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

==Basic Principles of the Experiment==
===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.


===Environmental Error Correction===

In precision metrology, the influence of environmental factors on the air refractive index <math>n</math> cannot be neglected. The actual wavelength in air,

<math display="block">
\lambda_{air} = \frac{\lambda_{vac}}{n}
</math>

directly determines the measurement accuracy. In this experiment, the refractive index of air is calculated using the Edlén formula as modified by Birch and Downs. This formulation accounts for the effects of temperature (<math>t</math>), atmospheric pressure <math>p</math> (Pa), and relative humidity <math>h</math> (%).

First, the refractive index under standard conditions <math>(n_s - 1)</math> is computed. Then, corrections are applied based on environmental parameters. The simplified expression is:

<math display="block">
(n_{tp} - 1) = \frac{p(n_s - 1)}{96095.43} \cdot \frac{1 + 10^{-8}(0.601 - 0.00972t)p}{1 + 0.0036610t}
</math>

In addition, a small correction due to the partial pressure of water vapor must be considered. Although the effect of relative humidity on the refractive index is small, it is not negligible. Since the refractive index of water vapor is lower than that of dry air, increasing humidity reduces the overall refractive index of air. Therefore, a correction term associated with the water vapor partial pressure <math>f</math> (in Pa) must be subtracted from <math>n_{tp}</math>:

<math display="block">
n_{tpf} = n_{tp} - f \times (3.7345 - 0.0401\sigma^2) \times 10^{-10}
</math>

Here, the water vapor partial pressure <math>f</math> depends on the ambient relative humidity <math>R_H</math> (%) and the current temperature.

===Thickness Reconstruction Algorithm Design===

The core task of thickness reconstruction is to identify a unique solution within the solution space defined by:

<math display="block">
L = (m_i + \varepsilon_i)\frac{\lambda_i}{2n_i}
</math>

that satisfies all wavelength constraints simultaneously.

First, the sample thickness is measured multiple times using a vernier caliper. The maximum and minimum measured values are used to define the search interval:

<math display="block">
[L_{min}, L_{max}]
</math>

Within this interval, for each wavelength <math>\lambda_i</math>, the possible range of integer orders <math>m_i</math> is given by:

<math display="block">
m_{i,start} = \left\lceil \frac{2n_i L_{min}}{\lambda_i} - \varepsilon_i \right\rceil,\quad
m_{i,end} = \left\lfloor \frac{2n_i L_{max}}{\lambda_i} - \varepsilon_i \right\rfloor
</math>

===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.