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 interferometry measurement model

The laser interferometry measurement of gauge block height is based on the interference principle of light waves. When monochromatic light is perpendicularly incident on the upper surface of the gauge block and the reference optical flat surface, two beams of reflected light produce interference, and their optical path difference Δ and the gauge block center length L satisfy the relationship Δ=2nL. According to the interference extremum condition, the gauge block length can be expressed as:

</math display="block"> L = \frac{\lambda}{2n}(m + \varepsilon) </math>

$H=\frac{{\lambda }_i}{2n_i}(N_i+{\epsilon }_i)$

In the formula, λ is the vacuum wavelength of the light source, n is the refractive index of air under the experimental environment, m is the integer order of the interference fringes, and ϵ is the fractional part measured from the interference pattern (0≤ϵ<1).

Because the non-ambiguous range of single-wavelength interferometry is only λ/2n, and the integer m cannot be directly determined, a phase ambiguity problem exists. Multi-wavelength interferometry technology utilizes different fractional parts (ϵ 1 ​

,ϵ 

2 ​

,...,ϵ 

k ​

) produced by different wavelengths (λ 

1 ​

,λ 

2 ​

,...,λ 

k ​

) under the same optical path difference as constraint conditions. According to the principle of the Method of Exact Fractions, the correct length L must simultaneously satisfy the equation sets of all wavelengths. By introducing multiple wavelengths, because the "synthetic wavelength" formed by their least common multiple is much larger than a single wavelength, the non-ambiguous measurement range is significantly expanded, achieving the absolute measurement of gauge block height.