Measurement of In-Plane Displacement in Two Orthogonal Directions by Digital Speckle Pattern Interferometry

: The measurement of in-plane displacement in two orthogonal directions is of considerable signiﬁcance for modern industries. This paper reports on a spatial carrier phase-shift digital speckle pattern interferometry (DSPI) for the simultaneous measurement of in-plane displacement in two orthogonal directions. The object is illuminated from a single direction and observed from four symmetrical directions simultaneously. One pair of the four observation directions is sensitive to in-plane displacement in one direction, and the other pair is sensitive to in-plane displacement in the perpendicular direction, resulting in the displacement in two directions being measured independently. The polarization property of light is used to avoid cross-interference between the two pairs of beams. Spatial carrier frequencies are generated by aperture misalignment, and the displacement in two directions is modulated onto the same interferogram. With a spatial carrier phase-shift technique, the displacement can be separated in the frequency domain and the phase can be evaluated from a single interferogram in real time. The capability of DSPI is described by theoretical discussions and experiments.


Introduction
Owing to the rapid development in the manufacturing industry, such as automobile and aerospace applications, measuring the surface displacement of complex mechanical structures and new materials is important. The surface displacement can be further translated into strain and stress, which is the key parameter for design, manufacturing, and quality control [1]. A dynamic full-field and highly sensitive measurement of displacement is the foundation of rapid and optimization design.
The measurement of a three dimensional displacement, along three mutually orthogonal directions, can be classified as out-of-plane and in-plane displacements. However, the in-plane displacement component is more important than the out-of-plane displacement component in many practical problems [2]. Information on in-plane displacement is helpful in determining Young's modulus and Poisson's ratio of materials [3,4]. In-plane rotations, which can be determined by in-plane displacement, is an essential component of the geometrical metrology [5][6][7]. Under similar testing conditions, residual stress, obtained from in-plane displacements, has higher precision than that obtained from out-of-plane displacement [8].
Optical measurement techniques have been widely used in displacement measurements because they are non-contact, full-field, and highly sensitive. Advanced optical methods include, digital image correlation (DIC) [9,10], Moiré interferometry [11], and digital speckle pattern interferometry

Materials and Methods
The optical configuration of the proposed DSPI system is illustrated in Figure 1. A coherent laser, with a wavelength of λ, is expanded to illuminate the object surface. The object is observed from four directions simultaneously and symmetrically around the z-axis. The four imaging beams are denoted as x 1 , x 2 , y 1 , and y 2 , and the corresponding wave vectors are property of light is used to solve the problem of cross interference. Theory derivation, spectrum analysis, and experiment results are shown in detail. The optical configuration of the proposed DSPI system is illustrated in Figure 1. A coherent laser, with a wavelength of  , is expanded to illuminate the object surface. The object is observed from four directions simultaneously and symmetrically around the z-axis. The four imaging beams are denoted as x1, x2, y1, and y2, and the corresponding wave vectors are

Materials and Methods
x K are located in the XZ plane, whereas 1 y K and 2 y K are located in the YZ plane. The angle between each of observation directions and the z-axis is . After being reflected by the single mirror and the four-sided mirror, each image passes through the corresponding aperture stop and polarizer, and is imaged onto the image surface by the same lens. The four aperture stops are spatially offset from each other, and a polarizer is placed in front of each aperture stop. The polarization directions of each beam, passing through four polarizers, are illustrated in Figure 2. After passing through the polarizers, the polarizing directions of imaging beams x1 and x2 are the same, and the polarization directions of the imaging beams y1 and y2 are perpendicular to those of beams x1 and x2. Therefore, imaging beam x1 interferes with x2, and imaging beam y1 interferes with y2. No cross interference occurs between the two pairs of beams. After being reflected by the single mirror and the four-sided mirror, each image passes through the corresponding aperture stop and polarizer, and is imaged onto the image surface by the same lens. The four aperture stops are spatially offset from each other, and a polarizer is placed in front of each aperture stop. The polarization directions of each beam, passing through four polarizers, are illustrated in Figure 2. After passing through the polarizers, the polarizing directions of imaging beams x 1 and x 2 are the same, and the polarization directions of the imaging beams y 1 and y 2 are perpendicular to those of beams x 1 and x 2 . Therefore, imaging beam x 1 interferes with x 2 , and imaging beam y 1 interferes with y 2 . No cross interference occurs between the two pairs of beams.
As shown in Figure 3, (ξ, 0), (−ξ, 0), (0, η), (0, −η) are the center coordinates of the four apertures of imaging beams x 1 , x 2 , y 1 , and y 2 , respectively. The phases of the waves, which pass through the apertures, have a spherical part plus a speckled part [28]. The four beams that pass through each aperture and reach the image plane have amplitudes given by Equation (1) [29][30][31]: where D is the distance from the image plane to exit pupil plane [32]. 2πi λD ξx and 2πi λD ηy are the spatial carrier frequencies introduced by aperture misalignment. apertures of imaging beams x1, x2, y1, and y2, respectively. The phases of the waves, which pass through the apertures, have a spherical part plus a speckled part [28]. The four beams that pass through each aperture and reach the image plane have amplitudes given by equation (1) [29][30][31]: where D is the distance from the image plane to exit pupil plane [32].  The interferogram on the imaging plane can be expressed as follows:  As shown in Figure 3,         ,0 , ,0 , 0, , 0,      are the center coordinates of the four apertures of imaging beams x1, x2, y1, and y2, respectively. The phases of the waves, which pass through the apertures, have a spherical part plus a speckled part [28]. The four beams that pass through each aperture and reach the image plane have amplitudes given by equation (1) [29][30][31]: , exp , where D is the distance from the image plane to exit pupil plane [32].  The interferogram on the imaging plane can be expressed as follows: The interferogram on the imaging plane can be expressed as follows: By applying Fourier transformation to the interferogram, Equation (2) can be transformed into the Equation (3), where ⊗ is the convolution operation, The schematic diagram of the spectrum after Fourier transformation is illustrated in Figure 4.
contains background information, which is located at the center of the spectrum. The terms U x1 ⊗ U * x2 and U * x1 ⊗ U x2 contain the information of in-plane displacement in the x-axis, and they are located at 2ξ λd , 0 and − 2ξ λd , 0 , respectively. The terms U y1 ⊗ U * y2 and U * y1 ⊗ U y2 contain the information of in-plane displacement in the y-axis, and they are located at 0,  x y x y      are the phase differences of the two pairs of mutually interfering beams. After displacement, the phase difference can be calculated using equation (5): By properly selecting aperture diameter and distance, the five spectrums on the Fourier domain can be separated. By selecting the term. U x1 ⊗ U * x2 , located at 2ξ λd , 0 and performing the inverse Fourier transformation, u x1 u * x2 can be obtained. In the same way, u y1 u * y2 can also be obtained. The phase term can be calculated by the following relation: where Im and Re denote the imaginary and real parts, respectively. ψ are the phase differences of the two pairs of mutually interfering beams. After displacement, the phase difference can be calculated using Equation (5): Through subtraction, the displacement can be evaluated from the relative phase difference. The relationship between the phase change and the displacement is given by [21], where → S x and → S y are the sensitivity vectors, defined as functions of the illumination and observation directions. For the DSPI proposed in this study, the sensitivity vectors can be expressed by Equation (7): The phase difference that corresponds to the two observation vectors, with a common illumination vector, does not depend on the illumination directions because the observation vector is removed. (6) and (7), in-plane displacement in two directions can be calculated by Equation (8):

Results
The experiments in this study have been designed to verify the DSPI. A 532 nm laser (Changchun New Industries Optoelectronics Technology Co., Ltd., continuous wave, 200 mW) is used as the laser source. A high-resolution camera (Basler ace, acA1600-20um, 4.4 µm × 4.4 µm pixel size, 1626 pixels × 1236 pixels) is used to record the speckle pattern images. The focal length of the imaging lens is 150 mm. The angle between each wave vector of observation direction and the z-axis is 4 • . The diameter of each aperture stop is 4 mm, and the distance of two aperture stops, through which light interferes, is 8 mm. This setting of aperture diameter and distance makes the different terms, shown in Figure 4, separates from each other. If the aperture is too large, or if the distance is too close, the different terms will alias. If the aperture is too small, the amount of light passing through will also be small, and the speckle noise will be large. By choosing an aperture size and distance appropriately, the different terms are evenly distributed to achieve better results.
The object under test is a metal plate with a diameter of 130 mm, which can be precisely rotated in-plane by a micro-head, as illustrated in Figure 5. In-plane displacement is produced by the rigid-body rotation.  A magnified portion of the recorded speckle pattern image is displayed in Figure 6, which reveals the two orthogonal spatial carrier fringes within the speckle. Figure 7 shows the spectrum distribution of the speckle pattern image after Fourier transformation, which is similar to that obtained theoretically as illustrated in Figure 4. The spectrum shown in Figure 7 is the logarithm of the original spectral intensity to make the figure clearer. The different terms, schematically presented in Figure 4, can be clearly separated in Figure 7. The two terms marked by a red circle and labeled with A and B are corresponding to  A magnified portion of the recorded speckle pattern image is displayed in Figure 6, which reveals the two orthogonal spatial carrier fringes within the speckle.  A magnified portion of the recorded speckle pattern image is displayed in Figure 6, which reveals the two orthogonal spatial carrier fringes within the speckle. Figure 7 shows the spectrum distribution of the speckle pattern image after Fourier transformation, which is similar to that obtained theoretically as illustrated in Figure 4. The spectrum shown in Figure 7 is the logarithm of the original spectral intensity to make the figure clearer. The  Figure 7 shows the spectrum distribution of the speckle pattern image after Fourier transformation, which is similar to that obtained theoretically as illustrated in Figure 4. The spectrum shown in Figure 7 is the logarithm of the original spectral intensity to make the figure clearer. The different terms, schematically presented in Figure 4, can be clearly separated in Figure 7. The two terms marked by a red circle and labeled with A and B are corresponding to U x1 ⊗ U * x2 , and U y1 ⊗ U * y2 , respectively. According to Equation (3) and Figure 4, the lower terms should be the conjugate of the upper ones. The negative displacement would be determined form the lower terms. A magnified portion of the recorded speckle pattern image is displayed in Figure 6, which reveals the two orthogonal spatial carrier fringes within the speckle. Figure 7 shows the spectrum distribution of the speckle pattern image after Fourier transformation, which is similar to that obtained theoretically as illustrated in Figure 4. The spectrum shown in Figure 7 is the logarithm of the original spectral intensity to make the figure clearer. The different terms, schematically presented in Figure 4, can be clearly separated in Figure 7. The two terms marked by a red circle and labeled with A and B are corresponding to   For both interferograms before and after rotation, by selecting the terms A and B, shown in Figure 7, and applying a windowed inverse Fourier transformation to each term, phase differences ∆ x and ∆ y , according to Equation (6), can be extracted. Only two interferograms are needed to evaluate the phase by using spatial carrier phase-shift technique. The phase map is smoothed by a low-pass filter algorithm as displayed in Figure 8(a1,a2). After phase unwrapping, the in-plane displacement in two directions, calculated using Equation (8), is displayed in Figure 8(b1,b2). For both interferograms before and after rotation, by selecting the terms A and B, shown in Figure 7, and applying a windowed inverse Fourier transformation to each term, phase differences x  and y  , according to equation (6), can be extracted. Only two interferograms are needed to evaluate the phase by using spatial carrier phase-shift technique. The phase map is smoothed by a low-pass filter algorithm as displayed in Figs. 8(a1) and 8(a2). After phase unwrapping, the in-plane displacement in two directions, calculated using equation (8), is displayed in Figure 8(b1) and 8(b2).
Generally speaking, the measurement accuracy and sensitivity of DSPI are mainly determined by the phase map quality and number of times the images are smoothed. Speckle interference is usually able to reach a measuring sensitivity of about 2π/20 to 2π/10 (2π corresponds one fringe) and a measuring accuracy of about 2π/5 to 2π/2. A detailed discussion regarding this topic can be found in reference [33]. In this research, the in-plane displacement is produced by the rigid-body rotation, which follows certain rules demonstrated by Sijin Wu [7]. The displacement reference value is obtained by fitting the measured in-plane displacements according to the rules. Then, the errors of the measured displacements, relative to the reference value are calculated, as shown in Figure 7(c1) and 7(c2). The root-mean-square (RMS) errors of in-plane displacements in the x-axis direction and y-axis direction are 0.071 μm, and 0.072 μm respectively, which exhibits high accuracy.  Generally speaking, the measurement accuracy and sensitivity of DSPI are mainly determined by the phase map quality and number of times the images are smoothed. Speckle interference is usually able to reach a measuring sensitivity of about 2π/20 to 2π/10 (2π corresponds one fringe) and a measuring accuracy of about 2π/5 to 2π/2. A detailed discussion regarding this topic can be found in reference [33]. In this research, the in-plane displacement is produced by the rigid-body rotation, which follows certain rules demonstrated by Sijin Wu [7]. The displacement reference value is obtained by fitting the measured in-plane displacements according to the rules. Then, the errors of the measured displacements, relative to the reference value are calculated, as shown in Figure 8(c1,c2). The root-mean-square (RMS) errors of in-plane displacements in the x-axis direction and y-axis direction are 0.071 µm, and 0.072 µm respectively, which exhibits high accuracy.
Because the displacement, shown in Figure 8, is produced by an in-plane rotation, the contours of the absolute in-plane displacement are essentially a set of concentric circles around the rotation center. The synthetic in-plane displacement vectors are illustrated in Figure 9, with the contours of the absolute in-plane displacement. The vectors are tangent to the corresponding displacement contours, satisfying the law of displacement introduced by rigid body rotation. The direction of rotation can be easily distinguished from the directions of the in-plane displacement vectors. Because the displacement, shown in Figure 8, is produced by an in-plane rotation, the contours of the absolute in-plane displacement are essentially a set of concentric circles around the rotation center. The synthetic in-plane displacement vectors are illustrated in Figure 9, with the contours of the absolute in-plane displacement. The vectors are tangent to the corresponding displacement contours, satisfying the law of displacement introduced by rigid body rotation. The direction of rotation can be easily distinguished from the directions of the in-plane displacement vectors.
Because only one interferogram is needed to measure the phase in this experiment, the interferogram can be continuously acquired during the displacement of the object. Thereby, the displacement at any acquisition time can be evaluated. Therefore, dynamic measurements can be made, and the measurement speed depends on the frame rate of the camera.

Discussion
This work presents a spatial carrier phase-shift DSPI for the simultaneous measurement of inplane displacement, in two orthogonal directions, by using a single camera. The carrier frequency is generated by aperture misalignment. Two pairs of observation beams are used to make the DSPI sensitive to in-plane displacement in two directions. Considering that the beams, whose polarization directions are perpendicular, do not interfere with each other, cross-interference between the two pairs of beams is avoided. Using the Fourier transformation procedure, the phase can be evaluated from a single interferogram by the spatial carrier phase-shift technique. Only two interferograms are required to simultaneously measure the displacement in two directions: The first image is acquired Because only one interferogram is needed to measure the phase in this experiment, the interferogram can be continuously acquired during the displacement of the object. Thereby, the displacement at any acquisition time can be evaluated. Therefore, dynamic measurements can be made, and the measurement speed depends on the frame rate of the camera.

Discussion
This work presents a spatial carrier phase-shift DSPI for the simultaneous measurement of in-plane displacement, in two orthogonal directions, by using a single camera. The carrier frequency is generated by aperture misalignment. Two pairs of observation beams are used to make the DSPI sensitive to in-plane displacement in two directions. Considering that the beams, whose polarization directions are perpendicular, do not interfere with each other, cross-interference between the two pairs of beams is avoided. Using the Fourier transformation procedure, the phase can be evaluated from a single interferogram by the spatial carrier phase-shift technique. Only two interferograms are required to simultaneously measure the displacement in two directions: The first image is acquired before the load and the second image is acquired after the load. The displacements in two directions are