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Abstract
As a symmetry-breaking system, bent metal nanowires (MNWs) are able to guide asymmetric plasmon modes, but are also subject to deteriorated waveguiding performance. Previous studies mainly focused on bending losses or longitudinal modes, while the inherent asymmetric nature of the transverse mode was often overlooked and its potential in high-performance manipulation has remained unlocked. Meanwhile, they also showed contradictory conclusions regarding the effect of the bend that need further study. Here, we investigate the previously neglected field-deformation effects to demonstrate their significance on mode behaviors, and propose manipulation strategies of asymmetric modes with excellent qualities that are even higher than their symmetric counterparts in straight MNWs. We show that the field distributions can be translationally, rotationally, and longitudinally reformed, allowing not only extra degrees of freedom in manipulations, but also possibilities to convert the bending into a favorable factor for mode quality. Even for sharply bent MNWs with a sub-wavelength bending radius, the quality of the asymmetric mode can be greatly enhanced, even enabling a figure of merit almost twice its symmetric counterpart in straight MNWs without bending. Our findings may greatly extend the capabilities of bent MNWs, offering new opportunities in high-performance plasmonic components and flexible devices.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As a typical one-dimensional nanostructure, metal nanowires (MNWs) are able to support guided modes of surface plasmon polaritons (SPPs) with both the deep-subwavelength confinement and the relatively low loss at visible and near-infrared regimes [1,2]. In the MNW system, asymmetric modes in terms of asymmetric transverse field distributions have recently attracted much attention due to their mode properties can be versatilely tuned with improved performance (e.g., reduced mode area [3], enhanced coupling efficiency [4], reduced plasmon damping [5,6], engineered zigzag/chiral/spin-dependent propagation [7–9]), offering numerous opportunities in applications including deep-subwavelength lasing [10], plasmonic circuits [11–13], and quantum optics [14,15]. Such asymmetric modes can be obtained by introducing symmetry breaking factors to lift the degeneracy of the system, enabling interactions and couplings between the originally symmetric modes, which can be practically realized by the material or the geometry configurations with dimensions ranging from the macroscale to the nanoscale, such as substrates [3–6,16], branches [17–19], bending corners [20], cross-sectional shapes [21], and nearby emitters [22–24].
Among these structures, bent MNWs have been widely used to form interferometers, ring resonators, couplers, and splitters, serving as crucial building blocks for plasmonic networks [1]. However, compared to other well-studied MNW systems, previous research in bent MNWs mainly focused on the bending loss [25,26] or the longitudinal mode distribution [20,27,28] with less discussion in their transverse mode characteristics and engineering. Meanwhile, some conclusions seem to be inconsistent — for example, whether the guided SPPs are affected [22,25,26] or unaffected [27,29] by the presence of the bend in MNWs. On the other hand, the asymmetric nature of these modes in bent MNW is usually overlooked by assuming the mode profile and the propagation constant to be the same as the ones in the straight MNW [25,26]. In fact, substantial changes occur with the severe bending. And as a result, the exclusion of the transverse field deformation may not only lead to inaccurate or even opposite predictions on the SPP behaviors, but also hinder realizing the full potential of the symmetry-broken system for mode engineering with performance improvement.
Motivated by these considerations, we systematically investigate characteristics of asymmetric modes in bent MNWs, providing clear insights on the previously overlooked field-deformation effect and the bending-dependent behaviors, as well as the optimized configuration for achieving high-performance waveguiding. Meanwhile, we propose various manipulation strategies to show that such asymmetric modes are highly versatile that can be tuned with extra degrees of freedom, enabling the translational, rotational, and longitudinal manipulation of their field distributions. Benefiting from the reformed field, it is even possible to convert the bending into a favorable factor for an improved energy confinement. More importantly, we also show that, by proper arrangement of asymmetric factors, one is able to greatly enhance the overall quality of the plasmon mode in sharply bent MNWs with the sub-wavelength radius of curvature, offering an excellent figure of merit (FoM) that is even superior to the symmetric mode in straight MNWs without bending. Our findings may extend the capabilities of bent MNWs in applications such as modulating the light-matter interaction at the nanoscale, and achieving both high performance and versatile tunability in a variety of plasmonic components and devices.
2. Mathematical model and mode assignment
By applying Marcuse’s method [30,31], the bent MNW with the geometric radius of curvature Rb can be described in a cylindrical coordinate system (r, θ, z) that locally follows the curved MNW axis, which is illustrated in Fig. 1(a)(i), and its cross-sectional and top views are also provided in Fig. 1(a)(ii-iii). The plasmon modes in the bent MNW have the azimuthal dependence and are expressed as [30–32]
where β is the propagation constant and ρ is the curvature radius of the plasmon mode. Note that, if the plasmon mode is symmetric with no field deformations, ρ is identical to Rb, however, the bend in the MNW may dramatically distort the mode profile, giving rise to a large deviation between ρ and Rb, which cannot be neglected and will be discussed later. By analogy with the straight MNW, Eq. (1) is understood in the way the SPP wave travels along the arclength of ρθ. Within this framework, the bent MNW can thus be mathematically modeled in the form of an eigenvalue problem of Helmholtz equation with the eigenvalue of -iρβ to be solved, which is in a similar manner to the straight MNW with an eigenvalue of -iβ. Here, we numerically solved the above eigenvalue problem using a finite element method under axial symmetry and further verified by a full 3D simulation implemented by the COMSOL Multiphysics. For calculation, Au MNWs with the permittivity from Johnson & Christy [33] working at the typical wavelength of 785 nm is selected as our model system.
Fig. 1. (a) Mathematical model: (i) a MNW is bent into a circular arc which is described in a cylindrical coordinate system (r, θ, z) that locally follows the curved MNW axis, and the corresponding unit vectors in the cylindrical coordinate system are er, eθ, and ez, respectively. The geometric radius of curvature is Rb. Red circle, an arbitrary cross section plane with its corresponding (ii) cross-sectional view. (iii) Top view of the bent MNW. The inner/outer side denotes the inner/outer side of the curve. (b) Mode assignment: three lowest-order modes from the higher order modes to the fundamental mode (from the left to the right panel) in (i-iii) a straight MNW and (iv-vi) a bent MNW with Rb = 1 µm. (c) Electric field intensity distribution of the bent MNW, where the fundamental mode dominates during the propagation. The MNW used here is 150 nm in diameter.
To reveal its asymmetric nature, we first start with the mode assignment in the bent MNW system. Figure 1(b) gives the mode profiles of the three lowest-order modes in a 150-nm diameter (D) Au MNW with Rb = 1 µm. For reference, the lowest three modes in straight MNWs are also given (from higher order to the fundamental modes in Fig. 1(b): (i-ii) doubly degenerate HE±1 modes and (iii) TM0 mode). As is shown, the symmetry breaking by the presence of the bending enables in-/out-phase couplings between the originally symmetric modes in straight MNWs, generating new hybridized modes with asymmetric field distributions (Fig. 1(b)(iv-v) hybrid-HE modes and (vi) hybrid-TM mode). The above mode hybridization is actually similar to the one in the substrate-MNW system [4], which agrees well with one of the bending theories in planner waveguides [34]: the bent waveguide can be approximated by a straight one with an increasing (decreasing) refractive index around the outer (inner) side of the curve. Note that, although the mode hybridizations are similar, waveguiding properties in bent MNWs are subject to influence by the field deformation (e.g., the travelling distance for SPP waves is ρ dependent and varies at the inner and outer sides of the curve), resulting in various intriguing properties for the bent MNW system which will be discussed later. Meanwhile, among the above asymmetric modes, the higher-order modes are less confined (e.g., Fig. 1(b)(v) vs. Figure 1(b)(vi)) with almost infinitely large mode areas once the MNW has a relatively small D, which cannot fully take the advantage of the confinement brought by SPPs. On the other hand, for real applications where single-mode operation is favorable, the fundamental hybrid-TM mode also dominates during the propagation in the bent MNW (Fig. 1(c)). Therefore, we will only focus on the hybrid-TM mode, and it will be generally referred as the fundamental mode for simplicity.
3. Free-standing bent MNWs with translational mode engineering
Due to the asymmetric nature discussed above, the fundamental mode is located towards the outer side of the curve with a field deformation at the r direction (Δr), which can be quantitatively characterized by the distance between the geometry center and the mode center:
Here, the second part in the Eq. (2) represents the position of the geometry center, and the first part indicates the mode center which is the average position of the total area of the mode weighted according to its time-averaged Poynting vector along the azimuthal direction (Pθ).
The calculated Δr of free-standing Au MNWs with D of 20, 60, 100, 150 and 200 nm are plotted in Fig. 2. Result shows, except for the very thin MNW (e.g., D = 20 nm), Δr gradually increases with the decreasing Rb (Fig. 2(b)), indicating a more asymmetric mode profile with an outward translational movement of the mode center from the geometry center in the r direction (e.g., Fig. 2(a), where the mode center is indicated by the blue plus symbol). For verification, we also calculated the dependence of the fractional energy inside the MNW (η) on Rb (Fig. 2(c), calculated using Eq. S1 in Supplement 1), which shows a decline trend in η with Rb, corresponding well to the lateral shift Δr in the mode profile. By precisely tuning the Rb, both Δr and η can be continuously modified. And for MNWs of typical diameters (e.g., D >= 100 nm), Δr can reach several tens of nanometers to over 100 nm with the Rb at the subwavelength scale, resulting in more energy (e.g., ∼95% of energy for D = 150 nm with Rb = 0.5 µm) being translationally pulled out of the MNWs compared to the straight ones, which is favorable in applications such as tactile sensing, evanescent-wave sensing, and evanescent-wave coupling.
Fig. 2. Field deformation in free-standing bent MNWs. (a) Mode profiles in terms of the time-averaged power flow in bent MNWs with (i) D = 200 nm, Rb = 2.5 µm, (ii) D = 200 nm, Rb = 0.5 µm, (iii) D = 60 nm, Rb = 2.5 µm, (iv) D = 60 nm, Rb = 0.5 µm, respectively. The blue plus symbols represent the positions of the mode centers under different Rb. The above deformation effect is characterized by the (b) distance between the geometry center and the mode center (Δr) with varying Rb, and verified by (c) their fractional energy inside the core (η). For reference, η of straight MNWs are also plotted in dashed lines and indicated by (str.) in the legend of the figure.
It is worth mentioning that, such substantial Δr in those bent MNWs cannot be neglected, otherwise it may lead to erroneous results about the waveguiding properties. To illustrate the necessity to consider the above deformation effect, we use the effective refractive index (neff) of the plasmon mode (neff = Re(β)/k0, k0 is the free-space wavevector) for demonstration. As is well known, neff is a key parameter for waveguiding structures, containing information about the SPP wavelength, the phase velocity and many other important properties (e.g., to determine whether the mode is bound or not in comparison to the surrounding index, and to be used as an important parameter to design plasmonic components and devices such as couplers). If we neglect the field deformation, Rb-dependent neff of free-standing MNWs can be obtained by assuming that ρ = Rb, which is plotted in Fig. 3 and neff of the same straight MNW is also provided (dashed line) for comparison. As is shown, neff increases with the decreasing Rb, resulting in a much higher neff at a sub-wavelength Rb compared to that of a straight MNW. However, since the phase velocity (vp) is inversely proportional to the neff (vp = c/neff, c is the free-space light speed), a higher neff implies a slower vp that is in conflict with the fact that most of the mode shifts to the outer side of the curve with a longer distance to go through (Fig. 2), and the phase fronts at where more distant from the center of the curve have to travel faster than the nearer ones to keep up with the mode propagation [35], which consequently requires a faster phase velocity. Meanwhile, an increasing and a higher neff than the index of the surrounding (n = 1, in this free-standing case) also wrongly indicates that the mode becomes highly confined and will always be a non-radiative bound mode regardless of the MNW diameter and the bending curvature, which is obviously in contrary to the experimental results that the mode leaks away from the MNW and becomes a radiating mode with serious radiation observed for MNWs with a radius of the curvature smaller than the diffraction limit [36].
Fig. 3. Illustration of the significance of the field deformation on the mode property. The calculated Rb-dependent effective refractive index (neff) under the assumption ρ = Rb exhibits a wrong trend with respect to the theory and experimental results. For comparison, neff of straight MNWs are also plotted in dashed lines and indicated by (str.) in the legend of the figure.
Therefore, it is of great importance to take the field deformation effect into account, where the Δr should not be neglected. Figure 4(a) gives the corrected Rb-dependent neff curves by applying the condition ρ = Rb + Δr, showing two different trends for MNWs of different D. On one hand, for the D of a moderate or a large value (e.g., 100 nm ∼ 200 nm), neff decreases from the value of the straight MNW with decreasing Rb, and the mode may thus evolve into a radiative mode eventually under a severe bending (e.g., neff = ∼0.99 for a 200-nm diameter MNW with Rb = 200 nm), which agrees with the trend in previous observation in MNWs (e.g., Ref. [36]). On the other hand, when the D is small (e.g., 20 nm or 60 nm), neff remains almost unchanged, which also coincides with the experimental result of thin MNWs from the EELS study (e.g., Ref. [27]). Such two distinct behaviors of MNWs with different D well explain the opposite results from the previous experiments, where the SPPs in thick MNWs (e.g., Ref. [25]) are greatly affected by the bend and the one in thin MNWs (e.g., Ref. [27]) are almost unaffected by the bend.
Fig. 4. Asymmetric modes in free-standing bent MNWs. (a) Effective refractive index (neff), (b) propagation length (Lm), (c) mode area (Am), and (d) FoM of the asymmetric modes as a function of Rb. Their corresponding counterparts in straight MNWs are also plotted in dashed lines and indicated by (str.) in the legend of the figure for comparison. Filling areas in (d) indicate the configurations for the optimized FoMs. Note that, when the Rb is extremely small with a very large Am, the calculated Am becomes less accurate since the mode area is slightly truncated by the symmetry axis in our simulation model, which is why the Am and FoM plots for D = 150 and 200 nm are cut at Rb = 300 nm and 400 nm, respectively.
Since the bending behavior is D- and Rb-dependent, it raises an interesting question about the optimized geometric parameter for guiding SPPs in the bent MNWs. For investigation, two key parameters — the loss and the confinement of the asymmetric mode are calculated in terms of the propagation length Lm (Fig. 4(b), calculated using Eq. S2 in Supplement 1) and the mode area Am (Fig. 4(c), calculated using Eq. S3 in Supplement 1). For the thick MNWs, the Lm decreases and the Am increases with the decreasing Rb significantly, resulting in a poorer quality of SPPs deteriorated by the sharper bend. While for the thin MNW, although it can preserve the Lm and the Am with a subwavelength Rb, its inherent propagation loss (i.e., the straight one) is already relatively large and mode tunability is relatively low, which may also limit its application. To find the optimized parameter, we use the FoM to characterize the quality of the asymmetric mode [4,37] as plotted in Fig. 4(d) (calculated using Eq. S4 in Supplement 1). The FoM curves exhibit a staircase behavior, which are divided into three regions with respect to the Rb. Within the range of interest discussed here, the highest FoM for Rb > ∼1.7 um, ∼0.5 um < Rb < ∼1.7 um and Rb < 0.5 um are ∼30-35 (achieved in MNWs with D = 100 nm), ∼24-29 (achieved in MNWs with D = 60 nm) and ∼23 (achieved in MNWs with D = 20 nm), respectively.
4. Substrate loaded bent MNWs with rotational mode engineering
Apart from the free-standing bent MNW, another typical application scenario is to place the bent MNW on a substrate (Fig. 5(a)), where two symmetry-breaking factors are incorporated. For the substrate loaded bent MNW with a small D (e.g., D = 20 nm), it is similar to the free-standing case that the mode is less affected with almost no change in characteristics (Fig. S1, see Supplement 1). While for the thicker MNW, it shows much higher tunability and mode variety. The substrate and the bend exert a combined effect on the plasmon mode with field deformations in both r and z directions. For quantification, the deviation of the mode center from the geometry center is characterized by the displacement (Δl) and the rotation angle (Δφ), which are expressed as
Fig. 5. (a) Schematic illustration of the structure. (b) Field deformation characterized by the displacement (Δl) and the rotation angle (Δφ) as a function of Rb of a bent MNW (D = 150 nm) on different substrates (MgF2, PDMS, PET, and SiN). (c) Mode profiles in terms of the time-averaged power flow of a MgF2-supported bent MNW with (i) Rb = 2.5 and (ii) Rb = 0.3 µm, respectively. The diameter of the MNW used here is 150 nm.
Figure 5(b)-(c) give calculated results and mode profiles of a 150-nm diameter MNWs on typical substrates including MgF2 (n = 1.37), PDMS (n = 1.41), PET (n = 1.56) and SiN (n = 1.89). It can be seen that, only slightly changes occur in Δl by reducing the Rb, while the mode tends to rotate around the MNW with a large Δφ. For example, for the MgF2-supported bent MNW with Rb ranging from 2.5 to 0.2 µm, the increment for Δl is only 3 nm (from ∼77 to ∼80 nm), whereas Δφ shifts from ∼9° to ∼51°. Such behavior enables another way to engineer the plasmon mode with reformed energy distribution in a rotational manner, apart from the translational manipulation. The reformed energy distribution may be used to increase the overlap between the plasmon mode and nearby substance (e.g., emitters) with the enhanced light-matter interaction.
For further characterization, the calculated Rb-dependent neff of the MNW on different substrates are plotted in Fig. 6(a), showing a decreasing trend compared to the straight one supported by substrates (dashed lines). For the MgF2 and PDMS cases, the original bound modes gradually become quasi-bound with the decrease in neff, and the transition points in Rb are 1.1 µm (neff = ∼1.369 for MgF2) and 1.6 µm (neff = ∼1.409 for PDMS), respectively. Meanwhile, for the PET and SiN substrate with high indices, the modes are already quasi-bound at Rb = 2.5 µm. As to the confinement and loss (Fig. 6(b)-(d)), the decreasing Rb results in decreasing Lm and increasing Am, exhibiting a smaller Lm and larger Am compared to the straight MNWs on substrate (except for the SiN-supported case). Notably, the Am in the SiN-supported bent MNW is ∼0.006 µm2, showing a much better confinement than the same straight MNW on the SiN substrate (∼0.033 µm2), which can be explained by the reform in the energy distribution. For the SiN-supported straight MNW, the mode is a quasi-bound mode (neff = ∼1.875) with most of the energy leaked into the high-index SiN substrate, while in the presence of the bend, more energy is dragged into the low-index air as a result of the rotation of Δφ in the mode profile, leading to less leakage with a consequently better confinement (e.g., Fig. 6(c) insets). More importantly, it also suggests that, compared to the straight counterpart, bending can be turned into a favorable factor with proper mode engineering. On the other hand, compared to the free-standing bent MNW with the same Rb (black solid line with pentagrams in Fig. 6(d)), substrate-supported bent MNWs can achieve a better FoM when Rb is small (e.g., Rb < 1 µm), indicating a possible way to improve the overall quality of SPPs by utilizing the synergetic effects of bending- and substrate-induced asymmetries.
Fig. 6. Asymmetric modes in substrate-loaded bent MNWs. (a) Effective refractive index (neff), (b) propagation length (Lm), (c) mode area (Am), and (d) FoM of the asymmetric modes as a function of Rb. For reference, properties of the substrate-supported straight MNWs on MgF2 (light red), PDMS (light orange), PET (light green), and SiN (light blue) are provided in dashed lines, and FoM of the free-standing bent MNW (black solid line with pentagrams in (d)) is also plotted. Insets in (c): Energy density distributions for the SiN-loaded (i) straight MNW and (ii) bent MNW (Rb = 2.5 µm). To clearly reveal the leakage, they are normalized and plotted in a color bar ranging from 0 to 0.06 with saturation.
5. MNWs bending around a curved medium with longitudinal mode engineering
Based on the above discussion, we find that, for the bent MNW with a small D (e.g., D = 20 nm), the plasmon mode suffers from inherent propagation loss with a consequently low FoM, as well as lack of tunability and mode variety. While for the bent MNW with a moderate or large D, the asymmetric mode is versatile for manipulation with a relatively low inherent loss but is subject to deterioration in the presence of a sharp bend (e.g., at the sub-wavelength scale). To address this issue and further explore the possibility to enhance the mode quality, we propose a system with rearrangement of asymmetric factors, which is realized by a MNW bending around a curved medium (e.g., flexible substrates or dielectric cylinders), as is shown in Fig. 7(a). For demonstration, we set the D and Rb to be 100 nm and 500 nm. Figure 7(b)-(c) give the mode profiles and electric field distributions with different refractive index (n) of the curved medium. As is shown, by introducing an index contrast between the inner and outer side of the bend, the factors of curved medium and the curved MNWs counteract against each other, enabling the possibility to manipulate the electric field distribution longitudinally along its propagation directions (Fig.7c) by tuning the interplay between these two asymmetric factors. For instance, the original outer-side biased electric field distribution (e.g., Fig. 7(b)(i) & 7c(i), n = 1, free-standing case) can be tuned into almost balanced distribution (e.g., Fig. 7(b)(ii) & 7c(ii), n = 1.3), and inner-side biased energy distribution (e.g., Fig. 7(b)(iii) & 7c(iii), n = 1.6) with the index of the curved medium from low to high, giving rising to intriguing waveguiding properties.
Fig. 7. Asymmetric modes in the bent MNW-curved medium system. (a) Schematic illustration of the structure, and the curved medium is located at the inner side of the bend of the MNW. (b) Cross-sectional mode profiles and (c) longitudinal fields in terms of the electric field intensity with the (i) outer-side biased distribution, (ii) balanced distribution, and (iii) inner-side biased distribution. (d) Effective refractive index (neff), (e) propagation length (Lm), (f) mode area (Am), and (g) FoM as a function of the refractive index of the curved medium (n). Purple lines with dots, mode characteristics of flat substrate-loaded bent MNWs for comparison. Black dashed line in (d), medium index.
For demonstration, Fig. 7(d)-(g) show dependence of waveguiding properties on the refractive index of the curved medium (red lines). For comparison, waveguiding properties of the same bent MNW on flat substrates (i.e. bent MNW-substrate system discussed in section 4) are also plotted in purple lines. Result shows that, unlike the flat substrate system, in which Lm is deteriorated by the increasing n, Lm in the curved medium system increases until n reaches ∼1.75, exhibiting a much smaller propagation loss (e.g., ∼2.61 µm vs. ∼ 0.77 µm in Lm, for the bent MNW-curved medium vs. the bent MNW-substrate system). When we further increase the n, its Lm starts to drop but is still much higher than the one in bent MNW-substrate system. This decline trend is mainly due to the emerging leakage loss since the mode is becoming quasi-bound with a higher index of the medium (as is shown in Fig. 7(d), where the neff is plotted). Besides the much smaller propagation loss, the Am in the bent MNW-curved medium system is also smaller (Fig. 7(f)), resulting in a much higher FoM compared to the bent MNW-substrate system (e.g., ∼52 vs. ∼ 11 in FoM at n = 1.7). Note that, the FoM of the same straight free-standing MNW is ∼36, while we can achieve a much higher FoM even with a sharp bend of a sub-wavelength curvature. Therefore, the quality of the plasmon mode in sharply bent MNWs can be greatly enhanced without sacrificing the tunability, and such high-quality mode represents the ability to bend the light with tight confinement at the nanoscale, which may have great potential for applications such as plasmonic interconnects, 3-D multi-level plasmonic circuits, and flexible devices.
6. Conclusion
In summary, we have illustrated the significance of the field deformation on the mode properties, elucidated the bending-dependent behaviors, and provided the optimized condition for high-performance waveguiding. Meanwhile, we have also proposed manipulation strategies for reforming the field distribution of the asymmetric mode in bent MNWs. We have shown that the field of the asymmetric mode can be translationally, rotationally, and longitudinally manipulated, offering not only high versatility for mode engineering with extra degrees of freedom, but also possibilities to improve its overall mode quality. Unlike the common beliefs that the bending is not favorable for the mode quality, the reformed asymmetric mode in sharply bent MNWs is possible to even outperform its symmetric counterpart in straight MNWs, greatly extending the capabilities of bent MNWs. Our findings may pave a way to achieve both high quality and versatility for plasmon mode in bent MNWs, which can be implemented in high-performance plasmonic components and devices such as sensors, ring resonators, routers and interconnects, multi-level plasmonic circuits, and plasmon-based flexible devices. Meanwhile, it also provides new opportunities for manipulating the highly confined field or modulating the light-matter interaction at the deep-subwavelength scale.
Funding
National Natural Science Foundation of China (62005031, 62005032); Fundamental Research Funds for the Central Universities (2021CDJQY-046, 2022CDJXY-018); Innovation Support Plan for Returned Overseas Scholars (cx2021058).
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 62005031 and 62005032), Fundamental Research Funds for the Central Universities (Nos. 2021CDJQY-046 and 2022CDJXY-018), and Innovation Support Plan for Returned Overseas Scholars (No. cx2021058).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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