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 block with a 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:

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

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 $L$, 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.

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 ${\lambda }_1=650\mathrm{n}\mathrm{m}$, a yellow laser with wavelength ${\lambda }_2=594\mathrm{n}\mathrm{m}$ and a violet laser with wavelength ${\lambda }_3=405\mathrm{n}\mathrm{m}$.

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.

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: $n_1=1.000268687558227179$ for red light, $n_2=1.000269432843863667$ for yellow light, and $n_3=1.000274643309471638$ for violet light. The corresponding wavelengths in air were then obtained as $0.6498254\mu \text{m}$ for red light, $0.5938400\mu \text{m}$ for yellow light, and $0.4048888\mu \text{m}$ for violet light.

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 $e$. 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.

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.

For each fringe-width condition, repeated measurements were carried out, and the adjacent fringe spacing $M$, the displacement $m$, and the corresponding fractional part $e$ 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.

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.