## 4 Discussion  
Theoretical estimations of modal volume for a step-index fiber using standard weakly-guiding approximations [19] would predict the support of 461-1614 modes at wavelengths of 532-983 nm and an air-clad sapphire fiber diameter of 6.5 \(\upmu\)m. Furthermore, to achieve single-mode behavior in air-clad sapphire at this wavelength range, fiber diameters would need to be less than 500 nm or so according to weakly-guiding assumptions. However, it is widely considered that the weakly-guiding approximation, which in large part is based on ray-optics assumptions, does not retain accuracy as conditions reach the single-mode cutoff, where wave considerations are dominant. This is especially true when a large refractive index difference exists between core and cladding (i.e., it is not considered to be \"weakly guiding\"); while these estimations are still useful and reasonably accurate at larger sapphire fiber diameters and correspondingly large modal volumes, they may not be reliable as core diameter and modal volume is reduced. That said, if parameters of the approximations (such as numerical aperture) that are theoretically derived using ray-optic assumptions are experimentally measured rather than calculated, the accuracy of these simple approximations could be improved in the low modal volume regime. In fact, it is frequently reported that the effective numerical aperture of sapphire fiber in particular varies greatly from its theoretical \(NA\) due to non-circular cross section (grown fibers are slightly hexagonal due to sapphire's crystal structure), attenuation factors, large refractive index between itself and its air cladding, and other parameters, some of which seem to become more impactful as fiber diameter decreases and core-cladding interactions become increasingly important. For instance, while the theoretical \(NA\) of air-clad sapphire as calculated by Equation (1) would be around 1.4 at the tested wavelengths, researchers and manufacturers have previously measured effective \(NA\)s of \(\sim\)0.35 at \(\lambda\) = 2.94 \(\upmu\)m for unclad sapphire fiber at a diameter of 300 \(\upmu\)m [6] and \(\sim\)0.45 at \(\lambda\) = 633 nm and a diameter of 125 \(\upmu\)m [20]. Replacement of the theoretical \(NA\) with a measured effective \(NA\) may therefore improve accuracy of weakly-guiding approximations predicting modal volume for air-clad sapphire of reduced diameter. \[NA=\frac{1}{n_{0}}\sqrt{n_{core}^{2}-n_{clad}^{2}} \tag{1}\]  
To evaluate this consideration in the context of the reported experimental observations, the measured effective numerical aperture of the 6.5 \(\upmu\)m-diameter sapphire fiber (\(NA_{\text{eff}}\) = 0.090) replaced the theoretical \(NA\) (\(NA_{\text{th}}\) = 1.4) in simple weakly-guiding approximations of modal volume for step-index fibers. The effect of this \(NA\) value replacement is contrastingly illustrated in Figure 9, where the calculated maximum radius for air-clad sapphire optical fiber supporting only a single mode is plotted versus input wavelength for both numerical aperture values (theoretical \(NA\) versus measured effective \(NA\)). This plot was generated by solving the \(V\)-parameter equation (Equation (2)) for radius, \(a\), where \(V\) = 2.405 is the single-mode cutoff. \[V=\frac{2\pi}{\lambda}a\cdot NA \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Analyzed** & **Avg. 4\(\sigma\) Beam** & \begin{tabular}{c} **Avg. 4\(\sigma\) Beam** \\ **Width (\(\upmu\)m)** \\ **D\({}_{2}\) = D\({}_{1}\) + 1 mm** \\ \end{tabular} & \begin{tabular}{c} **Avg. Vergence** \\ **Angle** \\ \end{tabular} & \
\begin{tabular}{c} **Avg. Effective** \\ **\(NA\)** \\ \end{tabular} &
\begin{tabular}{c} **Std. Dev.** \\ \end{tabular} \\ \hline SMF-28e & 3108.61 \(\upmu\)m & 3335.25 \(\upmu\)m & 6.49\({}^{\circ}\) & 0.113 & 0.029 \\ \hline Sapphire & 4513.64 \(\upmu\)m & 4697.16 \(\upmu\)m & 5.26\({}^{\circ}\) & 0.090 & 0.008 \\ \hline \hline \end{tabular}
\end{table}