# Model-Based Design and Simulation of Paraxial Ray Optics Systems
## 4 Results and Discussion
### Optical Fiber  
In an optical fiber, light travels through the core region and confined by total internal reflection (TIR) at the core-cladding interface. Transmission in multimode fibers can be treated by geometrical optics approach [24]; despite its simplicity, such model is useful for demonstrating basic concepts such as the effect of fiber geometry and material on light coupling and transmission [25], estimating waveguide losses [26], and modeling fiber sensors [27].  
Consider a fiber with core and cladding radii \(a\) and \(b\), respectively, wherein refractive indices of surrounding medium, core, and cladding are \(n_{0}\), \(n_{1}\), and, \(n_{2}\), respectively, as illustrated in Figure 4a. Input light refracts at the air-core interface (\(n_{0}<n_{1}\)) and travels in the \(yz\)-plane as a meridional ray with orientation \(\theta_{1}=\theta_{2}\) along the distance \(d_{2}=a/\theta_{1}\) (skewed rays are neglected in this model) until it reaches the core-cladding boundary (\(n_{1}>n_{2}\)). In the case of total internal reflection, light is totally reflected with \(\theta_{3}=-\theta_{2}\) and confined inside the core with consecutive transmissions along with \(d_{4}=2a/\theta_{3}\) and \(d_{6}=2a/\theta_{5}\). The TIR condition is only achieved if the orientation of input ray \(\theta_{1}\) is smaller than a maximum entrance angle \(\theta_{i}\), where NA = \(n_{0}\) sin \(\theta_{i}=({n_{1}}^{2}-{n_{2}}^{2})^{1/2}\), NA is called the fiber numerical aperture [24].  
Figure 3: Paraxial cloaking: (**a**) ray optics representation (adapted from [23]); (**b**) matrix optics model, T and L are transmission and thin lens components, respectively; (**c**) ray tracing obtained by simulation for \(-20\leq y_{0}\leq 20\) mm and \(\theta_{0}=0^{\circ}\), cloaked region is evidenced for \(200\leq z\leq 800\) mm.  
The implemented model is shown in Figure 4b assuming only two reflections for clarity, with \(y_{0}\) = 0 and \(2^{\circ}\leq\theta_{0}\leq 5^{\circ}\). As the core-cladding interface is parallel to \(z\)-axis, the reflection ABCD matrix must be adjusted to \(D=-1\) to correct the orientation of output light. Even though rays refracted into the cladding violate the paraxial approximation, the TIR condition can be verified by testing if \(\theta_{2}\leq\theta_{\text{max}}=\pi/2-\theta_{c}\), where \(\theta_{c}=\sin^{-1}(n_{2}/n_{1})\) is the critical angle [24]. Static simulations were conducted with \(a=25\)\(\upmu\)m, \(b=62.5\)\(\upmu\)m, \(n_{0}=1\), \(n_{1}=1.475\), and \(n_{2}=1.460\), so the fiber numerical aperture is NA = 0.21, yielding \(\theta_{i}=12.11^{\circ}\) and \(\theta_{\text{max}}\approx 8.18^{\circ}\), corroborating the analytical model. Ray tracing results for different orientations are plotted in Figure 4c, for example, \(\theta_{0}=5^{\circ}\) produces \(\theta_{2}=3.39^{\circ}\), satisfying the TIR. Besides, the fact of the light rays experience particular traveling times through the fiber is useful to introduce the modal dispersion among low-order and high-order modes, i.e., rays propagating straight or at steeper angles, respectively. Such a phenomenon is the limiting factor in the information transmission capacity of multimode fiber systems and results in a high bandwidth-distance product value [24].