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Abstract
Remote positioning by precise measurements of lateral displacements of laser beams at large distances is inevitably disturbed by the influence of atmospheric turbulences. Here we propose the use of optical vortices, which exhibit lower transversal variations at an experimentally validated range of 100 meters. We show the higher precision of the localization of vortex points as compared with standard centroid-based assessment of Gaussian beams. Numerical simulations and experimental measurements show further improvements by averaging of the positions of up to four secondary vortices forming a stable constellation when higher values of the topological charges are used.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser beams with optical vortices (vortex beams) attract scientists’ attention because of their unique optical features. The characteristic feature of vortex beam is its helical wavefront geometry, which means that in the transverse cross-section the equiphase lines converge to a single point (a vortex point). When going along a closed path around this point at the given observation plane, the phase undergoes 2πm changes, where m is called topological charge [1–3]. Since the phase at the vortex point is ambiguous/undefined, the point is dark, i.e. the light intensity is zero. Thus, the vortex beam spot has a doughnut like shape. Due to helical geometry of the wavefront the vortex beam possess non-zero angular orbital momentum [4–5], which means that there are stable phase structures inside the light field. Hence, the vortex point is a well-defined and stable dark point at the observation plane (or line in space) with a characteristic phase distribution in its neighborhood.
Several approaches have been made to use optical vortices in optical measurements. Having well-defined and stable singular points, they can serve as position markers and can be used in measurement of displacement [6–8] or deformation of the object wave in interferometry [9–10]. Some research concerning laser vortex beams is connected with their propagation over a large distance in atmospheric conditions [11–14]. Investigations made in this area show that a vortex beam offers additional benefits over a pure Gaussian beam, being more resistant to random inhomogeneities in the atmospheric refractive index. The goal of these works was to apply the vortex beam for evaluating the state of the atmosphere, which can be applied for example for improving adaptive optics methods, or improving the free-space optical telecommunication. We have focused on the other task. The question is if the vortex beam (i.e. beam carrying an optical vortex) can be used for improving the beam positioning. We can find the beam deviation by observing at the distant digital camera the displacements of a vortex point (due to atmospheric turbulence). This way of evaluating the necessary correction should be easier than observing the entire laser spot with no characteristic points inside. In this paper, we extend our preliminary work presented in [15]. A laser beam carrying a single vortex beam (m=1) or higher-order vortex beam (m=4) is propagated over a distance of 100 m (numerically and experimentally). In the case of a single vortex beam, its position can be determined by tracing the vortex point within the recorded beam spot. We show that a vortex beam is more resistant to beam degradation caused by atmospheric turbulence and thus its position can be found with better accuracy compared to a Gaussian beam. When high-order (m=4) vortex beam propagates, it splits into a constellation of four single charged vortices. Being shifted from the beam center, each vortex in the constellation “feels” the local disturbance within the beam. As a result, each vortex is shifted locally in a different way. Using this fact we can trace the vortices separately, find the change in the vortex constellation, and then position the beam using the constellation centroid. Hence we can think of measuring the intensity of local atmospheric variations and global ones simultaneously. When the vortex constellation geometry is changing, the atmosphere is highly turbulent and one has to be careful with the evaluation of position. When the entire vortex constellation behaves like a rigid body the state of the atmosphere is calmer. In the paper we show that a vortex constellation offers better accuracy in beam positioning than a Gaussian and single-charged vortex beam. Numerical modeling is confirmed by the experimental results.
2. Methodology
A laser beam having a Gaussian profile and the waist w0, traveling over a large distance in free space is diverged by the angle θ=λ/(πw0), where λ is a wavelength. When the beam encounters turbulence, some additional effects may occur [16]. Two scales of the turbulence eddies are usually considered. The eddies that are large compared to the beam diameter cause the beam deflection. The eddies that are small compared to the beam diameter tend to broaden the beam and change its symmetry. Consequently, if we observe the beam in a plane situated at a certain distance from the random medium, we will see beam wandering (caused by deflection) and at a shorter observation time – local beam deformation (caused by the random and local changes of the atmospheric refractive index).
If we want to use a laser beam for positioning over a larger distance, then it needs to be focused to make the spot smaller. Such a spot in the case of the ideal lens will have a size $d = \frac{{2.44 \cdot \lambda \cdot f}}{D}$, where f is a focal length and D is lens diameter. In the presence of aberrations, this spot size is enlarged. When we need to improve beam localization, we can use a laser beam with the embedded single optical vortex. Such a beam carries a characteristic point (vortex point, where the phase is singular) that can be accurately localized. Thus, the vortex point can serve as a position marker. If the vortex beam passes through the turbulent media, its symmetry is also broken. This will cause the change of the vortex position in the transverse beam section [17]. We propose to prescribe a more sophisticated internal phase structure within the beam in the form of the vortex constellation consisting of four vortices. Such a constellation can be generated as a beam a with basic optical vortex with a topological charge (m=4). Due to instability, it breaks into four single vortices, under influence of small phase disturbances. (Figure 1(a)) [18–20]. Being separated from the beam center, each vortex in the constellation will experience different turbulence (on a small scale) which should result in a deformation of the constellation shape. In order to determine the vortex points’ positions, various algorithms have been implemented and tested. We present them below.
Fig. 1. a) Exemplary intensity image (experiment) of the vortex beam (case m=4); b) four dark areas from Fig. 1(a) and their surroundings; c) pseudophase maps from the areas corresponding to b); d) vortex points positions(circles) found by the artificial neural network.
In the case of Gaussian beam (vortex free), its position can be determined by Center of Gravity (CoG) algorithm [21]. If we have a two-dimensional image I(x,y) with the discretized intensity values at coordinate Xij=(xi,yj) where i and j number the image pixels, then the centroid position (xc, yc) is given by:
Centroid detection becomes problematic in the presence of turbulences, which randomly deforms the beam shape and intensity distribution.In our case we localize the position of vortex point. Here variations of the whole beam spot play secondary role, but the question of vortex point localization method plays the key role. We adopted the known method based on vortex filtering. The detected intensity signal $I(x,y)$ was transformed to its complex analytic representation $\bar{I}(x,y)$ through the vortex (spiral) filtering [22,23]. This relation can be written in the form:
where $\phi (x,y)$ is called the pseudophase, ${\otimes}$ denotes the convolution operation and FV(x, y) is the Fourier transform of the vortex filter, which in spatial domain can be written as:Exemplary results showing the operation of this filter for m=4 are shown in Fig. 1. As we mentioned earlier, the higher-order vortex splits into the constellation of four single-charged vortices. Figure 1(a) shows the exemplary detected intensity image. In Fig. 1(b) the four vortex points and their surroundings are presented separately. The convolution (2) is performed on these separated intensity images using a numerically generated vortex filter (Eq. (2)). In result, we get a complex signal from which phase term can be extracted (pseudophase). Pseudophase maps, for each spot (Fig. 1(b)) are presented in Fig. 1(c). A specially trained artificial neural network [24] has been applied to these phase-maps. This allows us to localize vortex point positions with sub-pixel precision. In Fig. 1(a) the vortex positions are also plotted (circles). Having localized the vortex positions in the constellation, its centroid can be calculated (indicated by the cross in Fig. 1(a)). This centroid determines the beam position.
3. Numerical modelling of beam deformation from turbulences
To model the real situation we assumed that the laser beam is transmitted over a distance of one hundred meters. Any turbulence in the environment (caused by the refractive index fluctuations) in such distant propagation would affect the beam displacement and its intensity distribution variations. Refractive index fluctuations, which are commonly referred to as atmospheric turbulence, are caused by temperature fluctuations. Different parts of the laser beam passing through atmospheric turbulence will experience various phase shifts, which will result in beam deformations. To calculate the influence of the air turbulence on the laser beam, the series of a special masks must first be defined.
The power spectrum of the refractive index fluctuations that matches atmospheric turbulence [16,25] can be represented by:
To model the wave propagation through the medium, we can divide the extended random medium into slabs of thickness Δz, perpendicular to the direction of propagation. Refractive index irregularities are represented by the thin phase screens, each introducing independent random contribution to the wave phase. Thus the medium can be modeled by uniformly spaced phase screens ${\varphi _S}$. The relation between the phase spectrum of each screen and the refractive index power spectrum ${\Psi _n}$ is [26,27]:
In this work, we study the propagation of the vortex beam with m=1 and m=4 and compared it with the Gaussian beam under the same turbulence conditions. In order to perform the simulation the structure constant $C_n^2 = {10^{ - 14}}{\textrm{m}^{ - {\textstyle{2 \over 3}}}}$ was taken which corresponds to the conventional Kolmogorov turbulent index of refraction structure parameter and the wavelength of the laser beam was 0.633 µm.
The scheme of the beam propagation simulation is shown in Fig. 2. Propagation distance of 100 m was divided into 5 equally spaced distances. For each propagation distance, a random screen is generated, as described above. For a given initial laser beam, the beam is propagated from plane to plane using the scalar diffraction theory. At each plane the random phase mask representing accumulated turbulence multiplies the obtained optical field and then propagation to the next screen is continued. The simulations were repeated 100 times to simulate time-dependent changes detected on the CCD. Figure 3(a) illustrates the exemplary intensity distribution of the Gaussian beam after propagating through the turbulent medium. The beam position was determined by CoG method (see the previous section). The beam positions for 100 different random turbulence phase masks are shown in Fig. 3(d). We can observe here a scattering of results equal to dozen of pixels with the standard deviation equal to 13 pixels.
Fig. 3. Simulation of the Gaussian and vortex (m=1) beam propagation through the turbulent medium; a) Gaussian beam intensity distribution ; b) vortex beam intensity distribution; c) vortex beam phase map; d) centroid position (Gaussian beam) found for 100 simulations; standard deviation along OX and OY axis are sdX=12.8 and sdY=12.5 pixels respectively e) vortex point positions for 100 simulations (sdX=7.0 px; sdY=8.1 px);
Figure 3(b) and 3(c) present the vortex beam (m=1) with the same disturbance as in the case of the Gaussian beam. Here the beam shape (Fig. 3(b)) and also its phase map is deformed (Fig. 3(c)). Additionally one can observe the displacement of the vortex within the beam, caused by the local phase disturbances. The beam position was determined by the localization of the vortex point (see the Methodology Section) and the results are presented in Fig. 3(e). The dispersion of results is about 8 pixels which is smaller value than in the case of a Gaussian beam. Thus, localizing the beam position by inspecting the vortex point is more resistant to beam deformation.
Figure 4 presents the higher-order vortex beam (m=4) with the same disturbance as in the case of the Gaussian beam. Figure 4(a) shows the exemplary intensity and Fig. 4(b) the exemplary phase maps. Here also the beam shape and phase map are deformed. Additionally, the relative distances between vortices were changed. It is caused by the fact, that locally each vortex undergoes different phase shifts. As a result, each vortex is shifted to a different position. We can trace each vortex in the beam and find its positions (according to procedures described in Section 2).
Fig. 4. Simulation of the vortex beam propagation (m=4) through the turbulent medium, the same as in Fig. 3; a) intensity distribution; b) phase maps.
The results are shown in Fig. 5(a). Different vortices in the constellation are indicated by different colors. To each constellation a centroid can be assigned (black crosses), which determines the beam position. One can observe, that particular vortices in the constellations are dispersed by more than six pixels, however the centroid position is more resistant to the turbulence than single vortices and is equal to about 2 pixels along x and y directions. The deviation of the centroid position in the case of m=4 is smaller than in case of m=1. In Fig. 5(b) vortices in constellations are plotted for the centroid set at the common point. We can determine here a reference shape of the beam constellation and look how it is deformed under different turbulences (Fig. 5(c)).
Fig. 5. Simulation of the vortex beam propagation (m=4) through the turbulent medium; a) vortex point positions for 100 different turbulent propagations; b) reference shape of the vortex constellation; c) change of vortex constellation shape under different turbulences.
As was shown in Fig. 2, five phase screens were used in the simulations. There is no simple answer to the question of what number of screens should be used. Table 1 collects different simulations result for various screen numbers from 2 to 7. The clear conclusion is that more screens mean more disturbed beam. By increasing the number of screens one can mimic a more turbulent atmosphere. An interesting conclusion is that for large turbulence (7 screens) the vortex centroid sdX and sdY parameters for m=4 case are smaller than for two screens in the case of the Gaussian beam. This is due to the fact, that large turbulence cause local variations of intensity inside the light spot. The same is in the case of any single vortex, for which the sdX and sdY parameters are about 8 pixels. But these random jumps average into a smaller deviation of the centroid.
Table 1. Simulation results for different numbers of screens
In the next section, we show that experimental results confirm the character of results obtained by numerical simulations. However, we cannot claim that our numerical model reflects exactly the states of the atmosphere during the experiments. That was not the main goal. The phase masks used in numerical simulations were designed according to the well-known procedures described in the literature. That means they generate results at least similar to real atmospheric turbulence. With numerical simulation, we can control all beam parameters (like vortex positions) in a very precise way. Thus, when analyzing the calculated images with the method developed for the experiment we can check its accuracy. Our results confirmed the value of the proposed method for vortex points localization.
4. Experimental results
We verified conclusions obtained in the numerical simulation by the experiment. The experimental setup is presented in Fig. 6. A stabilized diode laser (λ=635 nm) with the Gaussian beam profile illuminated the spiral phase plate. The initial transverse size of the beam was 7 mm. The plate introduced into the beam the optical vortex with topological charge m=1 and m=4. The beam was observed at a far distance (100 m) at the CCD camera (pixel size 5.86 µm). During the propagation, the beam was diverged having the size of 30 mm. A lens of aperture 50 mm and focal length 200 mm was used to focus the beam on the CCD camera. In the case of a phase plate with m=1 one dark spot within the beam was observed. As was mentioned in the previous paragraph, the higher-order vortex beam (m=4) is unstable during propagation and splits into the constellation of four single-charged vortices of the same sign. The CCD camera registered the video (25 fps, 8 seconds duration) from which the separated frames have been extracted to analyze the results. The measurements were performed in a closed space with a calm atmosphere, which means low turbulence. To make turbulence larger, a bowl with hot water was put under the laser beam at a distance of 10 m from the CCD camera. So, two sets of data were recorded: the case of low turbulence and enlarged turbulence. The beam position on the extracted frames was determined according to the algorithms described in Section 2.
Fig. 6. Scheme of experimental set-up. Two different phase plates (FP) were used that introduced different vortex structure into the beam (m=1, m=4). Propagation distance between the laser and CCD camera was 100 m.
The results of the experiment were analyzed in the same way as it was shown in Section 3. Figure 7(a) illustrates the exemplary intensity distribution of the Gaussian beam, detected at a distance of 100 m, for different turbulences. Figure 7(b) and 7(c) show the calculated beam localization for both cases. Turbulence deforms the intensity distribution and changes beam localization. In the case of larger turbulence, the standard deviation is of the order of 40-50 px which is much greater than in the case of low turbulence (with the standard deviation equal to about 4 px).
Fig. 7. Experimental results in case of the Gaussian beam a) different beam shapes affected by the turbulence; b) calculated beam position for low turbulence (sdX=3.7px, stdY=3.8px), c) calculated beam position with large turbulence (sdX = 36 px, sdY=50 px)
The detected intensity distribution of the vortex beam (m=1) is shown in Fig. 8(a). We can observe here the intensity deformation but also a shift of the vortex point within the beam. Beam positioning here is less dispersive in both cases, with a clear improvement when the vortex beam passed through the medium with enlarged turbulence. The standard deviation of beam positioning, in this case, was a few times smaller and equal to 17 pixels.
Fig. 8. Experimental results in case of vortex beam with topological charge m=1; a) different beam shapes due to the turbulence; b) beam position for low turbulence;(sdX=1.7 px, sdY=1.8 px) c) beam position for enlarged turbulence (sdX=17 px, sdY=15 px)
Results concerning high-order vortex beam are shown in Fig. 9. Looking at the intensity distribution figures (Fig. 9(a)) we can confirm the numerical finding, that the geometry of the vortex constellation changes upon the turbulence. This is visible in Fig. 10, when we plotted particular vortices at the common centroid position. In the case of low turbulence, the vortex constellation behaves like a rigid body (Fig. 10(a)), while under enlarged turbulence particular vortices deviate from their original positions (Fig. 10(b)). Dispersion of the beam localization, indicated by constellation centroid, in case of low turbulence is similar to the vortex m=1 beam (Fig. 9(b)). However, when the beam passed through larger turbulence, the constellation centroid deviates by a lesser amount (Fig. 9(c)) compared to m=1 case. The experimental results confirm the earlier assumptions and numerical simulations.
Fig. 9. Experimental results in the case of vortex beam with topological charge m=4; a) vortex beam splits into the constellation of four single-charged vortices, local turbulences affects the constellation geometry; b) beam localization for the low turbulence, different colors pink, blue, red and green indicate position of single vortices in the constellation, with the black color the centroid is depicted, dispersion of results for all vortices are shown on the right; c) the same as in b) but with enlarged turbulence
Fig. 10. Vortex constellation (m=4) from Fig. 9 plotted with the centroid at common point; a) low turbulence, b) enlarged turbulence. In case a) one can see that the constellation shape is rigid. This allows for determination of the quadrangle reference shape, plotted in b); c) change of the constellation shape under enlarged turbulences.
5. Discussion
The beam with an optical vortex seems to be more stable in its shape than pure Gaussian beam, which can be noticed when comparing the Figs. 7 with Fig. 8 (experiment) and Fig. 3(d) with Fig. 3(e) (simulations). Here we want to apply the laser beam for measuring how much the laser beam deviates from the straight line while propagating through the atmosphere. The beam centroid is a good measure for the beginning. However, if the beam shape is changing by large factor the beam centroid may lead to larger errors. Better vortex beam stability leads to more trustful measurements. The additional bonus is the presence of vortex point, which can be precisely traced. Hence, our question is what the relation between the perfect straight line trajectory and the vortex point trajectory is?
Our experiments and simulations show that the dark point trajectory is more stable than the centroid of the pure Gaussian beam. The reason for that could be a fact that the vortex beam carries non zero angular momentum which must be preserved in its total value and direction. In the similar way the rotating bullet is less sensitive to air disturbance than the non-rotating one. The similarity between the behavior of vortex trajectories and rigid body has been confirmed in [28,29] in other context. The question if in our case an angular momentum is truly responsible for better beam stability needs more research. Even better results were obtained with higher order vortex beam having topological charge 4. Such a vortex structure splits during propagation into the constellation of four single-charged vortices. After splitting, the constellation of these four charges propagates (assuming perfect conditions) in a characteristic way which can be analyzed by the theory developed in [30–31]. At a short distance behind the generation the situation inside the beam is fixed. In the case of low turbulence the vortices propagate at their fixed relative positions, diverging with the beam and slowly rotating as a rigid body. However, in case of enlarged turbulence things are different. Each vortex passes through a different turbulence area and as a result each vortex is shifted by a different distance. We observe it as a change of a constellation shape. It is possible to trace each vortex separately and as well find the constellation centroid, that can be assigned to position of the beam center. Although the scattering of the individual vortices in the constellation is similar to that in the case of m=1 under the same turbulence, the centroid position itself is more resistant to turbulence than in the case of a single vortex imprinted into the beam. Moreover, by observing the stability of the constellation geometry, we can evaluate the state of the atmosphere. If there are strong local turbulence, the vortex constellation shape will vary in a visible way. This would mean that the accuracy of beam deviation measurements can be of lower quality, unless better theory is developed. On the contrary, if the vortex constellation geometry is stable, but it position is changing we can say that the turbulences have minor local significance and influence on the entire beam. Therefore the position of the constellation centroid is a good measure of beam deviation.
6. Conclusions
The laser beam with the prescribed vortex structure has been proposed and tested for the first time in laser beam positioning. We propagated the laser beam over a distance of 100 m. The vortex structure has been preserved and we were able to detect it on the CCD camera in turbulent conditions. We showed by the numerical simulation and in the experiment that strong refractive index turbulence disturbs the intensity distributions and additionally displaces the vortex within the beam. The vortex shift itself is more resistant than the intensity changes and its scattering is smaller than the scattering of the centroid of the disturbed Gaussian beam. When a higher-order vortex beam propagates, then any phase or amplitude disturbance separates the vortex structure into the characteristic constellation of single-charged vortices. We showed that it is possible to trace the position of these vortices within the distorted beam. Beam location determined by averaging vortices positions shows further improvement compared to low–order (m=1) vortex beam. It is also possible to evaluate the character of atmospheric turbulence. When the geometry of the vortex constellation changes in time the turbulence is high and estimating the variation of laser beam position is less reliable. When the vortex constellation behaves like a rigid body the atmospheric turbulence is low and the laser beam position can be determined more trustfully.
Funding
Narodowe Centrum Badań i Rozwoju (POIR.01.01.01-00-0953/19-00).
Acknowledgement
Sport Vision Technology Sp. z o.o Sp. k. and Szymon Radtke, Sebastian Konkol, Marek Pleskot for valuable contribution and discussion.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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