Extreme Nonreciprocity in Metasurfaces Based on Bound States in the Continuum

Nonreciprocal devices, including optical isolators, phase shifters, and amplifiers, are pivotal for advanced optical systems. However, exploiting natural materials is challenging due to their weak magneto‐optical (MO) effects, requiring substantial thickness to construct effective optical devices. In this study, it is demonstrated that subwavelength metasurfaces supporting bound states in the continuum (BICs) and made of conventional ferrimagnetic material can exhibit strong nonreciprocity in the Faraday configuration and near‐unity magnetic circular dichroism (MCD). These metasurfaces enhance the MO effect by 3–4 orders of magnitude compared to a continuous film of the same material. This significant enhancement is achieved by leveraging Huygens' condition in the metasurface whose structural units support paired electric and magnetic dipole resonances. The multi‐mode temporal coupled mode theory (CMT) is developed for the observed enhancement of the MO effect, and the findings with the full‐wave simulations are confirmed.


I. INTRODUCTION
In the race for the miniaturization of optical devices, a wide class of electromagnetic structures named metasurfaces arose in the last decade as one of the most prominent solutions.Optical metasurfaces are two-dimensional arrays of nano-inclusions with a sub-wavelength thickness that are engineered to manipulate light at will [1][2][3][4].Theoretically, if the inclusions, known as meta-atoms, are properly designed, almost any response that does not violate fundamental laws of physics could be achieved for these structures.One of the important industrial applications of metasurfaces is the design of nonreciprocal optical components such as isolators.In order to achieve optical isolation, a necessary condition is that the device must be nonreciprocal [5].Traditionally, nonreciprocity has been implemented using Magneto-Optical (MO) materials such as ferrites [6].Nevertheless, MO effects are very weak in the optical regime [7] and must be enhanced with proper techniques for designing compact nonreciprocal devices.Recently, there have been multiple techniques suggested to achieve an enhancement of MO effects at the nanoscale: magnetophotonic crystals [8][9][10][11], coupled surface-plasmon polaritons [12][13][14], magnetic Weyl semimetals [15], multiple layer systems [16], and resonant all-dielectric metasurfaces [17][18][19][20].Although these studies achieve a high enhancement, the resonant properties of the structures considered do not allow ultra-high quality factors that would enable higher interaction times between light and the MO material [18].
An attractive path toward designing compact optical nonreciprocal components is based on bound states in the continuum (BICs) which, if properly engineered, en- * lmmaeesp@upv.esable almost non-leaky resonances that increase the interaction time and the field concentration in the materials of the resonant structure.BICs in metasurfaces have been widely researched in the last few years [21].In electromagnetism, these bound states with infinite lifetimes are discrete solutions of the Maxwell equations embedded in a continuum of leaky modes and inaccessible by incident external radiation [22].Their useful realization comes in the form of quasi-BICs, which are perturbed BICs that have turned into accessible leaky modes with extreme quality factors (ultra-long but finite lifetimes) [23,24].Over recent years, metasurfaces that use these states have been proposed to enhance different effects: circular dichroism [25][26][27], absorption [28,29], and more.Furthermore, BICs can be classified into different categories depending on their formation nature [22].Particularly interesting and relatively easy to exploit are the Symmetry-Protected BICs (SP-BICs) [30], which have been precisely characterized by their group representations [31][32][33].Specifically, SP-BICs are inaccessible states that do not couple to the plane wave modes because of symmetry incompatibility between incident excitation and structural symmetries [24].Several metasurface designs with enhanced MO effect were recently proposed based on BICs [34][35][36].However, they were either limited to the THz range where magneto-optical effects are relatively strong even in natural materials [34], or based on nonrealistic materials (exhibiting no dissipation loss) [36], or exhibited relatively weak MO properties (in [35], the reported Faraday rotation was in the order of 12 mrad).More importantly, in none of these works, a comprehensive theoretical explanation of the MO enhancement mechanism was given, which is essential for optimizing the performance of the structure and reaching the maximum strength of the effect.
In this work, we present a metasurface made of MO ferrite material that simultaneously supports BICs with electric and magnetic nature.We delve into the phys-ical phenomena in the structure and develop a phenomenological multi-mode temporal coupled mode theory (CMT) [37,38] to serve as a recipe for the design of optical metasurfaces exhibiting maximum MO effects for the given materials being used.We demonstrate that by exploiting a pair of even and odd resonances in the metasurface unit cell (so-called Huygens' pair [39]), it is possible to obtain nearly unit magnetic circular dichroism (MCD) in a simple geometry with conventional material (Bi3YIG) being used.Using this recipe, we design a nonreciprocal metasurface with a near-unit MCD that behaves as a one-way nonreciprocal device for circularly polarized light, allowing a certain handedness to be transmitted when propagating in the forward direction and blocking it for backward illumination.If surrounded by two polarizers [40], such a structure would operate as an optical isolator in transmission.

II. METASURFACE DESIGN
The metasurface design will follow a similar topology to that of [40,41], where they described an array of dielectric nanodisks distributed in a squared lattice in a checkerboarded fashion with an asymmetry in diameter between nearest neighbors.However, in the present scenario, we consider the nanodisks to be made of MO ferrite material Bi3YIG experimentally characterized in [42].The metasurface geometry is shown in Fig. 1.Its geometrical parameters are l for the unperturbed diameter of the nanodisks, w for the squared lattice constant, h for the height of the nanodisks, and ∆ for the difference in diameter between nearest neighbors.The diameters for the nanodisks are defined as l a = l + ∆/2 and l b = l − ∆/2.For simplicity of the analysis, we assume the metasurface to be surrounded by air from both sides.It is worth noticing that the same physics applies to the scenario when the metasurface is immersed inside some background dielectric material (for that, merely the permittivity of the nanodisks should be scaled up accordingly).For the scenario where the metasurface is located on top of a dielectric substrate, the theory can also be straightforwardly extended by considering non-resonant reflections from the substrate.
The material of the nanodisks is magnetized by a bias static magnetic field applied along the +z-axis, which results in its antisymmetric permittivity tensor of the form [6] ε(ω, where ε xx = ε 1r − jε 1i , and ε xy = ε 2r − jε 2i are complex permittivities and H 0 denotes an external magnetic field along the z-axis.Here we use the e +jωt convention for temporal field evolution.Permittivity component ε xy is assumed to be a linear function of H 0 .Moreover, saturation is achieved in Bi3YIG with a field of 1.2 T [42].When ε xy = 0 and ε xx = ε 1r , the structure shown in Fig. 1 is known to hold quasi-BICs resonant modes at normal incidence due to the mismatch in diameter between nanodisks [40,41].The resonant modes of the metasurface that are those shown in Fig. 2(a) and (b), obtained using the commercial FEM simulator COMSOL Multiphysics .These two modes correspond to the scenarios where the nanodisks acquire electric and magnetic dipole polarizations, respectively.These modes remain inaccessible for normal plane-wave incidence in the metasurface without geometrical perturbations (i.e., ∆ = 0).They are symmetry-protected bound states in the continuum (SP-BICs) [30], and their origin is due to the symmetries of the structure and can be explained in several ways.The most intuitive explanation can be done in terms of induced dipoles.A checkerboarded array of nanodisks can be thought of as a superposition of two sub-lattices, in this case, each holding a magnetic/electric dipole moment, but with a π relative phase between wave oscillations in the two lattices.In the unperturbed case, the superposition of both lattices cancels the far-field radiation and forms an SP-BIC.In the perturbed case, the size of the nearest neighbor is altered, and the electric/magnetic dipoles are not identical.This geometrical mismatch induces a difference in amplitude and/or phase, enabling a non-perfect destructive superposition of scattered waves and creating a high-Q resonance, i.e., a quasi-BIC.When the perturbed metasurface supporting this type of resonant mode is illuminated by a linearly polarized wave, the interaction of the incident field and the resonant struc-ture will produce two peaks in the transmission spectra, as it is shown in Fig. 2(c).These two peaks correspond to the resonance conditions of the two sub-lattices of dipole moments.The positions and the widths of the transmission peaks are closely related to the asymmetry of the structure quantified by the parameter ∆.The resonant mode that appears at a lower wavelength is the electric dipole.As expected, the width of the resonant modes increases with the parameter ∆ as the structure becomes more asymmetric.It is also interesting to notice that because the parameter ∆ is directly related to the radius of the nanodisks, it has a larger impact on the position of the electric dipole.
In a more formal mathematical fashion, the BIC formation phenomenon can be explained by using group theory techniques.In the case of ∆ = 0, ε xx = ε 1r , and ε xy = 0, the cell and point symmetries are C 4v and C 2 with two additional mirror planes (see Supplementary Material [43], Section 2).As it is known, the eigenmodes of a periodic structure must satisfy the transformations given by an irreducible representation (IRREP) of the structure's symmetry group [32].There is a BIC when there is no coupling between the external excitation and the structure's eigenmode.The absence of coupling is due to the incompatibility between the symmetry restrictions of the resonant structure and the incoming plane wave's symmetries [44].Introducing a geometrical perturbation (i.e., ∆ = 0) removes the C 2 symmetry and one of the mirror planes, Fig. 3 in Supplementary material [43].Without them, the eigenmode symmetry is partially broken, and the coupling is enabled.The state is now referred to as a quasi-BIC.For this reason, the symmetry-protected label we used to refer to these BICs is now apparent.The structure behaves exactly the same way with linear y-polarized light or x-polarized light due to the C 4v symmetry that it still holds.Moreover, the cell symmetry C 4v explains the degeneracy of the electric and magnetic modes that belong to the same Γ 5 two-dimensional IRREP [31] (see Supplementary Materials [43], Section 2).Furthermore, these eigenmodes, shown in Fig. 2(a) and (b), transform according to the Γ 5 irreducible representation of the C 4v group [27].
With the introduction of a magnetic bias, but neglecting the losses (ε xy = ε 2r and ε xx = ε 1r ), both magnetic and electric dipolar resonances split.As an example of this behavior, Fig. 3 represents the transmission spectra for two different values of the parameter ∆ when the structure is biased by a magnetic field that brings the magnetic ferrite to saturation.Note that this analysis is done neglecting the losses in the material.The degeneracy that emerged from the two-dimensional Γ 5 IRREP is now broken by the non-equal off-diagonal components of the permittivity tensor (see Eq. 1).Due to the anisotropic and nonsymmetric nature of the material, mirror diagonal symmetries (σ d1 and σ d2 ) are broken.The point group of the structure is downgraded to C 4 , which is the group that has the 90 • rotation about the z-axis as its generator.The corresponding degener- ate resonances split as their two-dimensional IRREP (Γ 5 of C 4v ) splits into two one-dimensional IRREPs (Γ 3 and Γ 4 of C 4 ) [31] (see Supplementary Material [43], Section 2).
In addition, the introduction of losses leaves the structure's symmetry unperturbed but limits the quality factors of the quasi-BICs, as pointed out in [45].Moreover, for the same values of ∆, the quality factor and amplitude of the resonance will drop significantly.Nevertheless, losses in the material make possible such physical effects as the MCD and isolation.

III. COUPLED MODE THEORY DESCRIPTION
To better understand the physics behind the structure and as a guideline for the design of actual devices, we developed a theoretical model based on temporal CMT.More specifically, our model will consider the interaction of four ports (two ports for each linear polarization) and four resonant modes (two electric and two magnetic modes in the presence of the magnetic bias).Defining the ports of the structure as in Fig. 4, where ports 1 and 2 accounts for the x-polarization channel and ports 3 and 4 form the y-polarization channel.The C 4 symmetry of the structure ensures that the scattering matrix (expressed in the linear-polarization basis) of the device has the form (Supplementary Material [43]) where r and t represent the reflection and transmission amplitude coefficients when the polarization state of the incident light is preserved (co-polarized coefficients), while r xy and t xy are the reflection and transmission coefficients when the polarization is changed (crosspolarized coefficients).The scattering matrix of the metasurface in the circular-polarization basis with basis vectors e ± = (e x ∓ je y )/ √ 2 has the form (Supplementary Material [43]) (3) where the superscripts "f" and "b" indicate forward (+z) or backward propagation (−z), and the subscripts "L" and "R" indicate left and right-handedness of circular polarization, respectively.The elements of the scattering matrices in the two different polarization bases are related as: t + = t + jt xy , r + = r + jr xy , t − = t − jt xy and r − = r −jr xy .It is worth mentioning that in the circularpolarization basis, nonreciprocity does not lead to a nonsymmetric scattering matrix due to the way that the elements of the matrix are defined, as discussed in [46].Moreover, as Eq. 3 shows, the notation chosen embraces the mathematical definition of the circular-polarization basis, uniquely defined by the sign.Light handedness implies a choice in nomenclature and point of view, as pointed out and widely discussed in [47].Using the sign in the relative phase between components as the defining factor leaves no doubt about what we refer to.In any case, the convention used to relate both parts of Eq. 3 is the one commonly used in engineering, where the time dependence is defined as e jωt and the observer points towards the direction of propagation to declare the handedness.
The phenomenological CMT model is developed following the general coupled-mode equations [37]: where a is the vector of n × 1 size that represents the resonance amplitudes inside the metasurface and n = 4 is the number of resonant modes.Here the resonance amplitudes are normalized such that |a i | 2 corresponds to the energy stored in the i-th resonant mode.Moreover, in Eqs.4-5, D is the output m × n coupling matrix, m = 4 is the number of ports, C is the scattering matrix of the background (or the direct pathway coupling) that is described as an m × m matrix, K is the input coupling matrix of m × n size, Γ and Ω are n × n matrices which correspond to the decay rates and the resonance frequencies, respectively, and |s + and |s − are vectors of m × 1 size accounting for the output and input waves (see Fig. 4).
There are certain conditions that the elements in Eqs. ( 4) and ( 5) have to follow in order to preserve energy conservation [38].The restrictions are as follows: From the eigenmode analysis, we found that the four resonances are orthogonal and, therefore, ΓΩ = ΩΓ.Both operators share eigenvalues and eigenvectors as they commute.Moreover, because of the in-plane distribution of electric and magnetic fields, the pair of magnetic resonances present an odd field symmetry with respect to the z = 0 plane, while the pair of electric resonances have an even field symmetry.This will affect the coupling between the resonances and the excitation.Furthermore, it is known that the split in resonances is due to the offdiagonal components of the permittivity tensor, which now has circularly polarized waves as eigensolutions [48].This will make the magnetic/electric resonance split into two modes: one that couples with RCP light and one with LCP light.To model this coupling, a relative −π/2 phase is introduced between x− y components in the coupling matrices (D and K) in resonances 2 and 4.And a relative +π/2 phase is introduced between x − y components in resonances 1 and 3 (See Supplementary [43], Section 4).
In addition, the metasurface has geometric mirror symmetry with respect to the z = 0 plane.The coupling amplitudes to the resonances in the metasurface have to be the same for the forward and backward illuminations.Due to Eqs. 7-9, the elements of each of K and D matrices are related to one another.In terms of the D matrix, its complex elements defined as d ik = d ik e jθ ik have the following relations: Another constraint is found for the phases [37], where the mirror symmetry applies a constraint of well-defined parity, as in our case, and the phases have to comply where N is an integer.In the considered case, resonances 1 and 2 are even (the magnetic dipole resonances), while resonances 3 and 4 are odd (the electric dipole resonances).
Matrices Ω and Γ are assumed to be diagonal, , where the subscript refers to the nature of the resonance and its symmetry (magnetic and electric, respectively) and the superscript relates to which circular polarization the resonance couples after splitting due to the external magnetic field.In addition, a canonical phenomenological way to introduce losses in the model is to decompose the Γ matrix as Γ = γ + 1/τ , where the first matrix refers to the decay rates of the resonances coupling into the ports and the second matrix reflects the material losses.A simple and standard way to solve Eqs. ( 4) and ( 5) is first to solve the system of equations neglecting the material losses with all the previous constraints (1/τ = 0) and then introducing the change that accounts for the material loss as Γ ± m,e = γ ± m,e + 1/τ ± m,e .As a result, we obtain the final expressions for the S-matrix coefficients in the circularpolarization basis: In the derivation, the direct pathway (background) scattering matrix was modeled as where t d and r d are the transmission and reflection coefficients, respectively.The background pathway has been considered reciprocal and without losses, i.e., the C matrix is a unitary matrix.

IV. MAXIMIZATION OF NONRECIPROCITY AND MAGNETIC CIRCULAR DICHROISM
In this section, we derive the ideal conditions under which the metasurface with the aforementioned properties provides the highest (unitary) Magnetic Circular Dichroism (MCD) and the highest isolation.MCD is conventionally defined in terms of the difference in absorptions for the two circular polarizations of light [49]: In addition, we define the isolation ratio as ι = |t − /t + |.In the case at hand, achieving a high isolation ratio will lead to high MCD, as seen from Eq. 3.For instance, when illuminating one side (the "+z" direction) with RCP light, the response is t + , while LCP light yields t − .Conversely, illuminating from the opposite side ("−z" direction) with RCP light produces t − , and with LCP, t + .A high MCD means one polarization is mainly absorbed, and the other is not; the previously blocked polarization is not absorbed when the direction is reversed.Next, we look for the conditions to design a complete isolator for circularly polarized light, that is, a device that fully absorbs one polarization from one side and transmits it from the other.
We start by finding the conditions to achieve high isolation for circularly polarized light using the theoretical results obtained in the last section, that is, Eqs.12-15.Thus, we aim to ensure the full absorption and full transmission of light of a given circular polarization incident on the metasurface from the two opposite directions.Without loss of generality, we determine these two conditions for the right-handed circular polarization (as shown below, the dual conditions will be automatically satisfied for the opposite polarization).Mathematically, the conditions read as Naturally, they would lead to the scenario of maximization of the ι ratio.To simplify the derivation, in the following, the case where r d = 0 and t d = 1 is explored.Using Eqs.12-15 and Eq. 3, we can express absorption of the RCP light incident in the forward direction as ( To satisfy the first condition (maximization of A + ), the eigenfrequencies for the resonances of both parities have to be equal ω + m = ω + e = ω + , and the maximum will take place at the frequency ω = ω + .To reach precisely unitary absorption, we should ensure |r + | 2 = 0 and |t + | 2 = 0. We study both cases separately and then combine the conditions.At the resonant frequency, |r + | 2 = 0 is obtained if This constraint leads to the condition of a Huygens' metasurface [50].Furthermore, forcing separately that |t + | 2 = 0 at the resonance frequency will yield Combining both Eqs.20-21, we end up with the know condition of critical coupling that ensures full absorption of incoming energy at the resonance frequency: This result is widely known for reciprocal metasurfaces and can be interpreted as a balance between the losses induced by the material and the scattering losses [51].In contrast, in our case, the critical coupling condition holds at a given frequency only for one illumination direction.Substituting conditions 22 into Eq.19, we indeed obtain In what follows, we additionally require zero reflection condition |r + (ω)| 2 = |r − (ω)| 2 = 0 for all frequencies ω.Such a broadband Huygens' regime [52] has multiple practical advantages such as the possibility to cascade several metasurfaces and the minimization of parasitic interference.As is seen from Eq. 14, broadband suppression of reflections for illuminations from both sides is achieved when we choose ω − m = ω − e = ω − and additionally satisfy: Next, to satisfy the second condition in Eq. 18 at a given frequency ω, there are two possible limiting scenarios.The first one is to ensure full absorption for RCP light incident on the metasurface in the backward direction The second limiting scenario for satisfying the condition in 18 is when the metasurface is completely transparent at all frequencies (unitary transmission with an arbitrary phase) for RCP light incident in the backward direction.This scenario is presented in Fig. 5(b).It can be achieved if 1/τ − = 0.
As was mentioned above, at the frequencies where the isolation ratio ι = |t − /t + | is high, the MCD is large.In Fig. 5(c), we plot the MCD parameter for the two above scenarios.While for the first scenario, the MCD approaches to unit value at ω = ω + , in the second scenario, it is a unit for all the frequencies due to its definition.

V. FERRITE METASURFACE REALIZATION
Using a metasurface with the geometry described in Fig. 1 and ferrite material (Bi3YIG [42]) with a tensor given by Eq. 1, the maximization of MCD and isolation properties studied theoretically in the last section can be applied.In the case of the real structure, the mentioned configuration for the resonances must be achieved by tweaking the geometrical parameters.Studying the behavior of the structure when there is no bias and no losses (ε xy = 0 and ε xx = ε 1r ), it is apparent that the decay rates of the magnetic and electric resonances are intrinsically different, see Fig. 2.However, both can be changed by tuning the asymmetry parameter ∆.It must also be noted that the wavelength shift of the electric dipole resonances is noticeable when ∆ is changed.In contrast, the magnetic dipole resonances shift much slower, as is seen in Fig. 2(c) and Fig. 3.The bias will split each of the magnetic and electric dipole resonances into two, one coupling to the plus sign of the helical basis vectors and one for the minus sign.By introducing losses, the behavior of the resonances holds, but the resonances suffer a fall in quality factor and do shift.The final design uses optimization techniques for the geometrical parameters and the aforementioned knowledge to maximize the MCD.It should be noted that due to the material restrictions, we could not reach either of the two limiting scenarios shown in Figs.5(a) and (b).Nevertheless, characteristics of the designed metasurface are closer to those in Figs.5(a).
The simulated scattering parameters of the metasurface, when illuminated normally with circularly polarized light, are shown in Fig. 6(a).The CMT fitting is also shown in the same plot, accurately modeling the simulated data.A maximum for the MCD is obtained at λ = 648.5 nm, reaching |MCD| = 0.996, as shown in Fig. 6(b).At the same wavelength, |t + | is almost fully suppressed.Moreover, the amount of energy transmitted forward with LCP polarization reaches 73.90% while only a 0.12% of LCP light would be transmitted in the backward direction.The raw contrast between transmittances 625, which can be expressed as a 27.89 dB isolation ratio.The energy with a defined circular polarization can go through the device in one direction while being absorbed in the opposite direction.
Finally, in Fig. 6(c), we plot the Faraday rotation angle θ and ellipticity ǫ versus frequency for the designed metasurface.We use the conventional definition for the rotation angle θ = 1/2 arctan [2ℜ(χ)/(1 − |χ| 2 )], where χ = E tr y /E tr x .Where ℜ denotes the real part of a function.The ellipticity ǫ is defined as the ratio between the size of the major and the minor axis of the polarization ellipse of transmitted light.At λ = 648.5 nm the ellipticity reaches almost unity.This occurs because almost all of the transmitted light is circularly polarized with left-handedness.On the other hand, at λ = 649.05nm where ellipticity is equal to zero, the metasurface rotates linearly polarized light at an angle θ = 36.4• with the transmittance |t| 2 = 40.5%.The obtained values of the rotation angle and ellipticity are giant, taking into account the subwavelength thickness of the metasurface (h/λ = 0.21) and the realistic material parameters used.We can also estimate the effective Verdet constant of the metasurface V = θ/(µ 0 H 0 h) [6].Taking µ 0 H 0 = 0.66 T from [42], the Verdet constant reaches the maximum value of V = 7.12 × 10 6 rad/(T • m), which exceeds the value for thin-film of conventional MO materials by 3-4 orders of magnitude [6].Such giant enhancement of the MO properties is due to the use of quasi-BICs in our geometry.

VI. CONLCUSIONS
We have developed a metasurface made out of a magnetic ferrite and based on quasi-BIC resonances that, with the use of orthogonal pairs of resonances with electric and magnetic nature, can achieve high MCD combined with strong isolation for circular polarization.The device acts with this configuration as a one-way isolator, as the transmission for the non-blocked polarization is maximized.In addition, by matching the resonance amplitudes and phases, it is possible to use the quasi-BICempowered metasurface to enhance the Faraday effect of the structure.An enormous figure of merit is obtained using this approach.All of the above has been designed using a realistic material (Bi3YIG), experimentally characterized, and with noticeable losses.
The finite-element-method simulations have been supported with a coupled mode theory description of the metasurface.This phenomenological theoretical model has been useful in predicting and explaining the capabilities of the device.Furthermore, it supports the observed enhancement of the Faraday effect and proves the use of BICs as a path to control and enhance MO effects.The realization of a Faraday rotator can lead to the creation of an ultra-compact isolator using two linear polarizers in the famous Faraday isolator architecture.
In order to design a polarization-insensitive optical isolator, one needs to additionally break the inversion (parity) symmetry that is currently present in the metasurface design.This can be straightforwardly achieved using a conventional approach by introducing two circular polarizers (with the same handedness) around the meta-surface (e.g., the Supplementary Information in Ref. [40] and Section 6 in Supplementary material [43]).

FIG. 1 .
FIG. 1. Schematic of the nonreciprocal metasurface exhibiting one-way transmission for circularly polarized light.The inset depicts the unit cell with the in-plane geometrical parameters.The neighboring nanodisks have slightly different diameters la and l b and are arranged in checkerboard order.The LCP plane wave is labeled as "f" (forward) and "b" (backward).The external magnetic bias necessary to break reciprocity is shown at the top left of the figure.

FIG. 2 .
FIG.2.Resonant properties of the metasurface without external magnetic bias.Total field supported by the structure when ∆ = 5, 10 nm, l = 270 nm, w = 430 nm, h = 135.2nm, εxy = 0, and εxx = ε1r in the plane z = 0 (cuttingplane in the middle of the nanodisks).(a) Electric dipole mode at λ = 647.73nm and (b) magnetic dipole mode at λ = 649.30nm in the case where ∆ = 5 nm.The lattice cell is depicted with a dashed line.The dipole direction is shown with arrows.(c) Transmittance of the metasurface without magnetic bias and dissipation losses.The asymmetry in diameter is proportional to the width of the resonance.The magnetic resonance width is greater than the electric dipole resonance.There is a noticeable shift in the electric dipole resonance frequency when ∆ is changed.

FIG. 3 .
FIG.3.Resonant properties of the metasurface with external magnetic bias but without dissipation losses.The transmittance of the magnetically biased perturbed metasurface when ∆ = 5nm and ∆ = 15nm with the rest of parameters with values l = 270 nm, w = 430 nm and h = 135.2nm,.The electric dipoles p1 and p2 suffer a higher shift in wavelength than the magnetic dipole resonances m1 and m2.

FIG. 4 .
FIG. 4. Schematics of the ports and the resonant metasurface for the CMT analysis.

FIG. 5 .
FIG. 5. (a) Reflectances and transmittances for an ideal metasurface exhibiting maximal isolation for circularly polarized light of a given handedness.The metasurface behaves as a broadband-matched resonant absorber where the resonant frequency depends on the illumination direction.In this example, γ + = γ − = 1/τ + = 1/τ − = γ, t d = 1, and ∆ω = ω + − ω − = 30γ.The horizontal axis is normalized in such a way that its center corresponds to the average between ω + and ω − .At frequency ω + , transmission for the backward propagating RCP light reaches |t−(ω+)| 2 = 0.996.(b) The same for a metasurface that exhibits no material loss for RCP (LCP) light when illuminated in the backward (forward) direction.(c) MCD for the two limiting scenarios in (a) and (b).
occurring at a different frequency ω − = ω + .By decoupling the two resonance frequencies such that the spectral distance between them is much greater than their widths, i.e., |ω− − ω + | ≫ |γ + e,m + 1/τ + e,m |, we can asymptotically reach |t b R−R | ≈ 1 at the frequency of ω + .This scenario is depicted in Fig.5(a).Due to the condition in Eq. 23, reflections are zero at all frequencies, while the absorption peaks for two light propagation directions are spectrally separated.To have complete absorption A − (ω − ) = 1, one needs to additionally satisfy 1/τ − = γ − .