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

===Fractional Coincidence Method===

This algorithm adopts a “principal wavelength guided” strategy. The fundamental limitation of single-wavelength interferometry lies in phase ambiguity: the measurement only provides a fractional part within one fringe period and cannot directly determine the integer order.

Select one wavelength as the principal reference. Let the sample thickness be <math>L</math>, the air refractive index be <math>n_i</math>, the vacuum wavelength be <math>\lambda_i</math>, the measured fractional part be <math>\varepsilon_i</math> (<math>0 \le \varepsilon_i < 1</math>), and the integer order be <math>m_i</math>. For three wavelengths (red, yellow, and green), the following system of equations can be established:

<math display="block">
\begin{cases} 
\frac{2n_1 L}{\lambda_1} = m_1 + \varepsilon_1 \\
\frac{2n_2 L}{\lambda_2} = m_2 + \varepsilon_2 \\
\frac{2n_3 L}{\lambda_3} = m_3 + \varepsilon_3 
\end{cases}
</math>

Here, <math>\varepsilon_1, \varepsilon_2, \varepsilon_3</math> are directly measured experimental values, while <math>m_1, m_2, m_3</math> are unknown positive integers. The mathematical essence of this method is to search, within the estimated thickness interval, for a unique thickness <math>L</math> that satisfies all three equations simultaneously.

In principle, a synthetic wavelength constructed from any two wavelengths,

<math display="block">
\lambda_{syn} = \frac{\lambda_i \lambda_j}{|\lambda_i - \lambda_j|}
</math>

is sufficient to uniquely determine the thickness within a certain range.