Editing Sample Thickness Measurement via Multi-wavelength Laser Interferometry (section)

===Summary of thickness results===

To further compare the thickness solutions obtained under different principal-wavelength and fringe-width conditions, the optimal values selected from Table 2 were summarized, as listed in Table 3. When red, yellow, and violet light were respectively used as the principal wavelength, the mean thickness values obtained from the four fringe-width conditions were <math>2.000264\ \text{mm}</math>, <math>2.000250\ \text{mm}</math>, and <math>2.000245\ \text{mm}</math>, respectively. The maximum difference among these three mean values was only 19 nm, indicating that the results obtained using the principal-wavelength rotation strategy were highly consistent and that the multi-wavelength constraint effectively improved the stability of the final thickness solution.

A clear tendency can also be observed from the weighted mean results under different fringe-width conditions. The thickness values obtained under the wide, medium, narrow, and extremely narrow fringe conditions were <math>2.000229\ \text{mm}</math>, <math>2.000253\ \text{mm}</math>, <math>2.000250\ \text{mm}</math>, and <math>2.000274\ \text{mm}</math>, respectively. Among them, the result obtained under the wide-fringe condition was generally lower, whereas the result obtained under the extremely narrow-fringe condition was generally higher. In contrast, the thickness values obtained under the medium- and narrow-fringe conditions were much closer to each other and also more consistent with the overall mean value. This indicates that the effect of fringe width on the accuracy of fractional extraction is further transferred to the final thickness determination. Excessively wide fringes contain fewer effective fringe periods and are therefore unfavorable for precise localization, whereas excessively dense fringes are more susceptible to pixel discretization effects and image noise, which may introduce a certain systematic bias into the final result.

From the perspective of overall consistency, the thickness values obtained under the medium- and narrow-fringe conditions showed the best agreement among the three principal-wavelength solutions. This suggests that these two fringe-width conditions provide a better balance among fringe number, image resolvability, and the stability of fractional extraction. Therefore, compared with the wide- and extremely narrow-fringe conditions, medium to narrow fringes are more suitable for high-precision thickness determination in the present multi-wavelength interferometric measurement.

By combining the three principal-wavelength solutions with the weighted mean results, the sample thickness measured in this experiment at <math>21.9^\circ\text{C}</math> can be taken as approximately <math>2.000252\ \text{mm}</math>. This final value is in good agreement with the thickness values obtained under different principal wavelengths and fringe-width conditions, demonstrating that the established multi-wavelength interferometric measurement procedure and the method of exact fractions provide reliable and repeatable thickness determination results.

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