the determinant of the system (22) to zero. The result, after some manipulation, is ( The wavenumber k O of the first eigenstate (ordinary wave) is Apparently, this eigenstate does not interact with the toroidal medium and, propagates without attenuation provided absorption in the host dielectric is negligible [42]. Its polarization can be obtained by substituting equation (24) into either equation (22a) or (22b). The result is which corresponds to an elliptically polarized wave. Interestingly, its ellipticity is uniquely determined by the permittivity of the host medium. This eigenstate does not interact with the metamaterial because the e.m.f. induced by the toroidal and the magnetic dipole contributions cancel each other out, as can be verified by substituting equation (25) into equation (9). The second solution k E to equation (23) (extraordinary wave) is The polarization of the second eigenstate is given by which also corresponds to elliptical polarization but rotating in the opposite direction with respect to the first eigenstate. The ellipses of the two eigenstates are orthogonal to each other in the absence of losses.
Equations (24) and (25) suggest a possible application for the toroidal medium as a polarizer, similar in properties to a tourmaline crystal. Indeed, a wave with an arbitrary polarization entering the toroidal medium will excite the two polarization states (25) and (27). Since the second state is attenuated and the first one is not, a sufficiently thick metamaterial slab will transmit only the first polarization state thus operating as a polarizer and generating a polarization state given by equation (25) from an arbitrary input.

Permittivity sensing via polarization eigenstates
Given that the parameters of the polarization ellipse of the first eigenstate depend upon the permittivity of the host medium, one can detect changes in the latter by simply analysing changes in the polarization state of the wave exiting a thick slab of the toroidal metamaterial. While such a sensing mechanism is not uncommon and can be found in other types of artificial anisotropic medium such as, for example, the spiral medium [42], the toroidal response enables yet another, unique way of detecting permittivity changes via the medium's polarisation eigenstates.
This new mechanism exploits changes in the propagation characteristics of the second eigenstate and comes into play in thin slabs, where the first eigenstate emerges virtually unperturbed. In this case equation (26) can be used to calculate the ratio between the relative variation of the permittivity of the host material ∆ε/ ε and the resulting relative variation of the wavenumber of the second eigenstate (the extraordinary wave), ∆k E /k E . The result is The quantity G (ω, θ) is a measure of the sensitivity of the dispersion equation (26) to changes of the host permittivity. It is clear that in the absence of toroidal response G (ω, θ) = 1. This is easy to see by recalling that both l 2 and f (ω) are proportional to τ 2 and, hence G (ω, θ) → 1 in the limit τ → 0.
Equation (28) is plotted in figure 3. It can be seen that in the presence of toroidal response the sensitivity exhibits a resonant-like behaviour as a function of frequency and can become larger than one ( figure 3(a)). Moreover, large variations of the wavenumber could in principle result in relatively large variations of the transmission coefficient, as dictated by G ( figure 3(b)). This feature might be exploited in the construction of a permittivity sensor employed in e.g. medical testing [21].

Permittivity sensing via resonance frequency
It should be noted that the applicability of toroidal metamaterials for sensor applications has been considered in the literature (see, for example [21,29,39]). Those studies consider permittivity variation in the capacitive gaps of their respective metamolecules. In the framework of the model considered here this corresponds to variation of the resonance frequency ω 0 (see the definition of f 1 (ω) under equation (9)). In contrast, in equation (28) (and [18]) ω 0 is treated as a constant when differentiating with respect to ε. This suggests that the mechanisms described here, on the one hand, and in [21,29,39], on the other, are fundamentally different. Indeed, in the metamaterials that so far have been used to engage the toroidal dipole mode for sensing (which are two-dimensional (2D) structures) the engineered resonances are not purely of toroidal dipole type. They feature significant contributions from conventional multipoles. In particular, the toroidal dipole mode is always entangled with the magnetic quadrupole mode of a similar strength (unless special care is taken) since the current distributions resonantly induced in those metamaterials are 2D in nature [39]. Moreover, the higher sensitivity obtained for the toroidal dipole mode in [29] is a direct consequence of this mode having the highest resonant frequency among all other modes supported by the metamolecules (electric dipole, magnetic dipole, electric quadrupole etc). Indeed, in an LC oscillator (such as a split-ring resonator in a metamolecule) the variation of the resonant frequency f 0 with the permittivity ε of the material in the capacitive gap, ∆f 0 / ∆ε, is directly proportional to f 0 / ε in full accordance with the results reported in [29]. In other words, the highest-frequency resonance of a metamaterials sensor-regardless of whether it is toroidal or not-would always lead to the highest sensitivity for as long as the permittivity change occurs in the splits of the metamolecules. In contrast, equation (28) predicts a 'sensitivity' value equal to one in the absence of toroidal dipole moment and higher than one in its presence. Therefore, the toroidal dipole mode at its resonance should be more 'sensitive' to variations in the ambient permittivity than a magnetic dipole mode even if the toroidal dipole resonance is located at a frequency lower than the magnetic dipole one. Such behaviour, which does not follow the usual trend characteristic of conventional multipole resonances, is exclusive to the pure toroidal dipole mode and, therefore, can be regarded as genuine 'toroidal' sensing. Importantly, to observe this effect the permittivity variation must occur in the central section of each metamolecule, where the electric field is localised only in the case of a toroidal dipole excitation. Below, we conceptually demonstrate the proposed genuine 'toroidal' sensing by modelling numerically in Comsol Multiphysics the frequency shifts of toroidal and magnetic dipole resonances in the metamaterial that originally enabled the experimental observation of the toroidal dipole response [20]. As in the original work, we consider a metamaterial slab, which has a thickness of one metamolecule in the direction of wave propagation. The toroidal metamolecules are formed by a set of four diagonally placed square split 'rings' (see figure 4(a)) with the same dimensions as in [20]. The period of the metamaterial array is 7.5 mm in the vertical direction and 6.0 mm in the horizontal direction. The key advantage of such a metamaterial is that its electromagnetic response is well understood and it has an easy-to-implement design, which constitutes a practical simplification of the toroidal metamaterial we have discussed earlier (figure 2). In particular, the magnetic dipole and toroidal dipole resonances occur at different frequencies and this simplifies the problem. To distil the effect of toroidal sensing and further simplify its interpretation we assume that the metamolecules in our model are suspended in air rather than supported by a substrate. Such an assumption does not affect the nature of the metamaterial resonances and just shifts the resonance band from 15.2-16.5 GHz [20] to 21.5-23.5 GHz ( figure 4(b)). The latter features two reflection peaks corresponding to toroidal and magnetic dipole resonances, which in the case of a pristine metamaterial are centred at 22.0 GHz and 23.1 GHz, respectively.
The permittivity, ε, in our model varies in the range from 1 to 3 and the changes are confined in space to a cylindrical region in the centre of each metamolecule, which is as tall as the split rings and has a diameter about 10% smaller than the distance between the inner segments of the opposing rings (see figure 4(a)). We intentionally chose such a larger range of permittivity values in order to demonstrate more vividly the concept of permittivity sensing exclusive to the toroidal dipole mode. In practice, of course, a change in the permittivity does not have to be (and, typically, will not be) that large. The colour map in figure 4(c) shows how the reflection spectrum of the metamaterial evolves with increasing permittivity. While both reflectivity peaks are seen to red-shift in response to permittivity changes, the lower frequency peak (which is the toroidal dipole resonance) displays much stronger sensitivity. In particular, at ε = 3 the spectral shift of the toroidal resonance becomes as large as 0.5 GHz, which exceeds the magnitude of the shift exhibited by the higher-frequency magnetic resonance, 0.2 GHz, by more than a factor of two. The same effect is expected in transmission with the only difference being that the toroidal and magnetic dipole resonances will appear in the spectrum as dips. As we have pointed out earlier, such behaviour contrasts the usual capacitive sensing with LC resonances that has been employed so far under the term 'toroidal' , where larger spectral shifts can be achieved only by increasing the frequency of the involved resonances.
Practical implementation of the proposed sensing approach naturally requires the use of a substrate to structurally support metamolecules in the array, which will likely reduce the sensitivity of our toroidal metamaterial (as in many other schemes of dielectric sensing). For sensing with microwaves the substrate can be introduced in the form of dielectric strips encompassing the metamolecules, as it was done in [20]. The effect of the substrate can be partially mitigated by using dielectrics with small permittivity, as well as by physically removing the substrate material in the active areas of the metamaterial sensor at the fabrication stage. The latter will work particularly well for the above considered design taking advantage of the unique topology of the toroidal dipole mode. An analyte can be delivered via a dielectric tube/capillary integrated into the structure of the metamaterial, which will run through the centre of toroidal metamolecules.

Conclusions
It is shown that an array of toroidal solenoids is a reciprocal gyrotropic birefringent metamaterial. The polarization eigenstates have been derived explicitly in the quasistatic regime. Possible sensor applications are discussed and a new mechanism of permittivity sensing based on resonant toroidal response is demonstrated. It is shown that the new sensing mechanism is principally different from that exploited so far by the metamaterials community under the term 'toroidal' .

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).