# Overcoming Challenges in Large-Core SI-POF-Based System-Level Modeling and Simulation
## 3 Component Modeling
### SI-POF Modeling  
Two variant yet equivalent approaches to model SI-POFs are described here. Both use the power-flow differential equation to describe light transmission through the fiber, but one solves this equation in the temporal domain, obtaining the intensity as a function of time, while the other works in the frequency domain with its Fourier transform and uses matrices to describe fiber segments of different lengths. The basis of the first modeling solution for the SI-POF is the Glope power-flow equation in the time domain, which describes the flow of optical power inside the fiber as a function of internal angle \(\theta\)[20; 23; 29; 52] according to:  
\[\frac{\partial p(\theta,z,t)}{\partial z}+\frac{1}{v_{g}(\theta)}\cdot\frac{ \partial p(\theta,z,t)}{\partial t}=-\alpha(\theta)\cdot p(\theta,z,t)+\frac{ 1}{\theta}\cdot\frac{\partial}{\partial\theta}\bigg{(}\theta D(\theta)\cdot \frac{\partial p(\theta,z,t)}{\partial\theta}\bigg{)} \tag{1}\]  
where \(p(\theta,z,t)\) is the intensity as a function of internal angle, distance \(z\), and time \(t\). Furthermore, \(v_{g}(\theta)\) is an angle-dependent group velocity accounting for modal dispersion, \(\alpha(\theta)\) is an angle-dependent power attenuation accounting for differential mode attenuation (DMA), and \(D(\theta)\) is the diffusion function that accounts for power coupling during propagation. We define the angle-dependent group velocity as [29]:  
\[v_{g}(\theta)=\frac{c}{n_{core}}\cdot\cos(\theta) \tag{2}\]  
where \(c\) is the speed of light and \(n_{core}\) is the core index. For \(D(\theta)\), we use a sigmoid function [23]:  
\[D(\theta)=D_{0}+\frac{D_{1}}{1+D_{2}e^{\sigma_{d}^{2}\theta^{2}}} \tag{3}\]  
where \(D_{0}\), \(D_{1}\), \(D_{2}\), and \(\sigma_{d}\) are fitting parameters of our experimental data for a specific fiber type. Optionally, we may set \(D(\theta)=D_{0}\)[23], thereby neglecting any angular dependence. For \(\alpha(\theta)\), the angular dependence can be modeled quadratically using [20]:  
\[\alpha(\theta)=\gamma\cdot\left(1+A_{2}\theta^{2}\right) \tag{4}\]  
where \(\gamma\) is a power attenuation coefficient and \(A_{2}\) is a fitting parameter. Alternatively, following the treatment in [23], we may define \(\alpha(\theta)\) using:  
\[\alpha(\theta)=\gamma+\frac{1}{\theta Q(\theta)}\cdot\frac{\partial}{\partial \theta}\bigg{(}\theta D(\theta)\frac{\partial Q(\theta)}{\partial\theta} \bigg{)} \tag{5}\]  
where \(Q(\theta)\) is the SSD defined using a product of sigmoid functions:  
\[Q(\theta)=\frac{1+e^{-\sigma_{1}^{2}\theta_{1}^{2}}}{1+e^{-\sigma_{1}^{2}( \theta_{1}^{2}-\theta^{2})}}\cdot\frac{1+e^{-\sigma_{2}^{2}\theta_{2}^{2}}}{1 +e^{-\sigma_{2}^{2}(\theta_{2}^{2}-\theta^{2})}} \tag{6}\]and \(\sigma_{1}\), \(\sigma_{2}\), \(\theta_{1}\), and \(\theta_{2}\) are fitting parameters. As a first step to solve Equation (1), we assume that the intensity is defined up to some maximum internal angle \(\theta_{max}\) (which may be larger than the maximum as determined by the fiber's NA [23; 29]) and then, as described in [29], we discretize the \(\theta\) dimension by sampling over the interval [0, \(\theta_{max}\)] at \(N\) points, each of which corresponds to a discrete internal angle \(\theta\) whose intensity in the fiber is described by \(p(\theta,z,t)\). We then numerically solve the resulting equations to determine the output intensity \(p(\theta,z=L,t)\), where \(L\) is the fiber length. An iterative time domain split-step procedure can be applied [53; 54], wherein the algorithm alternates between propagation over a distance \(\Delta z\), and coupling/attenuation calculated via numerical integration [55]. Once the output intensity \(p(\theta,z=L,t)\) is obtained, any required information about the transmitted signal can be calculated.