# Overcoming Challenges in Large-Core SI-POF-Based System-Level Modeling and Simulation
## 3 Component Modeling
### Connector Modeling  
Representing optical signals as intensity functions of the propagation angle makes it possible to implement more computationally efficient models for other components within a large-core POF-based system. Of particular interest is an efficient model that represents connectors between optical fibers, which is essential for modeling a POF-based system. Our connector model follows an approach that is similar to the one used for the SI-POF Injection model discussed above and is based on work presented in [30; 50; 51]. The key assumption of the model is that for a particular connector, the output intensity at some angle \(\theta_{i}\) is a weighted linear summation of the input powers at all input angles. Similar to Equation (10), the output intensity as a function of the input intensities can be represented as:  
\[p_{out}(\theta_{i},t)=\sum_{j=0}^{N}m_{comm}(\theta_{i},\theta_{j})\cdot p_{in }(\theta_{j},t) \tag{17}\]where \(m_{com}(\theta_{i},\,\theta_{j})\) is a coupling coefficient between the input at \(\theta_{j}\) and the output at \(\theta_{i}\). Likewise, if we represent the intensities as vectors \(\mathbf{P_{out}}\) and \(\mathbf{P_{in}}\), which can be functions of either time or frequency, we can also represent this relationship as the matrix product:  
\[\mathbf{P_{out}}=\mathbf{M_{conn}}\cdot\mathbf{P_{in}} \tag{18}\]  
where \(\mathbf{M_{conn}}\) is a matrix with elements \(m_{com}(\theta_{i},\,\theta_{j})\). The advantage of this approach is that for any connector arrangement with arbitrary transverse, longitudinal, and angular misalignments, it is possible to measure an appropriate connector matrix which can be used in simulation. As was the case for the Injection model, and discussed in [30], in many cases, it is possible to fit this matrix to the Gloge power-flow equation [20; 23; 29; 30]. In this case, the interpretation of the Gloge power-flow equation is similar to that of the Injection model. We use measured data for \(\alpha\) (\(\theta\)), and a constant value for the power-coupling term \(D(\theta)=D_{0}\). It is also possible to model lateral and longitudinal misalignments between fibers via analytical expressions [50; 51]. This method is based on the hypothesis that each point in the fiber end surface acts as an independent uncorrelated source whose radiation pattern is given by \(g(\theta)\), as shown in Figure 3a. We assume this function is related to the fiber Far Field Pattern (FFP). To obtain the total radiated intensity at any point in space in cylindrical coordinates \(R(r,\,\varphi,\,z)\), the scalar contributions are added from all radiating sources that reach that point (Figure 3b). As all functions involved have circular symmetry, the zero-order Hankel Transform can be used to speed up and simplify calculations. Thus, the light intensity radiated from the end surface of this fiber is given by:  
\[R(r,z)=z^{2}\cdot\mathrm{H^{-1}}\left(\frac{a\mathrm{J_{1}}(2\pi a\rho)}{ \rho}\cdot G(z\rho)\right) \tag{19}\]  
where \(\rho\) is the transformed radial variable, \(\mathrm{H^{-1}}\) is the inverse zero-order Hankel transform, \(\mathrm{J_{1}}(2\pi\ a\ \rho)/\rho\) is the Airy pattern that is the transform of the circular aperture with radius \(a\) used to model the fiber surface, and \(\mathrm{G}(\rho)\) is the transform of \(g(r)\), which is the projection of the radiation pattern. As the radiated intensity has circular symmetry, the explicit dependency on \(\varphi\) has been omitted. Once the radiated intensity has been obtained, misalignment loss is calculated assuming that the intensity captured by the receiving fiber is the fraction of intensity radiated from the first fiber that overlaps with the core surface of the second fiber, as shown in Figure 4. In this case, the model produces  
Figure 3: (**a**) Point A in the fiber end surface has a radiation pattern given by \(g(\theta)\) and (**b**) each point in the fiber end surface (A, B, C, D, etc.) acts as an independent uncorrelated source.