# Simulation and Experimental Verification of the Thermal Behaviour of Self-Written Waveguides
## 3 Theoretical Investigation of Thermal Behaviour of SWWs  
The goal of this investigation is a better understanding of the interconnecting element, especially concerning the thermal response. We expect a separable change of the optical transmission of the fiber-SSW-fiber assembly in comparison to two fibers only due to its multi-material structure consisting of a transition from glass (fiber) to polymer with different material properties caused by the varying curing procedures. The required material parameters for the simulation are the TOC and refractive indices of the core and cladding. These measurements have been conducted for NOA68, with the setup depicted in Figure 4. The obtained results are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline
**T [\({}^{\circ}\)C]** & \(n_{core}\) & \(n_{cladding}\) & \(\Delta\) & **g** \\ \hline
25 & 1.51845 & 1.51767 & 0.000128 & 1.99969 \\
30 & 1.51826 & 1.51710 & 0.000190 & 1.99954 \\
35 & 1.51819 & 1.51690 & 0.000212 & 1.99949 \\
40 & 1.51808 & 1.51674 & 0.000221 & 1.99947 \\
45 & 1.518 & 1.5165 & 0.000240 & 1.99942 \\
50 & 1.51786 & 1.51632 & 0.000253 & 1.99939 \\
55 & 1.51777 & 1.51611 & 0.000272 & 1.99935 \\
60 & 1.51767 & 1.51590 & 0.000291 & 1.99930 \\ \hline \end{tabular}
\end{table} 
Table 1: Average values of the refractive index measurements (at \(\lambda=638\) nm) at different temperatures and their calculated \(\Delta\) and g-parameters required to calculate the refractive index of the SWW. Figure 5: Measurement of the refractive index distribution of a SWW embedded in the cured cladding. The refractive index profile corresponds to the cross section along the horizontal line in the false colour image. The y-axis provides the refractive index value. The measured refractive indices show a decreasing tendency with increasing temperature for core and cladding. The additionally shown \(\Delta\) values are defined by \(\frac{n_{core}^{2}-n_{slidding}^{2}}{n_{core}^{2}}\). The last column of Table 1 contains the calculated g-parameter. Both, g-values and \(\Delta\)-values allow the calculation of the refractive index distribution of the SWW based on the following:  
\[n(r)=n_{core}\sqrt{1-2\Delta(\frac{r}{a})g} \tag{2}\]  
for \(r\leq a\) where \(a\) is the size of the core. For the sake of completeness, the refractive index distribution can be calculated as \(n_{r}=n_{core}\sqrt{1-2\Delta}\) for \(r>a\), which results in a step-index distribution for \(g\rightarrow\infty\). G-values of \(\approx\)2 for all measurements result in a gradual change of the refractive index from the center towards the cladding. The calculated refractive index distribution and change with the temperature are shown in Figure 6.  
The expected refractive index distribution shown in Figure 6 was confirmed by measurements taken with the refractive index profilometer, being obvious in the shape of the core of the cross-sectional view of Figure 5.  
Further investigations aimed at the quantification of the change of the optical transmission during the heating of the SWW. The simulation was fed with the measured refractive index values for each temperature and the corresponding TOC. The obtained results here were in contrast to our expectations, where the transmitted optical intensity should decrease with the temperature and which was expected due to previous simulations based on the literature values and a constant TOC for core and cladding [30]. The results obtained in the simulations for the new case are shown in Figure 7.  
Figure 6: Refractive index distribution of core and cladding of the SWW during the heating process showing a decreasing refractive index with increasing temperature. Moreover, the refractive index difference between core and cladding increases at higher temperatures. An explanation of this behaviour lies in the different TOCs. These values are normally defined as \(dn/dT\) describing the change of the refractive index of a material with the temperature.