Microresonators enable efficient light manipulation through optical nonlinearity strongly enhanced by their high-quality-factor (Q) and small mode volume [1]. Moreover, precise controllability of the resonator geometry inherited from the microfabrication technique allows precise dispersion engineering to satisfy the phase-matching conditions required for the efficient optical nonlinear process over a wide wavelength range [2,3]. Therefore, light waves at multiple wavelengths over a wide range can be induced to emerge and strongly interact with each other in a microcavity, and in this way, practically useful light sources such as a soliton microcomb [4], entangled photons [5,6], and optical parametric oscillator [7] can be developed in a small footprint. To effectively access these light waves confined in the cavity from outside, and to control the spectral behavior of the light generation processes by changing loaded quality factors at specific wavelengths, it is beneficial to independently control the coupling strengths between the resonator and the waveguide at two different wavelengths.

A traditional coupler with one coupling point between the resonator and waveguide (one-point coupling method, OPCM) has been widely used, and its function was well-analyzed [8], but its coupling controllability is only guaranteed within a narrow wavelength range. When the waveguide and the resonator structures are determined, OPCM has just one degree of freedom to control the coupling efficiency, namely the gap between the resonator and the waveguide. Therefore, the spectral distribution of the coupling efficiency is passively determined by the spatial overlap between the evanescent fields of the waveguide and the resonator modes, which monotonically decays with frequency. To overcome this limitation and give additional degrees of freedom, different coupling geometries have been developed. A pulley scheme, where a waveguide wraps around a portion of the resonator, showed a more efficient extraction of shorter wavelength microcombs compared with an OPCM [9]. Two- or multiple-points coupling methods (TPCM or MPCM), where coupling occurs at two or even larger numbers of coupling points, have been demonstrated to overcome OPCM limitations. These approaches include varying coupling efficiency in a solid-state device [10], achieving better coupling efficiency at a single wavelength [11], tunable time delay by controlling the coupling efficiency [12], or modifying spectral behavior of cavity nonlinearity [13].

Here, for a TPCM, we focus on the controllability of coupling efficiency at multiple numbers of different wavelengths. For the simplest geometry of TPCM which consists of a resonator and a straight waveguide, an analytic model is developed to calculate its coupling efficiencies at various wavelengths. This is then experimentally verified using a flip-chip coupling scheme. The model and experiment show that the coupling strength oscillates with a different period depending on wavelength when the bus waveguide moves from the ring resonator edge to the direction of the ring center. Therefore, by properly introducing a phase difference between the waveguide and the resonator paths connecting two coupling points, various combinations of coupling efficiencies at different wavelengths can be achieved. In addition, it is confirmed that the trend of the periodic change of transmission rate according to the change of y can be categorized into two different regimes depending on the refractive index difference between the resonator and the waveguide.

A previously developed coupling model [8] was modified to analyze TPCM, which quantitively reveals the oscillatory changes in coupling strength originating from the constructive and destructive interference of two optical paths. The configuration and parameters of the TPCM are shown in Fig. 1(a), where coupling between the resonator and the waveguide occurs at two distinct points, A and B. At each coupling point, the light comes in and out through two paths. Parameters $({{a_1},\; {a_2}} )$ and $({a_1^{\prime},\; a_2^{\prime}} )$ are the amplitudes of the input and output light waves at point A, respectively. Likewise, the amplitude parameters $({b_1^{\prime},\; b_2^{\prime}} )$ and $({{b_1},{b_2}} )$ are defined at point B, respectively. Parameters $r,y,\theta $ are the radius of the resonator, the distance between the midpoints of the upper arc and the chord, and half of the central angle of the upper arc in radians, respectively. Here, ${\phi _1},{\phi _2},{\phi _3}$ are the phase changes when the light is transmitted through the chord, the upper arc, and the lower arc, respectively. When the effective refractive indices of the resonator and the waveguide are ${n_{re}}$ and ${n_{wg}}$, the phase changes are given by

 

Fig. 1. Configuration and parameters to explain the coupling mechanisms in the TPCM. (a) Configuration of the TPCM. The optical beam path and propagation direction in the resonator (the waveguide) are shown by the red (green) circles and the red (green) arrows, respectively. (b) Normalized transmission depending on the relative phase ${\phi _2} + {\phi _3}$. (c) ${T_{res}}$ for various $\mathrm{\Delta }\phi $ for the parameter $\alpha $ of 0.98.

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Also, $t,$ $\kappa $, and their complex conjugates ${t^\ast },{\kappa ^\ast }$ are the coupling coefficients at the coupling points A and B. Lastly, ${\alpha _1},\; {\alpha _2}$, and ${\alpha _3}$ are equivalent to $|{b_1^{\prime}/a_1^{\prime}} |$, $\; |{b_2^{\prime}/a_2^{\prime}} |$, and $|{{a_2}/{b_2}} |$, which are related to the attenuation rate of light intensity in the paths through the chord from A to B, the upper arc from A to B, and the lower arc from B to A, respectively. Here, $\alpha $ defined as ${\alpha _2}{\alpha _3}$ can be calculated by measuring the intrinsic quality factor $({{Q_{int}}} )$ of the resonator, which is given by Eq. (2) [9], where $L,\; {n_{eff}},\; $ and $\lambda $ are the circumference of the resonator, the effective refractive index of the resonator, and the wavelength of the light in a vacuum, respectively,

Meanwhile, coupling to resonator occurs under resonant wavelength, which satisfies the resonant condition given as ${\phi _2} + {\phi _3} = 2m\pi $ with integer $m.$ Therefore, the transmission at resonance, ${T_{res}}$, is smaller than the transmission in the non-resonant condition, which is shown in Fig. 1(b). By putting ${\phi _2} + {\phi _3} = 2m\pi $ into Eq. (3), ${T_{res}}$ is given as

It is noteworthy that ${T_{res}}$ is a periodic function of $\Delta \phi $ because it depends on $\textrm{cos}({\Delta \phi } )$. In addition, $\textrm{cos}({\Delta \phi } )$ is a function of $\theta $ and $y = r({1 - \textrm{cos}\theta } ).$ Therefore, ${T_{res}}$ periodically changes as y increases. By using the definitions of ${\phi _1}$ and ${\phi _2}$ in Eq. (1), the condition determining the coupling strength can be simplified as Eq. (10). Here, $\Delta n$ is defined as ${n_{re}} - {n_{wg}}$, and $\mathrm{\Delta }{l_{opt}}$ is defined as $2r({{n_{re}}\theta - {n_{wg}}\textrm{sin}\theta } )$ which is the optical pathlength difference between the upper arc and the chord. In Eq. (5), the maximum and minimum coupling occurs for even and odd number $m,$ respectively. Note that in the actual implementation of TPCM, because y variation is small with respect to $r,\; $ the approximation given as $\textrm{sin}\theta \approx \theta - {\theta ^3}/6\; $ can be used,

Figure 2 shows the combinations of ${T_{res}}$ at wavelengths 1.61 µm and 2.45 µm, which are examples of two different wavelengths corresponding to the laser wavelengths available for the following experiment. This figure can be attained while continuously changing y from 0 µm to 50 µm for a sample structure having parameters: $r = 5\; \textrm{mm},$ ${n_{wg}}$=2.375 at $\lambda = 1.61\; \mathrm{\mu}\textrm{m}$, and ${n_{wg}} = \; $2.298 at $\lambda = 2.45\; \mathrm{\mu}\textrm{m}$, $t = 0.99$, and $\alpha = 0.96$ to satisfy the critical coupling condition at $\mathrm{\Delta }\phi = 2m\pi .$ Here, ${n_{re}}$ is determined by the $\Delta n$ value marked on each sub-figure. Note that in general, $t,\; \alpha ,\; $ and Δn vary with wavelength. However, for simplicity, it was assumed that these values are fixed regardless of wavelength. However, the trace covers a wide range of the two-dimensional plane, which stands for all possible ${T_{res}}$ combinations, while its projections to both axes oscillate between 0 and 1 as expected.

 

Fig. 2. Distribution curves for all possible combinations of ${T_{res}}$ for two different wavelengths: $\lambda = 1.61\; \mathrm{\mu} \textrm{m}$ and $\lambda = 2.45\; \mathrm{\mu}\textrm{m}.$ Here, ${n_{wg}}$ is set as 2.375 and 2.298 for $\lambda = 1.61\; \mathrm{\mu}\textrm{m}$ and $\lambda = 2.45\; \mathrm{\mu}\textrm{m},$ respectively, and ${n_{re}}$ for each figure is determined by $\Delta n \equiv {n_{re}} - {n_{wg}}.$ The arrows indicate the direction in which y increases. Their length is proportional to the rate of transmission change per y movement. (a) $\Delta n = 0$. (b) $\; \Delta n = 0.004$. (c) $\Delta n = 0.008$. (d) $\Delta n = 0.012$.

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In Fig. 2, it is remarkable that a larger variable $\Delta n$ results in a broader area over which the trace spreads. This is because, for each wavelength, the phase difference between the arc and the chord, namely $\Delta \phi ,\; $ gets larger for the given y as $\Delta n$ increases. In practice, $\Delta n$ cannot be arbitrarily large owing to the phase-matching condition between the resonator and waveguide modes for efficient coupling. The allowed $\Delta n$ is mainly determined by the properties of the resonators and waveguides, such as mode volume, quality factor, and field distribution.

To validate the coupling model, ${T_{res}}$ was experimentally measured for various y values and compared with the results calculated by the model. The experiment was performed based on a flip-chip coupling scheme by which evanescent coupling can occur between two vertically faced chips as previously reported [14]. This scheme is shown in Fig. 3(a). The relative positions of the resonator and the waveguide in the vertical and horizontal directions can be precisely controlled by a piezo stage. The resonator and the waveguide were fabricated as a trapezoidal structure, which has a light-guiding core consisting of an $\textrm{A}{\textrm{s}_2}{\textrm{S}_3}$ film uniformly deposited on a bottom platform made of $\textrm{Si}{\textrm{O}_2}$ in a trapezoidal shape [15]. The radius of the resonator is 5 mm and the other geometry parameters are shown in Fig. 3(b).

 

Fig. 3. The geometry used to experimentally verify the properties of the TPCM. (a) Flip-chip coupling scheme. The resonator and the waveguide were fabricated on two separate chips. (b) The geometric parameters of the resonator and waveguide. Here, ${\textrm{w}_1},{\; }{\textrm{w}_2},\; \textrm{d}\; $ and $\varphi $ are $\lambda = 2.7\; \mathrm{\mu}\textrm{m},$ $\lambda = 9.3\; \mathrm{\mu}\textrm{m},$ $\lambda = 1.3\; \mathrm{\mu}\textrm{m},$ and 30°, respectively.

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The measured ${T_{res}}$ while varying y from 10 µm to 30 µm, shows the oscillatory property plotted as green dots in Fig. 4(a) and Fig. 4(b). The calculated ${T_{res}}$ from the analytic model are plotted as red solid curves.

 

Fig. 4. Comparing the measured results of ${T_{res}}$ and coupling Q (green dots) with the calculated results of ${T_{res}}$ and coupling Q based on the coupling model (red curves). The bottom x-axis denotes y, while the top x-axis denotes m number. The blue stars mark the points where ${T_{res}}$ is the minimum, namely when ${Q_c}$ is minimum at each wavelength, where y is the same. (a), (b) y versus ${T_{res}}$. (a) Case of $\lambda = 1.61\; \mathrm{\mu} \textrm{m}$. (b) Case of $\lambda = 2.45\; \mathrm{\mu} \textrm{m}$. (c), (d) y versus coupling quality factor. (a) Case of $\lambda = 1.61\; \mathrm{\mu} \textrm{m}$. (b) Case of $\lambda = 2.45\; \mathrm{\mu} \textrm{m}$.

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The parameters used for the analysis were determined as described below. The ${n_{re}}$ and ${n_{wg}}{\; }$ were numerically calculated from the cross-sectional geometries. Here, ${n_{re}}$, ${n_{wg}}$, and the corresponding $\Delta n$ were 2.385, 2.375, and 0.01 at $\lambda = 1.61\; \mathrm{\mu}\textrm{m}$, respectively, and were 2.315, 2.298, and 0.017 at $\lambda = 2.45\; \mathrm{\mu}\textrm{m}$. Despite their similar cross-sections, the effective refractive indices of the resonator and the waveguide were slightly different from each other owing to the difference in their top width and the curvature of the resonator. Here, $\alpha $ was calculated from ${Q_{int}}\; $ which was directly measured. The measured value of ${Q_{int}}$ at $\lambda = 1.61\; \mathrm{\mu}\textrm{m}$ and $\lambda = 2.45\; \mathrm{\mu}\textrm{m}$ were $5.01 \times {10^6}$ and $1.46 \times {10^6}$, respectively. Here, $|t |$ was calculated with the critical coupling condition observed experimentally, which is given as $\alpha = 2{t^2} - 1.$ In Figs. 4(a) and 4(b), $\alpha $ and t are set to $0.9705$, $0.9926$ at $\lambda = $1.61 $\mathrm{\mu}\textrm{m}$ and $0.9322$, $0.9829$ at$\; \lambda = 2.45\; \mathrm{\mu}\textrm{m},$ respectively. In Figs. 4(a) and 4(b), the results from the experiment and the analytic model show quite similar periods and amplitudes at two wavelengths. The local minima and maxima of the experimental data almost fit well with the even and odd numbers of m defined in Eq. (5), respectively, which proves that the analytic model explains TPCM well.

Meanwhile, these results can be represented as the plot for y versus coupling quality factor, shown in Figs. 4(c) and 4(d). The coupling quality factor, which is denoted as ${Q_c}$, can be calculated from the intrinsic quality factor denoted as ${Q_{int}}$ and the coupling parameter denoted as ${\rm K}$ by using the following equation: ${Q_c} = {Q_{int}}/{\rm K}$. Here, ${\rm K}$ can be calculated from ${T_{res}}$ in the coupling regime, using the following equation: ${\rm K} = \left( {1 - \sqrt {{T_{res}}} } \right)/(1 + \sqrt {{T_{res}}} $). As expected, ${Q_c}$ is minimum and maximum when ${T_{res}}$ is minimum and maximum, respectively.

The exact design of the coupler geometry for the targeted coupling efficiency can be obtained using the analytic model. For example, to have critical coupling at the two wavelengths in Fig. 4, y can be set at 18.8 µm, as marked with two blue stars, resulting in ${T_{res}} = $0.0146 and 0.0138 at $\lambda = 1.61\; \mathrm{\mu} \textrm{m}$ and $\lambda = 2.45\; \mathrm{\mu} \textrm{m}$, respectively. Further design rules to control the coupling efficiency at even more numbers of wavelengths than two can be understood from the distribution map for the transmission rate ${T_{res}}$ calculated for continuous wavelengths, shown in Fig. 5. Note that although the distance between the resonator and the waveguide is fixed, coupling strength increases as the wavelength of light increases because the evanescent field stretches out further at a longer wavelength. However, for simplicity, the coupling strength was assumed to be constant regardless of wavelength in this figure. Parameters used for the analysis model to draw Fig. 5 were the same as those used in Fig. 4 except for $\alpha $ and $|t |$, which were set as $\alpha = 0.98,\; |t |= 0.9950$ when $\lambda $ was in the range of 0.5 to 3.5 µm.

 

Fig. 5. Distribution map of$\; {T_{res}}$ for various y and$\; \lambda $. The white lines indicate the same m numbers and the white numbers indicate the corresponding m numbers. (a) Case of ${n_{re}} = 2.385,$ ${n_{wg}} = 2.375.$ (b) Case of ${n_{re}} = 2.375$, ${n_{wg}} = 2.385$.

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The TPCM shows completely different distributions of ${T_{min}}$ which suggest different practical benefits, depending on whether $\Delta n$ is positive [Fig. 5(a)] or negative [Fig. 5(b)]. If $\mathrm{\Delta }n$ is positive, $\mathrm{\Delta }{l_{opt}}$ monotonically increases as y increases. Therefore, ${T_{res}}$ determined by $\mathrm{\Delta }{l_{opt}}$ changes, at a fixed wavelength, in a relatively regular and short oscillation period with the increase of y. Therefore, various combinations of ${T_{res}}$ at different wavelengths are accessible in a limited range of y, which is beneficial for controlling coupling efficiencies at multiple wavelengths. In contrast, if $\mathrm{\Delta }n$ is negative, $\mathrm{\;\ \Delta }{l_{opt}}$ gets smaller, becomes zero at a certain y value, and finally gets larger as y increases. Therefore, in Fig. 5(b), ${T_{res}}$ shows a horizontal section where ${T_{min}}$ stays nearly constant while varying y at a fixed wavelength. This condition provides robustness of coupling efficiency for perturbations of the y value, which can arise from alignment fluctuations in a flip-chip coupling scheme, or fabrication imperfections in bonded or stacked solid-state structures. A more detailed explanation for two different regimes shown in Fig. 5 can be found in the supplementary information.

In conclusion, we have proposed a way to independently control coupling efficiencies at different wavelengths based on a two-point coupling scheme. By adjusting the horizontal position between a resonator and a waveguide, the phase difference between two beam paths bounded by two coupling points can be controlled. An analytic model to explain the properties of this TPCM was developed and verified by experimental results with a flip-chip coupling scheme. The overall behavior of the TPCM showed two different regimes, depending on the effective refractive indices of the waveguide and resonator.

Although the simplest geometry of a straight waveguide was considered for the TPCM in this paper, this approach can be applied to any other geometries having two or more numbers of spatially separated coupling points for controlling the coupling efficiency at different wavelengths. In addition, by introducing a phase tuner in a waveguide path between the coupling points, wavelength-dependent coupling efficiency, or equivalently coupling Q-factor, can be controlled without mechanical motion. TPCM is practically useful for efficient nonlinear processes over a broad spectrum and optical filters based on microcavities.