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Abstract
We have designed and numerically analyzed the theta-shaped dielectric arrays based on a single resonator per unit cell and generated non-radiative anapole resonance with an enhanced Q-factor. Relying on breaking the symmetry, it is shown that instead of leading to additional radiation losses, the Q-factor in theta-shaped Si arrays is enlarged about one order larger than that of the perfect disk arrays. And the magnetic near-field enhancements can be extended outside of the structures and are extremely enlarged. Further, the asymmetrirc magnetic field distributions can be observed by changing the nanorod position, providing a new way of indirectly manipulating the localized magnetic fields. The high Q-factor and strong magnetic near-field enhancements outside of the structures can be easily tailored by adjusting different geometric parameters and are achieved simultaneously, which furthermore provides a useful insight into their tuning behavior.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High Q-factor resonators are important for many applications, such as optical sensors [1,2], switches [3] and nonlinear optics [4]. Fano resonance provides an approach to realize the high Q-factor resonances and has attracted lots of research interest due to the asymmetric lineshape, sharp spectral profile and sensitivity to structural and environmental parameters [5–7]. Usually, such resonance appears in plasmonic systems characterized by a certain discrete (or narrow) state that interacts with the continuum (or broad) state through an interference effect [8,9]. However, owing to the high intrinsic absorption losses in plasmonic nanostructures, the Q-factor is difficult to improve at infrared and optical frequencies and severely limit their practical applications.
Recently, the high-index dielectric nanoparticles with low dissipative losses have been explored as a promising alternative for improving the Q-factor of Fano resonance [10–12]. Compared to the plasmonic counterparts, where resonances are often dominated by electric responses, high-index dielectric nanoparticles can support a series of Mie resonances with both electric and magnetic responses [13–15]. More importantly, the toroidal multipoles characterized by currents flowing on the surface of a torus can be excited as a possible realization of radiationless objects. [16–20]. In particular, when properly engineered, the toroidal dipole (TD) can be tuned to the same amplitude and out of phase with the electric dipole (ED), leading to the far-field scattering cancellation due to similarity of their far-field scattering patterns, and thus produce the so-called the nonradiating anapoles, which have attracted considerable attention [21–26].
Although nonradiative losses can be weakened in single dielectric nanoparticle, strong radiative losses are still obstacles to further enlarge Q-factor. Recently, the collective oscillations in the dielectric nanoparticle arrays have been developed to achieve higher Q-factor [27–31]. The resonators in the arrays interact through near-field interaction between the unit cells, resulting in collective oscillation of the resonators and suppression of radiative loss. As a result, the extremely high Q-factor can be observed due to the formation of sharp surface lattice resonances [25,27,32,33], but the magnetic field is confined effectively within the dielectric particles.
Strong near-field enhancements around plasmonic nanoparticles can interact with external environment, while the field enhancements inside of plasmonic nanoparticles are weak. Differing from the plasmonic nanoparticles, the external fields can be confined effectively within the dielectric particles, leading to the formation of relatively strong field enhancements inside of the dielectric nanoparticles [22,26,34]. However, the enhanced field inside of the nanoparticles can severely limit the interaction with external environment. Fortunately, near-field enhancements can be extended outside of dielectric nanoparticles by affecting the system so as to change the parameters of the resonance, but the Q-factor is reduced simultaneously by additional radiation losses [28]. Despite growing efforts to enhance near-field by using plasmonic/dielectric hybrid structures [35,36], it remains a formidable challenge for all-dielectric nanostructures to gain the high Q-factor and large local field enhancement outside of all-dielectric nanostructures, especially for the magnetic components.
In addition to the methods mentioned above, the structurally engineered nanoparticle arrays with symmetry breaking in the unit cells provide a new method to enhance the Q-factor. In this letter, we design the theta-shaped silicon (Si) arrays realizing the Q-factor enhancement with the anapole excitation while using only one dielectric resonator per unit cell. Instead of leading to additional radiative losses with the present of the theta-shaped Si arrays, the Q-factor of the anapole resonance is about one order larger than that of the corresponding disk arrays. Relying on breaking the symmetry of the arrays, the magnetic dipole resonance is excited and has the large Q factor due to the weak radiative and nonradiative decay in the z direction. And the magnetic near-field enhancements can be extended outside of the nanostructures and are extremely enlarged using the theta-shaped Si arrays. Furthermore, the asymmetry magnetic near-field enhancement can be observed due to symmetry breaking, which provides a new way of indirectly manipulating the localized fields. Therefore, this analysis confirms that high Q-factor and strong magnetic near-field enhancements outside of the nanoparticles can be achieved simultaneously, and provides a useful insight into their tuning behavior.
2. Structures description
Figure 1 shows the broken symmetry theta-shaped Si arrays consisting of Si ring and the nanorod. The internal and external radius is r and R, respectively. And the width of the nanorod is W. And the asymmetry parameter is defined as δ. The array’s period and thickness are P and T, respectively. The linearly polarized lights with a propagation vector along the z direction incident perpendicularly onto the plane of arrays. The response properties of the arrays were simulated using the finite element and finite-difference time-domain software package. The dielectric constants for amorphous Si are taken from the measured data [37].
Fig. 1. (a) Schematic illustration of the theta-shaped Si arrays. (b) Top view and geometric parameters of a unit cell in the Si arrays. The polarization (E) and propagation (k) directions of incident wave are denoted.
3. Simulation results and discussions
The solid lines in Fig. 2(a) show the transmission spectra of the proposed arrays with a symmetry geometry (δ = 0 nm) at different width W. The two resonances in Fig. 1(a) are moved to facilitate a direct comparison between the lines width. When W = 350 nm, that is, a perfect disk arrays, a Fano spectral feature is observed with a broad transmission dip around 1045 nm. Previous studies have demonstrated that such Fano line shape in the spectrum arises from the destructive interference of ED and TD moments, which leads to the generation of the anapole mode in dip and will be discussed below [22]. Although the radiative damping can be suppressed with the anapole mode and collective oscillations in arrays, the Q-factor of the Si disk arrays is still poor due to the strong scattering around the resonance position. When W = 55 nm, an extremely narrow Fano resonance with the modulation depths near 100% is excited around 869 nm, where the line width of the resonance is reduced distinctly. The magnetic field distributions of such typical anapole resonance can be observed in Fig. 5. The magnetic field profile around 869 nm forms a vortex in the middle as a result of the rotation of the induced magnetic field in the left and right sides of the ring, giving rise to the strong toroidal moments oriented parallel to the ring surface. And the enhanced magnetic field mainly happens in the air gaps and almost all of the field amplitude is extremely enlarged using the theta-shaped Si arrays. Furthermore, the magnetic field can be extended outside of the structures due to the air gaps, which can enhance interaction with the surrounding environment. Interesting, instead of leading to additional radiative losses with the present of the air gaps, the Q-factor is further enhanced in the proposed Si arrays.
Fig. 2. (a) Calculated spectra (1-Transmission) of the arrays (solid lines), where the W = 55 nm (red line) and W = 350 nm (blue line), R = 230 nm, r = 175 nm, T = 125 nm, P = 600 nm for the array with δ = 0 nm, and the dashed lines represent the fitted spectra with the oscillator model. (b) The extracted Q-factors and resonance positions as a function of the W.
The Fano-like spectral character of our design, besides this extremely narrow resonance shown in Fig. 2, can be further confirmed by a Fano model [28]. The calculated transmittance spectra are fitted to a Fano line shape given by:
where ${a_1}$, ${a_2}$ and b are constant real numbers, ${\omega _0}$ is the resonant frequency, and $\gamma $ is the damping rate of the resonance. It is found that the fitted spectra (the black dashed lines) agree well with that of the numerical calculated spectra in Fig. 2(a). In addition, such model can be used to quantify the line width and the Q-factor. The extracted Q-factors as a function of the W are depicted in Fig. 2(b), and the Q-factor of the resonance is calculated as Q = (${\omega _0}$/2$\gamma $). When the W = 350 nm, the corresponding Q-factor of the transmission dip around 1045 nm is about 49 (${\omega _0}$ = 1.186 eV and ${\gamma}$ = 1.220*10−2 eV). When the W decreases to 55 nm, the calculated Q-factor at the anapole resonance around 869 nm (${\omega _0}$ = 1.427 eV and $\gamma $ = 0.990*10−3 eV) reaches to about 720, which is about 15 times larger than that of the Si disk array. Furthermore, the resonance wavelength as a function of the W is represented in Fig. 2(b). The resonances blue shift rapidly from 1045 nm to 896 nm as the W decreases from 350 nm to 50 nm.To get insight into the anapole response of the theta-shaped Si arrays, we calculate the Cartesian multipole contributions into the transmission when W = 55 nm. The Cartesian electric dipole |P| and toroidal dipole |ikT| moments shown in Fig. 3(a) are the only dominating multipole contributions and have identical value at the dip 869 nm. In addition, they have opposite phases and the phase difference |φ(P) - φ(ikT)| is approximately equal to π around 869 nm in Fig. 3(b), indicating that the dip mostly arises from the destructive interference between the ED and the TD moments, giving rise to an anapole response. It must be noted that, magnetic dipole moment is substantially suppressed while the toroidal dipole moment is resonantly enhanced and plays a major part in the anapole resonance because of the symmetry of the structure.
Fig. 3. (a) Amplitude of the Cartesian ED moment |P|, MD moment |M|, and TD moment |ikT| around the TD resonance for W = 55 nm. (b) The phases and their difference of P and ikT. The other geometry parameters are identical as that of Fig. 2(a).
However, it is possible to “perturb” the asymmetry parameter δ to change the spectral positions and induce the mode mixing. In arrays with broken symmetry, two tunable Fano resonances appear: a high energy resonance (M1) and a low energy resonance (M2), as shown in Fig. 4(a). The two resonances are strongly modified by increasing the δ. A similar analysis shows that the origin of the M1 is analogous to the mechanism described above. When the δ = 60 nm, for instance, the opposite phase and equal amplitude of the P and ikT generate the destructive interference at the anapole resonance around 837 nm, as shown in Fig. 4(b) and (c). However, the strength of the ED moment P is severely weakened for δ = 60 nm, and the symmetry of TD moment ikT is broken and suppressed in the vicinity of the M2. Instead, the strength of the MD moment M (i.e., a z-directed magnetic dipole) increases remarkably and dominates all other multipoles by nearly 20 times of magnitude in the M2, as shown in Fig. 4(b). Although the phase difference between P and ikT is approximately equal to π, anapole condition is dissatisfied (P = − ikT), as depicted in Fig. 4(c). In fact, the M2 is a z-directed magnetic dipole and will be discussed in following. Figure 4(d) represents the extracted Q-factor of the M1 and M2, respectively. When the δ = 15 nm, the Q-factors of the M1 and M2 reach about 520 and 300, respectively. The Q-factors of the M1 and M2 are being further extended to about 815 and 455, respectively, when δ = 40 nm. Further enlarge the δ, the Q-factor decreases rapidly. As the δ is greater than 90 nm, the M1 vanishes, which means that interaction between ED and TD moments would be approaching to zero in this case.
Fig. 4. (a) Transmission spectra of the arrays at different δ. (b) Amplitudes of the ED moment |P|, MD moment |M| and TD moment |ikT|. (c) The phases and the difference of P and ikT. (d) The extracted Q-factors of the arrays as a function of δ. The other geometry parameters are identical as that of Fig. 3.
To further identify the mechanism of the resonances in Fig. 4, we calculates the magnetic field distributions at different resonances when δ = 60 nm, as shown in Fig. 5. The magnetic fields are normalized to the incident electromagnetic field. Symmetry breaking generates the asymmetric magnetic field distributions. Such asymmetric distributions are attributed to two distinct origins: the asymmetric magnetic field around 837 nm locates at small air gap arising from the typical anapole resonance, as depicted by the electric and magnetic field vector distributions. The circulating electric field seen around 986 nm is reminiscent of a magnetic dipole field pattern; however, differing from the usual in-plane magnetic dipole, the orientation of such dipole is out of the plane of the array. And the magnetic field enhancement concentrates in the large air gap. Such magnetic dipole resonance arises due to the different size of the two air gaps. Considering each part of the air gap as a separate polarizable dielectric region based on the approximate spatial decomposition, the two separated electric dipoles will exhibit slightly different dipole strengths, which lead to the z-directed magnetic dipole resonance with large Q-factor arising from the weak radiative and nonradiative decay in the z-direction of the array. Meanwhile, the enhanced magnetic field shown in Fig. 5 is extended outside of nanostructure and is mostly concentrated in the air gaps. The localized field intensity is enhanced as the δ increases. The maximum magnetic field enhancement around 837 nm reaches 70, which is about 8 times larger than that of the δ = 0 nm. The enhanced magnetic field around 986 nm can be larger than 30. It is possible to achieve magnetic modulation with strong near-field enhancements and high Q-factor simultaneously in theta-shaped arrays. Note also that such an excitation mechanism is unavailable for the symmetric resonator with the δ = 0 nm.
Fig. 5. a) Normalized magnetic field (Hz) distributions at different resonances in the x − y plane at the δ = 0 nm and 60 nm. The black cones represent electric field vector distributions. (b) Normalized magnetic near-field distributions |H/H0| and magnetic field vector distributions (white cones) at different resonances for the W = 55 nm at different δ.
In the following, we investigate the dependence of the transmission spectra on the different geometric parameters in Fig. 6. As shown in Fig. 6(a)–(d), it can be seen the M1 and M2 resonances redshift as the P, the W, the T and the R increase. As mentioned in Fig. 4(b), the z-directed magnetic dipole resonance at M1 and M2 dominates multipole contributions due to breaking. Along with the increase of the period P, as shown in Fig. 6(a), the radiation of the z-directed dipole is enhanced, which results in broad resonance line widths in the M1 and M2. Noting that the radiative decay of the z-directed dipole in the M1 can be further aggravated due to larger localized enhancement mediated by collective oscillations at the position of the resonator [38,39], which leads to significantly improved strength. In addition, the attenuated collective oscillations with the increase in period results in redshift of the M2. In addition, the increase in the W also lead to the sharp anapole responses in the M1, and the M2 is less sensitive to the W compared with that of the M1 in Fig. 6(b). As the T increase, the M1 can be excited more effectively and generates higher Q-factor due to the contributions from the ED and TD moments in Fig. 6(c). On the contrary, the anapole responses in the M1 are decreased by increasing the R in Fig. 6(d). The M2 is weakened and the reason is that interaction between the two separated electric dipoles is reduced with the increase of T and R. Thus, the spectral responses can be easily tailored by adjusting the geometric parameters.
Fig. 6. Dependence of transmission spectra of the arrays with δ = 60 nm on the different (a) P, (b) W, (c) T, (d) R. Other parameters are the same as the parameters used in Fig. 1(a) except the parameter shown in each Fig. 6.
4. Conclusion
In conclusion, we have shown how breaking the symmetry of the proposed theta-shaped Si arrays leads to the enhanced Q-factor with anapole excitation and magnetic field tunability that features only one resonator in the unit cell. It is shown that instead of leading to additional radiation losses, the Q-factor in the theta-shaped Si arrays is enlarged about one order larger than that of the perfect disk arrays. In addition, symmetry breaking can be used to adjust the Q-factor and leads to the asymmetry magnetic near-field enhancement which provides a unique approach to induce mode mixing with the large Q factor. Importantly, such asymmetric magnetic enhancements can be extended outside of nanostructures at the Fano resonant frequencies, which serve as an ideal platform for enhancing interaction with the surrounding environment. Combining high quality factor and large magnetic field enhancement simultaneously will open the possibility of using such arrays for a wide range of applications including bio-sensing, nano-antennas and optical modulation.
Funding
National Natural Science Foundation of China (NSFC) (11647102); Doctoral Research Fund of Dalian Polytechnic University (61020729, 71600160).
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