Sample Thickness Measurement via Multi-wavelength Laser Interferometry

Team Members

Hang Tianyi A0326626B

Ding Jiahao A0332300A

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 gauge block with a nominal 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.

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 $\mathrm{\Delta }$ is related to the central thickness of the sample $L$ by:

$$\mathrm{\Delta }=2nL$$

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

$$L=\frac{\lambda }{2n}(m+\epsilon )$$

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

Because the unambiguous range of single-wavelength interferometry is limited to $\lambda /2n$, and the integer order $m$ cannot be determined directly, phase ambiguity arises. Multi-wavelength interferometry overcomes this limitation by using multiple wavelengths (${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$) and their corresponding fractional parts (${\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_k$) under the same optical path difference as constraints. According to the Method of Exact Fractions, the correct thickness $L$ 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 $n$ cannot be neglected. The actual wavelength in air,

$${\lambda }_{air}=\frac{{\lambda }_{vac}}{n}$$

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 ($t$), atmospheric pressure $p$ (Pa), and relative humidity $h$ (%).

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

$$(n_{tp}-1)=\frac{p(n_s-1)}{96095.43}\cdot \frac{1+10^{-8}(0.601-0.00972t)p}{1+0.0036610t}$$

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 $f$ (in Pa) must be subtracted from $n_{tp}$:

$$n_{tpf}=n_{tp}-f\times (3.7345-0.0401{\sigma }^2)\times 10^{-10}$$

Here, the water vapor partial pressure $f$ depends on the ambient relative humidity $R_H$ (%) 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:

$$L=(m_i+{\epsilon }_i)\frac{{\lambda }_i}{2n_i}$$

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:

$$[L_{min},L_{max}]$$

Within this interval, for each wavelength ${\lambda }_i$, the possible range of integer orders $m_i$ is given by:

$$m_{i,start}=\lceil \frac{2n_i{L}_{min}}{{\lambda }_i}-{\epsilon }_i\rceil ,m_{i,end}=\lfloor \frac{2n_i{L}_{max}}{{\lambda }_i}-{\epsilon }_i\rfloor$$

Traditional Fractional Coincidence Method

This algorithm adopts a “primary 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 primary reference. Let the sample thickness be $L$, the air refractive index be $n_i$, the vacuum wavelength be ${\lambda }_i$, the measured fractional part be ${\epsilon }_i$ ($0\le {\epsilon }_i<1$), and the integer order be $m_i$. For three wavelengths (red, yellow, and green), the following system of equations can be established:

$$\{\begin{array}{l}\frac{2n_{1}L}{{\lambda }_1}=m_1+{\epsilon }_1\\ \frac{2n_{2}L}{{\lambda }_2}=m_2+{\epsilon }_2\\ \frac{2n_{3}L}{{\lambda }_3}=m_3+{\epsilon }_3\end{array}$$

Here, ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$ are directly measured experimental values, while $m_1,m_2,m_3$ are unknown positive integers. The mathematical essence of this method is to search, within the estimated thickness interval, for a unique thickness $L$ that satisfies all three equations simultaneously.

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

$${\lambda }_{syn}=\frac{{\lambda }_i{\lambda }_j}{|{\lambda }_i-{\lambda }_j|}$$

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