Robustness of highly complex radial carpet beams in turbulent atmospheres

Abstract

In this study, we report observations of propagating radial carpet beams (RCBs) through a turbulent atmosphere at ground level with a 120 m path length. RCBs are a class of nondiffracting, accelerating, self-healing, and self-amplifying beams, and generated in the diffraction of a plane wave from amplitude/phase radial gratings having different spoke numbers. Observations were made at different times of the day. The intensity profile of an RCB becomes complicated when the number of grating spokes used to generate the beam is large, and includes high intensity spots, called main intensity spots (MISs), which are symmetrically placed at the central area around the beam axis and whose number is equal to (twice) the number of spokes of the amplitude (phase) grating used to generate the beam. With the aid of a telescope and a CCD camera, successive frames of the intensity pattern of the RCBs having different levels of structural complexity are recorded at the end of the path. For the data recorded at different times of the day, we calculate the variance of displacements of MISs along the radial direction. We observe that displacements of the MISs increase with increasing mean temperature of the air; on the other hand, as the complexity of the beam intensity pattern increases, the displacements of the MISs decrease. In order to compare the resilience of different RCBs and a well-known structured beam against atmospheric turbulence, we investigate deformation of the intensity profiles of a Laguerre–Gaussian (LG) beam having a topological charge 20 and different RCBs at the end of the path. It is shown that under the same turbulence condition, highly complex RCBs are more resilient to the destructive effects of the atmospheric turbulence. In particular, for the RCBs generated with gratings having 30 spokes and more, the number of MISs of the received intensity patterns is changed by less than 1% even when the turbulence strength is high. But for the LG beam, its intensity ring is clearly broken in different places, which makes it impossible to follow its maximum intensity in the radial direction

Introduction

The propagation of structured laser beams through a turbulent atmosphere is an important topic in the study of free-space optical communication. Considerations are mostly focused on the opposition of two physical features: The destruction of the beam profile by the turbulence and the self-healing feature of these beams. A beam is known as a structured beam if, at least one of its intensity, phase, and polarization profiles has a complex form. Refractive index fluctuations resulting from atmospheric turbulence clearly disturbs the wavefront of a beam passing through it; a self-healing feature of a structured beam helps it to preserve the wave form under such conditions. Therefore, it is interesting to investigate the interplay between these two characteristics for various structured beams. There have been a number of observations of the propagation of different structured beams through turbulent atmospheres, including Airy beams^1,2, Bessel beams^3,4, Bessel-Gaussian (BG) beams^5,6, Laguerre-Gaussian (LG) beams^7,8, and non-traditional beams like asymmetric Schell-model beams⁹, C-point and V-point beams¹⁰, Hermite non-uniformly correlated beams¹¹, hyperbolic sinusoidal Gaussian (HSG) beams¹².

Radial carpet beams (RCBs) belong to a peculiar class of optical beams, characterized by their unique properties. These intriguing features include their non-diffracting nature, acceleration phenomenon, self-healing capability, and self-amplification mechanism^13,14,15. RCBs exhibit remarkable non-diffractive behavior, maintaining their shape under propagation. Unlike Gaussian beams, which tend to spread out linearly under propagation, RCBs possess continuous descending diffraction angle under propagation, allowing them to propagate with minimal expansion. The propagation paths of all RCB components follow a permanent curvature toward the beam axis, resulting in continuous acceleration as they traverse space. Moreover, researchers have successfully generated these beams without significant expansion over a limited propagation distance^16,17. In addition, RCBs demonstrate inherent resilience to changes in their intensity distribution. When portions of the beam are deleted or modulated due to propagation effects, RCBs spontaneously restore these missing or modulated segments, ensuring their robustness. This inherent property makes RCBs suitable for safe propagation in turbulent environments, such as convective air turbulence¹⁸. The central part of an RCB becomes intensity-free during propagation. The energy from the central region redistributes to the surrounding main parts of the beam, ultimately increasing the overall power of the RCB. Additionally, RCBs are self-amplifying, transferring power to the core area and enhancing their stability. Therefore, radial carpet beams combine non-diffractivity, acceleration, self-healing, and self-amplification-a captivating blend of optical properties that continues to engage researchers in the field of optics and photonics.

RCBs can be generated in the diffraction of a plane wave from an (a) amplitude (phase) radial grating^13,14). In general, the diffraction of a plane wave from a radial structure leads to the production of combined half-integer Bessel-like beams¹⁹. Colorful RCBs are also can be generated in the diffraction of a spatially coherent and collimated white light beam from amplitude and phase radial gratings²⁰. To increase the resistance of a beam against the destructive effects of turbulence, its intensity/phase/polarization structures can be suitably chosen.

The intensity profile of an RCB includes high intensity spots, called main intensity spots (MISs), which are symmetrically placed at the central area around the beam axis and whose number is equal to (twice) the number of spokes of the amplitude (phase) grating used to generate the beam. The level of complexity of intensity and phase profiles of an RCB is generally proportional to the number of grating spokes used to generate the beam; in particular, by increasing the radial grating spokes, the number of phase discontinuities on the generated RCB phase profile increases.

RCBs undergo self-amplification due to their normal radial expansion¹⁵. Recently it is examined that the power carried by an MIS of different RCBs produced by the diffraction of a plane wave from amplitude/phase radial gratings with sinusoidal/binary transmission profiles, and it is found that the core area of an RCB, which has a more complex intensity distribution and surrounds the central patternless area, increases in power as the beam propagates. This feature distinguishes RCBs from other beams and makes them self-amplifying beams. The power transfer to the core area also improves the stability of the beam in turbulent environments such as the atmosphere, making it a promising candidate for atmospheric optical communication.

In this work, we investigate the propagation of RCBs—having different levels of complexity—through atmospheric turbulence at different times of the day. We also compare propagation of LG beams and RCBs through atmospheric turbulence, and show that the RCBs are typically more resilient to the effects of atmospheric turbulence.

Figure 1

This diagram illustrates the experimental setup for our study. The left inset provides transmission functions for three distinct types of gratings: (a) a typical phase fork grating, (b) a phase radial grating, and (c) an amplitude radial grating. Meanwhile, the right inset depicts the relative position of the recording camera in relation to the telescope’s back focal plane.

Experimental setup

The experimental setup for studying how different RCBs and an LG beam with a topological charge of TC = l = 20 travel through atmospheric turbulence is illustrated in Fig. 1. The experiments were carried out in the Institute for Advanced Studies in Basic Sciences (IASBS) campus, Zanjan, Iran. The second harmonic of a diode-pumped Nd:YAG laser beam with a wavelength of 532 nm is filtered and collimated by a lens with a 20 cm focal length. The collimated beam is diffracted by an amplitude or a phase radial grating to produce the required RCB. The RCB travels through a turbulent atmosphere with a 120 m length at various times of the day to test different atmospheric conditions. A telescope and a CCD camera capture successive images of the RCB’s intensity pattern at the end of the path. The laser and telescope are placed 60 cm above the ground and the beams go through a layer of turbulence created over an asphalted area. Different RCBs are created by diffracting a plane wave from radial gratings with different numbers of spokes.

Theoretical background

In this section, we delve into the fundamental formulas governing radial gratings and the resulting RCBs. While the mathematical expressions remain consistent with our previous publication¹⁸, we have meticulously rephrased the surrounding text to ensure originality. In this work, we investigate RCBs under realistic atmospheric turbulence, contrasting with our prior study on indoor convective air turbulence¹⁸. We conducted an extensive series of experiments involving various RCB modes and atmospheric turbulence conditions. Our findings reveal new and interesting results that have not been reported previously. RCBs exhibit superior resilience to atmospheric turbulence, especially for high-order modes, making them promising for optical applications.

We can print the following transmission function on a transparent plastic or glass sheet to create a binary profile radial grating that has varying amplitudes:

$$\begin{aligned} t_a(r,\theta ) = {\frac{1}{2}}\{1+sgn[ \cos ({m{\theta }})]\}, \end{aligned}$$

(1)

where sgn is the sign function, and r, ${\theta }$, and m are the radial coordinate (which does not affect the grating), the azimuthal angle, and the number of spokes in the grating, respectively.

If a coherent plane wave passes through this radial grating, the complex amplitude of the light beam that is diffracted after traveling a distance of z can be expressed by¹³:

$$\begin{aligned} {\psi }(r,\theta ;z) =\frac{e^{ikz}}{2}\{1+R^\prime {\sum \limits _{s = 1}^{ + \infty } }{{\psi _s}[{J}_\frac{sm+1}{2}(R^2)+i{J}_\frac{sm-1}{2}(R^2)]\cos (sm\theta )}\}. \end{aligned}$$

(2)

The wave number k is the ratio of $2\pi$ to the wavelength $\lambda$. We define ${R} = r\sqrt{\frac{{\pi }}{2{\lambda }z}}$, $R^\prime =R{e^{iR^2}}$, $\psi _s=\sqrt{2\pi }(-i)^{\frac{ms}{2}+1}sinc(\frac{s\pi }{2})$. The Bessel function is denoted by J.

Equation (2) reveals that the complex amplitude of the light field varies with R, and the optical pattern it produces keeps its shape as it propagates. The diffraction pattern does not change for a given R, regardless of how the propagation distance, z, or the radial coordinate, r, vary. Since the light field in Eq. (2) has a shape-invariant distribution under propagation, we can regard it as a beam, that is, the RCB^14,19.

We can use a spatial light modulator (SLM) to create a phase radial grating that has the same structure as Eq. (1). We employed an SLM that we obtained from a video projector (LCD projector KM3, model no. X50) to generate the desired pure phase gratings.

We create a binary phase profile on the SLM that has this form:

$$\begin{aligned} t_p(\theta ) = e^{i\gamma sgn [{\cos (m{\theta }})]}. \end{aligned}$$

(3)

In this equation, $\gamma$ is the amplitude of the phase variation.

The phase structure lets a coherent plane wave go through it, and after the light beam travels a distance of z, we can write its complex amplitude as¹⁴

$$\begin{aligned} {\psi }(r,\theta ;z) =e^{ikz} \{cos(\gamma )+{R^\prime }\underset{odd}{\sum \limits _{s = 1}^{ + \infty } }{\phi _s[{J}_\frac{sm+1}{2}(R^2)+i{J}_\frac{sm-1}{2}(R^2)]\cos (sm\theta )\}}, \end{aligned}$$

(4)

where $\phi _s=\frac{2}{s}\sqrt{\frac{2}{\pi }}sin(\gamma )(-i)^{(\frac{m}{2}-1)s+1}$. The light field in Eq. (4) keeps its shape as it propagates, and we can call it an RCB. We used Fresnel’s integral^13,14 and wave equation¹⁹ to find out how these beams propagate. These beams follow the paraxial wave equation.

We create an LG beam with no radial index p = 0 by diffracting a plane wave from a phase fork linear grating. We also put an amplitude fork linear grating on the SLM that has this transmission function:

$$\begin{aligned} t_f(r,\theta ) = {\frac{1}{2}}\{1+sgn[ \cos ({\frac{2{\pi }x}{d}+l{\theta }})]\}, \end{aligned}$$

(5)

where d is the spatial period of the grating, and l is an integer that shows the dislocation of the grating lines. We call it the topological defect number of the grating.

We make a binary phase fork linear grating on the SLM that has this form:

$$\begin{aligned} t_{fp}(r,\theta ) = e^{i\gamma sgn[ \cos ({\frac{2{\pi }x}{d}+l{\theta }})]}. \end{aligned}$$

(6)

The complex amplitude of an LG beam after it travels a distance of z is

$$\begin{aligned} {\psi }(r,\theta ;z) =\frac{C_{LG(p,l)}{\omega }_0}{\omega (z)}(\frac{r\sqrt{2}}{\omega (z)})^{\mid l \mid } L_p^{\mid l \mid } (\frac{2r^2}{\omega ^2 (z)}) e^{\frac{-r^2}{\omega ^2 (z)}-\frac{ikr^2z}{2(z^2+z_R^2)}}e^{i(2p+\mid l \mid +1) \zeta (z)} e^{il\theta }, \end{aligned}$$

(7)

where $C_{LG(p,l)}=\sqrt{2p!/(\pi (p+\mid l \mid )!)}$ denotes a normalization constant, $L_p^l$ is the generalized Laguerre-Gauss polynomial of p (radial mode) and l (angular mode), $\zeta (z)=\arctan (z/z_R)$ shows the Gouy phase, $\omega (z)=\omega _0\sqrt{1+(z/z_R)}$ denotes the beam radius, $\omega _0$ shows the beam waist, and $z_R=\pi \omega _0^2/\lambda$ is the Rayleigh range.

The phase profile of the form $e^{il\theta }$ gives these beams orbital angular momentum (OAM). We make an LG beam with an OAM mode l = 20 and p = 0 by putting the transmission function of Eq. (6) on the SLM.

We use Eqs. (2), (4), and (7) to calculate the intensity and phase distributions of the field that is diffracted from the amplitude and binary phase gratings. Figure 2 shows these patterns. They have the intensity and phase profiles of the RCBs and LG beams.

Figure 2

The transmission profiles of different gratings are shown in the first column. The first row displays a phase fork grating that can generate an LG beam when applied to an SLM, using Eq. (6). The second and third rows show two radial gratings with pure amplitude profiles that are printed on transparent sheets, using Eq. (1). The fourth to seventh rows present four radial gratings with pure phase profiles that can produce RCBs when applied to an SLM, using Eq. (3). The value of l for the fork grating and values of m for each of the radial gratings are given on the left side. The second and third columns show the calculated intensity and phase distributions at a distance of z = 100 cm for each grating. The patterns in the first row are obtained by Eq. (7), the patterns in the second and third rows are obtained by Eq. (2), and the patterns in the fourth to seventh rows are obtained by Eq. (4).

Figure 2 shows that the RCBs have many abrupt changes in their intensity/phase profiles along both radial and azimuthal directions. This means that these beams have a very intricate structure. The structure intricacy of the RCBs ensures that these beams preserve their intensity distribution when they travel through a turbulent medium and are more robust to turbulence than LG beams.

We also discovered recently that the core area of an RCB, which has a more complicated intensity distribution and encloses the central area without any pattern, grows in power as the beam travels. This feature sets RCBs apart from other beams and makes them self-boosting beams. The power transfer to the core area also enhances the stability of the beam in turbulent environments like the atmosphere, making it a potential candidate for atmospheric optical communication¹⁵.

Measurement procedure

The amplitude and phase of a light beam are affected by random changes when it travels through atmospheric turbulence. This means that Angle of Arrival (AA) of the beam varies at different points over the receiving telescope’s entrance pupil, depending on how strong the turbulence is. Also, the RCBs and LG beam have almost dynamic forms at the entrance pupil of the receiving telescope, so their points will move around on that plane. These two effects, the AA changes and beam wandering, lead to image motion/image distortion at the image plan where a CCD camera replaced.

We generated one of the beams that we had introduced earlier and sent it through the medium with turbulence. Then, the beam was received by a Cassegrain-Schmidt telescope. We used a Nikon D7200 camera to record the intensity profiles of the beam at the image plane in video format at 60 frames/sec. The camera had a 1.56 cm $\times$ 2.35 cm sensitive area with 720 $\times$ 1280 active pixels at the movie recording setting. Each pixel was 21 μm long. The camera was placed 3 cm away from the focal plane of the receiving telescope, where the beam’s full cross section was covered by the camera’s sensitive area. At this distance, 640 $\times$ 640 active pixels of the camera covered the full area of the beam hitting it. The receiving telescope had a 20 cm entrance pupil and a 2 m focal length. We removed the imaging lens of the camera and recorded the beam intensity profile directly over the sensitive area. We recorded and analyzed 3600 successive frames for each data set using MATLAB software. The data was captured on September 03, 2020. The weather conditions reported by the Zanjan Meteorological Organization are shown in Table 1.

A structured and non-diffracting beam, such as an RCB, has a different physics of imaging through a turbulent medium than a star imaged by a telescope. The star image only moves due to the AA fluctuation of the beam on the telescope’s pupil, if the pupil diameter is smaller than the Fried parameter. If the Fried parameter is smaller than the pupil diameter, the star image becomes blurry and also shifts due to the AA fluctuation at the pupil. However, a structured beam can also have image motion/distortion because of the wandering of its different parts at the pupil, even if the AA of the beam does not change. This effect is not present in the star imaging, since the star’s plane wave has no structured intensity distribution on the transverse plane (see¹⁸ for more details).

Table 1 The reported weather condition for September 03, 2020 by the Zanjan Meteorological Organization.

Experimental results

The experimental results of LG and RCBs intensity profiles are presented in Fig. 3. The first row displays the intensity distributions of LG beam at different times of the day. The LG beam was generated by propagating a plane wave through a phase fork linear grating with a topological defect number of l = 20. The second and third rows show the RCBs that were generated by diffracting a plane wave through amplitude radial gratings with spoke numbers of m = 10 and m = 25, respectively. The fourth to seventh rows show the RCBs that were generated by diffracting a plane wave through phase radial gratings with spoke numbers of m = 10, m = 20, m = 30, and m = 40, respectively. The beams in the first to sixth columns were recorded at 3 PM, 4 PM, 5 PM, 6 PM, 7 PM, and 8 PM, respectively.

The intensity profiles of LG beam (produced by a phase fork grating with l = 20) and RCB (produced by a phase radial grating with m = 20) are shown in the Visualization data 1–12. These data illustrate how the beams change over time during a day (from 3PM to 8PM). The Visualization data 1-6 correspond to the LG beam, while the Visualization data 7–12 correspond to the RCB.

The results show that RCBs are more stable and less affected by turbulence when they have the same conditions, especially when their structures are more complex (i.e. made with radial gratings that have a larger number of spokes, m).

Figure 3

The first row shows the intensity profiles of an LG beam that was created by a phase fork linear grating (SLM-generated) with a topological defect number l = 20 and then propagated through atmospheric turbulence at different hours of the day: from 3 PM to 8 PM. The LG beam was formed by a plane wave that passed through the grating (see also the Visualization data 1–6). The other rows show the intensity profiles of RCBs that were generated by a plane wave diffracting from radial gratings with different spokes numbers m. The second and third rows show the RCBs from amplitude radial gratings (printed on transparent sheets) with m = 10 and m = 25, respectively. The fourth to seventh rows show the RCBs from phase radial gratings (SLM-generated) with m = 10, m = 20, m = 30, and m = 40, respectively (see also the Visualization data 7–12).

We measure how much four MISs of the RCBs move, which are the farthest ones on the top, bottom, left, and right of the dark area in the middle. We also find the distance from the center of the brightest spots on the donut-shaped ring of the LG beam at the same places. We use these measurements to calculate how much variation there is in the movements.

We measured the radial displacements of the MISs for the RCBs, and the radial displacements of the peak intensities at the same locations for the LG beam, in all the recorded intensity profiles. Figure 4 shows the time series of the radial displacements of the MISs of two RCBs produced by an amplitude radial grating with m = 10 and by a phase radial grating with m = 40. We recorded these data at 5PM and used 3600 consecutive images for each plot.

Figure 4

The displacements measured for an RCB produced by a radial grating with an amplitude profile and m = 10 (first column) and for an RCB produced by a radial grating with a phase profile and m = 40 (second column) at 5PM are shown. The four MISs chosen are labeled as top, left, right, and bottom based on their positions relative to the beam axis. The value of $\sigma _r ^ 2$ for each measurement is given at the top of the graph.

To compare how different beams are affected by turbulence, we plotted the displacements of the MIS at the bottom side of the beams for three RCBs in Fig. 5. These RCBs are generated by radial gratings with different values of m: an amplitude radial grating with m = 10 (first column), a phase radial grating with m = 20 (second column), and a phase radial grating with m = 30 (third column). The recording times were from 3 P.M. to 8 P.M., with one-hour intervals. The value of $\sigma _r ^ 2$ for each data set is shown at the top of the graph.

Figure 5

The measured displacements of the MISs of an RCB generated by a amplitude radial grating with m = 10 (first column), an RCB generated by a phase radial grating with m = 20 (second column), and an RCB generated by a phase radial grating with m = 30 (third column) at different recording times.

The variation of $\sigma _r ^ 2$ for the LG beam and RCBs in turbulent atmosphere at different times of the day is shown in Fig. 6. It is evident that the LG beam has a much higher value of $\sigma _r ^ 2$ than the RCBs. Moreover, the value of $\sigma _r ^ 2$ for RCBs decreases significantly as the value of m increases. This implies that the RCBs are more robust to the turbulence, especially when they have complex structures (those with a large number of MISs in the profile).

Figure 6

Calculated $\sigma _r ^ 2$ for an LG beam with l = 20 and different RCBs at different times of the day.

To visually demonstrate the high resilience feature of RCBs, we present the results obtained at 3 PM in Fig. 7a and d. In Fig. 7a, we traced the maxima of intensity over the LG beam ring at different azimuths using a white line. Additionally, white plus signs mark the center of MISs of an RCB generated by a phase radial grating with (m = 30) in Fig. 7d. The figure reveals that the MIS centers (intensity maxima) form a nearly circular pattern, while the white line representing the LG beam exhibits irregular and discontinuous behavior. The insets in Fig. 7b and c provide 2D intensity profiles over a portion of the LG beam located on the right side. This region corresponds to the spatial extension of an MIS of the RCB and is shown in two different forms. Similarly, the insets in Fig. 7e and f display the 2D intensity profile of the MIS located on the right side of the RCB, again in two different forms. In Fig. 7g, we present the 1D intensity profile of both beams along the half-circle indicated by capital letters ABC in Fig. 7a and d. Notably, the 1D intensity profile for the RCB exhibits a regular sinusoidal form with (m) repetitions over the ABC path, whereas the LG beam displays a highly irregular and discontinuous profile. Despite this difference, the values of maximum intensities remain consistent in both plots. Figure 7h shows 1D intensity profiles of the RCB and LG beam along the D-E path illustrated in Fig. 7a and d. Remarkably, both beams exhibit nearly identical sizes in the radial direction. Finally, in Fig. 7(i), we analyze the time series of maximum intensity values over the MIS of the RCB and a given azimuth for the LG beam. Both beams exhibit fluctuations in intensity over time. The standard deviations of these fluctuations are $(\sigma _I ^ 2)_{_{RCB}}$ = 39.8 for the RCB and $(\sigma _I ^ 2)_{_{LG}}$ = 66.3 for the LG beam. This difference in standard deviations underscores the superior resilience of RCBs to turbulence.

Figure 7

Comparative intensity spatial profiles and intensity time series analysis of RCBs and LG beams. (a) Maxima of intensity traced over the LG beam ring at different azimuths, revealing irregular behavior. Insets in (b) and (c) show the 2D intensity profiles of the LG beam on the right side, corresponding to the spatial extension of an RCB’s MIS. (d) Center of MISs (the white plus signs) in an RCB generated by a phase radial grating with (m = 30). Insets in (e) and (f) show 2D intensity profile of the MIS located on the right side of the RCB. (g) 1D intensity profiles along the half-circle (ABC) for both beams-RCB exhibits regular sinusoidal form, LG beam is irregular. (h) 1D intensity profiles along the D-E path-both beams nearly identical in radial size. (i) Time series of maximum intensity values-RCBs show superior resilience with lower standard deviations (39.8 vs. 66.3) compared to LG beams. For both patterns in (a) and (d), the data were recorded at 3PM.

The distinctive characteristics of the LG beam make it a suitable reference for our study. Both RCBs and LG beams share the same size and remain similar during propagation. Their divergence remains identical as well. Additionally, both exhibit the same polarization due to their common origin from the Gaussian beam. Notably, the intensity maxima over the MISs of RCBs and the donut ring of LG beam are comparable. Given nearly identical sizes during propagation, matching polarization, and close intensity maxima, RCBs and LG beams provide a robust foundation for comparison. Furthermore, both RCBs and LG beams belong to two sets of family modes, rendering them suitable for free space communication.

Now let us explain how our image displacement-based approach, which primarily relies on AA fluctuations, ensures accuracy and reliability. The impinging beams form an image on the sensitive area of a camera (Nikon D7200), which is installed near the focal plane of the telescope $(\text {MEADE}~ LX 200)$. The telescope aperture diameter is 20.32 cm. At the focal plane $(f~=~2~m)$, the images are smallest. By slightly shifting the camera outside the focal plane, enlarged images are obtained. Due to atmospheric turbulence, the positions of points in the images are displaced. Based on these displacements, we determine the radial component of AA fluctuations by calculating the x and/or y components of AA fluctuations relative to the center of the image. For simplicity, consider the region over the MIS of the RCBs and the intensity ring of the LG beam at zero azimuthal angle (x-direction). Between the two local AA components of the laser beam’s wavefront, the AA in the x-direction is identical to the radial component. It can be calculated as follows:

$$\begin{aligned} \alpha _{x}=\frac{l_x(x_{g}-\bar{x}_{g})}{f}, \end{aligned}$$

(8)

where, $l_x~=~ 21~\mu m$ is the pixel size in the x-direction on the image plane, $x_{g}$ is the x-component of the maximum intensity over the MIS of the RCB/LG beam intensity donut, $\bar{x}_{g}$ is the mean position of the maximum intensity in the x-direction, and f represents the telescope focal length. In our experimental setup, we found that the minimum measurable AA fluctuation was $10^{-6}$ rad when the CCD was positioned near the focal plane of the telescope. By leveraging sub-pixel accuracy for measuring image point displacements, we achieved this level of sensitivity. It is worth noting that by using the sub-pixel accuracy for the measurement of the image point displacements, the sensitivity of the measurements can be improved by one order of magnitude²¹.

Now we show another example of how RCBs can withstand the effects of atmospheric turbulence. In Fig. 8, the first and third rows display the percentage of MISs of various RCBs (created with different radial gratings with different numbers of spokes) that are preserved after passing through the turbulent medium. The exact percentages are given in the second and fourth rows. It can be seen that the MISs are barely affected by the turbulence. Especially, for the RCBs created with gratings with m = 30 spokes or more, the MISs of the received intensity patterns change by less than 1% even when the turbulence strength is high. This feature makes the RCBs a suitable choice for the free space transmission of information.

Figure 8

Ratio of the number of the MISs of the created RCBs that remain unchanged from the input to the output planes for different times of the day.

Method of measuring the number of preserved MISs of the RCBs and their displacement

We describe the procedure of counting the MISs of the RCBs and measuring their displacements. We used consecutive recorded intensity patterns of the RCBs for this purpose. We also compare these measurements with the radial displacements of the intensity maxima on the doughnut ring of the LG beam. To count the MISs of the RCBs, we applied image processing in MATLAB software, which is shown step by step in Fig. 9. First, we multiplied a ring mask (Fig. 9b) with the beam image (Fig. 9a), where all the MISs of the beam are inside the ring mask and subject to image processing (Fig. 9c). The mask size varies depending on the beam type. Then, we used an intensity filter to isolate the MISs from the original image. The filter’s threshold is defined by a critical intensity value $I_c=I_{max}-(\frac{I_{max}}{1.5})$, where $I_{max}$ is the maximum intensity over each spot area. As seen in Fig. 9d, some secondary spots remain within the ring mask after applying the intensity filter. To remove these spots, we calculated the average area of the main spots and then eliminated the spots with an area smaller than a certain factor of this area from the original image. This filter removed the secondary spots as seen in Fig. 9e. In the next steps, we counted the number of MISs and determined their centers. Figure 9f and g show the main image of MISs through their corresponding binary aperture for a better demonstration of the procedure.

Figure 9

Illustration of the counting method of detectable MISs at the end of the propagation and determination of the location of their maximum intensities. (a) Intensity pattern of an RCB; (b) ring mask with an average radius equal to the average of the radial distance of the MISs from the beam center with sufficient width to faithfully cover all MISs; (c) mask in (b) multiplied by RCB intensity pattern (a) in gray scale; (d) Binarizing of MISs and smaller secondary intensity spots through thresholding; (e) MISs after removal of secondary intensity spots; (f) multiplication of (e) and (a); (g) The red plusses show the position of the maximum intensity of each of the MISs. See text for further details.

Conclusion

In this study, we investigated the impact of atmospheric turbulence on RCBs with varying structural complexity. Observations were made as RCBs propagated through a turbulent atmosphere over a 120 m path length. The displacements of MISs were quantified along the radial direction. Our findings reveal that structurally complex RCBs exhibit remarkably small MIS displacements. Furthermore, when compared to LG beams under the same turbulence conditions, RCBs demonstrate greater resilience to atmospheric disturbances. Specifically, RCBs generated with gratings containing 30 or more spokes experience minimal changes in MISs, even in high turbulence. By contrast, the LG beam’s intensity ring is significantly disrupted, rendering it unsuitable for reliable free-space optical communication through stronger turbulent air. Additionally, we conducted a detailed analysis of MIS counting, further highlighting the robustness of RCBs. The resilience of RCBs to atmospheric turbulence is crucial for optical communication systems. When transmitting information through turbulent air, maintaining beam integrity is essential. RCBs, with their self-healing properties and minimal MIS displacements, offer a promising solution. Their ability to withstand turbulence ensures more reliable communication links, making them valuable candidates for practical applications in free-space optical communication under challenging atmospheric conditions.