# Fabrication Tolerances' Impact on an ODAC-Based PAM-4 Transmitter
## 2 Proposed Model  
An oDAC PAM-4 transmitter is presented in Figure 1, where the signal from a continuous wave (CW) laser is split into the two arms of the device through an input coupler with input coupling factor (\(K_{in}\)) and then recombined through an output coupler with output coupling factor (\(K_{out}\)). The non-return-to-zero (NRZ) electrical signals (NRZ\({}_{1}\) and NRZ\({}_{2}\)), which act as the most significant bit (MSB) and least significant bit (LSB), are applied to drive the modulators, as shown in Figure 1.  
Figure 1: Structure of an oDAC PAM-4 transmitter. Due to the sensitive nature of interferometers, imprecision in the implementation of this type of device can reduce their performance during signal detection. For instance, eye diagrams at different values of \(K_{in}\) are presented in Figure 2a-c. This is a challenge that requires further investigation. To model these effects, the FPPGA device developed by iPronics is used. The FPPGA comprises a set of MZMs or programmable unit core (PUC)s, which are hexagonally connected via a replicated geometric. Each PUC from the arrays is itself an MZM, as shown in Figure 3, which can be configured as a bar state with \(\kappa=0\), cross state with \(\kappa=1\), and as tunable with \(\kappa=\) any value between 0 and 1. The mathematical expression of the output signal of a typical PUC/MZM, as shown in Figure 3, can be expressed as  
\[\left[ \begin{array}{c}(E_{\text{O1}})_{n}\\ (E_{\text{O2}})_{n}\end{array} \right]= \left[ \begin{array}{cc}\sqrt{1-\kappa_{2}}&j\kappa_{2}\\ j\kappa_{2}&\sqrt{1-\kappa_{2}}\end{array} \right]\times\left[ \begin{array}{ cc}e^{-j\phi_{1}}&0\\ 0&e^{-j\phi_{2}}\end{array} \right] \tag{1}\] \[\times\left[ \begin{array}{cc}\sqrt{1-\kappa_{1}}&j\kappa_{1}\\ &\\ j\kappa_{1}&\sqrt{1-\kappa_{1}}\end{array} \right]\times\left[ \begin{array}{ cc}(E_{I1})_{n}\\ (E_{I2})_{n}\end{array} \right]\]  
Figure 3: PUC loaded with phase shifter and heater on both of its arms and its mode of implementation as the bar state, cross state and tunable coupler. Figure 2: Eye diagrams of an oDAC-PAM-4 signal: (**a**) \(K_{in}=0.2\), (**b**) \(K_{in}=0.36\), (**c**) \(K_{in}=0.4\). where \(V_{1}\) and \(V_{2}\) are the applied voltages to tune the phase shifter (PS) in order to create \(\phi_{1}\) and \(\phi_{2}\) phase changes, \(\kappa_{1}\) and \(\kappa_{2}\), PUC input and output coupling coefficients, \(V_{\pi}\), the half-wave voltage and \(\phi_{1|2}=\frac{\pi(V_{1|2})n}{V_{\pi}}\). A hexagonal array of the PUCs is shown in Figure 4a and it can be electrically tuned to provide functionalities such as filtering, optical interconnects and several other optical functionalities. Furthermore, our proposed model, in Figure 4b,c, developed from the PUC arrays in Figure 4a, comprises two MZMs acting as amplitude modulator (AM) (MZM\({}_{MSB}\) and MZM\({}_{LSB}\)), which are arranged in parallel to form the two arms just as another major MZM circuit. Light from a CW laser can then be split into the two arms of the outer MZMs via an input optical splitter (which itself is an MZM) by setting its \(K_{in}\) to 0.36, as recommended in [24]. In addition, phase modulator (PM)s  
equivalent to the proposed design before fabrication. From the results in Figure 5b-d, \(L_{00}\) remains unchanged despite increased \(\sigma_{n}\) since this level implies an absence of light. In addition, since rail to rail modulation is used, both \(L_{00}\) and \(L_{11}\) exhibit compression, as can be observed from the histograms. However, increased \(\sigma_{n}\) significantly impaired \(L_{01}\) and \(L_{10}\) due to overlapping, which may result in decision error. \(L_{11}\) maintains improvement when compared with \(L_{01}\) and \(L_{10}\). This is because \(L_{11}\) has the highest signal-to-noise ratio since the highest signal power occurred at this level. Nevertheless, the impact of increased \(\sigma_{n}\) can be seen to cause \(L_{11}\) to fold back towards \(L_{10}\). Next, we have optimized to ensure the equidistance of the amplitude levels of our PAM-4 signal.