1. Introduction

Benefiting from their ability to enhance light-matter interactions, high quality (Q) factor optical microcavities, in particular whispering-gallery-mode (WGM) microresonators [1], have been widely used in many applications ranging from ultra-high sensitive sensing [2–9 ] to low-threshold lasing [10, 11] to nonlinear optics [12–14 ]. They have been also employed as a platform for fundamental studies including quantum information [15, 16] and optomechanics [17,18]. Recently there has been an increasing interest in using two or more microresonators that are coupled with each other via their evanescent fields. Such interconnected microcavity clusters are referred to as photonic molecules (PMs), and they have been shown to modify and further enhance light-matter interactions by adjusting the coupling between adjacent microcavities [19].

In the formation of a PM, the resonance modes in each of the resonator hybridize leading to the formation of split modes. The frequency spacing of the split modes and their linewidths are controlled by tuning the coupling strength between the resonators. The tunability of coupling in PMs not only makes it possible to produce optical devices with improved performance or new functionalities such as single mode lasers [20] and optical delay lines [21] but also paves the way for exploiting novel physics. For example, the nontrivial physics around the exceptional point of a non-Hermitian system has been studied in PMs [22,23]. In a PM formed by a passive lossy microresonator and an active resonator with gain balancing the loss of the passive resonator, parity-time symmetry and its breaking have been demonstrated and nonreciprocal light transmission has been achieved [22]. Similarly, loss induced suppression and revival of Raman lasing has been demonstrated in a PM composed of silica toroid microresonators [23], and single mode laser was realized in laterally coupled microring resonators with InGaAsP quantum wells [20]. Tunable and controllable coupling between the resonators in a PM is undoubtedly the central issue for applications and for demonstrating their improved performance over a single microcavity [19]. Traditionally, the resonators in a PM are coupled horizontally, and the coupling strength decreases exponentially as the horizontal distance between the resonators increases [24].

In this Letter, we demonstrate the formation of a PM by vertically coupling two microdisk resonators. In this PM, decreasing the vertical distance between the planar surfaces of the microdisks monotonously increases the coupling strength and mode splitting; however, when one of the resonators is displaced laterally with respect to the other, mode splitting exhibits an oscillatory behavior indicating the oscillatory coupling between the modes. We also present a theoretical model based on coupled mode theory to explain this oscillatory behavior. The demonstration of horizontal oscillatory coupling in PMs with vertically coupled resonators enables realizing photonic structures with more functionalities and more flexibility, and manipulating the location of the PT-symmetric phase transition and exceptional points [22, 23].

2. Experimental results

In our experiments, we used two WGM microdisk silica resonators μR₁ and μR₂ with resonance wavelengths at 1553.7 nm to form the vertically coupled PM as shown in Fig. 1(a). The microdisks were fabricated in the same batch with silica on silicon wafer, and thus they were of the same radius (∼ 38 μm) and thickness (∼ 2 μm). The Q factors of the resonators were ∼ 1.6×10⁵. To have easy access for vertical and lateral coupling, and for precise manipulation of the relative positions of the resonators, we fabricated the resonators at the edges of two different wafers, as clearly shown in the optical microscope images of μR₂ taken in the transmission (Fig. 1(b)) and the reflection configurations (Fig. 1(c)).

 

Fig. 1 Coupled microdisk resonators. (a) Illustration of the vertically coupled microdisk resonators μR₁ and μR₂ forming a photonic molecule (PM). Light is coupled to the PM through μR₁ via a tapered fiber. (b) and (c) Optical micrographs taken in the transmission and reflection configurations, respectively, showing the top views of μR₂. The radius of μR₂ is 38 μm, which approximately equals that of μR₁.

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We studied the optical properties of the vertically coupled WGM resonators by coupling light from a 1550-nm band external cavity laser diode to μR₁ via a tapered fiber as shown in Fig. 1(a). The tapered fiber was coupled to μR₁ from its bottom side to avoid any direct coupling to μR₂. The transmission spectra of the coupled system were monitored by recording the transmission at the output port of the tapered fiber by a photodetector connected to an oscilloscope as the wavelength of the laser diode was scanned. A thermo-electric cooler was used to precisely tune the resonance wavelength of μR₁ to be exactly the same as that of μR₂. A manual rotation stage and two 3-axis piezo nanopositioning stages (100-nm precision) were employed to align μR₁ and μR₂ on parallel planes, and then to change the relative horizontal and vertical positions of μR₂ with respect to μR₁, which was kept fixed to ensure a stable coupling with the tapered fiber. Despite the fact that we could precisely (within 100 nm) tune the distance between the resonators, it was very challenging to measure the exact vertical distance between them. Using optical images taken by CCD cameras attached to optical microscopes, we roughly estimated the vertical distance between the microdisks to be about 1 μm before changing the distance by moving the second microdisk vertically using the nanopositioning stage. Since initially the resonance frequencies of the modes in the microdisks were not spec-trally overlapping with each other (i.e., non-degenarate), we could not observe mode splitting even if the resonators were within 1 μm with each other. We used a thermo-electric cooler to tune the resonance wavelength of μR₁ to be the same as that of μR₂. As the resonances started to overlap with each other, the modes started exchanging energy (i.e., coupled resonators) and mode splitting was observed in the transmission spectra. We continued the tuning process until the mode splitting was maximized, implying that the modes were maximally coupled (i.e., their central frequencies were the same and spectral overlap was maximized). After then we tuned the distance between the resonators as we monitored the mode splitting. Presence of mode splitting and the continuous increase in the splitting amount, as we decreased the distance between the resonators, implied that spectral overlap of the resonances did not change much to affect our results. For the following discussions and the theoretical model, we label the center of the top surfaces of μR₁ and μR₂ as O₁ and O₂, respectively, and set their spatial coordinates as (0,0,0) and (x,y,z). The top surface of μR₁ is on xOy plane, which is parallel to that of μR₂. Note that the top surface of μR₂ is close to that of μR₁ (see Fig. 1(a)). Therefore, z represents the vertical distance between the two resonators, and $\sqrt{x^2+y^2}$ is the transverse distance between O₁ and O₂.

We performed two sets of experiments. In both of the experiments, the transmission spectra were recorded as the position of μR₂ was varied while the position of μR₁ was fixed at (0,0,0). In the first set of experiments, we varied the vertical distance between the resonators (i.e., O₂ at different values of z) and kept x and y fixed (i.e., no lateral displacement). Figure 2(a) presents the intensity graph obtained from this experiment when the vertical distance between the resonators was reduced by 900 nm with a step resolution of 100 nm; each step-like change in the transmission spectra corresponds to a 100-nm change in the vertical distance. It is clearly seen that the mode splitting of the PM, which indicates the coupling strength between the resonators, increased as the vertical coupling between the resonators decreased. This dependence of mode splitting on the vertical distance can be explained as follows. The spectral and spatial overlaps of the involved WGMs in each resonator determine the coupling strength between the resonators. In our experiments, the selected WGMs were adjusted to have the same resonance frequency to maximize spectral overlap. Moreover, the amplitudes of the wavevectors were fixed at the time of the selection of the WGMs, and thus the angle between the wavevectors of the selected WGMs, which are in the tangential directions of the rims of the resonators, stayed the same because the transverse positions of the resonators stayed the same. With the spectral overlap and the wavevectors fixed, increasing the spatial overlap of the two WGMs in μR₁ and μR₂ by decreasing their vertical distance increased mode splitting in an exponential form as shown in Fig. 2(a). Theoretically, the mode splitting of coupled resonators increases exponentially as the gap between them decreases. However, as reported in the experiments for the coupled coplanar resonators [24], when the gap between the coupled resonators approaches zero, the mode profiles are significantly disturbed and the losses of the involved modes increase due to increased scattering losses. This significantly broadens the resonances and reduces effective coupling. As a result, resonance modes become indiscernible in the transmission spectrum.

 

Fig. 2 Dependence of mode splitting of vertically coupled resonators on the vertical gap between μR₁ and μR₂. (a) Normalized transmission spectra measured when the gap between the two microresonators was decreased by 0.9 μm with steps of 100-nm. (b) Simulation results showing the transmission spectra of the photonic molecule when the vertical distance between the resonators was varied by moving μR₂ along the z-axis (see Fig. 1(a)). The experimental results agree well with the simulation results predicted by our theoretical model.

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The dependence of the mode splitting of PMs consisting of two coplanar resonators on their transverse distance has been extensively investigated [19]. It was shown that mode splitting monotonously increased as one resonator approached the other [24]. Here, in the second set of experiments, we placed two WGM resonators on two parallel planes as illustrated in Fig. 1(a), and varied the transverse distance between the resonators by moving μR₂ in x direction while the vertical distance between them was kept fixed. We observed that the mode splitting spectra and the coupling strength between the resonators exhibited an oscillatory behavior as μR₂ was moved horizontally (Fig. 3(a)). This is in stark contrast with both the vertical-scan case (Fig. 2) and the transversely-scanned coplanar PM structures [19, 24] in which mode splitting and the coupling strength monotonously increased as the transverse distance was decreased.

 

Fig. 3 Oscillatory coupling of vertically coupled resonators in a photonic molecule. (a) Experimentally obtained transmission spectra when the microdisk μR₂ was moved along a line parallel to x axis (see Fig. 1(a)) while the vertical distance between the resonators was kept fixed. (b) Simulation results showing the transmission spectra of the photonic molecule when the position of μR₂ was varied only along the x-axis while the position in y and z was kept fixed. The experimental results agree well with the simulation results predicted by our theoretical model.

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3. Theoretical analysis

In the following, we introduce a theoretical model, which is based on coupled mode theory [25], to explain the oscillatory splitting spectra observed in the experiments. The geometric illustration used in developing the model is given in Fig. 4. For simplicity, we assume the coupling between μR₁ and μR₂ or the tapered fiber takes place only at points labeled by P_m with m being an integer. P_m is on the rims of the WGM resonators and gives the minimum distances between the rim of μR₁ and that of μR₂ or the waveguide. The coupling coefficient at P_m is κ _m, its corresponding transmission coefficient is t _m, satisfying |t _m|² + |κ_m|² = 1. As an example, κ ₁, t ₁ and their conjugates are marked in Fig. 4. The key point of this theory is that it takes into account the interference of light coupled into WGM resonators through multiple different points. Under this framework, the equations governing the coupling behaviors at the points P₁, P₂, and P₃ are

 

Fig. 4 Schematics of the theoretical model. The microdisks μR₁ and μR₂ forming the PM are located on parallel planes, their centers and diameters are indicated by O_1,2 and R _1,2, respectively. E _in and E _out are the amplitudes of the input and output light fields in the tapered fiber. Coupling between different optical components takes place through coupling points P_m with coupling coefficients κ _m and transmission coefficients t _m. E _m indicates the light field amplitude just after the coupling point P_m. E _m experiences amplitude changes α _m and phase changes ϕ _m, respectively, while propagating along different arcs separated by coupling points P_m. θ is the angle between the wavevectors of the coupled whispering gallery modes at the coupling points.

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By solving Eqs. (1)–(3), the normalized transmission of the tapered fiber coupled PM can be expressed as

The simulation results using Eq. (5) show a very good agreement with the experimentally measured data as shown in Figs. 2(b) and 3(b). In the simulations, we used the following parameters: R ₁ = R ₂ = 38 μm, α ₀ = 0.1688 cm⁻¹, n _eff = 1.38, γ = 0.3, κ ₁ = 0.0534i, κ ₂₀ ≈ κ ₃₀ = 0.0142i, and λ ₀ = 1553.7 nm for both cases, and (x,y) = (0,3.5) and (y,z) = (3.5,0.2) in units of μm for the simulations of Fig. 2(b) and Fig. 3(b), respectively. κ ₁ was set as a constant in consistent with the experimental setting where the tapered fiber and μR₁ were kept stable.

As presented in Figs. 2 and 3 that the experimental results agree very well with the theoretical results. Therefore, we can attribute the oscillatory coupling behavior of the vertically coupled PM in response to horizontal displacement of one resonator from the other to the interference between the light coupled into the WGM resonators through different coupling regions. It is well known that interference was determine by the amplitudes and phases of the involved light field. The former determines the contrast ratio of the interference pattern, and the latter impacts the spatial and temporal distributions of the interference pattern. In vertically coupled PMs, ϕ _m was determined only by the relative horizontal displacement between the coupled resonators. Thus the interference-induced oscillatory coupling only showed up when the resonators in PMs experienced relative transverse motion rather than a relative vertical motion. The ability to vary the coupling strength between the vertically coupled WGM resonators just by laterally displacing one of the resonators with respect to the other within distances of tens of micrometers provides more flexibility in constructing three dimensional (3D) photonic devices based on WGM or microring resonators.

4. Conclusions

In conclusion, we have studied both theoretically and experimentally the dependence of the coupling strength and the mode splitting spectra of vertically coupled PMs on the relative positions of the resonators forming the PM. We observed that, in contrast to the exponential dependence on the vertical displacement of the two WGM resonators, mode splitting and the coupling strength show an oscillatory dependence on the relative transverse displacement. This allows the coupled resonators to have an efficient coupling even if one of the resonators is laterally shifted by tens of μm. We also introduced a theoretical model which successfully explains the observed oscillatory coupling behavior by considering the interference of light coupled into the resonators from multiple coupling regions. We believe that the oscillatory coupling between vertically coupled resonators achieved by adjusting their relative horizontal displacement is useful in many applications of PMs, such as ultra-sensitive sensing, construction of 3D photonic devices, and manipulation of the exceptional point of PT symmetric resonators.