## 3 Results
### Compound Waveguides  
In order to achieve long, defect-free waveguides we optimized the fabrication parameters for simple waveguides to be \(1.78\times 10^{12}\) W/cm\({}^{2}\) nominal intensity and 3.2 mm/s writing velocity. This intensity is lower than the visually-determined threshold of \(1.93\times 10^{12}\) W/cm\({}^{2}\) but significantly reduces the probability of defects. At this lower fabrication intensity, \(\Delta n\) is further reduced and the crosstalk is more significant. Thus, the simple waveguides present a unique challenge of excessive crosstalk when being implemented in a high-density integration. Although increasing \(\Delta n\) is the most effective way to minimize the crosstalk, it is constrained by the current photochemistry of choice, and more fundamentally, by the intrinsic self-focusing and beam collapse leading to material damage and fabrication defects. Given the fundamental constraints and a goal of inter-waveguide spacing of 10 \(\upmu\)m at the length of 5 cm, the only feasible approach is to increase the waveguide width for better mode confinement. Since this was difficult to achieve with larger focal spot size because of the required light intensity for multi-photon absorption, we instead fabricated compound waveguides where multiple tracks go side-by-side in parallel with a very small interval spacing \(\delta\) (Figure 5). The spacing is so small that the potential barrier between the individual tracks is tunneled through easily and the multiple tracks effectively behave like a single, wider waveguide (a compound waveguide). This approach also calls for a lower fabrication intensity and a higher writing speed in order to avoid material damage caused by the refractive index modification carried by the adjacent core. Figure 5: Schematic of the writing of compound waveguides. The PDMS sample is displaced by a small distance \(\delta\) by means of a piezoelectric stage in the y direction. A long travel range mechanical stage moves the sample over centimeters in the x-direction. The compound waveguides were fabricated with a nominal focal intensity of \(1.78\times 10^{12}\) W/cm\({}^{2}\) and a writing velocity of 3.2 mm/s, which were optimized from the fabrication of simple waveguides. We first fabricated short double- or triple-track compound waveguides at an intra-waveguide interval \(\delta\) = 1.2 \(\upmu\)m for a pilot study at three different PI concentrations and measured the dimensions of the individual constituent simple waveguide. At the same fabrication parameters, both waveguide width and height increase when a higher concentration of photoinitiator is used (Figure 6a). At 1 wt% PI concentration, the lateral and axial dimension of the waveguides were measured to be \(\sim\)0.63 \(\upmu\)m and \(\sim\)2 \(\upmu\)m on average, respectively, and reached \(\sim\)0.69 \(\upmu\)m and \(\sim\)2.7 \(\upmu\)m, respectively, at a PI concentration of 5 wt%. The cross-sectional and side view of the compound waveguides acquired from a phase contrast microscope is shown in Figure 6b-e.  
We then fabricated 5-cm long, double- and triple-track compound waveguides at 5 wt% PI concentration and 1.2 \(\upmu\)m intra-waveguide interval for more comprehensive characterization. The refractive index contrast \(\Delta n\) of the compound waveguides was measured using the interferometric imaging system [24]. Both the double- and triple-track waveguides show a refractive index contrast on the order of \(\Delta n\) = 0.005, calculated from the phase we extracted from the interferometric image of the waveguides. The V-number ranges from 2.22 down to 1.68 over the spectral band from 535 nm to 710 nm when we consider the waveguide cross-section as a square of side 2.7 \(\upmu\)m. Figure 7a shows the waveguide output intensity profiles of a double-track (blue) and a triple-track (red) waveguide in the 535/43 nm spectral band, whose two-dimensional intensity distribution is shown in the inset.