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Here, we propose a hybrid approach to uniquely determine the elements of the circuit analogue of the terahertz metamaterial absorber that we previously reported. The proposed method is based on calculations, fitting, and physical mechanism of the absorption process interpreted by the model. In this work, the correlation between the model components and our designed absorber is comprehensively enlightened, and the dependence of the model elements to structural dimensions of the absorber is analyzed both qualitatively and quantitatively. By applying this approach on frequency selective surface (FSS) model, we are also able to interpret the polarization insensitivity of our designed absorber. The proposed model and approach is applicable for all metamaterial absorbers with any arbitrary FSS design.
© 2015 Optical Society of America
1. Introduction
Since the theoretical prediction of Veselago and pioneering work of J. Pendry metamaterials, have paved a new avenue to engineer novel optical devices [1–3]. Designed to render a desired optical characteristics at a given spectrum, metamaterials are promising structures to fill the so called terahertz gap (0.3 up to 10 THz) where natural materials are rarely available to exhibit a functional THz response. Their application spans from superlenses, cloaking, imaging, and spin Hall effect of light to modulators, switches, detectors, and sensors [4–12]. THz metamaterial perfect absorber is one of the engineered devices that has been recently of a great interest. A frequency selective surface (FSS) followed by a dielectric spacer and a metal backplane forms a metamaterial absorber. Since its first demonstration, it has been extensively developed into polarization independent, multiband, broadband, and tunable absorbers, and its applications as a THz sensor and detector have been recently demonstrated [13–23]. Owing to the subwavelength dimension of metamaterials, and since their spectrum mimics a Lorentzian line-shape, their functionality resembles an equivalent circuit model [24]. Creating a good correspondence between the elements of a model and a metamaterial structure is greatly helpful for better understanding and faster designing of optical devices. Several articles have been reported concerning the electric models of basic one layer metamaterials such as SRRs, filters, and FSSs in which they have considered metallic part containing oscillating current as an inductor in series with a resistor and the gap between metals as a capacitor [25–32]. In closely tight or multilayer structures of metamaterials where there is an interaction between unit cells, a mutual inductance between adjacent unit cells has been also considered [33, 34]. However, there are only a handful reports regarding electric analogs of the metamaterial absorber [35–37]. More recently, we proposed a dynamic circuit which models these kinds of absorbers based on the interference theory of the reflected waves [38, 39]. The model is composed of a series RLC band-pass circuit for the case of perfect absorption (100%) where L and C are the capacitance and inductance of FSS, while R is the resistance of FSS and metal backplane. In order to consider the effect of dielectric spacer in imperfect cases, another RLC circuit was embedded between the initial LC and R. The second RLC circuit (RPCPLP) which is accounted for the resonance inside the spacer is a series LPCP component parallel to the resistor RP. The corresponding components of the model were evaluated by fitting the squared magnitude of VO/Vi, the transfer function, to the simulated data where VO and Vi are input and output voltages, respectively, and therefore a circuit model with an equivalent response of the absorber was reported.
In this paper, we report a hybrid approach to determine unique values of the elements of this dynamic circuit model by using calculation, fitting, and absorption mechanism interpreted by the model. We express in more details the correlation between the model components and our designed absorber and analyze qualitatively and quantitatively the dependence of the model elements to structural dimensions of the absorber. The metamaterial absorber under study is designed such that it exhibits polarization insensitive response. Based on our evaluated parameters, we are able to explain the effect of the dimensions on THz response and polarization insensitivity of the absorber. This model with the associated hybrid approach is applicable for all metamaterial perfect absorbers with an arbitrary FSS design.
2. Design and simulation
The metamaterial absorber under study in this paper is illustrated schematically in Fig. 1. Figures 1(a) and 1(b) show three dimensional and front view illustrations of the absorber, respectively. To design the structure, COMSOL was utilized to solve Maxwell equations by using finite element numerical method (FEM). Cu was used as the metal and polyamide as the dielectric spacer in simulation. We used COMSOL’s material library data for polyimide and Cu conductivities which are 6.67 × 10−16 S/m, and 6 × 107 S/m, respectively. Also, polyimide permittivity was considered to be 3.15. A continuous plane wave source and periodic boundary conditions were applied in FEM simulation where the total number of free tetrahedral mesh elements was 349346.
Fig. 1 (a) Three dimensional schematic and (b) Front view illustration of the designed absorber composed of FSS, polyimide and metal backplane. The purple areas are Cu and the green areas are polyimide.
Absorption calculation was performed via A = 1 – R with A as the absorption and R as the reflection whereas the transmission through the structure is equal to zero. Table 1 summarizes the dimensions of the designed structure studied here. Absorber with 14 μm polyimide thickness gives rise to nearly perfect absorption (≈100%) at 0.476 THz while 11 μm and 17 μm polyimide thicknesses lead to imperfect absorption (<100%).
Table 1. Dimensions of the Designed Metamaterial Absorber
3. Absorption process and dynamic circuit model
The details of the general dynamic circuit model of metamaterial absorber have been reported in our previous work [38]. Here, we briefly explain the model and the underlying absorption process based on the interference theory to apply it for detailed calculations of the electric circuit elements. Figure 2(a) shows the electric field and current density of the absorber at resonance frequency of 0.476 THz where the incident polarization angle is 0°. When the electromagnetic wave illuminates the absorber, FSS is first excited. The FSS primarily selects the absorption resonance frequency dependent upon its configuration. An electric dipole at a given resonance frequency is created on the FSS with corresponding oscillating current density which has been shown by white arrows (J) in Fig. 2(a) for clarity. This excited dipole partially reflects and transmits the incident electromagnetic wave, and thereafter induces a dipole on the metal backplane with corresponding oscillating current density shown by red arrows (Ji) in Fig. 2(a).
Fig. 2 (a) Electric field and current density on FSS and backplane of absorber at 0.476 THz with 0° incident polarization. White and black arrows on FSS and backplane have added to indicate to currents for clarity. (b) Schematic illustration of absorption process based on interference theory. (c) Electric model of the metamaterial absorbers for the perfect absorption case (100%). (d) Realistic model of metamaterial absorbers by considering the effect of dielectric spacer to achieve perfect (≈100%) or imperfect absorption (<100%).
Obviously, J and Ji will be in opposite directions. Figure 2(b) depicts schematically these dipoles in which the excited dipole on FSS is shown by PFSS and the induced one by PInduced. The reflected wave out of the PInduced will go under multiple reflections inside the cavity made by FSS and backplane. If the thickness and permittivity of the dielectric spacer inside the cavity are chosen properly then the resonance frequency of the cavity and FSS will match, and the resulted compounded wave (RInduced) from the induced dipole can be made exactly out of phase with that of the excited dipole (RFSS), thus leading to a zero reflection (100% absorption) at the resonance frequency.
Since the absorption spectrum of the metamaterial perfect absorber mimics a Lorentzian line-shape [38], it can be modeled by a band-pass circuit shown in Fig. 2(c) in which the squared magnitude of VO/Vi also follows a Lorentzian distribution. At resonance frequency, LC will get short-circuit and the model will result in 100% absorption. However, this model works only for the perfect absorption case (100%), as it does not consider the effect of the dielectric spacer cavity that can change the frequency and the peak of absorption. The dynamic circuit model, which can imitate both perfect and imperfect absorption cases, has been illustrated in Fig. 2(d). Indeed, in order to take into account the dynamics of the resonance inside the cavity, a circuit loop consisted of LP, CP, and RP, representing the dielectric medium, is integrated between LC and R. In this model, L shows the self-inductance of FSS, C stands for the capacitance between excited electric poles on FSS, and R represents the resistance of FSS and backplane, whereas LP and CP represent the mutual inductance and capacitance between FSS and backplane and RP determines a loss due to the non-zero reflection. This circuit includes two resonances: LC which is associated with FSS, and LPCP which is related to the cavity. The resonance is dominantly specified by LC. When LC and LPCP resonances match, RP will get short-circuit and perfect (100%) absorption will be obtained. As a matter of fact in this case, the loop currents of IFSS and IIn will be equal but in opposite directions resulting in zero current in RP and thus zero reflection and perfect (100%) absorption. Contrarily, if two resonances are not identical, the loop currents will be no longer equal and the current passing through RP will be non-zero, and therefore the frequency and peak of absorption will slightly deviate from the perfect case. If we view currents IFSS and IIn, respectively as the reflected waves of the FSS and backplane, in the case of an imperfect absorption (<100%) some incident power will be dissipated as the reflection in RP.
4. FSS model and polarization insensitivity
The FSS model of this absorber has been also reported in our previous work, and the polarization independent response of the absorber has been discussed qualitatively [40]. However, here we give a brief explanation for this model in order to apply our hybrid approach on it, and thereby we quantitatively verify the polarization insensitivity of the absorber. Figures 3(a) and 3(b) show the electric field and current density distribution on FSS, respectively under 0° incident polarization. Figure 3(c) demonstrates the electric filed with current density of backplane at the same polarization angle, and Fig. 3(d) is a demonstration of electric model of FSS at the same polarization.
Fig. 3 Electric field and current density profiles on FSS, backplane and the corresponding electric model for 0° (a, b, c, and d), 20° (e, f, g, and h), and 45° (I, j, k, and l) incident polarizations. Black arrows indicate to current directions on FSS and backplane added for clarity. Red Closed curves are added to indicate to the effective dipoles created on FSS.
Correspondingly, the electric field and current density of FSS and backplane and the electric model of FSS have been depicted for incident polarization angle of 20° in Figs. 3(e)-3(h), and of 45° in Figs. 3(i)-3(l). Dependent upon the incident polarization angle, a variation is observed in the distribution of poles on FSS, and the corresponding poles on metal backplane. For clarity, we have used red closed lines to imply the effective excited poles on FSS in Figs. 3(a), 3(b), 3(e), 3(f), 3(i), and 3(j) and black arrows to indicate the direction of currents in Figs. 3(b), 3(c), 3(f), 3(g), 3(j), and 3(k).
When a current passes through a rectangular metallic bar in FSS, it creates a self-inductance in that bar. If we assume that the inductance is approximately proportional to the length of the bar, the FSS can be modeled as Figs. 3(d), 3(h), and 3(l) for associated polarizations. In these models, Li, (i = 1,2,3) represents the inductance of different bars of FSS. Dependent on the length of bars and proportionality of inductance to the length, their inductances will be approximately related to each other according to L1 ≈0.5L2 ≈0.25L3. The equivalent inductance of FSS for each polarization can be summarized as in Table 2. It is observable that the L0> L20> L45 is stablished between inductances which is due to the reduction in the length of current paths as the polarization angle is rotated from 0° to 45°, leading to a reduced inductance.
Table 2. Equivalent Inductance and Capacitance of FSS
Contrarily, the origin of the capacitances in FSS at different polarizations is quite subtle. In fact, those capacitances stem from the energy stored between the poles created on FSS such that the distribution of those poles at different polarizations changes the associated capacitance C in the model. From the comparison of distributions of the poles for 0°, and 20° in Fig. 3, it is observable that increasing the angle of polarization leads to an increase in the effective interaction area of poles, while the effective distance between them (red arrows in Figs. 3(b) and 3(f)) decreases. At 45°, the effect is so obvious that the corresponding capacitance stems from the gaps between legs (labeled V and H), which is the smallest possible distance between the poles (red arrows in Fig. 3(j)). Thus, the capacitance C45 should be larger than C20, and both should be larger than C0, or in other words C45> C20> C0. Therefore, any aforementioned drop in inductance L will be relatively compensated by a rise in capacitance C at different incident polarizations such that the resonance frequency fres = 1/2π√LC, will be preserved at all polarizations owing to the symmetry of structure. We will use this model and hybrid approach to quantitatively verify polarization independent response of the absorber.
5. Determining model elements
5.1 Self-inductance (L), capacitance (C), and resistance (R)
Inductance L and capacitance C of the model are considered as self-inductance, and capacitance of FSS. The self-inductance, L (nH), of a metal rectangular bar containing the resonating current at different polarization can be calculated by [41–43]
where l is the length in cm, w and t is width and thickness of metal, respectively. The capacitance C of FSS is the capacitance between two charge distributions on top and bottom parts of the FSS. However, calculating the capacitance C directly from FSS geometry is not straightforward, particularly in different polarization angles. Instead the capacitance C can be calculated through fres = 1/2π√LC where fres is the resonance frequency of perfect absorption case (100%) and L is obtained from the calculation described above.The value of resistance R relative to the total impedance of circuit, determines the amount of absorption, as well as the bandwidth of frequency response. Since Cu has a much larger conductivity and imaginary part of refractive index than polyimide, the majority of THz radiation will be absorbed by FSS and metal backplane as an oscillating electric current. Therefore, R should be approximately equal to the resistance of FSS and metal backplane in the realistic model (Fig. 2(d)). The resistance of FSS can be calculated through R = ρl/A where ρ and l are Cu resistivity and the length of FSS rectangular bars which contain resonating currents, respectively, and A is the effective cross-sectional area of those bars by considering the skin depth effect. It is worthy to mention that the resistance of FSS should be doubled since there exist two current flows in FSS, one associated with LC resonance of FSS and another with LpCp resonance inside polyimide. The first current in FSS comes from the coupling of incident electric filed directly to FSS which creates a current on front side of FSS which faces air. The second current in FSS is due to the resonance inside polyimide (between backplane and FSS) which excites current on the back side of FSS facing polyimide. In contrast to FSS, calculation of the resistance associated with metal back plane is not straightforward, as the current is distributed on its surface such that the effective cross-sectional area is hardly determined. However, another method to estimate the value of R is to use the model of the perfect absorption case shown in Fig. 2(c). The squared magnitude of transfer function for this model can be obtained through
where ω is the angular frequency. In this equation L and C are already determined and the only unknown R can be uniquely determined by fitting the absorption spectrum of the absorber to Eq. (2). We will show that this two methods of evaluation of R result in good agreement. The value of R obtained by fitting this model or by aforementioned calculation will be used as an estimation in determining R in the realistic model. This estimated value will be actually utilized to confine R between up and low limits while fitting the realistic model (Fig. 2(d)) to the absorber frequency response.5.2 Mutual inductance (LP), capacitance (CP), and resistance (RP)
To quantify CP, LP, and RP, their origin in the absorber should be first enlightened. In contrast to C which comes from the capacitance between the poles on FSS, CP originates from the capacitance between those poles on FSS and their corresponding induced ones on metal backplane. Figure 4 demonstrates the mutual capacitances, CP, between FSS and metal backplane for three different polarization angles of 0°, 20°, and 45°. In this figure, the effective poles on FSS are shown with red closed lines and the related bars are highlighted with red. The capacitance CP is depicted by blue capacitance symbols with values of CP/2. Any rectangular bar on FSS in conjunction with the metal backplane and polyimide between them is considered as a microstrip waveguide.
Fig. 4 Schematic illustration of mutual capacitance between FSS and backplane for (a) 0° (b) 20°, and (c) 45° incident polarization. Ci, i = 1, 2, and 3 are the capacitances between the metal bars and backplane as microstrip waveguides and CP is the total equivalent capacitance between FSS and backplane, (d) side view illustration of a microstrip waveguide composed of a metal bar, backplane, and polyimide between them.
By this assumption, Cp can be obtained by calculating the capacitance of the microstrip waveguides associated with poles on FSS (red bars in Fig. 4). These microstrip capacitances are labeled as C1, C2, and C3 in Fig. 4 for different polarizations. A side view of one of these microstrips is shown in Fig. 4(d). For this microstrip waveguide the capacitance per unit length between the strip and ground plane can be obtained by [44]
where Ԑr is polyimide permittivity in our calculations, W is polyimide thickness, and T is the metal thickness. Since all capacitances Ci, i = 1, 2, 3 are parallel to each other CP will be equal to their summation in each polarization.Another mutual parameter needs to be specified is LP. The origin of LP is similar to fishnet or paired nano-rod metamaterials where the magnetic response comes from antiparallel currents of parallel metal rods with incident magnetic field perpendicular to the surface between them [45, 46]. To visualize the mutual magnetic response in our absorber, a schematic representation of the absorber under 0° polarization has been shown in Fig. 5.
Fig. 5 (a) Schematic illustration of the absorber under 0° polarization with corresponding currents on FSS (blue arrows called IFSS) and backplane (green arrows called IIn). The created and induced dipoles are displayed by using red closed lines. (b). Schematic representation of mutual inductance created between FSS and backplane. LP originates from the surfaces S (enclosed with dotted lines) where the incident magnetic field is perpendicular to those.
Figure 5(a) shows the created and induced dipoles on FSS and backplane enclosed by red lines where the associated metal strips are highlighted in red on FSS. The created and induced currents on FSS and backplane are demonstrated by blue (IFSS) and green arrows (IIn), respectively. Figure 5(b) illustrates schematically where the mutual magnetic response comes from. Again in this figure created and induced currents on FSS and backplane are seen with corresponding colors and the magnetic field between FSS and backplane is demonstrated by red arrows (H). Like fishnet metamaterial, we believe that the incident magnetic field (H) couples to surfaces S enclosed by dotted lines where there exit antiparallel currents on FSS and backplane. The resultant magnetic flux created in any surface S mutually couples to each other and the one of adjacent unit cell leading to mutual LP between them. However, since the current distribution on backplane is sparse and the magnetic field (H) is not evenly distributed between FSS and backplane, particularly in different polarization angles (confirmed by simulation), LP can hardly be determined by direct calculation. To determine this mutual inductance in different polarization angles, we take the advantage of the model in two different cases of perfect (100%) and imperfect (<100%) absorption. In perfect case, the resonance frequency of LPCP matches with that of LC which is equal to the absorption resonance frequency. As we already calculated the value of CP, the inductance LP can be determined by fres = 1/2π√LPCP where fres is the resonance frequency of perfect absorption case. In imperfect case, the value of LP is determined by using the realistic model of the absorber depicted in Fig. 2(d). The squared magnitude of frequency response of this model is driven as
The three unknown variables in this equation LP, RP, and R can be uniquely evaluated by fitting Eq. (4) to simulated absorption spectrum of the absorber. It is noted that we already obtained an estimation for the value of R by using the calculation or the model of perfect absorption case. Thus in determining the value of RP through fitting the realistic model, we confine R between high and low limits around its estimated value. The following flowchart summarizes the algorithm to drive the values of elements of electric model of the absorber (Fig. 6).
6. Results and discussion
Table 3 summarizes the values of the model elements for absorber under normal incidence at 0° polarization for three different polyimide thicknesses of 11 μm, 14 μm, and 17 μm. Polyimide with 14 μm thickness gives rise to almost perfect (99.96%) absorption at 0.476 THz. Figure 7 shows the simulated data of the absorber (square dots) together with the model response (solid graphs) based on the values in Table 3. A closely perfect fitting between the simulated data and the response of the model is observable from the figure. Figure 7(a) compares the fitted absorption spectrum for polyimide thickness of 11 μm (red) and 14 μm (black), while Fig. 7(b) is a comparison of 17 μm (blue) and 14 μm (black) polyimide thicknesses.
Table 3. Evaluated Parameters of the Absorbers by Using the Hybrid Method at 0° Polarization
Fig. 7 (a) Simulated data of absorber with 14 μm (black dots) and 11 μm (red dots) polyimide thicknesses and their corresponding model responses with black and red solid graphs, respectively (b) Simulated data of absorber with 14 μm (black dots) and 17 μm (blue dots) polyimide thicknesses and their corresponding model responses with black and blue solid graphs, respectively.
As the FSS geometry and the polarization angle are same, the values of C and L show no change in Table 3 for all polyimide thicknesses. In contrast, by deviating polyimide thickness from the perfect case, the values of LP and CP will undergo a change. The value of CP for sep = 11 μm is larger than 14 μm and both are larger than 17 μm. Since CP is the capacitance of microstrip waveguides, by increasing the space between metal plates the associated capacitance will decrease similar to a parallel plate capacitor. However, LP increases when polyimide thickness gets larger. According to the origin of mutual inductance LP which comes from the magnetic coupling between surfaces S, any increase in polyimide thickness will increase the area S while the distance between surfaces S are fixed. Like a solenoid with N rings, where its mutual inductance is equal to Lrr = μN2S/d with S as the area of rings, d as the distance between them, and μ as the permeability, larger polyimide thickness results in a larger S and therefore increase in the value of LP.
RP goes under reduction in both cases of 11μm, and 17 μm relative to 14 μm. When polyimide thickness is 14 μm (perfect absorption) loop currents IIn and IFSS in Fig. 2(d) at resonance frequency will be equal but in opposite directions leading to zero series impedance (ZLpCp) of LP and CP and therefore zero current in RP. In contrast, at both imperfect cases of 11 and 17 μm, the value of RP decreases, and on the other hand the series impedance (ZLpCp) of LP and CP increases. This will make the currents IIn and IFSS unequal and therefore a current will flow through RP. This non-zero current flowing through RP is associated with non-zero reflection at resonance frequency resulting in imperfect absorption when the polyimide thickness is deviated from that of the perfect case.
The determined values of R in Table 3 are in almost a close agreement with estimated values by using calculation and perfect model (Fig. 2(c)). The calculated value of R (R ≈2 × RFSS) and the estimated value by fitting the perfect model turned out 13.5 Ω and 14.6 Ω, respectively. These values are less than the value found by realistic model. We believe that this difference is due to the absorption of the backplane which is not considered in calculation of R (R ≈2 × RFSS) and might not be accounted in the perfect model. Moreover, a slight reduction of values of R in Table 3 for both imperfect cases of 11 and 17 μm compared to perfect case of 14 μm is also observed. Since in imperfect absorption the LPCP resonance deviates from the resonance frequency of the perfect absorber (0.476 THz), the contribution of metal backplane and the back side of FSS in absorption process is diminished which becomes apparent in the reduced values of R in the model.
Table 4 summarizes the determined values of perfect absorber with 14 μm polyimide thickness but in three different polarizations. The parameters in Table 4 have been obtained by using the hybrid approach and by considering the electric current paths shown in Fig. 3, which interprets the polarization insensitivity of the absorber. According to the interpretation, the length of current path between poles on FSS decreases as the polarization angle goes up from 0° to 45° which leads to reduction in the value of L. Meantime, the capacitance C between poles on FSS rises to maintain the resonance frequency at a fixed value. The values of L and C in Table 4 for three different polarizations are consistent with this behavior. Additionally, we observe that the value of R is reduced by increasing the polarization angle. A drop in values of R also stems from a decrease in the length of current path on FSS when the incident polarization angle increases.
Table 4. Evaluated Parameters of Perfect Absorber by Using the Hybrid Approach of the Realistic Model at Three Different Polarizations
The mutual capacitance CP between FSS and backplane which is directly calculated by considering microstrip waveguides gets larger in the course of polarization changing from 0° to 45°. Indeed, when the polarization angle goes from 0° to 45°, the effective area of poles increases (from Fig. 3) while their distance (14 μm) remains fixed which leads to larger capacitance Cp similar to a parallel plate capacitor. Contrarily, by increasing the polarization angle, the values of mutual inductance LP drops. Figure 8 shows the magnetic field vector between polyimide and backplane at resonance frequency for 0° (Fig. 8(a)) and 45° (Fig. 8(b)) polarizations from a front view obtained by simulation. In this figure blue lines with label S indicate to the effective areas of surfaces responsible for magnetic response (LP) and the lines connecting dots with label LP imply the surfaces which may mutually couple to each other. As illustrated in the figure, when we move from 0° to 45° the effective area of mutually coupled surfaces S eventually reduces by half such that in 45° two surfaces S that contribute in creating mutual inductance LP less interact with each other, whereas in 0° polarization a larger surface S couples to parallel surfaces S of the same unit cell and that of adjacent unit cell. Thus, LP should decrease as the polarization angle changes from 0° to 45°.
Fig. 8 Front view illustration of magnetic field vectors inside polyimide at resonance frequency for the perfect absorber with polyimide thickness of 14 μm when the incident polarization is 0° (a) and 45° (b). Blue lines labeled S indicate to the areas that exhibit magnetic response and connected dots labeled LP shows the possible mutual surfaces that can couple to each other.
7. Conclusion
We presented a hybrid approach for the dynamic circuit model of metamaterial absorber to evaluate the components of the circuit model. The method is on the basis of calculations, fitting, and the absorption mechanism interpreted by the model. We studied in more detail the effect of the absorber’s dimensions on the effective parameters of the model both qualitatively and quantitatively. Based on the proposed FSS circuit models and the evaluated parameters in different polarizations, we explain well the polarization insensitivity of our designed absorber. The model with the associated hybrid approach is applicable for any metamaterial absorber with an arbitrary FSS design.
Acknowledgment
This work was partially supported by the NSF CAREER Award (ECCS- 0955160) and the NSF Award ECCS-1441947. M.P.H. acknowledges Graduate Council Fellowship from the University of Alabama.
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