## 3 Results
### Far-Field Analysis  
The far-field speckle patterns of the sapphire fiber were captured at diameter intervals from 125 \(\upmu\)m down to a final diameter of 6.5 \(\upmu\)m using laser wavelengths of 532 nm, 783 nm, and 983 nm, as seen in Figure 7. At the initial sapphire fiber diameter of 125 \(\upmu\)m, where nearly a million modes are theoretically supported at lower wavelengths, the speckle patterns familiar to highly multimodal projections were observed. As the diameter decreased, the number of speckles decreased and individual speckles broadened in diameter, a result of the reduction in supported modes causing the interference peaks and valleys to which the speckles correspond.  
Figure 5: The etch rate of the sapphire optical fiber (\(a\)-plane is the etched plane) exhibited an approximately linear relationship with etch temperature.  
Figure 6: (**a**) Etching occurs cleanly and consistently at regions where the sapphire fiber is fully immersed; (**b**) Deposits could appear near the air-acid interface during the etching process if not properly shielded.  
Once the sapphire fiber diameter was reduced to 6.5 \(\upmu\)m, low order LP\({}_{11}\) and LP\({}_{01}\) modes became visible at the tested wavelengths. As may be anticipated, the smaller 532 nm wavelength exhibited the higher-order LP\({}_{11}\) mode while only the fundamental mode was observed at wavelengths of 783 nm and 983 nm.  
Alterations were made to excitation conditions to assess the retention of low order mode dominance. First, the injection angle of the laser into the fiber was altered by 15\({}^{\circ}\); if the observed far-field patterns were truly stable, it could be expected that the patterns would not change significantly. This was shown to be the case, as seen at the bottom of Figure 7; the LP\({}_{11}\) mode was once again visible at a wavelength of 532 nm, and the fundamental LP\({}_{01}\) mode was observed at 783 nm. Only scattered light could be captured at the 983 nm wavelength, perhaps due to lack of collimation as the light entered outside the acceptance angle of the fiber, or simply due to alignment difficulties for this particular measurement.  
Figure 7: Far-field measurements were taken at varying diameters using laser wavelengths of 532 nm, 783 nm, and 983 nm. Low-order modes became visible at a sapphire fiber diameter of 6.5 \(\upmu\)m, and measurements were consistent when the light injection angle was altered by 15\({}^{\circ}\).  
Additional modifications to input conditions further bolstered the conclusion that the observed low-order modes were stable. The previous measurements were taken via direct input of laser light using an objective lens to focus light into the optical fiber. In Figure 8, the far-field projections observed through direct laser input are compared to those observed when 1 m and 2.5 m patch cables (SMF-28e and 50/125 step-index multimode fiber (MMF), respectively) were butt-coupled to the sapphire fiber, with the patch cables receiving the laser light themselves through direct input.  
No significant differences were observed in the far-field projections regardless of input method. The LP\({}_{11}\) mode was still observed at 532 nm whether direct, SMF, or MMF input was used, and the fundamental LP\({}_{01}\) mode was observed at 783 nm and 983 nm. Due to the smaller core of the SMF fiber, the intensity of the SMF-input signal was lower than others (preventing the capture of a far-field projection at 983 nm due either to the reduced intensity of the projection or sensitivity of the detector at this wavelength), but the modal patterns themselves did not change. Based on these results, the following may be stated with confidence: low-order modes, and even a single mode, are dominant in air-clad sapphire optical fiber at wavelengths of 532 nm and greater when the fiber diameter is 6.5 \(\upmu\)m or less.