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Abstract
In this paper, a 3D sub-wavelength graphene-coated nano-disk dimer (GDD) is proposed for multi-frequency giant near-field enhancement. We observed that the dual-band operation originates from the excitation of hybridized localized surface plasmons on top and bottom faces of the disks along with the mutual coupling from the adjacent particle. Due to the sub-wavelength nature of the disks, the excited localized surface plasmons on the sidewalls are weak but they still can affect the dual operating bands. On the other hand, the strength and resonance frequency of the enhanced fields can be simply modulated by tuning the relative distances of 2D graphene disks on top and bottom faces. Adjustable dual-band performance is hardly attainable using simplified 2D graphene disks, however, it naturally comes out through modal hybridization in the subwavelength 3D structure containing multiple resonant units. Our suggested configuration has better optical properties than its noble metal counterparts because of its higher field enhancement and lower ohmic losses. Moreover, the electromagnetic response is reconfigurable by varying the bias voltage. The influence of graphene quality, chemical potential, and dimer gap size on the electric field enhancement and the resonance frequency of the surface plasmons are investigated, as well. To further improve its performance, a double negative metamaterial core is considered. This mechanism of the performance improvement by the core material is feasible thanks to the 3D nature of the structure. Two possible applications of the presented design are in Surface-Enhanced Raman Spectroscopy (SERS) and optical absorbers.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Near-field enhancement by means of the metallic nanoparticles has been of great interest due to its broad range of applications including but not limited to biological sensing, blood immunoassays, drug delivery materials, in vivo optical contrast agents, and photo-thermal cancer therapy [1]. In this regard, optical nano-antennas have been efficiently used for investigation of the lipid and protein dynamics by breaking the diffraction limit [2]. Such arrays have also been fruitful for diffusion analysis of phosphoethanolamine (PE) and sphingomyelin (SM) in the plasma membrane of living cells [3]. Moreover, sensing and imaging capability of metallic nanoparticles makes them a promising candidate for probing their surrounding environment [4]. As another instance, heat-based therapeutic methods in cancer immunotherapy can facilitate from metallic nanoparticles [5]. In these structures, collective excitations of the electrons in the conduction band, known as surface plasmon resonances, produce a high-intensity electromagnetic field [6,7]. When a pair of such nanoparticles is put in the proximity of each other (known as a dimer), the optical response is altered under ordinary conditions because of the additional contribution of dipole-dipole interaction [8]. In dimers, the field enhancement can be caused by the formation of either the electric or magnetic hot spots at the gap [9]. For very small gaps, quadrupole and octopole modes are excited besides the dipole modes [10].
The near-field enhancement in dimers depends on the shape, size, and orientation of the nanoparticles [6]. Metallic nanoparticle with various shapes such as circular or ellipsoidal cylinders, spheres, disks, rods, and rings have been explored both theoretically and experimentally [11]. To account for the shape-dependent response of the nanoparticles, a dimensionless shape factor defined as the ratios of the surface area of a spherical nanoparticle and a nanoparticle in any shape, can be introduced [12]. Alternatively, Maxwell’s equations can be directly solved by numerical methods such as discrete dipole approximation (DDA), Finite-Difference Time-Domain (FDTD), and Finite Element Method (FEM) [13,14]. Moreover, to experimentally control the geometrical parameters, nanofabrication techniques like electron beam lithography, ion beam milling, and stamp-printing can be used [15]. Recently, combined use of planarization, etch-back and template stripping is proposed as a novel fabrication technique for improvement in overcoming the size limitation in conventional fabrication methods [16]. Regarding the effect of the particle orientation in the performance of the dimers, it is interesting to note that the field enhancement for the longitudinal polarization (electric field along the dimer axis) is stronger than transversal polarization (electric field perpendicular to the dimer axis) [17]. Furthermore, for the former, the resonance is slightly redshifted and broadened in comparison to a single particle [18].
The disk-shaped configuration which has high design flexibility in several ways is of interest in various applications. The possibility of obtaining high near-field enhancement with a large gap size is suggested by disk dimers in order to facilitate device fabrication [19]. Moreover, periodic assembly of the horizontally arranged gold disks have been successfully used for the Surface-Enhanced Raman Scattering (SERS) applications and vertically positioned hybrid Ni/SiO2/Au disks are promising candidates for the refractive index sensing applications [20,21].
Graphene, which is a 2D carbon material in a honeycomb crystal lattice, supports surface plasmon resonances and it can be considered as an alternative to the plasmonic metallic structures in various applications. Graphene-based devices can be designed from terahertz to the visible frequencies with potential applications in metamaterials, modulators, photodetectors, and sensors [22,23]. Moreover, graphene provides a platform for the gradually emerging topic of 2D modern optics to be used in optical circuits and meta-surfaces [24]. In a different context, the plasmonic coupling between graphene and localized surface plasmons of metallic particles is used to design a hybrid platform for SERS [25,26]. Specifically considering assemblies of graphene-based particles, a graphene-coated bow-tie nanowire dimer with the capability of controlling the optical response by adjusting the graphene chemical potential is proposed as a waveguide [27]. Moreover, localized surface plasmons of graphene-based Au nano-spheres are suggested to be used as an active tunable antenna [28]. Also, multiple invisibility regions are attained by a subwavelength trimer of graphene-coated circular cylinders [29]. It is worth noting that due to the van der Waals forces, particles with various morphologies can be coated by graphene [30]. Moreover, the experimental realization of graphene-wrapped nanowires with diameters ranging from tens of nanometers to several micrometers is feasible with current fabrication techniques [31].
In this paper, 3D graphene-coated nano-disk dimers (GDD) are used to achieve a giant dual-band near-field enhancement. In principle, graphene is a 2D material and mode coupling properties of a dimer consisting of 2D disks of graphene have been studied, previously [32,33]. Here, we are suggesting the sub-wavelength graphene-wrapped disks for the gap near-field enhancement. Our proposed graphene-based disk dimer fulfills the required near-field enhancement for SERS applications. The advantage of our proposed structure with respect to their 2D counterparts is the existence of two pairs of 2D disks simultaneously. The electromagnetic coupling of these pairs with each other and with the neighboring plasmons, supports two distinct frequency bands with the capability of active spectral tuning. At the same time, in this configuration, the optical response can be manipulated by the core material which is not feasible with 2D designs. Prior to this study, various SERS substrates using noble metals, metallic oxide and semiconductors, and two-dimensional materials have been designed [34]. Specifically, graphene material is chemically inert, highly resistant to oxygen, and strongly transparent to light [35]. Thus, graphene sheet has been successfully used as the substrate for Raman enhancement [36]. Graphene sheet has three functionalities in the SRES applications: 1) helpful to quench the fluorescence, 2) beneficial to protect the oxidation, and 3) an internal standard to normalize the Raman intensity [37].
The paper is organized as follows. In Section 2, dual-resonance coupled dipole modes of two graphene-covered hollow disks are investigated numerically to evaluate the performance in terms of near-field enhancement and metallic losses. Later, the underlying physics is clarified by discussing the absorption cross-section of GDD with various heights. In Section 3, the influence of graphene quality, applied bias field, and nano-gap size on the resonance frequency and field enhancement factor of GDD is explored. To further improve the performance, a metamaterial core is considered for the disks. Finally, concluding remarks are mentioned in Section 4.
2. Graphene-coated nano-disk dimer design
The structure under analysis is two identical graphene-coated hollow nano-disks in the close proximity of each other, as shown in Fig. 1. The radii of both disks are considered R = 100 nm throughout the paper. It is assumed that both disks are of identical heights of h = 50 nm. The minimum gap size is g = 2 nm to enable neglecting the nonlocal effects. Moreover, a linearly polarized plane wave with unit amplitude illuminates the dimer. The plane wave propagates along the z-direction with the electric field parallel to the gap axis.
Fig. 1. Graphene-coated nano-disk dimer under plane wave illumination, propagating along the disks axes and with the electric field parallel to the gap.
To investigate the performance of the proposed GDD numerically, it is simulated in the frequency domain solver of CST 2017 software package with tetrahedral legacy mesh cells. Fine local meshes along with the adaptive mesh refinement feature are used to increase the accuracy of the simulations. Moreover, perfectly matched layer (PML) boundary conditions are used in all the boundaries. Due to the extra-large absorption cross-sections, the dimensions of the bounding box are found by trying various sizes to ensure that the infinite space is simulated properly. In all the simulations, graphene is considered as a 2D sheet with surface conductivity calculated by the well-known Kubo formulas which in addition are available in the library of the software. The complex surface conductivity of graphene can be controlled by the temperature T, chemical potential μc, and relaxation time τ. Although the Kubo formulas are extracted for the planar graphene sheet, it can be used for the curved optical devices for the radii larger than ${\sim} 5\,nm$ [38].
2.1 Performance of the dimer
Figure 2 shows the normalized absorption cross-section (NACS) and the amount of metal losses for the graphene coating with τ=1.5 ps, and μc=0.4 eV. The normalization factor is the geometrical cross-section which equals $\pi {R^2}$. As Fig. 2 indicates, there are two sharp peaks in the NACS of GDD with negligible metal losses. The reason for investigating NACS is the fact that the strong extinction cross-sections result in the giant near-field enhancements [9]. Since the absorption cross-section dominates the scattering cross-section, in all simulations, we have focused on the former. Therefore, the more NACS exceeds the geometrical cross-section, the stronger the resulted near-fields. The high values of the NACS in Fig. 2 suggest that one may expect giant near-field enhancement in the associated peaks. Absorption cross-section and the losses of a gold dimer of the same dimensions are included in the aforementioned figure as well. In the simulation, well-known Johnson's model is used to account for the frequency dispersive permittivity of the gold. It is clear that in our specific application graphene is superior to its noble metal counterparts for various reasons. Due to the 2D nature of graphene, the graphene-based structure is dual-band while the metal-based structure is single band. Moreover, the resonance strengths for the graphene dimer are much higher than that of the gold dimer. Clearly, the resonance frequency of plasmons is different in the aforementioned structures. The resonances of the graphene-based structure are narrower than that of the gold dimer and for these specific dimensions (frequency), gold dimer has lower losses.
Fig. 2. (a) Normalized absorption cross-section (NACS) and (b) the amount of metal losses for the proposed graphene-based dimer with R = 100 nm, h = 50 nm, g = 2 nm, τ=1.5 ps, and μc=0.4eV. Low-loss dual-band performance is clearly illustrated. The normalization factor is the geometrical cross-sections of the top face and the absorption far exceeds it in the operating frequency spectrums. (c)-(d) The same parameters for a gold dimer of the same size. Please note that since graphene is modeled by its surface conductivity and it behaves as a metal, the software calls the corresponding losses as metal while since gold is modeled as a frequency dispersive dielectric, the dielectric losses are automatically calculated by the software.
Figure 3 illustrates the spatial electric field distributions in the resonances of NACS curves in Fig. 2. The provided field distributions confirm the giant field enhancement at the gap center respectively with the order of |E|=12851 and |E|=8900 with respect to the incident wave. By defining the enhancement factor M as the second power of the absolute electric field, it can be concluded that in comparison to plasmonic silver dimer with $M\sim {10^5}$ [6], our design has the minimum $M$ value of the order ${\sim} {10^7}$ in both operating frequencies. The required M for conventional SERS is around ${10^6}$ [6]. Consequently, the proposed configurations can be potentially used for SERS applications. Moreover, because plasmonic metallic particles suffer from large ohmic losses [9], this structure is superior to them due to employing the surface plasmon resonances of low loss graphene material. Importantly, our proposed GDD is sub-wavelength in thickness since the electrical heights at the first and second resonances with the wavelength of λ1 and λ2 are $h\sim 0.007\,{\lambda _1}$ and $h\sim 0.011\,{\lambda _2}$, respectively. In fact, among various types of SERS substrates including tip, gap, pore, and sphere, we are using the gap technique because of relatively easy fabrication process, possibility of single molecule detection, and high confinement of electromagnetic fields [39].
Fig. 3. (a) 3D and (b) 2D views of electric field distribution for the proposed graphene-based dimer with R = 100 nm, h = 50 nm, τ=1.5 ps, and μc=0.4eV at the first resonance for the illustration of giant near-field enhancement. (c)-(d) the same information at the second resonance.
As Fig. 2 shows, our proposed structure operates in the mid-infrared frequency range but Raman-excitation lasers are commonly operated at the wavelengths of 532, 633, and 785 nm [40]. Graphene material has a dual metal-dielectric property with the transition frequency depended on the chemical potential. On the other hand, by increasing the chemical potential of graphene, one can upshift the transition frequency and thus achieve plasmonic behavior at the near-infrared frequencies [41]. Moreover, another approach for achieving vis-NIR modes in graphene is reducing the geometrical dimensions of the structure [42]. To estimate the maximum frequency achievable as a function of size reduction with the current technological limits in order to achieve Raman spectroscopy, we have simulated a graphene-based nano-dimer with R = 30 nm, h = 15 nm, g = 2 nm, τ=1.5 ps, and µc=2 eV. The illustrated NACS in Fig. 4 shows a plasmonic resonance at 100.62 THz, which is far below the operating band of Raman laser. Interestingly, evidence for graphene plasmons in the visible spectral range is experimentally probed for planar structures [43] and hopefully it will be realized for the particles in future.
Finally, It is interesting to note that metal-based single disk configuration has been widely used as the SERS substrate because of its polarization-independent property [39]. But in our specific case, due to the use of dimer configuration, the response is polarization-dependent. In a quite general context, the maximum field enhancement is observed when the polarization direction of incident light is in the particle-particle connection direction. We have plotted the NACS and losses for the dimer of Fig. 2 by considering the illuminating electric field perpendicular to the gap axis. As Fig. 5 shows, we have a single-band performance in which the NACS is far less than that of the parallel polarization. The amounts of losses in both cases are identical.
Fig. 5. (a) Normalized absorption cross-section (NACS) and (b) the amount of metal losses for the proposed graphene-based dimer with R = 100 nm, h = 50 nm, g = 2 nm, τ=1.5 ps, and μc=0.4eV with illuminating electric field perpendicular to the gap axis.
2.2 Underlying physics
In order to show the origin of double resonance behavior, we have conducted other simulations by sweeping the height in each graphene-based cylinder. Specifically, simulations with the disk thickness of h = 0 nm, 25 nm, and 65 nm are conducted and the results of NACS are shown in Fig. 6. For the zero thickness, which resembles 2D circular graphene disks, only the first resonance appears. By increasing the height of the disks the second peak appears and it is observed that the magnitude and frequency of this peak can be modulated by proper choice of the height parameter. As it is observed, by decreasing the heights of the disks, both frequency bands experience a blue shift. It is expected since the smaller particles resonate in the larger frequencies. Based on these observations, it seems that the limiting structure with zero thickness might resonate in frequency that is larger than the first resonance frequency of the aforementioned structures. But it does not happen since graphene walls play a key role in determining the resonance frequencies.
Fig. 6. Normalized absorption cross-section (NACS) for the proposed graphene-based dimer with R = 100 nm, τ=1.5 ps, and μc=0.4eV by varying the heights of disks.
It can be concluded that the mode coupling between surface plasmon resonance of the top and bottom surfaces of the disks along with the near-field coupling of surface plasmons with the neighboring disk with nanometer gap results in the dual-band performance. The contribution of the latter factor will be clear by examining the data of Table 3. Interestingly, the cooperative effects of double plasmonic arrays constructed by two pairs of spherical particles on both sides of metallic films are used to achieve near-unity transparency. In these structures, the top and bottom arrays can be respectively considered as input and output couplers [44]. The same interpretation can be made for the top and bottom faces of our proposed structure.
It is worth noting that in the infinite length graphene-coated cylinders with an electric field along the cross-section of the cylinder, localized surface plasmons are excited [30,41]. Figure 7 shows NACS for the sub-wavelength graphene shell of the proposed dimer. The NACS of the top and bottom circles are included in the figure as well. As expected, we have two operating frequency bands in which absorption cross-section far exceeds the geometrical cross-section. Thus, we can conclude that these four discs are responsible for the dual-band response. Comparing this result with results of the disk dimer containing both sidewalls and circular disks reveals that the weakly localized surface plasmon resonances of the side walls affect both the resonance frequency and resonance amplitude. Therefore, we cannot simply remove them for the sake of easing the fabrication unless we consider other specifications for our problem.
Fig. 7. A graphene-based dimer consisting of two finite-length cylindrical shells and corresponding NACS. The optical and geometrical parameters are the same as Fig. 2.
The above discussion implies that the near-field enhancement is originated from the coupled localized plasmons of the top and bottom disks. For the graphene disks with the radii larger than 20 nm, the effect of edge plasmons on the optical performance can be neglected. Moreover, due to the 2D nature of graphene material very strong localized surface plasmon resonances can be observed in graphene nano-disk [45]. On the other hand, our structure performs based on the dipole resonances of four graphene circular disks. In fact, we have 4 planar graphene disks (2 on top and 2 down), so that hybridization is a complex process of all 4 disks. Similarities may be found in the performance of the proposed structure with that of the concentric nano-shells in which four states of hybridization are possible regarding the plasmons of the inner and outer plasmonic shells, i.e., ω+ +, ω+ −, ω− +, ω− − [46]. Moreover, the performance of the graphene-based dimer is different from the metal-based structure in a manner that in the former, localized plasmonic hybridization occurs due to the 2D nature of graphene, while in the latter localized surface plasmons are excited.
3. Study of the optical and geometrical parameters
In this section, the effect of various optical and geometrical parameters on the frequency and amount of the electric field enhancement is investigated in both operating frequencies. Specifically, graphene relaxation time, graphene chemical potential, nano-gap size, and core material’s constitutive parameters are studied. The choice of these parameters is based on physically realizable values. It is observed that the target operating frequency can be adjusted in several ways. The active tuning is also feasible via the chemical potential of graphene.
3.1 Graphene relaxation time and chemical potential
The aim of this sub-section is exploring the effect of graphene relaxation time and chemical potential on the optical response of the proposed dimer. The values as high as τ=3 ps and μc=2eV are respectively experimentally realizable for these parameters [33]. Moreover, the relaxation time of graphene is directly proportional to the quality of graphene material, i.e., higher relaxation times lead to lower transmission losses and thus better quality. To find the proper relaxation time, NACS for the structure of Fig. 1 is investigated for the relaxation times of τ=0.5, 1, 1.5, and 2 ps and they are provided in Fig. 8. Corresponding field enhancements are summarized in Table 1. It is confirmed that increasing the quality of graphene increases the amount of near-field enhancement but it does not affect the resonance frequency of the plasmons.
Fig. 8. The effect of quality of graphene on the amount and frequency of enhancement for the structure of Fig. 2 at (a) first and (b) second resonance.
Table 1. M1,2 for the GDD of Fig. 2 by considering various relaxation times
To further investigate the performance, the chemical potential of the graphene is varied from the μc=0.2 eV up to μc=0.6 eV. This parameter can be simply tuned by means of electrostatic biasing or chemical doping. The results of NACS for the first and second resonances are shown in Figs. 9–10 respectively, and corresponding field enhancements (M1 and M2) are provided in Table 2. It is observed that increasing the chemical potential blue shifts the resonance frequency of the plasmons and results in the highly confined electric fields in both operating frequencies thanks to the increase of charge carriers contributing to the plasmonic oscillations [47]. We did not further increase the chemical potential due to large simulation times required in the low wavelengths resulting from blue shifts of plasmon frequency. We have included the FWHM as a function of chemical potential in both frequency bands in Table 2. As expected, by increasing the chemical potential, the FWHM decreases in both frequency bands due to NACS enhancement.
Fig. 9. NACS for the structure of Fig. 2 at the first resonance for various chemical potentials of (a) 0.2 eV (b) 0.4 eV and (c) 0.6 eV.
Fig. 10. NACS for the structure of Fig. 2 at the second resonance for various chemical potentials of (a) 0.2 eV (b) 0.4 eV and (c) 0.6 eV.
Table 2. M1,2 and FWHM1,2 for the GDD of Fig. 2 by considering various chemical potentials
3.2 Nano-gap size
To study the effect of the dimer gap size on the amount of field enhancement and its resonance frequency, the parameter g is varied from g = 4 nm to g = 20 nm. With this choice of the gap dimensions, complex quantum and non-local effects can be neglected and only the hybridized dipole-dipole interactions should be considered. Similar gap thicknesses are used to investigate the invisibility states of graphene-coated dimers with cylindrical symmetry [41]. The simulation results for various gap thicknesses are provided in Table 3. As the table shows, increasing the gap size blue shifts the resonance frequency of the plasmons because of the reduction in the plasmonic coupling. Moreover, as expected, the amplitude of the enhanced field is inversely proportional to the gap thickness. Since the fabrication of small gaps is challenging, in the practical applications a compromise should be taken between the gap size and field enhancement factor. Also, for g = 14 nm, the second resonance does not fulfill the required field enhancement for the SERS and a single band operation is attained. For g = 20 nm, neither the first nor the second resonance is suitable for SERS applications. Therefore, for SERS applications, the nano-gap size cannot be arbitrarily large unless some other mechanism is exploited to enhance the performance. It should be noted that as Table 3 shows, a slight change in the gap size from 2 nm to 4 nm greatly reduces the field enhancement. This observation was previously reported for Au spherical dimers [48]. Spatial electric field distributions at the first resonance for gap thicknesses of g = 8 nm g = 14 nm are shown in Fig. 11 in order to further confirm the results of table3.
Fig. 11. Spatial electric field distribution for the structure of Fig. 2 at the first resonance for various gap thickness (a) g = 8 nm (b) g = 14 nm.
Table 3. M1,2 and fr1,2 (THz) for the GDD of Fig. 2 by considering various gap sizes (nm)
Importantly, the localized fields attained with the large gaps may be possibly used in the design of other novel optoelectronic devices by periodic pattering the proposed dimers. For example, in the optical absorbers, the confined electromagnetic fields resulting from charge accumulation in the pattered graphene-based structures help to enhance the coupling of the incident wave to the SPPs. Reflecting mirrors are commonly used in the design of optical absorbers for the purpose of blocking the wave transmission [47]. Interestingly, the same strategies can be used to provide an extra hybridization mechanism in the nanoparticle-mirror structures in substrate-mediated SERS [49]. Moreover, as SERS is a surface spectroscopy tool, a larger surface area with a periodic array of plasmonic hot spots can increase the detection of a greater number of molecules. Therefore, by residing periodic arrangement of dimers on top of the engineered substrates, the SERS signals can be enhanced considerably. Apparently, graphene plasmons can be potentially used in the design of SERS substrates and the coupling coefficient of it with the residing particles can be manipulated by varying the number of stacked layers [50].
3.3 Core material
The core material of the disks is another parameter that can be used to adjust the amount and resonance frequency of the field enhancement. In order to investigate the effect of core material on NACS, the dimer of Fig. 2 is simulated with a dielectric core with the permittivity of 2. As Fig. 12 shows NACS amplitude is in the same order of magnitude with the vacuum disk and only an up frequency shift has happened.
Fig. 12. NACS at the first resonance for graphene-coated disk dimers with dielectric core with the permittivity 2.
Scattering analysis of various geometrical shapes with the cores made of left-handed metamaterials (LHM) has been previously examined in the literature [51]. Specifically, graphene-coated spheres with metamaterial cores are examined by Lorenz-Mie scattering theory and it is observed that they have the capability of enhancing the extinction cross-section [52] and thus near-field enhancement. Therefore, two different simulations with (ɛr,μr) pairs of (−1,−2) and (−5,−2) are conducted and NACS results are provided in Fig. 13 at their first resonances. As it is expected, both of the cores provide better NACS than the hollow dimer in Fig. 2. It is clear that the core material will alter the resonance frequency of the maximum enhancement, as well. Since both resonances are of the same nature we did not conduct the simulations of the second resonances.
Fig. 13. NACS at the first resonance for graphene-coated disk dimers with different metamaterial cores. (ɛr,μr) pairs are (−1,−2) and (−5,−2).
4. Summary
Two identical graphene-coated nano-disks in the proximity of each other are proposed as a novel dual-band configuration for the near-field enhancement. By proper choice of the geometrical and optical parameters, a giant field enhancement is observed at the gap center. The dual-frequency bands of the selective signal enhancement emerge from the excitation and hybridization of the localized surface plasmons on the surface of the top and bottom cross-sections of the disks and can be modulated by tuning the relative distance of the disks. The designed structure is superior to its noble metal counterparts due to larger field enhancement, lower ohmic losses, and tunable optical response. Designed dimers fulfill the required field enhancement for the SERS applications. Moreover, by wrapping graphene around the metamaterial core, it is feasible to increase the field enhancement considerably and control the target frequency, as well.
Disclosures
The authors declare that there are no conflicts of interest related to this article
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