## 3 Design of a 3 dB Directional Coupler  
An integrated photonic 3 dB directional coupler is a crucial component in optical communication systems, facilitating efficient power splitting and combining two waveguides on a chip [24]. It achieves a balanced power division (50:50) through precise design and fabrication techniques, where half of the input power is transferred to each output port. This device operates based on the principle of evanescent field coupling, enabling the seamless transmission of optical signals with minimal loss and crosstalk. In general, the coupling length (\(\mathrm{L_{DC}}\)) can be calculated using the following formula:  
\[\mathrm{L_{DC}}=\frac{\pi}{2\beta_{\mathrm{eff}}}\]  
where \(\beta_{\mathrm{eff}}\) is the effective propagation constant of the coupled modes, and the effective propagation constant \(\beta_{\mathrm{eff}}\) depends on the waveguide parameters, such as the waveguide's refractive indices, the waveguide's width and height, and the wavelength of light.  
The transmission spectrum is plotted for a directional coupler based on an SOI platform with \(\mathrm{L_{DC}}\) varying between 0.5 \(\upmu\)m and 12 \(\upmu\)m. The transmission is calculated as \(\mathrm{P_{out}}/\mathrm{P_{in}}\), where \(\mathrm{P_{out}}\) and \(\mathrm{P_{in}}\) are the output and input power, respectively. When designing with the COMSOL Multiphysics tool (v.6.1), it considers several types of losses of the light propagation in optical waveguides. These include absorption losses, where the waveguide material absorbs light energy. The software also accounts for radiation losses from waveg  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Expression** & **Value** \\ \hline w & Width of the waveguide & 250, 260, 270 nm \\ g & The gap between waveguides of directional coupler & 100, 125, 150, 175 nm \\ R & Radius & 2 \(\upmu\)m \\ \(\mathrm{L_{DC}}\) & Length of directional coupler & 0.5 to 12 \(\upmu\)m \\ L & Length of MZI structure & 1, 3, 5, 7, 10 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters of LT-MZI structure.  
Figure 1: Schematic representation of LT-MZI structure embedded with PHMB polymer on the sensing arm for CO\({}_{2}\) gas sensing. [PERSON] presents the H-field distribution at 1550 nm in the cross-sectional view of the waveguide covered with PHMB polymer.  
uide bends and mode mismatches, as well as coupling losses at interfaces and junctions. Additionally, mode conversion losses occur when higher-order modes are present or when unintended mode coupling happens. By incorporating these factors, accurate modeling of waveguide performance and efficiency is obtained. Incremental changes in the width (w) of the waveguide core from 250 nm to 280 nm are made to examine their influence on the coupling length, as depicted in Figure 2a-d. Notably, the beat length of the directional coupler demonstrates an increase as w rises from 250 nm to 280 nm. In optical directional couplers, the beat length signifies the length over which the phase difference between the coupled modes changes by 2\(\pi\) radians (one full cycle). This metric is influenced by factors including the wavelength of light, refractive indices of the materials, and the physical dimensions of the coupler. Consequently, for w = 250 nm, achieving a 3 dB power distribution remains feasible even at an \(\mathrm{L_{DC}}\) as small as 3 \(\mathrm{\SIUnitSymbolMicro\meter}\). Figure 2e,f show the normalized H-field distribution in the directional coupler for \(\mathrm{L_{DC}}\) = 4 \(\mathrm{\SIUnitSymbolMicro\meter}\) and \(\mathrm{L_{DC}}\) = 3 \(\mathrm{\SIUnitSymbolMicro\meter}\), respectively.