Heterogeneous photonic molecules (HPMs) formed by coupling two or multiple photonic cavities serve as a versatile platform for studying phenomena such as strong coupling^[1,2], Rabi splitting^[3], and Fano resonances^[4–6], which are critical for developing advanced photonic devices. The interaction between photonic modes in coupled cavities of the HPM can result in the formation of a super mode^[7–9], creating new opportunities to tailor optical properties and enhance light–matter interactions or modify the far-field radiation^[10,11]. These advancements have profound implications for applications including low-threshold lasers^[12–14], optical switches^[15,16], and quantum information processing^[17,18].

Photonic crystal nanocavities (PCNCs) and Mie nano-resonators (MNRs) are two prominent dielectric nanocavities for achieving robust light confinement and diverse electromagnetic modes in the nanoscale. PCNCs fabricated in a dielectric slab are ideal for realizing strongly localized fields and high densities of photon states, leveraging their high-quality (Q) factor and small mode volumes (Vm)^[19,20]. MNR, which is formed by a dielectric nanoparticle with high refractive index, supports both electric and magnetic multipole resonance modes, providing versatile tools for manipulating light at subwavelength scales^[21,22]. The coupling between an MNR and a PCNC represents an innovative approach to constructing HPMs. This coupling harnesses the unique strengths of both nanocavity systems: the strong field confinement of PCNCs and the rich modal properties of MNRs.

In this work, we demonstrate the strong mode coupling in an HPM that hybridizes an MNR and a PCNC. The coupled mode theory (CMT) framework is developed for the HPM with an energy-level analogy to describe this mode interaction, which treats the MNR as a continuum-like state and the PCNC as a discrete bound state. This CMT-based model explains the emergence of hybrid near-field modes, Fano resonances, and Rabi splitting. We validate these theoretical predictions through numerical simulations that confirm the expected strong coupling behavior. Collectively, our theoretical and simulation results highlight implications of the proposed HPM for photonic technologies, including enhanced nonlinear optical effects, improved emission directivity, and new opportunities for nanoscale light sources, sensors, and quantum emitters^[23–25].

Figure 1(a) shows the diagram of the proposed HPM. The MNR is designed by a silicon nanosphere with various radii of r. The PCNC is formed by cutting two inner adjacent air holes into a D shape with a D′=0.8D in a silicon hexagonal photonic crystal lattice^[26]. The thickness of the PCNC slab is set as d=220nm with the lattice constant a=454nm, while the air hole diameter is D=0.56a. The MNR is laid on the surface of the PCNC center with a thin silicon oxide (SiO2) spacing layer to diminish the destructive perturbation caused by the MNR with a high refractive index. The thickness of the SiO2 layer could be designed to control the coupling strength between the two resonators. Figure 1(b) shows the resonance spectra of the PCNC and the magnetic modes of the MNR, which are denoted by the magnetic dipole (MD) and quadrupole (MQ), respectively. Note, there are other higher-order magnetic modes, such as the octupole (MO) at an even shorter wavelength, which is not shown in Fig. 1(b). Compared with the narrow linewidth of the resonance mode in the PCNC, the resonance modes of the MNR have a wider linewidth, indicating that there is more overlap in the coupling of each mode. We can achieve maximum coupling by tuning the radius of the MNR to make the resonance wavelength match the resonance wavelength of the PCNC.

To excite the modes of the HPM composed of the MNR and the PCNC, a linearly polarized light is set as the vertically incident light, and the coupled near magnetic field mode [see Fig. 1(c)] of the HPM is obtained by the interference between the MD mode (mode amplitude a1) of the MNR and the resonance mode (mode amplitude a2) of the PCNC, as shown in Figs. 1(d) and 1(e), respectively. According to the CMT, the HPM system is modeled as a coupled oscillator, as shown in Fig. 1(f), and the coupled equation can be expressed as [ddt−(jω1−γ1/2)−jκ12ddt−(jω2−γ2/2)−jκ21][a1a2]=[γ1,in·Sin0],(1)where γi, ωi, and γi,in denote the loss, frequency, and input loss, respectively, and i=1, 2 refers to the MNR and PCNC. Sin is the amplitude of the incident field. κ denotes the coupling coefficient between the MNR and PCNC (κ12≠κ21). It should be noted that γ1,in represents the coupling efficiency of the incident light energy into the system through this mode. In contrast, γ2,in is neglected because the mode of the PCNC cannot be effectively excited by the vertically incident linearly polarized light. This is attributed to the mode symmetry and phase matching conditions of the PCNC, which may not be satisfied by the vertically incident light^[27–29].

The above equation can be converted into algebraic form to solve the scattering response, and the complex amplitude expression about frequency can be obtained after a2 is eliminated: a1(ω)=F(ω)(ω2−ω−iγ2)(ω1−ω−iγ1)(ω2−ω−iγ2)−κ12κ21,(2)where F(ω) represents the equivalent driving force, which is related to the amplitude of the incident electromagnetic field. Analyzing the denominator of the system’s response function reveals that it is a quadratic expression in terms of the complex frequency variable. This implies the existence of two poles, corresponding to the complex frequencies at which the denominator equals zero. These poles represent the eigenfrequencies of the coupled resonance modes within the HPM system. The real parts of these poles determine the resonance frequencies of the hybridized modes, while the imaginary parts, associated with the damping factors γ1 and γ2, indicate the attenuation rates or linewidths of each mode.

In the case of weak coupling or when one of the modes has a low Q-factor in the HPM, the scattering spectrum will show the asymmetric line characteristic of Fano resonance. This asymmetric Fano lineshape is derived from the interference effect, that is, the amplitude of the broadband “background” scattering channel and the narrowband resonance channel is offset. In our proposed HPM, the loss of the MD mode of the MNR is usually large (Q is limited), making the resonance lineshape wide. Herein, the scattering resonance mode of MNR could act as a continuous background scattering. In contrast, the loss of the PCNC mode is very small, supporting a high Q-factor and a narrowband resonance linewidth. When the resonance modes of the MNR and PCNC are coupled and excited together, the background scattering field with a broad linewidth and the resonant scattering field with a narrow linewidth produce a long/destructive interference. This interference leads to peak enhancement on one side and suppression on the other side near the resonance frequency of the PCNC, resulting in an asymmetric Fano-type lineshape. The typical phenomenon is that there is a “dip” or asymmetric broken line near the resonance peak, rather than a simple symmetrical Lorentz-type peak. According to Eqs. (1) and (2), the scattering intensity spectrum of the Fano resonance can be expressed by the standard Fano line formula as I(ω)=(q+ϵ)21+ϵ2,(3)where ϵ=2Δ/Γ, Δ=ω−ω0, and Γ is the linewidth representing the loss. q is the Fano asymmetric parameter, which determines the degree of asymmetry in the spectral line. The value of q depends on the amplitude and phase difference of the two modes. When the cavity mode scattering of the PCNC is stronger than the MNR scattering (wide background) and they are in phase, q>0. If the relative phases of the two modes are opposite, q<0.

To evaluate the Fano resonances in the HPM, the scattering spectra are calculated for the HPMs with different SiO2 spacing thicknesses (coupling gap) between the MNR and PCNC, which could control the coupling strengths between the resonance modes. The results are displayed in Fig. 2(a). The nanosphere radius of the MNR is 197.5 nm, supporting an MD mode centered at the wavelength of λ0=1433nm. For the individual PCNC, the resonance wavelength is λ0=1433nm as well. In the HPM structures with different coupling gaps, the scattering spectra have two significant asymmetric scattering peaks, namely mode splitting, which is the typical result of strong coupling.

With the increase of coupling gaps, the splitting between the two peaks becomes smaller, and the scattering intensity decreases gradually. This phenomenon is caused by the coupling coefficients, coupling losses, and radiation losses of the resonance modes. According to the coupled mode theory, we derive the coupling coefficient κ between the MD mode of the MNR and the resonance mode of the PCNC: κi,j=ω02∭V[ε(r)−εbg]HiHj*dV,(4)where Hi and Hj are the magnetic fields of the MD mode in the MNR and the resonance mode in the PCNC, respectively. With the increase of the coupling gap, the mode overlapping integral between the MNR and PCNC gradually decreases, resulting in the decrease of κ. The blue curve in Fig. 2(b) displays the variation of κ12·κ21 according to Eq. (4), showing a decreased function with respect to the coupling gap. On the other hand, the distance between the two scattering peaks gradually decreases, which indicates that the coupling strength weakens, as shown in the red curve of Fig. 2(b). The calculated coupling coefficient from Eq. (4) is consistent with the splitting and trend obtained from the scattering spectra shown in Fig. 2(a).

In Fig. 2(c), we show the scattering spectra at several different coupling gaps. The resonance peak of the PCNC is fitted with Eq. (3), obtaining the value of the q parameters. As the coupling gap increases, the linewidth of the scattering peak from the PCNC gradually narrows, and its asymmetry reverses, indicating a gradual decrease in coupling strength. The background scattering of the MNR becomes dominant, as evidenced by the gradual increase in the left peak. Simultaneously, the mode phases of the MNR and PCNC shift from being opposite to being the same (as indicated by q changing from negative to positive). The results indicate that q is influenced by the coupling coefficients κ, suggesting that we can leverage the high sensitivity of the spectrum to the q parameter to achieve ultra-high sensitivity refractive index sensing or dynamic parameter characterization. For instance, when the ambient refractive index simultaneously modulates κ12 and κ21, its sensitivity is approximately κ21/κ12 times higher than that of a single resonator. In addition, the two-mode entangled state under the asymmetric coupling condition can be used to generate correlated photon pairs^[30], and the entanglement degree is related to the coupling asymmetry parameter, that is, κ21/κ12^[31,32]. It is possible to offer a new platform for applications such as on-chip integrated tunable quantum communication and quantum cryptography.

The asymmetric spectral responses not only highlight the tunability of HPMs for practical applications but also reveal the fundamental mode interactions driving such phenomena. In the case of strong coupling, these interactions result in significant mode hybridization and the characteristic frequency splitting of eigenmodes, known as Rabi splitting, which reflects the coupling strength between two cavities directly. This process gives rise to new eigenstates of supermodes in the coupled system, formed by coherent superpositions of the original, uncoupled cavity mode. According to Eq. (1), in the case of a steady state, the complex eigenfrequency Ω of the coupled system can be obtained by solving $$\begin{gathered} ΩS,A+jΓS,A=12[ω1+ω2−j(γ1+γ2)] \\ ±124κ12κ21+[ω1−ω2−j(γ1−γ2)]2, \end{gathered}$$(5)where Γ is the supermode loss and the symmetric (S, corresponding to bonding)/antisymmetric (A, corresponding to anti-bonding) supermodes corresponding to the plus and minus signs, respectively. As ω1≈ω2, we set g=κ12κ21, which yields the mode splitting, Δ=2ℏ4g2−(γ1−γ2)2,(6)which is related to the Rabi frequency. It should be noted that the condition of strong coupling regime is only when g>|γ1−γ2|/2. To explain the split in Fig. 2 and verify the coupling strength of the HPM, the scattering spectra are also calculated with the full wave method, as shown in Fig. 3(a), which plots the scattering spectra as a function of r of the MNR. As the r increases, the frequency of the magnetic resonance modes (such as MD, MQ, and MO) in the MNR matches with the frequency ω0 of the resonance mode of the PCNC, subsequently denoted by the white dashed line.

There are three obviously void crossings that emerge in the frequency, characterized by prominent mode splitting, denoted as Δ1≈50.8meV, Δ2≈44meV, and Δ3≈49.6meV. These mode splittings are typical signatures of the strong coupling of the HPM, implying that the formation of hybridized supermodes is neither purely localized in one cavity nor completely delocalized, but arises from coherent interaction between the resonators. The value of each splitting reflects the strength of coupling between the MNR and PCNC. A larger energy gap indicates stronger interaction and more efficient mode hybridization. The linewidth of the peak provides insight into the intrinsic losses of radiative leakage. The narrow peak corresponds to a high-Q mode with minimal dissipation, whereas broader lines imply higher losses. These spectral features provide a comprehensive picture of both the coupling dynamics and modal lifetimes.

To further elucidate the nature of the coupled states in the HPM, Fig. 3(b) presents the electric field distribution of the supermodes corresponding to each frequency splitting. Each row in the panel corresponds to one of the three mode splitting regions (labeled I–III), with left and right columns showing the anti-symmetric and symmetric modes, respectively. The anti-symmetric modes exhibit out-of-phase field distributions between the two nanocavities, indicating the destructive interference at the interface and the presence of a field node. In contrast, symmetric modes display in-phase field distributions, supporting constructive interference and strong localization at the mode coupling region. The symmetry of these modes plays a crucial role in their coupling to external fields and radiative channels. For instance, anti-symmetric modes often exhibit reduced radiation losses due to symmetry-protected interference, while symmetric modes may couple more strongly to free-space modes, leading to higher radiative decay. Such symmetry-dependent behavior not only governs the Q-factors of the supermodes but also impacts their utility in applications like sensing, filtering, or nonlinear optics, where mode confinement and field overlap are critical.

Although the present study is based on idealized theoretical simulations, we acknowledge that the proposed structure exhibits pronounced sensitivity to the size of the Mie-type resonator, as illustrated in Fig. 3. From a fabrication perspective, achieving precise control over the diameter of silicon nanospheres remains a significant challenge, particularly when employing femtosecond laser ablation techniques. To address this issue in future experimental implementation, we propose substituting the nanosphere with a silicon nanopillar, which can be accurately fabricated and deterministically positioned using well-established top-down nanofabrication techniques such as electron-beam lithography, inductively coupled plasma (ICP) etching, and dry-transfer processes. These methods allow for dimensional control with an accuracy on the order of ±10nm, which is compatible with the requirements for realizing strong coupling in the proposed hybrid photonic system.

The observations of mode splitting and the associated field distributions serve as direct fingerprints of the underlying coupling mechanisms between the resonators. By analyzing the energy separations, linewidths, and modal symmetries, one can obtain both quantitative and qualitative insights into the interaction dynamics of optical modes, the mechanisms of energy redistribution, and the formation of hybridized eigenstates in the case of strong coupling, which also reveals that our HPM system possesses the potential to achieve ultra-low threshold optical switching^[33,34] and generation of specific emission^[35]. The ultra-high-Q and asymmetric coupling of the PCNC can also be utilized to achieve efficient pumping of Mie particle gain media and reduce the threshold power of nonlinearity^[36,37].

Unequivocal demonstration of strong coupling requires the observation of Rabi oscillation in the time domain, in addition to spectral and mode splittings. For this purpose, we performed a calculation of the time evolution of energy in the two nanocavities. Based on Eqs. (4) and (5), it is evident that the parameter g, which is related to the coupling coefficient, will increase or decrease as ω1 is tuned by the change of r of the MNR. More intuitively, the field distribution, i.e., energy distribution, of the supermode will become increasingly localized to either one of the coupled nanocavities with decreasing g, indicating that the photon cannot be exchanged efficiently between the MNR and PCNC. However, due to the radiation loss ΓS,A of supermodels, it provides the channel for transferring and exchanging photons of supermodes between the two nanocavities. The loss of supermodes will directly affect the energy exchange between the two nanocavities, which can be confirmed by considering the intensity variation in either cavity with time as^[38]|Ei|2∝e−2ΓSt|ej(ΩS−ΩA)te−(ΓA−ΓS)t±1|2,(7)where i=1, 2 corresponds to the resonance modes of the MNR and PCNC, respectively. According to this expression, the exchange rate of energy is proportional to the coupling strength, and the efficiency of energy exchange is determined by the difference of loss in the supermode. As the mode loss ΓS=ΓA, the energy can be completely transferred between the two nanocavities of the HPM. From the previous results, it is known that the first anti-crossing point of the HPM occurs at r=197.5nm, where ΓS≈ΓA. By calculating the eigenfrequency of the HMP structure at r=192.5, 197.5, and 205 nm, the specific values of frequencies and losses corresponding to ΓS<ΓA, ΓS≈ΓA, and ΓS>ΓA can be obtained. Note that this result can also be obtained from the scattering spectra. Figures 4(a)–4(c) show the numerical results of the temporal evolution of energy between the MNR and PCNC in the three conditions.

In the time domain of the electromagnetic wave, the integrations of energy density in the MNR and PCNC are calculated, respectively. Figure 4(d) shows that the energy exchange between the MNR and PCNC is consistent with the numerical result of Fig. 3(b). The inset displays the evolution of the electric field distribution of the HPM, which visually demonstrates the evolution process of photon energy exchange in the two nanocavities within the picosecond time scale. Due to the presence of intrinsic losses, photons in the HPM system exhibit an exponential decay trend in energy. The photons can be trapped in one cavity with dynamics in time, which can be modulated with external methods.

High-directional radiation is highly desirable for applications in micro- and nano-photonics^[11,39–41]. For a single MNR or PCNC, the drawback of a divergent emission pattern limits its practical applications. When an MNR is coupled with a PCNC, the far-field emission properties of the HPM could be modulated by their near-field couplings. The precise emission pattern of the HPM can vary significantly with the state of coupling.

Figures 5(a1)–5(a5) show the electric field distributions of coupled HPMs when the MNR is located at different positions of the PCNC. There are obvious dipole modes in MNRs, presenting different dipole moments closely related to the locations of MNRs. Figures 5(b1)–5(b5) present the normalized x–y plane far-field emission patterns of the HPM as the MNR shifts from the PCNC center to the left, while Figs. 5(c1)–5(c5) depict the corresponding x–z plane far-field emission patterns. Notably, highly directional emission patterns appear in both the x–y and x–z planes when the x-coordinate of the MNR is −0.25a, −0.75a, and −a relative to the PCNC center. These emission patterns are governed by the dipole moment of the MNR, which is re-excited by the near field of the resonance mode in the PCNC. The ability to tune the far-field radiation direction offers a promising platform for quantum information processing and optical antennas. This tunability can enhance the radiation transition efficiency of quantum dots or defect color centers^[42,43], while precise control over photon emission direction can facilitate the development of high-purity light sources for quantum key distribution^[44].

We have demonstrated a novel HPM consisting of an MNR and a PCNC, exhibiting strong coupling effects and tunable far-field radiation characteristics. Our study highlights the formation of Fano resonances and Rabi splitting, which are key to understanding complex light–matter interactions within the coupled system. Through the control of coupling distance and structural parameters of the MNR, we achieved dynamic modulation of optical properties, including photon transferring and enhanced directional radiation. These results pave the way for advanced applications in quantum optics^[11,45–47], high-efficiency light sources, and ultra-sensitive sensing applications.