## 4 Discussion
Regardless of the exact distribution of \(\Delta n_{pDVB}\) and the extent of the polymerization region, the total phase change as measured in the interferometric imaging system, where the width of the polymerized structure is below the diffraction limit, is \(\Delta\Phi=k\Delta n_{pDVB}n\propto l_{0}^{2}\), where \(k=\frac{2\pi}{\lambda}\), \(\lambda\) is the laser wavelength used in the measurement of \(\Delta\Phi\). Figure 8 shows the plot of the measured \(\Delta\Phi\) as a function of \(I_{0}\), which fits well to a second-order curve while also revealing a threshold behavior. The properties of the polymerization process are largely determined by the parameters \(\eta_{p}\), \(\sigma_{2}\), and \(D_{PI*}\) in the coupled partial differential equation system. The polymerization coefficient \(\eta_{p}\) is a parameter dependent on the specific combination of PI and monomer and determines how probable the polymerization is initiated, propagated, and terminated given a specific \(c_{PI*}\) and \(c_{DVB}\). The parameter \(\sigma_{2}\) reflects how efficient two-photon absorption happens to generate PI radicals. Finally, the diffusion constant \(D_{PI*}\) controls how far the PI radicals move around, which eventually determines the dimensions of the waveguides. The effect of \(D_{PI*}\) is clear when we compare poly-DVB waveguides with poly-phenylacetylene waveguides [24]. In the poly-phenylacetylene system, the small-molecule monomer (molecular weight 102 g/mol) also serves as a PI, which has a relatively larger \(D_{PI*}\) resulting in a waveguide width of 1.3 \(\upmu\)m. In contrast, the OXE 02 used in this work as a PI is a much larger molecule (molecular weight 412 g/mol), so its \(D_{PI*}\) is much smaller yielding waveguides roughly 0.5 \(\upmu\)m wide. In a mixed PDMS/DVB system, the nonlinear refractive index is mainly of electronic origin, which usually falls within the order of \(10^{-15}\) cm\({}^{2}\)/W. This would project to a refractive index change on the order of 0.001 at the writing focus. In contrast, the refractive index change after polymerization often saturates at the level of 0.1 (refractive index difference between the resulting polymer and the monomer) and increases over time during the laser irradiation. Thus, beam collapse and material damage can happen at any intensity above the polymerization threshold. This is well manifested in the fabrication of  
Figure 8: Plot of the measured phase change in the waveguides as a function of the writing intensity. The error bar indicates the standard deviation among three different waveguides. Dashed line shows a quadratic regression. phenylacetylene waveguides. Indeed, the good quality of fabrication demonstrated in the present work benefits largely from the lower refractive index contrast. In theory, the self-focusing can be balanced with a well-controlled motion speed during fabrication, which is currently in practice. However, due to the nonlinearity the intensity can grow exponentially at small disturbances, and material homogeneity plays a key role in minimizing the fabrication defects. Unlike poly-phenylacetylene, which features extensive conjugated carbon-carbon double bonds, poly-DVB is joined with mostly single bonds in the backbone. We thus expect that poly-DVB is mostly non-absorptive like polystyrene. In the waveguides constructed with poly-DVB, the transmission properties would be Rayleigh scattering-dominated in the visible band due to the molecular weight fluctuation in the resulting poly-DVB. This is precisely confirmed by the slope of \(-4\) in the double-track waveguides in Figure 6b. The transmission loss of triple-track waveguides follows slope \(-4\) in the shorter wavelength region but drops quicker than \(-4\) in the longer wavelength region, which could indicate a constructive interference or resonance condition created by the more complex and larger structure.