## 4 Waveguide Characterization  
The first waveguides we fabricated, having ends curving upwards at a bend radius of 100 \(\upmu\)m while simultaneously tapering out for mode matching the waveguide ends to lensed fibers of 2.5 \(\upmu\)m mode field diameter, terminated at a 45\({}^{\circ}\) angle. The mode field diameter of the fundamental mode at the taper end was 3.1 \(\upmu\)m, overfilling the lensed fiber's 2.5 \(\upmu\)m MFD. This should theoretically add a 0.15 dB loss per facet but ease the alignment tolerance. Special fiber holders, aligned at 45\({}^{\circ}\) to couple in and out of the waveguides, were mounted on fine manual translation stages (resolution \(\sim\)10 \(\upmu\)m) to facilitate optical excitation with the lensed fibers. Multiple identical waveguides arranged along one dimension were fabricated, allowing rapid scanning of the waveguide samples by moving the substrate along one axis (shown in Figure 9a). After optimizing the coupling to each waveguide of the array, the optical transmission via the printed waveguides was assessed with a wavelength scan using an optical vector analyzer (Luna 5000 OVA). Five printed waveguides were assessed and exhibited \(\sim\)5.5 dB insertion loss variations (9 dB to 14.5 dB for five identical waveguides). The reason for the variation is unknown, but it may originate from random waveguide defects causing scattering loss, angular misalignment and end-facet imperfections impacting coupling loss, or alignment sensitivity exceeding the capability of our translation stages. Figure 9b shows a transmission spectrum with minimum insertion losses of \(\sim\)8 dB and \(\sim\)9.4 dB for the two principal polarization states. Before coupling to the upwards bent waveguide, we measured the lensed-fiber-to-lensed-fiber direct coupling and found the loss was \(\sim\)1.8 dB. Hence, the \(\sim\)200 \(\upmu\)m long DLW waveguide insertion loss was \(\sim\)6 dB (inclusive of all possible loss factors). Periodically varying the transmission spectrum indicated that the light was weakly reflected from the waveguide end facets, as expected, with the waveguide then functioning as a _Fabry-Perot_ cavity having a free spectral range (FSR) of \(\Delta\lambda\sim 3.75\) nm +/\(-\)0.004 nm (for both polarization states). The periodicity associated with the accumulated phase from a round trip of the waveguide length, L, is given by:  
\[\Delta\lambda=\frac{\lambda_{0}^{2}}{\text{n}_{\text{g}}2\text{L}} \tag{1}\]  
where \(\text{n}_{\text{g}}\) is the group index of the waveguide \(\text{n}_{\text{g}}=\text{n}_{\text{eff}}-\lambda\frac{dn_{\text{eff}}}{d \lambda}\). Substituting L = 197 \(\upmu\)m (the precise length of the waveguide), we obtained an estimated value of \(\text{n}_{\text{g}}=1.53\). We compared this value with the numerically simulated group index value for the printed  
Figure 8: (**a**) Fabricated spiral waveguide, in the process of scanning with AFM tip, (**b**) AFM image of the waveguide’s curved surface topography, and (**c**) two line profiles along waveguide direction revealing its surface roughness. waveguide cross-section, obtaining an \(\mathrm{n_{g}}\) value of 1.61, which is close to our experimentally measured value. Since the same FSR was measured for the two polarizations, we can say that no significant birefringence was observed, as expected for a 2 \(\mathrm{\SIUnitSymbolMicro m}\) diameter waveguide core. Propagation Loss EstimationA rough estimate of the waveguide's propagation loss can be obtained by analyzing the fringe visibility of the spectral scan of Figure 9b, with an assumption of Fresnel reflection loss at the two end facets of \(\mathrm{R}=\left(\frac{1-\mathrm{n}}{1+\mathrm{n}}\right)^{2}\cong 0.044\).