Autler–Townes splitting (ATS), a well-known quantum phenomenon observed in atomic and molecular systems [1], finds its analogs across diverse classical domains, including acoustic [2], optical [3], electronic [4], and mechanical [5] systems. While various optical methods have been employed for photon-mediated ATS [6,7], optical microresonators, particularly whispering gallery mode (WGM) resonators, have emerged as a fruitful platform for studying the all-optical analogs of ATS [8]. For the realization of mode splitting, these systems usually operate in the “strong-driving” regime, namely the resonances exchange the energy faster than that it leaks to the external space. The lifted degeneracy between two originally identical eigenmodes results in a doublet structure in the resonant spectrum, as well as a transparency window between the split modes [8]. Notably, coupled WGM microresonators with identical sizes and shapes, such as microspheres [9], microtoriods [10], microdisks [11], microtubes [12], and microrings [13,14], often termed “photonic molecules” due to their analogs of chemical molecule systems [8], have been extensively explored for the generation of ATS and ATS-empowered optical signal processing. Significantly, robust coherent excitation, such as microwave fields, enables the engineering of programmable photonic molecules utilizing the Pockels effect, resulting in high-order ATS with fine splitting into four energy levels [15].

ATS has been extensively studied in WGM resonator-based two-element photonic molecules spanning both optically passive [15,16] and active systems [17–19], as well as parity–time symmetric systems [20,21]. However, prevailing theoretical models often assume a simplistic two-level energy system. In practical scenarios, the idealized notion of pure reflectionless coupling between two cavities is rarely achievable, and the impact of backscattering inside each cavity or at the inter-cavity coupling region cannot be disregarded. Notably, ATS has been reported in single WGM cavities, such as microrings [22–25], microtoroids [26], and microtubes [27], owing to the contribution of backscattering-mediated mutual coupling between originally degenerate clockwise (CW) and counterclockwise (CCW) traveling modes. In such a single-cavity configuration, the chiral symmetry of resonant modes might be broken, in which the electro-optical control of ATS leads to on-demanding magnetic-free optical isolation [28,29]. If one revisits the system of coupled “imperfect” WGM resonators [30], in addition to inter-cavity coupling, the intra-cavity coupling emerges as a significant factor, further lifting the mode degeneracy. These two factors together contribute to high-order ATS with finer energy splitting without the mediation of microwave fields in prior works [15]. The experimental observation and the post-fabrication mediation of such high-order ATS have remained elusive.

In this paper, we report the observations of high-order ATS on a silicon photonics platform using waveguide-coupled microring dimers functioning as size-mismatched photonic molecules. The inherent inter- and intra-cavity coupling results in split eigenmodes in a four-level system. By exploiting the surface roughness as Rayleigh scatterers or designing local defects as Mie scatterers, the backscattering-induced splitting in two individual rings can be adjusted. Utilizing an integrated thermo-optic tuner, the resonance detuning between two rings leads to electrically tunable mode hybridization and reshapable resonant spectrum. In the strong coupling regime where the inter-cavity coupling strength surpasses the loss contrast of two rings, the distinct backscattering-induced intra-cavity ATS in individual rings gets balanced at the zero-detuning point. This is in contrast with the opposite effect of out-of-balance intra-cavity ATS in the weak coupling regime.

The high-order ATS effect can be understood using the Hamiltonian matrix taking into account both the inter- and intra-cavity coupling effects. Initially, for an ideal microring, one can find nice spectral degeneracy due to the absence of backscattering [see the left panel of Fig. 1(a)], which is also termed the “diabolic point” (DP). A two-element photonic molecule is formed through the arrangement of two microrings with a defined gap spacing; the Hamiltonian can be written as H=(H1ggH2),(1)where H1 and H2 describe the characteristics of the individual rings as discrete systems. The complex-valued cross-coupling coefficient g describes the evanescent coupling at the gap region. As illustrated in the middle panel of Fig. 1(a), for two microrings upon resonant coupling, the efficient inter-modal coupling leads to splitting into symmetric (S) and anti-symmetric (AS) optical modes [31].

For single microrings considering the fabrication-induced imperfections, one can write a generalized Hamiltonian on the two-dimensional traveling-wave basis [CCW, (1,0); CW, (0,1)] as H=(ΩABΩ).(2)

The diagonal elements describe the original modes, including the energy (real part) and the decay rate (imaginary part). A (B) describes the backscattering from the CW (CCW) to the CCW (CW) components [24,32]. Combining Eqs. (1) and (2), the complete matrix suggests a dimension of 4×4, yielding high-order ATS with four eigenvalues and corresponding eigenstates [see the right panel of Fig. 1(a)].

The matrix in Eq. (2) might be non-Hermitian [33], i.e., A≠B*. Here, the studies were conducted on microresonators with a regular ring shape and hence well-preserved structural symmetry, in which the mode chirality is usually very low according to previous studies [34] and also our experimental calibrations. Without loss of generality, the intra-cavity coupling coefficients, κ, describe the coupling between CW and CCW lightwaves due to inherent surface roughness-induced backscattering. The coefficient g describes the coupling strength between two rings. Assuming lossless coupling, the Hamiltonian can be written in a simplified form:H=(Ωoκog0κoΩo0gg0Ωiκi0gκiΩi).(3)

Considering a photonic molecule coupled with a bus waveguide in Fig. 1(a), the diagonal elements can be described as Ωo=ωo−i(γo+γc)/2, Ωi=ωi−iγi/2=ωo+δω−iγi/2, where ωo(ωi) and γo (γi) are resonance frequency and intrinsic loss (e.g., material absorption, scattering, radiation, and bending) of the outer (inner) ring, respectively, δω corresponds to the resonance detuning between two individual rings, and γc corresponds to the losses due to coupling between the waveguide and the ring (see Appendix A).

Leveraging the maturing silicon photonics technology, optically passive microring resonators with well-defined geometry can be readily obtained using a standard foundry-based process. Silicon nitride featuring a wide transparency window from visible to infrared wavelengths was adopted as the waveguiding material. Figure 1(b) presents the waveguide-coupled microring dimer. Differing from the common wisdom of photonic molecules containing identical cavities, here two microrings with a slight difference in cavity radius are adopted, as illustrated in Fig. 1(b). The small ring embedded inside the big ring results in efficient inter-cavity coupling. The Vernier effect in these size-mismatched photonic molecules has been investigated, opening up new modalities in lasing [35,36], comb generation [35], and spectrometry [14]. Furthermore, it enables the study of mode hybridization with distinct resonance detunings at various mode orders. Here an electrical phase shifter was employed and placed atop a portion of the inner ring, to perform post-fabrication resonance detuning.

Figure 2(a) presents a size-mismatched photonic molecule fabricated on a silicon-nitride-on-silica platform by a multi-project wafer (MPW) service (AN150, Ligentec). The process involves the deposition of silicon nitride film via low-pressure chemical vapor deposition (LPCVD) on a 6-inch wafer [37,38]. The geometry of the size-mismatched ring pairs was patterned by 248 nm DUV lithography process. To facilitate single-mode light propagation at ∼780nm, we selected a waveguiding SiN layer thickness of 150 nm, accompanied by a waveguide width of 500 nm. The aluminum-based phase shifter was integrated covering half of the inner ring, leaving the coupling region unaffected. For experimental characterizations, transverse-electric (TE)-polarized wavelength-tunable laser light (TLB-6712, Newport) was coupled into a single-mode fiber using free-space optics. The polarization was adjusted by a fiber-based polarization controller. The bus waveguide was probed via grating couplers and two single-mode fibers with the assistance of a high-precision aligning system and a feedback loop (AP-SSAS-SIP-SAXYZ, Apico). Transmission signals were measured using an optical powermeter (PM100D, Thorlabs).

Figure 2(b) provides an overview of the discerned resonance wavelengths from the measured transmission spectrum ranging from ∼765 to ∼780nm. Due to the ∼6% difference in cavity circumference and effectively the free-spectral range (FSR), the spectral offset between resonances of two rings varies over different azimuthal mode orders. At the longer-wavelength side [Fig. 2(c)], the large spectral offset between two microrings leads to very weak excitation of the inner ring, and therefore, inefficient mode hybridization. The spectral offset between identified supermodes Δλi−o can be extracted as ∼0.56nm. While the two resonances get further approached [Fig. 2(d)], the extinction ratio (ER) of the resonances of the inner ring increases accordingly. Notably, one can discern a pair of split modes with width Δλi of ∼8pm. Notably, such a mode splitting is scarcely evident for resonances in the outer ring. As resonances in the outer ring suffer from additional waveguide-induced coupling loss, their characterized loaded-Q factors (∼49,000) are significantly lower than those (∼80,000) in the inner ring. In practice, the broadened linewidth here hinders the characterization of any potential splitting with Δλo.

The crossing of resonances at two rings occurs at ∼769nm, indicating a minimal value of Δλi−o. At the short-wavelength side, the ER of the resonances of the inner ring gradually decreases, suggesting a decreased excitation efficiency [Fig. 2(e)]. In addition, the mode splitting vanishes upon a sufficiently large |Δλi−o| [see Fig. 2(f)].

In order to authenticate the essence of ATS and distinguish it from other well-known interference phenomena such as electromagnetically induced transparency (EIT) [10,39], one pragmatic way is to perform dynamic resonance detuning to examine the avoided-crossing characteristic [3,10,40]. Here resonance detuning was performed by an integrated phase tuner module on top of the inner ring. The bias current was applied via two electric probes using a source meter (S100, Precise). Using regular microrings on the same block as test structures, the calibrated tuning efficiency is ∼8.4×10−6 RIU/mW. Due to the heat dissipation effect, the spectral red shift was observed for both rings upon electrical injection. Nevertheless, the relatively higher tuning efficiency for the inner ring leads to efficient control over the deviation of original resonances between two rings δω=ωi−ωo (or δλ=λi−λo).

Figure 3(a) summarizes the evolution of the resonant spectrum upon thermo-optic tuning. At the original state (power of 0 mW), the spectrally detuned split mode pair of the inner ring resides considerably distant from the mode of the outer ring, the discernable transmission dips accounted for weakly hybridized resonances are highly asymmetric. Upon an elevated power, corresponding to a decreased |δλ|, the ERs of detuned resonances of the inner ring increase. Meanwhile, the splitting effect becomes weakened. At a power of 40 mW, discernable transmission dips become almost symmetric, indicating the approaching of the zero-detuning point (i.e., resonant coupling) and consequently the almost equal distribution within two microrings. Notably, the splitting of the inner ring can hardly be resolved. Upon a further increased power above 50 mW, δλ becomes a positive value, resulting in the revival of the split mode pair. Throughout the entirety of the detuning process, these discernable three dips never merge into a single one, which is a clear sign of avoided-crossing behavior in the strong coupling regime. The absence of Fano resonances here again solidifies the nature of mode splitting at the zero-detuning point due to strong coupling instead of pure interference.

Here temporal coupled mode theory (TCMT) [41] was employed to study the spectral responses by excitation from one side of the bus waveguide (see Appendix A). The input coupling coefficient between the waveguide and outer ring can be estimated from single-resonator-based systems as test structures. The parameters of the photonic molecule systems, including κi, κo, g, together with the intrinsic loss coefficient of two rings γo and γi can be estimated by extracting the values of linewidths, splitting width, and the splitting width between two major branches Δλi−o. Figure 3(b) summarizes the numerically modeled evolution of spectra upon resonance detuning. While δλ approaches 0, the gradually diminishing splitting width of the inner ring Δλi agrees well with the observation in Fig. 3(a). At the zero-detuning point, two spectrally separated supermodes can be discerned with Δλi−o reaching its minimum of ∼19.4pm. Such a characterized anti-crossing behavior is an indicator of strong coupling in this two-element photonic molecule.

To illustrate the evolution of intra-cavity ATS upon varying inter-cavity coupling conditions, Fig. 3(c) summarizes Δλi as a function of Δλi−o between two main branches. Upon a decreased |δλ|, Δλi, originally dominated by intra-cavity ATS, becomes mediated by inter-cavity coupling, diminishing to its minimal value of 3.4 pm at the zero-detuning point (indicated as the shaded area with the minimal value of Δλi−o of 19.4 pm). The disparity between modeling and experimental results in Fig. 3(c) is mainly attributed to the fixed values of κi and κo for simplicity of the modeling and their neglected dependence on wavelengths.

As a comparison, another round of characterization was carried out on a photonic molecule with a widened coupling gap between two rings (380 nm), while the rest of the structural parameters remained the same. The tracked evolution of Δλi varying between ∼11.6pm and 6.0 pm in Fig. 3(d) suggests a good consistency with that in Fig. 3(c), which is attributed to the comparable backscattering condition of the inner rings in the two cases. In contrast, the discerned minimal value of Δλi−o decreases from 19.4 pm to 13.0 pm, owing to the suppressed inter-cavity coupling.

To understand the general behavior of Δλi and Δλi−o as two key signs of high-order ATS, the eigenvalues of a four-level system were calculated based on the Hamiltonian model in Eq. (3). Here g and δω were utilized as two control knobs to manipulate the mode degeneracy and also the high-order ATS. Figure 4(a) summarizes the eigenvalue surfaces. According to the well-established theory of two-element photonic molecules [16], strong coupling can be realized while the inter-cavity coupling strength surpasses the loss contrast of two rings, i.e., g>|γo+γc−γi|/4 here [see Fig. 4(b)]. Here one can identify a clear spectral anti-crossing feature, while the coherent energy transfer between two mode pairs can be ascertained as four crossing points in the imaginary part of the eigenvalues at δω=−2.6, 0 (twice), and 2.6 GHz, respectively. The minimum splitting width between adjacent modes Δωi−o is discerned as ∼0.96GHz (corresponding to Δλ∼2pm). Without the efficient inter-cavity coupling, i.e., δω is far away from 0, there is a major contrast of backscattering strength for two original cavities (Δωi=5.75GHz and Δωo=0.65GHz). Notably, the intra-cavity ATS of the inner and outer rings gradually reaches a balanced state, converging to an averaged value of ∼3.2GHz (corresponding to Δλ∼6pm) at the zero-detuning point, which agrees well with the experimental characterizations in Figs. 3(c) and 3(d).

At g=1GHz [see Fig. 4(c)] indicating the transition point between strong and weak coupling (i.e., g=|γo+γc−γi|/4), the system reaches the situation where part of the supermodes coalesce, i.e., exceptional points (EPs), at moderately detuned resonances satisfying the following criteria:|ωi−ωo|=κi−κo.(4)

One can identify degeneracy for both real and imaginary parts at δω=−2.6 and 2.6 GHz, corresponding to the reoccurrence of two-fold EP. Interestingly, the original split mode pair of the detuned ring diverges from each other around two EPs. Such a curvature is induced due to the peculiar nonlinear response toward perturbations (e.g., δω) in the vicinity of EPs. The system enters the weak coupling regime while g further decreases [see Fig. 4(d)]. There are four crossing points identified in the real part of eigenvalues [see insets in Fig. 4(d)], while there is no crossing observed for the imaginary part. One can discern minor expulsion behavior in the intra-cavity ATS for both rings.

Typically, the backscattering condition in a regular microring resonator is determined by fabrication-resulted surface roughness acting as Rayleigh scatterers [30,42]. Here one can intentionally introduce local modifications such as notches onto the cavity boundary [34] and extra particles in the vicinity of the boundary [43] via lithography. These local defects with dimensions in the same order as the wavelength generate scattering in the Mie regime, so that the significantly influenced inter-modal coupling potentially results in strong mode chirality [43–45]. To avoid additional complexity, here four identically designed bulges acting as Mie scatterers were placed symmetrically along the outer boundary of the inner ring, respectively, so that the backscattering strengths along opposite directions, i.e., |A| and |B|, are roughly balanced. Through the out-of-plane elastic scattering, the effect of Mie scatterers was ascertained by the local scattering spots in Fig. 5(a), which are much brighter than those induced by the inherent roughness in other regions. From the discerned supermodes with large resonance detuning, Mie scatterers elevate the intra-cavity coupling strength κi and increase Δλi from originally ∼10.5pm [in Fig. 3(c)] to ∼19.0pm (corresponding to an increase of κi by ∼66%).

Compared with the original case without Mie scatterers, spectral anti-crossing is again observed upon electrical tuning. Δλi−o at zero-detuning point of ∼20.0pm verifies the comparable coupling strength g. In addition, Δλi is suppressed from 17.2 pm to 11.0 pm (at the zero-detuning point), and rises back to 19.0 pm. From the resonant spectrum measured at δλ∼0nm (at P=15mW), three transmission dips can be resolved instead of two in Fig. 3(a) (at P=40mW), indicating the reinforced high-order ATS as a result of lifted κi.

In this article, the high-order ATS in a two-element photonic molecule has been comprehensively studied, in which both the backscattering-induced intra-cavity coupling and evanescent field-based inter-cavity coupling are considered. Leveraging the mature silicon photonics technology, we have fabricated waveguide-coupled ring pairs with intentionally mismatched sizes, in which the contrast of backscattering and loss conditions between each other can be adjusted. Via the post-fabrication electrical tuning, the evolution of intra-cavity ATS in the individual microrings reflects the interplay between the inter-cavity coupling strength and also the loss contrast between two rings.

The elucidation of high-order ATS in such a four-level system offers an extended picture compared with previous studies of photonic dimers in a two-level framework, in which reflectionless coupling is assumed, and therefore advances the understanding of the non-Hermiticity in a multi-level system. The concept of size-mismatched photonic molecules can be extrapolated to explore high-order ATS in multi-element (N≥3) coupled microresonators [46,47], e.g., coupled-resonators optical-waveguides (CROWs), and side-coupled integrated spaced sequences of resonators (SCISSORs). By combining with quantum emitters embedded inside the structures [48,49], high-order ATS is envisaged to offer new programmable Purcell-enhanced channels for resonant excitation and on-demand generations of single photons or photon pairs. In general, the high-order ATS presents promising avenues to various on-chip optical signal processes in the classical and quantum domains, such as photon storage and retrieval [15], nonlinear signal generation [13,50], optical filtering [28,51], switching [25], and sensing [52,53].

Acknowledgment. J. Wang, J. Duan, and X. Xu acknowledge the support from the National Natural Science Foundation of China, Guangdong Basic and Applied Basic Research Foundation Regional Joint Fund, Talent Recruitment Project of Guangdong, and Science and Technology Innovation Commission of Shenzhen.