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

==Introduction==
Precision length measurement plays a pivotal role in scientific research, industrial manufacturing, and defense technology. Among various length measurement techniques, laser interferometry has become the "gold standard" in modern metrology due to its high precision and resolution. However, conventional single-wavelength interferometry is inherently a relative measurement method. Constrained by the periodicity of light waves, its unambiguous measurement range (UMR) is limited to only half a wavelength. When the measured length exceeds this limit, the measurement faces significant phase ambiguity and integer-cycle uncertainty, making it difficult to apply single-wavelength techniques directly to large-scale absolute length measurements.

To overcome these range limitations, Multi-Wavelength Interferometry (MWI) was developed. By introducing multiple light sources of different wavelengths, this technique utilizes synthetic wavelengths to extend the UMR. The Method of Exact Fractions is the most classic and effective algorithm for this purpose. First proposed by Benoit in 1898 and successfully used for the calibration of the International Prototype Meter, its core principle involves using the fractional parts of interference fringes at different wavelengths as constraints. By identifying the unique set of integer orders that satisfies the equations for all wavelengths, the absolute length can be determined.

Although this theory is well-established, practical applications are significantly influenced by wavelength stability, the precision of fractional part extraction, and fluctuations in the refractive index of air. Studies by Birch, Downs, and others on the revision of the Edlén formula underscore that precise environmental compensation is a prerequisite for interferometric accuracy. Furthermore, fringe quality (such as width and contrast) and the noise immunity of fractional extraction algorithms are critical factors in ensuring the robustness of the measurement system.

To address the aforementioned challenges, a high-precision multi-wavelength interferometry system based on the Twyman-Green configuration was developed, complemented by an in-depth investigation into robust reconstruction algorithms for complex environments. Three visible laser sources—red (650 nm), yellow (594 nm), and violet (405 nm)—were employed in the experimental setup. A block with a length of 2.00 mm (pre-measured by a vernier caliper) served as the measurement specimen. To mitigate wavelength fluctuations induced by environmental variables, the updated Edlén formula was implemented to provide precise compensation for the refractive index of air.