# Coupling of Modes in Step-Index Plastic Optical Fibers by Using D-Shape Technique
## 3 Material and Methods, Components, and Experimental Setup  
This work introduces a D-shaped design to filter out high-order modes effectively, as shown in Figure 2. The AB dimension represents the width of the D-shape of 20 mm, and the CD dimension represents its depth of 0.25 mm. The optical beam enters the POF, passes through the D-shaped region, and exits on the opposite side. Due to their steep propagation angles, high-order modes refract at the interface between the D-shaped core and surface. Meanwhile, low-order modes traverse the D-shaped area until they exit the fiber's end. For the evaluation of the behavior of modes along the POF, the power flow equation developed by [PERSON] was applied [26]. Based on the assumption that mode coupling in multimode optical fibers occurs between neighboring modes, [PERSON] formulated the time-independent power flow equation as follows:  
\[\frac{\partial P(\theta,z)}{\partial z}=-\alpha(\theta)P(\theta,z)+\frac{D}{ \theta}\frac{\partial}{\partial\theta}\bigg{(}\theta\frac{\partial P(\theta, z)}{\partial\theta}\bigg{)} \tag{1}\]  
P(\(\theta\),z) is the angular light power distribution in the POF, D is the mode coupling coefficient, \(\theta\) is the propagation angle in the POF core, and z is the reference point of the POF length relative to the light input. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Technique** & **Method** & **Reference** \\ \hline Mode Scrambler & This technique aims to create controlled disturbances in the propagation modes of light within a fiber. It does this by using specially made cylinders that bend the Plastic Optical Fiber (POF). This bending process facilitates the coupling or interaction between different modes of light within the fiber, enhancing the fiber’s performance for specific applications. & [24] \\ Airgap Filter & A spatial filter creates an air gap, reducing the numerical aperture and modal dispersion. & [17] \\ Microstructured POF & Designing the fiber with a specific pattern of air holes reduces the number of propagation modes, minimizing modal dispersion. & [7] \\ Strained POF & Applying strain to the fiber enhances mode coupling, reducing dispersion. & [25] \\  & This consists of wrapping a section of optical fiber around a cylindrical mandrel of a specified diameter for several turns. & \\ Mandrel Wrap & This creates a bend that induces additional losses for higher-order modes without significantly affecting lower-order modes. & [26] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 2: A schematic diagram of the D-shape in POF. To solve the differential equation proposed by [PERSON] (2018), we adopted the implementation of numerical methods based on explicit finite differences. This approach allowed us to develop a robust and efficient computational solution to the equation. As a result of implementing these methods, we derived a reformulated version of Equation (1):  
\[P_{i,j+1}=\left(\frac{\Delta zD}{\Delta\theta^{2}}-\frac{\Delta zD}{2\theta i,j \Delta\theta}\right)P_{i-1j}+\left(1-\frac{2\Delta zD}{\Delta\theta^{2}} \right)P_{i,j}+\left(\frac{\Delta zD}{2\theta i,j\Delta\theta}+\frac{\Delta zD} {\Delta\theta^{2}}\right)P_{i+1j} \tag{2}\]  
A dedicated computational tool was developed to apply the Gloge equation and investigate the behavior of the modes along the POF. The tool was created in MATLAB using the APP Designer, which enables the manual creation of a graphical user interface (GUI). As shown in Figure 1, this tool considers different angles of light entering the POF to analyze the angular light power distribution at the POF output from two perspectives: 1--the angle of light entering the POF; 2--the length of the POF. Figure 3 is a screenshot of the development of the application. It shows the angular light power distribution at the POF output, considering the idea in Figure 1.  
Figure 3a shows that it is possible to see that there is a length of the POF from which the convergence of the angular light power distribution at the POF exit becomes evident.