1. Introduction

Optical vortices are stable phase singularities within the optical field [1]. They already have been broadly applied in multiple fields, ranging from metrology [2–4], through fundamental interferometry [5], telecommunication [6] and astronomy [7,8], to cryptography [9]. In particular, they already established their importance in the area of optical tweezers [10,11] and Stimulated Emission Depletion (STED) microscopy [12,13], both awarded with the Nobel Prize. Due to some peculiar features of optical vortices, as for instance super-oscillations, they show the extreme sensitivity to any optical field disturbance. Therefore, it makes them especially capable to be applied in any metrology related sub-fields. One of such examples is aberrometry. This, being a general term, typically refers to the efforts of measuring the optical field deviation from its ideal reference. Recently, most of works that incorporated optical vortices for related purposes focused on using them as optical field quality markers, for instance to correct Spatial Light Modulators (SLMs) [14]. Similarly, vortices have been applied to correct optical tweezers setup in an iterative process, where some typical aberrations were eliminated at each step [15]. While it was not a direct outcome, a value of existing aberration could be indirectly determined using the same algorithm or its modified version. Another interesting approach analyzes the splitting of the higher-order optical vortices into single units [16]. In that work, the authors provide some insights into the topological aberration caused by the multiple field reflection but did not measure it directly. We believe that there is still room for an aberrometer that uses optical vortices and their features, which will provide detailed and direct information about the aberration existing within the examined optical setup. The further development of the vortex aberrometer is the overall purpose of this work.

The core of the proposed approach is to use vortex as a scanning tool and therefore examine an optical vortex trajectory - a path that vortex follows when shifted inside the beam. Such shift is caused by the translation of the vortex generating structure with the relation to optical beam. Any disturbance on the optical path leads to a significant modification of its trajectory. This approach, has initially been applied in super-resolution optical vortex microscopy [17,18] and later has been extended to the examination of the quality of the beam shaped by the spatial light modulator [19]. The obtained results paved the way for this technique to be applied in aberrometry, where the vortex trajectory could examine the quality of the optical setup and directly suggest the correction that has to be implemented. Here, we further developed that idea, starting with improving the localization procedure and re-applying it into the extended version of the experiment presented in [19]. Changing not only the localization procedure but also proposing new parameters describing vortex trajectory, that has been tested under two optical aberrations: defocus and astigmatism.

Tracking the optical vortex is a delicate task, especially when performed on experimental images. These will differ across various optical systems, having different dynamic ranges, resolutions, and contrasts. So far, two of the most popular approaches that were applied to internal vortex scanning experiments were based on either interferometric phase retrieval [18,20] or numerical analysis of the intensity images (without the interference) leading to the calculation of the weighted centroid [19]. Nevertheless, there is still room for improvement, especially in optical systems, where the environment can change rapidly during the experiment. The developed optical vortex aberrometer is one such example.

Phase retrieval methods aim to provide the phase of the object beam, containing an optical vortex and then localize the singular vortex point through the analysis of the edge of the 2$\pi$ step, a characteristic feature of the optical vortex phase map. These methods are fast and accurate, especially if supported by neural networks [21]. They can be easily automatized as long as the reference beam remains unchanged. Nevertheless, interference requirement becomes the main limitation, which vastly increases the complexity of the potential vortex aberrometer setup.

On the other hand, the weighted centroid does not require interference and can provide the position of a singular vortex point by just analysis of the vortex intensity image. The weighted centroid algorithm has been applied in our preliminary experiments [19] and worked efficiently for most cases. Nevertheless, when the optical vortex becomes strongly aberrated, its symmetric circular shape becomes distorted, providing a dark intensity leakage - the background annihilates the vortex core. This annihilation becomes a real challenge for the weighted centroid method. Potential automatizing requires an efficient algorithm to determine the weighted centroid threshold. If calculated globally (for the whole data set) increases the analysis time. On the other hand, locally (single data point) leads to inaccurate vortex trajectory.

Therefore, to support our work on designing an optical vortex aberrometer, we implemented the new method, which overcomes these limitations. Initially, the preliminary algorithm was reported by us in the conference proceeding [22], but was not fully automatized up to this point. The method uses the Laguerre-Gaussian (L-G) transform to provide the exact position of the singular point and can be treated as a compromise between interferometric and weighted centroid approaches. In this work, we analyze each of the mentioned vortex localization methods. For most cases, we will compare L-G transform with the weighted centroid for the apparent reason that both methods do not need a reference beam to perform.

Firstly, we introduce the theoretical background of the L-G transform represented as subsequent steps of a vortex point localization algorithm. Then, we analyze the performance of the L-G transform and eventually compare it with other methods. Lastly, we apply the presented algorithm to obtain vortex trajectories under artificially introduced aberrations and study their behavior, proving that the Laguerre-Gaussian transform method is versatile, accurate and easy to apply.

2. Vortex point localization algorithm using the Laguerre-Gaussian transform

The Laguerre-Gaussian transform was proposed to track the displacement of a speckle field [23]. Such a field is well known for containing the numerous phase singularities of irregular shape. The algorithm proposed there allowed precise localization and tracking. The latter was possible due to unique parameters given by the L-G transform, independent of rotation and translation. We extend the idea introduced in [23] to track a single optical vortex when shifted off-axis. In particular, we aim to take advantage of the L-G transform capability to work with irregular singularities, which resemble aberrated optical vortices. Below, we introduce the procedure to calculate the vortex point position.

Hence, the algorithm’s only parameter is the vortex core diameter, represented as the bandwidth $\omega$. This essentially simplifies the whole procedure. We found this algorithm to work efficiently even without any numerical image analysis. We note that the algorithm does not require to align the recorded intensity and the Laguerre-Gaussian function in the Eq. (2). However, the tight clipping, symmetrical to the beam’s center, can improve the algorithm’s performance and increase its speed, leading to less potential singular point locations. In the next section, we examine the performance of the presented algorithm in various experimental scenarios. However, the algorithm itself remains unchanged.

3. Localization performance

3.1 Bandwidth and its impact

As derived in the previous section, the only parameter that has to be controlled is the bandwidth $\omega$, which aims to match the size of the vortex core. It is worth noticing that the core definition is not standardized. For this work, the vortex core is a distance between two peaks, corresponding to the diameter of the dark hollow $\varphi _{core}$ (Fig. 1).

 

Fig. 1. a) The simulated intensity image of the optical vortex b) The intensity profile is taken along the dashed line shown in a) with a definition of a vortex core diameter $\varphi _{core}$

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Having only one parameter to control is both promising and challenging. One can easily automatize it, but it may strongly impact the algorithm’s precision if not correctly determined. Therefore, we analyzed the impact of the bandwidth $\omega$, concerning $\varphi _{core}$, on the calculated vortex position. We determined the position of the vortex point for a series of numerically simulated vortex trajectories with bandwidth as a variable. We shifted the optical vortex along the x-axis, which resulted in the straight movement along the y axis at the image plane. For clarity, the dependence of the y vortex point position on the vortex generating structure shift is presented in Fig. 2.

 

Fig. 2. The trajectory of an optical vortex calculated using various bandwidth determined as the $\%$ of $\varphi _{core}$ The trajectory represents the dependence of the y vortex point position when shifted off-axis.

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As long as the bandwidth $\omega$, is defined concerning $\varphi _{core}$ these results are independent of the parameters of numerical simulations. Some deviations from ideal linear dependence are still visible. Nevertheless, the detected positions for $\omega \in [10\%\varphi _{core},100\%\varphi _{core}]$, almost overlap, differing by less than 0.2 [px]. Thus, as long as the chosen $\omega$ is within this range, it has a minor impact on the localization precision, hence all detected positions are almost identical. The maximum error, defined as a deviation from the simulated position, was equal to 3.5 [px], while the STD of the deviation from the ideal trajectory was less than 1 [px]. These numerical artifacts are caused by a lack of proper definition of vortex position in the image plane. We expected a straight vortex trajectory based on analytical results, and this linear dependence is taken as a reference (ideal/simulated) example. According to our simulations, the detected position is strongly aberrated when the chosen $\omega$ value exceeds the $[10\%\varphi _{core},100\%\varphi _{core}]$ range. For the reference, the detected positions for $\omega =1250\%\varphi _{core}$ is included.

Hence, the only parameter that has to be chosen - bandwidth, can be automatically estimated and, even if misestimated, still leaves a high possibility to provide correct results. This stays in contrast to any method based on weighted centroid, where the threshold misestimation can have a remarkable impact on the vortex localization precision, especially when the vortex beam is highly distorted. This can happen especially for the highly aberrated beam, where the vortex will lose its ideal circular symmetry. As has been mentioned, the wide range $\omega$ values are acceptable. So even if the beam is disturbed (large aberrations) the proper bandwidth determination is still possible. In such a case, the bandwidth $\omega$ can be determined in relation to the beam diameter $\varphi _{beam}$. The user can estimate the $\varphi _{core}=0.4 \varphi _{beam}$. Certainly, the vortex should be placed on-axis (not shifted) to achieve the highest beam symmetry.

3.2 Localization in the strongly distorted beam

The ability to process and precisely localize the position of the vortex point even when distorted is crucial for the optical vortex aberrometer to work accurately. Hence, any aberration will quickly destroy the vortex circular symmetry. Thus, we compared the efficiency of two discussed methods: weighted centroid and L-G transform. An example of such a case is presented in Fig. 3 where three different images (200x200px each) representing off-axis optical vortex in three different scenarios are shown. These images were taken in the same experimental setup, during the same measurement.

 

Fig. 3. Positions of the vortex point determined through the weighted centroid marked by $x$ and the L-G method marked by $o$ for the a) Non-aberrated off-axis optical vortex. The difference between localized vortex position within both methods is less than 2px for each case. b) Aberrated off-axis optical vortex with a visible dark intensity leakage, characteristic for the aberrated vortex beam scenario. The determined vortex position differ by 3px for the first column and 6px for the second one. Insets present the intensity distribution in a grayscale.

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The position of the vortex point on each image was calculated using both the weighted centroid and L-G methods, both having their boundary parameters fixed, which means that the same parameters have been applied to calculate each image. For the weighted centroid, the threshold was determined using the Otsu method [27,28]. Then, for the L-G transform, the bandwidth $\omega =70\%\varphi _{core}$ was chosen. As presented in Fig. 3, the weighted centroid determined the vortex point position correctly for most examined cases. However, it led to wrong results when the beam was highly distorted ((Fig. 3(b)). This is caused by the dark intensity leakage, which is visible. On the other hand, the L-G method did not have a problem with the precise localization of the vortex point on the same image (Fig. 3(b)).

The proposed method arises from speckle metrology, which has already proved its efficiency in determining random phase singularities’ position. It should be noted that each image was slightly saturated only for visual purposes. An image saturation may give the weighted centroid method a slight advantage when the vortex shifted off-axis, reducing the dark intensity leakage. Additionally, we repeated the localization procedure, but with adaptive thresholding, [29] - calculated for each image separately. The vortex position for highly distorted vortices was still far from accurate.

The main reason for the advantage of the L-G transform method is an a priori assumption that an optical vortex exists in the beam. Therefore, it will try to locate and track the fictional vortex position for the bandwidth that matches the pseudo vortex core diameter. Therefore, we did not recommend the application of our algorithm to beams without phase singularities.

3.3 Contrast dependence

We examined the performance of the proposed algorithm in the low-contrast regime. To do so, we registered a series of images of the optical vortex with modified contrast. The contrast was controlled by the rotation of the linear polarizer which was responsible for the output intensity. Then, we detected the vortex position using the algorithm based on the L-G transform proposed in previous sections for each registered image. For each image, the boundary parameter remained unchanged, so that $\omega =40 [px]$. We repeated this process for three different scenarios presented in Fig. 4. Non aberrated optical vortex (Fig. 4(a)), optical vortex with manually introduced astigmatism (Fig. 4(b)), and finally, the same aberrated optical vortex, but shifted slightly off-axis (Fig. 4(c)).

 

Fig. 4. Localization of the vortex point using L-G transform method in low contrast regime. a) Almost ideal optical vortex b) Aberrated optical vortex (manually introduced astigmatism) c) Aberrated optical vortex (manually introduced astigmatism) shifted off-axis. The contrast value is placed in the top right corner of each image. For the visual purposes, the original grayscale image registered by the camera is placed as an inset in the bottom left corner.

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Each image has a size of 160x160 [px]. The calculated Root Mean Square of vortex point distance from its C=1 position is equal $RMS=(2[px], 2 [px], 3.5 [px])$ for subsequent a), b) and c) cases of Fig. 4. The main source of this error was beam stability. Therefore, we registered a series of 60 images each taken every 1s, which spanned over 60s. The same optical vortex localization procedure followed this procedure. Analogically, we calculated the Root Mean Square of vortex point distance from its original position. The calculated $RMS=1.2 [px]$ represents the fluctuation of the optical vortex center caused by mechanical vibrations and air fluctuations.

This analysis confirms a high precision even in the low-contrast. For most cases, the detected vortex point position remained within the sub-pixel distance from its reference position, if the stabilization error is considered. We were able to retrieve the position of the vortex for the contrast as low as C=0.16 for the non aberrated optical vortex (not shown), C=0.28 for the aberrated optical vortex on-axis (Fig. 4(b)), and C=0.23 for the aberrated optical vortex off-axis (Fig. 4(c)). In this analysis, we did not take into account any errors arising from the improper determination of bandwidth $\omega$, since the chosen value remained unchanged for each of the evaluated images.

3.4 Vortex trajectory

Finally, we would like to compare the L-G method with other methods, those that we have been using for vortex trajectory purposes up to this point. We run the experiment in the optical vortex scanning microscope setup [18]. The optical vortex was introduced into the beam and shifted off-axis in the x-direction (scan). The observation plane was placed close to the Fourier plane. Thus the movement of the vortex point was perpendicular to its off-axis shift, and the vortex moved along the y-axis. This relation is caused by the orbital angular momentum carried out by an optical vortex and has been studied both analytically and experimentally [17,18]. We run two such scans with and without the reference beam, registering the image either with or without interference. Images were registered after each step so that the whole trajectory would be well represented. Then, working on the same data (assuming that a separate scan with a reference arm provided the same vortex trajectory), we calculated the vortex position at each step independently, with 3 different methods: Fourier transform phase retrieval [20,30], weighted centroid [31], and finally, L-G transform. All trajectories are presented in Fig. 5. We do not provide detailed experimental parameters because that experiment is out of the scope of this work. The purpose of bringing these results is to provide a simple comparison between various methods applied to vortex trajectory retrieval in the same experimental environment.

 

Fig. 5. The trajectory of an optical vortex is calculated by 3 independent methods: Phase retrieval method (black circles), L-G transform (orange circles), and Weighted centroid (blue circles). The trajectory represents the dependence of the y vortex point position when shifted off-axis. The approximate position of the vortex in a particular section is visible on the inserted images.

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These trajectories were expected to provide a linear relation with the vortex generating structure shift for the correctly aligned optical system. The interferometric approach (phase retrieval) provides a quantized trajectory with a resolution of 1 [px]. This effect has been previously reported in [20] and depends on the system magnification. Nevertheless, the sub-pixel resolution is achievable when combined with artificial neural networks [21]. Two of the remaining methods: weighted centroid and L-G transform, provided relatively similar results of vortex trajectory, both providing a sub-pixel resolution. The maximum difference between the detected vortex point positions was equal to 4 [px], corresponding to approximately 8 [$\mu$m] at the CCD camera. Moreover, when a singular point was in the close vicinity of the beam center, the position determined through the L-G method was similar to the one determined by the phase retrieval method. Such accuracy in the central area is of significant importance, which acts in favor of L-G over the weighted centroid method.

Not less important is the computation time, required for each. The weighted centroid was 40$\%$ faster then L-G transform, which for the images of size 600x600 [px] corresponds to approximately 0.09 [s] vs 0.15 [s] per image. Given times strongly depend on the resolution of the images and the computer parameters. Nevertheless, when considering that the boundary parameters for the L-G transform can be used repetitively for a whole series of images taken in the same conditions, the time required to process a single image reduces to 0.04 [s]. Thus, retrieving the whole vortex trajectory, consisting of 200 images, took 10.27 [s] for the L-G transform method, which was about 40$\%$ faster than the weighted centroid (18 [s]).

Optical vortex localization through Laguerre-Gaussian transform offers high sub-pixel precision, simultaneously avoiding numerical errors. The risk of wrongly calculated vortex point position, associated with improper determination of the vortex core diameter $\varphi _{core}$ or, more directly, the bandwidth $\omega$, is relatively low. This fact makes this method especially interesting if the final goal is related to automatizing the whole process. More importantly, for the chosen $\omega$, the algorithm can provide the vortex position even when the beam is highly aberrated, and the vortex core leaks out of the beam. The computation time is comparable to the weighted centroid. Finally, the discussed method is also sufficient when applied to low contrast images without losing localization precision.

One of the critical features of the optical vortex aberrometer is its versatility. Such an algorithm should be easily applied in every system that uses either Liquid Crystals Spatial Light Modulator, Digital Micro-mirror Device, or any other beam shaping device. This sets up high post-processing requirements so that any localization method must offer similar performance no matter the experimental conditions. The proposed algorithm meets these criteria. Therefore, we report the first experimental results of vortex trajectories in the next section, obtained with the proposed localization algorithm.

4. Experimental verification

The experimental setup of the optical vortex aberrometer is based on the 4f optical system (Fig. 6). The Holoeye PLUTO LC-R2500 Spatial Light Modulator (SLM) is illuminated by the expanded He-Ne laser beam $\lambda =632 [nm]$. Polarization of the illuminating beam is rotated by the $\lambda /2$ waveplate (HWP) to match the SLM’s liquid crystals director and achieve high phase modulation. Additionally, the neutral density filter (ND Filter) controls the output intensity. The SLM is responsible for optical vortex shifting through the mathematical translation of the vortex generating hologram. The hologram is later superposed with diffraction grating to redirect the modulated light into the first order of diffraction. The 4f optical system consisted of an imaging lens of $f=150 [mm]$, aperture, and 10x microscopic objective spatially filters the beam with embedded vortex and images the beam on the CCD camera.

 

Fig. 6. The scheme of the optical vortex aberrometer experimental setup.

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In general, such a simple setup can be the source of multiple misalignment errors. Therefore, to reduce their impact, we applied optimization procedures, reported in [19]. Each experiment was composed of a series of vortex scans in both perpendicular directions (x and y). By vortex scan, we denote a series of hologram translations in the chosen direction The whole optical setup remained unchanged, the hologram translation was a purely mathematical operation. Thus, one scan consisted of 70 translations with 2 px steps in between. This corresponds to $39 \mu m$ increments across the $2660 \mu m$ range. The vortex intensity distribution at each step was registered by the CCD camera and further localized using the presented localization algorithm. The goal of the experiment was to examine vortex trajectories under particular optical aberrations - defocus and astigmatism. These two are ones of the main sources of optical setup misalignment [15]. Each of them can be artificially introduced by superposing the aberration hologram with the one being responsible for vortex generation and shifting. Thus, the evaluation of vortex trajectories under specific aberration was possible.

An aberration is represented by the Zernike polynomial coefficient, defined as $Z^0_2=2r^2-1$ in case of defocus and $Z^{2}_2=r^2cos(2/theta)$ in case of astigmatism. Both have been defined across an aperture of radius $R=5.13mm$. The coefficient value was changed from $-0.25\lambda$ to $0.25\lambda$ with non uniform steps: $-0.25\lambda$, $-0.15\lambda$, $-0.10\lambda$, $-0.05\lambda$, $0$, $0.05\lambda$, $0.10\lambda$, $0.15\lambda$, $0.25\lambda$.

Figure 7 presents the calculated vortex trajectories for three exemplary $Z^0_2$ and $Z^2_2$ values. Both share the same unaberrated trajectories depicted in the center. Additionally, the system’s stability was examined by the measurement of the vortex point fluctuation. The RMS of vortex position based on around 60 independent vortex images taken within the 60s time range was equal to $0.3px$, which corresponds to $2.4\mu$ at CCD camera. This error is mainly influenced by air fluctuations but can also be impacted by mechanical vibrations and numerical errors.

 

Fig. 7. Experimental vortex trajectories determined through the vortex localization algorithm based on the Laguerre-Gaussian transform. Each trajectory represents two vortex scans in both x and y direction (at the SLM plane) under artificially introduced defocus $Z^0_2$ and astigmatism $Z^2_2$. Subfigures present exemplary experimental results obtained for a) $Z^0_2=-0.25\lambda$ b) Without aberration (see Visualization 1 and Visualization 2) c) $Z^0_2=0.25\lambda$ d) $Z^2_2=-0.25$ e) $Z^2_2=0.25$

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The vortex trajectories are linear, no matter the introduced aberration. The linear regression returned the $R^2$ coefficient above $0.9$ for each analyzed case. Figure 7(a) and Fig. 7(c) show vortex trajectories, which rotate with the shift of the observation plane. It is worth noting that the single vortex trajectory is perpendicular to the shift of the vortex hologram when observed at the focal plane (Fig. 7(b)). The observation plane was unchanged in the current experiment, but the artificially introduced defocus shifted the focal plane along the z-axis, resulting in a similar effect - trajectories rotation. In contrast to the defocus, astigmatism caused two trajectories to incline towards each other (Fig. 7(d), f). These are in agreement with the results reported in [19]. Both x and y trajectories rotate in the opposite direction, which makes a similar case, reported therein.

To analyze trajectories rotation, we introduced two parameters, $\alpha$, and $\beta$, that determine the rotation angles of trajectories for x and y vortex scans, respectively. Each angle uses the same reference axis. The perfect, unaberrated vortex trajectory should produce $\alpha =90^o$ and $\beta =0^o$. The evolution of both $\alpha$ and $\beta$ under the defocus and the astigmatism coefficient value is presented in Fig. 8.

 

Fig. 8. a) The dependence of the x scan trajectory’s inclination $\alpha$ on the coefficient value b) The dependence of the y scan trajectory’s inclination $\beta$ on the coefficient value. Sub-figure’s insets present the way that both angles $\alpha$ and $\beta$ were calculated

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Defocus caused both x and y trajectories to rotate in the same direction. Both angles of rotation showed linear dependence under the introduced aberration strength. Such linear behavior of trajectories inclination has been studied analytically in [17]. It persists within the close vicinity of the optical system focal plane. This range corresponds to the small defocus coefficient value in the current experiment, which shifts the focal plane backward or forward, depending on the chosen coefficient sign. Therefore, the vortex trajectory behavior is confirmed by the experimental results.

Astigmatism, on the other hand, forced both trajectories to incline towards each other. This inclination can be seen both in Fig. 7(d,f) as well as in Fig. 8. The latter presents the evolution of angles $\alpha$ and $\beta$ under the introduced aberration strength. The x scan (vertical trajectory) inclined in the opposite direction than in the case of defocus. The angle $\alpha$ increased together with the astigmatism value, despite the same coefficient values. The angle $\beta$, determining the inclination of y scan (horizontal trajectory), decreased with the increase of the astigmatism coefficient value, presenting a similar, but less rapid tendency as in the case of defocus.

Furthermore, such analysis provides important information on the optical system quality. The unaberrated beam (coefficient value = 0), as seen in Fig. 8(a) returned $\alpha =86^o$. On the other hand, when the introduced defocus was equal to $Z^0_2=0.05\lambda$, the $\alpha =91^o$ which is very close to the theoretical value for the perfect, unaberrated beam ($90^o$). Similarly, the ideal $\alpha =90^o$ was provided by the $Z^2_2=0.1\lambda$. Analogically, unaberrated beam returned $\beta =2^o$, which also leaves room for further improvement, suggesting that the small defocus ranging between $Z^0_2=0$ and $Z^0_2=0.05\lambda$ exists within the optical system (Fig. 8(b)), together with possible astigmatism $Z^2_2=0.1\lambda$. This result show the optical vortex aberrometer’s potential in sensing the source of beam imperfection. However, at the current stage, the direct determination of Zernike coefficient values requires further analytical and experimental efforts.

Nevertheless, the improvement in vortex localization accuracy and it’s repeatably, paves the way to differentiate between the most common sources of imperfections in the typical optical system - defocus and astigmatism. Still, future efforts on developing the optical vortex aberrometer should further focus on exploring the vortex trajectory behavior on various aberrations introduced in the optical system. With the newly developed localization procedure, the more complex aberrations can be automatically analyzed, which should vastly improve the development of the presented concept of optical vortex aberrometer. We believe, that even at this point, the presented concept and early results of optical vortex aberrometer can be useful for the scientific society. For this reason, we made the proposed localization algorithm publicly available. It can be sufficiently used to evaluate optical vortex trajectory, supporting the aligning procedure of demanding optical systems and providing insights into two of the most common optical aberrations: defocus and astigmatism.

5. Conclusion

The overall goal of this work is to support our efforts to develop an optical vortex aberrometer, which incorporates optical vortex as a beam scanner, revealing any imperfections existing within the optical beam. We applied the L-G transform to track the position of the singular vortex point inside the beam and compared its performance with two other tracking methods - phase retrieval and weighted centroid.

We further examined its performance in the actual experimental scenario with different aberrations, which have been artificially introduced to optical beam. The obtained vortex trajectories and calculated parameters show that the algorithm can efficiently determine vortex trajectories and differentiate between two the most common types of aberrations - defocus and astigmatism of strength as low as $0.05\lambda$.

One of the main requirements that has been set for the optical vortex aberrometer is its versatility. It has to be easily applicable to any system that uses any spatial light modulator, either liquid crystal-based or digital micro-mirrors. Then, the boundary parameters have to be easily determined, paving the way for sufficient automatizing, even when the beam is highly distorted - a typical scenario for the strongly aberrated beams. Similarly, such method should be applicable to the low-contrast regime. Last but not least, the algorithm has to be accurate, providing the vortex position with sub-pixel precision. Ideally, these requirements would not impact the computation time. In this context, we compared each of the analyzed method. A brief summary of such comparison is presented in Table 1.

Table 1. Comparison of examined vortex localization algorithms.

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The obvious disadvantage of the phase retrieval (interferometric) method is the necessity to implement the reference beam. Even though it can lead to sub-pixel accuracy, the range of its application is limited. On the other hand, weighted centroid and L-G transform methods act directly on the vortex intensity distribution. The first one is a well-established but very general technique, which requires an individual approach. Its accuracy depends on numerical image processing, which can lead to incorrect vortex position if done carelessly. This becomes a real issue in the case of a highly aberrated beam due to the intensity leakage.

On the other hand, the L-G transform method relies on a single and objective parameter - bandwidth, that is determined in reference to the beam diameter. Moreover, our study has shown, that even if chosen incorrectly does not lead to major differences in vortex point positions. In the analysed case, the miscalculation error remains in the subpixel range for the $\omega \in [10\%\varphi _{core},100\%\varphi _{core}]$. This feature becomes more useful in the case of a highly distorted beam because no modification of the bandwidth is needed. Therefore, the user can use it to analyze the same data set, even in the low-contrast regime. This method does not require post-processing and tight image cropping, although the latter improves the computation time.

Even though we applied this method to track a single charge optical vortex, it can track a complex set of such singularities. However, the final vortex position criterion must be modified (step 8 of the algorithm). Similarly, one can apply this method to track higher-order optical vortices. We underline that they are highly sensitive and tend to split into a set of single charge singularities in the experimental environment. These all singularities will be detected by the proposed algorithm. In such a scenario, one can estimate the centroid of the whole set, which behaves as a single charge beam [32]. The code of the proposed algorithm is publicly available [33].