## 2 Design of the 1 \(\times\) 4 Channel Wavelength Demultiplexer
### MZI Based 1 \(\times\) 4 Channel Wavelength Demultiplexer  
Figure 4 shows the schematic of the proposed demultiplexer architecture. The demultiplexer is designed to filter four 40 Gb/s data rate channels, where the wavelength separation between the adjacent channels is 3.2 nm. Therefore, in the demultiplexer structure shown in Figure 4, each of the four output channels is designed to have \(4\times 3.2\) nm = 12.8 nm free spectral range (FSR) in its corresponding optical spectrum. The detailed design method of the MZI based optical demultiplexer/filter can be found in [15,16]. In addition to the extra delay lengths (_L1_, _L2_ and _L3_) each of the MZI filters contains two equal length s-shaped bend waveguides (= 25.13 mm long), with a bending radius of 4 mm. As shown in Figure 4, on top of these bend waveguides, 12 mm long heaters are situated for thermal heating. To achieve 12.8 nm FSR at the four output channels (_ch-1_, _ch-2_, _ch-3_ and _ch-4_) of the demultiplexer, the first MZI (MZI with the extra delay waveguide length, _L1_) is designed to have 6.4 nm FSR at its two output channels (_A1_ and _A2_), where the wavelength separation between the two channels is 3.2 nm. We have set the delay length, _L1_ = 249.5 um in the first MZI to obtain _FSR_ = 6.4 nm at its output channels (_A1_ and _A2_). The following equation is used to estimate the value of _L1_:  
\[L_{1}=\frac{\lambda^{2}}{FSR\ \times\ n_{g}} \tag{1}\]  
where, the group index, \(n_{g}\) = 1.505 at \(\lambda\) = 1550 nm wavelength (from Figure 2b). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{L_{MMI}}\) & \multicolumn{2}{c}{**Percentage of the Input**} & \multicolumn{2}{c}{**Input Power Coupled**} \\ (\(\mathbf{\mu}\)M) & **Power Coupled to the Two**} & \multicolumn{2}{c}{**to the Two O/P Waveguides**} & \\  & \multicolumn{2}{c}{**O/P Waveguides**} & \multicolumn{2}{c}{**Waveguides (dB)**} \\ \hline  & O/P 1 & O/P 2 & O/P 1 & O/P 2 \\ \hline
1850 & 42\% & 40\% & \(-\)3.76 & \(-\)3.97 \\
**1860** & **47.1\%** & **46.5\%** & \(-\)3.27** & \(-\)**3.32** \\
1870 & 43\% & 42.5\% & \(-\)3.66 & \(-\)3.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation values of the power coupled to the two O/P waveguides of the MMI coupler. Figure 3: (**a**) Schematic of the 2 \(\times\) 2 multimode interference (MMI) coupler with taper waveguide length, _Laper_ = 200 μm and access waveguide width, _Wa_ = 5 μm and (**b**) the electric field distribution in a 2D plane of the MMI coupler at 1550 nm wavelength. The values of the delay waveguide lengths, \(L_{2}=124.7\)\(\upmu\)m + \(L\
u z\) and \(L\)_s_ = 124.7 \(\upmu\)m, are chosen such that the corresponding MZIs containing these delay lines have an \(FSR\) = 12.8 nm. Note that an extra delay length, \(L\
u z\) = 0.257 \(\upmu\)m, is added to \(L\)_z_ in the second MZI, to induce a phase shift of \(\pi\)/2 in the optical signal of the corresponding delay arm. This extra \(\pi\)/2 phase shift is necessary to obtain the required wavelength separation of 3.2 nm and low crosstalk between the output channels _ch-1_ and _ch-2_. Figure 5a presents the simulation result of the overlapped optical transmission response of the four channels (_ch-1_, _ch-2_, _ch-3_ and _ch-4_) of the demultiplexer, where a \(\sim\)15 dB crosstalk between the channels is obtained. We find that it is important to implement thermal tunability in the delay arms of the MZIs to ensure low crosstalk as well as the required channel separation between the output channels. As an example, a 100 nm length offset error is added in the simulation to all the three delay arms (\(L_{1}\) to \(L_{3}\)) of the demultiplexer to assess the impact of fabrication error on the demultiplexer. From Figure 5b, it is clear that this error results in a significant degradation in the crosstalk performance of the demultiplexer (from 15 dB to 8 dB). The 100 nm length offset error induces 0.19\(\uppi\) of phase error.