Open Access
Add to BibSonomy
Abstract
A scanning-based second-order moiré method is proposed for high-accuracy deformation measurement in a large field of view (FOV) by analyzing the phase distribution of a single-shot scanning moiré fringe image using a spatial phase-shifting technique. In this method, the grating pitch can be as small as around one pixel in the scanning moiré image to ensure a wide FOV, while high-precision phase measurement is achievable. The strain measurement accuracy has been verified from simulations at different grating pitches, applied strains, and noise levels. The simulation results show that the closer the grating pitch is to the scanning pitch, the smaller the strain measurement error, and the recommended pitch ratio is $0.9 \sim 1.1$. Furthermore, the feasibility of this method has been verified from a tensile experiment on an aluminum specimen under a laser scanning microscope with scanning moiré images recorded. The microscale strains of aluminum measured at different tensile loads agree well with the strain gauge results. As an integration of the scanning and sampling moiré methods, this method has the advantages of a large FOV, high accuracy, strong noise immunity, and visualization of magnified deformation. Compared with the traditional phase-shifting scanning moiré method, this method only needs to record a single scanning moiré image and is suitable for dynamic deformation analysis.
© 2021 Optical Society of America
1. INTRODUCTION
Full-field deformation measurement is a vitally important topic for evaluating the performance of materials under external loads and predicting crack initiation as well as failure progress. At present, the commonly used optical techniques for deformation measurement include the moiré methods [1], the grid method [2], the digital image correlation method [3], geometric phase analysis [4] using Fourier transform or windowed Fourier transform (WFT) [5], and electronic speckle pattern interferometry [6], due to their noncontact and full-field properties. In this research, we focus on developing the moiré methods owing to the advantages of deformation visualization and strong noise immunity.
The traditional moiré methods fall into two main categories. The first type is to directly record moiré fringes, such as the microscope moiré or scanning moiré [1,7] (collectively called the scanning moiré) method, moiré interferometry [8,9], and the CCD or CMOS moiré method (abbreviated by the CCD moiré method). The second type is to record the grid (or grating) images and then digitally generate moiré fringes, such as the sampling moiré method [10–12], the digital moiré method [13–15], and the overlapped (geometric) moiré method. In these traditional moiré methods, it is challenging to balance the view field and the measurement accuracy.
The first type of moiré method has a large field of view (FOV) because the grating does not need to be observed. Typically, the scanning moiré method in a scanning electron microscope (SEM) [16,17], scanning transmission electron microscope (STEM) [18,19], and a laser scanning microscope (LSM) [20] has been widely used in micrometer/nanoscale displacement and strain measurement of various materials such as metals, polymers, composites, semiconductors, and butterfly wings. One analysis method for this type of moiré fringe is the fringe centering method [21]. However, the deformation measurement accuracy is not high, because only the centerlines of moiré fringes are used to calculate the deformation distribution. Although the temporal phase-shifting method [22], and the spatiotemporal phase-shifting method [23] can improve the measurement accuracy of the moiré methods, recording several moiré patterns or grid images is time-consuming.
The second type of moiré method can be combined with the spatial phase-shifting (SPS) technique to improve measurement accuracy. As a representative, the sampling moiré method has been extensively used in atomic defect detection [24], distortion calibration [25], residual strain evaluation [26], and deformation measurement from the microscale [11,27] to the meter scale [28–30]. Nevertheless, the FOV is relatively small because the grid must be observed, and the grid pitch should be greater than three pixels. The FOV here refers to the size of the area where deformation can be measured when the specimen grating pitch is fixed.
To balance the measurement accuracy and the FOV, a reconstructed multiplication moiré method using two-pixel sampling moiré [31] has been developed for deformation measurement. The grid image with a pitch of around two pixels is first recorded to generate two-step sampling moiré patterns, which are further combined to generate a multiplication moiré pattern to double the deformation measurement sensitivity. Furthermore, a second-order moiré method [32] has been reported to improve the deformation measurement accuracy by utilizing the SPS technique; meanwhile, the grid pitch on the recorded grid image can be as small as around two pixels to ensure a large FOV. This method has been used in the microscale strain measurement of carbon-fiber-reinforced plastics under three-point bending.
In this study, we propose a scanning-based second-order moiré method by integrating scanning and sampling moiré for micro/nanoscale deformation measurement to further expand the FOV (Fig. 1). The scanning moiré pattern is first recorded where the grid pitch is only around a single pixel, and then the SPS second-order moiré fringes are generated by performing downsampling and intensity interpolation to the scanning moiré pattern. The micro/nanoscale deformation can be measured from the phase information of the second-order moiré fringes. Simulations and an experiment are used to verify the strain measurement accuracy of this method.
Fig. 1. FOVs of different deformation measurement methods using phase analysis from a single microscope image, where ${{p}_{{s}\_{\min }}}$ is the mimimum specimen grating pitch on the image that can be analyzed, and px is the abbreviation of pixel.
2. MEASUREMENT PRINCIPLE
To achieve high measurement accuracy in a large view field, a novel scanning-based second-order moiré method for deformation measurement is proposed in this study. The core idea is treating the traditional moiré fringes as gratings that are used to generate phase-shifting second-order moiré fringes. The phases of the scanning moiré fringes before and after deformation are accurately measurable from the phases of the second-order moiré fringes analyzed by a SPS algorithm. The displacement and strain distributions can be determined from the phase difference of the scanning moiré fringes before and after deformation. As a result, it is possible to measure the deformation distribution over a large FOV accurately.
A. Scanning Moiré Image Recording
The scanning lines of a microscope act as a reference grating. When there is a small mismatch or misalignment between the specimen grating and the reference grating, scanning moiré fringes will emerge from the interference between the specimen grating and the reference grating [Fig. 2(a)]. The formation principle of the scanning moiré fringes is explained as follows.
The pitch of a specimen grating with the principal direction of ${y}$ and the pitch of the scanning lines of a microscope are, respectively, defined as ${p_{s}}$ and ${p_{r}}$ in the $y$ direction. The intensities of the specimen grating before deformation and the reference grating can be, respectively, written as
where $a({x,y})$ is the modulation amplitude, ${\varphi _0}$ stands for the initial phase, and $b({x,y})$ is the background intensity. ${\varphi _{s}} = 2{\pi}y/{p_s} + {\varphi _0}$ means the phase of the specimen grating before deformation.The scanning moiré can be considered as a multiplicative moiré [33], and the intensity of the scanning moiré image can be represented by
After the higher-frequency items are filtered out, the phase of the scanning moiré fringes before deformation can be expressed as
In the scanning moiré image, the grating pitch is only around one pixel, ensuring a large FOV. If the number of the scanning lines is $ N $, the height of the FOV is ${N}{p_{r}}$, where ${p_{r}}$ is the scanning pitch. In addition, the scanning moiré image can also be recorded using a “digital” zoom ratio of $Z$ if the microscope has this function. In this case, the digital pixel numbers of the grating pitch and the scanning moiré spacing will become $ Z $ times the original pixel numbers, but the “physical” size of the FOV will not change.
When the pitch of the specimen grating changes from ${p_{s}}$ to $p_{s}^\prime $ due to deformation and the pitch of the scanning lines remains unchanged, the phases of the specimen grating and the scanning moiré fringes after deformation will become $\varphi_{s^\prime}=2{\pi}y/p_{s}^{\prime}+\varphi_{0}$ and $\varphi_{m^\prime}({y} )=2\pi{y}( 1/{{{p}}_{s}^{\prime}}-1/{{{p}}_{{r}}})+\varphi_{0}$, respectively.
Thus, the phase difference of the scanning moiré fringes will be
From Eq. (5), the phase difference of the scanning moiré fringes is perfectly equal to the phase difference of the specimen grating before and after deformation.
The scanning moiré pattern is a kind of first-order moiré pattern. In addition, the moiré interferometric fringes, the CCD moiré fringes, or various multiplication moiré fringes can also serve as the first-order moiré fringes in the same manner.
B. Second-order Moiré Generation and Phase Measurement
Interestingly, in this study, we can treat the scanning moiré fringes as a grating image. We can perform downsampling at a pitch $T$ close to the spacing ${p_{m}}$ of the scanning moiré fringes, shift the starting point of downsampling by one pixel for $T - 1$ times, and carry out linear (first-order) or higher-order (i.e., second-order or third-order using B-spline function) intensity interpolation to each downsampling image. In that case, we can obtain $ T $-step SPS second-order moiré fringe images, as seen in Fig. 2(b).
The intensities of the second-order moiré images before deformation can be expressed as
The phase distribution of the second-order moiré image before deformation can be accurately determined by a SPS technique using a discrete Fourier transform algorithm,
From Eqs. (4) and (7), the phase distribution of the scanning moiré image is obtainable from the phase distribution of the second-order moiré image using the following equation:
As a consequence, the phase difference distribution of the scanning moiré images before and after deformation can be calculated by
where $\varphi _m^\prime (y)$ is the phase of the scanning moiré image after deformation, which can be obtained by Eqs. (7) and (8) using the downsampling pitch after deformation and the intensities of the second-order moiré fringes after deformation.In the case of small deformation, the downsampling pitch can remain the same during the deformation process. If the scanning moiré spacing changes greatly due to deformation, we can choose a different downsampling pitch ${{ T }^\prime}$ to make it always close to the spacing of the scanning moiré fringes after deformation.
Note that if the principal direction of the grating is near the ${y}$ direction but the principal direction of the recorded scanning moiré fringes is near the ${x}$ direction, the second-order moiré fringes will be generated in the ${x}$ direction, and the phase of the scanning moiré fringes can also be measured using Eq. (8) by replacing ‘$y$’ with ‘$x$’.
By the way, the phase of the scanning moiré fringes can also be obtained from WFT or other spatial phase analysis methods. Our previous study has found that the sampling moiré method is a faster version of WFT [34] with single-frequency analysis and has higher phase analysis accuracy than the well-established Fourier transform [35]. Therefore, the sampling moiré method is chosen to analyze the scanning moiré fringe phase in this study, due to its high accuracy, high speed, and simplicity of analysis.
C. Deformation Measurement Principle
The deformation measurement process is demonstrated in Fig. 3. The simulated gratings were generated using Eq. (1) with ${\varphi _{0}} = 0$ before deformation, and ${\varphi _{0}} = {k} \times {z}$ after deformation, where ${k} = 0.1$ and ${z} = 3{({1-{x}})^{2}} \times {\exp }[{- {{x}^{2}}-{{({{y} + 1})}^2}}]-10({{x}/5-{{{x}}^3}-{{{y}}^5}}) {\times}\, {\exp }({- {{x}^{2}}-{{{y}}^2}})-1/3 \times {\exp }[{- {{({{x} + 1})}^2}-{{{y}}^2}}]$.
Fig. 3. Deformation measurement process of the scanning-based second-order moiré method. (a) Grating; (b) recorded moiré image; (c) second-order moiré fringes; (d) second-order moiré phase; (e) phase difference of the recorded moiré; and (f) deformation distributions.
Assuming that $u$ expresses the displacement of the specimen grating in the $y$ direction, the phase of the specimen grating after deformation can also be expressed as ${{\varphi}_{s^\prime}}({y}) = 2\pi({y-u})/{{p}_{s}}$. Based on the phase of the specimen grating before deformation ${{\varphi}_{s}}({y}) = 2{\pi y}/{{p}_{s}}$, the phase difference of the specimen grating can be calculated by $ \Delta \varphi_s(y)=\varphi_{s^\prime}(y)-\varphi_s(y) =-2\pi u_y/p_s$. Consequently, the displacement is directly proportional to the phase difference of the specimen grating.
Because the phase differences of the specimen grating and the scanning moiré are equal from Eq. (5), the displacement of the specimen in the $ y $ direction can be determined by
The strain distribution of the specimen in the $y$ direction is measurable from the first-order differential of the displacement,
For the measurement of the displacement and strain distributions in the ${x}$ direction, we should rotate the specimen or the scanning lines by 90 deg to record the scanning moiré image in the ${x}$ direction. Then the deformation in the ${x}$ direction can be measured using the similar equations by changing ‘$y$’ to ‘$x$’.
3. SIMULATION VERIFICATION
When a grating with a pitch of around one pixel is generated, the image that appears is not a grating image but a moiré image that is caused by the interference between the grating and the pixel array with a pitch of one pixel. Therefore, we used the pixel array to simulate the scanning lines and generated digital moiré images to simulate the scanning moiré images. Linear interpolation was used when the phase-shifting second-order moiré fringes were generated in all the simulations.
A. Strain Measurement Accuracy at Different Grating Pitches
In this simulation, we generated moiré images formed by the interference between a series of oblique gratings and the pixel array. The length direction of these gratings was along the direction that was counterclockwise rotated by ${\theta} = 1^\circ$ from the horizontal direction [Fig. 4(a)]. The intensities of the simulated scanning moiré images before and after deformation were, respectively, represented by
Fig. 4. Simulated grating and moiré images. (a) Diagram of the oblique grating, and (b) simulated scanning moiré images with 30% random noise when the grating pitch ${{p}_{s}}$ is around one pixel.
The simulated grating pitch ranged from 0.70 pixels to 1.58 pixels with an increment of 0.04 pixels, and the simulated pitch of the scanning lines was one pixel. A preset strain of 1000 µ$\unicode{x03B5}$ was exerted on each grating. A random noise ${\sigma} = 30{\%}$ was added to each simulated moiré image before and after deformation. The simulated moiré images with a size of $30\;{\rm pixels} \times 300\;{\rm pixels}$ before deformation are shown in Fig. 4(b).
The closer the grating pitch was to one pixel, the greater the moiré spacing. These moiré spacings in the ${y}$ direction could be calculated by ${{p}_{m}} = | {{{{p}}_{s}}/({{{p}_{s}} - {\cos\theta}})} |$, where ${{p}_{s}}$ was the grating pitch and ${\theta} = 1^\circ$. The moiré spacings in the $ y $ direction varied from 2.33 pixels to 50.61 pixels. These moiré fringes were treated as gratings to generate SPS second-order moiré fringes. The sampling pitch was set to ${T} = {\rm round}({{{p}_{m}}})$ and ${T} = 3\;{\rm pixels}$ when ${{p}_{{m}}} \lt 2.5\;{\rm pixels}$ for each simulated moiré image, where the “round” function means taking the nearest integer. The adopted sampling pitch ranged from 3 pixels to 51 pixels. As the theoretical strain was only 1000 µ$\unicode{x03B5}$, the same sampling pitch was used before and after deformation for each simulated moiré image.
The phase differences of the simulated moiré images were calculated using Eqs. (7)–(9), and the strain distributions were measured using Eq. (11). No filters were used in the calculation process. To reduce the uncertainty caused by the random noise, the noise was separately added to each moiré image, and the noise addition and the strain measurement processes were repeated 500 times to get the average strain results. In addition, to remove the influence of the possible errors at edges, the evaluation area was set to $30\;{\rm pixels} \times 180\;{\rm pixels}$ in the image center, labeled by the red squares in Figs. 4(b) and 5(a). Taking the grating with the pitch of 1.02 pixels as an example, the displacement and strain distributions and the strain values along a vertical section line in the middle of the image are illustrated in Fig. 5. The absolute value of error in strain is within 50 µ$\unicode{x03B5}$, and the absolute value of the relative error in strain is within 5% without any filters. Although the measured strain has a periodic error, this error can be reduced by using filters to the phases and strains in the calculation process.
Fig. 5. Deformation distributions of a simulated grating with the pitch of 1.02 pixels when the noise level is 30% and no filters are used. (a) Simulated scanning moiré pattern and its intensity profile; (b) displacement; (c) strain; and (d) strain values along a vertical section line AA’.
Compared with the theoretical strain, the relative errors of the average strain values and the root mean square (RMS) errors in strain in the evaluation area along with the grating pitch are presented in Fig. 6. When the specimen grating pitch is within $0.9\sim1.1\;{\rm pixels}$, both the RMS error and the absolute value of the relative error in the measured strain are less than 0.5%. The closer the grating pitch is to one pixel, the smaller the RMS error in strain. Consequently, it is better to set the grating pitch to be $0.9\sim1.1$ times the scanning pitch, or the scanning pitch to be $0.9\sim1.1$ times the grating pitch to obtain high strain measurement accuracy.
Fig. 6. Simulation results of strain measurement from gratings with different pitches when the theoretical strain is 1000 µ$\unicode{x03B5}$ and the noise level is 30%. (a) Relative error and (b) RMS error.
Fig. 7. Simulation results of strain measurement at different theoretical tensile strains when the grating pitch is 1.06 pixels before deformation and the noise level is 30%. (a) Relative error and (b) RMS error.
B. Strain Measurement Accuracy at Different Tensile Strains
In the usual experiments to record scanning moiré images, the scanning lines’ pitch is generally made as close to the grating pitch as possible. Therefore, we chose a grating with a pitch of 1.06 pixels in this simulation to investigate the strain measurement accuracy under different deformation conditions. The oblique angle of the grating, the noise level, the moiré image size, and the evaluation area were the same as in Section 3.A.
We applied the theoretical tensile strains, ${{\varepsilon}_{y}}$, of 100, 300, 500, and $1000\sim5000\;\unicode {x00B5}\unicode{x03B5}$ with an increment of 500 µ$\unicode{x03B5}$ to the oblique grating to generate moiré images after deformation. The moiré spacings in the ${y}$ direction changed from 17.62 pixels to 16.28 pixels at these tensile strains. The second-order moiré fringes were generated using the sampling pitch of 17 pixels from each simulated moiré image. The random noise was added to each moiré image before and after deformation. The noise addition and strain measurement processes were repeated 500 times, and the averaged strain results were used. No filters were used in the strain measurement process.
Fig. 8. Simulation results of strain measurement at different random noise levels when the grating pitch is 1.06 pixels before deformation and the theoretical strain is 1000 µ$\unicode{x03B5}$. (a) Relative error and (b) RMS error.
The relative errors of the average strain values and the RMS errors in strain in the evaluation area at different tensile strains are displayed in Fig. 7. The absolute value of the relative error is less than 0.7% under each deformation condition. The farther the applied strain is from 500 µ$\unicode{x03B5}$, the greater the absolute value of the relative error in strain. The RMS error in strain under each deformation condition is around 0.2%, demonstrating the high strain measurement accuracy.
C. Strain Measurement Accuracy at Different Noise Levels
A grating with a pitch of 1.06 pixels bearing the tensile strain of 1000 µ$\unicode{x03B5}$ was used to study the strain measurement accuracy at different noise levels. The grating oblique angle, the simulated moiré image size, and the evaluation area are the same as in Section 3.A.
A random noise with a level of 0% to 100% with an increment of 10% was added to the moiré images before and after deformation. The moiré spacings in the ${y}$ direction are 17.62 pixels and 17.33 pixels before and after 1000 µ$\unicode{x03B5}$ tensile strain, and the sampling pitch of 18 pixels was used to generate second-order moiré fringes from each simulated moiré image. Similarly, the noise addition and strain measurement processes were repeated 500 times to get the average strain values, and no filters were used in the strain measurement.
The relative errors of the average strain values and the RMS errors in strain in the evaluation area at different noise levels are plotted in Fig. 8 and compared with the theoretical strain. The absolute value of the relative error is less than 1% under each noise condition. The RMS error increases with the increase of the noise level, and the greatest RMS error is less than 0.8%. It demonstrates that the strain measurement is effective even when the noise is heavy.
4. EXPERIMENTAL VERIFICATION
The deformation measurement of an aluminum (Al) specimen in a tensile test is demonstrated to show the validity and accuracy of the proposed method.
A. Specimen Preparation and Tensile Experiment
The length, width, and thickness of the middle part of the dumbbell-shaped Al specimen were 27, 6.3, and 0.5 mm, respectively. One surface of the specimen was polished with sandpaper and polishing solutions. A two-dimensional (2D) grating with a pitch of 3.0 µm was fabricated on the polished surface of Al [Figs. 9(a) and 9(c)] by ultraviolet nanoimprint lithography in an EUN-4200 device. The used nanoimprint resist was PAK01. One principal direction of the 2D grating is close to the length direction of the specimen. On the other surface of the specimen, a strain gauge was attached to the middle area [Fig. 9(b)] to validate the strain measurement accuracy.
Fig. 9. Al specimen and experimental setup. (a) Grating pattern and (b) strain gauge on the two surfaces of the specimen; (c) enlarged image of the 3-µm-pitch grating; and (d) the tensile experimental setup under an LSM.
The specimen was loaded by a self-developed tensile device under a LSM (Lasertec, Hybrid Optelics), as shown in Fig. 9(d). The tensile load was measured with a load cell (TCLB-200 L), and the tensile stress was calculated from dividing the load by the cross-sectional area.
When the magnification of the objective lens was $5 \times$ and the number of the scanning lines was 512, the pitch of the scanning lines was ${{p}_{r}} = 2.907\; {\unicode{x00B5}{\rm m}}$ measured from the recorded image, close to the grating pitch ${{p}_{s}} = 3.0\; {\unicode{x00B5}{\rm m}}$. The sample stage was rotated to make the length direction of the specimen be perpendicular to the scanning direction. Under this condition, the laser scanning moiré fringes caused by the interference between the specimen grating and the scanning lines were able to be observed. A digital zoom ratio of 2 was automatically used in the LSM, and the recorded image size was $1024\;{\rm pixels} \times 1024\;{\rm pixels}$. During the tensile process, a series of laser scanning moiré images were recorded when the strain gauge values were 154, 323, 531, 1030, 1590, 2160, 2471, 2797, and 3001 µ$\unicode{x03B5}$, respectively.
B. Strain Measurement of Al and Discussion
The recorded scanning moiré images were imported into a computer. The scanning moiré spacings could be estimated from theoretical analysis using Eq. (4). For example, the estimated scanning moiré spacing before deformation was $ p_m \,{=}\, 1/(1/p_s{-}\,1/p_r))\, {=} \,1/(1/3.0{-}\,1/2.907) \,{=}\, 93.75\,\unicode {x00B5}{\rm m}$. Because the image scale was 1.453 µm per pixel, the estimated scanning moiré spacing would be $93.75/1.453 = 64\;{\rm pixels}$ on the image. If the approximate strain of the specimen after deformation is unknown, it is better to estimate the scanning moiré spacing on the image. In this experiment, the estimated scanning moiré spacings ranged from 58 pixels to 64 pixels during the tensile test. The sampling pitch was chosen as 57 pixels for all the scanning moiré images for simplicity, i.e., the SPS number was 57.
Taking the case that the strain gauge value is 3001 µ$\unicode{x03B5}$ as an example, the deformation measurement process of the scanning-based second-order moiré method for the Al specimen under tension is shown in Fig. 10. The recorded laser scanning Moiré images before and after deformation were used to generate SPS second-order moiré fringes using downsampling with pitch of 57 pixels and linear intensity interpolation. The phases of the second-order moiré fringes before and after deformation were measured using Eq. (7) and filtered using a sine/cosine filter with a size of $31\;{\rm pixels} \times 31\;{\rm pixels}$. The phase difference of the scanning moiré image was then calculated using Eq. (9). To avoid the possible influence of the scanning distortion at the upper and lower edges, an area of $900\;{\rm pixels} \times 690\;{\rm pixels}$ ($1308 \times 1003\;{{\unicode{x00B5}{\rm m}}^2}$) in the image center was set as the evaluation area. The displacement and the strain distributions were measured using Eqs. (10) and (11), and the strain was filtered with an average filter with a size of $41\;{\rm pixels} \times 41\;{\rm pixels}$. The average value of the measured strain distribution in the evaluation area was 3087 µ$\unicode{x03B5}$, and the strain difference was 86 µ$\unicode{x03B5}$ compared with the strain gauge value (3001 µ$\unicode{x03B5}$).
Fig. 10. Measurement process on the Al specimen under tension when the strain gauge value is 3001 µ$\unicode{x03B5}$. (a) Laser scanning moiré images; (b) second-order moiré fringes; (c) second-order moiré phases; and (d) deformation distributions.
Similarly, the displacement and strain distributions at other tensile loads were measured (Fig. 11), and the average strain values in the evaluation area were obtained. The stress-strain curves of the Al specimen from the scanning-based second-order moiré method and the strain gauge method are presented in Fig. 12.
Fig. 11. Displacement and strain distributions on the Al specimen under different strain gauge values measured from the scanning-based second-order moiré method.
Fig. 12. Tensile stress-strain curves of the Al specimen where the red and black curves are the fitted curves of the strain data measured from the scanning-based second-order moiré method and the strain gauge method, respectively.
The strain values measured from the scanning-based second-order moiré method are very close to those from the strain gauge method, verifying the validity and the measurement accuracy of the proposed method. The slight difference in the strain values from the two methods may be caused by the angle error between the strain gauge direction and the tensile direction because the strain gauge is manually attached to the specimen.
5. CONCLUSIONS
A scanning-based second-order moiré method was developed for accurate deformation distribution measurement by integrating the scanning and sampling moiré techniques. The displacement and strain measurement principles were presented. Simulations demonstrated that the strain distribution can be accurately measured at different theoretical strains and noise levels, and the measurement error was smaller when the scanning pitch was closer to the specimen pitch. The strain measurement accuracy was also verified from a tensile experiment to Al under a LSM compared to the strain gauge results. This method has a wider FOV than the sampling moiré method, as the grating pitch can be as small as around one pixel, with a higher measurement accuracy than the scanning moiré method, owing to the use of the SPS technique. This method is expected to be used for deformation measurement and structure characterization of various materials under scanning-type microscopes or line scan image sensors.
Funding
Japan Society for the Promotion of Science (JP20H02038, JP20K04171).
Disclosures
The authors declare no conflicts of interest.
REFERENCES
1. S. Kishimoto, M. Egashira, and N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32, 522–526 (1993). [CrossRef]
2. M. Grediac, F. Sur, and B. Blaysat, “The grid method for in-plane displacement and strain measurement: a review and analysis,” Strain 52, 205–243 (2016). [CrossRef]
3. T. Chu, W. Ranson, and M. A. Sutton, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985). [CrossRef]
4. A. Pofelski, S. Woo, B. Le, X. Liu, S. Zhao, Z. Mi, S. Löffler, and G. J. U. Botton, “2D strain mapping using scanning transmission electron microscopy Moiré interferometry and geometrical phase analysis,” Ultramicroscopy 187, 1–12 (2018). [CrossRef]
5. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004). [CrossRef]
6. E. Hack and R. Brönnimann, “Electronic speckle pattern interferometry deformation measurement on lightweight structures under thermal load,” Opt. Laser Eng. 31, 213–222 (1999). [CrossRef]
7. H. Wen, H. Zhang, Z. Liu, C. Liu, S. Liu, X. Yang, F. Liu, and H. Xie, “Stress mapping of a strain superlattice using scanning moiré fringe imaging,” Appl. Phys. Lett. 113, 031905 (2018). [CrossRef]
8. D. Post and B. Han, “Moiré interferometry,” in Springer Handbook of Experimental Solid Mechanics (Springer, 2008), pp. 627–654.
9. S. Yoneyama, P. G. Ifju, and S. E. Rohde, “Identifying through-thickness material properties of carbon-fiber-reinforced plastics using the virtual fields method combined with moiré interferometry,” Adv. Compos. Mater. 27, 1–17 (2018). [CrossRef]
10. S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010). [CrossRef]
11. Q. Wang, S. Ri, H. Tsuda, M. Koyama, and K. Tsuzaki, “Two-dimensional Moiré phase analysis for accurate strain distribution measurement and application in crack prediction,” Opt. Express 25, 13465–13480 (2017). [CrossRef]
12. M. Fujigaki, A. Masaya, K. Shimo, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50, 101506 (2011). [CrossRef]
13. H. Xie, Z. Liu, D. Fang, F. Dai, H. Gao, and Y. Zhao, “A study on the digital nano-moiré method and its phase shifting technique,” Meas. Sci. Technol. 15, 1716 (2004). [CrossRef]
14. K. Patorski, M. Wielgus, M. Ekielski, and P. Kaźmierczak, “AFM nano moiré technique with phase multiplication,” Meas. Sci. Technol. 24, 035402 (2013). [CrossRef]
15. H. Xing, Z. Gao, H. Wang, Z. Lei, L. Ma, and W. Qiu, “Digital rotation moiré method for strain measurement based on high-resolution transmission electron microscope lattice image,” Opt. Laser Eng. 122, 347–353 (2019). [CrossRef]
16. Y. Li, H. Xie, Q. Wang, M. Zhou, M. Xu, Q. Luo, and C. Gu, “The residual strain measurement of thin conductive metal wire after electrical failure with SEM moiré,” Acta Mech. Solida Sin. 29, 371–378 (2016). [CrossRef]
17. B. Fan, J. Li, H. Xie, and X. Dai, “Development of solute-solvent separation soft lithography grating for measuring high-temperature in situ deformation using scanning electron microscope moiré method,” Exp. Mech. 59, 29–39 (2019). [CrossRef]
18. A. Ishizuka, M. Hytch, and K. Ishizuka, “STEM moiré analysis for 2D strain measurements,” Microscopy 66, 217–221 (2017). [CrossRef]
19. S. Kim, S. Lee, Y. Kondo, K. Lee, G. Byun, S. Lee, and K. Lee, “Strained hetero interfaces in Si/SiGe/SiGe/SiGe multi-layers studied by scanning moiré fringe imaging,” J. Appl. Phys. 114, 053518 (2013). [CrossRef]
20. H. Xie, Q. Wang, S. Kishimoto, and F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101, 103511 (2007). [CrossRef]
21. F. Dai and Z. Wang, “Automatic fringe patterns analysis using digital processing tehniques: I fringe center method,” Acta Photonica Sin. 28, 700–706 (1999).
22. C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: a review,” Opt. Laser Eng. 109, 23–59 (2018). [CrossRef]
23. S. Ri, Q. Wang, P. Xia, and H. Tsuda, “Spatiotemporal phase-shifting method for accurate phase analysis of fringe pattern,” J. Opt. 21, 095702 (2019). [CrossRef]
24. Q. Wang, S. Ri, H. Tsuda, M. Kodera, K. Suguro, and N. Miyashita, “Visualization and automatic detection of defect distribution in GaN atomic structure from sampling Moiré phase,” Nanotechnology 28, 455704 (2017). [CrossRef]
25. Q. Zhang, H. Xie, W. Shi, and B. Fan, “A novel sampling moiré method and its application for distortion calibration in scanning electron microscope,” Opt. Laser. Eng. 127, 105990 (2020). [CrossRef]
26. Q. Wang, S. Ri, and T. Enomoto, “Residual thermal strain distribution measurement of underfills in flip chip electronic packages by an inverse approach based on the sampling moiré method,” Exp. Mech. 60, 611–626 (2020). [CrossRef]
27. Q. Zhang, H. Xie, Z. Liu, and W. Shi, “Sampling moiré method and its application to determine modulus of thermal barrier coatings under scanning electron microscope,” Opt. Laser. Eng. 107, 315–324 (2018). [CrossRef]
28. X. Chen and C.-C. Chang, “In-plane movement measurement technique using digital sampling moiré method,” J. Bridge Eng. 24, 04019013 (2019). [CrossRef]
29. S. Ri, Q. Wang, H. Tsuda, H. Shirasaki, and K. J. S. Kuribayashi, “Displacement measurement of concrete bridges by the sampling Moiré method based on phase analysis of repeated pattern,” Strain 56, e12351 (2020). [CrossRef]
30. S. Ri, H. Tsuda, K. Chang, S. Hsu, F. Lo, and T. Lee, “Dynamic deformation measurement by the sampling Moiré method from video recording and its application to bridge engineering,” Exp. Tech. 44, 313–327 (2020). [CrossRef]
31. Q. Wang, S. Ri, and H. Tsuda, “Digital sampling Moiré as a substitute for microscope scanning Moiré for high-sensitivity and full-field deformation measurement at micron/nano scales,” Appl. Opt. 55, 6858–6865 (2016). [CrossRef]
32. Q. Wang, S. Okumura, S. Ri, P. Xia, H. Tsuda, and S. Ogihara, “Second-order moiré method for accurate deformation measurement with a large field of view,” Opt. Express 28, 7498–7514 (2020). [CrossRef]
33. H. Xie, H. Shang, F. Dai, B. Li, and Y. Xing, “Phase shifting SEM moiré method,” Opt. Laser Technol. 36, 291–297 (2004). [CrossRef]
34. Q. Kemao, N. Agarwal, S. Ri, and Q. Wang, “Sampling moiré as a special windowed Fourier ridges algorithm in demodulation of carrier fringe patterns,” Opt. Eng. 57, 100503 (2018). [CrossRef]
35. S. Ri, N. Agarwal, Q. Wang, and Q. Kemao, “Comparative study of sampling moiré and windowed Fourier transform techniques for demodulation of a single-fringe pattern,” Appl. Opt. 57, 10402–10411 (2018). [CrossRef]
