# Simulation and Experimental Verification of the Thermal Behaviour of Self-Written Waveguides
## 3 Theoretical Investigation of Thermal Behaviour of SWWs
This parameter, again, is defined (in linear approximation) as \(\frac{dn}{dT}=\left[\frac{\lambda}{2\Gamma\Delta T}\right]-n\alpha\)[31], with \(\lambda\) as corresponding wavelength, \(L\) as sample thickness, and \(\alpha\) as linear expansion coefficient. In particular, the latter is a material specific parameter which depends on the density, among others. As obtained from previous experiments, the refractive index and the density, respectively, correlate with the curing time and intensity [4]. Additionally, the fact that the monomer and cured polymer show a high absorption coefficient results in a refractive index gradient of the surrounding resin, which was cured with UV flood exposure. This enables that the surface of the resin is completely hardened, whereas the core-near region is also solid but less hardened, depending on the curing time and thickness of the resin applied. This, in sum, enables us to explain the achieved simulated results. The different material characteristics due to the varying types of curing result in an increasing refractive index difference between core and cladding as discernible from the values of Table 1. Subsequently, this fact results in an increase in the NA of the waveguide when heating is applied, as indicated in Table 2, based on the calculation NA = \(\sqrt{n_{core}^{2}-n_{cladding}^{2}}\). The obtained values shows an increase in the NA \(\approx\) 67% during the heating cycle up to 60 \({}^{\circ}\)C. This decreases the coupling losses from the silica fiber to the polymer structure. In comparison, the used optical fibers have a numerical aperture of NA = \(0.2\pm 0.015\) (OM2-fiber), which accepts a higher number of modes than the SWW. In order to start experimental validation with the setup shown in Figure 8, we studied the temperature distribution within the material numerically in order to verify that the temperature obtained correlates with the one of the SWW. Therefore, we simulated the \begin{table}
\begin{tabular}{c c} \hline \hline
**T [\({}^{\circ}\)C]** & **NA** \\ \hline
25 & 0.049 \\
30 & 0.059 \\
35 & 0.063 \\
40 & 0.064 \\
45 & 0.067 \\
50 & 0.068 \\
55 & 0.071 \\
60 & 0.073 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Change of the numerical aperture with increasing temperature calculated from the refractive indices of Table 1.  
Figure 7: Simulation of the change of the optical transmission with increasing temperature for the heating and cooling process applied to the SWW. distribution of the temperature by heating an SWW surrounded by the cured cladding using the COMSOL Multiphysics package. The result of this investigation is shown in Figure 9.  
For the simulation shown in Figure 9, the heater bottom layer (heater) applies the thermal energy, which distributes to different layers and the surrounding air is working as a heat sink. In the case of cooling, the heat input from the bottom layer was set to zero, and the material cooled down by heat exchange with the surrounding air. The simulation yields the temperature distribution inside the SWW for the case of heating the structure up to 60 \({}^{\circ}\)C. By using this simulation, we also examined the time it takes until the SWW reached the expected temperature. These results are listed in Table 3.  
Figure 8: Experimental setup for verifying the simulated thermal behaviour of the SWWs. The interconnecting structure is written in between two multi-mode silica fibers (OM2) on top of an aluminum plate, which is heated by a Peltier element (QC-32-0.6-1.2, Quick-Cool). The actual temperature is checked with an additional thermal sensor (Heraeus Nexensos M222). Figure 9: Simulated contour map of the temperature distribution of the cross-section in the middle of an SWW when the heater was at maximum temperature of 61.28 \({}^{\circ}\)C. The thickness of each layer is assumed as 50 \(\upmu\)m, which corresponds to the thickness of the fiber core we normally use for the SWW experiments.