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Abstract
This paper explores the potential of optical-based systems, specifically pseudo-non-diffractive beams, as an alternative for alignment. The study focuses on structured laser beams and hollow structured laser beams, which exhibit lower divergence and enhanced detection capabilities. The research objective is to analyze and compare centroiding algorithms in terms of accuracy and robustness to noise. The study compares the gamma-corrected and threshold-corrected center of gravity and correlation template matching. It also introduces a polarization-based algorithm.
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1. INTRODUCTION
A multi-point alignment of particle accelerator components is a challenging task due to the need for accuracy of tens of micrometers over hundreds of meters. The fundamental approach of these systems involves measuring the offset relative to a reference line. At CERN, the European Organization for Nuclear Research, line reference systems such as a wire positioning system combined with a hydrostatic leveling system were developed to meet tight accuracy needs [1]. While these systems exhibit high precision, they also have limitations, such as implementation complexities and component costs. Using an optical-based system as a reference line serves as a potential alternative. Several optical-based systems for aligning structures over long distances have been presented in various works [2–6]. One drawback of these systems is the relatively high divergence of optical systems caused by diffraction, which makes the straight-line reference measurement more challenging over long distances.
A laser light path is typically assumed to be a straight line. Systems with laser beams propagating in a vacuum, to minimize the effect of refraction caused by air fluctuations, have been developed for several accelerators. The Japanese High Energy Accelerator Research Organization (KEK) employs mechanical shutters with quadrant-photodetectors to detect the laser propagating inside a vacuum tube [2]. The estimated accuracy is 100 µm over 500 m. However, the quadrant-photodetectors have a drawback: if there is an irregularity in the laser spot shape, the centroid detection accuracy can be strongly affected [7].
The Compact Linear Collider (CLIC) [8] is a proposed accelerator that is currently under study at CERN. The project has very tight alignment tolerances of 10 µm ($1\sigma$) over 200 m. A LAMBDA sensor was developed for this task by Stern [9]. The sensor’s principle involves projecting a Gaussian beam (GB) propagating through a vacuum on a plate. The plate is placed on a mechanical shutter that can obstruct the path of the laser. The projection of the GB is then viewed with a camera. The center is calculated and measured using Gaussian function fitting and photogrammetry techniques. This system suffers from several drawbacks. After propagating over several tens of meters, most of the beam intensity is lost, and the ability to precisely detect the center is compromised due to the divergence. Another source of error arises from the photogrammetry technique used, which induces field deformation of the captured image due to lens aberrations. The correction of the field deformation requires additional calculation steps.
A structured laser beam (SLB) and hollow structured laser beam (HSLB) are pseudo-non-diffractive beams with a transversal intensity profile similar to a Bessel beam (BB) of a zeroth and first or higher orders, respectively [10–13]. They share the properties of the BB, such as lower divergence of the central core compared to a GB and the possibility to regenerate behind an obstacle. The generation of such beams relies on optical aberrations of the optical elements used. More about the generation principle and parameter assessment can be found in our previous works [14–17] and in the paper by Herman [18]. Propagation over 900 m was recently tested with the divergence of the central core under 0.01 mrad. Even after propagating for several hundred meters, the central core of the beam can fit within a reasonably sized CMOS or CCD chip, enabling direct image capture from the chip without the need for additional optics. Hence, there is a potential to minimize the drawbacks of the LAMBDA sensor and other optical systems. This makes a pseudo-non-diffractive beam-based straight-line reference system a promising candidate for the alignment of future colliders at CERN. These include the currently studied CLIC or Future Circular Collider (FCC) [19].
There are several reports on using pseudo-non-diffractive beams for alignment. The BB has been used for the alignment of lenses by Parks [7] where the center of gravity is used with a threshold to detect the centroid of the beam and comments on the disadvantage of using quadrant-photodetector for the BB centroid detection. Gale detects the center by observing a crossed wire with a projection of the BB to align large structures over 19 m [20]. None of these works provides a quantitative, in-depth analysis of algorithms that can be used to detect the BB, SLB, or HSLB centroid. Bridging this knowledge gap is critical for quantifying the error introduced by the algorithm in the alignment system.
Center of gravity and correlation template matching algorithms are well known, and their accuracy has already been studied for different applications such as Shack–Hartmann sensing and image processing [21–23]. Nevertheless, these algorithms have never been studied when applied to centroid detection of pseudo-non-diffractive beams. This work aims to provide an in-depth analysis of the accuracy of such algorithms on simulated data. Additionally, a novel method for the detection of the HSLB centroid using knowledge of the polarization distribution is presented. This is made possible by the generation principle of the HSLB that includes the non-classically polarized field of the input beam through radial, azimuthal, or spiral polarization. The resulting data give a perspective on the potential of pseudo-non-diffractive beams, particularly the SLB and HSLB, for alignment applications. Furthermore, the study highlights the pros and cons of the algorithms for centroid detection in specific conditions.
2. METHODOLOGY
A. Weighted Center of Gravity Algorithm
The weighted center of gravity (WCoG) algorithm, the so-called first moment weighted by intensity, is not computationally expensive and reaches sub-pixel accuracy. The centroid coordinates $\bar x$ and $\bar y$ are calculated as
The preprocessing of an image can reduce noise in the image and increase the accuracy of WCoG algorithms. This paper used thresholding and gamma correction, referred to as the TCoG and GCoG, respectively.
The TCoG algorithm uses the thresholding function $T(x,y)$, which sets the pixel value to zero when its intensity is lower than the threshold level $t$ after background subtraction:
In the case of GCoG, a non-linear relationship between pixel intensity levels using function ${G_{{\rm corr}}}$ was introduced. Hence, low-intensity values are lowered, and high-intensity values are enhanced:
As the value of the gamma parameter $\gamma$ increases, the curvature and the gradient of the curve become more significant, hence the more prominent the relative difference between the low- and high-intensity levels. In other words, higher $\gamma$ values make the bright parts even brighter and the dark parts even darker, emphasizing the differences. Lower $\gamma$ values have the opposite effect, making the image appear more uniform in intensity. Hence, in the gamma-corrected image, only the brightest pixels of the inner core will be visible [24].
B. Correlation Template Matching Algorithm
The correlation template matching algorithm (CORR) calculates a normalized 2D correlation function $C(u,v)$ between a squared 2D Gaussian function template $t(x,y)$ and the intensity distribution [25]:
Here $\bar t$ is the mean of the template intensity and $\overline {{I_{u,v}}}$ is the mean of the intensity distribution in the region under the feature. The input parameters of the algorithm are the size of the side of the squared template ${S_c}$ and the standard deviation of the Gaussian function ${{\rm STD}_c}$, which affects the shape of the function. A Gaussian interpolation is applied to obtain sub-pixel results. An in-depth explanation of the sub-pixel extraction and the formula can be found in the paper from Li et al. [23]. Note that for the HSLB, the Gaussian template has inverted intensity values to match the HSLB core.
C. Polarization Algorithm
If one assumes a radially polarized input beam to produce the HSLB, a polarimetric centroid detection algorithm (POL) developed in this paper can be used. Polarimetric measurement allows Stokes parameters to be obtained [26]. Knowing these parameters, a polarization ellipse orientation angle $\psi$ can be determined; see Fig. 1. For each interval (${\psi _i},{\psi _{i + 1}}$) spaced by $\frac{\pi}{{180}}$ rad, pixel coordinates $({x_p},{y_p})$ were found with the corresponding value of the polarization ellipse orientation angle:
Fig. 1. S0, S1, S2, and S3 represent normalized Stokes parameters of HSLB with radial polarization distribution. $\psi$ represents orientation angle of polarization ellipses.
Every set of ${P_i}$ coordinates was fitted with the linear equation
To preserve the stability of evaluation, the coefficient of determination ${R^2}$ was required to be higher than 0.996. Only fitted lines in the following interval are considered to fulfill those conditions:
The centroid coordinates $\bar x$ and $\bar y$ of the HSLB are found as the intersection of fitted lines fulfilling the condition in Eq. (8). The number of lines ${p_{{\rm lines}}}$ can be altered by changing the spacing of the interval.
D. Simulation Parameters
A comprehensive analysis of centroiding algorithms on simulated data is presented. The data used for the analysis were generated through numerical simulations utilizing custom-developed software. The simulation involved tracing rays through the generator, consisting of a 5 mm ball lens made of S-LAH79 Ohara glass with a refractive index of approximately two for a wavelength of 632.8 nm, which is the wavelength of the laser source used. A plano–convex projection lens, constructed with Schott N-BK7 glass, was utilized, having a diameter of 25.4 mm and a radius of curvature of 25.8 mm. The distance between the lenses was set to be ${L_{{\rm PL}}} = 96.8\;{\rm mm} $. The input field of the SLB was linearly polarized, while for the HSLB, it was radially polarized. The scheme of a generator can be seen in Fig. 2.
For long-distance applications, where the beam can travel hundreds of meters, detecting the centroid position at multiple points, each at a different distance, is necessary. Hence, it is crucial to evaluate the performance of the algorithms for beams with various core sizes due to divergence. A numerical calculation of a Fresnel diffraction integral was performed for the traced field directly behind the generator and propagated to distances from 2 to 102 m with a step size of 2 m to generate intensity distributions in different distances. This resulted in 51 intensity distributions that were subjected to subsequent analysis. The virtual camera chip had dimensions of ${10} \times {10}\;{\rm mm}$, with ${2001} \times {2001}$ pixels (px). The intensity distributions with the corresponding diagonal line intensity profiles and core sizes at 102 m can be seen in Fig. 3. For the SLB, the core size was defined as the radius of the first dark circle surrounding the central core. For the HSLB, it is the radius of the most intense part of the first ring.
Fig. 3. (a) Simulated SLB intensity distribution at 102 m. (b) Simulated HSLB intensity distribution at 102 m. (c) SLB’s diagonal line intensity profile with the core size definition indicated in red. (d) HSLB’s diagonal intensity line profile with the core size definition indicated in red.
A rectangular region of interest (ROI) measuring ${1501} \times {1501}\;{\rm px}$ was extracted from each simulated intensity distribution. The corresponding position of the centroid for both the ROI and the beam was then established. Subsequently, a sub-pixel shift of 0.31 px was introduced in both axes for the ROI centroid using bicubic interpolation to displace the beam and ROI centroid relatively. This means that the position of the beam centroid was never precisely in the middle of the pixel. Hence, the symmetry did not artificially improve the results. Applying these measures accounted for real measurement conditions. Consequently, the centroid coordinates for both axes were adjusted to 751.31 px. Shifting the beam to different positions within the ROI was tested without affecting the overall results.
The algorithms underwent various testing methodologies, including evaluations of accuracy, time span, robustness to intensity noise, and sensitivity to parameters such as $t$, $\gamma$, ${{\rm STD}_c}$, ${S_c}$, ${p_{{\rm lines}}}$. All parameter values were determined based on empirical observations derived from previous experiments.
The accuracy assessment involved defining an absolute error (AE) as the absolute difference between the known position of the beam centroid and its corresponding position detected by the algorithm. A mean absolute error (MAE) with a standard deviation of the AE values (${{\rm STD}_e}$) was calculated by averaging the absolute error values across all core sizes in both axes to obtain a measure of overall accuracy. Analyzing the time span entailed modifying the region of interest (ROI) size and evaluating the accuracy and time required to analyze a single intensity distribution by each algorithm. This process is referred to as windowing. The noise was introduced by adding Gaussian white noise, which was employed to analyze the noise robustness of the algorithms, characterized by a mean value and a variance denoted as ${{\rm var}_{{\rm noise}}}$. During actual measurements, the signal-to-noise ratio (SNR) values are high, meaning that the beam is easily visible due to the possibility of controlling the laboratory condition in our particular case. The values of ${{\rm var}_{{\rm noise}}}$ were chosen based on the empirical observation and are therefore relatively low. Note that there will be no investigation of algorithm failure due to noise since this requires more than an order of magnitude higher ${{\rm var}_{{\rm noise}}}$ values. The sensitivity to parameters $t$, $\gamma$, ${{\rm STD}_c}$, ${S_c}$, ${p_{{\rm lines}}}$ was compared by examining the mean and standard deviation of detected center positions (${{\rm STD}_p}$) with different parameter values.
3. RESULTS
A. Effect of the Core Size
In Figs. 4(a) and 4(b), the absolute error (AE) of the SLB and HSLB centroid detection for different algorithms and various sizes of the central core is presented. The values of the parameters $t$, $\gamma$, ${{\rm STD}_c}$ along with the mean absolute error (MAE) for the SLB algorithm can be found in Fig. 4(c). Similar values for the HSLB, including ${p_{{\rm lines}}}$, are provided in Fig. 4(d). It is crucial to note that ${S_c}$ was set to be equal to the core size. The image data used for analysis were free of noise.
Fig. 4. (a) AE of the algorithms detecting the centroid of the SLB in noiseless intensity distributions with the pixel size of a core. The change of the core size is caused by the increasing camera distance from 2 to 102 m. Zoom in on GCoG and CORR. (b) AE of the algorithms detecting the centroid of the HSLB in noiseless intensity distributions with the pixel size of a core caused by the camera distance varying from 2 to 102 m. Zoom in on GCoG, CORR, and POL. (C) MAE and $t$, $\gamma$, ${{\rm STD}_c}$ parameters used for SLB centroid detection. (D) MAE and $t$, $\gamma$, ${{\rm STD}_c}$, ${p_{{\rm lines}}}$ parameters used for HSLB centroid detection.
The AE generally goes down as the core size increases for all of the algorithms. The TCoG algorithm has a linearly decreasing trend with the MAE values for both beams in the order of hundredths of a pixel. The CORR is exponentially decreasing for both beams together with the GCoG for the SLB. The AE for CORR remains below 1e-2 px for a core size of 50 px and reaches the order of 1e-6 px for larger core sizes in both beams. The values repeat themselves for the GCoG in the case of the SLB. For the HSLB, the GCoG, and the POL, the AE has a slightly linearly increasing trend with core size with the MAE equal to 1.24e-4 and 1.51e-3 px, respectively.
For short-distance alignment applications, it is crucial to ensure that, due to the small size of the core, the resolution is adequate for the required level of accuracy. It is worth noting that the GCoG algorithm seems to have the lowest AE values for beams with core sizes smaller than 150 px for both beams and has an overall best performance for the SLB. The best-performing algorithm for the HSLB seems to be the CORR.
B. Windowing and Time Span Analysis
The AE of the SLB and HSLB centroid detection for a single intensity distribution at a distance of 102 m is shown for different sizes of ROI. The ROI sizes ranged from 15 to 750 px, while the core size of the SLB was fixed at 1056 and 811 px for HSLB. The parameters $t$, $\gamma$, ${{\rm STD}_c}$, ${p_{{\rm lines}}}$ used are similar to the ones in Figs. 4(c) and 4(d). If the ROI size was smaller than the core size, the parameter ${S_c}$ was set to be two pixels smaller. The image analysis was conducted on noiseless data.
For the SLB, as presented in Fig. 5(a), the AE decreases as the ROI size increases for all algorithms before hitting a plateau. The TCoG algorithm exhibits an AE in the order of tenths of a pixel, which drops rapidly after the ROI size reaches 200 px after which AE remains in the order of hundredths of a pixel. The GCoG algorithm demonstrates that the AE is in the order of tenths of a pixel until the ROI size exceeds 220 px where it reaches the AE order of 1e-6 px after hitting the plateau. The AE of the CORR algorithm is only affected the most among the algorithms by the changing ROI size and hits the plateau after 320 px.
Fig. 5. (a) AE of the algorithms detecting the centroid of the SLB in noiseless intensity distributions with ROI size varying from 15 to 750 px. (b) Time span of algorithms for different sizes of the ROI.
Fig. 6. (a) AE of the algorithms detecting the centroid of the HSLB in noiseless intensity distributions with ROI size varying from 15 to 750 px. (b) Time span of algorithms for different sizes of the ROI.
The time span analysis, shown in Fig. 5(b), indicates that the time spent calculating the centroid increases with the ROI size for all three algorithms. For ROI sizes smaller than 200 px, the time span of the CORR algorithm is below 0.05 s, below 5e-3 s for the GCoG algorithm, and below 5e-4 s for the TCoG algorithm. For the 750 px ROI, the AE of the CORR algorithm exceeds 1.64 s. For the GCoG, it reaches 0.13 s, while for TCoG, it goes up to 0.05 s.
Similar to the SLB analysis, the AE for HSLB decreases with increasing ROI size for all algorithms, as seen in Fig. 6(a). The AE of the TCoG algorithm reaches up to 0.95 px and slowly falls until it drops rapidly after the ROI size reaches 641 px. Afterward, the AE gets to 4.97e-3 px. The AE of the GCoG algorithm follows a similar trend, starting at 1.85 px, falling to 2.17e-4 px after the ROI size is equal to 691 px. The AE of the CORR algorithm starts at 0.29 px, reaching a plateau after the ROI size is equal to 400 px. The POL algorithm is almost unaffected by the ROI size and has the MAE value of 2.11e-3.
The time span analysis, shown in Fig. 6(b), exhibits similar trends and values for TCoG, GCoG, and CORR as observed in the SLB analysis. The POL algorithm is slower compared to the other algorithms. The time span starts at 1.10 s for an ROI size of 15 px and increases to 5.90 s.
The algorithms are susceptible to small ROI sizes, which may result in a higher AE due to the exclusion of a significant portion of the central core from the analysis. The ROI size can be reduced without sacrificing the accuracy of each algorithm by analyzing the point when the curve reaches the plateau phase. Otherwise, there will be a trade-off between the time and accuracy.
It is important to note that lowering the value of the ${p_{{\rm lines}}}$ parameter can also decrease the time span for the POL algorithm, but this change may increase the AE. Further analysis of this aspect will be conducted in the subsequent section. The results also suggest that the beam center can be larger than the chip size. However, the image’s contrast is significantly reduced, and the centroid position analysis can be challenging during an actual measurement.
C. Sensitivity to the Change of Algorithm Parameters
To determine the optimal algorithm parameter and evaluate its sensitivity to variations, a comparison between the mean detected centroid along the x-axis and the mean reference value, accompanied by the standard deviation ${{\rm STD}_c}$ was conducted. A Gaussian white noise mask with a mean of zero and a variance of 1e-4 was introduced, which was then applied to intensity distributions of beams with varying core sizes. This method ensures uniformity in the noise distribution across all images, emphasizing the algorithm parameter’s sensitivity in the outcomes rather than susceptibility to the inherent randomness of the noise. The known reference position of the centroid along the x-axis was established at 751.31 px.
Figures 7 and 9 depict the AE values corresponding to various core sizes. Additionally, Figs. 8 and 10 furnish the mean and standard deviation (${{\rm STD}_c}$) values for the detected x-coordinates of SLB and HSLB, respectively. In the case of the TCoG algorithm, the parameter $t$ was calculated as $t = {c_{\!t}} \cdot {I_{{\max}}}$, with values of ${c_{\!t}}$ being 0.65, 0.7, and 0.75 for SLB, and 0.4, 0.5, and 0.6 for HSLB. As for the GCoG algorithm, the parameter $\gamma$ was consistently set to 3, 4, and 5 for both types of beams. The template size $d{S_c}$ was varied by $\gamma \pm 4\;{\rm px}$, and ${{\rm STD}_c}$ was established at 50, 100, and 150 for both SLB and HSLB.
Fig. 7. Detected x-coordinate positions of the SLB for the beams with different core sizes with different values of the algorithm parameters. (a) TCoG with changing ${c_{\!t}}$, (b) GCoG with changing $\gamma$, (c) CORR with changing $d{S_c}$, and (d) CORR with changing ${{\rm STD}_c}$.
Fig. 8. Mean and STDp values of the detected x-coordinates for different values of the algorithm parameters. (a) TCoG with changing ${c_{\!t}}$, (b) GCoG with changing $\gamma$, (c) CORR with changing $d{S_c}$, and (d) CORR with changing ${{\rm STD}_c}$.
Fig. 9. Detected x-coordinate positions of the HSLB for the beams with different core sizes with different values of the algorithm parameters. (a) TCoG with changing ${c_{\!t}}$, (b) GCoG with changing $\gamma$, (c) CORR with changing $d{S_c}$, (d) CORR with changing ${{\rm STD}_c}$, and (e) POL with changing ${p_{{\rm lines}}}$.
Fig. 10. Mean and STDp values of the detected x-coordinates for different values of the algorithm parameters. (a) TCoG with changing ${c_{\!t}}$, (b) GCoG with changing $\gamma$, (c) CORR with changing $d{S_c}$, (d) CORR with changing ${{\rm STD}_c}$, and (e) POL with changing ${p_{{\rm lines}}}$.
For the SLB, the mean closest to the reference value for the TCoG was achieved with ${c_{\!t}} = 0.7$, resulting in a difference of the mean equal to 1.04e-3 px relative to the reference value, together with the smallest ${{\rm STD}_p}$ relative to the other values, equal to 2.03e-2 px. The mean closest to the reference value for the GCoG, with a difference of 1.01e-3 px and with the smallest ${{\rm STD}_p}$ of 5.94e-4 px, were both observed for $\gamma = 4$. For the CORR algorithm, the most precise results were obtained with the precisely measured core size together with ${{\rm STD}_c} = 100$, with a difference of the mean equal to 1.05e-3 px and the ${{\rm STD}_p}$ of 9.03e-3 px.
For the HSLB, the mean closest to the reference value for the TCoG, with a difference of 1.24e-3 px, was observed for ${c_{\!t}} = 0.4$, along with the smallest ${{\rm STD}_p}$ of 1.20e-3 px. For the GCoG algorithm, the mean with a difference of 1.12e-3 px, and the smallest ${{\rm STD}_p}$ of 9.20e-3 px were obtained with $\gamma = 4$. The most precise results for the CORR algorithm were obtained with the precisely measured core size, with a difference of the mean equal to 1.02e-3 px and the ${{\rm STD}_p}$ of 1.10e-2 px using ${{\rm STD}_c} = 100$. Increasing the ${p_{{\rm lines}}}$ parameter for the POL algorithm improves accuracy and reduces the ${{\rm STD}_p}$. When ${p_{{\rm lines}}} = 50$, the mean and reference value difference is 1.02e-3 px, with an ${{\rm STD}_p}$ value of 1.70e-2 px.
Using a sub-optimal parameter value for each algorithm can lead to elevated AE and a reduction in overall robustness. It is observed that the CoG algorithms appear to exhibit greater sensitivity to the parameter selection compared to the CORR algorithm for both types of beams, with the exception of the GCoG, which demonstrates the least susceptibility to the choice of the parameter in the case of the SLB. Regarding the POL algorithm, a reduction in the number of ${p_{{\rm lines}}}$ enhances the algorithm’s speed, offering advantages in high-speed applications. The time span of the algorithm was linearly increasing with the number of lines used from 1.2 to 5.9 s. However, this also results in an increase in AE. Using ${p_{{\rm lines}}} = 50$ yielded satisfactory results in terms of the AE.
D. Noise Influence
The impact of noise was investigated by systematically varying the ${{\rm var}_{{\rm noise}}}$ of Gaussian white noise while keeping the mean value at zero. Twenty intensity distributions were analyzed, and the mean of the AE was taken for each core size to mitigate the random effects of noise. The error bars are not shown in the graphs, as the standard deviation of the data was lower by one order of magnitude.
For the SLB, the results of TCoG, GCoG, and CORR algorithms with ${{\rm var}_{{\rm noise}}}$ values of 1e-4, 5e-4, and 10e-4 can be observed in Fig. 11. The mean absolute error (MAE) and standard deviation of error (STDe) values are presented in Fig. 12. As ${{\rm var}_{{\rm noise}}}$ increases, both MAE and STDe show an upward trend for all algorithms. This behavior is also evident in the graphs as the curves shift towards higher absolute error (AE) values and exhibit increased oscillations. The MAE is generally in the order of 1e-2 px for all algorithms. For core sizes larger than 500 px, CORR exhibits similar MAE values to GCoG. However, for smaller core sizes, both AE and oscillations increase significantly, especially for ${{\rm var}_{{\rm noise}}} = 10{\rm e} {\text -} 4$, where AE exceeds 0.1 px for several ROI sizes, representing the highest value among all algorithms.
Fig. 11. AE of the (a) TCoG, (b) GCoG, and (c) CORR detecting the centroid of the noised SLB intensity distributions with the ${{\rm var}_{{\rm noise}}}$ equal to 1e-4, 5e-4, and 10e-4 px and the size of the core in pixels caused by the camera distance increase from 2 to 102 m.
Fig. 12. MAE and STDe values for the (a) TCoG, (b) GCoG, and (c) CORR detecting the centroid of the noised SLB intensity distributions with the ${{\rm var}_{{\rm noise}}}$ equal to 1e-4, 5e-4, and 10e-4 px.
For HSLB, the results of TCoG, GCoG, and CORR algorithms with ${{\rm var}_{{\rm noise}}}$ values of 1e-4, 5e-4, and 10e-4 can be seen in Fig. 13. The MAE and STDe values are presented in Fig. 14. Like SLB, increasing ${{\rm var}_{{\rm noise}}}$ leads to higher MAE and STDe, as evident from the graphs with curves shifting towards higher AE values and increased oscillations. The MAE is generally in the order of 1e-1 px for all algorithms, except for GCoG, which achieves a minimum of 3.91e-3 px for ${{\rm var}_{{\rm noise}}} = 1{\rm e} {\text -} 4$, 7.92e-3 px for ${{\rm var}_{{\rm noise}}} = 5{\rm e} {\text -} 4$, and for CORR, which achieves a minimum of 5.32e-2 px for ${{\rm var}_{{\rm noise}}} = 1{\rm e} {\text -} 4$. Except for CORR, all algorithms perform significantly worse for small core sizes, particularly with ${{\rm var}_{{\rm noise}}}$ values of 5e-4 and 10e-4. In these cases, there is an exponential decrease in performance with core size.
Fig. 13. AE of the (a) TCoG, (b) GCoG, (c) CORR, and (d) POL for detected centroid positions of the noised HSLB intensity distributions with the ${{\rm var}_{{\rm noise}}}$ equal to 1e-4, 5e-4, and 10e-4 px and the size of the core in pixels caused by the camera distance increase from 2 to 102 m. Note that the y-axis scale in (d) is three times higher.
Fig. 14. MAE and STDe values for the (a) TCoG, (b) GCoG, (c) CORR, and (d) POL detecting the centroid of the noised HSLB intensity distributions with the ${{\rm var}_{{\rm noise}}}$ equal to 1e-4, 5e-4, and 10e-4 px.
In the majority of cases, algorithms have lower AE for bigger core sizes. The results indicate that for SLB, GCoG is the least affected by noise influence, while CORR performs as well as GCoG for core sizes larger than 500 px. TCoG is the most susceptible to noise among the algorithms. For HSLB, all algorithms, except CORR, exhibit significantly poorer performance for small core sizes. This makes CORR the most consistent algorithm, GCoG the least sensitive to noise, and the POL algorithm the most affected by noise.
4. DISCUSSION
The results indicate that the GCoG algorithm is the least affected by algorithm parameter choice and exhibits the highest overall accuracy for SLB. It also demonstrates robustness to noise. For the HSLB, the GCoG algorithm is relatively more sensitive to the choice of the $\gamma$ parameter. When selected appropriately, it remains the least susceptible to noise, except for core sizes smaller than 30 px. In this case, the CORR algorithm performs better, although its performance is slightly worse for larger core sizes. For SLB, the CORR algorithm shows comparable accuracy to GCoG for core sizes larger than 500 px but performs significantly worse for small core sizes. Additionally, the CORR algorithm consistently exhibits robustness to the choice of ${S_c}$ and ${{\rm STD}_c}$ parameters for both beams.
In comparison, the TCoG algorithm performs with lower accuracy than the GCoG and CORR for both beam types and is also sensitive to the choice of the $t$ parameter. It exhibits significantly poorer performance for HSLBs with small core sizes, especially for ${{\rm var}_{{\rm noise}}}$ equal to 10e-4. The POL algorithm is the most noise-sensitive and yields the lowest accuracy among the algorithms, particularly for small core sizes.
The CoG-based algorithms are considerably faster than CORR and POL, especially when working with high-resolution images. However, the speed of CORR and POL can be increased through windowing without compromising algorithm accuracy. The speed of the CORR algorithm can be enhanced to match the level of CoG-based algorithms. On the other hand, the POL algorithm exhibits significantly lower speed even for small ROI sizes. The speed of CoG-based algorithms can also be improved by windowing but requires a sufficiently large window size to maintain accuracy.
Overall, the centroid-detection-based algorithm errors are approximately one order of magnitude smaller for the SLB than the HSLB when considering noise. The induced error for SLB ranges around 1e-2 px for most algorithms and noise values, while for the HSLB, it is in the order of 1e-1 px. This discrepancy may be attributed to the higher contrast between the core and rings in SLB. CoG-based algorithms only utilize the core of the SLB for detection and rely on the first bright ring of the HSLB, resulting in a smaller area and reduced sensitivity to intensity noise. Even though the POL algorithm performs worst relative to the others, the results suggest that it can be successfully employed in low-noise conditions, where the polarization analysis for centroid detection is favorable, for example, when only part of the beam is visible on the chip or where the intensity is not uniform but where the polarization distribution is preserved.
These results indicate the potential of an SLB and an HSLB for long-distance accelerator alignment, where the required accuracy is typically in the micrometer or tens of micrometer range. It suggests that centroiding algorithms will not be the primary or decisive source of error in the case of relatively high-SNR applications such as accelerator alignment, where the measurement conditions can be controlled.
Funding
Technická Univerzita v Liberci (SGS-2023-3399); CERN (http://dx.doi.org/10.13039/100012470).
Acknowledgment
Portions of this work were presented at the 16th International Workshop on Accelerator Alignment (IWAA 2022), Detection of Structured Laser Beam Centroid and its Use for Alignment. The authors acknowledge the financial support provided by the Knowledge Transfer group at CERN through the KT Fund.
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.
REFERENCES
1. H. Durand Mainaud, S. B. Jimenez, T. Dijoud, et al., “HL-LHC alignment requirements and associated solutions,” in Proceedings 8th International Particle Accelerator Conference (IPAC) (2017), pp. 1893–1896.
2. T. Suwada, M. Satoh, S. Telada, et al., “Propagation and stability characteristics of a 500-m-long laser-based fiducial line for high-precision alignment of long-distance linear accelerators,” Rev. Sci. Instrum. 84, 093302 (2013). [CrossRef]
3. L. V. Griffith, R. F. Schenz, and G. E. Sommargren, “Magnetic alignment and the Poisson alignment reference system,” Rev. Sci. Instrum. 61, 2138–2154 (1990). [CrossRef]
4. D. Kämtner and J. Prenting, “Straight line reference system (SLRS) for the adjustment of the x-ray free-electron laser (XFEL) at DESY,” in 9th International Workshop on Accelerator Alignment, 26 –29 September 2006.
5. M. Beker, G. Bobbink, B. Bouwens, et al., “The Rasnik 3-point optical alignment system,” J. Instrum. 14, P08010 (2019). [CrossRef]
6. C. Zhang, M. Hasegawa, K. Kanda, et al., “Performance of the iris diaphragm laser alignment system of the spring-8,” in Proceedings of 13th International Workshop on Accelerator Alignment, Beijing, China, 2014.
7. R. E. Parks, “Practical considerations for using grating produced Bessel beams for alignment purposes,” Proc. SPIE 11816, 11–18 (2021). [CrossRef]
8. CERN, “Compact linear collider,” https://home.cern/science/accelerators/compact-linear-collider. Accessed 26 August, 2021.
9. G. Stern, “Study and development of a laser based alignment system for the compact linear collider,” Ph.D. thesis (ETH Zurich, 2016).
10. J. J. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]
11. J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988). [CrossRef]
12. S. N. Khonina, N. L. Kazanskiy, S. V. Karpeev, et al., “Bessel beam: significance and applications—a progressive review,” Micromachines 11, 997 (2020). [CrossRef]
13. S. N. Khonina, N. L. Kazanskiy, P. A. Khorin, et al., “Modern types of axicons: new functions and applications,” Sensors 21, 6690 (2021). [CrossRef]
14. J.-C. Gayde and M. Sulc, “An optical system for producing a structured beam,” (2019).
15. J.-C. Gayde, M. Sulc, and K. Polak, “Introduction to structured laser beam for alignment and status of the R&D,” (2022).
16. W. Niewiem, K. Polak, M. Dusek, et al., “Variation of structured laser beam pattern and optimization for an alignment reference line creation,” Opt. Express (to be published).
17. M. Dusek, J.-C. Gayde, and M. Sulc, “Wavefront reconstruction of a non-diffracting structured laser beam,” Opt. Express 31, 42099–42110 (2023). [CrossRef]
18. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991). [CrossRef]
19. A. Abada, M. Abbrescia, S. S. AbdusSalam, et al., “FCC-ee: the lepton collider: future circular collider conceptual design report volume 2,” Eur. Phys. J. Spec. Top. 228, 261–623 (2019). [CrossRef]
20. D. M. Gale, “Visual alignment of mechanical structures using a Bessel beam datum: practical implementation,” Proc. SPIE 8082, 1013–1024 (2011). [CrossRef]
21. A. M. Nightingale and S. V. Gordeyev, “Shack-Hartmann wavefront sensor image analysis: a comparison of centroiding methods and image-processing techniques,” Opt. Eng. 52, 071413 (2013). [CrossRef]
22. H. C. V. Assen, M. Egmont-Petersen, and J. H. C. Reiber, “Accurate object localization in gray level images using the center of gravity measure: accuracy versus precision,” IEEE Trans. Image Process. 11, 1379–1384 (2002). [CrossRef]
23. X. Li, X. Li, and C. Wang, “Improvement of correlation-based centroiding methods for point source Shack–Hartmann wavefront sensor,” Opt. Commun. 411, 187–194 (2018). [CrossRef]
24. S.-C. Huang, F.-C. Cheng, and Y.-S. Chiu, “Efficient contrast enhancement using adaptive gamma correction with weighting distribution,” IEEE Trans. Image Process. 22, 1032–1041 (2012). [CrossRef]
25. J.-C. Yoo and T. H. Han, “Fast normalized cross-correlation,” Circuits Syst. Signal Process. 28, 819–843 (2009). [CrossRef]
26. B. Schaefer, E. Collett, R. Smyth, et al., “Measuring the stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007). [CrossRef]
