Magnetic switching of Kerker scattering in spherical microresonators

1 Introduction

Following an increasing need for the deployment of active (tunable) photonic materials during recent years, magneto-optical-aided dielectric resonators have emerged as a valuable platform because of their unique and energy-efficient ability to control the propagation of optical waves. This is achieved through an external agent, mainly an external magnetic field, which alters the constitutive parameters of the magneto-optical material. Multi-fold enhancement of magneto-optical effects in Si nanodisks covered with thin nickel films, as well as in magneto-photonic array of nanodisks, has been experimentally demonstrated, for the development of active and nonreciprocal photonic nanostructures [1]. Magneto-optical materials have been successfully employed in cylindrical structures, such as infinite cylinders or arrays of such. In particular, their use was recently achieved in the context of active scattering, via the available Mie theory for infinite, two-dimensional circular gyroelectric/gyromagnetic cylinders whose solution can be separated into TE and TM modes [2]. Specifically, active control of directional scattering with a core–shell infinite cylinder driven by an external magnetic field, as well as in cylindrical reconfigurable meta-lattices, has been theoretically examined by breaking the degeneracy of the multipoles in the subwavelength regime, thereby allowing the splitted modes to interfere and lead to directional optical switching [3], [4]. In addition, a normally irradiated array of magneto-optical core–shell cylinders has resulted in tunable anomalous back-scattering, where the extinction mean free path is longer than the transport mean free path [5].

Magneto-optical effects have been studied for attaining all-optical modulation [6], by exploiting nonlinear optical effects in two-dimensional materials, including graphene which involves magneto-optical effects in its conductivity model [7], [8], in magneto-optical active plasmonic resonators [9], in the implementation of highly tunable terahertz filters with magneto-optical Bragg gratings [10], in high-index structures where a Si spherical nanoantenna encapsulates a magneto-optical garnet core giving rise to magnetic and electric resonances that dominate the long-wavelength spectrum [11], in spatially dispersive gyrotropic semiconductor spheres [12], in obtaining topologically robust ultraslow waves in the presence of absorption and scattering channels [13], in laser devices for Q-switching [14], in nickel–silicon metasurfaces [15], nickel-based nanodisk arrays [16], and in photonic crystals with magnetized epsilon-near-zero metamaterials [17] for achieving magneto-optical response enhancement. Although high-index dielectrics do not exhibit the intrinsic losses of plasmonic materials, magneto-plasmonic media may offer additional functionalities, such as e.g. tunable plasmon-driven Hall currents existing in silver-coated Bi:YIG spherical scatterers [18].

The objective of our present work is to theoretically and numerically establish the feasibility of obtaining functional directionality inversion in spherical particles in the THz regime with the aid of magneto-optical materials that obey a Drude–Lorentz dispersive model. A straightforward strategy to achieve tunable forward- and back-scattering functionality is to deposit a magneto-optical layer on top of a high-index material. In this way, the high-index core contributes the magnetic dipolar mode, while the magneto-optical shell can exhibit the necessary electric dipolar plasmon that interferes with the magnetic dipolar mode, thereby leading to the desired directionality inversion. By employing a rigorous VIE method developed for the electromagnetic scattering by highly inhomogeneous gyroelectric spheres [19], and by means of full-wave electrodynamic calculations, we demonstrate such a tunable scattering-directionality inversion.

2 Methods

The setup of the subwavelength spherical antenna that will enable active directionality inversion is depicted in Figure 1. It consists of a high-index isotropic dielectric core of radius R₁ and permittivity ϵc, coated by a magneto-optical concentric shell of outer radius R₂. The magneto-optical material exhibits gyroelectric properties in the presence of an external DC magnetic field Bzˆ, otherwise it is isotropic when the magnetic bias is null. The structure is embedded in free-space of permittivity ϵ0 and permeability μ0. The core and shell are nonmagnetic, i.e., their permeability is equal to μ0. The configuration is illuminated by a plane electromagnetic wave impinging along the positive Oz semi-axis with its electric field being x-polarized.

Figure 1:

Schematic view of the spherical setup consisting of a high-index dielectric core (radius R₁) coated by a Drude–Lorentz magneto-optical shell (outer radius R₂). The external magnetic bias B is z-oriented, while the incoming plane wave impinges along the positive Oz semi-axis.

A general way to describe the permittivity properties of the core–shell structure is by means of the inhomogeneous cartesian permittivity tensor

where r is the radial distance in the spherical coordinate system. Therefore, Eq. (1) comprises a two-branch function with ϵ1(r)=ϵ3(r)≡ϵcr=16 (ϵcr is core’s relative permittivity) and ϵ2(r)≡0, 0⩽r⩽R1 for the isotropic core, while ϵ1(r)≡1+[ωp2(ω02−ω2−iωγ)]/[(ω02−ω2−iγω)2−ω2ωc2], ϵ3(r)≡1+ωp2/(ω02−ω2−iγω) and ϵ2(r)≡ωp2ωωc/[(ω02−ω2−iγω)2−ω2ωc2], R1<r⩽R2 for the magneto-optical shell. The relative permittivity elements of the shell obey a Drude–Lorentz dispersive model [20] with plasma angular frequency ωp=6π×1012 rad/s, resonance angular frequency ω0=3π×1012 rad/s, cyclotron angular frequency ωc=eB/m* (e and m* are the charge and effective mass of electrons), and damping angular frequency γ=0.06π×1012 rad/s [3]. Casting the permittivity of the structure in the form of Eq. (1), allows us to employ the VIE method, developed in [19], for the calculation of the scattering characteristics, i.e.,

In Eq. (2), Einc(r) is the incident electric field, kb the background medium wavenumber—in our case the wavenumber of free space—g(r, r′) the free-space Green’s function, and ∥ the unity dyadic. The time factor e−iωt is assumed and suppressed throughout. For r∈V, i.e., r⩽R2, the electric field is written as an expansion of the form

where Mmnℓ, Nmnℓ and Lmnℓ are the Dini-type spherical vector wave functions (SVWFs) [19], [21], while Γmnℓ, Δmnℓ and Zmnℓ are unknown expansion coefficients. The values of the Dini-type wavenumbers kmnℓM, kmnℓN, kmnℓL are chosen as roots of proper eigenvalue equations [19], so that the vector sets Mmnℓ, Nmnℓ are fully orthogonal in V and Lmnℓ decouples from Nmnℓ. Following the procedure described in detail in [19] for the solution of Eq. (2), we finally arrive at the following expansion for the scattered field

where Mmn(3) and Nmn(3) are the SVWFs of the third kind and Dmn=in[(2n+1)(n−m)!/n/(n+1)/(n+m)!]1/2. The scattering properties can then be expressed in terms of the computed scattering coefficients CmnM, CmnN, through the scattering cross section

where λb is the wavelength of the background medium, and through the asymmetry parameter [22]

.

In Eq. (6), Re denotes the real part, the asterisk complex conjugation, while Fmn=PmnCmnN+QmnCm,n+1M+RmnCm,n−1M, Gmn=PmnCmnM+QmnCm,n+1N+RmnCm,n−1N, with Pmn=m/n,/(n + 1), Qmn=[n(n+2)(n−m+1)(n+m+1)/(n+1)2]1/2(2n+ 1)(2n+ 3)] and Rmn=[(n−1)(n+1)(n−m)(n+m)/n2/(2n−1)/(2n+1)]1/2. It should be noted that the asymmetry parameter g∈[−1, 1] can be employed for determining the scattering direction. In particular, if g<0, more radiation is scattered in the backward direction, whereas g>0 means that the scattering is directed more towards the forward direction. Therefore, examination of g through its sign change, reveals for which operational wavelength directionality inversion takes place. This will be made clear in the following Section.

3 Engineering functional spherical terahertz antennas for directionality inversion

The engineering of a setup for active directionality-inversion, is achievable by utilizing a magneto-optical material deposited on a high-index dielectric. Once the structure is manufactured, its geometrical parameters—i.e., radius R₂ and R1/R2 ratio—as well as its material, cannot be altered. Therefore, to realize functional directionality inversion, the application of an external agent—in the present case a magnetic bias—is necessary in order to alter the constitutive parameters of the coating and enable the switching between forward- and back-scattering. In the analysis that follows, we demonstrate a state which corresponds to forward-scattering in the absence of magnetic field, and to back-scattering with non-zero external magnetic bias.

Figure 2A depicts the normalized scattering cross section Qsc/(πR22) versus the normalized wavelength λb/R2, under null magnetic field, for outer radius value R2=10 μm, and R1/R2=0.9. In addition to the full-wave contribution to Qsc—i.e., all spherical harmonics and angular momentum terms, required for convergence, contribute in Eq. (5)—the magnetic dipolar (MD) and electric dipolar (ED) terms (with the former and the latter being obtained solely from the contribution of C1,1M and C1,1N, respectively), are also plotted. The resonant peak appearing at λb/R2=5.16 corresponds to the higher order magnetic quadrupolar (MQ) term, whose sole contribution—through the Cm2M expansion coefficient—is not depicted. The validity of our calculations is confirmed by comparison with HFSS commercial software. The blue dots depict results for Qsc/(πR22) obtained via HFSS simulations, thus establishing a good agreement with the VIE. It should be mentioned that an excellent agreement can be met, especially for the localized surface plasmon (LSP) appearing at λb/R2=9.92, but it would require a tremendous amount of CPU time for HFSS to extract the spectrum using its finite element solver. In qualitative contrast to single high-index dielectric resonators where the first subwavelength resonance (as we move from higher to lower wavelengths) is the MD [23], [24], an extra resonance observed at λb/R2=9.92 precedes the MD resonance at λb/R2=7.44. This is due to the magneto-optical shell, whose dispersive properties change drastically around the plasma frequency. In particular, for frequencies below or equal to 3.353 THz (or, equivalently, for λb/R2⩾8.941), the Drude–Lorentz dispersion exhibits plasmonic behavior, while, just above 3.353 THz, it yields near-zero permittivity values. The aforementioned plasmonic region is gray-shaded in Figure 2A. Therefore, the λb/R2=9.92 peak corresponds to an LSP of a cavity-like hybrid of the shell [25], [26]. This is also confirmed by the near-field plot of the normalized magnitude |E|/max(|E|) of the total electric field on xz-plane which, as it is shown in Figure 2E, is concentrated within the plasmonic shell. We note here, that the LSP has a purely ED character, as confirmed by the green line in Figure 2A.

Figure 2:

Scattering cross sections, asymmetry parameter, near field and polar (far field) plots to illustrate functional spherical antenna design for directionality inversion. (A) Scattering cross section spectrum of a core–shell particle of total radius R2=10 μm, with core radius R1=0.9R2, under null external magnetic field. Blue curve: scattering cross section, i.e., including all terms of Eq. (5) up to a truncation order that ensures convergence; blue dots: scattering cross section via HFSS; red solid line: MD contribution to the scattering cross section; green solid line: ED contribution to the scattering cross section; point (I): Kerker point of second kind (back-scattering) at λb/R2=8.93; point (II): Kerker point of second kind (back-scattering) at λb/R2=6.6; point (III): Emerged forward-scattering point at λb/R2=8.04. Gray region corresponds to wavelengths with plasmonic behavior of the shell material. (B) Scattering-asymmetry parameter within the given spectral window. Black: null magnetic field; gray: 0.9 T. (C) Scattering cross section spectrum for the same setup as in (A) but for external magnetic-field bias 0.9 T. Legends are the same as in (A). The small black/gray arrows indicate the mode splitting due to magnetic bias (see text). Black arrows: m=−1 non-degeneracy; gray arrows: m=+1 non-degeneracy. The large black arrow indicates the location of the Kerker point of second type which now, under 0.9 T magnetic field, coincides with point (III). Gray region corresponds to wavelengths with plasmonic behavior of the shell material. (D) Mapping of asymmetry parameter g versus B and λb/R2. (E) Normalized magnitude |E| of the total electric field on xz-plane for the plasmonic resonance at λb/R2=9.92 appearing in (A). (F) Polar/bistatic cross section (far field) for antenna operation at the preferential point λb/R2=8.04. Black: forward-scattering at 0 T; gray: back-scattering at 0.9 T. Curves: VIE; dots: HFSS.

Figure 2B depicts the scattering-asymmetry parameter g in the same spectral window as in Figure 2A. With reference to the black curve—which corresponds to B=0 T—three points in Figure 2A are of interest: point (I) at the intersection of MD–LSP resonances at λb/R2=8.93 marked as (I), point (II) at the crossing of ED–MD resonances at λb/R2=6.6, and the intermediate point (III) at λb/R2=8.04. The dipolar resonances within the spectral range up to λb/R2=8.9 are typical subwavelength resonances that stem from the high-index dielectric core, where the intersections between the ED–MD and MD–LSP modes correspond to Kerker points of the second kind, as defined in [23]. We point out here that, for a single dielectric sphere [23], the first intersection—as we move from higher to lower wavelengths—corresponds to the Kerker point of the first kind where directional forward-scattering takes place, while, the second crossing, corresponds to the second Kerker point where strong back-scattering occurs. However, in our setup, due to the presence of an extra electrical dipolar mode, i.e., the LSP of the shell, there is an extra Kerker point of the second type, which is marked as (I) in Figure 2A. To further clarify this, we examine the behavior of the dominant dipolar terms C1,1M, C1,1N versus λb/R2. Both points (I), (II) satisfy the conditions Im(C1,1N)=−Im(C1,1M) and |C1,1N|2=|C1,1M|2 [23] that should hold for a point to be of the second Kerker type. These properties are interpreted as two local minima in g—in the absence of magnetic bias—precisely at points (I) and (II), as can be seen in Figure 2B. Interestingly, g yields one local maximum, indicating directional forward scattering, at an intermediate point marked as (III). Although in this emerged point, the condition C1,1M=C1,1N [23] for Kerker points of the first kind, is now relaxed due to interference with the LSP, the resulted forward scattering is sufficient enough.

To further proceed in the design of active tuning, in Figure 2D we plot a two-dimensional mapping of g versus λb/R2—in the same spectral range as in Figure 2A, B—and versus B, with values of the latter varying from zero to 1.2 T. Focusing on the value λb/R2=8.04—point (III)—we observe a region around 1 T where g changes its positive sign and takes large negative values. The local minimum of negative g values is observed for B=0.9 T, as indicated by the arrow in Figure 2D. We then return to Figure 2B and plot g versus λb/R2 for fixed B=0.9 T (gray curve). This curve reveals a local minimum in the same position where g(B=0 T) has its maximum. To elucidate the physics behind this mechanism, in Figure 2C we plot the scattering cross section spectrum for the same setup as in Figure 2A, but now for B=0.9 T. It is evident that now, the intersection between MD–LSP modes (Kerker point of second kind) moves to point (III), introducing a strong directional back-scattering at this wavelength. We point out here, that the LSP mode continuously shifts to lower wavelengths (from Figure 2A–C), by gradually increasing the external magnetic field from B=0 to B=0.9 T.

To better elucidate the optical response at the MD–LSP intersection in Figure 2C, as well as the spreading of its spectral shape where multiple peaks appear, we examine all terms that play a role in Eq. (5). Particularly, when magnetized, the sphere exhibits both uniaxial (birefringent) and gyrotropic optical response, as evident from Eq. (1) for B≠0. In Ref. [27] it is shown that uniaxial metallic particles mix the electric/magnetic character and multipole order (angular momentum index n) between the modes, followed by a respective |m|-degeneracy lifting. This is also valid for spherical resonators made from a birefringent material [28], [29]. Moreover, the gyrotropy induced by the external magnetic field further lifts the degeneracy between modes with index ±m, resulting in an additional Zeeman-like splitting [30], [31]. However, in the present case, light couples only with the m=±1 terms, since it impinges along the z axis [32]. The small black arrows indicate the location of the m=−1 resonances when obtained solely from the C−1,1N term, while the small gray arrows correspond to m=1 modes, as obtained from C+1,1N the term. These CN terms are sufficient for this analysis since the spectral features associated with the parent LSP mode preserve a predominant electric-dipolar character. With these in mind, we proceed by examining the possible combinations between predominantly-electric and -magnetic dipoles, respectively, i.e., C+1,1N–C+1,1M, C−1,1N–C−1,1M, C+1,1N–C−1,1M, and C−1,1N–C+1,1M. Each combination fulfills the Kerker condition of second kind in the proximity of point (III). Therefore, the strong back-scattering at λb/R2=8.04, i.e., at the intersection between the two dipolar curves pointed out with the large black arrow in Figure 2C, is easily interpreted through a generalized (extension from the isotropic case) Kerker condition of the second kind.

In Figure 2F we plot the normalized (to unity) polar/bistatic (far field) scattering cross section σ(θ), 0⩽θ⩽180∘, on yz-plane, when the spherical antenna operates at λb/R2=8.04. Obviously, directionality inversion occurs through a switching of the external magnetic field from 0 to 0.9 T. Results from HFSS are plotted as dots in Figure 2F and fully coincide with the results obtained from VIE. This particular setup enables forward-scattering when the external bias is off, while when it is on, back-scattering occurs.

Next, the effect of R1/R2 ratio in the design of active tuning is studied in Figure 3. Each subfigure A, B, C examines the behavior of g and σ when R1/R2 takes the values 0.7, 0.8 and 0.9, respectively. The top subfigure in each case depicts the map of g versus λb/R2 and B, while the bottom subfigure depicts the normalized σ/(πR22) when the antenna operates at the respective to Figure 2 point (III). The spectral position of this point is also indicated by the black arrows/circles in each map of g. Judging from Figure 3A–C, in principle, directionality inversion can be achieved for various thin coating widths, nevertheless, the shell’s thickness cannot be increased too much, otherwise the scattering response becomes nugatory, as it is evident from the R1/R2=0.7 case. In addition, the core–shell ratio has an impact on the frequency and external magnetic-field bias values for which the preferential operation is feasible. From the maps in Figure 3A–C it is evident that the location of point (III) shifts to lower frequencies and to lower magnetic-field magnitudes as R1/R2 increases. This behavior is summarized in Figure 3D, E which plot the precise values of λb/R2 and B, for various core–shell ratios, for which directionality inversion takes place.

Figure 3:

Scattering-assymetry maps and polar/bistatic scattering cross sections (far-field plots) for multiple setups with various R1/R2, when R₂ is fixed at 10 μm. (A) R1/R2=0.7. Top: map of g. Bottom: σ/(πR22) at λb/R2=6.56. Red: forward-scattering at 0 T; blue: back-scattering at 1.42 T. (B) R1/R2=0.8. Top: map of g. Bottom: σ/(πR22) at λb/R2=7.34. Red: forward-scattering at 0 T; blue: back-scattering at 1.16 T. (C) R1/R2=0.9. Top: map of g. Bottom: σ/(πR22) at λb/R2=8.04. Red: forward-scattering at 0 T; blue: back-scattering at 0.9 T. (D) Operational wavelength versus R1/R2 for which the setup supports active directionality inversion. (E) Values of B versus R1/R2 for achieving preferential back-scattering.

Focusing on the forward-scattering operation depicted by Figure 3A–C (bottom subfigures/red curves), when the R1/R2 ratio increases, the absolute value of the σ(θ=0∘) point increases, but at the same time, the absolute value of the σ(θ=180∘) point also increases. The same is true for the respective points at the back-scattering operation depicted by Figure 3A–C (bottom subfigures/blue curves). This behavior instructs us to perform an optimization of the setup versus R1/R2 in order to achieve an equal directionality for both operation states. To this end, we examine the variation of the relative forward-to-back ratio Λf≡σ(0∘)/σ(180∘) versus R1/R2 during the forward-scattering operation, and the variation of the respective back-to-forward ratio Λb≡σ(180∘)/σ(0∘), during the back-scattering operation. The results are depicted in Figure 4A. Obviously, Λf, Λb cannot be simultaneously maximized, however there exists the special value of R1/R2=0.836 for which Λf=Λb, thus achieving equal directionality for the forward- and back-scattering operations. This point corresponds to λb/R2=7.6 while the back-scattering operation takes place for B=1.09 T. In Figure 4B we plot the normalized σ/(πR22) when the antenna operates at this specific point. The directionality inversion is evident and an excellent agreement with HFSS is met.

Figure 4:

Optimization of the setup versus R1/R2, while R2=10 μm, in both forward- and back-scattering operation to achieve equal directionality. (A) Ratios Λf, Λb versus R1/R2 for which the setup supports active directionality inversion. Red: Λf; blue: Λb. The black arrow indicates the value R1/R2=0.836 where Λf=Λb. This point corresponds to λb/R2=7.6 and B=1.09 T. (B) σ/(πR22) at λb/R2=7.6. Red: forward-scattering at 0 T; blue: back-scattering at 1.09 T. Curves: VIE; dots: HFSS.

Next we calculate the working bandwidth for our setup. For the designed device with R1/R2=0.9, from Figure 4A we deduce that Λf=2.8 and Λb=3.5 when directionality inversion occurs. We therefore seek in Figure 2D a common range Δf in the dark-red (B=0 T) and in the dark-blue (B=0.9 T) region of point (III), for which Λf and Λb maintain the same aforementioned levels. This yields a working bandwidth of Δf≈27 GHz around λb/R2=8.04 (corresponding to an operation frequency of 3.73 THz). We have verified that for all frequencies inside this range, directionality inversion indeed takes place, with far-field plots similar to the one in Figure 2F. To complement this study, we further examine how the working bandwidth is affected by R1/R2. Focusing on the range from R1/R2=0.8 to 0.9, Δf is maintained at the same level of 27 GHz for all ratios, but interestingly, it can be enhanced at the optimized point R1/R2=0.836 of Figure 4A, where the latter yields Δf≈41 GHz.

Up to this point, the setup is stimulated by a plane wave which impinges parallel to the orientation of the external magnetic field. In Figure 5 we examine how the switching performance is affected due to a relative orientation between Einc(r) and Bzˆ, when the former impinges with an angle θ0 with respect to the Oz semi-axis, while keeping R1/R2=0.9. To this end, the asymmetry parameter g needs to be computed under oblique incidence. To achieve this goal, we consider the primed coordinate system x′y′z′ that is rotated with respect to the original coordinate system xyz, so that Einc(r′) remains x-polarized and propagates parallel to the z′-axis. Then, by applying appropriate transformation formulas for the SVWFs [33], we obtain the expression of Esc(r′) in the rotated frame and proceed to compute g as given by Eq. (2). In Figure 5A we plot the scattering cross section ratios Λf, Λb versus incidence angle θ0. Independently from θ0, Λf is constant since when B=0 T, the coating is isotropic and the incidence orientation does not affect g. Contrariwise, Λb can reach a minimum and a maximum value at θ0=30∘ and θ0=90∘, respectively. The corresponding bistatic scattering cross sections are depicted in Figure 5C, D. From the bistatic scattering cross section plots we deduce that, for the given configuration, when 0∘⩽θ0<30∘, the forward-scattering point σ(θ0) when B=0 T, is always smaller than the back-scattering point σ(180∘−θ0) when B≠0 T. This can be confirmed by Figure 3C (bottom subfigure) where θ0=0∘, but also from other σ/(πR22), θ0<30∘ plots that have been computed and not given here. On the contrary, when θ0⩾30∘, σ(θ0)|B=0 T overpasses σ(180∘−θ0)|B≠0 T. This is evident from Figure 5C, D where, at the limiting case where Λb reaches its minimum—i.e., at θ0=30∘—σ(θ0)|B=0 T≡σ(180∘−θ0)|B=0.86 T, while at θ0=90∘, σ(θ0)|B=0 T>σ(180∘−θ0)|B=0.83 T. It should be noted that at oblique incidence, although the main lobe in the forward-scattering pattern points precisely at the θ0-direction, the back-scattering pattern has a tilt with respect to the (180∘−θ0)-direction. This can be observed from the θ0=30∘ case in Figure 5C where a 15° tilt exists in the back-scattering direction, making the main lobe of the far-field pattern to point at 165°, rather than 150°. Finally, Figure 5B depicts the values of B needed for achieving preferential back-scattering at the prescribed angles θ0. As we shift from parallel to normal orientation between Einc(r) and Bzˆ, the B values should be slightly decreased. In addition, λb/R2=8.04 for all θ0 since when B=0 T, the oblique incidence does not affect the asymmetry parameter g.

Figure 5:

Study of the switching performance under oblique incidence of the impinging plane wave for R1/R2=0.9, while R2=10 μm and λb/R2=8.04. The external magnetic field is always z-directed. (A) Ratios Λf, Λb versus θ0. Red: Λf; blue: Λb. (B) Values of B versus θ0 for achieving preferential back-scattering. (C) σ/(πR22) at θ0=30∘. Red: forward-scattering at 0 T; blue: back-scattering at 0.86 T. (D) σ/(πR22) at θ0=90∘. Red: forward-scattering at 0 T; blue: back-scattering at 0.83 T.

In Figure 6 we plot the near-fields and in particular the normalized magnitude |E|/max(|E|) of the total electric field on yz-plane. The field patterns in Figure 6 correspond to the R1/R2=0.9 ratio antenna when the latter operates at the preferential point λb/R2=8.04 under forward-scattering (Figure 6A) and back-scattering (Figure 6B) activity. The forward- and back-propagation in the two states can be clearly seen from field distribution in the background free-space medium. Figure 6C, D are zoom-in versions of A, B, respectively. The field pattern inside the sphere in Figure 6C is indicative of a MD resonance, although point (III) in Figure 2A does not precisely correspond to the MD resonance at λb/R2=7.44. Contrariwise, the field pattern in Figure 6D is more complicated owing to the gyroelectric nature of shell’s permittivity due to the external magnetic field. This yields a strongly localized field inside the shell, but inside the core the similarity of the pattern to a MD resonance is evident.

Figure 6:

Near-field images of the normalized magnitude |E| of the total electric field on yz-plane. The values of parameters are the same with the ones in Figure 3C and the antenna is stimulated at λb/R2=8.04. (A) Forward-scattering operation at 0 T. (B) Back-scattering operation at 0.9 T. (C) Zoom-in plot at 0 T. (D) Zoom-in plot at 0.9 T.

4 Scalability to the nano regime

Recently, it was experimentally demonstrated that static magnetic field-assisted resonant topological cavities, provide a platform for the implementation of photonic waveguides in the quantum regime [34]. In addition, it was theoretically and experimentally observed that graphene-based inhomogeneous strains can induce pseudomagnetic effects in the optical regime that are similar to the magnetic effects produced by the magneto-optical devices in the THz regime [35]. In particular, such effective magnetic fields can reach thousands of Tesla, exceeding standard laboratory values [36]. As a consequence, this would also increase the working bandwidth of the proposed device at the nanoscale, since the latter is linearly dependent on the magnetic field [20]. This rescaling observation led us to apply our analytical model in the THz regime, and export various intrinsic properties being scalable to shorter wavelengths.

5 Conclusion

In summary, we have shown that by judiciously designing a high-index spherical microresonator coated by an active magneto-optical material, full directionality inversion can be achieved. Based on analytical calculations and corroborating full-wave electrodynamic simulations, we have shown that by actively tuning the constitutive parameters of the magneto-optical coating by means of an external magnetic bias, tunable forward- and back-scattering functionality occurs. Our design is based on a Drude–Lorentz dispersive coating operating above its plasma frequency. The permittivity properties of the magneto-optic material change from those of an isotropic material, in the absence of magnetic field, to those of a gyroelectric one when the magnetic field is applied. Our results may allow the flexible and with low energy cost active switching of emerging metasurface and metalenses devices for applications like beam steering [37], as well as tunable nanophotonic structures such as magneto-metasurface devices for one-way transmission and sensing [38], or functional metamaterials like magneto-optical core–shell-based composed metamaterial devices for optical memory control [39].