Benefiting from low dissipative absorption, dielectric nanoparticles with high-refractive-index have become efficient platforms in producing magnetic dipole resonances and thus harvest functionality in enhancing spontaneous decay rate of optical magnetic dipole emitters, which is also named as magnetic Purcell effect. However, this effect is highly sensitive to emitter orientations, which brings difficulty in practical experiments. Here, we propose an effective method to suppress this orientation sensitivity based on an engineered silicon nanocavity. We will demonstrate that by balancing the resonant wavelengths and emission enhancement for two perpendicularly orientated MD emitters, the whole emission spectrum could be almost orientation-independent, thus accomplishing a completely isotropic magnetic Purcell effect. Further simulations show such effects could survive against a 10 nm spatial deviation of the emitter and will be slightly influenced by the presence of substrate. We anticipate the results of this paper could bring new possibilities in enhancing emission intensity from magnetic dipole transitions in experimental investigations.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Recently, intensive efforts have been devoted to modifying spontaneous decay rate of magnetic dipole transitions from quantum nanoemitters [1–6]. Compared with the widely investigated spontaneous emission from electric dipole (ED) transitions [7–9], that coupling from magnetic dipole (MD) transitions has always been neglected. Generally, MD transitions are considered to be ignorable due to its intrinsic weak strength, which is orders of magnitude weaker than ED transitions . Nevertheless, quantum emitters with strong MD transitions can be found in lanthanide ions (such as Eu3+ [11–13] and Er3+ ), transition metals (Cr3+)  and semiconductor quantum dots . Besides, optical magnetism has played an increasingly significant role in realization a series of novel functions [17–21]. Therefore, it will be of great significance to exploit a feasible way to manipulate the spontaneous emissions for MD emitters. Previously, a variety of metallic nanostructures, such as diabolo antennas  and nanosphere  and split rings [24,25], are employed to enhance the optical magnetic field and modify MD emissions. Yet these structures suffer from considerable ohmic loss induced by conduction current, leading to weak quantum yield. One promising candidate to extricate the dilemma of dissipation is the high-index dielectric nanoparticle, which supports Mie resonances with its inherently lossless property in optical region, as these nanoparticles produce MD resonances based on the displacement current loop rather than the conduction one. Up to date, dielectric nanoparticle has become a popular platform for enhancing and manipulating light−matter interactions in the magneto-optic domain.
In previous works, it has been presented from both theoretical [26–28] and experimental [1,29] perspectives that a silicon nanodisk or hollow nanocavity is capable of enhancing spontaneous emissions from MD emitters. By placing emitters at their magnetic hot spots coupled from the MD resonances, both Purcell factor and radiative decay rate of emitters could be enhanced by hundreds of times, simultaneously maintaining a high quantum efficiency (QE). However, almost all the aforementioned structures rely on the orientation of emitters , which is due to the intrinsic consistency between MD emissions from emitters and MD resonances excited in structures. In these cases, MD emitters should be placed at the position where the local magnetic field gets strongest to maximally promote the emissions from themselves. More importantly, the MD moment must be along the direction of surrounding magnetic field, otherwise the emission enhancement will be significantly impaired. One solution is to control the orientation of emitters. For example, specific molecules inside nanocavity can be accurately orientated by using the nanotechnology called host-guest chemistry [3,31]. Yet for many other commonly used emitters like quantum dots and ions, accurate rotation control is still difficult to realize . Therefore, despite the orientation control, developing an orientation-independent structure is highly desirable. Pioneering work has been done with the silicon nanocavity . It is found that if one can equate Purcell factor (PF) for two perpendicularly orientated emitters under one wavelength, the PF of all the other orientations will become identical for this wavelength. In this case, PF becomes isotropic for all directions, which is thus named as “Isotropic Magnetic Purcell Effect” (IMPE). In , the proposed IMPE operates at single wavelength, which requires a precise spectral overlap between the IMPE wavelength and emission. However, it is known to all that emissions from atoms and molecules always possess spectral linewidth, which originates from their intrinsic emission mechanism, as well as thermal motions or collisions between neighboring atoms . Besides, the targeted wavelength in  has to be off-resonance in order to achieve IMPE and thus weakens the emission enhancement. Therefore, it would be of great demand to develop a device supporting IMPE in a wide spectrum near the center wavelength of emission and keeping the maximum enhancement simultaneously.
In this paper, we propose a structured dielectric nanoparticles, i.e. silicon hollow nanocavity (SHNC). We will demonstrate an effective designing method for SHNC to support IMPE under magnetic dipolar excitations. After analyzing both forming principle and boundary conditions for MD resonances, a feasible way is found to balance both resonant wavelengths and emission enhancement for two perpendicularly polarized MD emitters. More importantly, we will show the IMPE in our engineered nanoparticle can be expanded to the whole emission spectrum, which includes the resonant wavelength. Thus, identical emission enhancement for arbitrary-polarized MD can be realized, which also releases the prerequisite of accurate spectral overlap between IMPE wavelength and emitter.
2. Description of magnetic dipole emissions inside silicon hollow nanocavity
According to Fermi’s golden rule, for a two-level system (TLS) with the transition MD moment m, the spontaneous emission rate γ is related to the magnetic local density of states (LDOS) ρm at r0 (position of the TLS) with transition frequency of ω0 and MD moment m. γ can be expressed as :
Here, is the dyadic Green function of magnetic dipole, n is the unit vector pointing the direction of dipole moment in the TLS. ε0, ћ and c are the reduced Planck constant, permittivity and speed of light in vacuum, respectively. Hence, the spontaneous emission rate γ of the MD emission could be effectively modified through altering the magnetic LDOS , which is approximately proportional to the local magnetic field intensity , i. e. In the context of this paper, strong can be produced through the scattered field by SHNC of the MD emissions from the emitter.
In order to characterize the emission enhancement, three factors are adopted, i.e. radiative decay rate enhancement (RE), non-radiative decay rate enhancement (NRE) and quantum efficiency (QE). RE and NRE are defined as the ratio between the radiative decay rate γr (or non-radiative decay rate γnr) in the structure and the spontaneous emission rate in vacuum. In particular, RE and NRE can be written as RE = Pr / P0 and NRE = Pabs / P0, where P0 is the radiated power by the MD emitter in vacuum with no surrounding structures, Pr is the radiated power into far field when MD emitter is embedded in SHNC, while Pabs stands for the power absorbed by SHNC in near field. Besides, QE is defined as the percentage of the radiated power in the total power emitted by the MD emitter, which can be written as .
3. Identical enhancement for magnetic dipole emissions
To begin with, the schematic of a standalone SHNC with oblique incidence is depicted in the inset of Fig. 1(a). The height (h), inner and outer radius (r and R) are set to 140, 15 and 105 nm, as shown in the lowest panel of Fig. 1(c). Here, the central hole is introduced based on two reasons. One is to fully take advantage of the optically induced magnetic hot spot, making the concentrated magnetic field readily accessible for emitters, which is commonly adopted in silicon nanoparticles [26–28]. The other is the hole can provide an additional dimension for tuning MD resonances . In this paper, numerical simulations are performed by a commercial FDTD software package . The permittivity of silicon is fitted from data of Palik . TE incidence is adopted through the paper, which means the wave vector k is always in YZ plane. The SHNC is illuminated by an x-polarized plane wave. Perfectly-matched layers are employed in all six boundaries of the simulation region to mimic a non-periodic nanoparticle suspended in vacuum. The minimum mesh size is set to 1 nm at the central hole to ensure the calculation precision. Operating wavelength is targeted from 500 nm to 1000 nm, in which Si has a relatively low intrinsic loss with a high index and thus a strong field confinement.
In Fig. 1(a), scattering (Qsca, solid lines) and absorption (Qabs, dashed lines) efficiencies of SHNC are investigated under frontal (θ = 0°, blue curves) and lateral excitations (θ = 90°, yellow curves, also depicted in Fig. 1(c)), which would help to identify the electromagnetic modes and provide further information on localized fields inside cavity. Qsca is calculated by integrating the scattering power on an enclosed surface containing the particle, then normalized to the incident intensity and the projected area of the particle in the plane perpendicular to k. For instance, the projected areas are πR2 and 2Rh for θ = 0° and θ = 90°.
From Fig. 1(a), several phenomena can be observed. Firstly, the SHNC shows a typical scattering behavior of a high-refractive-index nanoparticle under both excitations. In Qsca spectra, MD and ED resonances are both observed with the ED resonance appearing on the shoulder of MD resonance by the blue side, which is a commonly observed lineshape for silicon nanocylinders [27,29]. Secondly, the resonant wavelength (λr) of MD resonance is more sensitive to the incident direction compared with the ED resonances. As shown in Fig. 1(a), λr of the two MD resonances under frontal and lateral excitations are ~742nm and ~827 nm (marked as red dot and circle through Figs. 1 and 2). But λr for ED resonances stay nearly unaffected. This is due to the different origins of ED and MD resonances. Excitation of MD resonance depends on the displacement current loop inside the particle. Driving such a loop requires opposite electric fields at the top and bottom, which suggests the particle should support enough field retardation . In contrary, ED resonance results from the collective polarization of the material driven by the incidence wave and leads to a magnetic current loop perpendicular to the polarization. However, the magnetic loop under ED resonances is allowed to extend outside the particle profile due to the same permeability for both SHNC and vacuum. Therefore, sufficient retardation is necessary for MD resonance, but unnecessary for ED resonance. Since here 2R > h for SHNC, retardation for MD resonances increases accordingly from θ = 0° to 90°, resulting in an increased λr. Thirdly, magnetic quadrupole (MQ) resonance appears at ~537 nm in the Qsca spectrum under frontal excitation. This higher-order mode is composed of two MD-like modes oscillating in the reversed phase along y direction, as shown in Fig. 1(f). Fourthly, Qabs for both illuminations stays in a very low level, which is because the operating frequencies are below the semiconductor bandgap and the material absorption can be ignored.
In order to further identify the resonances in Fig. 1(a), normalized | H |2 (solid lines) and mode volumes (dashed lines) for both excitations are extracted from the numerical results and plotted in Fig. 1(b). The data of | H |2 are taken from the center of SHNC (origin point in the inset of Fig. 1(a)) and normalized by | H |2 at the same point without the particle. The normalized mode volume (NMV) is calculated through the definition below:Fig. 1(b), it clearly shows two minimums in each NMV curve. The one at longer wavelength corresponds to MD resonance, which could also be verified by the normalized | H |2. Generally, modes inside nanocavity exhibit stronger confinement around the pronounced MD resonance and thus leads to a higher | H |2. In Fig. 1(b), two MD resonances in normalized | H |2 locates at 746 nm and 830 nm (212 and 354 times for θ = 0° and 90°), while that in the normalized mode volume are 737 nm and 830 nm, which matches the result of Fig. 1(a) and further validates two MD resonances.
Next, Figs. 1(d) and 1(e) illustrate two magnetic field intensity distributions at their respective resonances in Fig. 1(a), shown as the red dot and circle. As illustrated, the magnetic hot spots are concentrated inside the particle for the both, but an evident difference is the hot spot in Fig. 1(d) splits into two parts, localized besides the hole in the dielectric side. This is due to the discontinuity of refractive index and the polarization difference in H field between the two MD resonances. The resonances in Figs. 1(d) and 1(e) are respectively dominated by Hy and Hz (as shown in Figs. 1(h) and 1(i)) which are induced by the displacement current loop in xz and xy planes. This implies the dipole moments for Figs. 1(d) and 1(e) are along y and z directions. Hy in Fig. 1(d) is normal to the two hole interfaces (indicated by red lines) and will be scattered by the refractive index discontinuity. On the other hand, Hz in Fig. 1(e) is tangential with respect to the interface and thus will not be scattered. Similar field distribution is also observed in . This also explains why field intensity inside the hole of Fig. 1(d) is lower than Fig. 1(e). Moreover, in Fig. 1(d), Hy inside dielectric could readily extend into the hole. This is attributed to boundary conditions of H fields. When extended into the hole, Hy needs to distribute across two interfaces (shown by two red lines), where Hy and Hz are respectively normal and tangential components and thus follow n • (μ2H2 – μ1H1) = 0 or n × (H2 − H1) = K. Here, n is the normal vector of interface, H2 and H1 are the field on both sides, K is the surface current along the interface. Since and K = 0 for dielectric, both Hy and Hz are continuous across the interface. Consequently, highly confined H field in Fig. 1(d) could extend into the hollow region. More importantly, the continuity in Hy under frontal excitation also implies the maximum of H field could be further improved by decreasing r, which is also demonstrated in Fig. 1(g). As r shrinks from 15 nm to 5 nm, normalized | H |2 for θ = 0° increase from 212 to 307 (36.6% of the average RE value). On the contrary, field intensity for θ = 90° is less sensitive, which only increases from 354 to 382 (7.6% of the average RE value). In addition, it has also been shown the resonance amplitude increases for smaller hole width , which is a result of the increased effective index of the particle and thus a stronger light confinement . Nevertheless, hole dimension provides a powerful way to eliminate the difference of enhancement between the two excitations. This is very important for building identical emission spectra for randomly polarized MD emitters, which will be demonstrated in Fig. 4.
After examining the properties under frontal and lateral incidence, we are ready to study if SHNC is capable of supporting MD resonances under oblique incidence. Figure 2(a) provides two spectra at θ = 45°, the normalized | H |2 (black curve) and Qsca (green curve). Compared with Fig. 1, MD resonances in both curves split into two with reduced peak values, resulting in two dual-band spectra. Moreover, two resonances in both spectra show up at two almost identical positions (as red dot and circle shown) as in Figs. 1(a) and 1(b), which are ~742 nm and ~827 nm. Consequently, this identity among spectra demonstrates MD resonances of and 90° are both preserved when illuminated obliquely. In addition, | H |2 distributions at the two resonances are illustrated in Figs. 2(c) and (d), which are also dominated by Hy and Hz (similar to Figs. 1(d) and (e)). Besides, Fig. 2(c) further proves fields available in the hollow region are extended from the hot spot in the dielectric side. As the property under plane wave excitation is studied, now it is more important to investigate the emission enhancement of SHNC for an MD emitter. The inset of Fig. 2(b) plots an ideal oscillating MD located at the center of SHNC and polarized with the angle θMD of 45° in respect of z axis. In order to investigate the performance of MD emitter, RE and NRE are calculated. It is remarkable that RE spectrum in Fig. 2(b) is almost identical to the normalized | H |2 in Fig. 2(a), indicating the importance of MD resonance for emission enhancement. Specifically, enhancing MD emissions requires spectral overlap between emitter and MD resonance. Furthermore, Fig. 2(e) calculates RE under arbitrary θMD, while the emitter is fixed at the maximum position of field intensity when excited by the plane wave, i.e. the center of SHNC. Figure 2(e) predicts that although θMD obviously affects RE, the two resonances at ~742nm and ~827nm are still preserved, which are represented by red circle and dot with black dashed lines. As θMD increases from 0° (corresponding to θ = 90° in Fig. 1(a)) to 90° (corresponding to θ = 0°), resonance gradually shifts from 827nm to 742nm. For angle around 45°, two peaks are excited simultaneously, forming the dual-band spectrum In Fig. 2(b). In Fig. 2(f), QE is also calculated as a function of θMD. Obviously, θMD shows very limited influence on QE, which stays at a high level above 600 nm. Besides, QE dramatically drops as λ moves towards blue light band, which is due to the increase of imaginary part of permittivity in silicon.
Since Fig. 2(e) shows λr strongly depends on θMD, it would be meaningful to develop an orientation-independent structure, which will be compatible for arbitrary θMD. To accomplish this destination, we need to tune the two resonances into an identical λr and then reduce the enhancement differences between the two peaks as much as possible. As aforementioned, MD resonance is dependent on displacement current loop and thus sensitive to the aspect ratio (AR), which will dramatically affect the loop . Hence, Fig. 3(a) shows a series of RE spectra with different AR (defined as 2R/h in this paper) when θMD = 45° and r = 15 nm. Numbers in brackets are respectively 2R and h of SHNC, which decrease or increase by 5 nm in each adjacent curve along the gradient-color arrows. By gradually reducing AR, it can be found the double-band behavior is gradually eliminated and resonances can thus be amalgamated, which thus provides the opportunity to optimize the emission spectrum for arbitrary-orientation MD sources, opening the possibility of constructing IMPE spectrum.
In order to fully understand how the two peaks merges together, it is necessary to review these peaks under plane-wave excitations. Former works have demonstrated that increasing both outer diameter and height of SHNC will cause redshift of MD resonances , which is due to the larger loops formed inside the particle. As observed in the normalized | H |2 in Fig. 2(a), there are originally two MD resonances, which have the same wavelengths as the peaks in Fig. 2(b) under magnetic dipolar excitation. In Fig. 2(a), the first resonance (shown by red dot) corresponds to frontal plane-wave incidence (θ = 0°, 742 nm). Its displacement current loop is in xz plane and its wavelength is thus sensitive to the particle height (h). On the other hand, the other resonance in Fig. 2(a) (shown by red circle) corresponds to lateral plane-wave incidence (θ = 90°, 827 nm). The current loop is in xy plane and its wavelength is thus sensitive to the outer diameter (2R). Hence, increasing h or decreasing 2R means a larger or smaller current loop for formation of these resonances, which consequently causes a redshift of the red-dot resonance or a blue shift of the red-circle resonance. Moreover, it is shown in section 2 the magnetic local density of states and spontaneous emission rate , which leads to an intrinsic consistency between normalized and RE spectrum (this is also indicated by black curve in Fig. 2(a) and red curve in Fig. 2(b)). Therefore, the evolving law above for resonances excited by plane wave can be applied to resonances by dipolar excitation (Fig. 3(a)) in a straightforward way. As increasing h and decreasing 2R simultaneously, the red-dot and red-circle peaks in Fig. 3(a) show respectively redshift and blueshift, which in turn merge with each other (as the three red dots and circles illustrated).
Afterward, Fig. 3(b) firstly calculates the normalized | H |2 and scattering efficiency of a (175 nm, 175 nm) SHNC excited by plane wave, which further confirms the MD resonance around 750 nm in both spectra, as blue star indicated. Then, RE is plotted as a function of θMD in Fig. 3(c), which evidently reveals λr becomes almost independent on θMD. The maximal and minimal λr appear at 0° and 90°, which are 758 nm and 742 nm, with only 16 nm deviation. This unique property is due to the elimination of difference in λr and will benefit SHNC in achieving IMPE spectrum. Figures 3(d)–3(f) further plot RE, λr and QE for SHNC by red lines under arbitrary θMD in polar coordinates. As a comparison, two ideal cases are listed aside, which are a silicon nanosphere with a 30 nm-diameter circular hole (green dashed lines) and a perfectly isotropic silicon nanosphere with a 30 nm-diameter hollow core (blue dashed lines, named as “ideal nanosphere” thereinafter). The diameters of the two nanospheres are 200 nm and 194 nm with λr both at ~754 nm when θMD = 0°. Obviously, the latter one holds a totally isotropic property due to its symmetry. Usually, λr for MD resonance in silicon nanosphere is related to its diameter as λr/nsi ≈2R . Since and , 2R thus should be 202nm, which is close to the diameter adopted here. In Fig. 3(e), it shows λr in left and central panels are almost perfectly isotropic, with only a slightly larger λr in vertical direction than horizontal. Considering the cylindrical symmetry of SHNC in respect of z axis, this λr will stay unaffected from any xy orientation of MD. Figure 3(f) further plots QE in three particles under their respective λr, which also shows an isotropic property. Unfortunately, this isotropy in SHNC fails in RE, as clearly depicted in Fig. 3(d). Due to difference of RE between and 90°, the whole RE curve is obviously anisotropic. This problem becomes even serious for the perforated silicon nanosphere, only the ideal nanosphere supports isotropy of RE.
In order to accomplish IMPE in SHNC as the ideal nanosphere, it is crucial to erase the difference in RE for θMD = 90° and 0°. As concluded Fig. 1(g), shrinking the hole will decrease the difference in | H |2, which thus implies the way for balancing RE between 0° and 90°, as the magnetic LDOS Consequently, RE spectra is calculated with decreased inner radius for θMD = 90° and 0°, as illustrated in Figs. 4(a) and 4(b). It is found that RE for 90° increases much faster than 0°. For r = 5 nm, the difference is only 15, which is 4% of the total enhancement factor. This thus allows us to construct an IMPE spectrum in SHNC. In order to verify the final effect, RE is then plotted under multiple θMD, as shown in Figs. 4(c) and 4(d). r, 2R and h are fixed at 5 nm, 175 nm and 175 nm, as illustrated in Fig. 4(e). Obviously, Fig. 4(c) demonstrates that as θMD gradually changes from 90° to 0°, the RE peak value increases within a tiny range (368~383, 3.9% of the average RE value). More importantly, RE become almost identical in a wide wavelength range. This indicates IMPE of the optimized SHNC is achieved in a wide spectrum. As a result, such an effective designing method will bring greater spectral tolerance for experiments. Moreover, Fig. 4(f) studies RE as function of θMD, which further proves this angle-independent property and the identical emission enhancement. Finally, RE, λr and QE are plotted by polar coordinates in Figs. 4(g)–4(i) as a comparison of the ideal nanosphere. It can be seen that the three quantities are almost identical as the ideal nanosphere. Compared with , the IMPE in this paper is achieved in almost the whole emission spectrum (from 550 nm to 1 μm, as plotted in Fig. 4(c)), which will be useful for practical magnetic light emission. Furthermore, the IMPE here covers λr of the RE spectrum, which means such a design of SHNC could support IMPE while keeping enhancement to the maximum extent.
In addition, although it could be challenging to experimentally realize the hole sizes adopted in Fig. 4 (r = 5, 10 and 15 nm), such a small hole has still been successfully fabricated on Si nanocylinders using electron beam lithography and reactive ion etching in .
Finally, it will be of great meaning for real applications to test if IMPE can be tuned to other wavelengths or supported under the presence of substrate. After shrinking both 2R and h down to 128 nm with a fixed r (5 nm), Fig. 5(a) depicts the wavelength of IMPE can be tuned to ~590 nm, which is also the well-known emission wavelength of Eu3+ (the 5D0→7F1 MD transition). Inset in Fig. 5(a) is an enlarged view of the spectral peaks. It clearly shows the difference of RE between θMD = 90° and 0° is 30, which is about 7.9% of the whole RE value, thus demonstrating the IMPE can be preserved for 590 nm. Besides, Figs. 5(b)–5(f) list two imperfections for IMPE, i.e. spatial deviation of emitters from the center and influence of the substrate. Firstly, Fig. 5(b) shows a 10 nm Δz will cause an enlarged difference of RE for and 90°, which is 22 and about 5.9% of average RE value. Again, this result prove IMPE in this designed SHNC is robust against 10 nm location deviation. Similar results have been observed in . But if further increasing Δz to 20 nm, this difference will keep increasing to 42 (12.2% of average RE factor). In addition, since the inner radius , the situation of is also calculated. It is found that the result is almost identical to that of (Fig. 4(c)) and thus not plotted for comparison. Next, influence of the substrate is also calculated in Figs. 5(d)–5(f). As clearly shown in Fig. 5(d), larger n brings a dramatically reduced peak values, gradually broadened and slightly redshifted resonance. This reduced peak value can be attributed to a decreased intensity formed inside the particle under dipolar excitation, which is similar to the situation by plane wave excitation. In plane wave cases, a high-refractive-index substrate will weaken the displacement current loop and thus the field intensity inside a nanoparticle . Figures 5(e) and 5(f) further plot RE spectrum for n = 1.5 and 2 under multiple θMD, the respective RE differences are 9.2% and 20.5% of RE value, which suggests n of the substrate should be as low as possible to protect IMPE. Moreover, two points should also be noted about the IMPE concept. One is that although IMPE has been achieved in a broad spectrum range, only the narrow band near the resonance could be utilized due to the low emission enhancement of the rest. The other is that the IMPE concept is about the unified spontaneous emission rates for arbitrary-orientation MD emitters. For practical applications, the induced absorption of emitters from external excitation should also be taken into account.
In summary, we have proposed an effective method to unify emission spectra for arbitrary-polarized magnetic dipole emitters, which thus broadens isotropic magnetic Purcell effect to the whole emission spectrum. By a numerical demonstration, we reveal the importance of equating both emission enhancement and resonant wavelengths for emitters in horizontal and vertical directions, which can be achieved by properly designing either the hole dimension or the aspect ratio of nanocavity. Further simulations demonstrate such isotropic emission property could survive on spatial deviation of 10 nm and a low-refractive-index substrate, which thus brings a concrete possibility of easy implementations. Altogether, the result of this work may facilitate the modification of magnetic dipole emissions with all-dielectric nanocavity, especially in increasing the proportion of emitters that can be actually enhanced, whose orientations are usually considered to be random in nanodevices.
National Natural Science Foundation of China (NSFC) (51871167 and 11504268).
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