1. Introduction

Magnetic materials have achieved much attention in optics as they possess the ability to turn the electromagnetic orientation of the polarized light, which can be characterized by the magneto-optical (MO) Faraday effects in transmission or Kerr effects in reflection. The rotation angle is restricted by the path length of light in the medium and a coefficient relative to the magnetic susceptibility and refractive index of the materials [1]. In order to improve rotation angle, the path length should be as long as possible, which will increase medium absorption, reduce light transmission and is badly detrimental to the Faraday effects. Recently, it was found that surface plasmons provide an optimal choice for enhancing MO effects of the magnetic materials [2–5]. When a magnetic film is covered by a noble-metal film with perforated subwavelength holes or in a designed nanostructure, known as magneto-plasmonic crystal (MPC), the resonant plasmons as induced by the incident light on the metal surface can localize light and alter the field distribution, which promotes the transformation of the transverse magnetic (TM) field and transverse electric (TE) filed of the polarized light and brings extraordinary optical transmission (EOT). For example, Chin et al. [6] reported a large enhancement of the Faraday effect over a rather broad bandwidth for the periodic gold nanowires on a bismuth iron garnet (BIG) thin film. The Faraday rotation is increased by up to 8.9 times compared with the bare film, while high transparency is maintained. In a bilayer system of a metallic film perforated with subwavelength hole arrays and a uniform dielectric film magnetized perpendicular to its plane, Belotelov et al. [7] also found a large Faraday rotation enhanced by an order of magnitude and high transmittance of about 35%. By using the Au antennas stacked on the bismuth-substituted yttrium iron garnet film, Kharratian et al. [8] yielded few degrees of rotation in a broad band with a maximum exceeding 6.5°. It is well known that the surface plasmons are primarily restricted by the nanostructure of noble metals. The proper choice of the gyrotropic nanostructure geometry allows acquiring the EOT and extraordinary Faraday rotation where the localized surface plasmons (LSPRs) and surface plasmon polaritons (SPPs) can be excited efficiently [7,9–12]. Nevertheless, in the previous research, much attention mainly focused on improving the MO properties of a specific MPC. The electromagnetic field distribution and MO characteristics of the MPCs in different morphology and magnitude were rarely studied and compared with each other, which may hinder the choice for the best performance.

Therefore, in this article, the MPCs composed of bilayered Au/BIG film and trilayered Au/BIG/Au film were constructed and analyzed by the finite-difference time-domain (FDTD) method. The optical and MO properties of the bilayered Au/BIG film with different equilateral polygon holes in the top Au film were firstly clarified, in addition to the comparison of the electromagnetic field distribution. Under the fixed structure of the polygon holes with best Faraday effects, the evolution of the light deflection and transmission with the size and period of hexagonal hole arrays, as well as the thickness of Au and BIG layers was illustrated. Finally, Faraday effect of the trilayered Au/BIG/Au film under various thickness of the bottom film, incident light polarization and orientation was examined. By systematically optimizing the geometric parameters of the crystal structure, we have obtained both high transmittance and large Faraday rotation of the MPC in the near-infrared range. The mechanisms underlying the MO characteristics of the specimens were also revealed in detail.

2. Methods

The proposed heterostructure is shown in Fig. 1. It is composed of a Au film perforated with equilateral polygon holes (here the hole in hexagonal structure is used as an example) in subwavelength scale on top of a uniform BIG film magnetized perpendicular to its plane (in -$\mathrm{\vec{Z}}$ orientation) and a Au film in the bottom. The thickness of top Au, middle BIG, and bottom Au films is designed to be h1, h2, and h3, respectively. The hexagonal holes seat in the middle of square arrays along x and y coordinates with a fixed period of d. A diagonal of the hexagonal holes is in x coordinate with a fixed length of 2r. For the normal incidence, the incident light with TM and TE modes of plane wave is normal to the surface of the BIG film. In the TM-polarized light, the electric field is parallel to the x coordinate (see Fig. 1), whereas the electric field is parallel to the y coordinate in the TE-polarized light. For the oblique incidence, the direction of the electric field keeps stable to the light orientation in the TM and TE modes, while the incident light is in the xz plane with an angle relative to the z coordinate. Except special note, the polarized light is in the normal incidence for the following study. At normal incidence, TM-polarized light has the electric field parallel to the x coordinate, whereas the electric field is parallel to the y coordinate for the TE-polarized light. By artificially changing the period and length of the holes, the thickness of Au and BIG films and the light polarization, the transmission spectra, deflection spectra and electric field distribution at the resonant wavelength of the heterostructure are collected through the monitoring of the FDTD frequency domain field and power. The wavelength of the simulated spectra is in the range of 650 ∼ 1200 nm, which is the best range for the BIG materials [9,12].

 

Fig. 1. Schematic diagram of the structure composed of a Au film perforated with sub-wavelength hexagonal hole arrays on top of a uniform BIG film magnetized perpendicular to its plane and a Au film in the bottom. The thickness of top Au, middle BIG, and bottom Au films is designed to be h1, h2, and h3, respectively. The hexagonal holes seat in the middle of a closed square array along x and y coordinates with a fixed period of d. A diagonal of the hexagonal holes is in x coordinate with a fixed length of 2r. At normal incidence, TM-polarized light has the electric field parallel to the x coordinate, whereas the electric field is parallel to the y coordinate for the TE-polarized light.

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To achieve physical properties of the infinite ordered array, periodic boundary condition is considered in the x and y directions, and perfectly matched layers boundary condition is applied along the z direction. The simulation area is 300×300×950 nm³. The grid size is refined to be 4 nm in xy plane and 2 nm in z axis in the crystals, but varies with a little larger value in other area. The heterostructure is seated in vacuum under a constant external magnetic field parallel to the light incidence. In such conditions, the dielectric tensor of the BIG film can be written as (here we assume that the magnetic medium is optically isotropic, and the second-order magneto-optical effect is ignored) [8],

3. Results and discussion

The MO properties of the bilayed MPCs in different equilateral polygon hole structures are first examined in comparison with the MPCs in circular holes and film. Figure 2(a) shows the light deflection of the heterostructures with the sizes of d = 300 nm, r = 60 nm, h1 = 40 nm and h2 = 120 nm under the TM light. In contrast to the smooth line with a low deflection angle of −005° ∼ 0.05° for the heterogeneous thin film, all samples with perforated holes show a similar trend with two resonant bands in the detected wavelength range, as observed in other MPCs with the subwavelength holes [9,13,14]. The first band locates at ∼690 nm (λ1), and the rotation angle is relatively small, below 0.20°; while the second band seats at ∼850 nm (λ2) with an obvious rotation angle larger than 0.25°. The full-width at half maximum (FWHF) of the two bands is 20 ∼ 30 nm. As there is no anomaly for the thin film, the two bands can be ascribed to the resonant coupling of light and surface plasmons excited by the interfaces of Au/air and Au/BIG for the perforated holes in the Au film. That is, the noble incident light can not induce surface plasmons on the smooth thin film, while for the thin film with perforation, it excites plasmons on the noble metal surface (SPP) and at the edge of the nanoholes (LSPR) as well as Bloch waves propagating along the x-coordinate. The resonant plasmons give rise to enhancement of the electromagnetic field in the BIG layer, which increases light-matter interactions and enhances MO effects [15,16]. The MPCs in quadrilateral, hexagonal and circular holes have deflection angle of more than 0.4° at λ2, while the weakest is triangle, but it also reaches 0.281°. As the rotation band at λ2 is due to the resonant coupling of light and surface plasmons, for the perforated holes in quadrilateral, hexagon and circle, the void structure expands the most in the direction of electric field oscillation (120 nm), the distance between holes in adjacent cells is smaller, and the surface plasma coupling is stronger, which results in the large MO effect. While for the perforated holes in triangle, the void dimension is limited in the direction of electric field oscillation (90 nm), which induces the weak MO effect. In light of field distribution in Fig. 3, it is found that the local field generated by quadrilateral, hexagonal and circular holes around the holes at λ2 is enhanced most obviously, followed by holes in heptagon, octagon, pentagon and triangle, which is in good agreement with the deflection development of the heterostructures.

 

Fig. 2. Spectra of Faraday rotation (a) and transmittance (b) of the bilayed MPCs in different equilateral polygon hole structures with the sizes of d = 300 nm, r = 60 nm, h1 = 40 nm and h2 = 120 nm under the TM light.

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Fig. 3. Sectional electric field intensity distribution of the bilayed MPCs in different equilateral polygon hole structures with the sizes of d = 300 nm, r = 60 nm, h1 = 40 nm and h2 = 120 nm at the resonant wavelength of 850 nm under the TM light: (a1-g1) schematic diagrams of the holes with the incident light; (a2-g2) air-Au interface in xy plane; (a3-g3) Au-BIG interface in xy plane; (a4-g4) cross section in xz plane. Color ranging from blue to red indicates the strength change of the electric field.

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Figure 2(b) displays transmission spectra of the MPCs in different hole structures. Similarly, in the detected wavelength range, the perforated holes produce two resonant bands near 700 nm ($\mathrm{\lambda }_1^\mathrm{^{\prime}}$) and 880 nm ($\mathrm{\lambda }_2^\mathrm{^{\prime}}$), which are also due to the resonant coupling of light and surface plasmons, but are red shifted compared with the resonant wavelength of Faraday deflection. Since the transmittance at $\mathrm{\lambda }_2^\mathrm{^{\prime}}$ is much larger than that at $\mathrm{\lambda }_1^\mathrm{^{\prime}}$, as that of Faraday rotation, we will focus on the MO characteristics at λ2 and $\mathrm{\lambda }_2^\mathrm{^{\prime}}$. The transmittance at the resonant position in Fig. 2(b), ratio of area covered by holes and normalized-to-area transmittance [17] of the perforated hole arrays in different hole structures are supplied in Table 1. Obviously, the periodic hole arrays also improve the transmission at the resonant wavelength. The transmittance of triangular holes is the lowest (8.5%), while that of circular holes is the largest (18.4%). Both are coincident with the open area of the perforated holes. However, the transmittance of quadrilateral (14.1%) and hexagon (16.4%) is larger than that of pentagon (12.7%) and heptagon (13.4%). It is speculated that the quadrilateral and hexagon supplies the smaller gap in adjacent hole in the direction of electric field oscillation (180 nm), which enhances the surface plasma coupling nearby and induces the EOT. Furthermore, all the specimens own a normalized-to-area transmittance larger than 120%, especially for the holes in triangle, quadrilateral and hexagon exceeding 150%.

Table 1. Transmittance at the Resonant Position in Fig. 2(b), Ratio of Area Covered by Holes and Normalized-To-Area Transmittance of the Perforated Hole Arrays in Different Hole Structures.

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In order to elucidate the MO characteristic, electric field distribution of the bilayed MPCs is supplied at λ2 under the TM light, as shown in Fig. 3. The color bar represents the field ratio, $\textrm{log}\left|{\frac{\textrm{E}}{{{\textrm{E}_0}}}} \right|$, where E and ${\textrm{E}_0}$ are the induced and original electric field intensity, respectively. It is found that the electric field at the edge of the hole is enhanced for the LSPR, reaching the strongest within a few nanometers of the inner circumference, up to 10⁵ or even higher, and then gradually weakened towards the center of the hole. The field intensity of the Au film away from the hole is slightly weakened along the direction of electric field oscillation and significantly reduced in the perpendicular direction. In addition, comparing columns 2 and 3 of each structure, at the inner circumference of the hole, the field intensity at the interface between Au and BIG is much stronger than that at the interface between air and Au, and the former is about 10³ times that of the latter. In column 4, the electric field is primarily localized at the edge of Au, especially the joint point between Au and BIG. This suggests that the LSPR can be amplified by the magnetic dielectric and in turn improves the MO effects. Moreover, the field intensity at the junction between the hole side and the light oscillation is stronger than the opposite for the holes in triangle, pentagon and heptagon. Therefore, if the radius of curvature is defined by the subwavelength hole, the larger the radius of curvature is, the electric field is enhanced more obviously, which is contrary to the nature of the nanoparticles, as the local field becomes much strong at the position with small radius of curvature [18,19]. In fact, in the fabrication of the MPCs, the sides of the polygon hole may not be straight, and the interior angles may become round for the manufacturing errors, which would induce light-matter interaction as the circle hole and affect Faraday rotation, especially for the polarized light passing through the deviation site.

In contrast to the circle, the polygon owns the anisotropic characteristic, which is beneficial for the MO modulation. Based on the requirement of Faraday effects for the large rotation and strong transmission, the MPC in hexagon possesses the best properties, with a relatively large Faraday rotation of 0.425° and a maximum transmittance of 16.4%, which are respectively 9.4 and 6.1 times enhancement of the 0.045° and 2.7% of the smooth Au/BIG film at the same wavelength. Therefore, the MPC with periodic hexagonal holes in the top Au film will be used for our following study.

Figure 4(a,b) presents Faraday rotation and transmittance of the bilayed MPCs with hexagonal hole arrays in the sizes of d = 300 nm, h1 = 40 nm, h2 = 120 nm, and different r under the TM light. By increasing the size of the hexagonal holes from 40 nm to 120 nm, the resonant bands at λ2 and $\mathrm{\lambda }_2^\mathrm{^{\prime}}$ red shift first and blue shift later, but the turning point is different, which requires d = 60 nm for Faraday deflection and d = 100 nm for transmission. The band shift may stem from the change of the plasmon intensity and plasmon interaction. Increasing the side length gives rise to extension of the electron oscillation length in the hole and weakness of the interaction of LSPR between adjacent sides, which results in the red shift of the resonant bands. Nevertheless, the distance between the holes nearby becomes shorter, the loss of SPP and Bloch waves reduces, and the plasmon interaction between adjacent holes becomes stronger, which induces the blue shift of the resonant bands. Therefore, it is conceivable that the former case plays a leading role in the short side range, whereas the later case dominates in the long side range for the MO effect of the MPC. The rotation maximum also increases first and reduces afterward, but the light transmittance is merely improved for rising area of the open holes. The specimen with r = 80 nm owns the maximum deflection angle (0.554°) and a relative large transmittance (32.2%). The hexagonal holes in this side length may be the best choice.

 

Fig. 4. Spectra of Faraday rotation and transmittance of the bilayed MPCs with hexagonal hole structure in various periodicity under the TM light: (a,b) d = 300 nm, h1 = 40 nm, h2 = 120 nm, and different r; (c,d) r = 80 nm, h1 = 40 nm, h2 = 120 nm, and different d.

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By changing the array period d, the spectra of Faraday rotation and transmittance of the MPC are also analyzed, as displayed in Fig. 4(c,d). The resonant bands at λ2 and $\mathrm{\lambda }_2^\mathrm{^{\prime}}$ red shift synchronously, just like the phenomenon found in a periodically connected slot antenna hole arrays in Au film [20]. The increase of period leads to the extension of the electron oscillation length at the adjacent nanoholes on the metal film, and the weakening of the plasma interaction between them, which brings the red shift of the resonance wavelength. On the other hand, as the period increases, the deflection angle at λ2 increases, while the transmittance at $\mathrm{\lambda }_2^\mathrm{^{\prime}}{\; }$decreases. For the perforated MPC, the TM-TE conversion of the light in the dielectric plate increases with increasing period, resulting in the augmentation of the deflection angle, but at the same time reducing the coupling between the local plasmons and the effective area of the holes, so the EOT is weakened and the red shift appears.

Under the fixed period of 300 nm and side length of 80 nm, the MO properties of the proposed heterostructure are studied by varying the thickness of metallic and magnetic films. Figure 5(a,b) exhibits the spectra of light deflection and transmission of the specimens with h1 in the range 0–60 nm and h2 = 120 nm. Except the specimen without Au film, all the specimens produce two resonant bands as caused by the plasmons at the air/Au and Au/BIG interfaces. By increasing the Au thickness, the resonant bands get a blue shift with reduced FWHM, and the shift is more obvious when the gold layer is thinner. In addition, the deflection angle at λ2 becomes larger and larger, while the transmittance at $\mathrm{\lambda }_2^\mathrm{^{\prime}}$ becomes smaller and smaller. For comparison, the maximum Faraday rotation and transmittance are 1.204° and 23.8%, respectively for the MPC with Au thickness of 60 nm, while they are 0.215° and 38.5%, respectively for the the sample in Au thickness of 20 nm. The development can be understood as follows. When the metal layer becomes thickened, the interaction distance for the incident light and the metal at the edge of the hole becomes larger, which leads to much TE-TM mode conversion and improves Faraday rotation. At the same time, the absorption of the Au layer increases, the generated plasmons (LSPR and SPP) at the Au/BIG interface become stronger, and the resonant bands blue shift. The thicker the Au film, the further the plasmons to the top surface, the weaker the blue shift of the resonant bands can be achieved. For the sake of application, the Au film in a thickness of 40 nm may be the best choice with the large polarization conversion (0.554°) and transmittance (32.2%).

 

Fig. 5. Spectra of Faraday rotation and transmittance of the bilayed MPCs with hexagonal hole structure in various thickness under the TM light: (a,b) d = 300 nm, r = 80 nm, h2 = 120 nm, and different h1; (c,d) d = 300 nm, r = 80 nm, h1 = 40 nm, and different h2.

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Figure 5(c,d) displays MO spectra of the specimens with h1 = 40 nm and different h2. For the light deflection, with increasing h2, the resonant peak at λ2 is first improved and then depressed, and the λ2 red shifts first and blue shifts later. Both reach the maximum in the thickness of 140 nm, where the deflection angle is 0.688° at λ2 = 867 nm. It is well known that the coupling between SPPs and the quasi-waveguide in a medium requires certain conditions. There may be the strong coupling for the Au/BIG heterostructure when the thickness of the BIG film is 140 nm, which is responsible for the largest deflection. As for the transmittance, the resonant peak at $\mathrm{\lambda }_2^\mathrm{^{\prime}}$ rises up and its position red shifts successively with increasing h2. Nevertheless, the transmittance in the long wavelength decreases continuously. This may result from the EOT of the BIG film [21]. The transmittance is improved drastically near the resonant band, while it is reduced for the absorption of the material far away from the resonance.

It has been reported that an addition metal film underlying the bilayered MPC can improve the MO effect [15,22]. Under the optimal parameters of the perforated Au/BIG heterostructure, d = 300 nm, r = 80 nm, h1 = 40 nm, h2 = 140 nm, a Au film is added in the back of BIG layer and the MO properties of the trilayered MPC are studied by varying the film thickness, as presented in Fig. 6(a,b). When h3 increases, the deflection band blue shifts with continuous enhancement, and the transmission band also blue shifts but becomes depressed. It is reasonable that when the underlying Au film is added, a part of light that is originally to directly transmit out of the BIG layer is reflected back, resulting in the decrease in transmittance. At the same time, on the model in which two spring oscillators are coupled to each other [6], more TM oscillating electric field will produce more violent oscillations, which induces more excitation of the TE mode by the coupling effect, so the deflection angle increases, and the peak position blue shifts. Nevertheless, the transmittance is improved successively in the wavelength range far larger than the $\mathrm{\lambda }_2^\mathrm{^{\prime}}$, which may stem from the strong plasmonic effect for the improved SPP traveling distance [5]. Since the underlying film reduces transmittance seriously in the resonant band, the Au film in 3 nm thickness may be optimal for a relatively high rotation (0.788°) and transmittance (30.0%) at resonant wavelength. In contract to the continuous Au/BIG film, these are respectively 17.1 and 11.1 times enhancement of the 0.046° and 2.7% at the identical wavelength.

 

Fig. 6. Spectra of Faraday rotation and transmittance of the trilayed MPCs with hexagonal hole structure in various thickness and light polarization: (a,b) d = 300 nm, r = 80 nm, h1 = 40 nm, h2 = 140 nm, and different h3 under the TM light. (c,d) d = 300 nm, r = 80 nm, h1 = 40 nm, h2 = 140 nm and h3 = 3 nm under the light in different polarization direction.

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In the above analysis, the MPCs are achieved on the basis of the maximum deflection angle with guaranteed high transmittance (≥ 30%) at the resonant wavelength, but they are in the TM polarized light. Due to the anisotropy of the hexagonal holes in the top Au film, different results may be achieved if the incident light is in the TE mode or other polarization. Therefore, MO properties of the final trilayered MPC are studied in the TE light in comparison with TM light, as displayed in Fig. 6(c,d). The deflection spectrum in the TE mode presents a positive band and a negative band in sharing with a line in the edge extension, as those found in other systems [23–25]. The center position of the positive and negative peaks nearly coincides with the deflection peak position at λ2 in the TM mode. Moreover, the transmission spectra in both the polarizations almost overlap each other in the long wavelength. The identical resonant wavelength for the TM and TE modes may be ascribed to the limited gap of the perforated hole, which leaves much area for the Au film and brings little difference of the surface plasmon under the light polarization in the x or y orientation. This can be verified by the field distribution at different interfaces, as supplied in Fig. 7. Under both the light polarizations, the electric field at the air/Au interface, the Au/BIG interface, and the cross section in xz plane at y = 0 is almost the same, and the field intensity is mainly amplified at the hexagon sides perpendicular to the light polarization. Therefore, in order to increase the anisotropy, except using the anisotropic hole in the MPC, the gap dimension of the subwavelength holes should be large and the plasmons with peculiar characteristic under different light orientation should be in a dominative role.

 

Fig. 7. Sectional electric field intensity distribution of the trilayed MPCs in hexagonal hole structure with the sizes of d = 300 nm, r = 80 nm, h1 = 40 nm, h2 = 140 nm and h3 = 3 nm at the resonant wavelength of 850 nm under the polarized light in different polarization: (a1-c1) TM mode; (a2-c2) TE mode, whereas (a1,a2) air-Au interface in xy plane; (b1,b2) Au-BIG interface in xy plane; (c1,c2) cross section in xz plane. Color ranging from blue to red indicates the strength change of the electric field.

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In order to disclosure the nature of the plasmons, the transmission spectra of the trilayered MPC are analyzed in the oblique incidence, as presented in Fig. 8. In the TM mode, the spectrum is resolved into four resonant bands. The position of the three bands below 900 nm is nearly stable in different incident angles, while the band above 900 nm red shifts with increasing incident angle. As the spectral position of the SPPs depends on the angle of light incidence [2,26], the phenomenon suggests that the resonant bands below 900 nm are dominated by the LSPRs, while that above 900 nm is governed by the SPPs. In the TE mode, the structure of the spectrum in different incident angles is the same as that in the vertical incidence. The resonant bands become depressed with stable position under enlarging the angle of light incidence, suggesting that they are originating from the LSPRs.

 

Fig. 8. Transmission spectra of the trilayed MPC in hexagonal hole structure with different incident angle of the polarized light in (a) TM and (b) TE modes.

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For comparison, Faraday rotation and transmittance with their resonant wavelengths of the Au/BIG heterostructure in different layer number and structure are supplied in Table 2. It is found that the rotation angle and transmittance of the MPCs can be improved as large as ∼21° and ∼85%, respectively by modulating the component and structure, indicating the convenient regulation of the heterostructures and their potential application in nanoscience and nanotechnology. However, the resonant wavelength of the light deflection is far away from that of the transmission for the grating and antennas with an open gap in the hole boundary [6,8], or the large rotation and transmission is just at a wavelength point for the waveguide structure [16], which are disadvantageous for their performance and operation. The MPCs with the closed perforated holes seems to be superior as both the resonant wavelengths are at the same point or close to each other and the big rotation and strong transmission coexist in a relatively long wavelength range. In fact, the MPCs can be further manipulated by altering the hole geometry and noble metal to improve the MO properties [25,27], or the magnetic materials to move the resonant bands in visible or far-infrared range [28,29].

Table 2. Faraday Rotation and Transmittance With Their Resonant Wavelengths of the Au/BIG Heterostructure in Different Layer Number and Structure.

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4. Conclusion

There are two resonant bands in the spectra of light rotation and transmission for the bilayered Au/BIG and trilayered Au/BIG/Au MPCs with perforated holes in the top Au film, which can be ascribed to the resonant coupling of light and surface plasmons excited by the interfaces of Au/air and Au/BIG for the perforated holes. The MO properties of the heterostructure can be manipulated by the structure, size and period of the holes, the thickness of the thin films, and the incident light polarization. The MPCs with plasmonic hexagonal holes own superior Faraday effects with distinctive anisotropy. For the optimized bilayered and trilayered hexagonal MPCs, the Faraday rotation reaches 0.688° and 0.788°, respectively with 15.3 and 17.5 times enhancement of 0.045° of the continuous Au/BIG film, while the transmittance achieves 32.7% and 30.0%, respectively with improvement of 12.1 and 11.1 folds of 2.7% for the later. Possible mechanisms underlying the EOT and Faraday effects are analyzed systematically. The study indicates that the MPCs with perforated holes in the plasmonic film may possess large Faraday rotation with strong transmitance in the nearly identical wavelength range, which is beneficial for their application in MO devices, such as biochemical sensors, optoelectronics, and phase modulators.