Typical plasmonic resonators，such as metallic Nano-Antennas（NAs），have features of large Scattering or Absorption Cross-Section（SCS or ACS）as well as strongly confined Electric Field（EF）due to the Localized Surface Plasmon Resonance（LSPR）. The large ACS makes them useful in photocurrent enhancement^［1］，photothermal therapy^［2］，upconversion emission^［3］，etc.，while the strongly confined EF makes them useful in Raman spectra enhancement^［4］，quantum information processing^［5］，nonlinear plasmonic imaging^［6］，etc.

Recently，plasmonic-photonic hybrid resonators have drawn much attention. Some previous works demonstrated that when plasmonic resonators were introduced into typical optical resonators，such as Fabry-Perot microcavities^［7-10］，Photonic Crystal（PC）microcavities^［11-19］ and whispering gallery modes microcavities^［20-24］，the Purcell factor Q/V（the ratio of quality factor to mode volume）of the hybrid resonators can be efficiently improved as compared to pure optical resonators，benefiting from the deep subwavelength confinement and ultra-small mode volume of LSPR in plasmonic constituents. For examples，ZHANG Hongyu et al. proposed a hybrid system consisting of a two-dimensional（2D）PC microcavity and plasmonic bowtie NAs with an ultrahigh Q/V of 8.4 × 10⁶ （λ/n）^-3［15］.

Different from previous works which are focused on the contribution of plasmonic resonators on optical resonators，in this paper，we are focused on the other side，i.e.，the contribution of optical resonators on plasmonic resonators. It is well-known that plasmonic resonators usually show a broad LSPR resonance in the ACS spectrum due to the damping loss of metal. Here we will demonstrate that much stronger and sharper ACS spectrum can be generated and modulated on purpose in the plasmonic-photonic hybrid resonators. Meanwhile，we will demonstrate that obviously enhanced EF intensity can be obtained in the hybrid system.

Fig. 1 shows the schematic diagram of the plasmonic-photonic hybrid resonator consisting of a 2D PC microcavity and Au disk NAs. The 2D PC microcavity is made of triangular-lattice arranged air holes on a 160 nm thick Si₃N₄ membrane（n=2.1）. Three air holes in the center are removed to construct an L3-type line defect. The lattice constant and diameter of air hole are a = 280 nm and D = 0.64a. Two air holes near the line defect are moved outward with a distance of d_x = 0.15a. The Au disks are placed on the top surface of the line defect. The radius of Au disks is represented by r. The distance between Au disks and the thickness of Au disks are fixed to be 24 nm and 30 nm，respectively. The relative dielectric constant of Au is fitted based on the experimental results of Johnson and Christy^［25］.

The Finite Difference Time Domain（FDTD）method executed on the commercial software ‘Lumerical FDTD’ is employed to calculate the ACS and EF distribution. In our simulation，the polarization of the dipole or plane-wave light source is along the y direction. The dipole light source is placed in the center of the PC microcavity and 80 nm away from the bottom of the Au NA. The plane-wave light source is placed 140 nm away from the top of the Au NA. The perfectly matched layer boundary conditions are applied along x，y，and z directions. A rough spatial mesh is set as 10 nm for the whole structure，while a fine spatial mesh is set as 0.5 nm near the Au disks and PC microcavity along all three directions. The simulation region is 3 200 nm along the x direction and 2 200 nm along the y direction，respectively. The high precision of the meshes we set ensures the accuracy of the calculation results. The dipole light source is used to calculate the spontaneous emission spectrum of the PC microcavity and the EF distribution of the system. It should be noted that the EF intensity calculated by using the dipole light source is usually much higher than using the plane-wave light source，while the obtained physical law is consistent. The plane-wave light source is used to calculate the ACS，i.e.，σ_ACS = P_abs/I₀，where I₀ is the incident light intensity and P_abs is the summed absorption power of the monitors around the structure.

We first studied the basic resonant properties of the PC microcavity and a single Au NA. As shown in Fig. 2（a），the spontaneous emission of the emitter in the L3 type PC microcavity is simulated with a dipole light source，and it has two resonant modes in contour map of spontaneous emission. The resonant wavelength of each eigen mode is proportional to the lattice constant of PC microcavity. As a comparison，under the plane-wave light source，the Au disk presents a single resonant mode in its contour map of ACS spectrum，where the resonance wavelength and linewidth are found proportional to the radius of Au disk［Fig. 2（b）］. If we set the resonant wavelength at 657 nm，the lattice constant of the PC microcavity and the radius of Au disk are 280 nm and 31 nm respectively. The corresponding spontaneous emission spectrum of the PC microcavity shows two sharp resonant peaks at 657 nm and 576 nm respectively［Fig. 2（c）］，while the corresponding ACS spectrum the Au disk shows a LSPR mode at 657 nm with a much broader linewidth.

Then we studied the resonant properties of the hybrid resonator constructed by the PC microcavity and a single Au NA located at the center of the line defect. It can be clearly seen from Fig. 3（a）that，when the PC microcavity is introduced，the hybrid resonator shows obviously enhanced ACS than a single Au NA. On one hand，we can observe a Fano-like lineshape with spectral splitting at 657 nm. We use the following Fano resonance model to fit the ACS spectrum σ_ACS^［26-28］

where Ω is the dimensionless frequency，q =（ω_LSPR-ω_P）/（ħΓ_LSPR）is the Fano parameter to determine the spectral shape，ω_LSPR = 1.887 eV（657 nm）and Γ_LSPR are the frequency and linewidth of the LSPR respectively，ω_P = 1.887 eV（657 nm）is the frequency of the PC microcavity's resonant mode，g_LSPR，P is the coupling strength between the LSPR and the PC microcavity's resonant mode. In our fitting，Γ_LSPR = 213 meV according to the ACS spectrum of single Au disk in Fig. 2（d）. g_LSPR，P can be obtained by least square fitting. The fitted result represented by the blue solid line in Fig. 3（a）is almost consistent with the simulated one. The slight difference is due to the fact that the mathematical model used for fitting only considers the resonant mode of 657 nm. In particular，the fitted coupling strength g_LSPR，P（20 meV）is much smaller than the linewidth of the LSPR，which means the Fano-like lineshape at 657 nm satisfies the weak coupling criterion ^［27］.

On the other hand，the hybrid resonator shows a very sharp ACS peak at 576 nm. Considering the LSPR is far away from this position，such sharp peak should be directly related to the PC microcavity's sharp resonant mode at 576 nm. In other words，the PC microcavity's eigen mode can be directly coupled into the Au disk's broad ACS spectrum although the PC microcavity shows much lower ACS than the Au disk. To the best of our knowledge，such sharp ACS peak contributed by PC microcavity has never been reported in previous works related to the PC-NA hybrid systems^［11-19］.

In order to gain more insight behind the ACS spectrum of the hybrid resonator，we used the dipole source to simulate the EF distributions of the hybrid system as well bare Si₃N₄ substrate，bare PC microcavity，and single Au disk for a comparison purpose. The results are presented in Figs. 3（b）~（f）. The EF of PC microcavity is mainly distributed in the centre of the line defect at 657 nm and at both ends of the linear defect at 576 nm，while the EF of single Au disk is mainly distributed in the upper and lower edges at 657 nm. Apparently，the PC microcavity's fundamental mode at 657 nm is more easily coupled with the LSPR than the higher-order mode at 576 nm due to the superposition of EF distribution. As shown in Fig. 3（f），when the LSPR and the PC microcavity's fundamental mode are coupled，the EF is still strongly localized near the Au disk with maximum EF intensity of 3 654，which is nearly 5 times that of single Au NA，48 times that of the bare PC microcavity，and 900 times that of the bare Si₃N₄ substrate.

We next studied the resonant properties of the hybrid resonator constructed by the PC microcavity and Au NAs. Fig. 4（a）and 4（b）show the ACS spectra of the hybrid resonator containing three and five Au NAs，respectively. As compared to the singe-NA model in Fig. 3（a），the ACS of the hybrid resonator is gradually enhanced with the increase of Au NAs，but the responses at 657 nm and 576 nm are different. On one hand，dip and Fano-like lineshape at 657 nm becomes more and more unconspicuous. Actually，due to the coupling among Au NAs，the linewidth of the Au NAs' ACS resonance will be gradually broader［see Figs. 3（a），4（a），and 4（b）］，which can result in the decrease of coupling strength g_LSPR，P at resonant match position 657 nm and thus destroy the formation of Fano-like lineshape in the ACS of the hybrid resonator. The fitted results show that the coupling strength g_LSPR，P of the three-NAs model（or five-NAs model）is reduced to about 10 meV（or 5 meV）that is only half（or quarter）of the single-NA model. On the other hand，the sharp peak becomes more and more obvious at 576 nm. This response is believed due to the superposition of each Au NA coupled with the PC microcavity's higher-order mode at 576 nm. On the other word，the coupling among Au NAs can also improve the coupling strength between the Au NAs and PC microcavity at non resonant match position 576 nm and thus improve the intensity of sharp peak in the ACS of the hybrid resonator.

The corresponding EF distributions at 657 nm of three-NAs and five-NAs models before and after coupling are presented in Figs. 4（c）~（f），from which we can find two interesting behaviours. On one hand，as compared to the single-NA model［Figs. 3（e）and 3（f）］，the maximum EF intensity and amplification times after coupling are gradually decreased as the number of Au NAs increases. For the three-NAs and five-NAs models，the maximum EF intensity after coupling is amplified by about 4 times and 2 times，respectively. On the other hand，the spatial distribution of EF in Au NAs is quite different before and after coupling. For the three-NAs model，the EF is mainly confined in the centre Au disk before and after coupling［Figs. 4（c）and 4（e）］. For the five-NAs model，the EF is mainly confined in the centre Au disk before coupling，while after coupling the main EF is found mainly confined in both the centre Au disk and its next nearest neighbour disks［Figs. 4（d）and 4（f）］. Such different spatial distributions of EF in different Au NAs before and after coupling are obviously determined by the EF's spatial distributions of the PC microcavity's fundamental mode at 657 nm［Fig. 3（d）］. The amplification and spatial distribution rules of EF after coupling at 576 nm are similar to above rules at 657 nm. The maximum EF intensity increases from 275 to 1 247 for the three-NAs model and from 288 to 1 030 for the five-NAs model. Meanwhile，it should be noted that although the ACS at 576 nm is larger than that at 657 nm in Fig. 4（b），the maximum EF intensity at 576 nm is still lower than that at 657 nm，which means there is no corresponding relationship between the total ACS and maximum EF for the hybrid resonator. This is because the total ACS is the sum of each NA's ACS while the maximum EF is not dependent on the number of NAs.

At last，we investigated the influence of the resonance wavelength detuning（Δλ）between the Au NA and PC microcavity on the coupling properties of the hybrid resonator. In particular，the resonance wavelength detuning is defined as zero at 657 nm and it can be changed by modulating the radius of the Au NA（r）. Meanwhile，we establish the following equation to describe the EF coupling efficiency of the hybrid resonator at 657 nm.

where E_NA，E_PC，and E_coupled represent the EF intensity of the Au NA，PC microcavity，and hybrid resonator at the same coupling position. k represents the EF coupling coefficient.

As shown in Fig. 5（a），for the single-NA model，it can be found that the coupling coefficient is not the highest one at Δλ = 0 nm. Actually，it will gradually increase as Δλ increases along the negative direction，which means a smaller Au NA corresponds to a higher coupling coefficient. For the three-NAs or five-NAs model，the Δλ-k relationship is similar to the single-NA model. Here we only present the coupling coefficient of different NAs when Δλ = 0 nm as illustrated in Fig. 5（b）. For the three-NAs model，the central NA shows a larger coupling coefficient than its two neighbor NAs，while for the five-NAs model，the central and its nearest two neighbor NAs show larger coupling coefficients than its two neighbor NAs. Considering the EF distribution of PC microcavity in Fig. 3（d），it is not difficult to make a conclusion that the EF coupling coefficient is mainly determined by the PC microcavity's EF intensity at the coupling position，i.e.，higher EF intensity at the coupling position will result in larger EF coupling coefficient.

Meanwhile，as shown in Fig. 5（a），for the single-NA model，it can be found that the maximum EF intensity is not the highest one at Δλ=0 nm. It can reach another peak（5 893）at Δλ=-27 nm when the radius of the Au NA is 21 nm. In particular，the EF enhancement factor at Δλ = -27 nm is three order higher than that of the single Au NA by calculating the fourth power of the maximum EF intensity ratio at 657 nm between the hybrid resonator and the Si₃N₄ substrate.

The corresponding normalized ACS spectrum of the hybrid resonator at Δλ=-27 nm is shown in Fig. 5（c）. As compared to the case of Δλ = 0 nm in Fig. 3（a），there is no longer a dip and Fano-like lineshape at 657 nm. Instead，we can observer two sharp peaks at both 657 nm and 576 nm. The dependencies of these two sharp peaks’ linewidth and intensity on the resonance wavelength detuning are shown in Fig. 5（d），from which we can observe almost linearly relationships between them，which are convenient for modulation on purpose in the practical application.

In conclusion，we have systematically studied the ACS and EF coupling properties of a plasmonic-optical hybrid resonator constructed by the 2D PC microcavity and Au NAs. This hybrid resonator has proved obviously enhanced ACS spectrum with Fano-like lineshape and sharp peak as well as much higher EF enhancement factor as compared to the pure Au NA. The sharp ACS peak and the maximum EF intensity of the hybrid resonator are found sensitive to the numbers of Au NAs and can be modulated on purpose by changing the resonance wavelength detuning between the Au NA and PC microcavity. These particular characteristics of the hybrid resonator provide new opportunities for the studies of spectral enhancement，photothermal therapy，nonlinear optics，etc.