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Abstract
Due to their ability to produce high electric field enhancements in relatively large nanoscale volumes with minimum absorption and nonradiating properties, anapole modes excited in high index dielectric nanostructures have attracted considerable attentions in these years. We propose a design strategy to simultaneously excite the anapole mode efficiently and maintain its resonant wavelength, which has been remained as a challenge in the conventional dielectric nanostructures. Based on analyzing the relationship between the field enhancement factor and scattering intensity of the electric and toroidal dipoles, we introduce two and four nanocuboids into the nil field intensity areas in the silicon disk system, respectively. The geometric volume of the system can be increased effectively and the electric field enhancement is boosted to be 190% and 250% while the resonant wavelength of the anapole mode is almost maintained constant. The systems combined with a slot in the strongest field intensity area also follow the same law, revealing that the design strategy can be easily extended to other geometric, material and frequency systems. Different from the design strategy to add new components into the areas with strong field intensity, the incorporations occurring at the minimum intensity area is another design scheme to engineer the properties of the resonant systems and can find broad applications in nano-device designs.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The excitation of strong field enhancements is one of the most important and fundamental requirements in light−matter interactions. In the past few decades, metallic nanostructures have been widely used to collect energy into nano volumes based on the interactions of plasmonic modes [1,2]. However, their applications are severely limited by inevitable intrinsic absorption losses and related strong heating. To avoid this effect, high index dielectric nanostructures with low dissipative losses have attracted considerable attentions in manipulating light as a suitable alternative [3].
It is well known that high index dielectric nanostructures can support diverse Mie-type resonances including electric and magnetic modes [4–9]. The interactions between these resonant modes can provide strong field enhancement [10]. Among them, anapole modes, which are originated from the destructive interference between the electric and toroid dipole resonant modes [11–14], have been investigated widely due to their strong field enhancement and nonradiative scattering properties. They have been utilized for a series of applications such as Raman scattering [15], metamaterials [16] and third harmonic generation [17–22] etc. To further increase the field enhancement factor, several methods like combining silicon disk with metallic nanoparticles [17], coupling different modes in J-Aggregates [20] and applying higher index materials such as germanium [19] have been proposed and demonstrated. Nonetheless, despite those continuous efforts to enhance the near-field intensity, achieving high local fields in all-dielectric nanostructures that have comparable strengths to those of the plasmonic analogues remains a challenge [22]. On the other hand, most of the investigations on improving the electric field intensity [17–21] are based on disk-shaped nanostructures.
According to the fundamental theory of anapole modes, the field enhancement result is strongly dependent on the destructive interference between the electric dipole and toroidal dipole [11]. At the same time, on the basis of the scattering theory, the electric and toroid dipoles are both correlated with the volume of the nanostructures [23]. It is thus beyond doubt that the derived anapole mode is also dependent on the effective size of the nanostructure. If the resonance can be efficiently excited, directly increasing the size of the nanostructure seems an intuitive method to improve the field enhancement to some extent. However, before doing that, some problems have to be solved. Firstly, the resonance wavelength of the anapole mode will be red-shifted and thus tuning is very comprehensive in the applications. What’s more, after changing the size of the nanostructure, the anapole mode may not be able to be excited efficiently.
In this paper, we propose a Φ-shaped dielectric nanostructure composed of a disk and four nanocuboids in the crystalline silicon to improve the field enhancement. The operating wavelength is selected to be larger than 800 nm to avoid material absorption [24]. By analyzing the field distribution at the anapole mode resonance of the single disk, we find that there are two special areas at the edges of the electric loops in which the field intensity almost reaches zero. We incorporate four nanouboids into these areas and find that only perturbation is induced into the resonance yet the effective size of the nanostructure is increased significantly. Under this condition, the electric field enhancement (|Emax/Eo|) is boosted to be ∼7.3, which is ∼250% compared to that of the disk shape. Furthermore, the resonance wavelength is almost maintained, which means that the far-field signature of the anapole mode is not degraded and the system is easy to be manipulated. If a properly designed slot is incorporated into the maximum of the electric field generated by the anapole mode of the Φ-shaped nanostructure, the electric field enhancement can be further increased to be ∼45. In addition, the low-order mode characteristic enables its high tunability. In contrast to the design strategy to introduce additional components or slots into the areas with strong field intensity [21], which red- or blue-shifts the resonant wavelength, the incorporations occurring at the minimum intensity area can remain the resonant wavelength almost unchanged.. It can be easily extended to other systems, either with different materials or geometric properties, or in different frequency ranges, thus can find broad applications in designing various nanoscale devices.
2. Theoretical background and simulation parameters
For a simple nanostructure, at the excitation of the anapole mode, the field enhancement factor ${E_{en}}$ can be expressed by ${E_{en}}({\lambda } )\propto {C_p}({\lambda } ),{\; }{C_T}({\lambda } )$ [9], where ${C_p}{\; {\textrm{and}}\; }{C_T}$ represents the scattering intensity of the electric dipole and toroidal dipole respectively. They are dependent on the wavelength λ and are significantly influenced by the effective size and shape of the nanostructure. Generally, if we introduce a new component into the original structure, not only the resonance wavelength but also the field enhancement factor are to be affected. This effect is normally hard to be controlled. The field enhancement factor for the complex nanostructure can then be expressed by $E_{en}^{\prime}({{\lambda } + {\Delta \lambda }} )\propto C_p^{\prime}({{\lambda } + {\Delta \lambda }} ),{\; }C_T^{\prime}({{\lambda } + {\Delta \lambda }} )$. Δλ stands for the wavelength shift of the resonance. Basically, the increase of the effective size of the nanostructure can lead to a higher field enhancement while the resonant wavelength is red-shifted at the same time [9]. Consequently, it is a formidable challenge to simultaneously enhance the field intensity and maintain the resonant wavelength of the nanostructure.
However, the challenge might be overcome if the new component only introduces a perturbation to the original field distribution of the anapole mode. According to the mode theory, one type of field distribution pattern corresponds to a specific mode. Supposing that the field distribution is only perturbed slightly, the mode would not be changed and it can still be excited effectively. Thus $E_{en}^{\prime}({{\lambda } + {\Delta \lambda }} )\approx E_{en}^{\prime}({\lambda } )$. Moreover, under the perturbation condition, the change of the resonance wavelength is negligible. On the other hand, as the effective size of the nanostructure with the new component is larger than that of the initial one, the scattering intensity for the electric dipole and toroidal dipole will become higher, i.e., $C_p^{\prime}({{\lambda } + \Delta {\lambda }} )\;>\;{C_p}({\lambda } )$ and $C_T^{\prime}({{\lambda } + \Delta {\lambda }} )\;>\;{C_T}({\lambda } )$. As a result, $E_{en}^{\prime}({{\lambda } + {\Delta \lambda }} )\;>\;{E_{en}}({\lambda } )$ and $E_{en}^{\prime}({\lambda } )\;>\;{E_{en}}({\lambda } )$ can be deduced, revealing that the field enhancement is possible to be raised while the resonant wavelength is maintained constant. The requirement is that the new component introduced into the original nanostructure only induces perturbation to the original system.
In order to rigorously analyze the interaction of the electric and toroidal dipoles, we utilize the method of Cartesian multipoles [25]. The Cartesian moment for the electric dipole p, toroidal dipole T, magnetic dipole m are described as follows:
All the numerical simulations were performed by the commercially available software finite difference time domain (FDTD) solution (Lumerical). The nanostructures are placed in vacuum with perfectly matched layers boundary conditions. The mesh size was set to be 5 nm.
3. Results and discussion
Figure 1 shows a schematic illustration of the proposed silicon Φ-shaped nanostructure under normally incident plane-wave illumination polarized along the x-direction. We select a silicon disk (with a diameter D = 400 nm and height H = 100 nm), which is the most common shape for exciting anapole modes in applications, as the starting nanostructure.
Fig. 1. Schematic illustration of a Φ-shaped silicon nanostructure under normal x-polarized incident illumination. The geometric parameters H, LA and LB represent the height, the overall length and width of the nanostructure respectively. D is the diameter of the central disk. WA and WB define the width of the two cuboids besides the central disk. The distribution of the bright hot spots in the central area corresponds to the excitation of the anapole mode.
The Cartesian multipole expansion results for the scattering spectra of the electric dipole (ED), toroidal dipole (TD) and the interference result between the electric and toroidal dipoles (TED) are plotted in Fig. 2(a). An obvious dip at 883nm combined with the corresponding electric and magnetic field distributions shown in Fig. 2(b) featuring an enhanced near-field profile inside and around the disk indicates the anapole configuration. The electric field enhancement factor is about 2.9 at the center. As pointed by the thick pink arrows, both the electric and magnetic field intensity are almost zero in the areas between the edges of the electric loops. Figure 2(c) demonstrates the schematic illustration of the main electric flow at the resonance of the anapole mode. The horizontal pink arrow inside indicates the effective length of the anapole pattern. Those areas with nil field intensity are highlighted by the pink dashed ellipses. Based on the above analyses, if we add new components into these areas, the field distribution would not be strongly influenced while the geometric size of the nanostructure could be increased effectively.
Fig. 2. (a) Cartesian multipole decompositions of the scattering spectra for the silicon nanodisk with D = 400nm and H = 100nm. ED, TD, TED represents the electric dipole, toroidal dipole and the interference result between ED and TD respectively. All the resonance wavelengths are marked by vertical black dashed lines in the paper. (b) Distributions of the magnetic (left panel) and electric (right panel) field for the corresponding anapole mode at the resonant wavelength of 883 nm. The flow of the field vectors is indicated by the white arrows. The nil field areas are pointed by the thick pink arrows. The contour of the disk is traced by the black dash lines. (c) Schematic illustration of the electric field for the anapole mode in which the three thick yellow arrows clearly point to the directions of the electric current. The nil field areas are marked by the pink ellipses. All the color bars show the field magnitude |E|/|Eo| except otherwise specified.
For simplicity, we introduce two cuboids (C1) with the same height into the nil field intensity areas in the disk system. The schematic illustration of the combined structure is plotted in Fig. 3(a). The width WA of C1 is set at 220nm, which equals to the width of the nil field area shown in Fig. 2(b). The relationship between the length LA of the whole structure and the electric field enhancement factor is shown in Fig. 3(b). With an increase of LA, the enhancement factor has a maximum value at LA=940nm. The occurrence of the optimal value indicates that LA cannot be increased too much. It can be understood from the field distributions at the resonance for two groups of geometric parameters (refer to Fig. 3(d) and 3(e)), as displayed in Fig. 3(g1) and 3(g2). Comparing these two figures, we can observe that when LA is increased to 1040nm, the newly introduced two C1 respond to the incident light strongly and the fields at the outer ends of the cuboids become quite intense. This reduces the excitation efficiency of the anapole mode and the original field distribution for the anapole mode is disturbed significantly. Consequently, the field enhancement is deteriorated. It is worth to note that when LA=940nm, LC=320nm and LR=310nm (refer to Fig. 3(a)). These two roughly equivalent values confirm that when the length of the newly introduced cuboid is larger than the effective length of the anapole mode, its response to the incident plane wave is dominant which strongly suppresses the excitation of the anapole mode. On the other hand, the resonant wavelength of the anapole mode is almost maintained constant when LA <940nm and only redshifts slightly even when LR is increased to be larger than the effective length of the anapole mode. This proves that the response of the new components added into the nil field intensity area only contribute mildly to the disk system although they can effectively increase its geometric volume.
Fig. 3. (a) Schematic illustration of the Disk + Cuboids (DCs) system. LC and LR correspond to the effective length of the anapole mode and the length of C1 respectively. (b) (c) The electric field enhancements as functions of the wavelength and the length (b) or the width (c) of C1. The magenta and green thick dashed lines highlight the optimal results. (d)(e)(f) The scattering cross sections for ED, TD and TED modes when the geometric size of C1 is different. (g) The corresponding electric field distributions at the resonance of the anapole modes as shown in (d)-(f). Please note (g3) and (g4) corresponds to the two resonances shown in (f).
We then fix LA to be 940nm and investigate the influence of the width WA of C1 on the anapole mode. As shown in Fig. 3(c), when the width is narrow, C1 almost does not interfere the pattern of the anapole mode. Thus we can observe a constant resonant wavelength while an increased field enhancement factor. The optimal value is obtained at WA=220nm, which corresponds to the width of the nil field intensity area as marked by the pink arrows in Fig. 2(c), revealing that C1 starts to connect with the electric loop at the central part. As the two lobes of the pattern (strong field areas) at resonance is wider than its central area, the extension along the vertical direction, i.e. the increase of WA can affect the field distribution more intensely. This can be confirmed by comparing the field distribution demonstrated in Fig. 3(g2) and 3(g3). As a result, the redshift of the resonant wavelength appears more pronounced than that for increasing LA when WA is further increased beyond the optimal value.
The field enhancement factor reaches to 5.6 for the combined structure (disk + two cuboids, shortened as DCs), which is boosted ∼90% compared to that of the original disk. From Fig. 2(a) and Fig. 3(d), we can find that the scattering cross sections of ED and TD at resonance in the DCs structure are much larger than that in the disk, indicating that their interference is stronger which generates more significant field enhancement ultimately. It is interesting to note that there is another strong TD peak at a wavelength of 790nm as shown in Fig. 3(f). The corresponding field distribution in Fig. 3(g4) demonstrates that two toroidal modes can be excited in the two newly introduced cuboids. Consequently, their scattering cross section is larger than that at 893 nm, at which only a single toroidal mode can be excited. This extra peak reveals that multiple toroidal modes are accessible in an individual nanostructure, which can extend its applications.
If we look at the field distributions demonstrated in Fig. 3(g1), another two nil field areas can be found as pointed by two thick red arrows. Furthermore, these two areas are not so close to the central disk. We then add another two cuboids (C2) at the outer edges of C1. The schematic illustration of the new design (Φ-shaped) is shown in Fig. 4(a). The influence of the length (LB) and width (WB) of C2 is shown in Fig. 4(b) and 4(c), respectively. The optimal length and width are found to be 760 and 90nm respectively. Please note that the optimal width equals to the distance between the nodes of the field and the outer ends of C1. The field enhancement factor of the Φ-shaped system reaches about 7.3, which is increased to be 250% of that of the disk system. If the width extends over the position of the node, C2 will not only strongly disturb the central disk, but also result in a more complex field distribution in the rectangle part as shown in Fig. 4(f2). Both of these two factors can reduce the excitation efficiency of the anapole mode.
Fig. 4. (a) Schematic illustration of the Φ-shaped nanostructure. LB and WB represent the length and width of C2. (b) (c) The electric field enhancements as functions of the wavelength and the length (b) or the width (c) of C2. The magenta and green thick dashed lines highlight the optimal results. (d)(e) The scattering cross sections for ED, TD and TED modes for different C2. (f) The corresponding electric field distributions at the resonance of the anapole modes as shown in (d) and (e).
The relationship between the field enhancement factor and the wavelength for the disk, DCs and Φ-shaped systems is summarized in Fig. 6(a). As anticipated, besides the boost of the enhancement factor, their resonant wavelength is close to each other.
When a slot is introduced into the all-dielectric system, the field enhancement can be boosted significantly due to the boundary conditions [21]. Here we introduce a narrow slot into the Φ-shaped system, as demonstrated in Fig. 5(a) (Please note that the scale is not real). Figure 5(b) and (c) display the influence of the width WS and length LS of the slot on the resonant wavelength and the field enhancement factor. Obviously, narrower slot can induce stronger field enhancement. There is also an optimal value for the length of the slot. When it is too short, the interaction with the field is not sufficient while a very long slot will interfere the edge of the electric loop of the anapole mode. The optimal length 250nm is about the distance between the two electric nodes which is consistent with the previous research [21]. After introducing the slot, the resonance wavelength of the anapole mode blue shifts gradually with an increase of its length. This can be attributed to the decrease of the effective volume size of the system. The field distribution in Fig. 5(d) shows that magnetic hot spots tend to split from two to four (refer to Fig. 2(a) also), indicating that a higher order mode is excited as more electric loops are formed. This can be confirmed by the two modes shown in Fig. 5(c) although the high-order one is quite weak.
Fig. 5. (a) Schematic illustration of the Φ-shaped nanostructure combined with a slot at the center with a length LS and width WS. (b) (c) The electric field enhancements as functions of the wavelength and the length (b) or the width (c) of the slot. The magenta thick dashed line highlights the optimal result. (d) Distributions of the electric (left panel) and magnetic (right panel) field for the anapole mode at the resonance.
Figure 6(b) further compares the field enhancement factors for three systems with the slot and shows that the trend is similar to those without the slot. This result proves that the design is also effective in a different system. It is predictable that our design strategy can also be extended to other material systems and other frequency ranges.
Fig. 6. (a)/(b) Electric field enhancement results for the disk, DCs and Φ-shaped nanostructures without/with the split. The two magenta dashed lines are guides for eyes.
4. Conclusion
To conclude, we have proposed a strategy to design all-dielectric nanostructures with high electric field intensity enhancement while constant resonant wavelength at the excitation of the anapole modes. We started with a conventional disk system and analyzed the scattering cross sections for its electric dipole (ED), toroidal dipole (TD) and TED modes and the corresponding field distributions at resonant. Then we raised its effective geometric size by introducing two cuboids into the nil field intensity areas (forming the combined structure (disk + two cuboids, shortened as DCs,), which only bring perturbation to its anapole mode and does not affect the resonant wavelength significantly. The field enhancement of the DCs system is increased by 90%. After adding another two cuboids to the nil field intensity area of the DCs system (forming the Φ-shaped system), the field enhancement is boosted to be 250%. Finally, we introduced a slot into the disk, DCs and Φ-shaped systems and found that the increase trend is similar. Therefore, our design strategy can be easily extended to other geometric systems. We can anticipate that it can also be extended to other material systems and other frequency ranges. This design strategy can find broad applications in resonant systems.
Funding
National Natural Science Foundation of China (11704133, 61675070).
Disclosures
The authors declare no conflicts of interest.
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