## 3 Component Design and Modeling
### Interleaved AMMI Multiplexer/Demultiplexer  
The wavelength division multiplexing function is carried out by an interleaved AMMI device [45; 46; 47], which is schematically shown in Figure 1a. Notably, the device contains a bidirectional grating coupler which functions as the perfectly vertical grating coupler as well as the power splitter/combiner. From the view of optical filtering, this device can be seen as a three-stage optical filter. At the first stage, the grating coupler provides a broadband wavelength window that allows all the wavelength signals to be coupled into the waveguide plane. As the second stage, the MZI functions as a comb filter with two complementary outputs that divides the optical signals into two groups with odd (\(\lambda_{1}\), \(\lambda_{3}\), \(\lambda_{5}\), \(\lambda_{7}\)) and even channel numbers (\(\lambda_{2}\), \(\lambda_{4}\), \(\lambda_{6}\), \(\lambda_{8}\)), respectively. Lastly, two \(1\times 4\) AMMIs cascaded with the MZI demultiplex the two signal groups into separate wavelength channels, respectively. The key of designing such a _WDM_ device is to achieve a perfect matching between the MZI filter and the AMMI filters, which means firstly the free spectral range (_FSR_) of the MZI should be equal to the channel spacing of the two AMMIs:  
\[\Delta\lambda_{\text{MZI}}=\Delta\lambda_{\text{AMMI}} \tag{13}\]  
and the design rules of the AMMIs can be expressed as the following equations:  
\[L_{i}=\frac{4N_{eff}W_{m}^{2}}{\lambda_{i}} \tag{14}\]  
\[\Delta\lambda_{min}>\frac{W_{a}+x_{min}}{4N_{eff}sin\theta}\bigg{(}\frac{ \lambda_{i}}{W_{m}}\bigg{)}^{2} \tag{15}\]  
\[N_{max}<\frac{2W_{m}cos\theta}{W_{a}} \tag{16}\]  
Equation (14) represents the inverted self-imaging axial positions (\(L_{i}\)) for different wavelength channels (\(\lambda_{i}\)) of the AMMIs, where \(N_{eff}\) represents the effective index of the multimode waveguide fundamental mode, \(W_{m}\) is the width of the multimode interference waveguide. Equations (15) and (16) represent the design limitations of the AMMIs on channel spacing and channel count. \(W_{a}\) represents the width of the input/output access waveguides of the AMMI devices. As \(W_{a}\) is usually a few micrometers in width, tapered waveguides are utilized for adiabatic transmission and mode conversion between the access waveguides and single mode waveguides. The former is mainly limited by the minimum waveguide separation \(x_{min}\) for avoiding the cross-coupling of adjacent waveguide channels, while the latter can be obtained from the available access waveguide window in the simulated field profile of Figure 10b. Following the design rules, the AMMIs are designed based on two considerations. Firstly, as the AMMIs are designed to be folded back to be alongside the MZI modulators, a footprint of 2-4 mm is acceptable for our transmitter design. Secondly, the insertion loss should be as small as possible while the required channel spacing and channel count are fulfilled simultaneously. As mentioned before, the 1-dB bandwidth of the apodized bidirectional grating coupler is about 40 nm. Thus, a channel spacing of 10 nm and channel count of 4 (halved to 5 nm and doubled to 8 by interleaved design) is preferred and chosen. To minimize the optical loss and crosstalk, the design parameters are fully explored by the Eigenmode Expansion (EME) [48] solver embedded in the commercially available software of Lumerical mode solutions. For fabrication convenience, all the waveguides of the AMMIs are designed with a ridge waveguide structure of 220 nm in height and 150 nm in etch depth, which is the same as that of the PN phase shifters. The optimized results for the design parameters of the AMMIs are listed in Table 4. Figure 10a shows the calculated field intensity profile of a single channel AMMI with an incident wavelength of 1550 nm. The high-fidelity self-imaging phenomenon can be clearly seen which indicates a low transmission loss. Figure 10c,d show the simulated optical transmission spectra of two\(1\times 4\) AMMIs.