Ideal magnetic dipole resonances with metal-dielectric-metal hybridized nanodisks

Yao Zhang; Peng Yue; Jun-Yan Liu; Wei Geng; Ya-Ting Bai; Shao-Ding Liu

doi:10.1364/OE.27.016143

1. Introduction

The realization of effective optical magnetism with nonmagnetic nanostructures is crucial for manipulating light-matter interactions at the nanoscale, which has facilitated many important research areas such as negative refraction [1], superlensing [2], and cloaking [3] etc. In these studies, single nanostructures are often used as a unit cell to form functional metasurfaces or metamaterials [4]. Therefore, it is important to design nanostructures to generate magnetic responses effectively [5,6]. To date, there are mainly four kinds of nonmagnetic nanoparticles (NPs) that possess strong magnetic resonances. The first one is by using metallic split-ring resonators (SRRs), where the conduction current loop within the SRRs that driving by incident fields leads to the formation of an effective magnetic dipole (MD) that perpendicular to the structural plane [7] The second one is by using metal-dielectric-metal sandwich NPs, where the dielectric layer is often composed of a low refractive index material [8–12]. In this case, a bonding resonance caused by plasmon hybridization has the character of a circulating pseudocurrent, which leads to an equivalent MD oriented perpendicular to the incident polarization. Several years ago, it is demonstrated that a circular displacement current loop caused by retardation effect can be generated with single high refractive index dielectric NPs, thereby resulting in a MD resonance [13–19]. Besides that, a strong MD resonance can be excited with plasmonic or dielectric NP clusters when the structural symmetry is broken [20,21]. Owing to the formation of the magnetic responses with these NPs, a lot of promising functions have been demonstrated such as far-field manipulating [22–24], enhanced nonlinear responses [25–27], high-sensitive biosensing [28], energy-harvesting [29,30], enhanced fluorescence [31–34], and magnetic mirrors [35].

Considering the important role of the optical magnetism in nanophotonics, it is crucial to achieve pure MD resonances to unambiguous study the magnetic light-matter interactions. However, the MD resonances generated with the above NPs are often spectrally overlapping with other multipole contributions. For example, a strong electric dipole (ED) is always excited around the gap area for metallic SRRs with the formation of MD resonances [36,37], there is a relatively strong parasitic electric quadrupole (EQ) scattering for the metal-dielectric-metal sandwich NPs [38], and the MD resonances generated with dielectric NPs and NP clusters suffer from the spectral overlapping with ED modes [39]. Although the interactions between the MD and other overlapping multipoles provide many interesting interference effects such as the directional scattering [40,41] the overlapping multipoles have to be suppressed to achieve ideal MD resonances. It has been shown that the multipole resonances can be independently tuned by adjusting the geometry of dielectric NPs, and the maximum of MD resonance can be obtained for the wavelength corresponding to the minimum of ED response with a cylindrical NP [42]. Besides that, an ideal MD resonance can be excited with metallic SRRs or dielectric NPs under cylinder vector beam excitation [43,44], but it highly relies on the alignment between the incident beam and the NPs. By careful engineering the relative positions of the NPs for an oligomer cluster, an ideal MD scattering can be possibly achieved when the ED scattering is totally suppressed [5], but it is challenging to fabricate the asymmetric clusters based on current nanolithography methods.

In addition to the pure metallic or dielectric nanostructures, hybridized nanostructures composed of metallic and high refractive index dielectric NPs have gained considerable attentions in recent years [45–47], where the hybridized NPs combine the advantages of the two type structures, that is, the relatively low thermal losses and the effective confinement of incident energy. It has been shown that the performance of many applications can be improved with hybridized nanostructures (e.g., Purcell factor enhancements [48], near-field reconfiguration [49], and enhanced nonlinear optical effects [50–52]). Besides that, strong magnetic responses also can be observed with hybridized NPs [53,54], which has been widely used for far-field scattering manipulating [55]. Not long ago, it has been proposed and theoretically demonstrated that an ideal MD scattering can be achieved with a gold core and silicon shell hybridized NP [56], where the ED scattering can be totally suppressed with the excitation of the nonradiating anapole mode [57], and ideal MD scattering is observed when the MD resonance is spectrally overlapping with the anapole mode [56]. Although the designed nanodisk core/shell structures are compatible with current nanofabrication technology, it would be still relatively complex to fabricate the hybridized NPs since different materials are involved in the same layer. In this study, we provide an alternative way to generate ideal MD scattering with metal-dielectric-metal hybridized nanodisks, which can be easier fabricated based on current nanofabrication technology. It is shown that strong magnetic and electric near-field enhancements can be generated at the same time caused by the plasmon coupling between the metallic NPs, and one can expect that light-matter interactions can be strongly enhanced with the hybridized NPs. Besides that, the realization of ideal MD scattering is based on a hybridized plasmon resonance, and there is a tiny mode volume for the hybridized nanodisks, which is useful to design compact nanophotonic devices.

2. Mechanism to achieve ideal MD resonances

The main idea to achieve a pure MD resonance with metal-dielectric-metal hybridized nanodisks is schematically shown in Fig. 1. In previous studies, the dielectric layer is often composed of a low refractive index material (e.g., SiO₂), and the optical responses are governed by the plasmon resonances of the metallic nanodisk dimer. As shown in Fig. 1(a), plasmon hybridization of the dimer results in a higher energy antibonding and a lower energy bonding resonance, where the equivalent dipoles of individual disks oscillate in-phase and out-of-phase, respectively. Therefore, the antibonding resonance possesses a strong scattering efficiency, and it is dominated by the ED scattering. On the other hand, the out-of-phase oscillating of the metallic disk plasmons for the bonding resonance leads to an pseudocurrent loop (Scheme II, Fig. 1(b)), thereby forming an equivalent MD that oriented perpendicular to the incident polarization (the red arrow, Scheme III, Fig. 1(b)), and a strong magnetic response is generated in this situation. Nevertheless, the ED moments of the two metallic disks may not equal to each other, which leads to residual ED scattering for the bonding resonance (the blue arrows, Scheme III, Fig. 1(b)). In addition, the bonding resonance is spectrally overlapping with the antibonding mode, and the ED contribution around the bonding resonance would be further enhanced. It is also worth to mention that due to the formation of strong near-field enhancements around the gap areas, there are displacement currents within the dielectric layer (the red arrows, Scheme II, Fig. 1(b)), but they are much weaker than the conduction currents (the blue arrows), and the out-of-phase oscillating ED of the metallic disks results in a parasitic EQ scattering (the blue arrows, Scheme III, Fig. 1(b)). As a result, the bonding resonance of the hybridized nanodisk is dominated by the above-mentioned multipoles, and the far-field scattering is the interference result among the MD, ED and EQ contributions (Scheme IV, Fig. 1(b)).

Fig. 1 Physical mechanism to achieve ideal MD resonances with metal-dielectric-metal hybridized nanodisks. (a) Plasmon hybridization scheme of a metallic nanodisk dimer, where a higher energy antibonding and a lower energy bonding resonance can be excited. (b) The scattering properties of the hybridized bonding resonance for metal-dielectric-metal hybridized nanodisks with a low and (c) a high refractive index dielectric spacer layer. Depending on the geometry parameters and the dielectric layer refractive index, directional forward scattering or an ideal MD resonance can be achieved, where (I) the NPs are excited with a x-polarized incidence that propagating along the z-axis, (II) the conduction currents (the blue arrows) and the displacement currents (the red arrows) within the hybridized nanodisks, (III) the generated equivalent multipoles, and (IV) the far-field scattering patterns.

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In order to achieve an ideal MD resonance, the ED and EQ contributions for the bonding resonance have to be suppressed. Since the dipole moment of a metallic NP is governed by its geometry parameters, the ED scattering caused by the unmatched dipole moments between the two metallic disks can be suppressed by adjusting their relative thicknesses (Scheme I, Fig. 1(c)), and it is expected that this portion of ED scattering can be eliminated when the dipole moments are approaching to each other.

As for the ED scattering caused by the overlapping with the antibonding resonance, a simple way to solve this problem is by reducing the resonance energy of the bonding mode, and this portion of ED scattering can by effectively suppressed when the two hybridized plasmon modes are far away from each other. For example, the capacitances around the gap areas of the hybridized nanodisks would be enlarged with the increasing of the dielectric layer refractive index (Scheme I, Fig. 1(c)), and the bonding resonance will red shift significantly, thereby resulting in a broadening energy gap with the antibonding mode, and the ED scattering caused by the mode overlapping can be suppressed.

The displacement current density within the dielectric layer,

J_{s p a} = - i ω ε_{0} ε_{r} E = - i ω ε_{0} n_{s p a}^{2} E

where ε₀ is the vacuum permittivity, ε_r and n_spa are, respectively, the relative permittivity and refractive index of the dielectric disk, and the dielectric material losses are neglected. Therefore, another important feature caused by the increasing of the refractive index is that the displacement current (density) can be strongly enhanced. When the magnitudes of displacement current are approaching to the conduction current, a perfect current loop can be generated (Scheme II, Fig. 1(c)), and the parasitic EQ scattering would be effectively suppressed. In this case, it is expected to achieve ideal MD scattering for the bonding resonance (Scheme III, Fig. 1(c)), and a doughnut shaped far-field scattering pattern would be observed (Scheme IV, Fig. 1(c)).

3. Suppressing ED scattering with asymmetric dimers

Figure 2 shows the effect on the suppressing of ED scattering by adjusting the thickness of the lower metallic disk T_dn, where the dielectric layer is supposed to be SiO₂ with n_spa = 1.45, silver is used for the metallic disks since it possesses low material losses. The near- and far-field optical responses of the metal-dielectric metal hybridized nanodisks were calculated with finite element method (FEM), where the measured complex dielectric constants of silver were used in the calculations [58], In the simulations, the incident x-polarized plane wave is propagating along the z-axis, and perfectly matched layers (PML) around the nanostructures were used to simulate the open space. When T_dn = 20 nm, the black line in Fig. 2(a) denotes the scattering spectrum of the hybridized nanodisk. There are two pronounced resonances around 600 and 730 nm, which are the antibonding and bonding hybridized modes, respectively. The multipolar decomposition is a powerful tool for the study of light-matter interactions at the nanoscale [59], which can help to identify the multipole contributions to the resonances, and the decomposition results for the scattering spectrum are also demonstrated in Fig. 2(a), which can be calculated with the induced polarization inside the hybridized NPs [60],

P = ε_{0} (ε_{p} - ε_{d}) E

where ε₀, ε_p and ε_d are the vacuum dielectric constant, the relative dielectric permittivity of the NPs, and the surrounding medium relative dielectric permittivity, respectively. The total electric field (E) inside the NPs are calculated with the FEM simulation. To get the multipole expansion of the light-induced polarization,

P (r) = \int P (r^{'}) δ (r - r^{'}) d r^{'}

the Dirac delta function δ(r - r′) is expanded in a Taylor series with respect to r′ around the origin. Then one can get the irreducible representations individual multipoles, and the total scattering can be written as,

\begin{array}{l} P_{s c a} = \frac{k_{0}^{4}}{12 π ε_{0}^{2} v_{d} μ_{0}} {| p + \frac{i k_{d}}{v_{d}} T |}^{2} + \frac{k_{0}^{4} ε_{d}}{12 π ε_{0} v_{d}} {| m |}^{2} + \frac{k_{0}^{6} ε_{d}}{1440 π ε_{0}^{2} v_{d} μ_{0}} \sum_{α β} {| Q_{α β} |}^{2} + \\ \frac{k_{0}^{6} ε_{d}^{2}}{160 π ε_{0} v_{d}} \sum_{α β} {| M_{α β} |}^{2} + \frac{k_{0}^{8} ε_{d}^{2}}{3780 π ε_{0}^{2} v_{d} μ_{0}} \sum_{α β γ} {| O_{α β γ} |}^{2} \end{array}

where the five terms on the right side denote the ED, MD, EQ, magnetic quadrupole (MQ) and electric octupole (EO) contributions, respectively. The expressions for individual multipole moments can be found in the literature [60], and the total scattering are calculated by including up to the third-order multipoles as shown in Eq. (4).

Fig. 2 Suppressing ED scattering for the bonding resonance by manipulating the lower metallic nanodisk thickness T_dn. (a) The scattering and multipolar decomposition results for the hybridized nanodisk with T_dn = 20 nm, (b) 30 nm and (c) 50 nm, where the insets show the normalized current density distributions at the center cross section of the xz plane around the bonding resonance. The dotted-lines represent the scattering contribution of the TD for the hybridized structure, and the dashed-lines denote the ED scattering contribution from the dielectric layer, which are scaled by a factor of 10. (d - f) The corresponding electric (the upper panels) and magnetic (the lower panel) near-field enhancement distributions at the center cross section of the xz plane for the bonding resonances, where the number on the lower-left corner is the maximum enhancement factor. (g - i) The normalized three- (the left panels) and two-dimensional (the right panels) far-field scattering patterns for the three hybridized nanodisks. (j) The scattering and multipolar decomposition results for the hybridized nanodisk when T_dn is in the range of 20 – 80 nm (the upper four panels), and the MD contribution to the total scattering (the lowest panel). The dielectric layer is supposed to be SiO₂ with n_spa = 1.45, silver is used for the metallic disks, the NPs are supposed to be embedded in air, the diameter of the disks D = 160 nm, the thickness of the dielectric layer T_spa = 60 nm, the thickness of the upper metallic nanodisk T_up = 30 nm, and the incident x-polarized plane wave is propagating along the z-axis.

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The red line in Fig. 2(a) denotes the scattering contribution from the MD, where it is very weak for the higher energy antibonding resonance, while it plays an important role for the bonding resonance. On the other hand, the ED contribution denoted by the blue line dominates for the antibonding resonances, while its scattering intensity is almost equal to the MD at 730 nm, and there is an ED scattering peak around the bonding resonance. Besides that, a relative strong EQ scattering is also observed for the bonding resonance (the green line, Fig. 2(a)), which is a parasitic factor with the excitation of the MD for plasmonic NP dimers. The rest higher-order multipole scattering such as the MQ and EO are very weak, which can be neglected for the hybridized nanodisks (the cyan and the magenta lines, Fig. 2(a)). The term T in Eq. (4) denotes the toroidal dipole (TD) moment, and it has been shown that the destructive interference with the Cartesian ED moment would result in the formation of the nonradiating anapole mode [57]. The cyan dotted line in Fig. 2(a) shows the scattering contribution from the TD, which is scaled by a factor of 10, and it is found that the TD scattering is also negligibly weak around the bonding resonance for the sandwich NP.

When the thickness T_dn equals to that of the upper disk (30 nm), the scattering spectrum is similar as that of T_dn = 20 nm except for the wavelength shift (the black line, Fig. 2(b)). However, the multipole contributions to the bonding resonance change significantly. It is found that the MD scattering is strongly enhanced for the bonding resonance (~700 nm). On the contrary, the ED scattering is suppressed, and the scattering peak can no longer be observed around this spectral range (the blue line, Fig. 2(b)). The MD contribution can be further enhanced when T_dn is enlarged to 50 nm (the red line, Fig. 2(c)). At the same time, an ED scattering dip appears around the bonding resonance, indicating a further suppressing of the ED contribution (the blue line, Fig. 2(c)).

To understand the variations of the multipole contributions, the near-field distributions around the bonding resonance for the three hybridized nanodisks are shown in Figs. 2(d) – 2(f), respectively. In all cases, there are strong electric near-field enhancements around the gap areas caused by the plasmon coupling between the two metallic disks (the upper panels). The out-of-phase oscillating of the plasmons for the two metallic disks leads to an equivalent MD that oriented along the y-axis (the lower panels), and it is found that the magnetic field enhancement factor is weaker than that of the electric component. Although the overall near-field distributions are similar as each other for the hybridized nanodisks, the difference can be more obvious by investigating the current density distributions. When T_dn = 20 nm, the current density of the lower metallic nanodisk is much stronger than that of the upper one (the inset, Fig. 2(a)), which means that the ED moments are not matched with each other. Although the ED of the two metallic disks are cancelling with each other for the bonding resonance, a relative strong residual ED is generated caused by the mismatched dipole moments between the two metallic disks. And there is an ED scattering peak around the bonding resonance. The ED scattering of the bonding resonance is mainly attributed to the plasmonic nanoparticles, where the blue dashed line in Fig. 2(a) shows the ED scattering contribution from the dielectric layer, and it is negligibly weak compared with that of the total ED scattering of the hybridized structure. To further confirm the above observations, the ED moments of the two metallic nanodisks are calculated, and the intensity ratio |p_up|/|p_dn| = 0.47, which means that there is indeed a large difference between the ED moments when T_dn = 20 nm. When T_dn is enlarged to 30 nm, the relative current density in the upper disk is enhanced (the inset, Fig. 2(b)), and the intensity ratio |p_up|/|p_dn| is enlarged to about 0.64. Further enlarge T_dn to 50 nm (the inset, Fig. 2(c)), the intensity ratio |p_up|/|p_dn| = 0.85, where the ED moments of the two metallic disks are comparable with each other, the ED scattering can be effectively suppressed, and a ED scattering dip appears in the spectrum.

The far-field distributions are consistent with the multipolar decomposition results. When T_dn = 20 nm, the far-field scattering pattern is governed by the interference between the MD, ED and EQ (Fig. 2(g)), and the forward scattering is much stronger than the backward scattering. Previous studies have shown that in addition to the Kerker condition that governed by the interference between the ED and MD, directional forward scattering also can be achieved due to the interference between the MD, ED and EQ [38]. In our case, the directional forward scattering indeed can be observed when the incidence is about 747 nm (Scheme IV, Fig. 1). The ED scattering is suppressed when T_dn = 30 and 50 nm. Nevertheless, due to the residual ED and the parasitic EQ contributions, the far-field scattering patterns are still diverged from a doughnut shape of an ideal MD (Figs. 2(h) and 2(i)).

To better show the variations of the optical responses, the upper four panels of Fig. 2(j) represent the calculated scattering spectra and the multipolar decomposition results when the thickness of the lower disk T_dn is in the range of 20 – 80 nm. As indicated by the dotted lines, the bonding mode blue shift with the increasing of T_dn (the first panel, Fig. 2(j)), and the corresponding MD contribution is enhanced at the same time (the second panel, Fig. 2(j)). On the contrary, the ED scattering is relatively strong when T_dn < 30 nm (the third panel, Fig. 2(j)), and its contribution decreases with the increasing of T_dn caused by the cancelation of the dipole moments. Besides that, there is a relatively strong parasitic EQ contribution for the bonding resonance, and its scattering intensity is slightly enhanced with the increasing of T_dn (the forth panel, Fig. 2(j)). To quantify the MD contribution, the ratio between the MD and the total scattering intensity are calculated as shown in the lowest panel of Fig. 2(j), and the MD contribution indeed increases with the increasing of T_dn. The red circular points in Fig. 3 show the variation of the maximum MD contribution around the bonding resonances, where it is less than 50% when T_dn = 20 nm, and it is enlarged to about 75% when T_dn = 50 nm. On the contrary, the ED contribution decreases with the increasing of T_dn, and it is only about 7.4% when T_dn = 50 nm (the blue points, Fig. 3).

Fig. 3 The contribution of individual multipoles at the bonding resonance by adjusting the lower metallic nanodisk thickness T_dn, which is calculated as the ratio between the intensity of individual multipoles and the total scattering, and the geometry parameters for the hybridized nanodisks are identical with that of Fig. 2.

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Although the ED scattering can be suppressed by adjusting T_dn, the contribution has not been eliminated (Fig. 3). Besides that, the parasitic EQ contribution is slightly enhanced with the increasing of T_dn. By investigating the current density distributions shown in the insets of Figs. 2(a) – 2(c), it is found that the displacement currents within the dielectric layer are much weaker than that of the conduction currents within the metallic disks, and the current distributions are diverged from a perfect current loop. One can expect that the EQ scattering can be eliminated when a perfect current loop is generated, and an ideal MD scattering can be achieved in that case.

4. Realization of ideal MD resonances

According to Eq. (1), the displacement currents can be enhanced by enlarging the refractive index of the dielectric layer n_spa. Because the ED scattering can be effectively suppressed when T_dn = 50 nm, the geometry parameters of the hybridized nanodisks are kept the same as that of Fig. 2(c) in the following studies. When n_spa = 2.0, the scattering spectrum and the multipolar decomposition results are demonstrated in Fig. 4(a). Compared with that of n_spa = 1.45 (Fig. 2(c)), the bonding resonance red shift to about 825 nm, and there is a larger energy gap with the antibonding resonance. It is interesting that the MD contribution is dramatically enlarged to about 90.0% around the bonding resonance (the red line, Fig. 4(a)), while the scattering contributions of the ED and EQ are further suppressed (the blue and green lines, Fig. 4(a)). The electric field enhancement distributions for the bonding resonance shown in the upper panel of Fig. 4(d) reveals that the field enhancements within the gap areas are comparable with that of n_spa = 1.45 (Fig. 2(d)). However, the displacement current density is proportional to the square value of the refractive index (Eq. (1)), and the displacement current density is enhanced when n_spa = 2.0 (the inset, Fig. 4(a)). We have calculated the currents flowing through the center cross sections of the two metallic and the dielectric nanodisks, which are denoted as I_up, I_dn an I_spa, respectively (Scheme II, Fig. 1(b) and 1(c)). Please note that the displacement currents at the two sides of the dielectric disk (I_spa) equal to each other due to the structural symmetry.

Fig. 4 The realization of ideal MD resonance by manipulating the refractive index of the dielectric layer n_spa for the hybridized naodisks. (a) The scattering and multipolar decomposition results for the hybridized nanodisk with n_spa = 2.0, (b) 3.2 and (c) 5.6, where the insets show the normalized current density distributions at the center cross section of the xz plane around the bonding resonance. The black and red dotted lines in (c) show, respectively, the total scattering spectrum and the MD contribution for oblique incidence, where the incident angle is 45°, and the polarization is along the x-axis. (d - f) The corresponding electric (the upper panels) and magnetic (the lower panel) near-field enhancement distributions at the center cross section of the xz plane for the bonding resonances, where the number on the lower-left corner is the maximum enhancement factor. (g - i) The normalized three- (the left panels) and two-dimensional (the right panels) far-field scattering patterns for the three hybridized nanodisks. (j) The scattering and multipolar decomposition results for the hybridized nanodisk when n_spa is in the range of 1 – 5.8 (the upper four panels), and the MD contribution to the total scattering (the lowest panel), where the ED and EQ spectra are scaled by a factor of 2 and 10, respectively. The diameter of the disks D = 160 nm, the thickness of the dielectric layer T_spa = 60 nm, the thickness of the upper silver nanodisk T_up = 30 nm, and the lower silver nanodisk thickness T_dn = 50 nm, the NPs are supposed to be embedded in air, and the incident x-polarized plane wave is propagating along the z-axis.

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When n_spa = 1.45 (the inset, Fig. 2(d)), the displacement current is much weaker than the conduction currents, and the ratio I_up/I_spa and I_dn/I_spa are about 2.04 and 2.21, respectively. When n_spa = 2.0, the current ratio I_up/I_spa and I_dn/I_spa are reduced to about, respctively, 1.56 and 1.65, which means that a better current loop is generated in this case, and there is a stronger MD response for the bonding resonance (the lower panel, Fig. 4(d)). One can find that the maximum magnetic near-field enhancement factor is about 1.5 times as large as that of Fig. 2(f). At the same time, the far-field scattering is still affected by the residual ED and EQ contributions, and the scattering pattern is diverged from an ideal doughnut shape (Fig. 4(g)).

When n_spa = 3.2, the bonding resonance red shift to about 1178 nm (the black line, Fig. 4(b)). Besides that, the antibonding resonance is modified compared with that of Fig. 4(a), which can be attributed to the enhanced scattering intensity of the dielectric layer, and it can no longer be neglected with a high refractive index [61]. Nevertheless, the overall near-field distributions for the bonding resonance is unchanged. The electric fields around the gap areas are enhanced (the upper panel, Fig. 4(c)), and the displacement current density within the dielectric layer can be further enlarged (the inset, Fig. 4(b)). The calculated current intensity ratio I_up/I_spa and I_dn/I_spa are reduced to, respectively, 1.31 and 1.30, which means that the displacement currents are approaching to the conduction currents. In this case, the magnetic near-fields are dramatically enhanced (the lower panel, Fig. 4(e)), which is about one time larger than that of n_spa = 2.0. As a result, the MD contribution for the bonding resonance is enlarged to about 98.3% (the red line, Fig. 4(b)), and the ED and EQ are further suppressed with the increasing of n_spa (the blue and green lines, Fig. 4(b)). At the same time, the far-field scattering pattern is approaching to a doughnut shape (Fig. 4(h)).

Further enlarge n_spa to 5.6, the scattering spectrum indicates that the bonding resonance red shift significantly to about 1969 nm (the black line, Fig. 4(c)), and the MD scattering is almost overlapping with the total scattering around this spectral range (the red line, Fig. 4(c)). The ED and EQ contributions are negligibly weak, and the MD contribution is as large as 99.3%. Compared with that of n_spa = 3.2, the electric near-field enhancements are reduced. Nevertheless, considering that the current density is proportional to the square value of the refractive index, the displacement current within the dielectric layer are still comparable with that of the conduction currents (the inset, Fig. 4(c)), and the current intensity ratio I_up/I_spa and I_dn/I_spa are further reduced to about 1.21 and 1.19, respectively. An almost perfect current loop is generated within the hybridized nanodisk, and magnetic near-field enhancements are further enhanced compared with that of n_spa = 3.2 (the lower panel, Fig. 4(f)). In addition, Fig. 4(i) reveals that the far-field scattering pattern of the bonding resonance is symmetric, the scattering intensities along the MD axis are almost zero (0° and 180°), and an almost ideal doughnut shape scattering pattern is achieved in this situation, which validate a pure MD response [56]. Please note that the excitation of the bonding resonance is governed by the retardation effect, and the resonance would be weakened under oblique incidence. The black and red dotted lines in Fig. 4(c) represent, respectively, the total scattering spectrum and the MD scattering around the bonding resonance when the incident angle is 45°, where the polarization is also along the x-axis. It is found that the bonding resonance is at the same spectral position, and the scattering is indeed weaker than that of the normal incidence. Nevertheless, the bonding resonance is far away from the antibonding mode, and the MD contribution still dominates for the total scattering, where its contribution is larger than 98.4% for the oblique incidence.

To better show the realization of the ideal MD resonance, the scattering spectra and the multipolar decomposition results when n_spa is in the range of 1 – 5.8 are represented in Fig. 4(j). For the bonding mode indicated by the dotted lines, the resonance red shift with the increasing of n_spa (the first panel, Fig. 4(j)). The MD scattering intensity is obviously weaker than the total scattering when n_spa < 2.4 (the second panel, Fig. 4(j)), and the MD contribution dominates for the total scattering by further enlarging n_spa. At the same time, a scattering dip is also observed for the ED scattering around the bonding resonance, and the intensity decreases with the increasing of n_spa (the third panel, Fig. 4(j)). As for the EQ contribution, it decreases rapidly with the increasing of n_spa (the fourth panel, Fig. 4(j)), and the scattering is negligibly weak when n_spa > 3.6. With the suppressing of the ED and EQ, the MD contribution is strongly enhanced at the same time (the lowest panel, Fig. 4(j)).

The variations of the intensity ratio I_up/I_spa and I_dn/I_spa at the bonding resonance are calculated to confirm that the overlapping multipoles are suppressed when the current distribution is approaching to an ideal current loop (Fig. 5(a)). One can find that the ratios between the conduction and displacement currents decrease rapidly with the increasing of n_spa. Although I_up/I_spa and I_dn/I_spa are larger than one even when n_spa > 3.2, the conduction current is approaching to the displacement currents, and the parasitic EQ can be suppressed effectively by then (Fig. 4(j)). Another important feature is that I_up/I_spa almost equals to I_dn/I_spa when n_spa > 2.6, which means that the dipole moments of the two metallic disks are comparable with each other, and it is an important relationship for the suppressing of the ED scattering as we have mentioned before. The blue and green points in Fig. 5(b) show the variations of the ED and EQ contributions around the bonding resonance, where their scattering contributions decrease with the increasing of the refractive index of the dielectric layer. On the contrary, the MD contribution is significantly enhanced at the same time (the red points, Fig. 5(b)), where the contribution is larger than 99% when n_spa = 4.2, and it is approaching to an ideal MD scattering by further enlarging n_spa.

Fig. 5 The variations of the current intensity ratios (a) and the contribution of individual multipoles (b) at the bonding resonance by adjusting the dielectric layer refractive index n_spa, where the geometry parameters for the hybridized nanodisks are identical with that of Fig. 4.

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Please note that in the above studies, the material losses of the dielectric layer are neglected, and the realization of ideal MD resonances may be affected with the present of the losses. Fortunately, many dielectric materials with high refractive index possess weak losses around the near-infrared spectral range. For example, when the dielectric layer is composed of silicon, the optical responses of the hybridized NPs would be comparable with that of n_spa = 3.2 (Fig. 4(b)). In this situation, the MD contribution for the bonding resonance can be larger than 98.3%. Although the bonding mode is slightly diverged from an ideal MD resonance, the electric and magnetic near-fields can be strongly enhanced at the same time (Fig. 4(e)), which is promising for enhanced light-matter interactions at the nanoscale. For example, it has been reported that when a silicon nanodisk is surrounded by a metallic nanoring, or is placed between a metallic nanodisk and film, the confined energy within the silicon disk caused by the excitation of the anapole mode can be strongly enhanced, and it is useful for enhanced third-harmonic generation (THG) [61,62]. In our case, the magnetic response is distinct from that of the anapole mode. However, the incident energy also can be effectively confined compared with that of a pure dielectric NP [25], and one can expect that the hybridized NPs with a high refractive index dielectric layer are useful for enhanced THG. Besides that, the strong electric and magnetic near-fields are spatially overlapped with each other around a large area (Fig. 4(e)), which means that a strong Lorenz term can be generated with the hybridized NPs, and it can be possibly used as a nonlinear source to enhance second-harmonic generation (SHG) [63]. Another important feature for practical applications is the formation of high quality factor resonances, and it has been reported that free space giant magnetic field enhancement can be achieved mediated by bound state in the continuum for dielectric NP arrays [64], which is useful for sensing applications. As a result, one can expect that the resonance quality factor can be enlarged by using periodic arrays composed of the designed hybridized NPs, and the near-fields can be further enhanced in this case. Although the magnetic fields of the hybridized NPs are mainly within the structure, the nonlinear responses are expected to be further enhanced for the periodic arrays with the formation of high quality factor resonances.

5. Conclusion

In conclusion, we propose and theoretically demonstrate that metal-dielectric-metal hybridized nanodisks can be used to achieve ideal MD resonances, where the overlapping multipole scattering around the bonding resonance can be effectively suppressed by adjusting the geometry parameters and the refractive index of the dielectric layer. Compared with that of the core/shell structures, the hybridized NPs can be easier fabricated with current nanolithography methods. The realization of ideal MD scattering is based on a hybridized plasmon resonance, the structure footprint can be reduced effectively, and it is useful to design compact nanophotonic devices. In addition, due to the strong plasmon interactions and the formation of an equivalent current loop, the electric and magnetic near-fields are strongly enhanced at the same time, and the incident energy can be confined around the structures effectively, which makes the hybridized nanodisk a promising platform for enhanced magnetic light-matter interactions. Considering that directional scattering can be achieved by adjusting the multipole contributions, the hybridized nanodisks with different refractive index of the dielectric layer are useful for the manipulation of far-field scattering.

Funding

National Natural Science Foundation of China (NSFC 11574228 and 11874276), Natural Science Foundation of Shanxi Province (201601D021005), and San Jin Scholars Program of Shanxi Province.

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Ideal magnetic dipole resonances with metal-dielectric-metal hybridized nanodisks

Abstract

1. Introduction

2. Mechanism to achieve ideal MD resonances

3. Suppressing ED scattering with asymmetric dimers

4. Realization of ideal MD resonances

5. Conclusion

Funding

References

Cited By

Figures (5)

Equations (4)

Optics Express