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

==Analysis of Errors and Discussion==

Although the multi-wavelength interferometric method can effectively resolve the phase ambiguity in single-wavelength measurement, the final thickness result is still affected by several uncertainty sources. In the present experiment, the main sources of error can be attributed to environmental fluctuation, fractional extraction uncertainty, optical alignment error, and the finite resolution of the imaging system.

The correction accuracy of the refractive index of air directly affects the measurement result. Since the sample thickness is determined by <math>L=\frac{\lambda}{2n}(m+e)</math>, any small variation in the refractive index <math>n</math> will be transferred directly to the final thickness value. In this experiment, the refractive indices were corrected according to the measured temperature, pressure, and relative humidity. However, slight fluctuations in the laboratory environment during image acquisition and data processing may still introduce a certain systematic deviation. From the present results, the contribution of environmental factors appears to be relatively limited under stable indoor air-conditioned conditions, and its effect is generally smaller than the uncertainty arising from image processing and fractional extraction.

The extraction error of the fractional part <math>e</math> is one of the dominant uncertainty sources in this experiment. The fractional extraction depends on the accurate determination of the fringe spacing <math>M</math> and the fringe displacement <math>m</math>, while the fringe-center position can be influenced by local grayscale fluctuation, insufficient fringe contrast, residual background nonuniformity, and slight fringe curvature. When the fringes are too wide, the number of effective periods available for localization and fitting in the field of view is limited, so the extracted fractional value is more easily affected by local features. When the fringes are too dense, pixel discretization, enhanced noise, and local blurring may reduce the fringe-center localization accuracy. This is consistent with the results summarized in Table 3, where the thickness values obtained under the medium and narrow fringe conditions are more stable, whereas larger deviations appear under the wide and extremely narrow fringe conditions.

Residual optical alignment error may also influence the final result. Although the three wavelengths were adjusted to be as coaxial as possible and the measurement region and imaging position were kept consistent, slight differences in beam position, spot displacement, or imaging overlap may still exist among the wavelengths. Such errors may lead to small differences in the measured fringe displacement near the sample boundary, thereby affecting the coincidence among the three wavelengths. In addition, if the sample boundary is not sufficiently sharp in the recorded image, the determination of the corresponding fringe positions in the reference and sample regions may also introduce an additional bias.

According to the principal-wavelength rotation results, the maximum difference among the mean thickness values obtained using red, yellow, and violet light as the principal wavelength is only 19 nm. This indicates that the multi-wavelength constraint provides good overall stability and can effectively suppress large errors caused by incorrect integer-order determination. At the same time, this difference also shows that, once the integer order is correctly identified, the remaining nanometer-level variation in thickness is still mainly governed by the precision of fractional extraction. In other words, the present method is reliable in determining the correct integer order, whereas the ultimate precision of the result depends more strongly on interferogram quality and the accuracy of the fractional extraction procedure.

The summarized thickness results under different fringe-width conditions also show that fringe morphology has a direct influence on the final measurement. The thickness obtained under the wide-fringe condition is generally lower, whereas that obtained under the extremely narrow-fringe condition is generally higher. In contrast, the results under the medium and narrow fringe conditions are closer to each other and show better consistency. This indicates that the choice of fringe density itself is an important experimental factor in improving measurement accuracy. In the present experiment, medium to narrow fringes provide a better balance among fringe number, image clarity, and fractional extraction stability, and are therefore more suitable for high-precision thickness measurement.

Overall, the experimental results demonstrate that the multi-wavelength method of exact fractions can provide reliable absolute thickness measurement of the sample. The good agreement among the solutions obtained by principal-wavelength rotation confirms the validity of the integer-order determination, while the comparison among different fringe-width conditions highlights the importance of interferogram quality in controlling the final uncertainty. Further improvement in measurement accuracy may be achieved by optimizing fringe contrast, increasing imaging resolution, and refining the fringe-center extraction algorithm, thereby further reducing random error and systematic deviation in the experiment.