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

==Experimental Details==
===Experimental Setups===
The experimental optical setup consists of three main subsystems: three laser sources with different wavelengths, a beam-conditioning module, and an interferometer.

The light sources are selected as follows: a red laser with wavelength <math>\lambda_1 = 650\,\mathrm{nm}</math>, a yellow laser with wavelength <math>\lambda_2 = 594\,\mathrm{nm}</math> and a violet laser with wavelength <math>\lambda_3 = 405\,\mathrm{nm}</math>.

To improve the contrast and signal-to-noise ratio of the interference fringes, the beam first passes through a spatial filter composed of a microscope objective and a pinhole. This stage reshapes the wavefront by removing higher-order transverse modes and speckle noise. The filtered beam is then collimated into a parallel beam using an achromatic lens.
Next, the beam is directed through a beam splitter (BS) and normally incident onto both the standard sample surface and the reference mirror in optical contact with it. The reflected beams recombine at the beam splitter and interfere on the target plane of a CCD camera.
Since the sensitive area of the CCD is much smaller than the actual interference pattern, a beam-converging (imaging) system is incorporated into the optical path to ensure that the interference pattern can be captured more completely.

[[File:EXP_SETUP.jpeg|1000px|center]]

===Data Preprocessing===

Because laser interferometric measurement is essentially a comparative measurement that uses the wavelength of light as a ruler, even slight fluctuations in the refractive index of air will directly introduce systematic errors. This experiment was carried out under strictly controlled laboratory conditions, and the air parameters at the time of measurement were recorded in real time using an environmental monitor: ambient temperature 21.90°C ± 0.05°C, pressure 1009 hPa, and relative humidity 55%. According to the Edlén correction formula described above, these environmental parameters were substituted into the calculation, yielding the corrected refractive indices: <math>n_1 = 1.000268687558227179</math> for red light, <math>n_2 = 1.000269432843863667</math> for yellow light, and <math>n_3 = 1.000274643309471638</math> for violet light. The corresponding wavelengths in air were then obtained as <math>0.6498254\ \mu\text{m}</math> for red light, <math>0.5938400\ \mu\text{m}</math> for yellow light, and <math>0.4048888\ \mu\text{m}</math> for violet light.

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

The thickness of the sample, as measured with a vernier caliper, was between <math>1.98</math> and <math>2.02\ \text{mm}</math>.

[[File:Rulerfig.jpeg|300px|center]]

===Interference pattern acquisition===

To ensure the accuracy of the subsequent fractional extraction and length determination, interferograms of the sample were acquired after completing the coaxial alignment of the multi-wavelength optical path and optimizing the stability of the system. The primary requirement of interferogram acquisition was to obtain raw images with high contrast, continuous and clear fringes, a clearly identifiable sample boundary, and good consistency among different wavelengths, thereby providing a reliable data basis for subsequent image processing and parameter calculation.

Red (650 nm), yellow (594 nm), and violet (405 nm) lasers were employed for the measurements. Under each wavelength condition, the measurement region, imaging position, and basic optical geometry were kept as consistent as possible so that the interferograms obtained at different wavelengths remained comparable. Each interferogram contained both the reference region and the sample region. A relative displacement of the fringes appeared near the sample boundary, and this displacement reflected the variation in optical path difference between the sample surface and the reference surface. It therefore served as the direct basis for the subsequent extraction of the fractional part <math>e</math>. During image acquisition, particular attention was paid to ensuring that the fringes near the sample boundary were clear and continuous, so as to avoid errors in fringe-center localization caused by local blurring, brightness saturation, or noise interference.

[[File:Three color.jpeg|600px|center]]

To investigate the influence of fringe width on the measurement results, the density of the interference fringes was controlled by finely adjusting the posture of the reference mirror and the system angle. Under each wavelength, four typical fringe conditions were obtained, namely wide fringes, medium fringes, narrow fringes, and extremely narrow fringes, as shown in the figure. With the variation of the reference surface tilt angle, the fringe spacing in the field of view gradually decreased, while the spatial frequency of the fringes gradually increased. By repeatedly acquiring interferograms under different fringe-width conditions, the effect of fringe morphology on the stability of fractional extraction and the final length determination results could be systematically analyzed.

[[File:wide.jpeg|800px|center]]

For each fringe-width condition, repeated measurements were carried out, and the adjacent fringe spacing <math>M</math>, the displacement <math>m</math>, and the corresponding fractional part <math>e</math> were recorded. The statistical results are listed in Table 1. To avoid taking up too much space on the page, a selection of the statistical results is shown here.

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

The acquisition results show that the quality and measurability of the interferograms differ significantly under different fringe-width conditions. Under the wide-fringe condition, the number of fringes was relatively small; although the interferogram was visually intuitive, the number of effective periods available for localization and statistical analysis was limited. Under the extremely narrow-fringe condition, although more fringe periods were contained within a unit field of view, the excessively small fringe spacing imposed higher requirements on imaging resolution and system stability, making the measurement more susceptible to noise and pixel discretization effects. In comparison, medium to relatively narrow fringes provided a better balance among fringe number, boundary clarity, and resolvability, and were therefore more favorable for subsequent fringe-center localization and fractional extraction.