## 4 Discussion  
To obtain further insight into the optical process in the photopolymerization in our experiments, we consider the polymerization kinetics [30], noting that the irradiation time is generally brief (on the order of sub-one ms) in our system and a steady state of the reaction is not reached. We also note from our measurements that the polymer concentration is low and highly localized in the writing region, implying that the monomer concentration remains nearly constant. The final concentration of the polymerized DVB, \(c_{pDVB}(\mathbf{r})\) where \(\mathbf{r}\) is the spatial coordinate, is directly proportional to the time-integration of the concentration of the PI radicals \(c_{p\mathbf{r}_{i}}(\mathbf{r},t)\) during the laser irradiation, i.e., \(c_{pDVB}(\mathbf{r})=\int\eta_{p}c_{p\mathbf{r}_{i}}(\mathbf{r},t)dt\), where \(\eta_{p}\) is the polymerization coefficient depending on the initiation efficiency, propagation constant, termination constant, and monomer concentration of the specific PI-monomer combination. The instantaneous concentration of the PI radicals generated through two-photon absorption during the laser irradiation is expressed as \(\frac{2}{dt}c_{p\mathbf{r}_{i}}(\mathbf{r},t)=\sigma_{2}I^{2}(\mathbf{r}-vt)c_{p\mathbf{I}}(\bm {r},t)\); where \(\sigma_{2}\) is the two-photon absorption cross-section of the photoinitiator molecule in Goeppert-Mayer units; \(I(\mathbf{r})\) is the irradiation intensity expressed in terms of number of photons; \(v\) is the velocity of writing; and \(c_{p\mathbf{I}}(\mathbf{r},t)\) is the concentration of the remaining photoinitiator. Furthermore; due to the small region of writing, PI* is \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Transmission Loss (dB/cm)**} \\ \hline
**Wavelength Range (nm)** & **Single-Track** & **Double-Track** & **Triple-Track** \\ \hline
535/43 & \(14.2\pm 2.9\) & \(6.5\pm 2.0\) & \(12.4\pm 0.3\) \\
561/14 & \(13.5\pm 3.9\) & \(5.6\pm 1.7\) & \(10.8\pm 0.9\) \\
592/43 & \(13.2\pm 3.4\) & \(4.5\pm 1.0\) & \(7.5\pm 0.7\) \\
609/54 & \(12.6\pm 3.0\) & \(4.5\pm 0.9\) & \(6.6\pm 0.6\) \\
679/41 & \(12.6\pm 2.3\) & \(2.6\pm 0.5\) & \(2.3\pm 0.4\) \\
710/10 & - & \(2.2\pm 1.7\) & \(0.1\pm 0.4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of transmission loss for the different waveguide types in different wavelength ranges in dB/cm. Figure 7: (**a**) Output intensity profile of a double-cored (blue) and triple-cored (red) waveguides (wgs), at 535/43 nm spectral band. Waveguide outputs are displayed in the image inset; (**b**) Transmission loss in dB/cm as a function of wavelength for simple (green), double- (blue) and triple- (red) track waveguides. Dot lines indicate the linear fit with \(-4.0\) slope, for both double- and triple-track waveguides, typical of pure Rayleigh scattering. subject to a diffusion process as \(\frac{\partial}{\partial t}n_{PI*}(\mathbf{r},t)=D_{PI*}\
abla^{2}n_{PI*}(\mathbf{r},t)\), where \(D_{PI*}\) is the diffusion constant. The system is implicitly highly nonlinear, since the intensity \(I(\mathbf{r}-vt)\) at point \(\mathbf{r}\) will be modulated by the refractive index profile \(\Delta n_{pDVB}(\mathbf{r})\) it creates, which can be modeled with the unidirectional pulse propagation equation [34]  
Heuristically, the diffusion equation in the coupled equation system causes \(c_{PI*}\) to expand spatially from the source term once the PI molecules are radicalized. In the one-dimensional writing of a waveguide, this expansion is two-dimensional in the height and width. Therefore, the refractive index profile of the fabricated waveguide is jointly determined by \(\sigma_{2}\) and \(D_{PI*}\).