# Compact SOI Dual-Mode (De)multiplexer Based on the Level Set Method
## 2 Design Principle  
### Design Target  
The (de)MUX is designed on the SOI platform with a top silicon layer thickness of 220 nm, and equally thick buried oxide and cladding layers measuring 2 \(\upmu\)m each. The device in the design region, as shown in Figure 1, measures 9.4 \(\upmu\)m \(\times\) 3.6 \(\upmu\)m. The input waveguide has a width of 0.9 \(\upmu\)m to support both TE\({}_{0}\) and TE\({}_{1}\) modes. The output ports solely support the TE\({}_{0}\) mode, with a width of 0.5 \(\upmu\)m. The gap between two outputs is 0.8 \(\upmu\)m to reduce the mode coupling between them.  
To obtain the desired structure of dual-mode (de)MUX, we first initialized a simple shape in the design region to prepare for the subsequent iterations. To iterate the initial shape towards the optimal structure, we can consider the entire process as a constrained optimization problem. This is because the propagation of mode lights and modal transformation within the waveguide are governed by the constraints imposed by [PERSON]'s equations,  
\[\begin{array}{rl}\text{max }FOM&=f(\mathbf{E},\mathbf{H},\varepsilon_{r})\\ s.\ t.\ g_{i}(\mathbf{E},\mathbf{H},\varepsilon_{r})&=0,\ i=1,\ 2\dots\\ h_{j}(\varepsilon_{r})&\leq 0,\ j=1,\ 2\dots\end{array} \tag{1}\]  
where we define the figure of merit (_FOM_) to assess the performance of a device in terms of the electromagnetic field (\(\mathbf{E},\mathbf{H}\)) and the relative permittivity (\(\varepsilon_{r}\)). The _FOM_ is established by ensuring the fulfillment of [PERSON]'s equations \(g_{i}(\mathbf{E},\mathbf{H},\varepsilon_{r})=0\), which describe the spatial distribution of \(\mathbf{E}\) and \(\mathbf{H}\). The constraint equations \(h_{j}(\varepsilon_{r})\leq 0\) impose limitations on the materials and sizes of the designed structure.  
In our analysis, we calculate the propagation of the electromagnetic field in the structure for each of the two modes separately. We then multiply their respective _FOM_ to obtain the overall _FOM_ of the device, rather than simply adding the individual _FOM_ together.  
Figure 1: The basic structure of dual-mode demultiplexer. The red line indicates the propagation of the input TE\({}_{0}\) mode, which finally outputs at the first port, while the green line, TE\({}_{1}\), outputs at the second port.  
Multiplication offers the advantage of maintaining a balanced transmission efficiency between the two modes. The expanded form of the total \(FOM\) is as follows:  
\[\begin{array}{ll}FOM_{total}&=f_{1}(\mathbf{E}_{1},\mathbf{H}_{1},\mathbf{\varepsilon}_{1}) \cdot f_{2}(\mathbf{E}_{2},\mathbf{H}_{2},\mathbf{\varepsilon}_{2})\\ &=\prod\limits_{i=1}^{2}\frac{\left|\int_{S_{int}}(\mathbf{E}_{i,t,\mathbf{x}}\times \mathbf{H}_{i,\mathbf{x}}+\mathbf{E}_{i,\mathbf{x}}\times\mathbf{H}_{i,\mathbf{x}})\cdot dS\right|^{2} }{4\left|\int_{S_{int}}(\mathbf{E}_{i,t,\mathbf{x}}\times\mathbf{H}_{i,\mathbf{x}})\cdot dS \right|\left|\int_{S_{int}}(\mathbf{E}_{i,t,\mathbf{x}}\times\mathbf{H}_{i,\mathbf{x}})\cdot dS \right|}\end{array} \tag{2}\]  
where the lower subscript 1 represents the corresponding parameters of TE\({}_{0}\) mode, while the lower subscript 2 indicates TE\({}_{1}\) mode. \(\mathbf{E}_{in}\) (\(\mathbf{E}_{out}\)) represents the electromagnetic field propagating from the input (output) port as TE\({}_{0}\) (\(i=1\)) or TE\({}_{1}\) (\(i=2\)), while the target field, denoted as \(\mathbf{E}_{tar}\), is the distribution of the target electric field on the corresponding port.  
To solve the constraint optimization problem, the adjoint method and level set method are employed.