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Abstract
The simultaneous realization of high Q-factor resonances and strong near-field enhancements around and inside of dielectric nanostructures is important for many applications in nanophotonics. However, the incident fields are often confined within dielectric nanoparticles, which results in poor optical interactions with external environment. Near-field enhancements can be extended outside of dielectric nanostructures with proper design, but the Q-factor is often reduced caused by additional radiation losses. This paper shows that the obstacles to achieve high Q-factor, that is, the radiative losses can be effectively suppressed by using dielectric nanodisk arrays, where the Q-factor is about one order larger than that of the single disks associated with the nonradiating anapole modes and the collective oscillations of the arrays. When the resonance energies of the electric dipole mode and the subradiant mode are degenerate with each other, the destructive interference produces an effect analogous to electromagnetically induced transparency. Furthermore, the Q-factor can be extremely enlarged with dielectric split nanodisk arrays, where the present of the split gap does not induce additional losses. Instead, the coupling between the two interfering modes is modified by adjusting the gap width, which makes it possible to achieve high Q-factor and strong near-field enhancements around and inside of the split disks simultaneously. It is shown that the Q-factor is approaching to 106 when the gap width is about 110 nm, and the near-field enhancements around and inside of the split disks are about two orders stronger than that of the single disk.
© 2017 Optical Society of America
1. Introduction
Plasmonic and dielectric nanostructures have gained considerable attention due to their various potential applications in nanophotonics [1,2]. With the excitation of localized surface plasmon resonances (LSPRs), there are strong near-field enhancements, and the resonances are extremely sensitive to the environment surrounding the plasmonic nanostructures [3]. However, the field enhancements inside of metallic nanoparticles are weak, and LSPRs are suffering from the inevitable ohmic losses. On the contrary, ohmic losses can be eliminated for dielectric nanostructures, and the incident fields can be confined within the particles, leading to the formation of relatively strong field enhancements inside of the dielectric nanostructures [4,5]. Besides that, magnetic resonances are directly excited caused by retardation effect [6–8], which is useful for many applications such as directional emission [9,10], harmonic generation [11,12] and waveguiding [13]. Previous studies have shown that the field enhancements around dielectric nanostructures are often weaker compared with that of plasmonic nanostructures, and this problem can be solved by using plasmonic/dielectric hybrid structures [14], where there are hybridized resonances with suppressed losses, and strong field enhancements around and inside of the hybrid structures can be achieved [15,16].
The performance of many applications are governed by the Q-factor of the resonances, where a high Q-factor indicates an efficient energy confinement, enhanced near-fields around the nanoparticles can be achieved, and there are strong optical interactions with external environment and the structure itself [3]. Unfortunately, strong radiative and nonradiative losses result in poor Q-factors for LSPRs [17,18]. Although nonradiative losses can be weak for dielectric nanostructures, strong radiative losses are obstacles to further enlarge the Q-factor [19,20].
In addition to the bright modes that possess strong dipole moments, there are many dark subradiant modes for plasmonic and dielectric nanostructures, and they cannot be directly excited by external fields because the corresponding dipole moments are approaching to zero [21]. Nevertheless, dark modes can be indirectly excited through near-field coupling with bright modes, and the destructive interference often leads to the formation of Fano resonances [22–25]. When the resonance energies of the two interfering modes are degenerate with each other, the coupling is enhanced, which produces an effect analogous to electromagnetically induced transparence (EIT) [26,27]. In this way, radiative losses are effectively suppressed due to the subraidant nature of the dark modes. Various studies have demonstrated the improved Q-factor with the excitation of Fano resonances in plasmonic nanostructures [28–31], and there are larger Q-factors for dielectric Fano structures caused by the lower intrinsic nonradiative losses [32–36].
Another method to improve the Q-factor is by using nanoparticle arrays, where collective oscillations mediated by near-field coupling between unit cells lead to an effective suppressing of radiative damping [37–39]. For example, the coupling between bright [40] or dark LSPRs [41] and Rayleigh anomaly results in the formation of sharp surface lattice modes in metallic nanoparticle arrays, and the Q-factors are easily exceeding 102, which is only limited by nonradiative losses [42,43]. For dielectric nanoparticle arrays, extremely high Q-factors are realized because of the suppressed radiative and nonradiative losses [44–46]. It has been shown that the Q-factor is exceeding 104 for dielectric metasurface composed of bright and dark nanoresonators, where the incident fields are confined effectively inside of the dielectric resonators [44,45]. Although the near-field enhancements can be extended outside of the resonators by introducing split gaps, the Q-factor decreases rapidly caused by additional radiation losses [44]. While for the spindle-shape nanoparticle arrays, there are strong field enhancements outside of the metasurface, but the electric field enhancements inside of the structures are relatively weak, and the geometry is also relatively complex for the nanofabrications [46].
Not long ago, the so called anapole modes have been proposed and demonstrated with dielectric nanoparticles [47–50], where the anapole modes are generated due to the destructive interference of the radiation fields between the electric dipole and the toroidal dipole modes, and nonradiating resonance can be possibly achieved with single nanoparticles [51–53]. Due to the nonradiating nature and the efficient energy confinement, the anapole modes are useful for enhanced nonlinear effects [54–56], nanolasers [57], the realization of ideal magnetic scattering [58] and broadband absorption [59]. Very recently, it has been shown that extremely high Q-factor (~106) and near-field enhancements (~104) can be achieved with the excitation of anapole mode in metamolecule arrays composed of metallic split ring resonators [60]. However, the metasurface is operated in the gigahertz spectral range, and it would be hard to extend to the optical spectral range because of the strong nonradiative losses. In this study, we show that radiative and nonradiative losses are suppressed associated with the nonradiating anapole modes and the collective oscillations in dielectric nanodisk arrays, thereby leading to the generation of a subradiant resonance. EIT-like responses are observed in the near-infrared when the interfering electric dipole and the subradiant modes are degenerate with each other. Furthermore, the Q-factor and near-field enhancements inside and outside of the structures are extremely enlarged using split nanodisk arrays. Instead of leading to additional radiative losses with the present of the split gap [44], the coupling can be adjusted by manipulating the gap width. Therefore, high Q-factor (~106) and strong near-field enhancements around and inside of the nanoparticles are achieved simultaneously, and the dielectric split nanodisk arrays could be useful for many applications such as biosensing, nanolaser, and enhanced nonlinear effect.
2. Optical responses of single silicon nanodisk
The solid line in Fig. 1(a) shows the extinction spectrum of a single silicon disk calculated with the finite-difference time-domain method. In the calculations, the incident fields are propagating along the z-axis, and the polarization is along the x-axis, the disk radius R = 247 nm, the thickness h = 265 nm, the refractive index of the surrounding medium is supposed to be 1.46, and the optical constants of silicon are taken from the measured data [61]. A pronounced extinction dip, that is, a Fano-like spectral feature is observed around 1270 nm. The near-field enhancements and the field vector distributions at the structural cross sections when the incident wavelength is at the extinction dip are represented in Fig. 1(b). It is found that there are opposite circular displacement currents on the up- and the down-sides of the disk (the upper panel, Fig. 1(b)), and the generated circular magnetic moment is perpendicular to the disk surface (the lower panel, Fig. 1(b)), which leads to the formation of a toroidal dipole moment oriented parallel to the disk surface. Previous studies have demonstrated that the overall electromagnetic radiation of the toroidal dipole mode is identical with that of the electric dipole mode (the upper-right inset, Fig. 1(a)), and the destructive interference of their radiation fields leads to the generation of the anapole modes, thereby forming the Fano-like line shape in the spectrum [47–50].
Fig. 1 Optical responses of a single silicon nanodisk and a disk array. (a) Extinction spectrum of the single silicon disk and (d) the disk array (solid lines), where the radius R = 247 nm, the thickness h = 265 nm, the periodicity p = 600 nm for the array, and the dashed lines represent the fitted spectra with the oscillator model. (b) Electric (upper panel) and magnetic (lower panel) field enhancements and field vector distributions around the extinction dips for the single disk and (e) the disk array. (c) Multipole expansion results of the scattering spectrum in the Cartesian coordinate system for the single disk and (f) the disk array, where P, T, M, QE and QM denote Cartesian electric dipole, toroidal dipole, magnetic dipole, electric quadrupole, and magnetic quadrupole, respectively.
In order to get deeper insight, the Cartesian multipole contributions into the scattering for the single nanodisk is calculated based on the induced current inside of the nanoparticle (Fig. 1(c)) [47,49,50]. In this way, the multipole contributions for the optical responses can be identified. The calculation results indicate that although the Cartesian electric dipole and toroidal dipole contributions are not crossing with each other, the cancellation of the scattering in the far field still occur around 1200 nm, and the total scattering of the disk is suppressed around this spectral range. In addition, it is found that there are strong magnetic quadrupole contributions, where previous studies have shown that the magnetic quadrupole responses cannot be eliminated for planar metasurfaces with the excitation of toroidal dipole resonances [60]. As a result, the total scattering is still relatively strong even with the far field cancelation between the Cartesian electric and toroidal dipoles. By reducing the disk thickness, the contributions from the magnetic and electric quadrupoles can be suppressed, and the anapole mode of a single disk can be excited more effectively [47–50].
In nanophotonics, the Q-factor of nanoresonators is important for the performance of many applications. In order to consistently quantify the Q-factor, the extinction spectra E(ω) = |e(ω)|2 are fitted with an analytical Fano interference model [62],
where ar is the background amplitude, Γj, bj, φj and ωj characterize, respectively, the radiative damping, amplitude, phase and resonant energy of the j different oscillators that representing the interfering modes. A two-oscillator model is used to fit the calculated spectrum, and they are representing for the electric dipole mode (j = 1) and the subradiant mode (j = 2), respectively. It is found that the fitted spectrum agrees well with that of the numerical calculated spectrum (the dashed and the solid lines, Fig. 1(a)), where the resonance energy and the radiative damping of the subradiant mode are ω2 = 1.040 eV and Γ2 = 0.186 eV, respectively. The radiative damping can be used to characterize the line width, and the Q-factor of the subradiant mode of the single silicon disk is calculated as Q = ω2/Γ2 ≈6.3. EIT and improved Q-factor with nanodisk arrays
Although radiative damping can be suppressed with the excitation of the anapole mode, the scattering intensity around the spectral dip is still relatively strong due to higher-order multipolar scattering [47,60]. As a result, the Q-factor of the single silicon disk is poor, and the electric field enhancement of the subradiant mode is weaker than that of plasmonic nanostructures (Fig. 1(b)). The Q-factor can be enlarged by using nanoparticles arrays, where radiative damping is minimized due to the collective oscillations mediated by near-field coupling between the unit cells, and resonances with high Q-factor are realized for both plasmonic and dielectric nanoparticle arrays [37–46].
Figure 2 represents the transmission spectra of the silicon disk arrays by adjusting the radius of individual particles, where the periodicity p = 600 nm, and the other geometry parameters are identical as that of the single disk. When the radius R = 277 nm (the black line), two resonance modes around 1350 nm and 1425 nm are observed, and the line width of the resonance with lower energy is narrower than that of the other one. In addition, it seems that there is a destructive interference between the two resonances, and a sharp and asymmetric Fano resonance around 1425 nm appears in the spectrum. By reducing the disk size, the sharper resonance blue shifts more rapidly compared with that of the broad one. When R = 247 nm (the red line), the two resonances are almost degenerate with each other, and there is an EIT-like line shape in the transmission spectrum. The corresponding extinction spectrum (1 - transmission) is shown in Fig. 1(d) (the solid line), and the near-field distributions around 1360 nm reveal that the broad resonance is caused by the excitation of the electric dipole mode (upper-right inset, Fig. 1(d)). On the other hand, near-field distributions around the EIT spectral position indicate that there is a strong toroidal dipole response, which may interfere with the electric dipole to generate the anapole mode (Fig. 1(e)). The electric dipole mode and the subradiant modes are almost degenerate with each other in this case, and the destructive interference leads to the EIT-like resonance. Besides that, there are indeed strong near-field coupling between the unit cells for the disk array compared with that of the single disk (Fig. 1(b) and 1(e)), radiative damping is further suppressed due to the collective oscillations, and the line width of the subradiant mode is reduced for the disk array.
Fig. 2 Transmission spectra of the perfect silicon disk arrays with different radius, where the other geometry parameters are identical as that of Fig. 1(d).
The Cartesian multipole contributions into the scattering calculated based on a single nanodisk in the array are represented in Fig. 1(f). It is found that the Cartesian electric and toroidal dipoles are crossing with each other. However, the contributions from the magnetic quadrupole and magnetic dipole are relatively strong, and there is a strong scattering peak around the extinction dip position (Fig. 1(d)). Considering that the multipole expansion is performed with a single disk in the array, the coupling between the unit cells is not considered in this case. Therefore, in addition to the interference between the Cartesian electric and toroidal dipoles, the strong radiative couplings between the unit cells play a very important role for the realization of the sharp resonance [38,39,44,45]. For example, the magnetic quadrupole moment is strong according to the near-field distributions shown in Fig. 1(e). However, it is oscillating perpendicular to the array plane, and the coherent coupling between the unit cells leads to the formation of a surface wave, which only scatter at the edges of the array. Consequently, the incident field is efficiently trapped into the array, thereby leading to a sharp and subradiant resonance [38,39,44,45].
The two-oscillator model is then used to fit the calculated extinction spectrum of the disk array (the dashed line, Fig. 1(d)), where the fitting parameters for the resonance energy of the subradiant mode ω2 = 0.947 eV, and the corresponding radiative damping Γ2 = 0.011 eV. The calculated Q-factor is about 86, which is about 14 times larger than that of the single disk. As a result, the incident energy is confined more effectively around the disks, and the electromagnetic field enhancements are more than one times stronger than that of the single disk (Figs. 1(b) and 1(e)).
4. High Q-factor with split nanodisk arrays
The radiative decay can be effectively suppressed by using the silicon disk arrays. However, the Q-factor is far more smaller than that of the metasurfaces reported in previous studies [44–46,60]. It has been shown that in addition to the radiative damping of the subradiant mode of the dielectric EIT metasurface, the Q-factor is governed by the coupling coefficient between the two interfering modes, and the Q-factor increases monotonically with the reduction of the coupling [44].
For the silicon disk arrays, the subradiant mode and the electric dipole mode are excited with the same resonator, and there is no geometry parameters such as the commonly used gap detuning to modify the coupling strength [44]. However, the near-field distributions indicate that the electric fields are spatially overlapping with each other for the electric dipole and the subradiant modes (Fig. 1), and the coupling can be possibly modified by introducing a split gap along the x-axis (the lower inset, Fig. 3). Previous studies have shown that the present of split gaps may lead to strong near-field enhancements outside of the dielectricnanoparticles, and the optical interactions with external environments are strongly enhanced, but the Q-factor is reduced caused by additional radiation losses [44]. On the contrary, the coupling of the interfering modes is expected to be adjusted by manipulating the gap width for the split disk arrays, and improved Q-factor and strong near-field enhancements outside of the disk can be possibly achieved simultaneously.
Fig. 3 Evolution of the transmission spectra of the split silicon disk arrays against the gap width, where the inset shows the schematic view of the split disk, the disk radius R = 277 nm, and the rest geometry parameters are identical as that of Fig. 1(d).
The transmission spectra of the split disk arrays with different gap width are represented in Fig. 3, where the radius R = 277 nm, and the periodicity p = 600 nm. When the gap width G = 10 nm, the overall optical responses are similar as that of the perfect disk array, where there is a minor blue shift (the black lines, Figs. 2 and 3), and an asymmetric Fano resonance around 1420 nm appears in the spectrum due to the large energy detuning between the subradiant and the electric dipole modes. The two modes are strongly modified by increasing the gap width, where the resonance wavelengths shift to the higher energies simultaneously, and the subradiant mode shifts more rapidly compared with that of the electric dipole mode. When the gap width is about 90 nm (the red line, Fig. 3), the two interfering modes are almost degenerate with each other, and an EIT-like resonance appears in the spectrum. By further enlarging the gap width (G > 90 nm), the energy detuning increases, and there are asymmetric Fano resonances.
In addition to the resonance energy, the biggest difference with that of the perfect disk arrays is that the line widths are far more narrower for the split disk arrays. When the gap width is small (e.g. G = 10 nm), there is a minor change of the line width compared with the perfect disk array (the black lines, Figs. 2 and 3). However, the line width is reduced significantly with the increasing of the gap width. This result indicates that the coupling between the interfering modes can be modified by adjusting the split gap, and resonances with high Q-factor can be achieved with the dielectric split disk arrays.
In order to better show the reduction of the line width, the several calculated extinction spectra around the Fano resonances are magnified and shown in Fig. 4(a) (the open points).
Fig. 4 (a) Several magnified extinction spectra (open points) and fitted spectra (solid lines) around the subradiant mode for the split disk arrays. (b) Electric (upper panel) and magnetic (lower panel) field enhancements distributions around the EIT peak spectral positions of the split disk array with gap width G = 90 nm and (c) G = 110 nm.
The line width is less than 1 nm when G = 90 nm, and it decreases rapidly with the increasing of the gap width, where the line width is less than 0.01 nm when G = 110 nm. Then, the line width increases by further enlarging the gap width when G > 110 nm. The reduction of the line width indicates that the incident energy can be confined more effectively by using the split disk arrays. Figure 4(b) shows the near-field enhancement distributions around the EIT peak for the split disk array with G = 90 nm. The maximum electric field enhancement can be larger than 50. When the gap width is enlarged to 110 nm (Fig. 4(c)), the near-fields are further enhanced due to the improved energy confinement, and the enhancement is more than one order stronger than that of G = 90 nm. What’s more important is that the enhanced electric fields are extended outside of the dielectric nanoparticles for the split disk arrays, and the optical interactions with external environments are expected to be strongly enhanced, which would be useful for many applications.
Next, the oscillator model is used to quantify the line width and the Q-factor of the subradiant mode for the split disk arrays, and the several fitted spectra are shown in Fig. 4(a) (the solid lines). The blue and red circular points in Fig. 5(a) represent the evolution of the line width and the Q-factor by adjusting the gap width, respectively. The line width is more than 7 nm when the gap width G = 10 nm, and the corresponding Q-factor is about 190. The line width decreases to about 1 nm when G = 70 nm, resulting in a Q-factor of about 1217. Further enlarge the gap width, the line width decreases rapidly, where it is less than 0.002 nm when G = 110 nm, and the corresponding Q-factor is approaching to 106, which means that the coupling of the two interfering modes would be approaching to zero in this case. When G > 110 nm, the line width increases with the increasing of the gap width. For example, the line width increases to about 1 nm when G = 150 nm, and the corresponding Q-factor is reduced to about 1277.
Fig. 5 (a) Variations of the line width (blue points) and the Q-factor (red points) of the subradiant mode versus the gap width for the split disk arrays. (b) Maximum near-field enhancements for the split disk arrays with different gap width. (c) Electric field enhancements along the x- and (d) the y-axis indicated by the red lines in the insets.
Figure 5(b) represents the variations of the maximum near-field enhancement. As expected, the enhancement is relatively weak when the gap width is small, e.g., the maximum electric field enhancement is about 10 when G = 10 nm. With the improved energy confinement, the near-fields are enhanced dramatically with the increasing of the gap width. In accordance with that of the Q-factor, the near-field enhancement reaches the maximum when G = 110 nm, where the maximum electric and magnetic field enhancements are about 654 and 2508, respectively. In addition to the maximum field enhancement, the spatial variations of the field enhancements are small around and inside of the split disks. Figure 5(c) and 5(d) show, respectively, the electric field enhancements along the x- and y-axis (see the red lines in the insets of Figs. 5(c) and 5(d)), and the electric field enhancements around the gap area are strong and uniform. These results indicate that high Q-factor and strong near-field enhancements around and inside of the nanoparticles can be realized by using the dielectric split disk arrays.
In the above studies, the disk arrays are supposed to be embedded in a homogenous medium. Nevertheless, the sharp EIT-like resonance still can be excited with the present of a substrate. Figure 6 represents the transmission spectrum of a silicon split disk array that is placed on a semi-infinite silica substrate (nsub = 1.45) and embedded in air (nair = 1.00), where the periodicity p = 690 nm, the disk radius R = 300 nm, the thickness h = 280 nm, the gap width G = 90 nm, and the incidence propagates from the side of the substrate. A sharp EIT-like resonance appears around 1337 nm, the line width is about 2.7 nm, and the corresponding Q-factor is about 495. The Q-factor can be further enlarged by careful engineering the geometry parameters.
Fig. 6 Transmission spectrum of a silicon split nanodisk array placed on a silica substrate and embedded in air, where the inset shows the magnified spectrum around the EIT-like resonance.
5. Enhanced THG with split disk arrays
The nonlinear source for harmonic generation is proportional to the field enhancement at the incident wavelength [11,12,46,54–56]. Since the incident energy can be effectively confined within the dielectric nanoparticles associated with the nonradiating anapole modes and the collective oscillations of the arrays, it is expected that the dielectric disk arrays could be used for enhanced nonlinear effects, and the third harmonic generation (THG) with the designed structures will be investigated in this section.
The solid and dashed lines in Fig. 7(a) represent, respectively, the nonlinear transmission and reflection spectra of the split disk arrays with different gap width. The third order nonlinear susceptibility χ(3) of silicon is supposed to be 2.7910−18 m2/V2 [45], a Gaussian pulse is used as the incidence, the pulse length is 150 fs, the peak field amplitude is 1.55108 V/m, and the center wavelengths match with the EIT peaks of each arrays. It is found that there are relatively strong third harmonic emissions at the corresponding nonlinear wavelengths, and the emission intensity is governed by the gap width.
Fig. 7 (a) TH emission spectra of the disk arrays with different gap width, where the incident wavelengths match with the subradiant modes of each arrays, and the solid and dashed lines represent the transmission and reflection spectra, respectively. (b) The calculated THG conversion efficiency against the gap width of the split disk arrays.
The THG conversion efficiency can be quantified by using,
where PTH is the power around the TH wavelength (with the summation of the transmitted and the reflected fields), and PInc is the power of incident field. The evolution of the THG conversion efficiency versus the gap width is represented in Fig. 7(b). It is interesting to find that although there is an extremely high Q-factor for the split disk array with G = 110 nm, the THG conversion efficiency is weaker than the arrays with a narrower gap width, where the conversion efficiency is about 2.610−7 when G = 110 nm, and it increases to about 1.610−5 when G = 10 nm. This result could be understood by considering the wavelength dependent field enhancements. Although the field enhancements are very strong when the incident wavelength exactly matches with the subradiant mode, the enhancements decrease rapidly for the off-resonance conditions. For example, the maximum electric field enhancement is larger than 600 for the subradiant mode when G = 110 nm (Fig. 4(c)). However, the maximum enhancement decreases to about 5 when the wavelength is about 0.2 nm away from the subradiant mode (not shown here). Since the pulse width is 150 fs, there is a relatively broad wavelength span for the incidence (~15 nm). As a result, the nonlinear source for the THG can only be strongly enhanced around a very narrow spectral range when G = 110 nm. On the contrary, the maximum field enhancement for the array with G = 10 nm is reduced, but the overall field enhancements around a broad spectral range can be stronger than that of G = 110 nm, which may lead to the stronger THG conversion efficiency.6. Conclusion
In conclusion, this study shows that high Q-factor resonances and strong near-field enhancements around and inside of the nanoparticles can be achieved simultaneously by using dielectric split disk arrays. Due to the collective oscillations mediated by near-field coupling between the unit cells, radiative damping can be suppressed by using perfect disk arrays, which leads to an enlarged Q-factor for the subradiant mode compared with that of the single disk. When the resonance energies of the interfering electric dipole and the subradiant modes are almost degenerate with each other, EIT-like response is observed in the spectrum. Furthermore, it is shown that instead of leading to additional radiation losses, the coupling between the two interfering modes can be modified by introducing a split gap. The Q-factor of the subradiant mode is significantly enlarged by adjusting the gap width, and the maximum Q-factor is approaching to 106. As a result, there are strong near-field enhancements around and inside of the dielectric nanoparticles caused by the improved energy confinement, and it is expected that the dielectric split disk arrays could be useful for many applications such as enhanced nonlinear effects, biosensing and nanolasers.
Funding
National Natural Science Foundation of China (NSFC) (11574228, 61471254, and 11304219); Natural Science Foundation of Shanxi Province (201601D021005); Project of International Cooperation of Shanxi Province (2015081025); Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi; San Jin Scholars Program of Shanxi Province.
Acknowledgments
We thank Prof. Yuncai Wang and Prof. Xudong Fan in Key Lab of Advanced Transducers and Intelligent Control System for the helpful discussions.
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