## 2 Materials and Methods
### FTIR Analysis of Polycrystalline PMN-PT Sample
Specific aperture configurations for each FTIR spectrometer are summarized in Table 2.  
In addition to the prepared PMN-PT sample, a commercially available LiNbO\({}_{3}\) single crystal thin sample was characterized as a control, and subsequent refractive index calculations were be compared to literature values to help support the validity of this methodology (Figure 4). The LiNbO\({}_{3}\) sample used for this purpose is professionally polished and has a measured thickness of 508 \(\upmu\)m.  
Figure 4: The LiNbO\({}_{3}\) single crystal control sample (**a**) and prepared PMN-PT sample (**b**) arranged with millimeter scales for reference. Figure 3: Reducing the aperture diameter, thereby reducing the maximum angle of incidence, serves to amplify the magnitude of the interference fringes. Figure 2: A comparison of the incident ray bundles during unrestricted FTIR analysis (**a**) and restricted FTIR analysis (**b**). Using Equation (7), it was estimated that the interference fringes of the prepared PMN-PT sample have a period of approximately 5 cm\({}^{-1}\). With this in mind, the FTIR spectrometers were configured with a scanning resolution of 0.25 cm\({}^{-1}\) to produce well-defined fringes in the final transmission spectrum. Additionally, the transmission values were averaged over thirty-two scans to increase waveform smoothness. While a scan resolution of 0.25 cm\({}^{-1}\) is sufficient to capture the fringe waveforms, for the purpose of calculating the refractive index, the fringe peak positions must be known more precisely. To that end, the transmission spectra were mathematically characterized with a polynomial fit to more precisely interpolate the location of the fringe peaks. Peak locations measured using this method were expected to have an accuracy of at least two decimal places (0.01 cm\({}^{-1}\)\(\rightarrow\) 0.0001 m\({}^{-1}\)), supporting an index of refraction calculation out to four decimal places. An example of the fringe peak fitting is depicted in Figure 5.  
Once the fringe positions have been measured using the polynomial fit method, it is still necessary to assign absolute fringe order numbers. The absolute fringe order for each peak is extrapolated via a linear fit to the zeroth-order. Fringe peak spacing in the frequency domain is nearly linear such that plotting the relative fringe order numbers against the fringe frequency position allows for an accurate determination of the offset from zero (i.e., 0 cm\({}^{-1}\)). This extrapolation process is depicted in Figure 6.  
Of all the critical values necessary for accurate refractive index calculations, proper assignment of the fringe order numbers has the largest impact on the dispersion curve [32]. Successful determination of the fringe order numbers is aided by a large data set; it is at this point that the near-IR and mid-IR collections are integrated, extending the reach of the extrapolation closer to zero. If the assignment is trusted to be accurate, the index of refraction at each fringe peak location is calculated by rearranging Equation (7) as  
\[T(\
u)=\frac{1}{1+F\ \sin^{2}(2\pi\
u n(\
u)d)} \tag{8}\]0  
where \(\
u_{m}\), \(m\), and \(d\) are the known values. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Spectrometer** & **Aperture Diameter** & **Aperture Distance** & **Maximum Angle of** \\  & **(mm)** & **(mm)** & **Incidence (\({}^{\circ}\))** \\ \hline Thermo Scientific & 6.0 & 60.0 & 2.862 \\ Perkin Elmer & 5.0 & 54.75 & 2.614 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Aperture size and spacing used for each FTIR spectrometer to amplify interference fringe amplitude. Figure 5: The accuracy of interference fringe peak location measurements is increased by interpolation via a polynomial fit.