1. INTRODUCTION

Uniaxial bi-anisotropic media display, in general, both linear and circular birefringence. The term “uniaxial” refers to materials, such as calcite, quartz, or certain types of liquid crystals, among others, that have a single symmetry axis, known as the optical axis, along which the dielectric constant differs from those associated with the directions perpendicular to it [1]. As a result, when a linearly polarized wave propagates through a uniaxial material off-axis, it experiences different refractive indices in different directions, leading to a phase shift that alters its polarization state. This effect is known as linear birefringence and is a hallmark of uniaxial materials. The term “bi-anisotropic” on the other hand refers to magneto-electric coupling resulting from chirality arising from the lack of mirror symmetry at the molecular scale. For the most general medium, this coupling differs between the transverse and longitudinal field-components of an electromagnetic wave [2], and it manifests as circular birefringence whereby the plane of polarization of a linearly polarized light is rotated upon its transmission, as its composite right and left circular polarizations propagate at different speeds [3].

In the case where the transverse components of the electric field are coupled with the transverse components of the magnetic field, we have the so-called transverse bi-anisotropy [4], which can be conceptualized by a uniaxial medium hosting randomly scattered chiral helices with their axes being perpendicular to the optical axis. By contrast, axial bi-anisotropy refers to the coupling between the parallel electric and magnetic field-components in which the aforementioned chiral helices have axes aligned parallel to the optical axis. Due to their unique electromagnetic properties, the problem of electromagnetic wave propagation in uniaxial bi-anisotropic media received substantial attention two decades ago, with several authors addressing either the case of axial bi-anisotropy [5,6] or transverse bi-anisotropy [7,8], or the coexistence of both [9,10] in a uniaxial bi-anisotropic medium hosting, e.g., $\Omega$-particles. This effort was not in vain as several interesting applications occurred: from polarization-control [11] and beam-steering [12] photonic devices to wearable sensors [13], negative refracting materials [14], and optical vortices [15]. The key ingredient to all these implementations is the possibility to “trip the light fantastic,” which is partially plausible due to the degree of freedom that chirality offers.

Optical activity, which manifests as circular birefringence and circular dichroism, is known to be affected by electromagnetic fields, irradiance, and temperature, but in natural media interacting with longer than their dimensions’ wavelengths, it is typically very weak. For instance, in [16], a ${{\rm Bi}_{12}}{{\rm GeO}_{20}}$ crystal has an optical rotatory power of 42°/mm at about 480 nm, whereas at the same wavelength the sugar solution ${{\rm C}_6}{{\rm H}_{12}}{{\rm O}_6}$ of [17] achieves a still lower value of 1.84°/mm, even for a concentration close to saturation. Nevertheless, giant chirality has been experimentally observed in artificial metamaterials at optical [18,19], THz [20,21], and GHz [22,23] frequencies. Apparently, THz meta-media seem to serve as preferable candidates for achieving giant chirality due to the fact that longer wavelengths render the fabrication of more sophisticated three-dimensional structures easier. Notwithstanding, the higher-order multipoles excitation technique, recently introduced in [24] for planar nanostructures, inspired several researchers to investigate metasurfaces exhibiting giant chirality in the visible spectrum (see, e.g., [25,26]).

In view of the recently demonstrated meta-media with giant chirality, which can be, moreover, controlled via externally applied electric or magnetic fields [27], or temperature [28], or via mechanical re-configurations [29], we proposed in [30] a medium that exhibits what we have christened as a “uniform Bragg phenomenon.” We showed that a uniform circularly and linearly birefringent medium achieves a “Bragg-like” response that relies on resonant tuning of the chirality parameter to the medium’s average refractive index. Such a tuning resembles the usual Bragg condition in periodically modulated Bragg gratings [31] or in structurally chiral media [32]. However, by contrast to these, there is not any fabrication requirement, and the tuning occurs by matching the parameters of the medium, rather than by matching the wavelength of the incident light to the spatial period of a periodically modulated refractive index profile or of a helicoidal structurally chiral scaffold. Hence, such a “Bragg-like” mechanism is necessarily very broadband. Apart from wavelength-independent, our suggested medium was also found to be polarization-selective, with potential applications in wavelength division multiplexing [33], optical modulation [34], ameliorated electron transport in organic semiconductors [35], enhanced efficiency of OLED displays [36], or even in photonic devices with asymmetrical transmission [37].

This intriguing medium encounters three manufacturing challenges: first and foremost, it is based on the assumption that the chirality has approximately equal magnitudes in both birefringent axes. Although not fundamentally forbidden, achieving such a behavior in a meta-medium would be a highly ambitious task. Second, the required chirality must be comparable to the medium’s average refractive index. Finally, even if it is, the two must disperse almost equally for a wavelength range broad enough so that the resonance’s fractional bandwidth outperforms those achieved via current technology. In this communication, we address these issues by demonstrating that an axially bi-anisotropic uniaxial medium does also exhibit such a response. In particular, we show that non-axial wave propagation not only relaxes the previously identified hard-to-achieve resonant condition but also offers control over the location of the resonance and the corresponding bandwidth via the inclination angle, formed between the direction of wave propagation and the optical axis. Furthermore, anomalous wave propagation is identified at a singular point, in the proximity of which a giant refractive index can be achieved, similar to that of [38] but with improved spectral characteristics, being intrinsically broadband.

The manuscript is planned as follows: in Section 2, we employ the usual Tellegen model to describe uniaxial bi-anisotropic media, and in Section 3, we analytically solve the problem of non-axial wave propagation in the considered medium, discussing also the interesting case of anomalous wave propagation. In Section 4, we focus on the case of axial bi-anisotropy and demonstrate that a parameter window, i.e., a “bandgap,” can be formed in the medium’s parameters domain rather than that of wavelength. The polarization states of the supported eigenmodes are examined and an interesting analogy to structurally chiral media is outlined. In Section 5, the predicted electromagnetic response is simulated for a finite slab of the medium, and in Section 6, the identified resonance’s quality is assessed for some particular examples of recently proposed meta-media.

2. CONSTITUTIVE RELATIONS FOR UNIAXIAL BI-ANISOTROPIC MEDIA

As phenomenologically derived in [4], the temporal frequency domain constitutive relations for a uniaxial bi-anisotropic medium, expressed via the so-called Tellegen formalism, read

As we will be solving Maxwell’s equations in propagation-coordinates, in lieu of material-coordinates, we have introduced the subscript “${o}$” to indicate tensorial components in the basis of the latter. Evidently, the form of the matrix-representation of the permittivity tensor, signature of uniaxial crystals ($\epsilon \ne {\epsilon _z}$) of the classes 32, 422, and 622 [39], is inherited to all the other tensors. Indeed, considering symmetry arguments and extending those of [40], this can be traced back to the forms of the polarizability tensors. We note, though, that this is not broadly true for all classes of uniaxial crystals (see, e.g., the forms of the gyration pseudo-tensor in Table I of [41]).

For a reciprocal medium, the cross-coupling tensors satisfy the relation ${{\boldsymbol \zeta}_{o}} = - {\boldsymbol \xi}_{o}^T$, with $T$ indicating transpose, whereas for a non-reciprocal medium in the absence of dissipation, the symmetry constraints of Eqs. (6.14a) and (6.14b) in [42], which are independent of any of the medium’s structural symmetries, dictate that ${{\boldsymbol \zeta}_{o}} = {\boldsymbol \xi}_{o}^\dagger$, where the dagger denotes Hermitian conjugation. In such case, the reality of the energy density function $({{{\textbf E}^*} \cdot {\textbf D} + {{\textbf H}^*} \cdot {\textbf B}})/2$ requires that $\xi = \chi - i\kappa$, ${\xi _z} = {\chi _z} - i{\kappa _z}$, $\zeta = {\xi ^*}$, and ${\zeta _z} = \xi _z^*$, where the asterisk denotes complex conjugation, $\kappa ,{\kappa _z} \in \mathbb{R}$ are the transverse and axial chirality parameters, respectively, and $\chi ,{\chi _z} \in \mathbb{R}$ are the Tellegen coefficients expressing non-reciprocity in the corresponding directions. Likewise, ignoring dispersion, the positive-definiteness of the energy density function leads to the condition designating the negative refraction (due to chirality) regime [43]: $|{\xi _z}| \gt {\rm Re}[{({{\epsilon _z}{\mu _z}})^{1/2}}]$ (respectively, $|\xi | \gt {\rm Re}[{({\epsilon\mu})^{1/2}}]$) for axial (respectively, transverse) bi-anisotropy. Although we will be dealing with realistic scenarios that consider absorption, whenever we make the assumption of a loss-free medium it will be explicitly mentioned.

To simplify the notation, we may define the auxiliary fields ${\textbf h} = {\eta _0}{\textbf H}$, ${\textbf b} = ({\eta _0}/{\mu _0}){\textbf B}$, and ${\textbf d} = \epsilon _0^{- 1}{\textbf D}$ that will assure that all fields are similarly dimensioned [32]. Consequently, the constitutive relations of Eq. (1) are rewritten as

Under this notation, in a source-free space and for a monochromatic excitation with an ${e^{- i\omega t}}$ harmonic oscillation, Faraday’s and Ampère–Maxwell’s macroscopic curl relations in the $({{x_{o}},{y_{o}},{z_{o}}})$ Cartesian coordinate system take the forms

3. REFRACTIVE INDICES SUPPORTED BY UNIAXIAL BI-ANISOTROPIC MEDIA

A. Analytic Solution for Non-axial Wave Propagation

Although we will begin our analysis by solving the eigenproblem of non-axial plane wave propagation in an infinite uniaxial bi-anisotropic medium, it is insightful to highlight the distinction between “material-coordinates” and “propagation-coordinates,” as this will be helpful later when considering a slab of such a medium (Section 5). Figure 1 illustrates a finite slab of the considered medium, surrounded by an isotropic dielectric and equipped with the Cartesian coordinate system $({{x_{o}},{y_{o}},{z_{o}}})$ of Section 2. For normal incidence of a plane electromagnetic wave at the front interface of the slab, it is convenient to define a Cartesian coordinate system $({x,y,z})$ in such way that the $z$-axis, with an associated unit vector ${\boldsymbol {\hat z}}$, is perpendicular to the slab’s interfaces. Hence, the wave within the complex medium will be propagating along the ${\boldsymbol {\hat z}}$-direction, forming an angle $\theta = {\rm arccos} ({{\boldsymbol {\hat z}} \cdot {{{\boldsymbol {\hat z}}}_{o}}})$ with the optical axis. In these propagation-coordinates, if all tensors of Eq. (2) are diagonalizable in the same directions (see [44]), their components will be given by the transformation [45]

 

Fig. 1. Schematic of a finite slab of a lossy uniaxial bi-anisotropic medium with base parameters $\epsilon = 5 + 0.44i$, ${\epsilon _z} = 1 + 0.088i$, and $\mu= {\mu _z} = 1$, surrounded by an isotropic dielectric characterized by ${\epsilon _I}$ and ${\mu _I}$. The material-coordinate system is delineated by the blue solid lines, whereas the propagation-coordinate system by the red dotted ones. The optical axis forms an angle $\theta \in [{0,\pi /2}]$ with the direction perpendicular to both interfaces, and the slab’s finite dimension extends between $z = 0$ and $z = L$. The wavevector ${{\textbf k}_i}$ corresponds to a normally incident plane electromagnetic wave, which will be propagating inside the medium, along the $z$-axis, at an angle $\theta$ with the optical axis. This angle shall be referred to as the “inclination angle.”

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Partitioning the permittivity, the permeability, and the magneto-electric coupling tensors as

Here, $\tilde \tau = \tau {\cos}^2 \theta + {\tau _z} {\sin}^2 \theta$ and ${\tilde {\tilde \tau}} = ({\tau - {\tau _z}})\cos \theta \sin \theta$, with $\tau = \{\epsilon ,\mu ,\xi ,\zeta \}$ and ${\tau _z} = \{{\epsilon _z},{\mu _z},{\xi _z},{\zeta _z}\}$ in this precise ordering.

We may then decompose the electric field as ${\textbf E} = {{\textbf E}_ \bot} + {E_\parallel}{\boldsymbol {\hat k}}$ and the auxiliary field of the magnetic excitation as ${\textbf h} = {{\textbf h}_ \bot} + {h_\parallel}{\boldsymbol {\hat k}}$, where ${\boldsymbol {\hat k}}$ is a unit vector in the direction of the wavevector ${\textbf k}$ being parallel to ${\boldsymbol {\hat z}}$ and perpendicular to both ${{\textbf E}_ \bot}$ and ${{\textbf h}_ \bot}$. Taking the projection of Eq. (4a) yields

Combining now Eqs. (7b) and (8b), together with Eq. (5), we obtain

For a plane wave propagating as ${\textbf E}{e^{i({kz - \omega t})}}$, substituting Eqs. (9) and (10) into Eqs. (7a) and (8a) leads to the system

As a validity check, looking at the case of a bi-isotropic medium, all the “double-tilde” variables are zero and the “hat” elements of the diagonal matrices in Eq. (12) reduce to $\epsilon ,\mu ,\xi ,\zeta$, respectively, as anticipated. Moreover, in the absence of absorption where the Hermiticity relation between the cross-coupling tensors holds, it is straightforward to verify that $\hat \zeta = {\hat \xi ^*}$.

Bringing together both rows of Eq. (11), provided that $\hat \mu \ne 0$ so that the matrix ${\boldsymbol {\hat \mu}}|$ is invertible, we can derive the vectorial Helmholtz wave equation

To obtain non-trivial solutions, we set the determinant of the characteristic matrix in Eq. (13) equal to zero, by which solving for the refractive indices $n = k/{k_0}$ yields the bi-quadratic equation

Hence, the refractive indices supported by the medium will be given by the solutions to Eq. (14):

Characteristic equations for the refractive indices, similar to Eq. (14), are signatures of numerous optical systems, such as magneto-photonic crystal layers (quartic equation in [46]) or media that combine structural chirality with magneto-optic activity (bi-quadratic equation in [47]). In these examples, the non-reciprocal nature of the media, induced by the Faraday effect, can be readily identified by direct inspection of the characteristic equation. In our case, however, we see that for a solution to Eq. (14) $n$, ${-}n$ is also a solution. This is, of course, a necessary but not sufficient condition to determine whether or not a system is reciprocal, as this would normally require implementation of Lorentz’s reciprocity theorem (see, e.g., Sec. 2.4 of [4] or Sec. II C of [48]).

B. Anomalous Wave Propagation at a Singular Point

Anomalous wave propagation in uniaxial bi-anisotropic media has been examined in the context of the so-called Dyakonov–Voigt surface waves (see, e.g., [49]), spatial degeneracies [50], and as special cases for specific sets of the medium’s parameters: for axially bi-anisotropic media in [51] and for transversely bi-anisotropic media in [8]. In particular, [8,51] showed that under corresponding preconditions, two out of the four modes with associated refractive indices given by Eq. (15) can be suppressed. It is now established that these regimes are precisely those whereby we enter the negative refraction due to chirality regime [52,53]; the curious reader is referred to the comparison of Eq. (8) of [51] or of Eq. (15) of [8] with Eq. (3) of [30] for $\alpha = \bar n$.

Since Eq. (15) was derived under the condition that ${\xi _s}{\zeta _s} \ne {\epsilon _s}{\mu _s}$, in this section, we investigate the implications of this singular condition being met. Henceforth, for the sake of simplicity, we will be concerned with the scenario in which $\mu= {\mu _z} \Rightarrow {\boldsymbol \mu} =\mu \mathbb{I}$, i.e., the considered medium’s permeability tensor is deemed isotropic. Consequently, if ${\xi _s}{\zeta _s} = {\epsilon _s}\mu$, combining Eqs. (7b) and (8b) gives

Then, we may directly express ${{\textbf h}_ \bot}$ and ${h_\parallel}$ in terms of ${{\textbf E}_ \bot}$ and ${E_\parallel}$, through Eqs. (7a) and (7b), respectively, and upon substitution into Eqs. (8a) and (8b), we find the characteristic eigensystem for the singular case:

Here, ${n_c}$ is the refractive index associated with the singular point, defined as ${n_c} = {\lim}_{{\xi _s}{\zeta _s} \to {\epsilon _s}\mu} n$, where $n$ is the refractive index given by Eq. (15) and the subscript-“$s$” parameters are the scalars of Eq. (6b). We note that under the assumption of an isotropic permeability tensor, it is trivial to express the auxiliary magnetic excitation field-components as function of ${{\textbf E}_ \bot}$ and ${E_\parallel}$, and hence derive the row “missing” from Eq. (13).

To provide some intuition, we shall proceed by distinguishing some special cases.

1. Equal Chirality Parameters in Both Birefringent Axes

For $\xi = {\xi _z}$ and $\zeta = {\zeta _z}$, the singularity condition becomes $\xi \zeta = {\epsilon _s}\mu$ (n.b., for a bi-isotropic medium, this reduces to the condition mentioned but not investigated in [54]). For general off-axis propagation inside the medium, since ${\epsilon _s}$ is a function of $\theta$ [as per Eq. (6b)], we can solve $\xi \zeta = {\epsilon _s}\mu$ for $\theta$ and obtain

Remarkably, this angle resembles the well-known critical angle after which total reflection occurs at the interface between two conceptual media with refractive indices ${n_1} = {({\epsilon\mu- {\epsilon _z}\mu})^{1/2}}$ and ${n_2} = {({\xi \zeta - {\epsilon _z}\mu})^{1/2}}$. This affinity is indubitably meaningful as long as ${\rm Re}({{n_1}}) \gt {\rm Re}({{n_2}})$, which connotes that ${\rm Re}({\xi \zeta}) \lt {\rm Re}({\epsilon \mu})$, i.e., the regular refraction regime. On the contrary, the condition ${\rm Re}({\xi \zeta}) \gt {\rm Re}({\epsilon \mu})$ leads to the negative refraction regime (see Sec. IV B of [32]). For $\xi \zeta = {\epsilon _z}\mu$ we have ${\theta _{c}} = 0$, whereas for $\xi \zeta = \epsilon \mu$ we have $\theta = \pi /2$. Equating the determinant of the matrix in Eq. (18) with zero yields

Regarding the eigenmodes, from the last row of Eq. (18) [or equivalently from Eq. (16)], we get that ${E_y} = 0$. Thence, substituting Eq. (20) into Eq. (18) leads to

If ${\theta _c} \ne \{0,\pi /2\}$, setting ${E_x} = {E_0}$, the Cartesian components of the eigenmodes are

At this singular point, the polarization is, therefore, linear while the ${E_x}$ and ${E_z}$ components are in quadrature, i.e., out of phase by $\pi /2$, which is apparent from the requirement that $\zeta$ is the complex conjugate of $\xi$. Although it appears that two out of the four modes associated with the regular regime are suppressed [cf. Eq. (15) with Eq. (20)], this is not accurate. What actually happens is that when $\xi \zeta = {\epsilon _s}\mu$, the forward- (and backward-) propagating eigenmodes are superposed to give a net linear polarization, as will become apparent in Section 4.B.

For an isotropically magnetic, reciprocal, and loss-free medium, the imaginary parts of $\hat \xi$, $\hat \zeta$ are everywhere finite, $\hat \mu= 1$, and $\xi \zeta = \kappa _c^2$. Then, the singularity condition is equivalently written as ${\kappa _c} = \pm {({{\epsilon _s}\mu})^{1/2}}$ and for, say, the positive root, as ${\kappa _c}$ approaches ${\epsilon _s}\mu$ from the right (respectively, left), $\hat \epsilon$ tends to positive (respectively, negative) infinity. This discontinuity leads to a giant modulus of the refractive indices ${\pm}{n^{(-)}}$ in the proximity of ${\kappa _c} \in ({{\kappa _c} - \delta {\kappa _c},{\kappa _c} + \delta {\kappa _c}})$, where $\delta {\kappa _c}$ is an infinitesimal. Specifically, for $\kappa \in ({{\kappa _c},{\kappa _c} + \delta {\kappa _c}})$ we have giant real parts of ${\pm}{n^{(-)}}$ while their imaginary parts are zero, whereas for $\kappa \in ({{\kappa _c} - \delta {\kappa _c},{\kappa _c}})$ the real parts are zero with the imaginary parts being now giant. This abrupt change signifies the entrance to the negative refraction regime, and in Section 4.A, it will be linked to the upper edge of the “Bragg-like” photonic bandgap. Since no wavelength appears in the resonant condition, this in principle refers to all wavelengths, predicting extraordinarily high refractive indices at a very broad wavelength range, controllable via the inclination angle $\theta$. Naturally, the maximum achievable values will be dictated by the absorption characteristics of the medium.

2. Equal Ratios of the Permittivity and Chirality Parameters

To offer a sense of understanding about the $\epsilon /{\epsilon _z} = \zeta /{\zeta _z}$ condition, i.e., for the admittance in Eq. (17) to vanish, we may consider a reciprocal loss-free medium. In such case, we have $\zeta = i\kappa$ and ${\zeta _z} = i{\kappa _z}$, and consequently, the ratio $\epsilon /{\epsilon _z} = \kappa /{\kappa _z}$ expresses the degree of the medium’s uniaxiallity. Indeed, if the medium is bi-isotropic, the relation is identically fulfilled, whereas if the medium is bi-anisotropic, the ratio gives a measure of the deviation from the isotropic case.

Turning back to the general case, Eq. (16) now gives ${h_y} = 0$, provided that $\xi \ne {\xi _z}$, which is discussed in Section 3.B.1, and that $\theta \ne \{0,\pi /2\}$, which is already covered by Section 3.A. In this example, solving ${\xi _s}{\zeta _s} = {\epsilon _s}\mu$ for the inclination angle $\theta$, under the condition that $\epsilon /{\epsilon _z} = \zeta /{\zeta _z}$, yields

Combining the first two rows of Eq. (18), we can express all components of the electric field as functions of the ${E_x}$-component, which can be anew arbitrarily chosen. Thus, we have

4. PARAMETER VARIATION IN THE CHIRALITY DOMAIN

A. Conditions for Evanescent Waves in the Chirality Domain

Considering just axial bi-anisotropy, i.e., setting $\xi = \zeta = 0$ in Eq. (2), we are left with ${\xi _z} = {\chi _z} - i{\kappa _z}$ and ${\zeta _z} = {\chi _z} + i{\kappa _z}$, where ${\chi _z},{\kappa _z} \in \mathbb{R}$. Under the ${\boldsymbol\mu} =\mu \mathbb{I}$ assumption and after some algebraic manipulations, the cumbersome Eq. (15) simplifies to

As anticipated, Eq. (23) is equivalent to Eq. (50) of [55] and Eq. (25) of [2] (n.b., there is an erroneous sign in the latter, appearing correct in the same authors’ Eq. (8.48) of [4]). Although Eq. (23) was directly derived from Eq. (15), when we enter the negative refraction regime, the direction of phase propagation and the handedness of counter-propagating modes are reversed (see App. A of [32]), and this subtlety must be considered when assigning a refractive indices branch to a specific expression. Henceforward, whenever we refer to ${\pm}{n^{(\pm)}}$, it will be that of Eq. (23).

Careful examination of Eq. (23) shows that in the ideal loss-free case, where all parameters are real, the refractive indices become purely imaginary whenever the axial chirality parameter (squared) lies in the regime

Hence, Eq. (25) gives the centers of resonance in the “${\kappa _z}$-domain,” but such evanescence could be equivalently achieved if ${\kappa _z} = 0$ in the “${\chi _z}$-domain.” Crucially, both ${\kappa _z}$ and ${\chi _z}$ control the location of the resonances, alongside the inclination angle $\theta$. The corresponding bandwidths are

Since the chirality parameter can be controlled via a plethora of methods (see Section 6), we may set ${\chi _z} = 0$ and plot the refractive indices of Eq. (23) as ${\kappa _z}$ varies. The dependence of the supported refractive indices on the (axial) chirality parameter for, say, $\theta = \pi /6$ is illustrated in Fig. 2, where the loss-tangent delta has been taken to be twice the value measured experimentally in [28], equal in both birefringent axes, and the evanescent regions have been placed within windows. The plots of Fig. 2 may be interpreted as a “chirality-domain dispersion,” where by contrast to the usual “wavelength-domain dispersion,” the wavenumbers of the supported eigenmodes, or, equivalently, their refractive indices, vary with ${\kappa _z}$ instead of $\lambda$.

 

Fig. 2. The ${+}{n^{(\pm)}}$ refractive indices of Eq. (23) as functions of the axial chirality parameter ${\kappa _z}$ for general propagation in a lossy axially bi-anisotropic uniaxial medium. The parameters are those of Fig. 1, the Tellegen coefficient is ${\chi _z} = 0$, and $\theta = \pi /6$. The peaks occur for the values of $\kappa \equiv {\kappa _c}$ corresponding to the solutions of Eq. (18) for ${\xi _z}{\zeta _z} = {\epsilon _s}{\mu _s}$, and the windows mark the two bandgaps, as per Eq. (24), where the refractive indices ${+}{n^{(-)}}$ become purely imaginary in the ideal loss-free case where all the parameters are real. The ${-}{n^{(\pm)}}$ branches are mirror-symmetrical to those depicted above with respect to the $n = 0$-axis.

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Fig. 3. Comparison of the proposed medium with the circularly and linearly birefringent medium of [30] with ${{\boldsymbol \epsilon}_ \bot} = {\rm diag}({\epsilon ,{\epsilon _z}})$: (a) axial chirality-domain center of resonance $\kappa _0^{{{{\rm res}}_ +}}$, as per Eq. (25), and (b) bandwidth $\Delta \kappa _0^{{\rm res}}$, as per Eq. (26), both as functions of the inclination angle $\theta \in [{0,\pi /2})$ for increasing values of the Tellegen coefficient ${\chi _z}$. The horizontal lines stand for the resonance and the bandwidth associated with axial propagation in the medium of [30]. The vertical line corresponds to the angle of Eq. (27), marking the upper-limit of the inclination angles’ regime whereby the identified mechanism outperforms in terms of the required chirality that of [30]. The scenario is that of Fig. 2, including absorption; hence we have taken the real part of each parameter.

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Manifestly, Eq. (25) bears a remarkable resemblance to the usual Bragg condition of conventional Bragg gratings [31] or of structurally chiral media [32]. However, in our identified mechanism, there is not any kind of coherent superposition but rather a synchronous and alternate sampling of two different refractive indices by an elliptically polarized propagating wave. With no Bragg wavelength as such in the tuning condition, this “Bragg-like” response is necessarily very broadband in the spectral domain, provided that the medium’s parameters disperse equally, since it is the differential dispersion among ${\kappa _z}$, $\epsilon$, ${\epsilon _z}$ (and ${\chi _z}$ if the medium is non-reciprocal) that actually matters.

Upon observation of Eqs. (25) and (26), we are led to the following remarks: first, it is apparent that for $\theta = 0$, i.e., for axial propagation, $\Delta \kappa _0^{{\rm res}} = 0$, which means that no bandgaps will occur. This is reasonable as the co-propagating eigenaxes will then be effectively sampling two equal refractive indices. Second, the $\theta$ dependence of $\kappa _0^{{{{\rm res}}_ \pm}}$ relaxes the previously identified resonant condition $\kappa = \pm {[{({\epsilon + {\epsilon _z}})\mu /2}]^{1/2}}$ of [30], associated with axial propagation in a medium that combines circular and linear birefringence with ${{\boldsymbol \epsilon}_ \bot} = {\rm diag}({\epsilon ,{\epsilon _z}})$. Indeed, as illustrated in Fig. 3(a), setting ${\chi _z} = 0$, the centers of resonance for $\theta \in ({0,{\theta _{\rm m}}})$, where

B. Polarization States of the Eigenmodes

Substituting the refractive indices of Eq. (23) into Eq. (13) and solving for ${E_x}/{E_y}$ yields the general expression

 

Fig. 4. Polarization states of the supported eigenmodes by a lossy axially bi-anisotropic uniaxial medium: (a) polarization states of the eigenmodes associated with the refractive indices branch ${+}{n^{(+)}}$ and (b) those associated with the ${+}{n^{(-)}}$ branch, both inside and outside the ${\kappa _z} \gt 0$ axial chirality-domain bandgap. The scenario is that of Fig. 2, and the superscript signs “${+}$” (respectively, “${-}$”) indicate values of ${\kappa _z}$ being slightly greater (respectively, less) than the arguments in the parentheses. For ${+}{n^{(+)}}$, we have left-handed elliptically polarized modes whose handedness reverses when the sign of ${\kappa _z}$ changes. For ${+}{n^{(-)}}$ and inside the bandgap, the modes become linearly polarized due to the superposition of left- and right-handed, forwardly and backwardly propagating elliptically polarized waves.

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Accordingly, Fig. 4(b) illustrates the polarization states of the eigenmodes associated with the refractive indices branch ${+}{n^{(-)}}$, where it is noticeable that inside the bandgap, the polarizations are linear. This is due to the fact that when the axial chirality ${\kappa _z}$ exceeds its resonant value, the waves are evanescent and the forward-propagating mode must match the backward-propagating one, with the two having opposite-handed polarizations adding up to produce a net linear polarization. Such an interference of elliptically polarized modes cannot be achieved if the medium is non-reciprocal, since the superpositioned counter-propagating waves will then be having unequal amplitudes, as implied by scrutinizing Eq. (28) for ${\chi _z} \ne 0$.

The apparent similarity between the circular Bragg phenomenon identified here and that of structurally chiral media, such as sculptured thin films or cholesteric liquid crystals, warrants further exploration. In a right- (respectively, left-) handed structurally chiral medium with a helicity $h = + 1$ (respectively, $h = - 1$), characterized by an average refractive index $\bar n$ and a helical pitch ${L\!_p}$, incident right- (respectively, left-) circularly polarized light will be strongly backscattered when the Bragg condition $\lambda _0^{{\rm Br}} = \bar n{L\!_p}$ is met. At this wavelength, the opposite-handed polarization will be highly transmitted, depending though on absorption. To make the analogy to our example unambiguous, we will assume that ${\chi _z} = 0$ and that $\theta = {\theta _{\rm m}}$ so that the resonant condition of Eq. (25) can be approximated as $\kappa _0^{{{{\rm res}}_ \pm}} \approx \pm \bar n$, where $\bar n = {[{({\epsilon\mu+ {\epsilon _z}\mu})/2}]^{1/2}}$. Note that if $\theta \in ({0,\pi /2}) - \{{\theta _{ m}}\}$, we can always choose an appropriate $\epsilon = \epsilon ({\epsilon ,{\epsilon _z},\theta})$. We may then postulate that the axial chirality parameter can be virtually mapped to the ratio between an arbitrarily chosen wavelength and a helical pitch, multiplied by a factor ${-}h$, i.e., ${\kappa _z} \to - h({\lambda /{L\!_p}})$. Here, the choice of sign in front of $h$ respects the convention for the sense of the polarization rotation introduced by the constitutive relations of Section 2. Thereafter, as ${\kappa _z}$ scans the positive (respectively, negative) axis of the real numbers, the ratio $\lambda /{L\!_p}$ also varies with the numerator and the denominator taking various (continuous) values independent of each other. Hence, for $h = - 1$ (respectively, $h = + 1$), there is a point where ${\kappa _z}$ becomes equal to $\bar n$ (respectively, ${-}\bar n$) and the Bragg condition for the hypothetical left- (respectively, right-) handed structurally chiral medium is fulfilled. As we have mapped ${\kappa _z}$ to the ratio $\lambda /{L\!_p}$, the equivalent matching condition ${\kappa _z} = \kappa _0^{{{{\rm res}}_ \pm}}$ will be satisfied for all wavelengths, since different wavelengths $\lambda = \{{\lambda _1},{\lambda _2}, \ldots \}$ become resonant for different pitches ${L\!_p} = \{L{\!_p}\!\!^1,L{\!_p}\!\!^2, \ldots \}$, but the on-resonance ratio $\lambda _0^{{\rm Br}}/{L\!_p}$ is always equal to $\bar n$, forcing each tuned wavelength to match with its corresponding pitch. This offers a heuristic view of our proposed mechanism’s polarization-sensitive nature.

5. ELECTROMAGNETIC RESPONSE OF A FINITE SLAB

The transverse part of the total electric field within the medium is given by the sum of the supported eigenmodes, multiplied by their associated $z$-independent amplitudes:

We can now consider the reflection and transmission characteristics of the slab of Fig. 1, when the complex medium is axially bi-anisotropic and the surrounding medium is an isotropic dielectric with a relative permittivity ${\epsilon _I}$ and a relative permeability ${\mu _I}$. Field-matching at both boundaries requires that

We can then construct the global reflection and transmission coefficient matrices as

With the reflection and transmission coefficients accessible via Eqs. (30a) and (30b), respectively, we may define the intensity reflectances as ${R_{\textit{ij}}} = |{r_{\textit{ij}}}{|^2}$ and the intensity transmittances as ${T_{\textit{ij}}} = |{t_{\textit{ij}}}{|^2}$, where $\{i,j\} = \{R,L\}$ indicates reflection or transmission of the $i$ polarization for incident $j$ polarization. In Fig. 5(a), the intensity reflectances are plotted as functions of the axial chirality parameter where it is noticeable that the maximum values achieved are below unity, due to absorption, as befits. The co-reflectances’ profiles (${R_{\textit{RR}}}$ and ${R_{\textit{LL}}}$) follow the typical behavior observed in conventional scalar Bragg gratings and structurally chiral media: increasing the slab’s thickness $L$ results in steeper curves, while an increase in the linear birefringence $\Delta \bar \epsilon$, being equivalent to the depth of the refractive index modulation amplitude, widens the bandgaps. Accordingly, the intensity transmittances are plotted in Fig. 5(b), where the coinciding cross-transmittances (${T_{\textit{RL}}} \equiv {T_{\textit{LR}}}$) are at a much lower level compared to the co-transmittances (${T_{\textit{RR}}}$ and ${T_{\textit{LL}}}$). This is due to the relatively small length of the slab $L \approx {\lambda _0}/2\pi$, where ${\lambda _0}$ is the design central wavelength, chosen deliberately to be typical for metasurfaces (see, e.g., [56]).

 

Fig. 5. Electromagnetic response of a finite slab of a lossy axially bi-anisotropic uniaxial medium, surrounded by an isotropic dielectric: (a) intensity reflectances and (b) intensity transmittances, both in the the axial chirality domain. The complex medium’s parameters are those of Fig. 1, and the surrounding dielectric has a relative permittivity ${\epsilon _I} = ({\epsilon + {\epsilon _z}})/2$ and a relative permeability ${\mu _I} = 1$ so that impedance-matching is approximately achieved. The slab’s (normalized) thickness is set to ${k_0}L = 1$.

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In the absence of absorption, we have confirmed that the energy conservation relations ${R_{\textit{LL}}} + {R_{\textit{RL}}} + {T_{\textit{LL}}} + {T_{\textit{RL}}} = 1$, ${R_{\textit{LR}}} + {R_{\textit{RR}}} + {T_{\textit{LR}}} + {T_{\textit{RR}}} = 1$, and ${r_{\textit{LL}}}r_{\textit{LR}}^* + {r_{\textit{RL}}}r_{\textit{RR}}^* + {t_{\textit{LL}}}t_{\textit{LR}}^* + {t_{\textit{RL}}}t_{\textit{RR}}^* = 0$ of [57] are satisfied. However, in that case, abrupt oscillations at the upper (respectively, lower) edge of the positive (respectively, negative) bandgap occur in the vicinity of the singular points occurring when solving Eq. (18). These arise because as we approach the singular points from regions outside the two bandgaps, the imaginary parts of the refractive indices ${\pm}{n^{(-)}}$ tend to infinity, while their associated real parts drop from a high value to zero. As a consequence, the optical path is precipitously changed, resulting in violent oscillations. These oscillations are not visible in Fig. 5 as the multiple-pass nature of the Fabry–Pérot interference occurring in our realistic lossy medium is enough to diminish the wave reflected at the rear interface of the slab.

While Fig. 5 illustrates “Bragg-like” spectral features in the case of approximate impedance-matching, where the surrounding medium is simply an isotropic dielectric with ${\epsilon _I} = (\epsilon + {\epsilon _z})/2$ and ${\mu _I} = 1$, such an optical response would still be observable even if impedance-matching was significantly compromised (for instance, with a surrounding medium of vacuum). In such a scenario, the curves of the intensity co-reflectances would be less smooth and would have lower maximum values compared to those seen in Fig. 5, while the levels of the cross-reflectances would increase. These effects would arise from the interference of multiple waves being backscattered at the interfaces of the non-impedance-matched medium, where the handedness changes at each reflection. Furthermore, if we were to explore a semi-infinite slab scenario, we would observe that the resonances become sharper, further substantiating our claim of a “Bragg-like” mechanism. Specifically, in the impedance-matched case and in the absence of absorption, the co-reflectances would approach unity, with the cross-reflectances becoming essentially zero. In this context, the medium behaves like a metal, despite being a dielectric, with the added benefit of polarization-discriminatory optical response that is wavelength-independent.

6. META-MEDIA ROUTE TO THE IDEAL OPTICAL MODULATOR

Two decades ago, when interest in bi-anisotropic media was at its peak, it would have seemed preposterous to consider chirality parameter values as high as those comparable to the refractive index of regular materials. Nonetheless, as is often the case in scientific research, a few authors had the foresight to incorporate extreme chirality values into their simulations; e.g., [2] used a $\kappa = 2$, [5] a ${\kappa _z} = 1$, [51] a ${\kappa _z} = 1.58$, and so forth. With the subsequent rapid development of artificial media, it is now established that giant chirality can indeed be experimentally realized in various regimes of the electromagnetic spectrum. In fact, at optical frequencies, [18] demonstrated a value of $\kappa \approx 0.02$ at 640 nm in two-dimensional gratings of chiral gold nanostructures, with [24] achieving an almost 10-fold increase of $\kappa \approx 0.15$ at 540 nm in gammadion nanostructures. In [25], an optimized metasurface with pairs of vertically offset dielectric bars exhibited a $\kappa \in ({2.98,3.02})$ in the range 596–604 nm, while [26] reached to a value as high as $\kappa \in ({4.79,4.88})$ in the non-visible wavelengths 975–995 nm with an all-dielectric toroidal dipole metasurface. At THz frequencies, [20] measured a value of $\kappa \approx 2.45$ at 0.27 mm for a gold-based three-dimensional chiral meta-medium. At longer wavelengths, [23] estimated a $\kappa \in ({- 2.65, - 8.75})$ at around $3.97 - 4.62\;{\rm cm}$ in a bi-layer cross-wire metamaterial, with [22] obtaining a $\kappa \in ({- 0.62, - 2.65})$ at 5.08–6.52 cm for a four-layer metamaterial consisting of rosettes in its unit cells.

Although all of the aforementioned cases exhibit extreme values of optical rotatory power, its controllability over a broad wavelength range was not a primary concern. Only recently, the now mature manufacturing techniques sparked exciting experimental demonstrations of giant and, moreover, controllable chirality. In particular, the three-dimensional triple-helical platinum nanowires metamaterial of [58] achieved a $\kappa \in ({0.024,0.032})$ within a band 750–1000 nm, controlled via the orientation angle of the sample. Inconveniently, at such wavelengths, platinum’s refractive index varies as ${n_{{\rm Pt}}} \in ({0.51,0.78})$, being an order of magnitude greater than $\kappa$. This in not discouraging, though, as the average refractive index could be lowered via orthogonal wire inclusions that present a lower effective refractive index through control of the plasma frequency (see the founding meta-medium of [59]). At a range of 0.24–0.62 mm, [60] measured a voltage-controlled $\kappa \in ({- 0.29,0})$, while at 0.25–0.37 mm, [61] achieved a $\kappa \in ({- 1.59,0.43})$ controlled via the so-called pneumatic force. Values of $\kappa \in ({0,1.71})$ were found at 0.37–1.5 mm in the piezoelectric-controlled $R$-kirigami modulator of [27], Ref. [62] measured a $\kappa \in ({- 0.37,0.67})$ at 1.52–1.75 µm for a metamaterial whose optical activity is controlled by an external electric field, and the conductivity-controlled chiral medium of [28] gave a $\kappa \in ({- 0.82,2.89})$ at 0.4–5 mm.

In order to evaluate the quality of the resonance identified in this manuscript, we need to analyze it in the wavelength domain. If the fractional bandwidth is ${\rm FBW} = \Delta {\lambda _0}/{\lambda _0}$, we can make a straightforward comparison with, e.g., the ${\rm FBW}$ of typical cholesteric liquid crystals. For instance, in [63], assuming positive birefringence, the ordinary (respectively, extraordinary) refractive index is ${n_{o}} = 1.5$ (respectively, ${n_{ e}} = 1.7$) yielding a ${\rm FBW} = 2| {{n_{e}} - {n_{o}}} |/| {{n_{ e}} + {n_{0}}} | \approx 12.5\%$. Table 1 displays such a FBW, as this could be achieved via current technology, for a selection of meta-media that roughly meets the required parameters. Evidently, our proposed mechanism easily exceeds the indicative value of ${\rm FBW} \approx 10\%$.

Table 1. Approximate Assessment of the “Bragg-like” Resonance’s Fractional Bandwidth (FBW) in the Spectral Domain, Currently Attainable in Various Meta-media Proposed in the Literature

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Regarding the Tellegen coefficient, in natural media, $\chi$ is so small (${10^{- 5}}$ for ${{\rm Cr}_2}{{\rm O}_{3}}$ in [64]) that it can be rightfully neglected. Nevertheless, the most recent advances in non-reciprocal sub-wavelength optical devices [65], non-linear non-local metasurfaces [66], and in space–time modulated meta-media [67–69] indicate that enhanced non-reciprocal behavior should soon be expected in an increasing number of artificial materials. Within the theoretical framework of [70], where axion-coupling is applied to time-modulated crystals, we can safely say that the giant values appearing in Fig. 3 are well within reach of current technology (actually, [70] predicts an extreme value of $|\chi | = 0.59!$). In a time-varying medium though, a usual Helmholtz wave equation, like our Eq. (13), valid for monochromatic waves, is not applicable. However, for an instantaneously responding medium, the constitutive relations of Eq. (3) will still hold (see [71]). Interestingly, the treatments outlined in [72,73] seem to provide the necessary tools for analyzing such time-dependent media and an extension of our wavelength-independent Bragg reflector is therefore expected. We hypothesize that a time-dependent version of our “Bragg-like” effect will be uniform with respect to the photon energy, rather than momentum, opening up rich possibilities for light manipulation in futuristic time-varying systems. As a final remark, a duality transformation in [74] showed the equivalence between a non-reciprocal bi-isotropic Tellegen nihility medium [described by setting $\epsilon = 0$, $\mu= 0$, and $\kappa = {\kappa _z} = 0$ in Eq. (2)] and a reciprocal isotropic medium with $\epsilon = \chi$ and $\mu= - \chi$. While duality transformations that can map chiral media into non-chiral do not exist, Eq. (24) shows that for ${\kappa _z} = 0$, the phenomenon identified here can be replicated in the $\chi$-domain, with the Tellegen coefficient being controlled, e.g., via the polarization currents, which will determine all of the complex medium’s effective parameters, and thereupon its overall response.

7. CONCLUSIONS

Although media that combine circular and linear birefringence have been well studied for some time, it was only recently shown that they can exhibit a broadband circular “Bragg-like” phenomenon, exceptionally arising in a uniform medium. This alluring possibility, however, requires chirality with equal magnitudes in both birefringent axes, while also necessitating giant values to match the medium’s average refractive index. In this work, we demonstrated that such a mechanism can also be achieved in axially bi-anisotropic uniaxial media. Off-axis propagation appears to relax the previously identified resonant condition, offering control over the chirality-domain bandgaps. Furthermore, incorporating non-reciprocity in the model further alleviates the required values of chirality. Additionally, we identified anomalous wave propagation at a singular point, leading to extraordinarily high values of the refractive index across a broad wavelength range. Recent demonstrations of meta-media with giant and controllable chirality create a road-map towards a practical implementation of our medium, whereby externally modulating its parameters, a highly efficient polarization divider/combiner is feasible.