## 2 Structure and Design  
The microring resonator with MMI coupler is illustrated in the Figure 1a. The light is launched in through the input waveguide and then coupled into the microring by the MMI coupler. The resonance will take place at the periodic and specific wavelengths, which are named resonance wavelengths. The cross-section of the polymer-based ridge waveguide is shown in Figure 1b. The polymer, OrmoCore, was utilized as it has a low absorption loss at short wavelengths [28]. The refractive index (RI) is 1.543 at the wavelength of 890 nm. The under cladding is SiO\({}_{2}\) with the RI of 1.453. The upper cladding is assumed as water solution with the RI of 1.33 for biochemical sensing applications. In the imprinting process, the core layer cannot be completely squeezed out and is finally left as the residual layer. It will increase the bend loss of the microring and further decrease the Q factor and ER. The width and height of the waveguide were designed with the same value 1 um in order to support the single-mode condition. The Q factor is a key parameter for the microring resonators and low loss will lead to a high Q factor. Therefore, the bend loss of the microring needs to be optimized. The relationship between bend loss and the microring radius, R, was simulated with Rsoft, and the result is shown in Figure 2. It can be seen that the bend loss will decrease sharply and then tend to be 0 gradually with increasing radius. Figure 2 also shows that there is almost no extra bend loss when R = 220 \(\upmu\)m, which was chosen as the radius of the microring resonator. The MMI coupler is based on the self-imaging effect and its operation principle can be described as below. The input optical field excites the modes in the multimode waveguide at the junction between input waveguide and multimode waveguide. At periodic intervals along the propagation direction, the modes will overlay with each other and finally form single or multiple images of the input optical field [29]. By placing output waveguides at the positions of the images, couplers with different coupling efficiencies can be obtained. The structure of the MMI coupler is shown in Figure 3. L and W are the length and width of the multimode waveguide. D is the separation distance between the output ports. Figure 1: (**a**) The schematic configuration of microring resonator with multimode interference (MMI) coupler; (**b**) The cross-section of the polymer-based ridge waveguide. Figure 2: The relationship between bend loss and the radius of the microring. For a 3 dB 2 \(\times\) 2 MMI coupler, the length of the multimode waveguide can be calculated as:  
\[L=p\frac{L_{\pi}}{2}\quad p=1,\ 2,\ 3,\dots \tag{1}\]  
where \(L_{\pi}\) is the beat length of the two lowest-order modes and can be expressed as:  
\[L_{\pi}=\frac{\pi}{\beta_{0}-\beta_{1}}\approx\frac{4 nW_{e}^{2}}{3\lambda_{0}} \tag{2}\]  
where \(\beta_{0}\) and \(\beta_{1}\) are the propagation constants of the fundamental and the first-order lateral modes, respectively, \(\lambda_{0}\) is the free-space wavelength, \(n\) is the effective refractive index, and \(W_{e}\) is the effective width of the multimode waveguide, which is approximately equal to \(W\)[29]. The separation between the input/output ports is \(D=W\)/3. The width of the multimode waveguide was designed as 9 \(\upmu\)m, and the length could be calculated as 98 \(\upmu\)m through Equation (1). In order to calculate the exact value of the multimode waveguide length, the optimization was carried out through the beam propagation method (BPM). The relationship between the powers at the two output ports and the multimode waveguide length are illustrated in Figure 4.  
It can be seen that both of the output powers increased and then decreased with increasing multimode waveguide length, and that a maximum value existed. The two output powers had the same value at \(L=107\)\(\upmu\)m, which is not at the point of maximum values. However, low loss is more important than equal splitting ratio in our work.