Open Access
Add to BibSonomy
Abstract
In this paper, we firstly propose a method to measure the topological charges (TCs) of a circular Bessel Gaussian beam with multiple vortex singularities (CBGBMVS) by utilizing cross phase. Based on theory and experiment, the cross phase is utilized to realize the TCs measurement of the CBGBMVS in free space with different situations, such as different singularity number, TCs and singularity location. Especially, the TCs measurement method is also investigated and verified in atmosphere turbulence. Our work provides an effective and convenient way to realize the TCs measurement of multiple singularities embedded in abruptly autofocusing host beams which has plenty of potential application in optical communication.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical vortex singularity known as optical phase singularity is a point phase defect at which the phase is indeterminate, and amplitude is zero in the beam center. Vortex singularity, for carrying orbital angular momentum (OAM), provides one of the promising solutions for enhancing the capacity of data multiplexing to meet an unprecedented growth in big data and internet traffic information. Meanwhile, the unlimited OAM states supply new degrees of freedom for light, which also can be applied to many other fields such as optical manipulation, microscopic imaging, rotation measurements, and enantiomer-selective sensing [1–3]. In practical applications, vortex singularities are usually embedded in one of the host beams, for instance, the Gaussian vortex beam [4], Airy vortex beam [5], and Bessel vortex beam [6]. Topological charges (TCs) of vortex singularities are one of the vital physical quantities for characterizing these vortex beams, so it is significant to measure the TCs.
Up to now, various approaches have been developed to measure the TCs of vortex beams (VBs). Based on interference and diffraction, here several methods have been presented for successful measurement. For instance, the examination of the interference pattern of a vortex beam incident on a single and double slit [7–9] are utilized to detect TCs. Interference of VBs with a plane wave [10] and the use of a cylindrical lens [11] have also been examined the TCs of VBs based on the interference principle. The diffraction methods show unique properties by illuminating diffraction apertures of different geometric with VBs, such as a single slit [12], isosceles triangular [13], square [14], pentagonal aperture [15], hexagonal aperture [16], and sectorial screen [17]. Furthermore, the circular and annular diffraction apertures were also developed to directly measure TCs [18,19]. In addition, the method based on the Fourier transform of the intensity (or diffracted intensity) of VBs was also investigated [20]. However, this method usually needs a computing process, which means it is not intuitive when we do measurement experiments in the lab. In addition to the above methods, the phase modulation methods have also been put forward to measure the TCs of VBs, which is a relatively convenient way mainly by utilizing a spatial light modulator. Here, several modulation phases commonly used are twisting phase (cross phase) [21], lens phase [22], and sinusoidal phase [23], respectively.
Instead of being confined to a single vortex singularity, the case of multiple vortex singularities embedded in a host beam has stimulated the interest of researchers in the last few years, due to the need to improve communication capacity, manipulate multiple particles, and other applications. Therefore, it is necessary and important to explore the TCs measurement method for multiple vortex singularities. However, it is a pity that very little research work has been carried out. Zhao et al. [24] proposed a simple method which utilizes a cylindrical lens to realize the TCs measurement of multiple vortex singularities embedded in the Gaussian beam. Besides, Gu et al. [25] introduced the tilted lens method to detect not only the magnitudes and signs of two TCs of the off-axis double vortex beam but also the spatial distribution of the TCs. Abruptly autofocusing (AAF) beams, such as circular Airy beams [26], circular Pearcey beams [27] and circular swallowtail beams [28], have vital potential applications in optical communication [29] and optical tweezers [26]. Embedding multiple vortex singularities into AAF beams will open up new applications, so the TCs measurement of such beams is an important research content.
In this paper, the TCs measurement of multiple vortex singularities embedded in AAF beam is explored by utilizing cross phase (CP) for the first time. Here the circular Bessel Gaussian beams, as a kind of typical AAF beam, are chosen as the host beam where singularities are embedded. The evolution characteristics of circular Bessel Gaussian beam with multiple vortex singularities (CBGBMVS) are investigated by numerical simulation in free space. Meanwhile, the TCs measurement method for this beam via CP is also elaborated. Based on simulation and experiment, the TCs measurements under different singularity distributions are performed and discussed. Especially, the applicability and robustness of this method is also investigated in atmosphere turbulence. This work provides an efficient and convenient method to achieve the TCs measurement of multiple vortex singularities embedded in AAF beam.
2. Theory
Without loss of generality, the optical field of the CBGBMVS in cylindrical coordinates can be expressed as:
According to the Fresnel diffraction integral, the electric field at the propagation distance z can be expressed as [30]:
In this paper, the feasibility of TCs measurement for the CBGBMVS by utilizing CP modulation are explored by simulation and experiment. The mathematical expression of CP in the cartesian coordinates can be written as:
Figure 1 shows propagation dynamics of the CBGBMVS and corresponding parameters are set as follows: N = 2, l1 = 2, l2 = 2, r1 = r2 = 0.15 mm. The side view of the CBGBMVS propagation is shown in Fig. 1(a). Figures 1(b1-b5) present the transverse intensity profiles corresponding to different distance z which are marked by the white dotted lines in Fig. 1(a), respectively. Figures 1(c1-c5) show the phase distributions corresponding to the Figs. 1(b1-b5), respectively. What can be noticed that the light intensity of the CBGBMVS first converges and then gradually diverges due to the autofocusing characteristics. Meanwhile, the two singularities embedded into the CBGBMVS at the initial plane evolve into four singularities with TC = 1, which could be attributed to the instability of higher order vortex singularities [34].
Fig. 1. (a) Side view of the CBGBMVS propagation, l1 = l2 = 2; (b1-b5) transverse intensity distributions corresponding to the white dotted lines in (a), respectively; (c1-c5) phase distributions corresponding to (b1-b5), respectively.
The physical mechanism of the TCs measurement for single vortex singularity via CP is considered to be the effective anisotropic diffraction, which causes the rapid splitting of vortex singularities [35]. Based on this, we boldly speculate that the anisotropic diffraction effect of CP can also be used to the TCs measurement of multiple vortex singularities. To verify this hypothesis, we use the CP to modulate the CBGBMVS at the initial plane and study its propagation characteristics, as shown in Fig. 2. Here the corresponding beam parameters are consistent with the preceding ones. The side view of the modulated CBGBMVS propagation is presented in the Fig. 2(a). Figures 2(b1-b5) and Figs. 2(c1-c5) show the transverse intensity and phase distributions corresponding to different distance z which are marked by the white dotted lines in Fig. 2(a), respectively. Compared with Fig. 1, it can be clearly found that there are obvious changes in the intensity distribution after the autofocusing focus. Due to the mode-conversion property induced by the CP, the TCs information hidden in CBGBMVS is presented in the form of a set of separated nodes, where the number of gaps between nodes is equal to the sum of the TCs number of the CBGBMVS. Therefore, the CP modulation method can still be used to measure the TCs of multiple vortex singularities theoretically. In the following part, we will investigate the TCs measurement effect of this method for different singularity distributions by simulation and experiment.
Fig. 2. (a) Side view of the propagation of the CBGBMVS modulated by CP; (b1-b5) transverse intensity distributions corresponding to the white dotted lines in (a), respectively; (c1-c5) phase distributions corresponding to (b1-b5), respectively.
3. Experiments and simulations
3.1 Experiment set
Figure 3 shows the experimental optical path setup for measuring the TCs of CBGBMVS through the CP modulation method. A Gaussian beam of λ = 632.8 nm emitted from the He-Ne laser passes through an attenuator, a beam expander (BE), and a linear polarizer (LP). And then, the beam reflected by the mirror passes through the beam splitter (BS) and is incident on the spatial light modulator (SLM, pixel size 4.5 µm, pixel resolution (H) 1920 × (V) 1080 pixels). The beam modulated by the SLM passes through the BS and then through a 4f system consisting of two lenses (L1 and L2) and an aperture (AP). The AP is used to choose the beam’s first diffraction order. The intensity distribution of the desired CBGBMVS at different propagation distance can be obtained by adjusting the position of the charge-coupled device (CCD).
Fig. 3. Schematic diagram of the experimental setup. BE: beam expander; LP: linear polarizer; BS: beam splitter; SLM: spatial light modulator; L1, L2: lens (focal length = 150 mm); AP: aperture; CCD: charge-coupled device.
3.2 TCs measurement of the CBGBMVS in free space
The simulated intensity and phase distributions of the CBGBMVS modulated by CP at the initial plane are shown in Figs. 4(a1-a5) and Figs. 4(b1-b5), respectively, where N = 2, r1 = r2 = 0.15 mm. Here five different scenarios are considered: (a1) l1 = 1, l2 = 2; (a2) l1 = 2, l2 = 2; (a3) l1 = 2, l2 = 3; (a4) l1 = 4, l2 = 3 and (a5) l1 = 4, l2 = 5. Figures 4(c1-c5) and Figs. 4(d1-d5) demonstrate the numerical simulation and experimental results at the propagation distance z = 375 mm, respectively. It can be noticed that the experimental measurement results match well with the simulation results. Similar to the Fig. 2, the number of gaps between the nodes is equal to the sum of the TCs carried by the two embedded vortex singularities. Here the number of gaps between the nodes in each situation is 3, 4, 5, 7 and 9, respectively, which corresponds exactly to the total number of singularities TCs under the different cases. Therefore, the measurement method is proved to be feasible under the condition of constant number of singularities but varying TCs.
Fig. 4. (a1-a5) Initial intensity distributions of the CBGBMVS modulated by CP under the conditions of different TCs; (b1-b5) phase distributions corresponding to (a1-a5), respectively; (c1-c5) intensity distributions of the CBGBMVS with CP at the plane of z = 375 mm, respectively; (d1-d5) experiment results corresponding to the (c1-c5), respectively.
The validity of the TCs measurement method is also discussed under the condition of more embedded singularities. For convenience, all embedded singularities are symmetrically distributed on a circumference of radius r = 0.15 mm, where its center is located at the center of the beam and the TCs of each singularity are equal to 2. Figures 5(a1-a5) and Figs. 5(b1-b5) illustrate the simulated intensity and phase distributions corresponding to different singularity number (2 ∼ 6) at the initial plane, respectively. With the increase of singularity number, the intensity profile will change to some extent, and the phase singularities in the central region become denser. Figures 5(c1-c5) and Figs. 5(d1-d5) show the simulation and experimental results of the TCs measurements at the plane of z = 375 mm, respectively. It is not difficult to see that the simulation results are agreement with the experimental results. The number of gaps between nodes is 4, 6, 8, 10, 12, respectively, corresponding to the different singularity number case (2 ∼ 6). Here the number of gaps between nodes in each subgraph is also exactly equal to the total TCs carried by the CBGBMVS.
Fig. 5. (a1-a5) Initial intensity distributions of the CBGBMVS modulated by CP under the condition of different singularity number; (b1-b5) phase distributions corresponding to (a1-a5), respectively; (c1-c5) intensity distributions of the CBGBMVS with CP at the plane of z = 375 mm; (d1-d5) experiment results corresponding to the (c1-c5), respectively.
Compared Figs. 4(a1-a5) with Figs. 5(a1-a5), the conclusion can be deduced that when the TCs distribution of the CBGBMVS is not balanced, the intensity distribution of the measured nodes in the TCs measurement will no longer be symmetrical. It implies that the CP modulation can be used to measure or map the singularities absence of the CBGBMVS in the process of propagation. Taking initial four singularities as examples, we investigate the simulation and experiment results of each singularity carrying the same and different TCs. Firstly, we set the TC value of each singularity to 2 and investigate the simulation and experimental measurement results where the embedded singularity occurs absence in the case of different locations and numbers. The first column in Fig. 6 shows the simulation and experimental results without the absence of singularities. The second and third column in Fig. 6 present the absence of single singularity at different locations, respectively. Similarly, the fourth and fifth column in Fig. 6 show the absence of two singularities at different case. It can be seen that under the condition of different absence singularity number, the gap number between all nodes for the measured intensity distribution is different. In addition, when the number of absence singularities is the same, the absence singularities at different locations will also lead to changes in the overall intensity profile of the measurement nodes. However, the number of gaps between the nodes is still equal to the total TCs carried by all remaining singularities. Combined with the specific change of intensity profile and the gap number of measured nodes, the missing locations and number of the singularities embedded in the beam can be identified theoretically.
Fig. 6. (a1-a5) Initial intensity distributions of the CBGBMVS modulated by CP under the different conditions of absence singularities; (b1-b5) phase distributions corresponding to (a1-a5), respectively; (c1-c5) intensity distributions of the nodes at the propagation distance z = 375 mm; (d1-d5) experiment results corresponding to the (c1-c5), respectively.
Next, we investigate another case of absence singularities, where the TC of each singularity embedded in the initial beam is different (l1 = 1, l2 = 3, l3 = 2, l4 = 4). The first column in Fig. 7 shows the simulation and experimental results without the absence of singularities. As we can see that the annular intensity distributions of the beam at the initial plane are uneven. Moreover, the intensity distribution of each node obtained is also different due to the uneven distribution of TCs. Comparing the second and third column in Fig. 7, there are significant differences in the intensity distributions of the measured nodes for TCs detection, under the condition of different number of missing singularities and the same total number of missing TCs. The remaining singularities and their corresponding TCs can be seen in the text added from the graph in the first row, respectively. Furthermore, comparing the fourth and fifth column in Fig. 7, there are also significant differences in the intensity distributions of the measured nodes for TCs detection, under the condition of the same number of missing singularities and the same total number of missing TCs. This difference is more reflected in the bending direction of the intensity profile of the measured nodes. Therefore, theoretically, the method based on CP modulation can accurately detect the absence of the beam singularities from the point of intensity demodulation.
Fig. 7. (a1-a5) Initial intensity distributions of the CBGBMVS modulated by CP under the different conditions of absence singularities; (b1-b5) phase distributions corresponding to (a1-a5), respectively; (c1-c5) intensity distributions of the CBGBMVS modulated by CP at z = 375 mm; (d1-d5) experiment results corresponding to the (c1-c5), respectively.
The node intensity distribution for TC measurement is also affected by the location of the initial embedded singularities. To verify this result, this assumes that all embedded singularities are located at a circle with equal radius rp from the beam center. Here the TCs measurement in three different radius cases is examined based on numerical simulation and experiment. As shown in Fig. 8, the radii rp are set to 0.15, 0.2, and 0.3 mm, respectively, corresponding to the first, second, and third row. Each column of simulation and experiment graphs in Fig. 8 correspond to different numbers of embedded singularities and the TCs carried by them, respectively. The specific position distribution of the embedded singulars can be seen from each simulated phase distributions which are inserted into the lower right corner of each subgraph in Fig. 8. In each column, the specific TC values (l1, l2, l3, l4) for each embedded singularity have also been shown using the text description in the subgraph at the first row, respectively.
Fig. 8. (a1-a4), (b1-b4), (c1-c4) Simulated intensity distributions of CBGBMVS modulated by CP at the propagation distance z = 375 mm under the different conditions of singularity distributions, respectively; (d1-d5), (e1-e5), (f1-f5) experimental results corresponding to (a1-a5), (b1-b5), (c1-c5), respectively.
In the same rp case, it can be found that when the TCs distribution is balanced and unbalanced with respect to the beam center, there is a difference in the intensity distribution uniformity of the measured nodes. The balance can be simply understood as the distribution uniformity of TCs on a circle of radius rp, which is mainly affected by the number and location of singularities, and the TCs carried by each singularity. The better the balance, the better the intensity distribution uniformity of the measured nodes, which is conducive to the TCs measurement in optical communication. It is not difficult to find that, comparing each row in Fig. 8, the intensity profiles of the measured nodes will change with the radii rp increasing. Especially, the more unbalanced the distribution of TCs, the more obvious the variation of the intensity profile, and vice versa. As seen from Fig. 8, the numerical simulation results agree well with the experimental results. According to our further simulations, the regularity of TCs measurement is still satisfied when the radii rp is further increased to a certain extent.
3.3 TCs measurement of the CBGBMVS in turbulent atmosphere
In order to further explore the applicability of the TCs measurement method for multiple vortex singularities, the performance of proposed method in atmospheric turbulent environment is investigated based on numerical simulation and experimental testing in laboratory. Here we use the power spectrum inversion method based on fast Fourier transform to generate a random phase screen for emulating the disturbance effect of atmospheric turbulence [36]. The classical Kolmogorov spectrum is selected, and its function expression can be described as:
where C2 n is the structure constant of the refractive index, ĸ is the spatial wavenumber. Under the Markov approximation, the phase spectrum can be expressed as: where k = 2π/λ, and z can be regarded as the propagation distance in the turbulent atmosphere. Hence the phase screen function can be described as [37]:The TCs measurement method for multiple vortex singularities is tested in laboratory based on a SLM. The optical system used in the experiment is the same as that shown in Fig. 3, but the hologram loaded onto the SLM integrates the hologram phase of the CBGBMVS generation, the CP, and the phase of turbulent screen compensated [39,40]. In order to emulate a real outdoor path using the laboratory bench, the Fresnel numbers $F = \omega _\textrm{0}^\textrm{2}/({{\lambda_\textrm{0}}{z_\textrm{0}}} )= \omega _b^\textrm{2}/({{\lambda_b}{z_b}} )$ and scaling factors $C = {\omega _0}/{\omega _b} = {L_0}/{L_b}$ should be maintained [39], where ω0 is the waist width of the beam, z0 is the propagation distance in the outdoor turbulent environment. Here we set C = 50, λ0 = 632.8 nm, ω0 = 1.5 cm, z0 = 1800 m. The phase screen size L0 and the sampling number M are set to 0.4 m, 800, respectively. The above parameters are used to the simulation. The subscript b indicates the benchtop parameter, where λb = λ0, zb = z0/C2 = 0.72 m and other parameters can also be calculated according to the conditions, which are not listed one by one here. In our simulation and experiment, the number of singularities carried by the CBGBMVS is 2 and 4 respectively, and each singularity carries the TCs 2. All singularities are symmetrically distributed near the center of the beam. The beam propagates under the two different turbulence strengths, the structure constant of the refractive index C2 n of which is 1 × 10−15 and 1 × 10−16 m−2/3, respectively. In the first two columns of the Fig. 9, the number of embedded singularities is two. Contrastively, the number of embedded singularities is 4 in the third and fourth columns. Figures 9(a1-a4) show the random phase screens of atmospheric turbulence, which are used in simulations and experiments. Figures 9(b1-b4) and Figs. 9(c1-c4) show the simulation and experimental results for the TCs measurement, respectively. It can be found that the simulation results agree well with the experimental results, which proves the applicability of the CP modulation method to measure the TCs of multiple vortex singularities in turbulent environment.
Fig. 9. (a1-a4) Random phase screens of different atmospheric turbulence strengths; (b1-b4) simulation results for the TCs measurement; (c1-c4) experimental results for the TCs measurement.
4. Conclusion
In summary, we have extended the traditional method based on CP modulation to measure the TCs of a single vortex singularity to the TCs of multiple vortex singularities. The measurement effect of the total TCs of the CBGBMVS under different singularity distributions (different singularity number, location and TCs value) were investigated by numerical simulation analysis and experimental verification. In essence, the distinguishability of the gap between the measured nodes is closely related to the unbalanced distribution of TCs carried by multiple vortex singularities. It should be noted that we only investigated the case where the TCs carried by the embedded singularities are all positive integers in this work. However, this method is also applicable for the other cases where the TCs are all negative integers or integrating the positive and negative integers according to our in-depth simulation verification. The investigation results in atmospheric turbulence demonstrated the feasibility of TCs measurement for multiple vortex singularities based on CP modulation, which is of great significance in the field of optical communication.
Funding
National Natural Science Foundation of China (62105049, 62305232, 12204082); Natural Science Foundation of Chongqing (cstc2021jcyj-msxmX1119); Science and Technology Research Project of Chongqing Education Commission (KJQN202100618); Chongqing Postdoctoral Research Fund Project (D63012022051); Doctoral program sponsored by Chongqing University of Posts and Telecommunications (E011A2022310); Guangdong Basic and Applied Basic Research Foundation (2023A1515010054); Natural Science Foundation of Top Talent of SZTU (GDRC202313).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]
2. C. Zhang, C. Min, L. Du, et al., “Perfect optical vortex enhanced surface plasmon excitation for plasmonic structured illumination microscopy imaging,” Appl. Phys. Lett. 108(20), 780–782 (2016). [CrossRef]
3. T. Cheng, W. Wang, J. Li, et al., “Rotational Doppler effect in vortex light and its applications for detection of the rotational motion,” Photonics 9(7), 441–461 (2022). [CrossRef]
4. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, et al., “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22(5), 849–861 (2005). [CrossRef]
5. H. Dai, Y. Liu, D. Luo, et al., “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35(23), 4075–4077 (2010). [CrossRef]
6. S. Orlov, K. Regelskis, V. Smilgevičius, et al., “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209(1-3), 155–165 (2002). [CrossRef]
7. J. P. C. Narag and N. Hermosa, “Probing higher orbital angular momentum of Laguerre-Gaussian beams via diffraction through a translated single slit,” Phys. Rev. Appl. 11(5), 054025 (2019). [CrossRef]
8. H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31(7), 999–1001 (2006). [CrossRef]
9. O. Emile and J. Emile, “Young’s double-slit interference pattern from a twisted beam,” Appl. Phys. B 117(1), 487–491 (2014). [CrossRef]
10. J. Vickers, M. Burch, R. Vyas, et al., “Phase and interference properties of optical vortex beams,” J. Opt. Soc. Am. A 25(3), 823–827 (2008). [CrossRef]
11. S. N. Alperin, R. D. Niederriter, J. T. Gopinath, et al., “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41(21), 5019–5022 (2016). [CrossRef]
12. D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Lasers Eng. 47(1), 123–126 (2009). [CrossRef]
13. L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, et al., “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018). [CrossRef]
14. P. H. Mesquita, A. J. Jesus-Silva, E. J. Fonseca, et al., “Engineering a square truncated lattice with light's orbital angular momentum,” Opt. Express 19(21), 20616–20621 (2011). [CrossRef]
15. C. Acevedo, Y. T. Moreno, and Á Guzmán, “Far-field diffraction pattern of an optical light beam with orbital angular momentum through of a rectangular and pentagonal aperture,” Optik 164, 479–487 (2018). [CrossRef]
16. Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011). [CrossRef]
17. R. Chen, X. Zhang, Y. Zhou, et al., “Detecting the topological charge of optical vortex beams using a sectorial screen,” Appl. Opt. 56(16), 4868–4872 (2017). [CrossRef]
18. C. Guo, L. Lu, and H. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34(23), 3686–3688 (2009). [CrossRef]
19. N. Daryabi and S. G. Sabouri, “Intersecting of circular apertures to measure integer and fractional topological charge of vortex beams,” Opt. Express 31(17), 28459–28469 (2023). [CrossRef]
20. S. Prabhakar, A. Kumar, J. Banerji, et al., “Revealing the order of a vortex through its intensity record,” Opt. Lett. 36(22), 4398–4400 (2011). [CrossRef]
21. D. Shen and D. Zhao, “Measuring the topological charge of optical vortices with a twisting phase,” Opt. Lett. 44(9), 2334–2337 (2019). [CrossRef]
22. X. Liu, L. Guo, M. Wang, et al., “Measuring the topological charge of a helico-conical vortex beam via the lens phase,” Appl. Phys. Lett. 123(17), 174101 (2023). [CrossRef]
23. H. Pu, O. Sasaki, T. Suzuki, et al., “Measurements of phase distributions of optical vortices based on the sinusoidal phase modulation method,” Opt. Continuum 1(11), 2287–2297 (2022). [CrossRef]
24. Y. Peng, X. Gan, P. Ju, et al., “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32(2), 024201 (2015). [CrossRef]
25. M. Guo, W. Le, C. Wang, et al., “Generation, topological charge, and orbital angular momentum of off-axis double vortex beams,” Photonics 10(4), 368–381 (2023). [CrossRef]
26. W. Lu, X. Sun, H. Chen, et al., “Abruptly autofocusing property and optical manipulation of circular Airy beams,” Phys. Rev. Appl. 99(1), 013817 (2019). [CrossRef]
27. C. Sun, D. Deng, G. Wang, et al., “Abruptly autofocusing properties of radially polarized circle Pearcey vortex beams,” Opt. Commun. 457, 124690 (2020). [CrossRef]
28. H. Teng, Y. Qian, Y. Lan, et al., “Abruptly autofocusing circular swallowtail beams,” Opt. Lett. 46(2), 270–273 (2021). [CrossRef]
29. X. Yan, L. Guo, M. Cheng, et al., “Controlling abruptly autofocusing vortex beams to mitigate crosstalk and vortex splitting in free-space optical communication,” Opt. Express 26(10), 12605–12619 (2018). [CrossRef]
30. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]
31. H. Gong, G. Li, Q. Chen, et al., “Study of an arithmetic for fast computing transmission of light field,” Acta Opt. Sin. 36(4), 405001 (2016). [CrossRef]
32. C. Wang, Y. Ren, T. Liu, et al., “Measurement and shaping of circular Airy vortex via cross-phase,” Opt. Commun. 497, 127185 (2021). [CrossRef]
33. X. Liu, M. Wang, L. Guo, et al., “Measuring the photonic topological charge of power-exponent-phase vortex beam via cross phase,” Appl. Phys. Lett. 123(3), 034104 (2023). [CrossRef]
34. F. Ricci, W. Löffler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20(20), 22961–22975 (2012). [CrossRef]
35. Y. Li, G. Liang, and X. Hu, “Controllable splitting and bidirectional rotating of optical vortices due to cross phase,” J. Mod. Opt. 69(13), 734–740 (2022). [CrossRef]
36. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 0074, 225–233 (1976). [CrossRef]
37. M. J. Erik and T. G. Donald, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994). [CrossRef]
38. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves in Random Media 2(3), 209–224 (1992). [CrossRef]
39. X. Xiao, D. G. Voelz, I. Toselli, et al., “Gaussian beam propagation in anisotropic turbulence along horizontal links: theory, simulation, and laboratory implementation,” Appl. Opt. 55(15), 4079–4084 (2016). [CrossRef]
40. X. Wang, Q. Li, W Zhong, et al., “Drift behavior of Airy beams in turbulence simulated by using a liquid crystal spatial light modulator,” Chin. J. Laser 40(12), 1213001 (2013). [CrossRef]
