The microring resonator has various applications that range from wavelength division multiplexing add/drop filters [1–3], biochemical sensors [4–6], Kerr frequency comb generation [7, 8], mode-locked pulse generation [9], optical squeezing [10, 11], and entangled-photon-pair generation with high quality (Q) factor [10]. Whereas most microring resonators are formed from single-mode waveguides [12, 13], multimode waveguides are preferable in many photonic integrated circuits. Multi-mode waveguides have various advantages, such as low losses [14, 15], high data capacity [16], and improved device integration [17]. A multi-mode waveguide provides enhancement of the Q factor by reducing scattering losses, which are caused by sidewall roughness [18]. However, multimode waveguides inevitably support undesirable higher-order modes, which cause problems such as low Q factor and a different free spectral range. The higher-order modes in a microring resonator interrupt the characterization of the fundamental mode with high Q factor. Furthermore, a multimode waveguide interrupts the evanescent wave coupling from bus waveguide to microring resonator to satisfy a critical coupling condition due to the stronger mode confinement than in a narrow waveguide. There have been a lot of investigations of the evanescent wave coupling in a microring resonator [1, 3, 19–23]. The most common method for controlling the evanescent wave coupling in a microring resonator is by changing the gap distance between the bus waveguide and the microring waveguide [1, 3], but even a small gap distance can enhance parasitic loss due to sidewall roughness and the strong modal perturbation that deteriorates the coupling ideality [24]. The effect of the bus waveguide’s geometry on evanescent wave coupling is not well studied. In this paper, we investigate the dependence of evanescent wave coupling on the bus waveguide’s width. We propose an analytic calculation method based on coupled-mode theory (CMT) for the dependence of evanescent wave coupling on the bus waveguide’s width. We compare the analytic calculation using CMT to a full three-dimensional finite-difference-time-domain (FDTD) simulation. A silicon nitride microring resonator is fabricated by e-beam lithography, and its characteristics are analyzed.

Figure 1 shows a schematic diagram of the microring resonator. Both waveguides are made of silicon nitride. The bottom oxide layer is SiO₂ of thickness 2 μm on a silicon substrate. The width of bus waveguide is variable, from 0.7 to 2.5 μm while the width of the microring waveguide is fixed at 2 μm. Both waveguides are 600 nm high, and the sidewall angle is 75 degrees. The two waveguides stand 400 nm apart. The bending radius of the microring resonator is 170 μm. The fundamental TE₀₀ mode of the bus waveguide is launched into it with a center wavelength of 1550 nm. We calculate evanescent coupling by CMT for the ring-shaped resonator structure with radius of 175 μm, width of 2 μm, and thickness of 600 nm, without various resonator geometry [25, 26]. The coupled-mode equations for each waveguide as follows [27, 28]:

Figure 1. Schematic of the microring resonator.

(1)$$\begin{array}{l}\frac{d{E}_{bus}(x)}{dx}=-i{\kappa }_{12}(x)E_{ring}(x)-i{\beta }_{ring}{E}_{bus}(x),\\ \frac{d{E}_{ring}(x)}{dx}=-i{\kappa }_{21}(x)E_{bus}(x)-i{\beta }_{ring}{E}_{ring}(x),\end{array}$$

where E_bus and E_ring are the electrical field of bus and microring waveguides respectively, κ_ij is the mode-coupling coefficient, and β_i is the propagation constant of the i^th waveguide. The mode-coupling coefficient is expressed as [27, 28]

(2)$${\kappa }_{ij}(x)=\frac{\omega {\epsilon }_0{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}[n^2(y,z)-n_j^2(y,z)]}}{\overrightarrow{E}}_i^{*}\cdot {\overrightarrow{E}}_{j}dydz}{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\widehat{x}\cdot ({\overrightarrow{E}}_i^{*}\times {\overrightarrow{H}}_i+{\overrightarrow{E}}_i\times {\overrightarrow{H}}_i^{*})dydz}}},$$

where ω is the frequency of the light, ε₀ is the permittivity of free space, n(y, z) is the refractive-index distribution of the overall structure, and nj(y, z) is the refractive index distribution of only waveguide j with cladding. To calculate the power coupled from the bus waveguide to the microring waveguide by the CMT approach, we divide the microring waveguide into finite elements. Sufficiently small elements of a microring waveguide with sufficiently large radius of curvature can be approximated by a straight waveguide. The coupling coefficient is calculated with Eq. (2). From every set of coupling coefficients and mode profiles, the power coupled into the microring waveguide can be calculated with Eq. (1).

Here the upper cladding is SiO₂, the width of the bus waveguide is variable from 0.7 to 2.5 μm, and the width of the microring waveguide is fixed at 2 μm. Figure 2(a) shows the effective refractive index of both waveguides. The propagation length is the length that the input light travels through the bus waveguide. When both waveguides are closest, the propagation length is 22.5 μm. Figure 2(b) shows a color map of the power coupled into the microring waveguide, which is a function of the position of the microring waveguide and the bus waveguide’s width. The red dot line shows the position dependence of the coupled power of the TE₀₀ mode in the microring waveguide when the width of the bus waveguide is 1.40 μm. The blue line shows that the coupled power of the TE₀₀ mode in the microring waveguide when the propagation length is 45 μm, where the waveguide coupling ends. This result shows that the coupling power depends on the width of the bus waveguide. The maximum power coupling is seen when the bus waveguide is about 1.40 μm wide. It is important to note that the widths for the phase-matching condition (Δβ = 0) and for maximum power coupling from the bus waveguide to the microring waveguide are different. Figure 2(c) shows the power in both waveguides depending on propagation length when the bus waveguide’s width is 1.40 μm. Figure 2(d) shows the power in the bus waveguide at the smallest gap (propagation length 22.5 μm) and at the end of the bus waveguide (propagation length 45 μm).

Figure 2. CMT calculation results with various bus waveguide’s width. (a) Effective refractive index of the ring waveguide and the bus waveguide with varying width. (b) Calculated coupled power of the fundamental TE₀₀ mode in the microring waveguide, for various widths of the bus waveguide and positions of the microring waveguide, from CMT. (c) Evanescent-field intensity of both waveguides depending on propagation length. (d) Power in the bus waveguide.

To compare the CMT approach to simulation results, we implement FDTD simulations (Lumerical, Vancouver, Canada) [29]. The FDTD simulation numerically calculates the light propagation by solving the time-dependent Maxwell’s equations. We calculated evanescent coupling in two different geometries of ring resonators by FDTD. One is the same as the geometry in the CMT calculation. The other geometry features a resonator radius of 450 μm, waveguide thickness of 400 nm, and gap distance of 150 nm. The boundary condition is set as a perfectly matched layer, to prevent reflection in the simulation. The frequency-domain field and power monitor is placed on the green monitor plane in Fig. 1. This monitor records the transmission of the coupled power. Furthermore, we discriminate the fundamental TE₀₀ and higher-order modes at the green monitor plane by using a mode-expansion monitor (Lumerical FDTD Solutions). As shown in Fig. 1, the fundamental TE₀₀ in the bus waveguide is launched at the left end of the bus, with a center wavelength of 1550 nm.

Figure 3 shows that the power coupled from the fundamental TE₀₀ mode in the bus waveguide to the TE₀₀ and TE₁₀ mode in the microring waveguide depends on the bus waveguide’s width. Figures 3(a) and 3(b) are the results for the same geometry as in the CMT calculation. Figure 3(c) is the FDTD simulation result calculated with the different geometry, which is resonator radius of 450 μm, waveguide thickness of 400 nm, and gap distance of 150 nm. The power coupled into the TE₁₀ mode in the microring waveguide dominates for widths of the bus waveguide between 700 nm and 1.3 μm, whereas the power coupled into the TE₀₀ mode in the microring waveguide starts at 1 μm width of the bus waveguide in Fig. 3(a). Figures 3(a) and 3(c) do not provide maximum coupling strength under the phase-matching condition, which is 2 μm of bus waveguide width. When the width of the bus waveguide is less than 2 μm, the power coupled in the TE₀₀ mode with the microring resonator is maximized in the FDTD simulation. It is remarkable that this result shows that the phase-matching width does not offer maximum power coupling. The FDTD simulation result agrees with the result of the CMT approach in Figs. 3(a) and 3(b). The CMT approach has advantages, such as low computational resources and structural scalability. The mode-profile calculation for the CMT approach is more computationally efficient than FDTD simulations. In the CMT approach, the coupling power at different gap sizes is easily calculated by shifting the mode profiles corresponding to the gap size. However, the FDTD simulation has to be performed under different conditions. The phase-matching condition limits the maximum value of the coupling strength, and the weak mode confinement in a narrow waveguide provides stronger power coupling, which shows that the width of the bus waveguide should be narrower than the phase-matching width to achieve stronger evanescent wave coupling. In addition, the wider bus waveguides decline the coupling ideality, due to the degradation of the Q factor [22]. It is also important to note that when the bus waveguide is wider than 1.3 μm, only the TE₀₀ mode is dominantly excited in the resonator waveguide. Even though the resonator waveguide supports a higher-order mode, the number of the resonance-mode family can be only one, because the initial excitation to the higher-order mode is suppressed. However, if the resonator is not perfect, due to e.g. sidewall scattering and the mode mismatch between the bus waveguide and microring waveguide, there can be chance for mode interaction exciting the higher-order modes.

Figure 3. Calculated coupled power of TE₀₀ and TE₁₀ modes in the microring resonator for various widths of the bus, by CMT and FDTD simulation.

The micro ring waveguide had a diameter of 340 μm, width of 2 μm, and thickness of 600 nm, and is 400 nm away from the bus waveguide. The bus waveguide has a thickness of 600 nm and width of 1.5 μm, which is the optimized width calculated by FDTD simulation. Stoichiometric silicon nitride is deposited on the thermally oxidized silicon substrate by low-pressure chemical vapor deposition. The waveguide is patterned by e-beam lithography and etched by RIE. Top cladding is deposited with low-temperature oxide. To characterize the microring resonators, the transmission spectrum is measured using a continuous-wave tunable laser with low power, to prevent high-power thermal frequency drift. The quasi-TE mode is set by the fiber polarization controller. Fiber-to-chip coupling uses a lensed fiber with a 5-axis stage. Output power is measured by a power meter, and the spectrum is recorded by an optical spectrum analyzer.

Figure 4 shows the transmission spectrum of the microring resonator. The loaded quality factor is 0.64 million, which was measured by the Lorentz curve fitted to the transmittance spectrum. The intrinsic quality factor Q_i was 1.3 million, which is retrieved from the loaded quality factor Q_L and the extinction ration T_min as shown below in Equation (3) [30]:

Figure 4. Experimental results. (a) The normalized transmission spectrum of the microring resonator with a Lorentz fitted curve, for calculating the loaded quality factor. (b) Experimental and FDTD simulation data for the coupled power.

(3)$$Q_i=\frac{2Q_L}{1+\sqrt{T_{\min }}}.$$

The free spectral range (FSR) is 139.7 GHz around 1550 nm, from Fig. 4. The group index n_g is 2.01 which is calculated from the FSR of 139.7 GHz. The coupled power of the TE₀₀ mode i n the microring waveguide is given by

(4)$${\kappa }^2=1-r^2$$

where r is the self-coupling coefficient of the bus waveguide and κ is the coupling coefficient between bus and ring waveguides [3]. The self-coupling coefficient of bus the waveguide is calculated by Eq. (5):

(5)$$T_{\min }=\frac{{(r+a)}^2{(1-ra)}^2}{{(1+ra)}^2{(1-ra)}^2},$$

(6)$$wherea=\sqrt{1-\frac{2\pi r{n}_g}{{\lambda }_0{Q}_i}}$$

Figure 4 shows the coupled power in the microring waveguide, includeing both experimental (dots) and FDTD simulation (line) results. The experimental results show a similar trend to those of the FDTD simulation. The difference between simulation and experiment is caused by inevitable excitation of the higher-order modes in the resonator due to imperfections of the resonator, such as sidewall scattering and mode mismatch of the two waveguides. In addition, the difference between the refractive index of the real waveguide and the FDTD simulation causes a difference in the coupled power.

We have investigated whether the evanescent-coupling strength relies on the bus waveguide’s geometry. The optimal width of the bus waveguide was calculated by FDTD simulation and CMT (analytic calculation). The phase-matching condition did not provide the maximum evanescent-wave coupling. Stronger evanescent-wave coupling occurred when the bus waveguide was narrower than the microring resonator. Not only the phase-matching condition but also the mode confinement influenced the evanescent-wave coupling. The FDTD simulation and CMT could predict the experimental coupled power. The optimized bus-waveguide geometry results in an intrinsic quality factor of up to 1.3 × 10⁶. This study provides reliable guidance for the design of microring resonators, depending on various applications.

This research was supported by the “Ultra-short Quantum Beam Facility Program” and “GIST Research Institute Program” through grants provided by the Gwangju Institute of Science and Technology in 2021. This research is also supported by a National Research Foundation of Korea (NRF) grant, funded by the Korean government (No. 2021R1A2C1007130), and the Energy AI Convergence Research & Development Program through the National IT Industry Promotion Agency of Korea (NIPA), funded by the Ministry of Science and ICT (No. S1602-20-1009).