Abstract
1. INTRODUCTION
Before the proposal of Bohr’s model, in the early days of electron discovery, several models, such as non-radiating sources that do not radiate energy into the far-field, had been considered to explain the atomic structure and address the paradox of unstable atoms [1–3]. While Bohr eventually addressed this problem, the idea of non-radiating sources extended to other branches of science, such as the physics of elementary particles, quantum field theory, nuclear science, and astrophysics [4–12]. Despite considerable efforts, only recently an optical non-radiating state, known as anapole (from Greek “ana,” “without,” thus meaning “without poles”), has been experimentally demonstrated in high refractive index engineered dielectric nanoparticles, known as meta-atoms [13–19]. In particular, contrary to the plasmonics nanoparticles, all-dielectric scatterers, and their two-dimensional (2D) periodic arrangements, known as metasurfaces [20–26], provide an alternative, low-loss route to light-matter interaction at the nanoscale and have enabled various remarkable phenomena and applications, including Kerker, anti-Kerker, transverse Kerker effects [27–33], lattice resonances [34–36], beam steering [37–39], holography [40–42], invisibility cloaking [43–46], optical force manipulation [47–50], absorption engineering [51–54], and nonlinear harmonic generation [55–60].
The multipole decomposition of electromagnetic fields can characterize light interactions with all-dielectric nanostructures [29]. In particular, such a theoretical framework provides an opportunity to introduce the so-called toroidal contributions [61–65], enabling a platform to achieve unprecedented optical phenomena such as the formation of non-scattering anapole [15–19,66–68] and super-scattering [69–74] states, which are two opposite extreme regimes of light-matter interactions. In the scattering cross-section (SCR) spectra, each multipole corresponds to an independent scattering channel with an upper bound known as the single-channel limit, which is defined by , wherein represents the order of the multipole. However, the constructive interference of at least two resonant modes can surpass this limit resulting in a phenomenon of super-scattering, and enabling a plethora of applications, including solar cells and sensing [75–78], to name a few. Contrary to the super-scattering regime, anapole states form as a result of the destructive interference between primitive moments and their toroidal contributions [13–19]. Vanishing scattering accompanied by strong energy confinement in high-index dielectric meta-atoms supporting electric dipole anapole (EDA) states has recently attracted much research interest in the areas of strong exciton coupling [79,80], second- and third-harmonic generation [81–85], Raman scattering [86,87], photothermal effect [88], energy guiding [89,90], and lasing [91]. While to date, most of the works in this field of research are limited to the studies of EDAs, other types of nonradiative states, such as magnetic dipole anapole (MDA) and electric and magnetic quadrupole anapoles (EQA and MQA, respectively) remain largely unexplored. However, such higher-order anapole states can enhance radiation suppression and confinement of electromagnetic energy if designed to spectrally overlap with the corresponding dipolar moments [92]. The intriguing potential of non-scattering regimes—resulting from the simultaneous destructive interference of electric and magnetic Cartesian multipoles with their toroidal contributions—has garnered attention in several studies in recent years [92–97]. Notably, the experimental observation of the hybrid anapole (HA) was recently achieved in the seminal work by Valero
Figure 1.Schematic representation of the formation of a hybrid anapole state within an all-dielectric cuboid meta-atom with the width and height of
2. UNDERLYING PHYSICS OF HYBRID ANAPOLES
The optical response of nonmagnetic meta-atoms can be described using the multipole expansion approach, which can be performed using either a spherical or Cartesian basis [98–102]. The former is based on the multipole decomposition of electromagnetic fields in terms of the spherical harmonic coefficients, leading the total scattering cross-section to be defined as a series of coefficients as , where and represent the spherical electric and magnetic multipole scattering coefficients, respectively [101]. On the other hand, the second approach is based on the multipole decomposition of the Cartesian components of induced current density within the meta-atom as , with denoting a tensor of rank that corresponds to different Cartesian multipoles [102]. According to the theory of irreducible multipoles, the scattering cross-section of a subwavelength meta-atom can be expressed as (see Appendix A)
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As can be seen from Eqs. (1)–(3), the direct consequence of irreducible Cartesian multipole expansion is the possibility of distinguishing higher-order toroidal contribution within the SCR spectrum, thus providing an opportunity to induce various anapole states by satisfying the following conditions:
Equation (4) provides the conditions for exciting any type of higher order anapole state (up to MQA) at a particular frequency . Note that, while the destructive interference between the first and second type of electric toroidal dipole moments [] leads to so-called toroidal anapoles, we are not considering this case here due to the negligible strength of (see Appendix B).
3. DYNAMICS OF HYBRID ANAPOLE
The excitation of HAs within a polycrystalline silicon cuboid meta-atom has been studied using the numerical simulations based on the finite-element method (FEM) implemented in the commercial software COMSOL Multiphysics (see Appendix C for details). The cuboid scatterer is assumed to have a square cross-section with a width of in and directions, and a height of in the direction, as shown in Fig. 1. To investigate the effect of the geometry of the cuboid meta-atom on its scattering characteristics, first, we fixed the meta-atom width to and changed its height from 300 to 500 nm, while in the second scenario, the height was set to , and the width was changed from 100 to 310 nm over the wavelength range of 850–1150 nm. The contributions of the various multipolar moments were obtained by integrating the displacement current induced within the meta-atom using Eqs. (2) and (3), and their corresponding normalized total scattering cross-sections are shown in Figs. 2(a) and 2(b) in logarithmic scale for the former and latter cases, respectively. As can be seen from these figures, the scattering response of the meta-atom changes significantly as a result of varying the height and width of the meta-atom. In particular, when the height and width of the scatterer change in the range and , significant suppression of the peaks in the optical response of the meta-atom can be observed, marked by dashed lines, with the minimum value of .
Figure 2.Normalized total scattering cross-section of the cuboid meta-atom as a function of (a) height and (b) width and an operating wavelength under plane-wave illumination in logarithmic scale. The contribution of ED, MD, and EQ multipolar moments excited in the meta-atom as a function of (c) height and (d) width with respect to wavelength. By changing the topology of the meta-atom, several resonances emerge and disappear at certain spectral positions, leading to the formation of resonant branches that shift toward longer wavelengths as the size of the meta-atom increases.
To clarify the physical origin of such a significant suppression of far-field radiation, the contributions of each higher-order multipolar moment (i.e., primitive + toroidal) are plotted in Figs. 2(c) and 2(d) as functions of height and width over the same wavelength range of 850–1150 nm, respectively. As can be seen from these panels, upon varying the geometrical dimensions of the scatterer, several branches of resonances, marked by black dashed lines, emerged within the scattering spectra. In contrast to the optical response of other multipoles, two resonance scatterings can be seen upon varying the dimensions of the meta-atom, which shift toward longer wavelength as both the height and width of the scatterer increase. In particular, changing the height of the meta-atom results in the shift of the minima within the MD and EQ spectra, whereas the ED and EQ resonance scatterings shift as the result of varying the width of the meta-atom (see Appendix D for simulation results of MQ). Therefore, by changing the dimensions of the meta-atom, the spectral position of the minima in its scattering spectra can be overlapped at the same operating wavelength, leading to the regions with suppressed radiation, as shown in Figs. 2(a) and 2(b).
Despite the existence of spectral regions with suppressed far-field radiation, such an optical response does not necessarily indicate the excitation of the HA state and the satisfaction of conditions given by Eq. (4). Indeed, the suppression of the far-field radiation pattern may be attributed to the near-zero values of the induced displacement currents within the meta-atom [i.e., ], which yields the simultaneous suppression of both primitive [Eq. (2)] and toroidal [Eq. (3)] moments. However, such an optical behavior differs from the HA response as it does not lead to the confinement of energy within the meta-atom. According to Eq. (4), to excite an anapole state of any type, its corresponding primitive and toroidal contributions should destructively interfere with one another, that is, , where is the intensity of a particular moment, while () and denote the amplitude (phase) of its primitive and toroidal contributions, respectively. The intensity of a particular channel vanishes in the far-field, if the following two conditions are simultaneously satisfied: and . To study the origins of the emerged dips within the scattering spectra, we evaluate the explicit contributions of both primitive and toroidal moments as functions of height () over the wavelength range between 900 and 1200 nm (see Appendix E for the results of the corresponding study of the moments as functions of width). Next, we identify the regions in the parameter space wherein the conditions of inducing anapoles and are satisfied with respect to the operating wavelength and height of the meta-atom as it is shown in Fig. 3. It should be mentioned that, since the incident electric field is polarized along the direction, we have only considered the contributing multipoles of , , and , for primitive and toroidal, respectively.
Figure 3.Explicit contributions of amplitude and phase of primitive and toroidal moments as functions of height and wavelength for (a), (b) ED; (c), (d) MD; and (e), (f) EQ. An anapole of any kind is excited in regions wherein both conditions are simultaneously satisfied. The vertical dashed line corresponds to the dimension at which the HA state can be excited up to the EQA at the operating wavelength of
As can be seen from Figs. 3(a) and 3(b), while several branches of resonances emerge within the amplitude spectra of the ED multipole, the phase difference condition () is only fulfilled for two branches, leading to the excitation of electric dipole anapoles that are red-shifting toward longer wavelength once the height of the meta-atom increases from 400 to 500 nm. On the other hand, despite the fact that individual amplitude and phase conditions for the MD moment occur in the broader spectral range [see Figs. 3(c) and 3(d)], the magnetic dipole anapole is excited in a narrower range compared to its electrical counterpart since both necessary conditions are fulfilled simultaneously in a narrow spectral region. Moreover, the same scattering response as that of the EDA can be found for the electric quadrupole anapole, as shown in Figs. 3(e) and 3(f). In particular, in contrast to the recent study that demonstrated the excitation of HA within all-dielectric scatterer [92], both the amplitude and phase conditions for exciting ideal EQA are satisfied in the spectral range of interest for the cuboid geometry, making it an appropriate platform for confining electromagnetic energy at the nanoscale. To the best of our knowledge, this is the first prediction of the possibility for exciting an ideal electric quadrupole anapole (both conditions are satisfied exactly as and ) in an all-dielectric subwavelength scatterer. It should be noted that, despite the excitation of three higher-order anapoles in the spectral window of interest, the MQA conditions are not satisfied within this range, which is mainly attributed to the fact that the phase difference between and is much smaller than . Nevertheless, for , the MQA is excited, which appears to be the first theoretical observation of such a type of HA (see Appendix D for more details). From Fig. 3, we choose the dimensions of the cuboid meta-atom to be and , shown with a dashed white vertical line, to excite HA states at around (indicated with a horizontal line) up to the electric quadrupole anapole. To explicitly demonstrate the formation of the HA within such a meta-atom, we calculate its optical response with respect to the operating wavelength, as shown in Fig. 4.
Figure 4.Calculated amplitudes and phase differences between the primitive and toroidal contributions of a cuboid meta-atom with
Figures 4(a)–4(c) demonstrate the amplitude of the primitive (solid line) and toroidal (dotted curve) contributions together with their relative phase difference up to electric quadrupole multipole as a function of the operating wavelength. As can be seen from these panels, at the operating wavelength of , the general conditions of Eq. (4) are satisfied ( and ), leading to the formation of an HA state within the meta-atom. In particular, by carefully choosing the dimensions of the cuboid meta-atom, it is possible to excite the anapoles of all the contributing moments in close vicinity of each other, leading to a strong suppression of radiation in the far-field, as shown in Fig. 4(d). Note that, despite the presence of a dip in the spectrum of , it does not correspond to the formation of MQA as the condition for the phase of MQA is not satisfied at this wavelength (see Appendix D for more information). In addition, the sum of the contributing moments [] and the total scattering cross-section directly computed from COMSOL are in excellent agreement, proving that only the first four multipoles (up to MQ) are sufficient for explaining the optical response of the cuboid meta-atom. The energy confinement at different spectral positions within the scattering spectra of the meta-atom is also shown in Fig. 4(e). For this purpose, we evaluated the total electromagnetic energy, , integrated over the meta-atom volume, and normalized it to the trivial case of a transparent particle (same geometry as that of the cuboid but with the refractive index of ) with the total energy of . As shown with the purple color, at the condition of the minimum scattering, the calculated energy within the particle is non-zero and is almost 20 times larger than that of the trivial case (), which indicates the significant localization of energy distribution inside the meta-atom. It should be noted that, while the same phenomenon has been recently observed in a recent study conducted by Valero
The quality factor (-factor) serves as a vital metric to gauge light-matter interaction strength. To determine the -factor of the presented scatterer, we implemented the quasi-normal mode (QNM) expansion framework [103,104]. In this perspective, the -factor of the dominant QNMs is calculated to be with the mode lifetime of , which is higher than that of the traditional Mie-type resonances and can be a suitable alternative for material-intrinsic applications, such as nonlinear harmonic generation [58,105,106], nanolasers [107], and active meta-photonics at nanoscale [108,109]. Note that such simultaneous energy storage and minimized scattering (non-radiative) regime occurred in the steady state, which is preceded by the transient period during which the energy is being accumulated in the cavity. It is also important to emphasize that the non-radiating anapole state, whether it is an EDA or HA, does not naturally arise as an eigenmode of the scatterer [110]. In other words, the anapole state cannot sustain itself as a self-oscillating cavity mode satisfying boundary conditions independently of any incident field. Instead, it represents a resonant distribution of induced currents that can be excited through a well-designed interplay between the incident beam’s topology and the scatterer’s geometry in the steady state as long as the incident beam interacts with the meta-atom [110]. However, once the light beam has completely passed through the scatterer, the anapole field distribution, which cannot independently satisfy the boundary conditions on the scatterer’s surface, begins to radiate and rapidly dissipates its stored energy over time [110–113]. As was mentioned earlier, while the majority of the previous studies in the field of non-radiating states were limited to the excitation of electric dipole anapoles, recently, improved suppression of radiation and higher energy confinement as compared to the EDAs have been demonstrated at the HA state excited within a cylindrical meta-atom [94]. Therefore, we have also compared the optical response of our proposed cuboid meta-atom with that of a cylindrical particle that supports first- and second-order EDAs (see Appendix F). As counterintuitive as it can appear, despite the significant difference between the volumes of the two meta-atoms (), the scattering intensity of the HA state is found to be suppressed more than 23 times compared to its first-order EDA counterpart, whereas its stored energy is almost three times higher. We also note that the HA state corresponds to the 48 times better suppression of radiation to the far-field than its higher-order electric dipole anapoles, while its stored energy exceeds 2.5 times of second-order EDAs. Figure 4(f) shows the distribution of the electric field within the cuboid meta-atom at the corresponding wavelength of the HA state in three different planes of , , and . Contrary to the field distribution in the plane, which renders the well-known EDA field profile with a vortex-like distribution of the displacement current shown with black arrows, the internal distributions of the fields in and planes exhibit more complicated profiles that are attributed to the mixture of electric and magnetic multipole moments. As the final remark, we note that, contrary to the concept of dark mode in plasmonics, where a resonance with null net dipole moment cannot be excited by a planar incident wave due to symmetry reasons, the provided HA state can be efficiently excited by a plane wave (or Gaussian beam), simplifying the future experimental studies.
4. FROM NON-SCATTERING TO SUPER-SCATTERING
It has been previously shown that the optical response of a subwavelength particle strongly depends on its orientation with respect to the direction of illumination such that certain spectral features emerge or get suppressed [114]. Such spectral dependence stems from the changes in the components of multipolar moments and the variations in the coupling efficiency of a particular Mie resonant mode as a function of illumination direction. Therefore, tuning the illumination angle can be used as another degree of freedom for controlling the spectral position of the higher-order anapoles in the spectral window of interest. Here we investigate how higher-order anapoles change as a function of illumination angle. First, we fixed the dimensions of the cuboid particle to be and and swept the illumination angle from to over the wavelength range of as it is schematically demonstrated in the inset of Fig. 5.
Figure 5.(a) Normalized total scattering cross-section of the cuboid meta-atom under plane wave incidence with the angles
Figure 5(a) shows the total scattering cross-section of the cuboid meta-atom, which exhibits a symmetric response with respect to the angle of incidence due to the symmetric topology of the scatterer in the plane of incidence. Depending on the illumination direction, resonant peaks emerge or get suppressed leading to the formation of a region where the radiation to the far-field is strongly suppressed (HA state). As opposed to the conventional EDAs, the excitation of such a non-radiating state is possible in a wide range of incident angles from (rather than only at ) as shown with purple dashed line in Fig. 5(a). The multipole decomposition results shown in Figs. 5(b) and 5(c) show that tuning the angle of incidence yields the excitation/suppression of new resonances. In particular, several dips in the scattering spectra of the meta-atom are observed under oblique incidences, which are attributed to the destructive interference between the primitive and toroidal moments of the contributing Mie resonances as long as they are overlapping with the black dashed lines that are representing the regions wherein the phase difference conditions for inducing anapole states () are satisfied. Taking the electric dipole response as an example, we note that only three lines coexist with the emerged resonant dips, indicating that, for these particular branches, both the amplitude and phase conditions are satisfied, and the EDA can be obtained and tuned with respect to the incident angle. For the magnetic dipole and electric quadrupole anapoles, such behavior becomes more complicated, and the anapole conditions are satisfied in the two regions shown in these plots. Interestingly, the higher-order anapole states can also be obtained at , making them suitable platforms for on-chip applications. Figure 5(e) shows the optical response of the meta-atom corresponding to the incident angle changes from frontal () to lateral () illumination in the steps of 30°. While the overall scattering response of the scatterer changes significantly once the illumination angle varies, the optical behavior of the meta-atom at the operating wavelength of gradually transitions from a non-radiating hybrid anapole state (for ) to a radiative mode corresponding to a magnetic quadrupole (for ), as shown with the blue dashed curve. On the other hand, the scattering response of the same meta-atom can be tuned at from the radiative to the non-radiative state upon the continuous change of the illumination angle from to , respectively. We note that the observed optical phenomena are not related to the common shifts in the scattering spectra studied previously [29] and result from the change in the coupling efficiency of light to a particular Mie-type resonant mode once the direction of illumination is changed. It should be remarked that, unless only one multipole determines the light scattering, the value and accuracy of moments depend on the choice of the origin of the coordinate system and the position where the multipoles are assumed to be located. As shown previously [99–102], for arbitrary shaped particles, it is convenient to choose the origin of coordinate as the center mass of the scatterer. However, when the meta-atom in question consists of a cluster of scatterers, the system center of mass does not necessarily coincide with the origin of the coordinate, and the moments are assumed to be shifted with respect to the origin [115–117]. Recently, the pioneering work proposed by Ospanova
So far, we have discussed the excitation of an HA state within a cuboid meta-atom possessing inversion symmetry along the axis. In particular, the inversion symmetry of the optical scatterer results in the eigenmodes consisting of either only even (MD, EQ, etc.) or only odd (ED, MQ, etc.) multipoles, which can constructively/destructively interfere with one another as schematically shown in Fig. 6(a). However, as it has been recently shown by Poleva
Figure 6.Schematic demonstration of achieving bianisotropic responses via breaking the inversion symmetry of the meta-atom. For (a) symmetric particles, the eigenmodes consist of either only even or odd multipoles, while for (b) asymmetric meta-atoms, the optical response consists of multipoles of mixed parities. (c) Calculated response of the pyramid meta-atom with respect to the top-to-bottom width ratio and operating wavelength. Four various points corresponding to different values of
As can be seen from the calculated response, a strong super-scattering regime () can be obtained for a near-perfect inversion symmetric meta-atom () in a narrow spectral range, whereas an asymmetric meta-atom () can support super-scattering optical response in a broad spectral range of 900–1200 nm with lower amplitude of 1.6. Such an optical response is a result of breaking inversion symmetry, which not only leads to a bianisotropic response but also achieves super-scattering behavior that exceeds the single-channel limit by a factor of 2.3. The contributions of each moment (up to MQ) are also calculated and provided in Appendix G. In Fig. 6(d), we demonstrate the far-field intensity of the meta-atom for [marked by stars in Fig. 6(c)] at the operating wavelength corresponding to the maximum value of the super-scattering spectrum marked in Fig. 6(c). As can be seen from these radiation patterns, while for all the proposed cases directional scattering can be observed in the super-scattering regime, the magnitudes of the radiation patterns increase once the meta-atom geometry changes from asymmetric to inversion symmetric topology. Such a variation in the directivity of the meta-atoms is due to the spectral overlap of all the resonant modes for , whereas for other cases () fewer multipoles are overlapping with one another. These results demonstrate the role of inversion symmetry in realizing the super-scattering regime. Although recently it has been argued that the SCR of a scatterer with arbitrary size and shape does not have a maximum limit [123], the scattering limitations defined for spherical scatterers () are remained in the literature as a benchmark for establishing super-scattering regime [69–74]. We note that the transition from the non-scattering to super-scattering states can be explained in the perspective of Fano-like profiles. Fano resonance typically emerges when a discrete quantum state interferes with a continuum of states. This interference can be described in the scattering spectrum, by the modified Fano formula as , wherein represents the energy; is the Fano parameter; , with and denoting the resonance energy and linewidth; and is the phase shift between the resonant mode and the background contribution coming from other modes and nonresonant scattering [124,125]. In particular, in the modified Fano formula, denotes the smooth peak envelope and represents the background contribution of the scatterer’s nonresonant modes. While it has been shown that the spectrum of Mie scattering can be presented in the form of an infinite series of the Fano profiles [126], here we just limit ourselves to the spectral regions wherein the desired optical response, being non-scattering or super-scattering, is presented. In the context of the HA within (corresponding to ) and under normal incidence, the optical response exhibits Fano-like asymmetric behavior, with the Fano parameter of representing minimized radiative losses due to the interference between primitive and toroidal contributions. The formation of the super-scattering state can similarly be described via the Fano profile. In particular, by varying from 0.1 to 1, the Fano parameter changes from to 1.49, leading to different scattering profiles, with the alternation of scattering enhancement compared to the single channel limit (at their corresponding wavelengths) from 1.26 to 2.3. Interestingly, the provided results indicate that breaking the out-of-plane symmetry allows for on-demand tuning of the asymmetry factor (q), yielding the manipulation of the meta-atom’s quality factor and field confinement at nanoscale. We also note that the same optical phenomenon can be observed under different angles of illuminations as shown in Fig. 5. In particular, once the incident angle deviates from , the non-radiating state tends to be radiative, which manifests itself in the change of the Fano parameter value and, consequently, the scattering spectrum. Such a change in the Fano parameter value is a manifestation of variation in the phase shift () between the dominant resonant mode and the background contribution coming from other modes. As a final remark, we note that, while the exploration of hybrid anapoles has been pursued before, our paper uniquely navigates two contrasting regimes of light-matter interaction on a unified platform. We unravel the intricacies of transitioning from non-scattering to super-scattering and the pivotal role illumination angles play in achieving specific radiative features. Our findings dive deep into the dynamics of various anapole types, shedding light on how geometry and symmetry intricately shape the optical response and energy confinement within a meta-atom.
5. CONCLUSION
As opposed to the conventional EDAs, in this work, we predicted and demonstrated new non-radiative states supported by cuboid meta-atoms, formed due to the destructive interference between electric and magnetic primitive and toroidal contributions. In particular, we theoretically predicted and numerically showed that a cuboid meta-atom can support higher-order non-radiative states up to the electric quadrupole moment at the same operating wavelength, leading to significant suppression of the far-field radiation and a remarkable 20-fold enhancement of energy stored inside the subwavelength dielectric particle. We also explored the role of the illumination angle in the optical response of such a non-scattering state and demonstrated that, depending on the orientation of the meta-atom, its scattering response switches from non-radiating to radiating states. Moreover, we revealed that tuning the topology of the meta-atom from a symmetric to an asymmetric one leads to the super-scattering state, which is an opposite regime of light-matter interaction as compared to its non-scattering counterpart, with 130% enhancement in its scattering behavior. The discovered concept of the hybrid anapole state as well as the anapole to super-scattering transition is likely to enable a plethora of potential applications ranging from remote sensing relying on the shape and orientation of meta-atoms to the enhancement of nonlinear conversion efficiency and establishing strong coupling between photonic and excitonic platforms.
APPENDIX A: MULTIPOLE DECOMPOSITION
Recently, several theoretical frameworks, which are based on the irreducibility of Cartesian tensors with respect to the group, have been proposed to unambiguously describe the toroidal contributions in the scattering spectra of optical scatterers [
APPENDIX B: ELECTRIC TOROIDAL DIPOLES OF THE FIRST AND SECOND KINDS
The destructive interference between the first and second types of electric toroidal dipole moments [] can also lead to the toroidal anapole state. However, on account of the negligible strength of , compared to its first-type counterpart, the observation of such a non-radiating state has remained within the theoretical framework, and to the best of our knowledge, it has not been shown numerically or experimentally so far. Figure
Figure 7.Calculated amplitudes and phase differences between the electric primitive and first- and second-type toroidal multipoles of a cuboid meta-atom with
As can be seen from this figure, the amplitude of the second-type electric toroidal moment (shown with blue circles) is one order of magnitude lower than that of its first type in a broad spectral window of interest. However, at the shorter wavelength, the amplitude of the toroidal contributions becomes comparable, providing an opportunity to achieve a toroidal anapole once its corresponding condition is satisfied [i.e., and ]. In particular, such a criterion is fulfilled at the spectral position of , as highlighted with a green bar in Fig.
APPENDIX C: NUMERICAL SIMULATIONS
The numerical simulations have been carried out using the finite-element method (FEM) implemented in the commercial software COMSOL Multiphysics. In particular, we utilized the Wave Optics Module to solve Maxwell’s equations in the frequency domain together with proper boundary conditions. We used a spherical domain filled with air and having a radius of as the background medium, while perfectly matched layers of thickness were positioned outside of the background medium to act as absorbers and avoid undesired scattering. The tetrahedral mesh was also chosen to ensure the accuracy of the results and allow numerical convergence. As we have recently shown in Ref. [
Figure 8.Real (blue line) and imaginary (red curve) parts of polysilicon refractive index.
APPENDIX D: OPTICAL DYNAMICS OF MAGNETIC QUADRUPOLE
To investigate the effect of meta-atom geometry on the magnetic quadrupole moment, we follow the same procedure outlined earlier. In the first case, we fix the meta-atom width to and change its height from 300 to 500 nm, while in the second scenario, the height is set to , and the width is changed from 100 to 310 nm over the wavelength range of 850–1150 nm. The dynamics of the MQ moment are studied by integrating the displacement current induced within the meta-atom using Eqs. (
Figure 9.Contribution of MQ moment excited within the cuboid scatterer as a function of (a) height and (b) width with respect to wavelength. Changing the height of the meta-atom has a negligible effect on the MQ response, while the variation of its width can significantly shift the scattering response. (c), (d) Contributions of both primitive and toroidal MQ multipole moments as functions of height and wavelength. At the spectral position wherein the optical response of the meta-atom is mainly governed by higher-order anapole states up to EQA (shown with black dashed lines), the conditions for satisfying magnetic quadrupole anapole are not fulfilled.
As seen from Figs.
APPENDIX E: EXPLICIT CONTRIBUTIONS OF PRIMITIVE AND TOROIDAL MOMENTS AS FUNCTIONS OF WIDTH
To study the effect of width on the optical response of the cuboid meta-atom and clarify the origins of the emerged dips within the scattering spectra shown in Fig.
Figure 10.Explicit dynamics of (a) amplitude and (b) phase of primitive and toroidal contributions as functions of height and wavelength for ED, MD, EQ, and MQ. An anapole of any kind is excited in regions wherein both conditions are simultaneously satisfied. The vertical dashed line corresponds to the dimension at which the HA state can be excited up to the EQA at the operating wavelength of
As can be seen from these figures, while three branches of resonant dips emerge within the amplitude spectra of the ED multipole, the phase difference condition () is merely fulfilled in two branches, leading to the excitation of electric dipole anapoles. On the other hand, by changing the width of the meta-atom, typically for , the magnetic dipole anapole can be obtained in a broader spectral range of 900–1000 nm. Interestingly, the EQA response is significantly altered by changing the meta-atom width, and both the amplitude and phase conditions are satisfied in a broad spectral range.
APPENDIX F: OPTICAL RESPONSE OF HYBRID ANAPOLE COMPARED TO CONVENTIONAL EDA
Here, we compare the optical response of a cylindrical meta-atom supporting higher-order anapole (HAOM) states [
Figure 11.(a) Evolution of the scattering cross-section under the continuous change of the diameter-to-height aspect ratio. (b) Multipole decomposition and (c) stored energy spectra for a cylindrical particle with
Figure 12.Calculated total scattering cross-section (blue color) and stored energy (red color) of two meta-atoms supporting HA (solid line) and first-order EDA (dashed curve) as a function of the operating wavelength. The scattered field’s
As a final remark, according to the results provided in Fig.
APPENDIX G: CALCULATED MULTIPOLE MOMENTS AS FUNCTIONS OF TOP-TO-BOTTOM RATIO
As shown in the main manuscript, breaking the scatterer’s out-of-plane inversion symmetry makes changing its optical response from a non-scattering to a super-scattering state possible. Here, we demonstrate how the excited multipole moments within the scatterer are altered as a function of the top-to-bottom width ratio () and operating wavelength as shown in Fig.
Figure 13.Calculated scattering cross-sections of the cuboid meta-atom for (a) ED, (b) MD, (c) EQ, and (d) MQ multipole moments as functions of
As can be seen from these figures, we conclude that, once the out-of-plane inversion symmetry is broken (), the scattering response of the excited moments within the cuboid meta-atom varies significantly, yielding the emergence and suppression of new resonances for various values. Such a variation in the optical response of the subwavelength scatterer suggests that the asymmetric parameter can be used as another degree of freedom for the scattering manipulation at the nanoscale. In particular, for , the strong magnetic quadrupole anapole state emerges within the scatterer’s optical spectra, shifting toward a longer wavelength as the asymmetric parameter increases.
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