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

==Results==
===Results of the principal wavelength rotation analysis===

After obtaining the mean fractional values under the three wavelengths and four fringe-width conditions, the traditional method of exact fractions was further applied to determine the integer orders. To examine the robustness of the solution, red light (650 nm), yellow light (594 nm), and violet light (405 nm) were respectively used as the principal wavelength, while the fractional deviations <math>\Delta \xi</math> of the other two auxiliary wavelengths were used to screen the candidate integer orders. The results are listed in Table 2.

When red light was used as the principal wavelength, the candidate order <math>m_1 = 6156</math> gave the auxiliary fractional deviations closest to zero overall. Under the wide, medium, narrow, and extremely narrow fringe conditions, the corresponding thickness values were <math>2.000239\ \text{mm}</math>, <math>2.000259\ \text{mm}</math>, <math>2.000267\ \text{mm}</math>, and <math>2.000289\ \text{mm}</math>, respectively. At this integer order, the absolute values of the fractional deviations for yellow and violet light remained within a relatively small range, indicating that this solution satisfied the multi-wavelength coincidence condition well. It can therefore be identified as the optimal integer order under the red-principal-wavelength condition.

When yellow light was used as the principal wavelength, the optimal integer order was determined to be <math>m_2 = 6736</math>. Under the four fringe-width conditions, the corresponding thickness values were <math>2.000227\ \text{mm}</math>, <math>2.000254\ \text{mm}</math>, <math>2.000246\ \text{mm}</math>, and <math>2.000273\ \text{mm}</math>, respectively. As shown in Table 2, the fractional deviations of red and violet light at this order were also closest to zero, indicating that the solution obtained with yellow light as the principal wavelength was in good agreement with the other two wavelengths and that the selected integer order was reasonable.

When violet light was used as the principal wavelength, the optimal integer order was found to be <math>m_3 = 9880</math>. Under the wide, medium, narrow, and extremely narrow fringe conditions, the corresponding thickness values were <math>2.000225\ \text{mm}</math>, <math>2.000247\ \text{mm}</math>, <math>2.000242\ \text{mm}</math>, and <math>2.000265\ \text{mm}</math>, respectively. Similar to the previous two cases, the auxiliary fractional deviations at the row corresponding to <math>m_3 = 9880</math> were generally small, indicating that the integer order determined under the violet-light condition also achieved good coincidence with the fractional parts measured at red and yellow wavelengths.

A comparison of the results obtained by rotating the principal wavelength shows that, although the selected principal wavelength was different, the resulting thickness values were all concentrated around <math>2.0002\ \text{mm}</math>, and the differences among them were only on the order of several tens of nanometers. This demonstrates a high degree of consistency among the solutions. It also indicates that, with the principal-wavelength rotation strategy, the traditional method of exact fractions can provide mutually consistent integer orders and thickness values under different wavelength conditions, thereby reducing the accidental dependence on the choice of a single wavelength.

In addition, the calculated results still exhibited a certain dependence on fringe width. In general, the thickness values obtained under the wide-fringe condition were slightly lower, whereas those obtained under the extremely narrow-fringe condition were slightly higher. The results under the medium and narrow fringe conditions were relatively closer to one another. This tendency suggests that the influence of fringe width on the accuracy of fractional extraction is further transferred to the determination of the integer order and the final thickness solution. Since the results obtained from principal-wavelength rotation show a high level of consistency among the three wavelengths, the integer-order combinations listed in Table 2 can be regarded as reliable and can serve as the basis for the subsequent summary and weighted averaging of the thickness results.

[[File:Tabletwo.png|1000px|center]]

===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.

[[File:Table tree.png|800px|center]]