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.

The thickness of the sample, as measured with a vernier caliper, was between $1.98$ and $2.02\text{mm}$.

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.

Results

Results of the principal wavelength rotation analysis

After obtaining the mean fractional values under the three wavelengths and four fringe-width conditions, the traditional method of exact fractions was further applied to determine the integer orders. To examine the robustness of the solution, red light (650 nm), yellow light (594 nm), and violet light (405 nm) were respectively used as the principal wavelength, while the fractional deviations $\mathrm{\Delta }\xi$ of the other two auxiliary wavelengths were used to screen the candidate integer orders. The results are listed in Table 2.

When red light was used as the principal wavelength, the candidate order $m_1=6156$ gave the auxiliary fractional deviations closest to zero overall. Under the wide, medium, narrow, and extremely narrow fringe conditions, the corresponding thickness values were $2.000239\text{mm}$, $2.000259\text{mm}$, $2.000267\text{mm}$, and $2.000289\text{mm}$, respectively. At this integer order, the absolute values of the fractional deviations for yellow and violet light remained within a relatively small range, indicating that this solution satisfied the multi-wavelength coincidence condition well. It can therefore be identified as the optimal integer order under the red-principal-wavelength condition.

When yellow light was used as the principal wavelength, the optimal integer order was determined to be $m_2=6736$. Under the four fringe-width conditions, the corresponding thickness values were $2.000227\text{mm}$, $2.000254\text{mm}$, $2.000246\text{mm}$, and $2.000273\text{mm}$, respectively. As shown in Table 2, the fractional deviations of red and violet light at this order were also closest to zero, indicating that the solution obtained with yellow light as the principal wavelength was in good agreement with the other two wavelengths and that the selected integer order was reasonable.

When violet light was used as the principal wavelength, the optimal integer order was found to be $m_3=9880$. Under the wide, medium, narrow, and extremely narrow fringe conditions, the corresponding thickness values were $2.000225\text{mm}$, $2.000247\text{mm}$, $2.000242\text{mm}$, and $2.000265\text{mm}$, respectively. Similar to the previous two cases, the auxiliary fractional deviations at the row corresponding to $m_3=9880$ were generally small, indicating that the integer order determined under the violet-light condition also achieved good coincidence with the fractional parts measured at red and yellow wavelengths.

A comparison of the results obtained by rotating the principal wavelength shows that, although the selected principal wavelength was different, the resulting thickness values were all concentrated around $2.0002\text{mm}$, and the differences among them were only on the order of several tens of nanometers. This demonstrates a high degree of consistency among the solutions. It also indicates that, with the principal-wavelength rotation strategy, the traditional method of exact fractions can provide mutually consistent integer orders and thickness values under different wavelength conditions, thereby reducing the accidental dependence on the choice of a single wavelength.

In addition, the calculated results still exhibited a certain dependence on fringe width. In general, the thickness values obtained under the wide-fringe condition were slightly lower, whereas those obtained under the extremely narrow-fringe condition were slightly higher. The results under the medium and narrow fringe conditions were relatively closer to one another. This tendency suggests that the influence of fringe width on the accuracy of fractional extraction is further transferred to the determination of the integer order and the final thickness solution. Since the results obtained from principal-wavelength rotation show a high level of consistency among the three wavelengths, the integer-order combinations listed in Table 2 can be regarded as reliable and can serve as the basis for the subsequent summary and weighted averaging of the thickness results.

Summary of thickness results

To further compare the thickness solutions obtained under different principal-wavelength and fringe-width conditions, the optimal values selected from Table 2 were summarized, as listed in Table 3. When red, yellow, and violet light were respectively used as the principal wavelength, the mean thickness values obtained from the four fringe-width conditions were $2.000264\text{mm}$, $2.000250\text{mm}$, and $2.000245\text{mm}$, respectively. The maximum difference among these three mean values was only 19 nm, indicating that the results obtained using the principal-wavelength rotation strategy were highly consistent and that the multi-wavelength constraint effectively improved the stability of the final thickness solution.

A clear tendency can also be observed from the weighted mean results under different fringe-width conditions. The thickness values obtained under the wide, medium, narrow, and extremely narrow fringe conditions were $2.000229\text{mm}$, $2.000253\text{mm}$, $2.000250\text{mm}$, and $2.000274\text{mm}$, respectively. Among them, the result obtained under the wide-fringe condition was generally lower, whereas the result obtained under the extremely narrow-fringe condition was generally higher. In contrast, the thickness values obtained under the medium- and narrow-fringe conditions were much closer to each other and also more consistent with the overall mean value. This indicates that the effect of fringe width on the accuracy of fractional extraction is further transferred to the final thickness determination. Excessively wide fringes contain fewer effective fringe periods and are therefore unfavorable for precise localization, whereas excessively dense fringes are more susceptible to pixel discretization effects and image noise, which may introduce a certain systematic bias into the final result.

From the perspective of overall consistency, the thickness values obtained under the medium- and narrow-fringe conditions showed the best agreement among the three principal-wavelength solutions. This suggests that these two fringe-width conditions provide a better balance among fringe number, image resolvability, and the stability of fractional extraction. Therefore, compared with the wide- and extremely narrow-fringe conditions, medium to narrow fringes are more suitable for high-precision thickness determination in the present multi-wavelength interferometric measurement.

By combining the three principal-wavelength solutions with the weighted mean results, the sample thickness measured in this experiment at $21.9^{\circ }\text{C}$ can be taken as approximately $2.000252\text{mm}$. This final value is in good agreement with the thickness values obtained under different principal wavelengths and fringe-width conditions, demonstrating that the established multi-wavelength interferometric measurement procedure and the method of exact fractions provide reliable and repeatable thickness determination results.

Analysis of Errors and Discussion

Although the multi-wavelength interferometric method can effectively resolve the phase ambiguity in single-wavelength measurement, the final thickness result is still affected by several uncertainty sources. In the present experiment, the main sources of error can be attributed to environmental fluctuation, fractional extraction uncertainty, optical alignment error, and the finite resolution of the imaging system.

The correction accuracy of the refractive index of air directly affects the measurement result. Since the sample thickness is determined by $L=\frac{\lambda }{2n}(m+e)$, any small variation in the refractive index $n$ will be transferred directly to the final thickness value. In this experiment, the refractive indices were corrected according to the measured temperature, pressure, and relative humidity. However, slight fluctuations in the laboratory environment during image acquisition and data processing may still introduce a certain systematic deviation. From the present results, the contribution of environmental factors appears to be relatively limited under stable indoor air-conditioned conditions, and its effect is generally smaller than the uncertainty arising from image processing and fractional extraction.

The extraction error of the fractional part $e$ is one of the dominant uncertainty sources in this experiment. The fractional extraction depends on the accurate determination of the fringe spacing $M$ and the fringe displacement $m$, while the fringe-center position can be influenced by local grayscale fluctuation, insufficient fringe contrast, residual background nonuniformity, and slight fringe curvature. When the fringes are too wide, the number of effective periods available for localization and fitting in the field of view is limited, so the extracted fractional value is more easily affected by local features. When the fringes are too dense, pixel discretization, enhanced noise, and local blurring may reduce the fringe-center localization accuracy. This is consistent with the results summarized in Table 3, where the thickness values obtained under the medium and narrow fringe conditions are more stable, whereas larger deviations appear under the wide and extremely narrow fringe conditions.

Residual optical alignment error may also influence the final result. Although the three wavelengths were adjusted to be as coaxial as possible and the measurement region and imaging position were kept consistent, slight differences in beam position, spot displacement, or imaging overlap may still exist among the wavelengths. Such errors may lead to small differences in the measured fringe displacement near the sample boundary, thereby affecting the coincidence among the three wavelengths. In addition, if the sample boundary is not sufficiently sharp in the recorded image, the determination of the corresponding fringe positions in the reference and sample regions may also introduce an additional bias.

According to the principal-wavelength rotation results, the maximum difference among the mean thickness values obtained using red, yellow, and violet light as the principal wavelength is only 19 nm. This indicates that the multi-wavelength constraint provides good overall stability and can effectively suppress large errors caused by incorrect integer-order determination. At the same time, this difference also shows that, once the integer order is correctly identified, the remaining nanometer-level variation in thickness is still mainly governed by the precision of fractional extraction. In other words, the present method is reliable in determining the correct integer order, whereas the ultimate precision of the result depends more strongly on interferogram quality and the accuracy of the fractional extraction procedure.

The summarized thickness results under different fringe-width conditions also show that fringe morphology has a direct influence on the final measurement. The thickness obtained under the wide-fringe condition is generally lower, whereas that obtained under the extremely narrow-fringe condition is generally higher. In contrast, the results under the medium and narrow fringe conditions are closer to each other and show better consistency. This indicates that the choice of fringe density itself is an important experimental factor in improving measurement accuracy. In the present experiment, medium to narrow fringes provide a better balance among fringe number, image clarity, and fractional extraction stability, and are therefore more suitable for high-precision thickness measurement.

Overall, the experimental results demonstrate that the multi-wavelength method of exact fractions can provide reliable absolute thickness measurement of the sample. The good agreement among the solutions obtained by principal-wavelength rotation confirms the validity of the integer-order determination, while the comparison among different fringe-width conditions highlights the importance of interferogram quality in controlling the final uncertainty. Further improvement in measurement accuracy may be achieved by optimizing fringe contrast, increasing imaging resolution, and refining the fringe-center extraction algorithm, thereby further reducing random error and systematic deviation in the experiment.