## 2 Optical Waveguide Cross-Section Design and Simulation  
The design considerations and tradeoffs of the air-cladded optical waveguides supported by a nano-fin are described below and optimized for sensing and photonic inter-chip connects. Figure 2a shows a cross-section of the waveguide structure under consideration, with its geometrical definitions. Here \(h\) is the fin height from the substrate, \(w\) is the fin  
Figure 1: Simulated mode field profiles of nano-fin supported, single mode waveguides (vertically polarized) for core diameters of (**a**) 1.6 \(\upmu\)m, (**b**) 1.9 \(\upmu\)m, and (**c**) 2.2 \(\upmu\)m. The ability to fabricate such structures is demonstrated in the cross-section image of our fabricated waveguide (after FIB sectioning, shown in (**d**)), viewed at a 52\({}^{\circ}\) tilt from vertical. width, and \(d\) is core diameter of the waveguide. The waveguide material was the polymer IP-DIP, optimized for Nanoscribe (having a reported refractive index of 1.532 [25]); the substrate was fused silica glass (\(n\) = 1.454); and the surrounding cladding was air. We considered a case where the operating wavelength was 1.55 um (in a vacuum). We varied the design parameters and calculated the supported mode's effective refractive index and confinement factor using mode solver software (Lumerical Mode). Figure 2b shows the variation of the effective refractive index of the guided mode with the core diameter of the waveguide, with and without a fin of height 1.5 um and width 0.3 um, for the fundamental LP\({}_{01}\) mode for both polarizations. The cutoff core diameters for single-mode support were 0.6 um and 0.8 um for LP\({}_{01\text{x}}\) and LP\({}_{01\text{y}}\) polarizations, respectively, with the mode leaking to the substrate via the evanescent field. The LP\({}_{01\text{x}}\) was slightly more robust (it had a lower cutoff core diameter than LP\({}_{01\text{y}}\)) as the x-directed polarization was less influenced by the nano-fin supporting the core from below (in the y-direction). The LP\({}_{01\text{y}}\) mode leaked out through the fin at smaller core diameters. The effective refractive indices of a circular waveguide core with and without the fin were nearly identical, especially for x-polarized light (i.e., not in the direction of the nano-fin), with deviation emerging close to the guiding cutoff due to the presence of the fin, demonstrating that the impact of the supporting nano-fin on the model characteristics of our waveguide at diameters larger than 1 mm was negligible. When the core diameter increased beyond 3.4 um, the waveguide started supporting a second spatial mode, of type LP\({}_{11}\). The second-order mode was guided only at a substantially large core diameter (corresponding to a V-number of 8), due to the presence of the nearby substrate and not the nano-fin. Further, we optimized the fin parameters to prevent the interaction between the evanescent field of the circular core waveguide and the glass substrate. Figure 3a shows the variation of effective refractive index with fin height for core diameters of 1.5 and 2 um. The effective index increased with decreasing the fin height for both polarizations and there was negligible birefringence for a 2 um core diameter. The minimum fin height was 0.3 um for \(d\) = 1.5 um to support a single mode. The mode field profile is strongly confined with larger core diameters, which isolates it from the substrate with a shorter fin height. Figure 3b shows the variation in fin width for a fixed fin height of 1.5 um. The maximum fin width with a fixed fin height of 1.5 um supporting a core mode was 0.8 um for core diameters 1.5 um, beyond which value the mode leaked to the glass substrate. At our designed geometry of 1.5 um/0.3 um fin height/width, the simulated value of loss was negligible. Figure 2: (a) Cross-section of the single mode waveguide supported by a nano-fin to the substrate, demonstrating the waveguide definitions of its diameter, \(d\), fin height, \(h\), and nano-fin width, \(w\).