3.2.2. Experimental Results

Firstly, the ellipsometric parameters of the Au sample were measured by the RPAE at an incident angle of 70◦ in a wavelength range of 300–800 nm, with δ*θ*<sup>P</sup> varying from 0–20◦ , as shown in Figure 7a. The spectra of *ψ* and ∆ demonstrated that the measured results deviated significantly from the standard spectra with the increasing of the azimuthal error of the polarizer. Similarly, the ellipsometric measurements were performed with δ*θ*<sup>A</sup> varying from 0–20◦ , as shown in Figure 7b. The spectra of *ψ* deviated differently compared with those of δ*θ*P. Since the analyzer and polarizer rotated at an angular velocity ratio of 1:2 in measurement, the values of δ*θ*<sup>A</sup> and δ*θ*<sup>P</sup> were set at a group of angles with the same ratio. The value of δ*θ*<sup>A</sup> varied from 0–20◦ , compared with that of δ*θ*<sup>P</sup> in a range of 0–10◦ . Hence, the azimuth of the analyzer was always twice that of the polarizer during rotation. The same procedure was repeated to get the spectra of *ψ* and ∆ (Figure 7c). *Crystals* **2021**, *11*, x FOR PEER REVIEW 9 of 12

**Figure 7.** Measured ellipsometric parameters with various azimuthal errors of (**a**) the polarizer, (**b**) analyzer, and (**c**) both at a ratio of 1:2. **Figure 7.** Measured ellipsometric parameters with various azimuthal errors of (**a**) the polarizer, (**b**) analyzer, and (**c**) both at a ratio of 1:2.

**Figure 8.** Experimental data (points) and fitted curves (lines) of MAREs versus azimuthal errors

The polarizing elements were rotated in the opposite direction with the same absolute values of azimuthal errors to study the directional dependence on accuracy. The spectra of *ψ* and Δ, measured for a series of the same absolute values, showed a high consistency for both ± δ*θ*P and ± δ*θ*A, as observed in Figure 9. Consequently, the accuracy of the RPAE was found to be less dependent on the direction of the azimuthal error in terms

(for δ*θ*A:δ*θ*P = 2:1, the horizontal axis represents δ*θ*A).

of the acquired spectra.

The MAREs of *ψ* and ∆ were calculated and fitted into curves by applying the polynomial fitting method for three azimuthal error modes, as displayed in Figure 8. The comparison revealed the effect of the azimuthal error on the accuracy quantitatively. For the MARE of *ψ*, the three curves behaved similarly with small azimuthal errors, while the value increased significantly with a large value of δ*θ*P. As for ∆, the value increased slightly for the curve of δ*θ*A, while rapidly for that of δ*θ*<sup>P</sup> and δ*θ*P:δ*θ*<sup>A</sup> = 1:2. Therefore, the azimuthal error of the polarizer was indicated to more seriously affect the accuracy of the RPAE.

The polarizing elements were rotated in the opposite direction with the same absolute values of azimuthal errors to study the directional dependence on accuracy. The spectra of *ψ* and ∆, measured for a series of the same absolute values, showed a high consistency for both ±δ*θ*<sup>P</sup> and ±δ*θ*A, as observed in Figure 9. Consequently, the accuracy of the RPAE was found to be less dependent on the direction of the azimuthal error in terms of the acquired spectra. **Figure 7.** Measured ellipsometric parameters with various azimuthal errors of (**a**) the polarizer, (**b**) analyzer, and (**c**) both at a ratio of 1:2.

*Crystals* **2021**, *11*, x FOR PEER REVIEW 9 of 12

**Figure 8.** Experimental data (points) and fitted curves (lines) of MAREs versus azimuthal errors (for δ*θ*A:δ*θ*P = 2:1, the horizontal axis represents δ*θ*A). **Figure 8.** Experimental data (points) and fitted curves (lines) of MAREs versus azimuthal errors (for δ*θ*A:δ*θ*<sup>P</sup> = 2:1, the horizontal axis represents δ*θ*A). *Crystals* **2021**, *11*, x FOR PEER REVIEW 10 of 12

**Figure 9.** Ellipsometric parameters measured with azimuthal errors of the same absolute values in the opposite direction. **Figure 9.** Ellipsometric parameters measured with azimuthal errors of the same absolute values in the opposite direction.

We presented a method to implement the experimental error analysis of azimuth and the incident angle specifically for the RPAE in this work. The dielectric functions of bulk Au were determined from the measured *ψ* and Δ at incident angles in a range of 55°–80°, with an interval of 5°, to study the effect on the instrument. The results acquired at an incident angle close to the principal angle were observed to exhibit higher accuracy and better stability, according to the discrepancies between the two solutions, and the values of the MARE. The azimuthal error analysis was performed experimentally with the initial orientations of polarizing elements deviating from the zero azimuth. The fitting curves of the MARE versus azimuthal error suggest that δ*θ*P more seriously affects the accuracy of the RPAE. The demonstrated error analysis reveals the relationship between the acquired data and experimental conditions, which gives access to achieving accurate and reliable measurement by using the RPAE, and is easily generalized to other rotating element el-

**Author Contributions:** Conceptualization, H.T. and Y.Z.; methodology, H.T.; validation, Y.S., Y.C., and H.Z.; formal analysis, H.T. and Y.S.; investigation, Y.C. and H.Z.; data curation, Y.Z., Y.L. and L.C.; writing—original draft preparation, H.T.; writing—review and editing, Y.Z., R.Z., S.W., J.L., Y.L. and L.C.; supervision, Y.Z. and L.C.; project administration, Y.Z. and L.C.; funding acquisition, Y.Z. and L.C. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China, grant number 61775042 11674062 and the Fudan University CIOMP Joint Fund, grant number FC2017-

**Acknowledgments:** We would like to express our sincere appreciations to the anonymous referees

**Institutional Review Board Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable.

for valuable suggestions and corrections.

**4. Conclusions** 

lipsometers.

003.
