1. Introduction
With the rapid development of modern industry, the performance requirements for guideways in mechanical systems have become increasingly demanding. Traditional metal guideways are progressively unable to meet the needs for high precision, heavy loads, and operation under special environmental conditions. In response to these challenges, ceramic materials have gained attention as viable candidates for high-performance guideway applications. Ceramics such as alumina (Al
2O
3) and silicon nitride (Si
3N
4) offer desirable properties including high hardness, wear resistance, thermal stability, corrosion resistance, and rigidity, making them suitable for use in fields such as mechanical engineering, semiconductor manufacturing, and medical devices [
1,
2,
3]. In semiconductor manufacturing in particular, ceramic guideways have become indispensable components in critical equipment such as lithography systems, due to their advantageous physical and chemical characteristics. For large-scale, elongated structural ceramic components analogous to guideways, performance depends not only on the intrinsic material properties but also on the machining accuracy [
4]. Among the relevant parameters, flatness is a critical indicator of machining quality for ultra-precision ceramic components. It directly affects the motion accuracy and stability of planar motion pairs and is essential to the high-precision performance of mechanical equipment. Therefore, controlling flatness with high accuracy is crucial during the fabrication of such components.
Flatness refers to the deviation of the macroscopic height variation of a substrate surface from an ideal plane. It quantifies the amount by which an actual surface departs from perfect flatness and is used to describe the degree of surface uniformity. Common instruments for flatness measurement include the knife-edge [
5,
6], coordinate measuring machine (CMM) [
7], laser interferometer [
8], phase-shifting laser interferometer [
9], electronic level [
10,
11], autocollimator [
12,
13], and step gauge [
14]. The selection of a specific method depends on the size and required accuracy of the surface under inspection, as different techniques are suited to different measurement scenarios. The knife-edge method, with a measurement accuracy on the order of micrometers, is suitable for preliminary assessment of small surfaces but offers low sampling density. Coordinate measuring machines (CMMs) provide sub-micrometer accuracy and wide measurement ranges, making them suitable for high-precision flatness evaluation of various parts, with sampling density adjustable as needed. Laser interferometers also offer micrometer-level accuracy and are suitable for large-area surfaces, with flexible sampling intervals. Electronic levels typically offer an accuracy ranging from tens of micrometers to several millimeters per meter. Although applicable to surfaces of various sizes, multiple measurements are often required for large surfaces, resulting in relatively low sampling density. Autocollimators achieve micrometer-level accuracy and are appropriate for medium to large surfaces, with moderate sampling density. Step gauges, also with micrometer-level accuracy, are suitable for small to medium-sized surfaces, with sampling density determined by the length of the bridge plate. Compared to the instruments listed above, phase-shifting laser interferometers offer superior performance in both accuracy and sampling resolution. They can achieve root-mean-square (RMS) repeatability at the nanometer level and sampling intervals as fine as 0.1 mm. Depending on the aperture of the interferometer, they are suitable for measuring surfaces of various sizes and can acquire data from millions of measurement points simultaneously. However, due to their high sensitivity to surface roughness, such instruments are generally applicable only to polished surfaces.
The measurement of large-scale, elongated surfaces presents significant challenges. First, for ultra-precision flat guideway components, the flatness tolerance often needs to reach the sub-micrometer or micrometer level, rendering low-precision, small-range methods such as gap-based techniques unsuitable. Second, these ultra-precision ceramic components require accurate surface profile correction during grinding and polishing. As a result, measurement methods with low sampling density, limited repeatability, and high operational complexity—such as laser interferometry, autocollimation, electronic leveling, and step gauges—are inadequate for providing effective guidance in surface correction processes. Furthermore, for ultra-precision ceramic components with sizes exceeding 1 m, the measurement accuracy of methods such as CMMs, laser interferometers, autocollimators, and electronic levels tends to degrade with increasing measurement range. Additionally, the required measurement time increases proportionally with sampling density, making high-resolution measurements time-consuming. Moreover, during the grinding stage, due to relatively poor surface roughness, ceramic materials such as Al2O3 and SiC are generally unsuitable for measurement using phase-shifting laser interferometry.
To address these challenges, many researchers have proposed various approaches. Lu et al. [
15] segmented long surfaces into overlapping short sections for separate measurement and stitching. Using a custom-built system with a laser displacement sensor and a reference plate, they measured the straightness profile of a 500 mm-long machine tool guideway with a flatness deviation of approximately 92 μm. The maximum variation among three measurements was about 4.47% of the average dimensional tolerance. However, this method is limited to straightness measurement and does not provide comprehensive surface profile information. Li et al. [
16] developed a flatness measurement machine based on a high-precision reference mirror and an error separation system. The system achieved a repeatability better than 0.1 μm and was used to reduce the flatness deviation of a 700 mm-long metal guideway surface from PV 2.617 μm to PV 0.825 μm. Zhou et al. [
17] applied a stitching-based measurement approach using a high-precision flat mirror and displacement sensors, combined with a coordinate transformation algorithm. They measured the flatness of a 1400 mm × 500 mm metal optical surface and guided its error convergence from PV 7.54 μm to PV 2.98 μm. However, both methods rely on long glass reference surfaces with sub-micrometer or tens-of-nanometers flatness, which are difficult to manufacture and measure [
18]. Li et al. [
19] also proposed a laser interferometry-based method that utilizes the angular stability between adjacent guideway segments for long-range straightness stitching. The deviation from direct measurement was less than 10 μm over a 30 m length, and about 10 μm compared with an electronic level when applied to a 72 m guideway. This method is suitable for very long guideways, but its accuracy is limited. Rui et al. [
20] proposed a grazing-incidence stitching interferometry method with an astigmatism-based self-calibration algorithm for measuring rotationally symmetric circular polishing pads. While their approach achieved high precision (0.17 μm flatness error) in stable workshop environments, it exclusively addressed axisymmetric surfaces and relied on rotational scanning, thus lacking applicability to high-aspect-ratio elongated components. Furthermore, it concentrated on single measurements and did not experimentally demonstrate rapid detection during surface convergence processes. Lai et al. [
18] designed and manufactured a K9 glass guideway for use in a precision profilometer. The guideway was fabricated using computer-controlled optical machining and magnetorheological polishing. Flatness measurements of surfaces with dimensions 420 mm × 80 mm and 420 mm × 170 mm were conducted using a 24-inch Zygo Verifire MST interferometer (Zygo Corporation, Middle field, Middlefield, CT, USA), and the reference surface achieved a PV flatness of 98 nm. However, this approach is only applicable to optical surfaces and does not address the measurement of non-optical ceramic surfaces. Despite their effectiveness in specific applications, these techniques are not suitable for achieving high-precision and repeatable measurements on unpolished elongated ceramic surfaces.
To address the challenge of rapid surface flatness measurement for low-reflectivity, large-scale and high-aspect-ratio ceramic parts, this paper presents a measurement method that combines oblique-incidence laser interferometry with sub-aperture stitching. The oblique-incidence laser interferometry technique enhances the interference signal strength on low-reflectivity ceramic surfaces, thereby improving measurement accuracy. The sub-aperture stitching technique integrates high-precision measurements from multiple sub-apertures to enable a comprehensive and accurate assessment of large surfaces. This combined approach overcomes the limitations of conventional measurement techniques, such as insufficient accuracy, low sampling density, and the inability of standard laser interferometers to adapt to the full ultra-precision machining process. It provides a novel and effective solution for high-precision flatness measurement of large-scale, elongated components, such as ultra-precision ceramic guideways.
The structure of this paper is organized as follows:
Section 2 provides a detailed explanation of the fundamental principles of the proposed measurement method. In
Section 3, a mathematical model is established and numerical simulations are conducted to verify the feasibility and effectiveness of the method.
Section 4 present the design and implementation of a series of measurement experiments, in which the flatness of a silicon carbide (SiC) ceramic guideway with a length of 1050 mm and a width of 130 mm is evaluated to assess the performance of the proposed method. The results are compared with those obtained using conventional measurement techniques. Finally,
Section 5 summarizes the research findings and discusses the advantages and application prospects of the proposed measurement approach.
2. Method
As shown in
Figure 1, the proposed measurement method is primarily based on sub-aperture stitching technology [
21] and oblique-incidence laser interferometry [
22]. Its core component is a horizontal laser interferometer, which serves as the fundamental measurement instrument. By integrating an additional reflective mirror, the system enables oblique-incidence measurements. A single-axis translation mechanism is employed to move the workpiece, facilitating sub-aperture stitching for large-area surface measurement.
First, based on the dimensions of the measured surface and the aperture size of the interferometer, the required number of sub-apertures and the optimal angle of oblique incidence are precisely calculated. This step ensures the completeness andw3 accuracy of the measurement. Subsequently, an appropriate stitching path is planned, and the oblique incidence angle is adjusted accordingly. Interference fringe data are then sequentially acquired at each sub-aperture position using the oblique-incidence measurement approach. These fringe patterns serve as the foundation for subsequent phase information retrieval. Once the interference data are obtained, the phase information is extracted using the Phase Shifting Interferometry (PSI) algorithm in combination with a phase unwrapping algorithm [
23]. This step is critical to the measurement process, as it enables the conversion of fringe patterns into surface height data. It is worth noting that both the PSI and phase unwrapping algorithms are widely applied in optical metrology and are known for their ability to enhance measurement precision and stability. After acquiring the data for all sub-apertures, sub-aperture stitching is performed to integrate the local measurements into a complete surface dataset. This process enables the transition from local to global surface characterization. Finally, the compressed surface profile data along the longitudinal direction are reconstructed to recover the full surface topography. This final step achieves the overall objective of the measurement, providing a comprehensive and accurate surface profile of the measured component.
2.1. Principle of Oblique-Incidence Laser Interferometric Measurement
2.1.1. Method of Oblique-Incidence Laser Interferometric Measurement
In conventional laser interferometric surface measurement, the measurement beam is incident perpendicular or nearly perpendicular to the test surface. Due to subtle variations in the surface topography—such as local height differences—optical path differences (OPDs) are introduced during beam propagation. These OPDs result in the formation of interference fringes when the measurement beam is combined with a reference beam on the reference surface.
In contrast, the process of oblique-incidence laser interferometric measurement is more complex. As illustrated in
Figure 2a, the measurement beam is projected onto the test surface at an incidence angle θ, forming an elliptical measurement footprint. The reflected beam from the test surface is subsequently directed to a mirror, which then reflects the beam back to the reference surface along its original path. As shown in
Figure 2b, the resulting elliptical measurement area—with a minor axis equal to the interferometer aperture
, a major axis of
, and a total area of
—is ultimately compressed into a circular fringe pattern of diameter D on the interference image. During this process, the measurement beam interacts with the test surface twice. Variations in the surface profile cause differences in the optical path between the test and reference beams, generating an optical path difference d at each corresponding position, as illustrated in
Figure 2a. According to Equation (1), the relationship between the surface height difference h and the optical path difference d can be expressed as Equation (2). Furthermore, the relationship between the optical path difference and the interference phase leads to an expression linking the surface height h and the phase difference Δφ, as shown in Equation (3).
Therefore,
where
is the optical path difference (OPD) between the actual beam path and the nominal reference path;
h is the height deviation between the measured point and the nominal reference surface;
n0 is the refractive index of the gas in the interferometric cavity;
lr and
ls represent the geometric lengths of the actual and nominal beam paths, respectively, that contribute to the optical path difference.
The phase-shifting interferometer obtains nine fringe intensity images with successive phase differences of 90° by using a piezoelectric transducer (PZT). These images are processed using the Phase-Shifting Interferometry (PSI) algorithm and a phase unwrapping algorithm [
24], yielding phase difference data at all sampling positions. Based on the oblique incidence factor determined by the incident angle, the absolute surface height profile of the measured surface as Equation (4):
where
Mh denotes the height data of each point on the measured surface,
λ0 is the laser wavelength,
n0 is the refractive index of the interferometric cavity, and
represents the phase difference at each point obtained through the PSI and phase unwrapping algorithms.
2.1.2. Influence of the Incidence Angle on the Reflectivity of the Measured Surface in Oblique
The two coherent beams that generate interference fringes are the reference beam reflected by the reference surface and the measurement beam reflected back to the reference mirror. For rough surfaces, under normal incidence, most of the measurement beam will be scattered, resulting in a decrease in the intensity of the measurement beam and ultimately a reduction in the contrast of the interference fringes. As a result, the acquired interferogram is not clear enough, making it difficult for subsequent Phase Shifting Interferometry (PSI) and phase unwrapping algorithms to extract accurate phase information from the grayscale values. In practical engineering applications, when using a vertical incidence plane laser interferometer for high-precision flatness detection, the method to reduce astigmatism is to perform optical polishing on the product to reduce the surface roughness of the measured surface and improve the product’s reflectivity of the measurement beam. However, the oblique incidence interference stitching detection method proposed in this paper does not require polishing the product surface. According to the Fresnel equations shown in Equation (5), a relationship between reflectivity and the incident angle θ can be established: the larger the incident angle, the higher the reflectivity. This enhances the intensity of the desired interference signal relative to the scattered noise, thereby increasing the contrast of the interference fringes.
Of which
represents the refractive index of the incident medium, and
represents the refractive index of the measured reflecting surface. If the measured material is a light-absorbing material, then
is expressed as
, where
represents the extinction coefficient. When the incident angle
θ is equal to zero—that is, under normal incidence—the baseline reflectivity
can be obtained.
Using the Schlick approximation:
of which
denotes the reflectivity of the test surface at an incident angle
θ, the reflectivity of the test surface under oblique incidence can be derived from the baseline reflectivity and the incident angle. As shown in
Figure 3a, the reflectivity of the test surface increases significantly with the incident angle in the range of 80° to 90°.
2.1.3. Error Analysis of Oblique Incidence Laser Interferometric Measurement
The oblique-incidence laser interferometry method significantly extends the effective measurement range compared to conventional normal-incidence plane interferometry. However, this extension introduces two primary sources of systematic error: (1) mirror-induced system error, which arises from the amplified surface figure deviations of the reference and return mirrors [
25]; and (2) spatial resolution degradation along the ellipse’s major axis.
Mirror-induced system error also exists in conventional normal-incidence laser interferometry and is typically embedded within the phase map obtained by the interferometer. However, since the reference mirror generally exhibits a surface flatness much higher than that of the test surface, its influence is often negligible. In oblique-incidence laser interferometry, by contrast, the phase error arises from both the reference mirror and the return mirror, as shown in Equation (7), of which
means the phase difference data corresponding to all measurement pixels within the measurement range,
means the phase difference data generated by the measured surface,
means the phase difference data generated by the reference surface,
means the phase difference data generated by the reflecting surface. When the incidence angle exceeds 60°, the obliquity magnification factor significantly amplifies mirror-induced errors. Based on Equations (4) and (7), the overall mirror-induced system error, denoted as
, can thus be derived [
26].
As shown in
Figure 3b, when the incidence angle exceeds 80°, the magnification factor increases sharply, introducing significant wavefront system errors that affect the final measurement results. Furthermore, during the subsequent sub-aperture stitching process, sub-apertures are aligned and merged based on the slope of the overlapping regions on the test surface. Consequently, the slope characteristics of the error matrix are inherently incorporated into each sub-aperture, leading to a tilted compensation surface during stitching and resulting in stitching-induced errors.
The spatial resolution loss error arises from the projection effect in oblique incidence interferometry, where the originally elliptical wavefront region on the test surface is compressed into a circular wavefront of diameter D. In conventional normal-incidence measurements, each CCD pixel corresponds to a square sampling area with a resolution of . However, under oblique incidence, this sampling area becomes elongated in one direction, resulting in an effective resolution of . This resolution degradation leads to local averaging of the wavefront data: regions originally represented by pixels are merged into fewer data points, causing peaks to be suppressed and valleys to rise. Consequently, the peak-to-valley (PV) value of the measured surface tends to decrease, while the root-mean-square (RMS) deviation is relatively unaffected.
2.2. Principle of Sub-Aperture Stitching Measurement
The sub-aperture stitching measurement method consists of three main components: single-aperture measurement, stitching path planning, and stitching algorithm. The single-aperture measurement step involves selecting an appropriate method for acquiring point cloud data of the surface, such as white-light interferometry, spherical laser interferometry, or plane laser interferometry. The choice of measurement technique depends on the surface characteristics of the object under test and the required measurement accuracy. In this process, the surface profiles acquired from each sub-aperture are independent and typically contain errors such as lateral displacement and tilt.
To obtain full-aperture data, it is necessary to unify the coordinate systems of the sub-apertures through stitching path planning. The selection of the stitching path depends on the surface geometry of the object under test and must ensure sufficient overlap between adjacent sub-aperture measurement regions. For circular surfaces, a concentric path strategy can be adopted, whereas for rectangular surfaces, a grid-based row-column path is preferred to minimize the number of stitching operations. Ultimately, a global surface map covering all sub-apertures is generated based on the aperture configuration and the selected stitching path [
21].
On the global surface map, the overlapping regions between sub-apertures are identified based on their aperture size and spatial coordinates. A stitching algorithm is then applied to calculate the relative translation and tilt between sub-apertures, and the data are merged after compensation.
2.2.1. Sub-Aperture Stitching Algorithm
The primary objective of the sub-aperture stitching algorithm is to align the slope and positional deviations of data acquired at different locations. To ensure reliable registration, each sub-aperture must share an overlapping region with its adjacent sub-apertures, covering no less than one-quarter of the aperture area [
27].
Each sub-aperture is assigned an index based on the measurement path. The measurement data from the sub-aperture with index 1 is used as the reference to establish a global coordinate system. In this coordinate system, the relationship between the measured surface data
of each sub-aperture and the ideally stitched data
is given by the following Equation (9):
of which
represents the ideally stitched data after compensation;
is the measured surface profile data from the sub-aperture with index
i;
are the coordinates of the i-th sub-aperture data;
is the compensation function for slope and translation relative to the reference sub-aperture, where
and
are the tilt coefficients in the
X and
Y directions, respectively, and
is the translation coefficient in the
Z direction.
As illustrated in
Figure 4, the stitching process involves compensating for the displacement along the X, Y, Z-axis and the tilts along the X and Y axes in the overlapping regions of the sub-aperture data. The displacement along the X and Y axes corresponds to the motion of the translation stage and can be determined from the mechanical coordinates provided by the stage’s displacement sensors. The compensation for the Z-axis displacement and angular tilts along the X and Y axes is performed using a global optimization algorithm based on the least squares method, as given in Equation (10) [
21]:
of which
and
represent the measured surface profile data of two arbitrary sub-apertures, while
and
denote their respective compensation functions. Based on the least squares method, the slope compensation functions for each sub-aperture are obtained by solving for the parameters
,
, and
. Taking the partial derivatives of Equation (10) and setting them to zero yields:
of which
denotes the partial derivatives of V with respect to
;
represents the difference in the overlapping region between
and
;
;
=
; and
=
. By simplifying the expression into matrix form, Equation (12) can be obtained:
By applying the above derivation to the surface profile fitting process, we obtain Equation (13):
where
X and
Y denote the horizontal and vertical coordinate matrices, respectively, and
represents the mask matrix of the overlapping region between
and
, with values of 1 in the overlapping region and 0 elsewhere. The above equation can be simplified as:
of which
In order to compute
and
, we define:
To construct the matrix equation:
of which
Since sub-aperture No. 1 is chosen as the reference plane, its compensation coefficient vector is set as . By substituting matrices A and B into the objective function, the compensation coefficient vector K can be obtained by solving Equation (15), thus determining the compensation plane function for each sub-aperture. Once the slope terms are removed using these compensation functions, the complete surface profile can be accurately reconstructed by stitching the sub-aperture data. Each pixel measurement point within the overlapping regions typically contains data from two or more sub-apertures. After the processing, these data points may not be exactly consistent due to random measurement errors and wavefront systematic errors. We calculate the average of these data points and use this mean value as the final sub-aperture measurement result for that point.
2.2.2. Path Planning for Sub-Aperture Stitching
Different from classical sub-aperture planning, the actual measurement area of each sub-aperture in the proposed method is enlarged into an elliptical shape due to the oblique incidence measurement technique. Therefore, when planning the stitching paths, the actual elliptical measurement area should be considered as the effective measurement region for each sub-aperture when calculating the coverage, as shown in
Figure 5. Additionally, it is essential to ensure that the overlapping region between adjacent sub-apertures contains a sufficient amount of data to participate in the stitching process, thereby ensuring stitching accuracy.
During the preparation of this manuscript, the authors used ChatGPT-4o (version: 2024.05) to enhance the coherence and accessibility of the Abstract and Summary sections. The authors have reviewed and edited the output and take full responsibility for the content of this publication.
5. Conclusions
This study addresses the critical challenge of high-precision flatness measurement for large-scale, low-reflectivity ceramic surfaces by proposing a novel integrated approach combining oblique-incidence laser interferometry with sub-aperture stitching technology. The key innovations and contributions are summarized as follows:
- (1)
Methodological Advancements:
The oblique-incidence configuration significantly enhances interference signal contrast on rough ceramic surfaces by optimizing the incident angle, overcoming the reflectivity limitations of conventional normal-incidence interferometry.
Sub-aperture stitching extends the effective measurement range beyond the physical aperture of the interferometer, enabling full-surface characterization of elongated components while maintaining micrometer-level resolution.
- (2)
Validation and Performance:
Numerical simulations demonstrated robust reconstruction accuracy for surfaces with flatness values spanning 1–20 µm, revealing systematic errors dominated by mirror figure inaccuracies.
Experimental validation on a 1050 mm × 130 mm SiC guideway achieved complete surface mapping with PV 2.76 μm and RMS 0.59 μm. The method exhibited high consistency with traditional techniques in polished regions with PV deviation of 0.33 μm and RMS deviation of 0.04 µm.
- (3)
Practical Significance:
The system enabled quick monitoring of a 39-h lapping process, capturing flatness convergence from an initial PV 13.97 μm to a final PV 2.76 μm. This capability provides critical in-process feedback for ultra-precision manufacturing, reducing reliance on post-hoc metrology and accelerating production cycles.
Unlike existing methods limited to polished surfaces, this technique accommodates heterogeneous roughness distributions, making it uniquely suited for guiding multi-stage machining workflows.
- (4)
Industrial Applications and Outlook:
The method holds immediate promise for semiconductor manufacturing equipment, precision machine tools, and aerospace systems where large ceramic guideways demand sub-micrometer flatness control. Its adaptability to rough surfaces further supports quality assurance in the additive manufacturing of ceramic components.
Future work will focus on minimizing mirror-induced errors through mirror calibration or computational compensation and extending the framework to other large-scale surface metrology.