## 4 Waveguide Characterization
The fringe visibility estimation is not precise, due to the slope trend across the spectrum, yet using \(-8.2\) dB and \(-8.8\) dB for the maximum/minimum transmission, we found that:  
\[\mathrm{v}=\frac{\mathrm{P_{max}}-\mathrm{P_{min}}}{\mathrm{P_{max}}+\mathrm{ P_{min}}}=\frac{\frac{\mathrm{P_{max}}-1}{\mathrm{P_{min}}}-1}{\frac{\mathrm{P_{ max}}}{\mathrm{P_{min}}}+1}=\frac{0.6\mathrm{dB}-1}{0.6\mathrm{dB}+1}=\frac{1.15-1}{1.15+1}=0.07 \tag{2}\]  
With the fringe visibility and Fresnel reflection we found the transmission efficiency of the waveguide, \(\mathrm{G}=\frac{\mathrm{v}}{2\mathrm{R}}\), had a value of \(\sim\)0.8 or \(-1\) dB (see Appendix A). Since the waveguide was only 0.2 mm long, we estimate a loss of 5 dB/mm. This loss calculation is quite imprecise, as the printed waveguide length was mostly allocated to the waveguide tapers, whereas the propagation loss would be highest in the region with the narrowest diameter, which was only 40 \(\mathrm{\SIUnitSymbolMicro m}\) long in our implementation. Furthermore, the assumed end-facet reflection value can also deviate from the ideal value due to surface roughness. Finally, there could be a local printed waveguide defect which impacted the Fabry-Perot cavity loss, and hence the fringe visibility, and thus our loss estimate. In attempt to obtain a better propagation loss estimation, we designed a second-generation waveguide with significant propagation length at a constant core diameter of 2 \(\mathrm{\SIUnitSymbolMicro m}\). Working around the writing footprint limit of Nanoscribe and capitalizing on the 3D nature of DLW, we designed a folded waveguide following a rectangular spiral geometry, with the inner waveguide end exiting the structure by being bent upwards and passing over the spiral waveguides (Figure 10). Five different waveguide lengths were printed, all with the same upward-bending ends for coupling, and with twelve 90\({}^{\circ}\) bends of 25 \(\mathrm{\SIUnitSymbolMicro m}\) radius (making three spiral revolutions at a 10 \(\mathrm{\SIUnitSymbolMicro m}\) waveguide pitch), with varying lengths of the straight segments. The overall waveguide lengths following this design were \(1.528:0.1:1.928\) mm, and five identical copies of each waveguide length were printed. All 25 waveguides were identically analyzed by coupling with lensed fibers oriented at 45\({}^{\circ}\) and performing a spectral scan with the optical vector analyzer. Figure 9: **(a)** Experimental setup to couple to the waveguides using lensed fibers, with magnified view of the lensed fiber coupling on the left. **(b)** Transmission spectrum of printed waveguide for the two principle polarization states. Unfortunately, the loss variation for each identical design persisted. We measured the same structure multiple times and found that the insertion loss values were consistently nearly equal for the same waveguide structure. Figure 11a shows the five spectral scans for the 1.728 mm long waveguide, with losses varying from 9 dB down to 14.5 dB. Such large loss variations for identical waveguides are likely due to either coupling in/out loss variations or random waveguide defects/roughness that are unique to each printed sample. We resorted to the fringe visibility analysis technique, as it is independent of the coupling losses (an internal waveguide cavity process), and attached a fringe visibility value, \(\upsilon\), to each trace (see inset values next to each trace). The fringe visibility values continued to show large variations, indicating that printed defects/roughness were present. (We assumed that end-facet reflection variations were contributing to a lesser degree, since we significantly lengthened the waveguides.) For our loss analysis, we adopted the largest fringe visibility value of the five printed waveguides of identical length, as it represents the waveguide sample that had the fewest printing defects, and hence we obtained a 'best value' for each of the five waveguide lengths.