# Overcoming Challenges in Large-Core SI-POF-Based System-Level Modeling and Simulation
## 3 Component Modeling
### SI-POF Modeling
Thus, the temporal signal at a fiber length \(L\) is obtained by integrating over the solid angle:  
\[p(z=L,t)=\int\limits_{\varphi=0}^{2\pi}\int\limits_{\theta=0}^{\theta_{max}}p (\theta,z=L,t)\sin(\theta)d\theta d\varphi=2\pi\int\limits_{\theta=0}^{\theta_ {max}}p(\theta,z=L,t)\sin(\theta)d\theta \tag{7}\]  
where \(\varphi\) and \(\theta\) are the polar and azimuthal angle, respectively, and circular symmetry is assumed (that is, the function is independent on the angular coordinate \(\varphi\)). On the other hand, the angle-dependent intensity can be obtained by integrating the signal over time:  
\[p(\theta,z=L)=\int\limits_{t=-\infty}^{+\infty}p(\theta,z=L,t)dt \tag{8}\]  
Finally, to obtain the total power carried by the signal up to a length \(L\) requires performing both integrals. There are a number of important considerations to keep in mind as a result of the above approach. First, the input signal is converted to an equivalent intensity representation; therefore, any phase information in the signal is lost. Second, we typically neglect the effects of Fresnel reflection at the input, which may be small [26], and if not, can be accounted for via inclusion of an optical attenuator during simulation. Third, the fiber can be configured to reject any portion of the input signal that resides at an angle outside of the NA. Finally, if the input signal consists of different wavelength and polarizations, \(X\) and \(Y\) polarizations are treated separately, as are different wavelengths, and therefore, a separate Glope power-flow equation will be solved in each case. In the second modeling methodology, the Glope power flow equation is solved in the frequency domain. Equation (9) shows the Fourier transform of Equation (1):  
\[\frac{\partial p(\theta,z,\omega)}{\partial z}=-\bigg{(}\alpha(\theta)+j\omega \frac{1}{v  
where \(\mathbf{P}(z,\omega)\) are vectors, whose components are the intensity values for each discretized angle and are different at each \(z\) and for each frequency \(\omega\). Although it is usually called a propagation matrix, \(\mathbf{M}(\omega)\) is, in fact, not only one matrix but a set of tri-diagonal matrices that are calculated from the angular diffusion and attenuation functions as defined in Equations (3) and (5). To fully characterize the fiber, we need the complete set of matrices \(\mathbf{M}(\omega)\), one for each frequency, which have complex values except for \(\mathbf{M}(0)\). For the typical case where \(z_{1}\) is at the input source, \(\mathbf{P}_{\mathbf{in}}(\omega)=\mathbf{P}(0,\omega)\) and the output is at the end of a fiber segment of length \(L\), \(\mathbf{P}_{\mathbf{out}}(\omega)=\mathbf{P}(L,\omega)\), Equation (11) describes how the output intensity at the end of that fiber segment can be obtained from the input intensity:  
\[\mathbf{P}_{\mathbf{out}}(\omega)=\mathbf{M}_{\mathbf{L}}(\omega)\cdot\mathbf{ P}_{\mathbf{in}}(\omega) \tag{11}\]  
where the set of matrices \(\mathbf{M}_{\mathbf{L}}(\omega)\) are obtained as powers of the basic tri-diagonal matrices, \(\mathbf{M}(\omega)^{L/\Delta z}\), and account for the effect of the fiber on the input optical intensity. Therefore, to obtain the intensity after propagation through a given length \(L\), it is necessary to calculate the matrix product in Equation (11) for each frequency. The resulting output intensity \(\mathbf{P}_{\mathbf{out}}(\omega)\) contains the information required to obtain all the important spatial and temporal parameters related to fiber transmission. Figure 1a illustrates the operation described in Equation (11) with a set of vectors at the input (right) and another at the output (left), which is obtained by performing the corresponding matrix product of the former with the set of matrices. The absolute values of matrices \(\mathbf{M}_{\mathbf{L}}(\omega_{i})\) are shown in Figure 1b as an example for the case of a commercial SI-POF of 25 m and three different values of frequency.