Turning whispering-gallery-mode responses through Fano interferences in coupled all-dielectric block-disk cavities

Xiao-Jing Du; Xu-Tao Tang; Bo Xie; Lin Ma; Ma-Long Hu; Jun He; Zhong-Jian Yang

doi:10.1364/OE.500562

1. Introduction

In recent years, high-refractive-index dielectric resonators have been widely investigated due to their efficient optical responses with low material losses [1–3]. Besides, dielectric resonators not only exhibit electric resonances as that in plasmonic nanoparticles but also readily support magnetic resonances, which originate from excited circular displacement currents of electric fields [4–11]. These features have enabled various nanophotonic applications such as metasurfaces [12–15], enhanced nonlinear optics [16], and optical nanoantennas [17]. Near-field couplings between dielectric resonators can occur when they are close enough [7,18–23]. The corresponding far-field and near-field responses can be significantly modified. Fano resonances have been observed in many coupled dielectric resonators such as oligomers and nanodimers [10,19,21,22,24–28] as well as in single particles [29–31]. Fano resonances usually result from the interactions between a bright broad mode and a dark narrow mode [32–34]. The resonant modes of (sub)wavelength-scale dielectric resonators are usually low-order electromagnetic modes such as electric/magnetic dipolar resonances [11,18,20], toroidal modes [8,35], and supercavity modes [36].

The quality (Q) factors of common electromagnetic modes are relatively low (∼10¹), while supercavity modes can achieve a higher value (∼10²). Recently, it has been shown that subwavelength dielectric resonators of high refractive index (∼3.5) can support whispering gallery modes (WGMs) of high enough Q factors (∼10⁵) and the corresponding electromagnetic near-field enhancements can reach more than ∼10² [37]. WGMs are well-known in microcavities, and they have wide applications such as lasers [38–44] and sensors [45–47]. In microcavities, the refractive index is low or/and the thickness of a disk is much smaller compared with its diameter. Thus, it usually requires high enough order of WGM to achieve a high Q factor. This will make the structure much larger than its working wavelength. For subwavelength cavities, by choosing a high refractive index (∼3.5) and optimized geometries one can also obtain a WGM of high enough Q factor (>10⁵) [37,48,49]. In nanophotonics, the high Q feature of subwavelength dielectric resonators can find applications in enhanced light-matter interactions [48,50]. Moreover, they can also provide a unique platform for investigations of cavity quantum electrodynamics [51,52], topology and exceptional points [39,53–60]. The resonant near-field enhancements of high-Q WGMs increase with the Q factor. However, the total optical responses of WGMs, namely optical responses integrated over frequency, usually become relatively weaker with the Q factor [37,50]. Furthermore, the high-Q feature is usually induced with strict requirements for the geometric and excitation symmetries. For example, WGMs in dielectric disks can be excited by a transverse magnetic (TM) or transverse electric (TE) plane wave. On the other hand, it can be verified that no WGM or only very weak WGM can be excited when the incident direction is changed. Especially, no WGM of a disk can be excited with normal incidence. Thus, it is impossible to excite the WGMs of a disk by a normal incident plane wave in experiments.

In this work, we will show theoretically that WGMs in subwavelength dielectric cavities can be significantly turned through their interactions with other electromagnetic modes. We consider a coupled cavity composed of a block and a disk by numerical simulations. The block supports broad low-order electric/magnetic multipole modes which can be easily excited by normal incidence, while the disk holds narrow WGMs that cannot be excited by the same source. Fano resonances occur in the coupled systems due to the near-field interactions between the block and the disk. Correspondingly, efficient energy transfers occur from the block to the WGMs of the disk. The responses of WGMs can be tuned by varying the geometries of the coupled system. The WGM responses of coupled systems can even exceed that of the individual disk. A coupled oscillator model (COM) is also applied to analyze such a system. The results based on COM agree well with those from the direct simulations.

2. Method

Numerical simulation methods. The cross section spectra were obtained by a commercial time-domain electromagnetic solver (Ansys Lumerical) based on the finite difference time domain (FDTD) method. The excitation source is a total-field scattered-field plane wave. The mesh size was determined using the default Lumerical conformal mesh technology that supports subcell features. The nanogap region was overridden with a uniform mesh (Δx, Δy, Δz) = (10 nm, 10 nm, 2 nm) to improve the spatial resolution. The refractive index of the dielectric resonator is n = 3.5. The surrounding index is n = 1 for simulations. Additionally, a frequency-domain solver (COMSOL Multiphysics) based on the finite-element method (FEM) was used to calculate the electromagnetic near fields of high-Q whispering gallery modes. Free tetrahedral meshes were employed in the FEM simulation region while mapped and swept meshes were used in the perfectly matched layer. In order to obtain accurate results, the maximum mesh size was set to be 100 nm. In both the FDTD (FEM) simulations, perfectly matched layers were set in the x, y, and z directions.

The advantage of the finite-difference time-domain (FDTD) method is that it can obtain the simulation results of a wide spectrum with one calculation. The disadvantage is that when the structure is a high-Q cavity, the electromagnetic field decays slowly in the FDTD simulation, so it takes a long time to simulate with an auto shutoff setting (auto shutoff value 10⁻⁶ in our calculations). The finite element method (FEM) can calculate the optical responses at a single frequency much faster than FDTD for a high-Q mode. On the other hand, it also takes much longer time to obtain the spectral position of a high-Q mode with only the FEM method as this requires lots of calculations with different frequency points. If the spectral position of a high-Q WGM is known, it takes no more than several hours to obtain the optical responses around this mode. Therefore, the two methods can be combined. First, in the FDTD simulation, a broad spectrum can be obtained with a low calculation conversion accuracy (auto shutoff value 10⁻⁴). This accuracy is not enough for the near-field and far-field properties of the high-Q WGMs while the resonance positions can be obtained. Then, on this basis, the FEM was used to calculate the near-field enhancements and scattering cross-sections around the high-Q WGMs.

Multipole decomposition method (MDM). The different multipolar modes can be calculated using the multipole decomposition method [35,61] and their contributions to the scattering spectra can be obtained. Cartesian multipole moments can be expressed using the induced currents ${\hat{J}_\omega }({\hat{r}} )\; $as electric dipole moment:

(1)$${p_\alpha } ={-} \frac{1}{{i\omega }}\left\{ {\smallint {d^3}\hat{r}J_\alpha^\omega {j_0}({kr} )+ \frac{{{k^2}}}{2}\smallint {d^3}\hat{r}[{3({\hat{r} \cdot {{\hat{J}}_\omega }} ){r_\alpha } - {r^2}J_\alpha^\omega } ]\frac{{{j_2}({kr} )}}{{{{({kr} )}^2}}}} \right\},$$

magnetic dipole moment:

(2)$${m_\alpha } = \frac{3}{2}\smallint {d^3}\hat{r}{({\hat{r} \times {{\hat{j}}_\omega }} )_\alpha }\frac{{{j_1}({kr} )}}{{kr}},$$

electric quadrupole moment:

(3)$$\begin{aligned} Q_{\alpha \beta }^e &={-} \frac{3}{{i\omega }}\smallint {d^3}\hat{r}[{3({{r_\beta }J_\alpha^\omega + {r_\alpha }J_\beta^\omega } )- 2({\hat{r} \cdot {{\hat{J}}_\omega }} ){\delta_{\alpha \beta }}} ]\frac{{{j_1}({kr} )}}{{kr}}\\ &- \frac{{6{k^2}}}{{i\omega }}\smallint {d^3}\hat{r}[{5{r_\alpha }{r_\beta }({\hat{r} \cdot {{\hat{J}}_\omega }} )- ({{r_\alpha }{J_\beta } + {r_\beta }{J_\alpha }} ){r^2} + {r^2}({\hat{r} \cdot {{\hat{J}}_\omega }} ){\delta_{\alpha \beta }}} ]\frac{{{j_3}({kr} )}}{{k{r^3}}}, \end{aligned}$$

and magnetic quadrupole moment:

(4)$$Q_{\alpha \beta }^m = 15\smallint {d^3}\hat{r}\{{{r_\alpha }{{({\hat{r} \times {{\hat{J}}_\omega }} )}_\beta } + {r_\beta }{{({\hat{r} \times {{\hat{J}}_\omega }} )}_\alpha }} \}\frac{{{j_2}({kr} )}}{{k{r^2}}}$$

where $\omega $ is the angular frequency, k is the wavenumber, c is the speed of light, r is the location, and $\alpha ,\beta = x,y,z.$ The induced electric current density is obtained by using ${\hat{J}_\omega }({\hat{r}} )= i\omega {\varepsilon _0}({{\varepsilon_r} - 1} ){E_\omega }({\hat{r}} )$, where ${E_\omega }({\hat{r}} )$ is the electric field distribution, ${\varepsilon _0}$ is the permittivity of free space, and ${\varepsilon _r}$ is the relative permittivity of the disk and the block. The induced current has a harmonic time dependence exp(−iωt), which is omitted. We used the FDTD simulation to obtain the electric field distributions ${E_\omega }({\hat{r}} )/{E_{inc}}$. ${E_{inc}}$ is the electric field of the incident wave. The ${j_1}({kr} )$, ${j_2}({kr} )$ and ${j_3}({kr} )$ are the spherical Bessel functions of the first, second, and third kinds, respectively. The scattering cross section produced by these multipole moments can be written as:

C_{sca}^{total} = C_{sca}^p + C_{sca}^m + C_{sca}^{{Q^e}} + C_{sca}^{{Q^m}} + \cdots = \frac{{{k^4}}}{{6\pi \varepsilon _0^2{{|{{E_{inc}}} |}^2}}}

(5)$$\left[ {\mathop \sum \limits_\alpha \left( {{{|{{p_\alpha }} |}^2} + \frac{{{{|{{m_\alpha }} |}^2}}}{c}} \right) + \frac{1}{{120}}\mathop \sum \limits_\alpha \left( {{{|{kQ_{\alpha \beta }^e} |}^2} + {{\left|{\frac{{kQ_{\alpha \beta }^m}}{c}} \right|}^2}} \right) + \cdots } \right]$$

where ${p_\alpha }$, ${m_\alpha }$ are the electric dipole moments (ED) and magnetic dipole moments (MD), respectively. $Q_{\alpha \beta }^e$, $Q_{\alpha \beta }^m$ are the electric quadrupole moments (EQ) and magnetic quadrupole moments (MQ), respectively.

Coupled-oscillator model. We consider the dynamics of a pair of classical oscillators coupled by a weak spring [62,63]. These oscillators y₁ and y₂ represent the modes of the block and disk, respectively. They are coupled together through the electric near field with a coupling strength g. The equations of motion for the two oscillators are thus

(6)$${\ddot{y}_1}\left( t \right) + {\gamma _1}{\dot{y}_1}\left( t \right) + \omega _1^2{y_1}\left( t \right) - {g^2}{y_2}\left( t \right) = {F_1}\left( t \right)$$

(7)$${\ddot{y}_2}\left( t \right) + {\gamma _2}{\dot{y}_2}\left( t \right) + \omega _2^2{y_2}\left( t \right) - {g^2}{y_1}\left( t \right) = {F_2}\left( t \right)$$

where F₁ and F₂ represent the normalized forces driving motion of the coordinates due to the external electromagnetic field, and γ₁ and γ₂ represent the losses associated with the modes of the block and disk, respectively. If a WGM of a disk cannot be directly excited, the F₂(t) can be set to be F₂(t) = 0. For incident light with a given frequency ω, the driving force is F₁(t) = Re(F₁e^−iωt). At a steady state, both y₁(t) and y₂(t) will follow this driving frequency as

(8)$${y_1}(t )= \frac{{({\omega_2^2 - {\omega^2} - i{\gamma_2}\omega } ){F_1}(t )}}{{({{\omega^2} - \omega_1^2 + i{\gamma_1}\omega } )({{\omega^2} - \omega_2^2 + i{\gamma_2}\omega } )- {g^4}}}, $$

(9)$${y_2}(t )= \frac{{{g^2}{F_1}(t )}}{{({{\omega^2} - \omega_1^2 + i{\gamma_1}\omega } )({{\omega^2} - \omega_2^2 + i{\gamma_2}\omega } )- {g^4}}}\; .$$

3. Results and discussion

Figure 1(a) shows the schematic of a coupled structure consisting of a dielectric block and a dielectric disk. In the numerical simulations, the incident field of a plane wave is propagating along the z-axis, and the polarization is along the y-axis. The radius and height of the disk are R = 620 nm and h = 720 nm, respectively. The length, width, and height for the block are l = 550 nm, a = 500 nm, and b = 200 nm, respectively. The refractive index of dielectric is taken to be silicon (Si) with n = 3.5. In our work, we chose the geometries to make the corresponding working wavelength fall in the region where Si has a negligible material loss for wavelengths above 1200 nm and its refractive index is about n = 3.5. The surrounding medium is vacuum n = 1. The surface gap distance between the disk and the block is 20 nm.

Fig. 1. (a) Schematic of a coupled structure excited by a plane wave under normal incidence. The polarization of the incident wave is along the y-axis. (b) Optical response spectra of individual and coupled structures. (b-I) The scattering of an individual Si block and the contributions from different multipole modes including ED, MD, EQ, and MQ. The length, width, and height of the Si block are 550, 500, and 200 nm, respectively. (b-II) The scattering spectra of an individual Si disk under excitations along the z-axis (black) or x-axis (red). The disk has a radius of 620 nm and a height of 720 nm. (b-III) The scattering and near field spectra of a coupled system, where the gap between the block and disk is 20 nm. (b-IV) |y₁|² and |y₂|² spectra obtained by the COM. (c) Magnetic near-field distributions of an individual block and a coupled system. The two wavelength positions of the coupled system are marked on the spectrum in (b-III).

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Figure 1(b-I) shows the scattering spectra of an individual Si block (black). The multipole contributions to the scattering of the individual Si block are also calculated. The sum of all the contributions from the different multipole modes [the red line in Fig. 1(b-I)] agrees well with the directly simulated results. The scattering spectrum is dominated by the ED mode, and the other contributions can be ignored. Figure 1(b-II) shows the scattering spectrum of an individual disk with different excitations of a plane wave. A WGM is excited around λ = 1707nm when a z-axis polarized plane wave propagates along the x-axis. Near field results show that the azimuth index of this WGM is m = 5. Similar results have been reported previously [37]. However, no WGM can be excited in our considered region under normal incidence. Note that the resonant wavelength seems counterintuitive based on the well-known estimation formula 2πnR = mλ. The main reason is that the equation should be replaced by 2πnR_m = mλ for a subwavelength cavity, where R_m is effective radius of a WGM mode. In a microcavity, the mode index is high and R_m is close to R. For a subwavelength cavity with a relatively smaller mode index m, the mismatch between R_m and R becomes more obvious (R_m < R). This is verified by the near field distributions shown in Fig. 1(c). As a result, the resonance wavelength blueshifts with decreasing the R_m. This blueshift effect is dependent on the mode index m and the geometries of a disk, namely, its thickness and diameter [37].

Now let us turn to the coupled structures of a block and a disk. Figure 1(b-III) shows the scattering spectrum of a coupled structure. A Fano line shape can be seen on the spectrum indicating effective couplings between the block and disk. The excited WGM in the coupled system is confirmed by the near field distribution (Fig. 1(b-III)). The red line in Fig. 1(b-III) shows the electric field intensity as a function of wavelength for the WGM. Figure 1(c) shows the near field distributions of the individual block and the coupled structure. The magnetic field pattern of the individual block corresponds to the broad ED mode. For the coupled structure, the coupling between WGM and ED mode varies from destructive to constructive interference depending on the wavelength. This is a typical Fano resonance and similar phenomena have been widely reported in nanophotonic systems [27,32].

To further understand the coupling effect of the coupled system, we carried out analytical studies based on a coupled-oscillator model (COM). It is found that this model can well reproduce the coupling phenomenon in the coupled system (Fig. 1(b-IV)). In this model, y₁ and y₂ are utilized to represent the mode responses of block and disk, respectively. The frequency and linewidth of the WGM of disk are input parameters that can be obtained by the FDTD calculations, and they are ω₂= 0.724 eV and γ₂ = 0.1677 meV. Similarly, the frequency and linewidth of the block can also be calculated by FDTD as ω₁ = 0.813 eV and γ₁ = 0.447 eV. The only remaining variable parameters in the model are the coupling strength g and excitation factor F, where F does not affect the lineshape of the system. The spectrum lineshape of |y₂|² agrees well with that of the field intensity |E/E₀|² of the coupled WGM, where one of the most important characters is the linewidth. The linewidth varies with the coupling strength g, and a proper g can be obtained to match the linewidths of |y₂|² and the |E/E₀|².

The magnitude and linewidth of a WGM can be greatly modified by varying the coupling strength between the modes of block and disk. This can be obtained by decreasing the gap between the block and disk in the coupled system. Since the spectral lineshape is determined by the coupling strength g, one can obtain the g as a function of the gap based on the simulation results as shown in Fig. 2(a). The coupling strength g increases with decreasing the gap as expected. Figure 2(b) shows the simulated peak electric field intensity (|E/E₀|²) as a function of the gap (black). The peak value of |y₂|² can be obtained by substituting g into the COM. The results obtained from the two methods agree well with each other. The peak |E/E₀|² first increases and then decreases with the gap size. The peak |E/E₀|² reaches a maximal at a critical coupling strength. Although the peak value of |E/E₀|² may decrease with the coupling strength g, this does not mean the energy transfer from block to disk becomes weaker. In fact, the spectral width becomes larger with g, and the integrated |E/E₀|² becomes larger with g (Fig. 2(c)). Here, the Δλ in the expression |E/E₀|²×Δλ (Fig. 2(c)) corresponds to the full-width-half-maximum of a |E/E₀|² spectrum.

Fig. 2. (a) Simulated Δλ with gap varying from 0 to 150 nm. The corresponding g obtained by the COM is also shown. (b) Simulated peak |E/E₀|² (black solid line) and calculated peak |y₂|² as a function of Gap. (c) |E/E₀|²×Δλ (black solid line) and integrated |y₂|² (red solid line) with Gap varying from 0 to 150 nm.

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Besides the coupling strength g, the response strength of the block (y₁) can also affect the integrated |E/E₀|² of the disk (|y₂|²). The resonant responses of a block can be turned by varying its height. This can be done based on the fact that other resonant modes besides the above ED can also interact with WGM through near field couplings, and the responses of all these modes vary with the height of a block. On the other hand, by varying the height of the block, the coupling strength g also changes and different modes of the block have near-field phase differences between each other. These facts also bring complexes for turning the response of WGM with the height of the block. Figure 3 shows a case with varying the height of a block. The scattering cross section and its corresponding multipole decompositions of an individual block at the working wavelength as a function of its height b are shown in Fig. 3(a). Note that there is a little mismatch between the results of MDM and the directly simulated results. This mainly comes from the fact that the near field distributions inside a cavity used for the MDM are calculated based on the FDTD method. The mesh sizes around a cavity are of finite sizes which induces the mismatch between the two methods. Figure 3(b) shows the peak value of intensity |E/E₀|² and the integrated |E/E₀|² as a function of b. The other geometry parameters are the same as that in Fig. 1. The trend of the WGM response does not always follow the total scattering of the block. A typical point is the one with b = 300 nm. The scattering of the block is very high here, and it is about 3 times larger than that with b = 200 nm. However, the integrated |E/E₀|² is similar to that with b = 200 nm, which is not as large as expected. This can be explained as shown in Fig. 3(c). The block has two main contributions, namely, an MD and an ED. Near field shows that the ED corresponds to a higher order form, namely, electric toroidal dipole (ETD) [61]. The magnetic fields associated with the ETD and MD modes are out of phase, and their induced responses of WGMs will cancel with each other. Another typical point is the one around b = 800 nm, where the main contributions are the combined ETQ and MTD modes (Figs. 3(a) and 3(d)). Similar mode combinations have been reported in other dielectric nanostructures [61]. The bottom part of the magnetic field of the mixed mode is larger than the top one, which is due to the retardation effect. This is a favorable factor for enhancing the WGM response. Thus, it is found that various modes of a block can excite WGM modes through near field interactions, while the phase difference between different modes can also largely affect WGM responses.

Fig. 3. (a) The contributions from multipole modes to the scattering (black) of the individual Si block with different heights. They are ED (red), MD (blue), EQ (green), and MQ (purple). The working wavelength corresponds to the WGM resonant position. (b) Electric field intensity |E/E₀|² (black) and |E/E₀|²×Δλ (red) as a function of the height of the block. (c) Top: magnetic field distribution of ED and MD modes of the block. The height is b = 300 nm. Bottom: Schematic illustration of the overlapping of the magnetic fields associated with the ED (ETD) and MD modes. (d) Electric and magnetic field distributions of the mixed mode composed of ETQ and MTD of the individual Si block. The height is b = 800 nm.

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In terms of the WGM response strength in the coupled system, an efficient way is to provide an effective response from the block (y₁). However, this is not intuitive as indicated by the results in Fig. 3, where different modes in a block may have destructive contributions to the response of a WGM. On the other hand, calculations show that by choosing a smaller width, the electric and magnetic multiples do not show destructive contributions. The whole effective response of the block increases with its height (Fig. 4(a)). Thus the response of a WGM can continuously increase with the height of a block (Fig. 4(b)), and it can easily exceed that of an individual disk with a TM excitation.

Fig. 4. (a) The scattering of an individual Si block and the contributions from different multipole modes including ED, MD, EQ, and MQ. The length, width, and height for the Si block are 600, 220, and 2000nm, respectively. (b) Electric field intensity |E/E₀|² (black) and |E/E₀|²×Δλ (red) as a function of the height of the block.

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The properties of WGMs can also largely affect their responses in the coupled system. Figure 5(a) shows the results with varying the height of the disk from 550 to 900 nm. The considered WGM is the same as before, and the corresponding resonant position varies from 1640 to 1744nm. This variation is much smaller than the spectral width of the block. Thus, the contributions from block (or y₁) can be approximately taken the same here. The response of the WGM reaches maximal when its height is around the radius H = R. This is mainly due to the fact that the coupling g between WGM and the block varies significantly and reaches maximal around H = R. There are two factors that affect the coupling strength g in this system, namely, the relative field concentration of WGM and the distance between the WGM field and block. For the case of H > R, the distance between the WGM field and the block grows significantly. For the case of H < R, the field concentration of WGM becomes weaker, which is reflected by the relatively wider spatial distribution of the field (Fig. 5(b)). The combination of these two factors makes maximal coupling strength g around H = R. In addition to the coupling strength g, the decay rate of a WGM γ₂ can also affect the response of WGM. Figure 5(c) shows the WGM response as a function of γ₂ obtained by using the COM, where the coupling strength g is fixed for simplicity. The response of WGM decreases with the γ₂. It is known that under the situation of H < R, the γ₂ increases with decreasing H. On the other hand, the γ₂ keeps almost the same when the H is H > R [37]. Thus, the γ₂ also affects the response of WGM in the coupled system under the situation of H < R.

Fig. 5. (a) Simulated electric field intensity |E/E₀|² (black) and |E/E₀|²×Δλ (red) as a function of the height of a disk. The radius is 620 nm. (b) The magnetic field of an individual disk as a function of distance d from the surface. The heights H are 550, 720, and 900 nm. The inset shows the distance d from the disk surface. (c) The calculated peak value and integrated of |y₂|² vary with the full-width-half-maximum of the γ₂ spectrum. The coupling strength g is fixed. The red area denotes the variation of decay rate corresponding to the disks in (a).

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4. Conclusions

In summary, we have investigated an all-dielectric block-disk coupled system. Efficient near field interactions between broad low-order of electric/magnetic multipole modes and narrow WGMs can occur and induce Fano resonances. Both simulation and analytical methods show that the corresponding WGM responses can be largely modified by turning the modes of a block and their relative phases, the coupling strength g, and the decay rate of WGM γ₂. The coupling strength g is highly dependent on the disk-block gap size and the geometries of the disk and block. The WGM responses in terms of integrated field intensity of the coupled system can exceed that of the individual disk. In addition, such a configuration will also facilitate the experimental excitation of WGMs by a normal incident plane wave. This is due to the fact that the excitation of WGMs of an individual disk requires strict configuration and they cannot be excited by normal incidence. Note that the responses of WGMs are enhanced with broadened spectra here, which means the decreasing of the Q factor of a coupled WGM. This may be a disadvantage for some applications that requires high enough Q factors. Our analytical results also show us how to further enhance the responses of a WGM in other similar systems. For example, if one wants to optimize the value of resonant |E/E₀|² (not the integrated |E/E₀|²) in a coupled system, the best way is to keep a critical coupling strength and increase the strength of y₁. Our results could find many photonic applications, including enhanced light-matter interactions.

Funding

National Natural Science Foundation of China (11704416); Natural Science Foundation of Hunan Province (2021JJ20076).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Turning whispering-gallery-mode responses through Fano interferences in coupled all-dielectric block-disk cavities

Abstract

1. Introduction

2. Method

3. Results and discussion

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (10)

Optics Express