## 4 Conclusions  
In this work, we extend the analysis conducted in [30], placing the NRW relation in the context of the global analytic functions and the Riemann surfaces, thus providing a rigorous mathematical framework for handling the NRW relationship. In this context, the branch ambiguity problem finds its correct interpretation in terms of the number of Riemann surface sheets \(\mathbb{S}(\mathcal{L})\) on which the right-inverse \(h(\cdot)\), the solution of the NRW Equation (1), lies. Furthermore, we exploit this framework to derive a couple of phase unwrapping-based analytic continuation algorithms: the first, named PUNW\({}_{KK}\), is based on K-K relations, and the second, named PUNW\({}_{HT}\), is based on the Hilbert transform; both are capable of solving the NRW equation and avoid any ambiguity. To validate their numerical performances, some numerical experiments conducted on a couple of theoretical negative refractive index (NRI) slabs have been carried out. The close agreement obtained between the recovered and the exact refraction index provided by the PUNW\({}_{HT}\) approach demonstrates its superiority over the other recovery methods considered in this study and confirms the validity of our theoretical analyses. Although the results provided by the PUNW\({}_{KK}\) were less satisfactory than those given by PUNW\({}_{HT}\), the use of K-K relations merged with the phase unwrapping approach enhances the results these relations provides. Finally, to conclude, we point out that the theory we have given in this work rigorously demonstrates some algorithmic procedures empirically proposed in the literature [20; 21], broadening their meaning.  
Conceptualization, G.A.; methodology, G.A. and M.V.; software, G.A.; validation, M.V.; formal analysis, G.A.; investigation, G.A. and M.V.; resources, M.V.; writing--original draft preparation, G.A. and M.V. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement**: Not applicable.  
**Informed Consent Statement**: Not applicable.  
**Data Availability Statement**: Not applicable.  
**Conflicts of Interest**: The authors declare no conflicts of interest.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline \(n_{s}\) & \multicolumn{2}{l}{**PE**} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ \(\mathbf{\mathrm{PUNWOS}}\) & \multicolumn{1}{c}{**K-K**} & \multicolumn{1}{c}{**PUNW\({}_{KK}\)**} & \multicolumn{1}{c}{**PUNW\({}_{HT}\)**} \\ \hline
1024 & 206.03 & 6.86 & 3.90 & \(5.76\times 10^{-4}\) \\
2048 & 503.37 & 6.51 & 3.19 & \(5.62\times 10^{-4}\) \\
4096 & 464.54 & 6.48 & 2.84 & \(5.81\times 10^{-4}\) \\
8192 & 485.42 & 6.72 & 3.93 & \(7.73\times 10^{-4}\) \\
16,384 & \(8.85\times 10^{-4}\) & 6.86 & 3.86 & \(8.85\times 10^{-4}\) \\ \hline \(n_{s}\) & \multicolumn{2}{l}{\(t\) (s)} & \multicolumn{1}{c}{} & \\ \(\mathbf{\mathrm{PUNWOS}}\) & \multicolumn{1}{c}{**K-K**} & \multicolumn{1}{c}{**PUNW\({}_{KK}\)**} & \multicolumn{1}{c}{**PUNW\({}_{HT}\)**} \\ \hline
1024 & 0.30 & 0.41 & 0.37 & 0.36 \\
2048 & 0.31 & 0.93 & 0.50 & 0.37 \\
4096 & 0.38 & 3.18 & 1.30 & 0.27 \\
8192 & 0.40 & 11.13 & 3.84 & 0.35 \\
16,384 & 0.37 & 45.04 & 14.97 & 0.33 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Percentage error (PE) and computational time t for the slab with \(d_{eff}=300\) nm.