# Wideband E\({}_{00}\)-E\({}_{10}\) Silicon Mode Converter Based on 180 nm CMOS Technology
## 2 Device Design  
In MMI waveguides, the self-imaging effect could reproduce the input field profile in single or multiple images at periodic intervals along the propagation direction. Two scenarios exist in the MMI effect. One is the general interference, another case is the restricted interference. The beat length \(L_{\pi}\) can be described by  
\[L_{\pi}=\frac{\pi}{\beta_{0}-\beta_{1}}\approx\ \frac{4n_{r}w^{2}}{3\lambda_{0}} \tag{1}\]  
where \(\lambda_{0}\) is the operation wavelength, \(n_{r}\) is the refractive index of the core waveguide, and \(w\) is the width of the MMI coupler [27]. When one input waveguide is placed at the center of the MMI coupler, the distance \(L\) of \(M\)th \(N\)-fold images can be defined as  
\[L=\frac{M}{4N}(3L_{\pi}). \tag{2}\]  
For the MMI coupler with a certain length, and when only the fundamental mode or first-order mode is input, we have \(M=N=1\). The length of the MMI coupler \(L_{\text{MMI}}\) is  
\[L_{\text{MMI}}=\frac{3}{4}L_{\pi}. \tag{3}\]  
For instance, for the MMI coupler shown in Figure 1 below, two E\({}_{00}\) modes with \(\pi\) phase difference are launched into the MMI section through two single modes. The interference between two E\({}_{00}\) modes with different phases in the MMI waveguide leads to the E\({}_{00}\)-E\({}_{10}\) mode conversion. The first-order E\({}_{10}\) mode will be excited at the output port on the right side of the MMI waveguide. The proposed silicon E\({}_{00}\)-E\({}_{10}\) mode converter that is constructed by three sections of symmetric Y-branch 3 dB splitter, phase shifter (PS) waveguide, and 2 \(\times\) 1 MMI coupler is shown in Figure 2a. Firstly, two E\({}_{00}\) modes with the same mode power and phase are input into the PS waveguides via the 3 dB splitter. The \(\pi\)-phase difference would emerge between two E\({}_{00}\) modes after passing through the PS waveguide constructed by cascaded tapered waveguides with different lengths. The interference between two E\({}_{00}\) modes with different phases in the MMI coupler leads to the E\({}_{00}\)-E\({}_{10}\) mode conversion. Figure 2b shows the cross-sectional view of the silicon waveguide. When the silicon core thickness is fixed to 220 nm, the variation in effective indices (\(n_{\text{eff}}\)) of two modes with the silicon core width is illustrated in Figure 3a. To support the E\({}_{00}\) mode, the width of the input waveguide (\(w_{1}\)) is chosen as 0.4 \(\upmu\)m. We set the width of the output waveguide (\(w_{3}\)) as 0.8 \(\upmu\)m to support both the E\({}_{00}\) mode and E\({}_{10}\) mode. For getting \(\pi\)-phase difference, it is necessary to strictly manipulate the dimensional parameters of PS waveguides. With FD-BPM calculating, the width of the Y-branch 3 dB coupler (\(w_{1}\)) is set to be \(w_{1}\) = 0.4 \(\upmu\)m, and PS waveguides (\(w_{2}\)) are set to be \(w_{2}\) = 0.28 \(\upmu\)m to support E\({}_{00}\) mode. Additionally, the lengths of different tapered waveguides \(l_{1}\), \(l_{2}\), and \(l_{3}\) are selected to be 3.6 \(\upmu\)m, 12.4 \(\upmu\)m, and 8 \(\upmu\)m, respectively. Figure 1: Diagram of MMI as E\({}_{00}\)-E\({}_{10}\) mode converter. Here, the width of MMI coupler \(w_{\text{MMI}}\) is chosen to be 3.3 \(\upmu\)m, thereby, the beat length \(L_{\pi}\) is calculated to be 32.5 \(\upmu\)m at 1550 nm according to Equation (1). The length of MMI coupler that can be calculated by Equation (3) is optimized to be 25.3 \(\upmu\)m for a compact size. With above parameters, the mode field distribution of the proposed mode converter at 1550 nm is shown in Figure 3b. To be mentioned, the length of the Y-branch is chosen to be 25 \(\upmu\)m, the length of the taper that connects the Y-branch and PS is chosen to be 8 \(\upmu\)m, and the length of the straight waveguide that connects PS and MMI is chosen to be 5 \(\upmu\)m. These parameters are compromise choices after comprehensive consideration of optical loss and device size.