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Abstract
Digital shearing speckle pattern interferometry (DSSPI) is a powerful interferometric technique used to visualize the slope contours undergoing static and dynamic deformations. Precise determination of the shear amount is crucial for quantitative analysis in DSSPI. However, accurately measuring the shear amount is often challenging due to factors such as optical device dimensions, deflections, aberrations, and misalignments. In this paper, we propose a novel method utilizing optical vortices deflection in pseudo-phase for shear measurement. This method eliminates the need for attaching calibration objects and replacing the light source, making it applicable to inaccessible or fragile samples. Experimental results demonstrate the effectiveness and accuracy of the proposed method in determining shear amounts in DSSPI. The method can be easily automated and integrated into existing setups, offering broader application prospects.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Digital shearing speckle pattern interferometry (DSSPI) is an interferometric technique that allows the visualization of fringe patterns illustrating the slope contours of object surfaces undergoing both static and dynamic deformations [1–3]. The precision of computing the slope contours using this method relies, in part, on accurately measuring the amount of shear introduced during recording. The shear amount not only impacts the sensitivity and spatial resolution of the DSSPI system but, more importantly, precise determination of the shear amount is crucial for establishing an accurate strain field [4–6]. By precisely quantifying the shear amount, DSSPI facilitates quantitative analysis of strain distribution, dynamic response, or thermal gradients under various loading conditions, offering valuable insights into material characteristics, defect identification, and long-term structural health monitoring. The co-path and self-referencing characteristics of DSSPI contribute to its strong resistance to interference and high repeatability during practical measurements when shear amount calibration is appropriately performed [7]. Nevertheless, in practice, accurately measuring the precise geometrical dimensions and deflection of optical devices, such as the shearing angle of the Michelson interferometer, in the imaging system can be challenging or not directly feasible. Additionally, inherent issues like aberrations and misalignments of components during the optical alignment process further complicate the accurate evaluation of shear amount based on direct referencing of the optical path arrangement and principles [8]. Such an approach inevitably introduces significant errors, leading to deviations in the shear amount and severely compromising the accuracy and repeatability of the detection [4,9].
Currently, the widely employed method for evaluating shear amount is the tape target method [10]. Several enhancements, rooted in the same fundamental principles, have been identified to enhance the accuracy and robustness of shear amount measurements [4,8,11]. This methodology necessitates attaching or placing a known target object onto the surface of the test specimen to measure the optical misalignment caused by the deflection of the shearing device within the optical system. Consequently, the measurement accuracy is contingent upon the attachment conditions. However, it is impractical to adhere or place calibration objects onto samples that are inaccessible (such as under high-temperature or high-pressure environments) or excessively fragile (such as artworks or thin films). Furthermore, conventional approaches require replacing the laser light source with white light illumination to enhance the signal-to-noise ratio (SNR). This process proves arduous and cumbersome in highly integrated detection devices or projects that demand frequent adjustments of the shear amount to attain different scale sensitivities for detection. Lastly, the traditional method of quantifying shear amount involves measuring the distance between two shear components in terms of the number of pixels using tape targets on the object. Its precision remains confined to a single-pixel range, rendering it inadequate for applications that require precise shear amount measurements.
In this paper, a novel method employing optical vortices offset in pseudo-phase is demonstrated. With this method, there is no need to substitute the illumination and place the target on the object surface. In addition, it achieves precise and accurate shear measurements.
2. Principle and method
2.1 Digital shearing speckle interferometry principles
Let us represent the intensities at a specific point on the detector before and after object deformation as ${I_1}$ and ${I_2}$ respectively
where ${a_1}$ and ${a_2}$ represent the amplitudes of the scattered light, $\phi$ denotes the random phase between the two sheared beams and $\varDelta $ signifies the relative phase change resulting from deformation. The fringe information is extracted by the differencing operation between ${I_1}$ and ${I_2}$ , resulting in a subtracting intensity ${I_r}$ denoted asAs the frequency of $\phi$ exceeds the limit of visual detection, sinusoidal patterns consisting of alternating dark and light bands manifest solely when changes in $\varDelta $ occur across the observed field. The specific value of $\Delta $ has been previously established through earlier investigations [12,13] as
Upon analyzing Eq. (4), it becomes evident that the shear amount s exhibits an inverse proportionality with the slope components $\frac{{\partial u}}{{\partial x}}$, $\frac{{\partial v}}{{\partial x}}$ and $\frac{{\partial w}}{{\partial x}}$ while holding the value of $\Delta $ constant. This observation implies that as s increases, smaller changes in surface slopes are required to achieve an equivalent fringe interval. Considering that surface strains can be directly derived from the slope components, it follows that higher values of s correspond to heightened sensitivity in measuring surface strain.
Equation (4) further reveals that the precise determination of the slope components $\frac{{\partial u}}{{\partial x}}$, $\frac{{\partial v}}{{\partial x}}$ and $\frac{{\partial w}}{{\partial x}}$ from $\Delta $ relies on the accurate measurement of the amount of linear shear denoted by s, which highlights the criticality of attaining a high level of precision in quantifying s to ensure the accuracy of determining surface strains through DSSPI. Moreover, given the direct relationship between the slope components and surface strains, meticulous assessment of s is imperative for achieving precise and reliable results in strain analysis using shear metrology.
2.2 General shear measurement techniques
The shear amount exerts a direct impact on the sensitivity and SNR in shear measurement, and the accuracy of shear amount itself directly affects the precision of the DSSPI system. Consequently, before conducting formal measurements, it is of paramount importance to obtain an accurate shear amount for the entire optical shear device. One approach is to determine the shear amount directly through a simple correlation of the beam intensity distribution in the deflected and undeflected images. However, this method requires the removal of the beam expanding mirror and is not commonly employed due to its considerable error margin [4,14]. Currently, the widely adopted method for evaluating shear amount is based on tape target, wherein markers are placed on the object to calculate the distance between two shear images in terms of pixel count, as depicted in Fig. 1(a). It entails attaching or placing a known reference object on the test specimen to measure the optical deflection caused by the misalignment of the shear device [10].
Fig. 1. General shear amount measurement method. (a) schematic diagram of the principle; (b) acquired image and calibration example.
In Fig. 1(b), a calibration object of a known size, such as a black 3 M tape with a width of $W$($W$=18 mm =562pix), is used on an aluminum plate sample to calibrate the pixel indices at regions of the high gradient in the intensity distribution. In this case, ${i_1}$=860, ${i_2}$=894, ${i_3}$=956, ${i_4}$=1002, ${i_5}$=1422, ${i_6}$=1456, ${i_7}$=1516, ${i_8}$=1560. Subsequently, the expected value of the optical deflection at the region of intensity mismatch can be obtained by averaging two points, such as ${i_{12}} = {{({i_1} + {i_2})} / 2}$. Finally, by detecting the pixel positions ${i_{12}}$, ${i_{34}}$, ${i_{56}}$, and ${i_{78}}$, the linear shear amount s on the object's surface can be determined by
The expression for the maximum relative uncertainty associated with the utilization of a constant shear amount at the origin [4] can be formulated as follows
The uncertainty of the calibrated shear amount exhibits a decreasing trend as the shear amount increases; however, it remains primarily reliant on the spatial resolution of the camera utilized, where the maximum potential error is 1 pixel. For example, in Eq. (5), employing ${i_{34}} - {i_{12}}$ instead of ${i_{78}} - {i_{56}}$ results in a disparity of 3 pixels. In this case, the shear amount is determined to be 99 pixels with an uncertainty of 1.01%.
While conventional methods generally deliver relatively accurate calibration results under ideal conditions, they encounter numerous challenges in practical applications. Firstly, the need to replace the light source with a white light source to enhance the signal-to-noise ratio and achieve more precise calibration outcomes proves intricate and laborious, especially in highly integrated detection devices or situations necessitating frequent adjustments of shear amounts for detecting different scales of sensitivity. Moreover, the conventional calibration approach requires the affixing of a calibration object onto the target being tested, which becomes impractical in cases where samples are difficult to access due to extreme conditions or exhibit excessive delicacy, as in the case of artwork or thin films. Lastly, the precision of traditional methods in measuring shear amounts is confined to a single-pixel range, rendering them inadequate for applications that demand precise shear amount measurements. In summary, despite their relatively accurate calibration results under ideal conditions, conventional methods encounter significant challenges when applied to real-world scenarios.
Since the advent of laser technology, the understanding of speckle fields has gradually deepened, and various techniques have been developed for widespread applications, such as experimental mechanics and biospeckle [15,16]. So far, most research on speckle metrology has focused on the cross-correlation of speckle pattern intensity, with little attention given to applications based on phase field information. Although the complex representation of real-valued speckle patterns does not introduce new information, this transformation effectively utilizes existing information by providing new approaches for analyzing and processing speckle patterns recorded by cameras through the introduction of pseudo-amplitude and pseudo-phase related to complex analytic signals. Moreover, the pseudo-phase information has multifunctionality, enabling its application beyond known laser speckle metrology [17]. Moreover, the phase field exhibits phase singularities, which are topological defects of wavefronts. In information optics, phase singularities serve as optimal encoders for marking positions, as they exhibit minimal photon noise at positions where the amplitude is zero and demonstrate robustness in the decorrelation process of speckle fields [18,19]. Phase singularities also act as ideal markers due to their well-defined geometry and the ability to locate them with arbitrary search accuracy. This has led to the exploration of a new possibility known as Optical Vortex Metrology (OVM), which utilizes phase singularities in speckle pattern analysis. OVM has demonstrated applications in high-resolution measurements of nanoscale displacements, flow field velocity measurements, and high-precision measurements of full-field angular displacements [20,21]. Building upon these findings, we observe that the shear device plays a critical role in the misalignment and differentiation of the two shear components during optical field transmission, as per the principles of shear imaging. Furthermore, the components between the two optical paths of the Michelson interferometer exhibit the same optical field structure and phase field information before interference occurs, with only spatial deflection present. By calculating the pseudo-phase representation between these components and analyzing the unique structure of phase singularities through tracking, we can obtain the statistical characteristics of full-field shear amounts using the statistical variation of optical vortices.
2.3 Pseudo-phase representation based on Laguerre-Gaussian filter
Following the introduction of optical vortex detection advancements, this section further delves into the detection and measurement of optical vortices using the pseudo-phase representation method based on the intensity field. Initially, a concise introduction to the two-dimensional isotropic complex signal representation of speckle patterns is presented. Utilizing complex signals to represent real signals is a common practice in physics and engineering. The concept of analytic signals for one-dimensional signals was introduced by Gabor in the 1940s in the field of communication theory [22]. Typically, the real part of an analytic signal is obtained by removing the mean value from the original signal, while the corresponding imaginary part represents the Hilbert transform of the real part. When applying the Hilbert filter to two-dimensional speckle patterns, the resulting analytic signal exhibits obvious anisotropy due to the introduction of different spectral bandwidths in the x and $\textrm{y}$ directions by the partial Hilbert filters. To obtain the two-dimensional isotropic analytic signal of speckle patterns, Wang et al. proposed using Riesz transform [23] and vortex transform [24] as a substitute for Hilbert transforms and demonstrated its effectiveness through experiments. On the other hand, considering that the Riesz transform possesses a vortex structure in the frequency domain with pure spiral phase functions, without utilizing the amplitude information of the Fourier spectrum, the use of Laguerre-Gauss (L-G) transform instead of Riesz transform or vortex transform was proposed to exploit its bandpass filter characteristics [20,25] and obtain stable and optimally distributed phase singularities. Building upon the outstanding nature and stability of the L-G transform, this paper adopts the approach of generating pseudo-phase using L-G filters.
Let $I(x,y)$ represent the original intensity distribution of the speckle pattern, whose Fourier spectrum is $\Im ({f_x},{f_y})$. The association between $I(x,y)$ and its isotropic complex signal, denoted as $\tilde{I}(x,y)$, can be achieved using the L-G filter. The definition of the two-dimensional isotropic complex signal representation of speckle patterns is as follows
The phase singularities ${n_t}$ in pseudo-phase are defined in terms of the topological charge, denoted as
where $\nabla \hat{\theta }(x,y)$ denotes the local phase gradient. Furthermore, the contour integral is evaluated along path $\vec{l}$, which forms a closed loop c encircling the vortex. It is evident that, except for phase singularities where the phase is undefined, the pseudo-phase $\hat{\theta }(x,y)$ remains continuously differentiable when ${n_t} = 0$. Since $\hat{\theta }$ is a continuous function with continuous first-order derivatives, it can be derived using Stokes’ theoremFig. 2. L-G pseudo-phase representation of the speckle pattern and the optical vortex over it. (a) Speckle pattern; (b) corresponding pseudo-phase image generated by the L-G filter; (c) positive vortices and negative vortices detected in the pseudo-phase, represented by blue dots and red dots respectively.
2.4 Shear measurement based on optical vortex core structure
Similar to the scenario of coherent light illumination on a rough object's surface, the speckle patterns reflected onto the object surface exhibit random distribution of phase singularities [26]. These phase singularities can be visualized through the pseudo-phase. By recording the speckle patterns on the two mirrors of the shearing device separately, both the deflected and undeflected phase images can be obtained. Leveraging the position information of phase singularities before and after shearing, the deflection of the optical field induced by the shearing device can be accurately measured.
To achieve the stated objective, the initial step involves identifying the positions of phase singularities in both the reference mirror and post-shearing mirror images. Subsequently, it is essential to establish correspondences between phase singularities in the deflected and undeflected phase maps. When the shearing distance is small, or prior information is available, the search range can be limited to phase singularities with the same topological charge and nearest neighbors. However, for large or unevenly distributed shearing distances without prior information, uniquely determining the corresponding phase singularities can be challenging. To address this issue, this section proposes incorporating additional information related to the structural characteristics of phase singularities as supplementary conditions for the search, aiming to enhance specificity. Similar to the presence of optical vortices in the true phase of a random speckle field, the pseudo-phase variation around phase singularities is also non-uniform, and typical core structures around phase singularities exhibit noticeable anisotropy. Figure 3 illustrates an actual example of reconstructed amplitude contours and pseudo-phase structures near the phase singularities based on the complex signal representation of speckle patterns.
Fig. 3. The core structure surrounding the phase singularities with zero-crossings in the real and imaginary components of the complex signal represented by the speckle pattern. (a) Amplitude contour pattern and zero-crossing lines; (b) pseudo-phase three-dimensional structure.
The presence of phase singularities can be observed at the center of the elliptical contour of the amplitude, specifically at the zero-crossing points of the real and imaginary components of the complex signal $\tilde{I}$, exhibiting a characteristic spiral structure with a 2π phase, its three-dimensional structure is displayed in Fig. 3(b). As depicted in Fig. 3(a), the eccentricity of the contour ellipse e and the zero-crossing angle between the real and imaginary parts ${\theta _{RI}}$ possess invariance, unaffected by the phase singularity displacement caused by object translation and rotation. These two geometric parameters can be utilized to describe the local properties of the phase singularity. Furthermore, each phase singularity possesses its own topological charge (positive vortex ${q^ + }$ or negative vortex ${q^ - }$) and vorticity $\vec{\Omega } \equiv \nabla \{{\textrm{Re}[\tilde{I}(x,y)]} \}\times \nabla \{{\textrm{Im}[\tilde{I}(x,y)]} \}$, quantifying the intensity and direction of the optical vortex. These properties remain invariant during the process of deflecting the optical field with the shearing device. No two phase singularities exhibit identical local properties, as each one possesses distinct structural characteristics, including eccentricity e, zero-crossing angle ${\theta _{RI}}$, topological charge q, and vorticity $\Omega $. It is precisely this uniqueness of core structures that enables accurate identification and tracking of the complex motion of phase singularities. Typically, the real and imaginary components of the complex signal representation of the two-dimensional speckle pattern near the phase singularities can be expressed as follows:
According to the geometric definition, the geometric and physical parameters representing the phase singularity can be expressed as fitting coefficients
The coefficients ${a_r},{b_r},{c_r}$ and ${a_i},{b_i},{c_i}$ may undergo variation due to the optical field's deflection caused by the shearing device. However, the local properties characterizing the pseudo-phase singularity remain relatively stable, exhibiting invariance under translation and rotation, akin to properties of optical vortices. Notably, the local properties of the phase starting point in the speckle field correspond to the local three-dimensional distribution of the tested rough surface. When focusing on the object surface, the shearing device-induced displacement aligns with the measurement field of the rough surface, effectively determining the local displacement of the object surface. Consequently, the overall phase singularity displacement directly correlates with the shearing device's deflection. By matching the corresponding phase singularities using their core structures as keys before and after shearing, we can estimate the shearing distance induced by the device through statistical analysis of phase singularities’ motion. Besides the geometric and physical parameters $q$, e, ${\theta _{RI}}$ and $\Omega $ utilized in this study, various combinations of constraints can be applied to uniquely identify the matched phase singularity [27–29]. To ensure a reliable match between phase singularities before and after shearing, reasonable tolerances for these key parameters should be set during the matching process. These conditions can be expressed as follows:
After calculating the merit function for each pair of phase singularities, we can identify the correct corresponding points based on the minimum value of E. Therefore, the shear amount caused by the shear device can be estimated based on the coordinate variation $(\Delta x,\Delta y)$ of each pseudo-phase singularity within the detection region.
2.5 Preliminary
During the recognition process, assuming no prior knowledge of the shear direction and shear amount of the shear device, we conducted a search throughout the entire image area. Figure 4(a) presents an actual example of a coarse search for the motion of phase vortices in the speckle field acquired by occluding mirrors. The positions of the phase singularities before and after the two acquisitions are represented by circles (○) and (●), respectively, which are still used for subsequent images for clarity. As expected, most phase singularities can find corresponding ones in the right direction. By connecting the correct pairs of phase singularities with short straight lines, we can use only these particular phase singularities to estimate the optical deflection caused by the shear device based on the differences in their coordinates. However, it is also observed that some phase singularities do not have correct corresponding points. These points usually form long straight lines extending in random directions, becoming potential sources of error in optical vortex measurements.
Fig. 4. Statistical analysis after a coarse search of phase vortices displacement in the pseudo-phase pattern. (a) Search result pairs of phase vortex points on the pseudo-phase: ○ is the vortex position at the first acquisition; ● is the vortex position at the second mid-acquisition matching the vortex feature at the first acquisition; blue represents positive vortex; red represents negative vortex. (b) Histogram of the change in x-direction coordinates of the vortex points. (c) Histogram of the change of vortex point coordinates in the y-direction.
The observed result can be attributed to three reasons. The first reason pertains to changes in the temporal and spatial conditions caused by the interference of coherent light during the two-stage acquisition, resulting in variations in speckle patterns. In other words, as time progresses, the decorrelation of speckles occurs, leading to distortions in the core structure of phase singularities. Occasionally, new phase singularities are introduced or existing ones are annihilated in the pseudo-phase map. The distortion of phase singularities makes it more challenging to identify and match these singularities’ core structures, and the newly generated or vanished phase singularities lack corresponding points in the pseudo-phase.
Another reason is the interference caused by parasitic reflections within the optical device when the reflected light field enters it, generating speckle field noise and distortion. Consequently, the core structures of some singularities undergo changes or become extinct. It becomes impossible to identify the displacement between two acquired speckle patterns by tracking individual vortex variations.
Lastly, the flow of phase vortices along the boundaries of the detection area causes some phase singularities to cross the boundaries (or move out of) the detection area. This also contributes to the absence of corresponding points in other pseudo-phase patterns. To address the issues, one could adopt a statistical approach to analyze the overall characteristics of the optical vortex displacement. For instance, one can estimate the disharmony induced by the shearing device by simply calculating the mean values of the $\textrm{x}$ and $\textrm{y}$ coordinates based on the peak positions of the histograms. Figure 4(b) presents the histogram of $\textrm{x}$-coordinate variations, while Fig. 4(c) shows the histogram of $\textrm{y}$-coordinate variations. In this measurement, the mean displacement along the $\textrm{x}$ is 30.3 pixels, with a standard deviation $\sigma $ of 17.9 pixels, indicating a significant level of uncertainty in one-dimensional displacement estimation based on individual singular points. Similarly, there is a prominent peak in the $\textrm{y}$-direction, with an average value of 0.3 pixels and a corresponding standard deviation of 17.4 pixels. Due to the deviation direction of the set shearing device, shear only occurs in the x-direction, resulting in a significantly larger mean displacement in the x-direction than in the y-direction, and the y-direction is close to the zero point, aligning with the deviation direction of the shearing device. Consequently, the lateral shearing distance in the x-direction can be roughly estimated as 30.3 pixels. Additionally, on both sides of the main histogram peak, long tails can be observed in both histograms with an approximate distribution, indicating measurement error regions caused by the failure to find the correct corresponding points for the phase singularities. It is worth noting that despite the diffusion of phase singularities in the histogram, most of the $\textrm{x}$-coordinate differences of the phase singularities remain concentrated within an extremely narrow columnar distribution. This high degree of concentration enables an accurate estimation of the shearing distance. In this measurement, the standard deviation of $\Delta e$ is 0.0233, and the standard deviation of $\Delta {\theta _{RI}}$ is 0.0850, serving as stability metrics for the core structure of pseudo-phase singularities during the shearing calibration process.
Based on the initial coarse search of shearing distance and prior knowledge of the lateral shearing interferometer, the search region of the vortex core structure can be constrained to enhance search accuracy and reduce search time. Firstly, according to the principles of the Michelson shearing interferometer, all optical vortices can only undergo displacements on the same side. In the context of our sequential acquisitions, the vortices experience an overall rightward shift. Secondly, as a lateral shearing interferometer, we aim to generate consistent shear displacements in either the x or y direction. Based on these two conditions, we restrict the search region for the core structure of the singularities within a small window measuring 60 × 5 pixels to the right of the initial vortex position. Figure 5(a) illustrates the results of performing local searches within the aforementioned restricted region for phase vortices’ displacements. The histograms depicting the coordinate changes along the x and y directions for the phase vortices are shown in Fig. 5(b) and 6(c), respectively. As expected, all phase vortices exhibit uniform displacements on the same side.
Fig. 5. Statistical analysis after a refined search of phase vortex points on the pseudo-phase. (a) Search result pairs of phase vortex points on the pseudo-phase. (b) Histogram of the change in x-direction coordinates of the vortex points. (c) Histogram of the change of vortex point coordinates in the y-direction.
By computing the peak positions from the histograms, the average displacement along the $\textrm{x}$-direction is determined to be 30.01 pixels with a standard deviation $\sigma $ of 1.72 pixels, exhibiting a significant improvement compared to the initial estimation. Similarly, the mean for $\Delta \textrm{y}$ is 0.06 pixels with a standard deviation of 0.35 pixels. Meanwhile, the standard deviations for variables $\Delta e$ and $\Delta {\theta _{RI}}$ are 0.0189 and 0.0422, respectively. It should be noted that the complex signals generated by the speckle pattern actually represent an isotropic nature, hence the two histograms of phase vortices exhibit similar peak widths. With these mean values, the shear amount s of the shearing device can be calculated as $s = {\left( {{{\langle \Delta x\rangle }^2} + {{\langle \Delta y\rangle }^2}} \right)^{{1 / 2}}} \approx 30$ pixels by taking the square root of the sum of the squares of the mean displacements in the $\textrm{x}$ and $\textrm{y}$ directions. Figure 5 serves as an experimental confirmation of the effectiveness of the proposed shear amount measurement technique and provides further insights for precise full-field shear amount detection. In comparison to the previous calibration methods, the method proposed in this section does not require attaching calibration objects to the surface of the object under test, hence eliminating the need to consider the properties or positions of the samples being tested. Additionally, it does not necessitate changing the light source for shear amount measurement.
In summary, we can outline the entire detection process of this method as follows: First, the shearing mirror ${\textrm{M}_2}$ is shielded, and the speckle pattern in front of the reference mirror ${\textrm{M}_1}$ is measured as the initial frame before shearing. Subsequently, the reference mirror ${\textrm{M}_1}$ is shielded, and the speckle pattern on the shearing mirror ${\textrm{M}_2}$ is measured to obtain the second frame after shearing. In the second step, the pseudo-phase of the two speckle patterns is generated using an L-G filter, and the positions of the optical vortices within them are identified, along with computing the parameters characterizing the core structure of the optical vortices. The third step involves conducting a coarse search based on the core structure parameters and a set threshold. In the fourth step, we perform a fine search with a search range set at 1.2 times the detected shear amount determined from the coarse search. This allows us to search for successfully matched optical vortex displacements and calculate their average value to obtain the final accurate shear amount.
3. Analysis of shear determination method based on optical vortex search
To validate the proposed method for shear measurement in DSSPI utilizing optical vortices, we present the calibration results for shear amount using a Michelson shear device, as depicted in Fig. 6. The accuracy of the proposed method is verified and compared with the baseline approach. A single longitudinal mode green laser (MSL-III-532-50 mW, CNI) with a wavelength of 532 nm is employed as the light source. It is expanded and directed onto the specimen's surface. The diffused light from the specimen is collected by an imaging lens ($f$= 100 mm) and directed to the shearing device, which consists of a beam splitter (BS), the shearing mirror ${\textrm{M}_2}$, and the reference mirror ${\textrm{M}_1}$. Upon determining the shear amount, the piezoelectric transducer (PZT) positioned behind ${\textrm{M}_1}$ facilitates phase measurement using multi-step phase shifting to quantify the strain. The shearing wavefront is captured by a camera (acA2440-20gm, Basler) with pixel dimensions of 3.45 × 3.45µm. Throughout the experiment, a standard 2 mm thick aluminum plate serves as the specimen. The baseline calibration method, as described in [10], involves deflecting the shearing mirror ${\textrm{M}_2}$ at a specific angle along the x-direction to set the shear amount as a standard measurement of k pixels laterally. It should be noted that the optical vortices-based method quantifies the displacement generated by the shearing device on the image plane in units of pixels (pix). However, to obtain the actual shear amount on the object plane, it needs to be multiplied by the magnification factor $\textrm{R}$ of the optical system, where the overall magnification of the shearing system on the object plane is measured in mm/pix. The system magnification, which is dependent on the optical configuration and observation distance, has been calibrated using a conventional method, ensuring its constancy during the tests. Therefore, all shear amounts in the subsequent tests are measured in pixels.
3.1 Influence of the window size of the L-G filter
The investigation of the Laguerre-Gaussian filter's bandwidth $\omega $ influence on shear amount calibration accuracy is imperative. The $\omega $ value of the L-G filter governs the average speckle size generating the pseudo-phase, thereby regulating the density of phase vortices. Figure 7(a1)-(a4) exemplify the pseudo-phase images acquired from speckle patterns with filter window sizes of 0.1, 0.2, 0.4, and 0.8, alongside the observed vortex displacements resulting from consecutive measurements. Analysis of pseudo-phase images at different bandwidths reveals that higher $\omega $ values lead to increased phase vortex density per pixel. Conversely, smaller filter window sizes ($\omega $=0.1 and 0.2) exhibit reduced correctly identified phase vortex displacements, potentially impacting shear amount recognition accuracy and reducing calibration robustness. Comparing shear amount errors to the baseline method, where calibrated shear amount is 24 pixels, Fig. 8 displays the trend of measurement errors in relation to filter bandwidth. Window sizes less than 0.3 result in shear amount errors exceeding 5%. As the window size increases, the measurement error decreases until reaching a window size of 0.6, after which the error starts to enlarge. This behavior may be attributed to excessively small speckle sizes and dense phase singularities generated by large filter windows, making them overly sensitive to the decorrelation effect between consecutive measurements. Striking a balance between measurement accuracy and shear amount calibration robustness is crucial when selecting the spatial filter window size. In this study, a filter window size of 0.4 was utilized. Notably, the $\omega $ value is not universally applicable and varies with changes in working distance, focus camera, and other system configurations. We recommend selecting an L-G filter bandwidth resulting in a vortex density of approximately 1.6${\times} $10−6 per unit pixel for enhanced calibration accuracy across varying shear amounts. However, further validation under different configurations is necessary to confirm this selection strategy.
Fig. 7. Search results and histograms of vortex $\textrm{x}$-direction deflection on the pseudo-phase pattern for different Laguerre-Gauss filter bandwidths $\omega$. (a) $\omega$=0.1; (b) $\omega$=0.2; (c) $\omega$=0.4; (d) $\omega$=0.8.
Fig. 8. (a) Laguerre-Gauss filter selection for different bandwidths with the shear amount calibrated by the optical vortex method. (b) Relationship between filter bandwidth and phase error ratio.
3.2 Influence of shear amount on the accuracy of calibration
The measurement accuracy of optical vortex detection methods was further explored under various shear amounts. By adjusting the tilted mirror ${\textrm{M}_2}$ in the Michelson shear interferometer and using the tape target method as a baseline, calibration was performed for shear amounts of 5, 10-100 pixels (with a 10-pixel interval) to extract the optical vortex displacements along the x-axis, with 14 repeated acquisitions and calculation for each shear amount. Figure 9(a1)-(a4) display the pseudo-phase of the speckle field and the examples of shear amounts obtained through optical vortex searching at shear amounts of 5, 30, 60, and 100 pixels, respectively. Corresponding Fig. 9(b1)-(b4) present histograms of the vortex displacements in the $\textrm{x}$-direction. Figure 10(a) compares the proposed vortex method with the shear measurement data obtained through the tape target method. Figure 10(b) demonstrates the residuals corresponding to the shear amounts measured by the baseline method, with a blue box indicating a 95% confidence interval, a horizontal line within each box representing the mean value, and a red box indicating the mean value plus or minus one standard deviation (SD). It is evident that there is a close correlation between the shear amounts obtained by these two techniques. The results shown in Fig. 10(b) indicate that the standard deviations obtained from multiple measurements under different shear amounts are consistent, implying that the optical vortex method is capable of calibrating different shear amounts as the shear amount increases. Furthermore, compared to the baseline measurement, there is no systematic bias observed in the obtained mean values. It is worth noting that the traditional tape target technique does not employ interpolation, whereas our proposed vortex method uses four-fold super-resolution, enabling shear calibration up to two decimal places. Therefore, these errors may be limited by the detection accuracy of the traditional method, while our method demonstrates significant improvement in accuracy and is closer to the real situation. Additionally, we observed a decreasing trend in the logarithm of successfully matched optical vortices with an increase in shear amount, as shown in Fig. 11. This could be attributed to the longer length of phase vortices causing more phase singularities to cross the boundary and fall outside the measurement range.
Fig. 9. Search results and $\textrm{x}$-direction deflection histograms using the optical vortex approach after calibrating different shear amounts s as baseline using tape targets. (a) $s = 5pix$; (b) $s = 30pix$; (c) $s = 60pix$; (d) $s = 100pix$.
Fig. 10. Comparison of shear measurements obtained using the proposed optical vortex method and the tape target method for a total of 14 measurements. (a) Comparison between the mean values of shear amounts obtained by the tape target technique and the optical vortex measurements. (b) Residual statistics of shear measured by the optical vortex. Blue boxes indicate the 95% confidence intervals and the mean is marked with the horizontal line in each box. Red boxes denote mean ± 1SD.
Fig. 11. The number of well-matched optical vortex pairs in the search range of the optical vortex core structure varies with the shear amount. Blue boxes indicate the 95% confidence intervals and the mean is marked with the horizontal line in each box. Red boxes denote mean ± 1SD
In summary, the results of this study demonstrate the efficacy of the proposed vortex method for determining shear amounts in DSSPI. Despite the multiple steps involved in the measurement process, automation is easily achievable. For instance, incorporating an LCD screen as a light-blocking plate in front of the shear device's mirrors allows for automated control of light transmission. Moreover, the shear amounts can be conveniently adjusted by tilting the mirror through a PZT, enabling flexible modification of measurement sensitivity and resolution. The integration of prior information into the retrieval window size determination further improves the feasibility of the measurement process. Nevertheless, it is essential to consider the imaging system's position susceptibility, as the diaphragm of the imaging system affects the spatial spectrum. When the imaging system position is situated behind the shear device, the decorrelation of speckle fields from deflected and undeflected paths distorts the core structure of the pseudo-phase optical vortex, impeding the search for corresponding vortices and leading to divergent outcomes.
4. Conclusion
This study presents a novel approach employing the optical vortex method for determining line shear amounts in DSSPI. Experimental findings reveal a strong correlation between results obtained by this method and those acquired through the baseline approach, ensuring accurate and consistent measurements of line shear amounts. Moreover, this technique effectively operates under laser illumination on the object surface, obviating the need for adhesive calibration targets and overcoming limitations posed by the surface properties of the test specimen, thus broadening its potential applications. Importantly, the proposed method seamlessly integrates into existing digital shear speckle interferometry setups without necessitating fundamental system modifications. Future work will focus on conducting stability analysis of the optical vortex method in dynamic speckle fields, exploring its potential and precision in determining shear amounts.
Funding
National Natural Science Foundation of China (51975116); Natural Science Foundation of Shanghai (21ZR1402900); Fundamental Research Funds for the Central Universities and Graduate Student Innovation Fund of Donghua University (CUSF-DH-D2021057).
Acknowledgments
The authors thanks Natural Science Foundation of China and Natural Science Foundation of Shanghai for help identifying collaborators for this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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