1. Introduction

Speckle metrology, a widely used, long known, non-contact measurement method, was first introduced in 1970 for interferometric measurement of rough surfaces [1]. In 1982, Peters and Ranson suggested cross-correlating two speckle images for displacement measurement [2]. Since then, this technique has been a popular research topic with various applications, including the measurement of deformation [3], stress and strain [4], surface roughness [5], velocity [6, 7], and analyzing vibration [8]. It is given different names in numerous literature including Digital Image Correlation (DIC) and Digital Speckle Photography (DSP). Displacement measurement using DIC is possible by capturing a sequence of objective or subjective speckle patterns and finding the shift of the sequence of the patterns with respect to a reference pattern using cross correlation. Under certain conditions, the surface shift has a linear relationship with the pattern shift. The ratio of the pattern shift to the surface shift (scale factor) is dependent on the illumination and imaging specifications [9]. Subpixel resolution is achievable using interpolating or non-interpolating techniques [10, 11]. Three dimensional displacement measurement is possible by applying a steroscopic DSP system [12], and by using the dependency of the speckle correlation characteristics on the spatial frequency [13]. Micro and nano scale deformation measurements are also possible by coupling DIC with different microscopy methods [14–16].

In spite of various applications and capabilities, DIC has some limitations. The measurement accuracy strongly depends on the stability of the parameters that control the scale factor. As a result, this method requires a high quality imaging system; usually sample surface must be flat and remain parallel to the image plane during measurement; the out-of-plane displacement of the sample can change the imaging magnification, which further introduces additional in-plane spurious displacements [17]. Using a telecentric lens to account for this problem increases the cost of the system. Placing the camera far from the sample to approximate a telecentric imaging system prevents a compact design. The computational effort of DIC can also be a limiting factor, especially for applications that require high speed, real-time measurements [18]. Creating and maintaining a large database of speckle patterns allows using a basic correlation coefficient (cc) to enable precise alignment of work pieces with minimum relocation uncertainty of 7.6 µm [19]. It also allows using a simple setup for absolute position measurement with lateral resolution of less than ± 20 nm [20]. However, speckle patterns are sensitive to environmental disturbances, such as temperature fluctuations, deviation of the system's elements from their initial position, and any contamination of the sample. It is quite challenging to create a set of database patterns only once and expect the measurement system to operate for a long period of time. Using a precise stage as a part of the system in order to renew the database regularly, significantly increases the cost of the system and decreases its effectiveness.

This paper introduces a method for displacement measurement using speckle correlation that does not require a predetermined set of database patterns and is robust to environmental disturbances. It is based on the relative correlation of the speckle patterns generated by two parallel, overlapping laser beams with an identical spot size and different wavelengths. The accuracy of this method depends mainly on the stability of the beams' offset. The design is compact and the imaging requirements are low. The computational effort for determining the cc is significantly less than that for cross correlation and its Fourier transform. The unique characteristics of this method suggest it may be useful for high speed, non-contact, sub-micrometer, robust displacement measurement.

2. Theory

Just like fingerprints, speckle patterns can identify specific positions of the sample. If the time between capturing two speckle patterns is too short for any mechanical or environmental disturbances to degrade the correlation, it is possible to capture two speckle patterns under similar conditions with high correlation. Under certain conditions, they can even have different wavelengths, but still highly correlate. Figure 1 shows two parallel, overlapping laser beams with an identical spot size and different wavelengths (red and green) illuminating an optically rough surface, such that the red and the green speckle pattern created under similar conditions correlate. Lehmann et al. explains the conditions under which two speckle patterns with different wavelengths have high correlation, using the principles of speckle elongation [21]. Based on this reference, achieving high correlation between two speckle patterns of different wavelengths requires small wavelength separation, large speckle size, and precise alignment. In order to generate large speckles, the illuminating beam spot should be small and the imaging lens should have a large focal length (in case of a subjective speckle pattern). Lehmann et al. demonstrate high correlation between two speckle patterns with different wavelengths using approximately 20 nm wavelength separation, 650 µm spot size, 200 mm imaging lens focal, and samples with RMS roughness (R_q) from 0.1 to 1 µm.

 

Fig. 1 Location of the red and the green spot at different sample positions.

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In Fig. 1, the sample shifts in incremental steps, such that the area of the sample that the red beam initially illuminates, approaches the area that the green beam illuminates at each step. As the sample shifts, the correlation between the green speckle pattern captured at each step (green_i) with the red pattern captured at the beginning of the experiment (red0) increases, until the sample shifts by an amount equal to the beams' offset (d), which is the distance between the center of the two spots. This is when the cc between the two patterns, cc(red0,green_i), reaches its maximum. At this point, red0 is replaced by the red pattern captured at the correlation peak position, and cc(red0,green_i) is monitored up to the next correlation peak.

Figure 1(a) shows the initial position of the sample with a red and a green spot illuminating its surface. Figure 1(b) shows how the area of the surface initially illuminated by the red beam approaches the area that the green beam illuminates at each step. In this figure, the sample shifts to the left by the amount dx, while the beam spots are stationary. Figure 1(c) shows that if the sample shifts to the left by the amount equal to the beams' offset, the area initially illuminated by the red beam will completely overlap the area that the green beam illuminates at that step. This is where the maximum value for cc(red0,green_i) occurs.

The correlation method used here to compare the two speckle patterns is based on the Pearson cc as follows [22]:

$cc=\frac{{\displaystyle {\sum }_{ij}(a_{ij}-\overline{a})(b_{ij}-\overline{b})}}{\sqrt{{\displaystyle {\sum }_{ij}{(a_{ij}-\overline{a})}^2{\displaystyle {\sum }_{ij}{(b_{ij}-\overline{b})}^2}}}}$ (1)

In this equation, a_ij and b_ij denote the pixel intensities, $\overline{a}$and $\overline{b}$denote the mean values of the two captured patterns. Pearson cc is a measure of the linear dependence between two variables. It is a value between + 1 and −1 inclusive, where 1 is total positive correlation, 0 is no correlation, and −1 is total negative correlation. Because the pixel intensity values are always positive, the cc of two speckle patterns is a value between 0 and 1.

Figure 2 shows the correlation behavior associated with Fig. 1 assuming that the beams' offset is 30 µm, the sample shifts in 1 µm steps, and the red and the green pattern captured under the exact same condition have a very high correlation. In this figure, the star markers show the cc(green0,green_i) values, where green0 is the green pattern captured at the beginning of the experiment; the circle markers show the cc(red0,green_i). As soon as cc(red0,green_i) reaches its maximum, the red pattern captured at the correlation peak replaces red0. The distance between the two correlation peaks is equal to the beams' offset.

 

Fig. 2 Expected correlation behavior for dual wavelength speckle correlation.

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It is important to note that after the sample shifts a distance equal to the beams' offset, although the green spot illuminates the exact same area initially illuminated by the red spot, the green_i is shifted in the image plane with respect to red0. As a result, before determining cc(red0,green_i) values, red0 should be digitally shifted only once in the image plane such that the center of red0 coincides with that of green_i after the sample shifts equal to the beams offset. This is because the correlation technique used here compares the corresponding pixels of the two patterns. If the offset between two identical patterns is larger than the speckle diameter, the cc between them will be negligible. The required digital pattern shift is equal to the beams' offset if the magnification is 1, and is applied using phase correlation. Figure 3 clarifies this explanation. Figure 3(a) shows the initial position of the speckle pattern that a specific area of the sample generates in the image plane of the camera. Only after the sample shifts a distance equal to the beams offset, that exact speckle pattern is generated, this time by the green laser, but as Fig. 3(b) shows, green_i, that now highly correlates with red0, is shifted in the image plane of the camera with respect to red0. Digitally shifting red0 toward the green_i in Fig. 3(b) compensates for this problem. One advantage of the method introduced here over similar methods that use a single wavelength and a beam spot [18] is that even if minor fluctuations of the scale factor occurs, the correlation peak still happens after the sample displacement is equal to the beams' offset. By increasing the decorrelation of green0 with respect to red0 due to the beams offset, higher robustness to the fluctuations of the scale factor is achievable.

 

Fig. 3 The shift of the speckle pattern in the image plane of the camera due to the sample shift. a) The position of a specific speckle pattern before the sample shift. b) The position of the same speckle pattern after the sample shift.

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Figure 4 shows the expected correlation behavior over 97 µm range for 32 µm beam offset. In this figure, cc(red0,green_i) reaches its maximum every 32 µm. Although cc(red0,green_i) values decrease for the shifts greater than the beams' offset, as soon as a decrease in cc(red0,green_i) occurs, the red pattern captured at the correlation peak replaces red0. This is the reason that cc(red0,green_i) sharply drops right after each peak. Calibrating the system only once by determining the beams' offset allows determining the relative motion of the sample by monitoring the correlation behavior.

 

Fig. 4 Expected correlation behavior, cc(red0,green_i), for dual wavelength speckle scale.

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Displacement measurement in between the correlation peaks is possible using a curve fitting method. The curve used for this purpose can be based on the autocorrelation function (ACF) of the red or the green speckle pattern, which can be different from each other. One requirement for this purpose is that the half width of the curve should be larger than the cc(red0,green_i) peak separation. Otherwise, if the position of the sample is farther than the half width of the curve, the corresponding pattern will have no correlation with red0, which is the only reference pattern available. This sets the requirement for minimum speckle size.

Using the curve fitting method requires determining the suitable curve while calibrating the measurement system. For displacement measurement, the peak of this curve coincides with the peak of cc(red0,green_i) values. At each sample position, cc(red0,red_i) in case of using the red speckle pattern ACF for curve fitting, and cc(green0,green_i) in case of using the green speckle pattern ACF, defines a horizontal line (y = cc in a x-y plot). The amount of the sample shift from the peak of the curve to the point where the horizontal line crosses the curve, gives the distance of the sample from the reference pattern. Figure 5 shows an example where the reference pattern is at 930 µm. The cc(red0,red_i) at an unknown sample position is 0.6643. The horizontal line, y = 0.6643, crosses the curve at 18.2 µm from the peak of the curve, which shows the sample position determined by the curve fitting is 948.2 µm.

 

Fig. 5 Displacement measurement using curve fitting in between the correlation peaks.

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A description of the method and the details of different steps is explained so far. Figure 6 summarizes the method using a flowchart. In this flowchart, the assumption is that when (i) changes to (i + 1), the stage shifts the sample 1 µm and the camera captures a new pattern.

 

Fig. 6 The flowchart of the dual wavelength method.

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3. Experimental setups and results

The first experimental setup aims at verifying the correlation between two speckle patterns created with different wavelengths under certain conditions. Figure 7 shows the ideal schematic setup for this purpose. In this figure, a red and a green laser are coupled into a fiber and collimated such that the red and the green spot on the sample are completely overlapping. The color CCD captures the red and the green pattern simultaneously but separately, which allows determining the correlation between the two patterns.

 

Fig. 7 Schematic setup that helps to verify the correlation between a red and a green speckle pattern.

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The experimental setup shown in Fig. 8, uses a black and white CCD instead of a color camera. As a result, instead of capturing the red and the green pattern simultaneously, the camera captures them in black and white, one after another, while one of the lasers is off at each camera shot. In this setup, a HeNe laser with λ = 543.5 nm and a diode laser with λ = 657.0 nm are coupled into a single mode fiber and collimated using a lens. The spot size on the sample is about 1 mm. The sample has a ground silicon surface with 0.75 µm area RMS (S_q). A beam splitter reflects the speckle patterns toward the imaging lens. A black and white CCD captures the patterns in the focal point of a plano-convex lens (f = 200 mm).

 

Fig. 8 Experimental setup to verify the correlation between a red and a green speckle pattern.

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Analyzing the speckle patterns shows that the red pattern has larger speckles and higher angular dispersion relative to the green pattern. The cc between the red and the green is approximately 0.42. One way to increase the correlation between the two patterns is to take the effect of dispersion into account, which is possible by scaling the red pattern with a factor of λ_green/ λ_red. The function used for this purpose reconstructs a continuous signal from the original discrete signal, applies a low-pass anti-aliasing filter to the continuous signal, and then re-samples the resultant signal at the desired new sampling rate to get the output. After scaling the red pattern, the cc between the two patterns increases to 0.78.

The second experiment aims at displacement measurement using the dual wavelength method introduced in section ‎2. Figure 9 shows the schematic setup for this method. A green and a red collimated beam illuminate a beam splitter at 90° angle with respect to each other in order to create two parallel, overlapping laser beams with an identical spot size and different wavelengths. The beams illuminate the sample, and a second beam splitter reflects the speckle patterns to the imaging lens and the color camera.

 

Fig. 9 Schematic setup for displacement measurement using dual wavelength speckle correlation.

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Figure 10 shows the experimental setup for dual wavelength speckle correlation. This setup consists of a red (λ = 635 nm) and a green (λ = 520 nm) pigtailed laser diode. Two fiber collimators collimate the lasers. The green laser collimator is mounted on a precision translation mount (Thorlabs LM1XY), which allows adjusting the beams' offset by shifting the green collimator in x and in z direction. Here, x direction is parallel to the optical table and perpendicular to the green beam, while z is perpendicular to the optical table. Every full revolution of the adjusters of the translation mount is equivalent to 250 µm linear translation. A pellicle beam splitter brings the beams to overlap, and another pellicle beam splitter reflects the beams toward the sample. A plano-convex lens with 100 mm focal length (f) images the speckle pattern on the screen of a color CMOS camera (Thorlabs DCC1645C) with 3.6 µm square pixel size. The sample is ground steel with vibrational finish and 0.2 µm S_q, which is attached to a motorized stage with 4 mm range. The distance between the sample and the imaging lens is 2f. The distance between the imaging lens and the camera is also about 2f. This results in magnification equal to 1.

 

Fig. 10 Experimental setup for displacement measurement using dual wavelength speckle correlation.

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The first step in calibrating the dual wavelength setup is to determine the beams' offset. Although the offset is adjusted using the translation mount, error in tuning the adjuster and any misalignment in the setup can introduce error to the beams' offset. A more accurate way to evaluate the beams' offset is to determine the correlation peak separation for different positions of red0 and use it as an estimate for the beams' offset. The difference here is that the required digital shift for red0, which is related to the beams' offset, is not known. The sample shifts in incremental steps, such that the area initially illuminated by the red beam approaches the area that the green beam illuminates at each step, over a range that is about twice the beams' offset adjusted by the translation mount. This assures that the beams' offset is in the range of the sample shift. At every 1 µm step, the subpixel shift between red_i and red0 is determined using a suitable algorithm [23], digitally applied to red0 such that it is shifted toward green_i, and cc(shifted red0,green_i) is obtained. The sample shift associated with the maximum cc(shifted red0,green_i) shows the beams' offset. This process is for determining the beams' offset and red0 is not replaced at any point. Also, the amount of the required digital shift for red0 is updated at every 1 µm. Figure 11 shows an example for this process. The translation mount shifts the green laser approximately 31 µm in positive x direction, and the sample shifts from 0 to 80 µm in 1 µm increments. A second degree polynomial is fit to the experimental data in order to determine the position of the peak (32.6 µm), which is about a 5% deviation from the adjustment of the translation mount. The corresponding digital shift for red0 is −10.14 pixels in x direction and 0.66 pixels in z direction. Repeating the process for different positions of red0 brings the error due to the surface inhomogeneity into account.

 

Fig. 11 Estimating the beams' offset using cc(shifted red0,green_i).

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Surface inhomogeneity can introduce error to displacement measurement using dual wavelength speckle correlation. One way to identify a uniform area on the sample in order to evaluate the measurement method is to shift the sample in incremental steps over a long range (2 mm), and determine the correlation of the red pattern with the green pattern at every step. Figure 12(a) is an example of cc(red_i,green_i) at every 10 µm step over a 2 mm range. It shows that cc(red_i,green_i) from 930 to 1130 µm is high and the fluctuation (the difference between the maximum and the minimum cc in this range) is only about 0.008. Figure 12(b) shows the details of this uniform range.

 

Fig. 12 Identifying a uniform region of the surface for evaluating the dual wavelength method. a) cc(red_i,green_i) at every 10 µm over 2000 µm range. b) cc(red_i,green_i) at every 10 µm from 930 to 1130 µm.

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Evaluating the uncertainty of the beams' offset requires determining the beams' offset, using the method that generates the results of Fig. 11 at different positions of red0 over the uniform range of the sample. The ten selected red0s are at every 5 µm, ranging from 930 to 975 µm. The average of the correlation peak separation among the ten different sets of cc(shifted red0,green_i) values, which is the value used for the beams' offset, over this 200 µm range is 33.05 µm and the standard deviation (std) is 0.42 µm.

The next step in calibrating the dual wavelength system is to determine the curve to be fit in between the correlation peaks. Figure 13 compares the red and the green correlation distribution at 930 µm over a 32 µm range. The red distribution is cc(red0,red_i) with red0 at 930 µm and red_i changing from 930 to 962 µm. Determining the green distribution is similar to the red. The green distribution drops faster than the red, because the green speckle size is smaller than the red.

 

Fig. 13 Comparing the red and the green correlation distribution. The red distribution is cc(red0,red_i) and the green distribution is cc(green0,green_i) every 1 µm over 32 µm.

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A red distribution, which is a set of cc(red0,red_i) values, can be represented by a vector. In Fig. 13, this vector has 33 elements and is 32 µm long. Every position of red0 can define a new red distribution vector, where red_i positions change in 1 µm steps from red0 position to that plus 32 µm. Ideally all these vectors should be identical, but in practice, they are different. These deviations introduce a displacement error when using the curve fitting method. In order to estimate this error, 10 different vectors of cc(red0,red_i) values are generated using 10 different positions of red0. The average of these vectors is a new vector with 33 elements (avg_dist). The maximum absolute deviations of all the 10 vectors from avg_dist is a vector that shows the maximum cc(red0,red_i) deviations (Δcc) from their average values every 1 µm. Having these vectors, an estimate of the displacement error vector (Δx) is as follows

Here, the 10 red0s are selected at every 15 µm from 930 to 1065 µm. The maximum value of Δx, which is the maximum possible displacement error due to the curve fitting, is 1.35 µm for the red distribution. This process is similar for the green distribution and results in 1.56 µm maximum possible displacement error. This shows that the red distribution is more suitable that the green for the curve fitting process. In order to determine the correlation curve, a cubic spline interpolates the red distribution in 0.01 µm steps.

After calibrating the system, the dual wavelength speckle scale is ready for relative displacement measurement. The system is evaluated over a 200 µm uniform range of the sample, from 930 to 1130 µm, by comparing the sample shift determined using the dual wavelength method to the stage reading. The sample shifts in incremental steps to avoid any possible error that improper timing between capturing the speckle patterns and recording the stage reading might introduce. Figure 14 shows the correlation results. The curve derived from interpolating cc(red0,red_i) values is scaled such that its peak coincides with that of the curve associated with cc(shifted red0,green_i) values, because the maximum cc(red0,red_i) is 1, while here, the maximum cc(shifted red0,green_i) is less than 0.8.

 

Fig. 14 Correlation results of the dual wavelength method.

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The first red0 and the green-i are both at 930 µm, corresponding to cc(shifted red0{930 µm}, green_i{930 µm}). At 931 µm stage position, cc(shifted red0{930 µm}, green_i{931 µm}) is larger than cc(shifted red0{930 µm}, green_i{930 µm}), showing that the second cc peak has not been reached, and red0 remains at 930 µm.

At 931 µm, cc(red0{930 µm}, red_i{931 µm}) crosses the cc(red0, red_i) curve at 0.84 µm from its peak. The position of this peak coincides with the position of red0, here at 930 µm. As a results, the position determined by the curve fitting at 931 µm stage reading is 930.84 µm, which shows a 0.16 µm deviation from the stage reading. This process is the same for the stage position from 931 to 962 µm.

At 963 µm, cc(shifted red0{930 µm}, green_i{963 µm}) is smaller than cc(shifted red{930 µm}, green_i{962 µm}). This shows that the sample has shifted the amount equal to the beams' offset (about 32 µm), and the first correlation peak, as described in Fig. 4, has been reached. At this point, red0 is replaced with red_i at 962 µm and digitally shifted. Then, cc(red0{962 µm}, red_i{963 µm}) defines a horizontal line that crosses the cc(red0, red_i) curve at 1.94 µm from its peak at 962 µm. This means the position determined by the curve fitting is 963.94 µm, which shows a 0.94 µm deviation from the stage reading. Again, the process is the same up to 994 µm, where the second correlation peak happens, and at 995 µm, red0 is replaced with red_i at 994 µm and digitally shifted. Repeating this process from 930 to 1130 µm at every 1 µm results in 0.5 µm average absolute deviation from the stage reading and 0.8 µm std. Figure 15 shows the linear relationship between the stage position and the sample position determined using the dual wavelength method. Ideally, the slope of the best fit line should be equal to 1 and the y-intercept should be 0. Uncertainty in measurement causes the deviation of these parameters from the ideal values.

 

Fig. 15 Stage position vs. sample position determined using the dual wavelength method.

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Here, the dual wavelength method is only evaluated in one direction by monitoring cc(red0,green_i). Displacement measurement in the opposite direction is possible by considering cc(green0,red_i) at the same time. Before moving toward the opposite direction, green0 should be replaced with green_i at every step until the direction changes. This is when cc(green0,red_i) increases. When the sample shifts in the opposite direction, green0 is only replaced at every correlation peak, while red0 is replaced at every step until the direction changes one more time.

4. Summary and future work

This paper introduces a robust displacement scale based on speckle pattern correlation. Relative displacement measurement is possible by illuminating the sample with two parallel, overlapping laser beams with an identical spot size but different wavelengths (red and green), and monitoring the correlation between the red and the green patterns. The results show that over a 200 µm uniform range of a ground steel sample with 0.2 µm S_q, relative displacement measurement is possible with 1 µm resolution, where the average absolute deviation from the stage reading for 200 data-points is 0.5 µm with 0.8 µm std. This method does not require a database of speckle patterns, which makes it robust to environmental disturbances. Because the speckle patterns are captured under the same conditions, most imaging errors do not affect the results. The accuracy of measurement mainly depends on the stability of the beams' offset.

Unwanted stage motion in the other in-plane axis, which is perpendicular to the displacement direction, decreases the correlation peak and the peak separation. If this unwanted displacement is larger than the speckle size, the correlation peak cannot be reached. The setup is less sensitive to the out-of-plane motions of the sample. One reason is that the speckle size is larger in this direction [13, 24]. Also, due to the large focal length of the imaging lens (f = 100 mm), the effect of the out-of-plane motions on the magnification is not significant. Roll and pitch can introduce error to the system, because they can alter the shape of the curve used for curve fitting. The quantitative evaluation of these parameters is among the future work.

The method introduced in this paper requires high correlation between two speckle patterns with different wavelengths. This is possible if the roughness of the sample is smaller than the wavelength (R_q ranging from 0.1 to 1 µm). Under this condition, the statistics of speckle patterns including the speckle size become roughness dependant [25]. As a results, the accuracy of the proposed method depends on the uniformity of the surface roughness. This paper evaluates the dual wavelength method over 200 µm due to a limited access to a surface with uniform roughness over a longer range. Generating a suitable surface is possible using methods such as lapping and polishing. The future work aims at evaluating the method over longer ranges. Also, using a surface with a different R_q may require recalibration of the system. This might seem to be a limitation for this method. However, it is possible to use the dual wavelength method to create a displacement scale for machine tools and coordinate measuring machines, or to apply it to displacement stages for alignment and positioning purposes. In these applications there is no need to frequently change the length scale, and the system requires calibration only once. The future work also aims at quantifying and decreasing the sensitivity of the system to surface roughness using a modified imaging system or by redesigning the method to use a single wavelength.

The sampling rate used here (one speckle pattern per 1 µm) limits the measurement resolution to 1 µm. Increasing the sampling rate to sub-micrometer steps will increase the resolution. Higher sampling rate also allows averaging the data at the correlation peak and determining its position more precisely. The limitations of the reference scale used here does not justify increasing the measurement resolution. The current method is also capable of measuring continuous displacement of the sample. For continuous displacement, the speed of the scale depends on the resolution of measurement and the speed of the camera. For example, for 1 µm resolution, the camera should capture a pattern at least every 1 µm. If the speed of the camera is 1000 fps, the measurement speed will be 1 mm/s. Increasing the resolution will decrease the measurement speed if increasing the camera speed is not possible. The camera pixel size ultimately limits the resolution of the method.

Expanding the dual wavelength technique to two dimensions is possible by using three wavelengths (red, green, and blue) to illuminate the sample in an arrangement shown in Fig. 16. The red and the green spot are used for displacement measurement in x direction, while the blue and the green spot are used for that in y direction.

 

Fig. 16 Arrangement of beam spots for two dimensional relative displacement measurement

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Other than expanding this technique to two dimensions, the future work also aims at enabling velocity measurement and increasing the compactness of the setup. This can make the dual wavelength scale a valuable tool for industrial displacement measurement.