Boosting an anapole mode response through electromagnetic interactions beyond near-field limit in individual all-dielectric disk-ring nanostructures

Anapole modes of all-dielectric nanostructures hold great promise for many nanophotonic applications. However, anapole modes can hardly couple to other modes through far-field interactions, and their near-field enhancements are dispersed widely inside the nanostructures. These facts bring challenges to the further increasing of the response of an anapole mode. Here, we theoretically show that an anapole mode response in a dielectric nanostructure can be boosted through electromagnetic interactions with the coupling distance of a wavelength scale, which is beyond both the near-field and far-field limits. The all-dielectric nanostructure consists of a disk holding an anapole mode and a ring. Both analytical calculations and numerical simulations are carried out to investigate the electromagnetic interactions in the system. It is found that the electric dipoles associated with the fields of the anapole mode on the disk undergo retardation-related interactions with the electric dipoles associated with the ring, leading to the efficiently enhanced response of the anapole mode. The corresponding near field enhancement on the disk can reaches more than 90 times for a slotted silicon disk-ring nanostructure, where the width of the slot is 10 nm. This enhancement is about 5 times larger than that of an individual slotted disk. Our results reveal the greatly enhanced anapole mode through electromagnetic couplings in all-dielectric nanostructures, and the corresponding large field enhancement could find important applications for enhanced nonlinear photonics, near-field enhanced spectroscopies, and strong photon–exciton couplings.


Introduction
During the last several years, a new branch of nanophotonics has been growing that aims at manipulating the strong electric and magnetic Mie resonances in high-refractive-index dielectric nanostructures [1][2][3]. Compared to their plasmonic counterparts, dielectric nanostructures can also exhibit strong electromagnetic resonances while they have low material losses and are more compatible with the well-established semiconductor fabrication techniques. Thus, lots of applications enabled by dielectric nanostructures have been demonstrated such as metasurfaces [4,5], color printing [6], lasing [7], biosensing [8,9], and quantum optics [10,11]. Strong electric and magnetic resonances of high-index dielectric nanostructures can be excited not only in spheres [12][13][14][15][16][17] but also in many other geometries including spheroids [18], disks and cylinders [19][20][21], and rings [22,23]. The excited dipole or multipole electromagnetic modes can further interact with each other, which provides new routes for achieving strongly enhanced optical responses. Many optical phenomena induced by mode couplings have been reported, such as hybridization [24], Fano resonance [25][26][27], the magnetic field boosting [28], the formation of toroidal dipole (TD) modes from basic electric or magnetic dipole modes [29][30][31], and the quasi bound states in the continuum based on the strong coupling between Mie-like and Fabry-Perot-like modes [32,33].
Recently, nonradiating electromagnetic states in all-dielectric nanostructure have attracted lots of research interests [34,35]. Among these modes, an electric optical anapole corresponds to destructive interference of the electric and electric TD moments results in a peculiar, low-radiating optical state [36,37]. Owing to its nontrivial nonradiating feature, an anapole mode provides minimal far-field scattering and relatively high near-field enhancement among the common Mie resonance modes [37][38][39][40]. It has thus been widely investigated in many aspects of nanophotonics such as local field enhancement [41][42][43][44], nonlinear optical effects [45][46][47][48], nanolasers [49], ideal magnetic dipole generation [50], photon-exciton coupling [21,51], hybrid dielectric-plasmonic antennas [52][53][54], metamaterial [55] and absorption enhancement [56,57]. The further increasing of the response strength of an anapole mode plays a crucial role in making better use of it for many of the above applications. Currently, the relevant investigations are quite limited although there are a few reports in dielectric-plasmonic hybrid systems [45] and dielectric structures with complex geometries [42]. The challenges arise from the far-field and near-field characteristics of an anapole mode. Due to the canceled far-field response of an anapole mode, it is unlikely to utilize far-field couplings to enhance the anapole response. As for near-field couplings, the field of an anapole mode is widely distributed inside the whole volume, which brings challenge to the further enhancing of an anapole response through near-field interactions in an all-dielectric system.
In this work, we show theoretically that an anapole mode can be boosted through electromagnetic interactions with the middle coupling distances in an all-dielectric nanostructure, where such distances are beyond both the near-field and far-field limits. The all-dielectric nanostructure consists of a silicon (Si) nanodisk and a Si ring. The disk has an anapole mode response, where the corresponding electric dipole moments associated with the electric fields of the anapole mode have a certain distance. The distance between the separated dipoles of the disk can induce retardation-related interactions with the dipoles associated with the ring, leading to the efficiently enhanced response of the anapole mode on the disk. Analytical calculations are carried out to understand the electromagnetic couplings and the results agree well with the direct simulations. The corresponding electric near-field enhancement can reaches more than 90 times for a slotted Si disk-ring structure, where the width of the slot is 10 nm. This enhancement exceeds 5 times higher than that of an individual slotted disk. The geometric effects on the field enhancement will also be discussed.

Methods
Finite-difference time-domain (FDTD) Simulation method. A commercial FDTD software (Lumerical FDTD) is used for the numerical simulations. The excitation source is total-field scattered-field plane wave. To simulate an individual structure placed in an infinite space, perfectly matched layer boundary conditions were used. The dielectric constants of the Si are taken from Palik's book [58]. The surrounding index is n = 1 for simulations. The mesh size is 5 × 5 × 5 nm 3 for the region around the structures, while the mesh size is 1 × 1 × 1 nm 3 around the slot region.
Multipole decomposition method. The different multipolar modes can be calculated by using the multipole decomposition method [59,60] and their contributions to the scattering spectra can be obtained. Here we repeat the method for completeness. Cartesian multipole moments can be expressed by using the induced currentsĴ ω (r) as [60] electricdipolemoment : where ω is the angular frequency, k is the wavenumber, c is the speed of light,r is the location, and α, β = x, y, z. The induced electric current density is obtained by usingĴ ω (r) = iωε 0 (ε r − 1)E ω (r), where E ω (r) is the electric field distribution, ε 0 is the permittivity of free space, and ε r is the relative permittivity of the disk and the ring. The induced current has a harmonic time dependence exp(−iωt), which is omitted. We used the FDTD simulation to obtain the electric field distributions E ω (r) /E inc . E inc is the electric field of the incident wave. The j 1 (kr), j 2 (kr) and j 3 (kr) are the spherical Bessel functions of first, second and third kinds, respectively. The scattering cross section produced by these multipole moments can be written as: where, p α , m α are the electric and magnetic dipole moments, respectively. Q e αβ , Q m αβ are the electric and magnetic quadrupole (MQ) moments, respectively.

Results and discussion
Before considering the coupled disk-ring structure, let us first investigate the optical properties of individual dielectric nanostructures (figure 1). We choose the widely investigated Si as the material and its dielectric constants are taken from Palik's book [58]. The index of surrounding medium is taken to be 1. The numerical simulations were carried out by using commercial FDTD software (Lumerical FDTD). Figure 1(a) shows the scattering cross-section spectrum of an individual Si ring. The ring is excited by a normal incident plane wave and the polarization of the incoming beam in the y-direction. A peak appears at λ = 1111 nm on the scattering spectrum. The parameters of the ring is chosen to make the coupling with the disk strong enough, which will be discussed later. In order to understand this peak on the scattering spectrum, we calculate the contributions from different multipole modes to the spectrum and the near field distributions. The multipole mode contributions are calculated by using the multipole decomposition method. Note that in our decomposition method, each contribution mode consists of the multi-order responses [60]. Thus, an electric dipole term (ED) includes a common ED(− 1 iω d 3r J ω α ) and higher-order responses, for example, a toroidal dipole (TD,− 1 iω k 2 [60]. This is different from some literatures where an ED term means only the first order ( 1 iω J ω α d 3r ) [31,61]. Figure 1(b) shows the multipole decomposition results for the scattering spectrum of the ring. The sum of all the contributions from the different multipole modes [the black line in figure 1(b)] agrees well with the direct calculation of the scattering by FDTD. The peak on the scattering is mainly contributed from the MQ mode which shows a peak feature. A contribution from the ED mode is also seen, and it shows a dip feature. More calculations show that the ED mode here contains two contributions. One is a common electric dipole and the other one is a higher-order response which is a TD. These spectral lineshape features are similar to that of a common anapole mode in many dielectric nanostructures [37,41,62]. However, the scattering spectrum is a peak here while it is a dip for a common anapole mode. As an anapole mode corresponds to a minimal scattering, it is unusual to take this scattering response as an anapole mode. But if we consider only the ED spectrum, it is rational to take its spectral feature as an ED-related anapole response. Thus, we here take the scattering response of the ring as a 'modified anapole' mode. In fact, this scattering response is indeed caused by the modification of the geometries. It can be easily verified that the scattering spectral response gradually changes back to that of a normal anapole mode with decreasing the inner radius of the ring. We plot electric near field distributions on the z = 0 (left) and y = 0 (right) plane of the ring in figure 1(c). The wavelength is λ = 1111 nm corresponding to the scattering peak. The electric field distribution of this mode is also similar to the common anapole mode [37,41,59,62]. At the same time, an enhanced field distribution also appears above the ring. It can be verified that the near field profile in the 750 ∼ 1100 nm range are similar to that of the peak. This field distribution is important for the strong couplings in the disk-ring structures, which will be discussed later.
The radius and height of the Si nanodisk are 230 and 100 nm, respectively. With the chosen geometric parameters, a resonance dip is found around λ = 964 nm [ figure 1(d)]. The bottom inset shows the electric field distribution on the z = 0 plane, which indicates that the dip at λ = 964 nm corresponds to an anapole mode. Similar results have been reported in many other works [37,41,59,62]. Here, the parameters of a disk are chosen to make the anapole mode appear at a given wavelength. Now let us turn to the coupled system as shown in figure 2(a). The coupled system consists of the Si nanodisk and ring discussed above. The disk is placed at (0, 0, 100 nm), which is 100 nm above the center of the ring. The index of surrounding medium is also 1. The excitation configuration is shown in figure 2(a). Figure 2(b) shows the scattering spectrum of the coupled system. A sharp peak appears at λ = 947 nm on the spectrum, which is close to the anapole resonance of the individual disk. Multipole decomposition results of the coupled system are shown in figure 2(c). Their lineshapes around the resonance peak λ = 947 nm are similar to that of the individual disk, except that the contribution of the MQ mode to the scattering is increased a lot. The electric field distributions on the planes of z = 100 nm and y = 0 nm are shown in figure 2(d), respectively. Compared to the individual case, the field enhancement becomes ∼3 times higher while the field profile of the disk still remains the same. It can be understood by the fact that such a field distribution on the disk corresponds to both the anapole and MQ mode responses, which has been confirmed in other similar systems [31]. An increasing of the field corresponds to the enhanced responses of both modes. But the anapole mode does not show far-field scattering. Thus only the enhanced MQ response appears on the scattering spectrum. The near field enhancement here (∼9) is larger than that achieved by a qausi-BIC mode (∼6) [32]. The giant modifications of the spectrum and field enhancement indicate that strong electromagnetic coupling occurs in the disk-ring system.
In order to understand the strong coupling of the disk-ring system, a series of analytical calculations and numerical simulations are carried out. It is found that this coupling is induced by the interactions of the electric dipole moments associated with the anapole mode of the disk and the 'modified anapole' mode of the ring, where the coupling distance is beyond the near-field limit and the retardation effect plays an important role. The electric field distribution of the individual ring on the y = 0 plane is shown in figure 3(a). Due to the symmetry reason, we consider only the right part of the ring. The field distribution of the ring can be divided into three different regions R1, R2, and R3, where each one corresponds to an electric dipole moment response. The field profiles in these regions still keep almost the same under couplings [ figure 3(a)]. The fields of the disk can be divided into three different regions D1, D2 and D3, where each also corresponds to an electric dipole moment response. Among them, R1 (R3) and D2 are in (out of) phase, both D1 and D3 are out of phase with D2, and the phase delay between R2 and D2 is π/2 as shown by the simulation results. It is well known that the electric field E induced by an electric dipole satisfies [63]: where ω is the angular frequency, c is the speed of light,r is the location, and ε 0 is the permittivity of free space. P is the dipole moment, which is expressed as P = P(r)e −iωt . With equation (1) one can obtain the electric fields generated by R1, R2 and R3 at the location of disk. Here, the location r for the disk region is approximately taken to be perpendicular to the P. Thus, equation (1)  c + π P D2 · P R2 , where r 22 represents the distance between P D2 and P R2 . Thus, the interaction energy produced by the electric dipoles associated with the ring and disk can be written as E int = E int = − 3 i,j=1 P Di · E Rj , where Rj represent the region R1, R2, or R3. It should be pointed out that the vanishing ED of an anapole does not conflict with the fact that its corresponding different electric dipoles (for example P Di ) could be coupled to other modes. The reason is that the vanishing ED response only means its far-field scattering characteristic while it has significant near field distributions that can couple to other mode beyond the far-field limit. Here it should be pointed out that a normal anapole mode in a dielectric nanostructure usually shows a typical near field distribution pattern as demonstrated by figure 1(d). This distribution pattern is also accomplished by an MQ mode response [31]. Thus strictly speaking, the coupling also involves an MQ mode of the disk.
For numerical calculations, it is approximately taken that |P D2 | = 2 |P D1 | and |P R1 | = |P R2 | = |P R3 | at a given wavelength, which is indicated by the field distributions [figures 3(a) and 1(d)]. Let us start with a simple case, where the electric dipole moment of the ring is assumed to be the same for all the working (resonant) wavelengths, and the distances between the dipoles are the same for all the working wavelengths, respectively. Figure 3(c) shows the normalized interaction energy E int /P D1 of the disk-ring structure, where the geometry of the ring is the same as that in figure 2. The distance between the coupling dipoles is around kr ∼ 1, where k is the wave vector. Such a distance means that neither near-field limit nor far-field limit can be used. Now let us consider the effect from the dipole moments of the ring. Electric near field distributions associated with the ring show that the field enhancement is almost unchanged in the wavelength ranged considered, while the size of the enhanced field increases with wavelength. Thus, the dipole moments also increase with wavelength as it approaches the resonance peak of the 'modified anapole' mode [figure 3(c)]. Taking this factor into account, one obtains the normalized interaction energy E int /|P D1 | of the coupled system as a function of working wavelength [figure 3(d)]. Such a disk-ring structure can be realized in the FDTD simulation by considering a structure where the size of the disk is the same while its anapole response wavelength is tuned by varying the refractive index of the disk. The geometries of the disk and ring are the same as that in figure 2, respectively. We define the relative field enhancement of the disk as its field enhancement in the coupled disk-ring structure divided by that of the individual disk. Then, the relative field enhancement of the disk is proportional to the interaction energy normalized by the dipole moment of the disk (E int /|P D1 |). Figure 3(d) shows the direct FDTD calculation results of the relative field enhancement of the disk in the disk-ring system. The analytical and directly-simulated results agree well. Note that the relative field enhancement at resonance of the ring (∼1100 nm) is weaker. This can be understood by the results in figure 3(c), where the contribution from the normalized interaction energy E int /P D1 decreases faster than the increasing of the dipole moment of the ring with the working wavelength. The decreasing of E int /P D1 is due to the increasing of working wavelength which obeys equation (1).
We also consider more realistic cases in experiments where the disk and the ring have the same material Si while the radius of the disk is changed. It can be verified according to equation (1) that the interaction energy decreases when the distance between D1 (D3) and D2 becomes smaller, which corresponds to that the size of disk becomes smaller. This is rational because a smaller distance between D1 (D3) and D2 make the responses closer to an ideal anapole mode, where an ideal anapole mode will not be coupled to other modes. Taking the geometric factors of disk into account, one expects that there is a disk size for optimal the relative interaction energy, namely, the relative field enhancement of the disk. This is confirmed by the direct FDTD simulation results as shown in figure 3(e). It is seen that the relative enhancement also becomes weaker when the working wavelength approaches the resonance of the ring. The explanation is also similar to that in figure 3(d), where both the distance and the working wavelength varies here while the decreasing of E int /P D1 is still faster than the increasing of the dipole moment of the ring. Note that the working wavelength means the resonance enhancement on the spectrum, which is highly dependent on the anapole mode of the disk. The enhancement at a working wavelength does not necessarily mean the largest value at this wavelength. The strongly enhanced anapole response indicates that large electric field enhancement could be achieved in such a system by introducing a small slot in the disk [41,42]. The small slot has a little effect on the scattering spectrum of the coupled system, while the electric near field inside it can be further enhanced a lot depending on the size of the slot. Furthermore, a slot can also make the enhanced electric field easily accessible to other objects. Figure 4(a) shows the coupled system with a slotted Si nanodisk, which is the same as that in figure 2, except that there is a slot in the center of the disk. The slot has a length of L = 260 nm and a width of W = 10 nm. Figure 4(b) shows the resonant electric field (λ = 889 nm) distributions on the z = 100 nm and y = 0 planes. They are similar to that of solid ones in figure 2(d), respectively, except that a huge electric field enhancement appears in the slot. Figure 4(c) provides the electric field enhancement calculated at the center of slotted disk in the coupled system. The resonant electric field enhancements |E|/|E 0 | for the coupled system and individual case are around 96 and 19, respectively. The field enhancement of the coupled structure is about 5 times higher than that of the individual slotted disk. Here, it is noted that the resonance is blueshifted to 889 nm due to the appearance of slot in the disk. The relative field enhancement is increased more (∼5) compared to the solid case (∼3). This is inconsistent with the results in figure 3(d) as the working (resonant) wavelength is blueshifted and the corresponding relative enhancement is larger. Considerable value of the magnetic near field enhancement |H|/|H 0 | of the coupled disk is also seen as expected [ figure 4(d)].There is almost no magnetic field enhancement at the center of the disk, which leaves a pure electric near-field hot spot at the center of the disk. The effect from the size of the slot is illustrated by figures 4(e) and (f). The field enhancement decreases with the slot width as expected. As for the slot length, the field enhancement first increases and then decreases with the length L. This is associated with the fact that the anapole resonance of the disk varies with L. Thus it shows a peak value around a given L as the ring is fixed. This is similar to the case of figure 3(e). Note that the electric-field enhancement under such an interaction still shows a relative large spectral width especially compared to the ones that achieve enhanced response by engineering the quality (Q) factors, for example, the qausi-BIC mode. This is spectral behavior is important for some applications such as the fluorescence enhancement where the fabrication accuracy needs to be high enough if the spectral width of the structure is too small.
Next, we investigate the effects from the geometry of coupled system. Figure 5(a) shows the resonant field enhancement as a function of the location Z of the disk. The other parameters are the same as that in figure 4. An optimal resonant field enhancement occurs at Z = 100 nm. This can be explained by the discussion in figure 3. The three dipoles of R1, R2 and R3 all contribute to the field at the location of the disk, and the variation of Z induces the change of their corresponding r in equation (1). There is an optimal Z where the field contributed from the three dipoles is the largest. For the case of Z = 0 nm, the electric field enhancement of the coupled system reaches 81 times. The geometry of the ring is also an important factor that affects the value of field enhancement. The effect of outer radius R out of ring is investigated (from 430 nm to 590 nm) while the inner radius is kept at R in = 390 nm. We set the relative location to Z = 0 nm as this is experimentally more feasible. The resonant electric field enhancement of the coupled system shows an optimal value around R out = 505 nm [ figure 5(b)]. This is due to the fact that with increasing R out , the optical response of the ring increase while its resonant position moves further away from the working wavelength, and the relative distance between the dipoles of disk and the ring increases. The combination of these factors induces the optimal R out . Similar behavior appears if R in varies with fixing the R out , and the explanation is the same as that in figure 5(b). Figure 5(c) shows the results with varying the size of Si ring, where the R out − R in is restricted to be R out − R in = 100 nm for each case. One can also find an optimal electric field enhancement at R s = 500 nm. This can be explained by the factors that by increasing the size of the ring, the optical response of the ring increases while the resonance position also moves away from the working wavelength and the distance between the disk and ring also increases. The combination of these factors leads to the optimal phenomenon, which is similar to that in figure 5(b).

Conclusion
In conclusion, we have shown that the electric dipole moments associated with the optical anapole mode in dielectric nanostructures can induce significant electromagnetic interactions with other modes in the middle distance regime (kr ∼ 1). We investigate a disk-ring coupled system, where the disk has an anapole mode. Strong electromagnetic interactions are found and the anapole response is significantly enhanced. This is also confirmed by analytical calculations. The near field enhancement of a coupled Si disk-ring structure can reach more than 90 times with a 10 nm-width slot in the disk, which is more than 5 times larger than the individual slotted disk. Our results reveal the rich strong mode couplings in individual all-dielectric nanostructures, and the near field enhancement could be further increased through more careful design based on the electromagnetic couplings in the middle distance regime. The strongly enhanced near-field response with considerable spectral width could find many applications such as near-field enhanced spectroscopies [64] and strong photon-exciton couplings [65][66][67].

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).