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

===Multi-Wavelength Interferometric Measurement Model===

Laser interferometric measurement of the sample thickness is based on the interference principle of light waves. When monochromatic light is incident normally on the upper surface of the sample and the reference optical flat, the two reflected beams interfere. The optical path difference <math>\Delta</math> is related to the central thickness of the sample <math>L</math> by:

<math display="block">
\Delta = 2nL
</math>

According to the condition for interference extrema, the sample thickness can be expressed as:

<math display="block">
L = \frac{\lambda}{2n}(m + \varepsilon)
</math>

Here, <math>\lambda</math> is the vacuum wavelength of the light source, <math>n</math> is the refractive index of air under experimental conditions, <math>m</math> is the integer fringe order, and <math>\varepsilon</math> is the fractional part obtained from the interference pattern (<math>0 \le \varepsilon < 1</math>).

Because the unambiguous range of single-wavelength interferometry is limited to <math>\lambda / 2n</math>, and the integer order <math>m</math> cannot be determined directly, phase ambiguity arises. Multi-wavelength interferometry overcomes this limitation by using multiple wavelengths (<math>\lambda_1, \lambda_2, \dots, \lambda_k</math>) and their corresponding fractional parts (<math>\varepsilon_1, \varepsilon_2, \dots, \varepsilon_k</math>) under the same optical path difference as constraints. According to the Method of Exact Fractions, the correct thickness <math>L</math> must simultaneously satisfy all corresponding equations. By introducing multiple wavelengths, the effective “synthetic wavelength,” determined by their least common multiple, becomes much larger than any single wavelength. This significantly expands the unambiguous measurement range and enables absolute measurement of the sample thickness.