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Abstract
Beam splitting of high-order Gaussian (HOG) beams increases the channel capacity and improves the processing speed of the incoming information. Here a novel all-optical tunable multi-port HOG beam splitter under a periodic dielectric atomic structure is proposed and demonstrated. The original HOG beam is replicated in the output beams. A distinguishable five-port output beam is observed in the experiment, which is beneficial for high-speed optical communications. By tuning the optical properties of this periodic dielectric structure, the spatial position and intensity distribution of each output port are precisely controllable. The splitting ratio δ can be finely adjusted in the range 0 – 4.8. This work provides a new approach for multi-port HOG beam splitters and the basis for all-optical communication.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-order Gaussian (HOG) mode beams have been widely studied in modern photonics and used in numerous optical applications, such as optical tweezers , microscopy and imaging, high-order quantum entanglement, and optical communication [1–5]. Among the various HOG modes, the most representative ones are the Laguerre–Gaussian (LG) and Hermite–Gaussian (HG) modes [6]. Owing to the nonzero orbital angular momentum (OAM) and infinite topological charge of LG beams [7–9], as well as to the similarity as the communication modes of square apertures in the free-space propagation of HG beams [10,11], both LG and HG beams have important applications in the optical communication field.
The transformation of a single HOG beam into multiple beams can significantly improve the speed of optical communications by increasing the channel capacity. Indeed, various beam splitter proposals have been put forward to realize multi-port HOG beams, such as vortex sensing diffraction gratings [12,13], gradually-changing-period gratings [14], Dammann gratings [15], and geometric phase gratings [16]. However, these devices based on solid-state structures not only contain impurities but are also designed with fixed structural parameters. A pure HOG beam splitter with flexible adjustment features would be more suitable for high-speed all-optical communications.
Propagation of light in nonlinear dielectric media with a periodically-varying refractive index is known to exhibit many novel features, which do not occur in homogeneous nonlinear materials [17,18]. Recently, an artificial periodic dielectric atomic structure has been proposed as a promising candidate for all-optical HOG beam splitter owing to its flexible nonlinear optical properties [19,20]. Such structure was constructed by applying a periodically modulated optical field into an atomic medium, where the input beam is split into a multi-order pattern [21,22]. Owing to the tunable optical properties of coherent multilevel atomic systems, it has been shown that the dynamic behaviors of this dielectric structure can be conveniently controlled by the optical fields, and output beams with tunable intensity and spatial distribution have been realized [23,24]. Such method of processing optical signals without using electric signals has drawn considerable attention toward this structure for its application to novel optical devices, such as all-optical modulator [22], all-optical switching and routing [25], quantum storage [3], and optical diode [26]. Furthermore, the dielectric atomic structure has also been used to characterize topological matter, such as edge solitons, chiral edge currents, and flat bands [27–29]. Indeed, coherently prepared multilevel atoms with tunable optical properties are attractive systems for use in exploring novel all-optical beam splitter based on the atomic coherence effect and four-wave mixing process [30–33]. A novel all-optical beam splitter for HOG beams is here proposed to combine the advantages of this periodic dielectric atomic structure with HOG beams. To the best of our knowledge, similar investigations have not yet been conducted.
In this work, an all-optical tunable HOG beam splitter is implemented using a periodic dielectric atomic structure established via two identical control lasers crossing with a small angle in an $^{85}$Rb atom vapor. Compared with the traditional solid-state structure, this atomic structure beam splitter constructed via optical fields is more convenient to regulate and can accurately feedback the split beams in real time. The splitting ratio of this HOG beam splitter is finely adjusted by the control laser power, and the splitting result that 0th-order splitting intensity is much lower than that of $\pm$1st-order is firstly achieved in experiment to the best of our knowledge. The intervals of all output ports can be controlled by tuning the grating constant of the constructed structure as well. A five-port output perfect replica of the original HOG beam is observed in the experiment, which can be achieved as a $1\times N$ beam splitter. The proposed beam splitter has potential applications in all-optical devices and will promote the use of HOG beams in all-optical communication.
2. Experimental setup
A V-type three-level configuration of the $^{85}$Rb atom was employed to realize the HOG beam splitter, as shown in Fig. 1(a). The input and control lasers excite the atoms from the $5S_{1/2}(F = 2)$ level to the $5P_{3/2}(F = 3)$ and $5P_{1/2}(F = 3)$ levels, respectively. Taking the LG beam as an example, the corresponding schematic diagram and experimental implementation of the beam splitter are depicted in Figs. 1(b) and 1(c), respectively. Two independent external cavity diode lasers (DL pro, Toptica) with Gaussian profile were used to provide the input and control laser beams. Furthermore, the saturation absorption spectroscopy (SAS) method was used to lock the frequency of these two lasers after they pass through a double-pass configuration based on an acousto-optical modulator (AOM). The elliptical control laser shaped by an anamorphic prism (AP1) is split into two beams which have the same profile and power. The resulting two beams interfere with each other at a small angle ($2\varphi$) at the center of the vapor to form a periodic dielectric atomic structure, which corresponds to the functional area of the HOG beam splitter illustrated in Fig. 1(b)
Fig. 1. (a) Relevant energy levels of the V-type configuration of the $^{85}$Rb atom system. (b) Schematic diagram of the principle of the beam splitter. (c) Sketch of the experimental setup. AP, anamorphic prism; AOM, acousto-optic modulator; BB, beam block; BS, beam splitter; CCD, charge-coupled device; EIT, electromagnetically induced transparency; HWP, half-wave plate; M, high reflection mirror; PBS, polarization beam splitter; QWP, quarter-wave plate; SAS, saturation absorption spectroscopy.
Another external cavity diode laser is also split into two beams. The first beam serves as the input beam passing through a Q-plate to generate an LG beam with OAM [34], this beam, with radius of $\sim 160$ $\mathrm{\mu}$m in the center of the vapor cell, is then split into multiple orders by the beam splitter shown in Fig. 1(b). The second beam serves as the detection beam, it is shaped by the AP2 and is used for interfering with the input beam to measure the topological charge of the output beams. The output beams are observed with a charge coupled device (CCD), which is $\sim 10$ cm from the center of the vapor cell, shows the spatial and intensity distribution of these beams in real time.
3. Results and discussions
For deeper understanding of the performance of this HOG beam splitter, the theoretical analysis is first presented. In contrast with the beam splitter designed in terms of a solid-state structure, the HOG beam splitter proposed here depends on a three-level $V$-type atomic system, as shown in Fig. 1(a). The susceptibility of this atomic structure is described as [22]
In the following, the LG beam is taken as the input beam passing through this HOG beam splitter. The beam waist is $\omega _{0}$, and $\omega (\xi )=\omega _{0}(1+\xi ^{2})^{1/2}$ defines the radius at which the electric field intensity decreases to $e^{2}/2$ of its maximum value, where $\xi =z/z_{R}$, and $z_{R}=\pi \omega _{0}^{2}/\lambda _{i}$ is the Rayleigh length. The field of the input LG beam can be expressed as [35]
Figure 2 shows the normalized splitting pattern distribution of $l=1$ LG beam for different control fields. The Rabi frequencies of the control fields in Figs. 2(a-c) are $0.8\Gamma _{2}$, $1.7\Gamma _{2}$, and $2.9\Gamma _{2}$, and the detunings of the two laser fields are $\Delta _{c}=-3.6\Gamma _{2}$ and $\Delta _{i}=2.8\Gamma _{2}$, respectively. A phase modulation is introduced owing to such far detuned driving fields, which can effectively suppress the nonlinear absorption and enhance the refractivity of the atomic medium [19]. It can be seen that, under the introduction of a phase modulation, an increasing amount of laser energy is transferred from the 0th order to the $\pm$1st and $\pm$2nd order directions upon increasing the Rabi frequency of control laser. This suggests that a desired splitting ratio can be obtained by selecting an appropriate Rabi frequency of the control laser in this beam splitter when the other parameters (such as detuning, grating constant, and atomic density) are suitable and remain constant.
Fig. 2. Theoretical map of the normalized splitting pattern distribution of the LG beam, where the Rabi frequencies of the coupling fields from (a) to (c) are $0.8\Gamma _{2}$, $1.7\Gamma _{2}$, and $2.9\Gamma _{2}$, respectively. The other parameters are $l=1$, $\Delta _{c}=-3.6\Gamma _{2}$, $\Delta _{i}=2.8\Gamma _{2}$, $\Gamma _{1}=0$, and $\Gamma _{3}=0.95\Gamma _{2}$.
Figures 3(a-d) illustrate the output profile of an LG$_{10}$ ($l=1$ and $p=0$) beam acquired via the CCD. The powers of the control and input lasers are 20.0 and 2.6 mW, and their corresponding frequency detunings are about -180 and 150 MHz. The angle of two control lasers is $2\varphi \approx$0.40$^{\circ }$, resulting the grating constant $d=\lambda _{c}/2sin(\varphi )\approx 110$ $\mathrm{\mu}$m. Additionally, the atom density remained $\sim 8.78\times 10^{12}$ $cm^{-3}$ throughout the experiment. In the absence of a periodically modulated optical field, the output beam shown in Fig. 3(a) is an LG beam, which has the same topological charge ($l=1$) as the input beam shown in Fig. 3(b). In Fig. 3(c), a distinguishable three-port output beam is observed with the beam splitter. The interference pattern shown in Fig. 3(d) indicates that three perfect replicas of the input beam are generated. Furthermore, the input beam was changed to a $l=2$ LG beam, for which the grating constant is chosen as $d\approx 80$ $\mathrm{\mu}$m (2$\varphi \approx$0.56$^{\circ }$)and the power of the control laser was set to 50 mW. The experimental results shown in Figs. 3(e-f) indicate that the output split beams have the same topological charge ($l=2$) as the input beam. In order to further verify the detected results, a cylindrical lens is used to detect the output beam as well [36–38], which show the same viewpoint as the interference results [see Fig. 3(i)]. Thus, the beam splitter can effectively work for both the $l=1$ and $l=2$ LG beams, and it can be reasonably assumed that the beam splitter can work for all $l=n$ LG beams.
Fig. 3. (a) and (c) illustrate the output profiles of a weak LG$_{10}$ beam without and with the periodically modulated optical field, respectively. (b) and (d) illustrate the corresponding interference patterns obtained through the detection beam. (e-h) display the case of the LG$_{20}$ beam. (i) The measured topological charge of output beam by a cylindrical lens.
In existing reports on the design of solid-state structure beam splitters, a fixed splitting ratio is obtained using the fabricated arrays with an unchangeable interval. By contrast, beam splitters with an adjustable splitting ratio are required in many applications, such as holography, interferometers, and optical information processing. As the theoretical prediction in Fig. 2 shows, a continuously variable beam splitting ratio can be obtained as a consequence of phase modulation by changing the control laser power of the input LG beam.
The influence of the control laser power on the splitting ratio of the HOG beam splitter is here analyzed in detail, as shown in Figs. 4(a-e). Here, the power of control laser was chosen as 5, 15, 25, 35, 45, and 55 mW. In Figs. 4(a) and 4(b), the intensity distribution of the output beams is mainly concentrated in the 0th order for a relatively low power of the control laser. The $\pm$1st orders gradually become clearer with the increase of the control laser power, and the beam energy of the three output ports is almost the same when the power reaches 25 mW [see Fig. 4(c)]. As shown in Fig. 4(d), the weak $\pm$2nd orders are observed when the power exceeds 35 mW, which indicates that the proposed scheme can be used at least as a $1\times 5$ HOG beam splitter. By further increasing the control laser power, an increasing amount of energy transfers from the 0th to the $\pm$1st and $\pm$2nd orders under the conditon of phase modulation, and the energy distribution tends to be saturated when the power is sufficiently high [see Figs. 4(e) and 4(f)]. These results of low 0th-order and high $\pm$1st-order intensities are extremely difficult to achieve with conventional solid-state structure beam splitters. The experimental results are consistent with the theoretical predictions shown in Fig. 2.
Fig. 4. (a-f) Output splitting beams for different control laser powers. (g) Corresponding 0th- and +1st-order diffraction efficiency for different control laser powers.
To specifically quantify the splitting ratio of the proposed beam splitter, taking the intensity of the $+$1st and 0th orders as an example, the adjustable range of the splitting ratio of the beam splitter is investigated. Figure 4(g) shows the corresponding 0th- and $+$1st-order diffraction efficiency with the variable control laser power, which are defined as $\eta _{0}=I_{0}'/I_{0}$ and $\eta _{1}=I_{1}'/I_{0}$, where $I_{0}$ is the intensity of input beam passing through the vapor without the periodically modulated optical field, and $I_{1}'$, $I_{0}'$ are the intensities of $+$1st- and 0th-order diffraction beams with the periodically modulated optical field, respectively. The error bars are the standard deviation of three experimental measurements. As the control laser power increases, the $+$1st-order diffraction efficiency exhibits a nonlinear trend which first increases and then saturates at a power of 35 mW, while the 0th-order diffraction efficiency has the opposite change. Thus, the adjustable range of the splitting ratio between $+$1st-order and 0th-order, which is obtained by $\delta =\eta _{1}/\eta _{0}$, is about 0-4.8 in the experiment.
The position and intensity of the splitting beams can be affected by the grating constant $d$ of the periodic dielectric atomic structure, which according to the theory increases as the angle between the two control lasers decreases. Figure 5 shows the output splitting beams for the two control lasers angle 2$\varphi$, where $\varphi$ varying from 0.16$^{\circ }$ to 0.38$^{\circ }$. In this case, the power of control laser is 50 mW. A compact five-port output beam with a low energy 0th order can be observed at an angle $\varphi$ of about 0.16$^{\circ }$, as shown in Fig. 5(a). Upon increasing the grating constant, not only the intensity of the 0th order increases while that of the $\pm$1st and $\pm$2nd orders decreases, but also the intervals of the splitting beams increase at the same time. The intervals between each output port increase clearly when the grating constant is changed from 0.24$^{\circ }$ to 0.30$^{\circ }$, as shown in Figs. 5(d-f). However, when the angle is too large [see Fig. 5(h)], the intervals between the 0th and $\pm$1st orders become so large that the splitting phenomenon is hard to detect via the input beam. Thus, the position of all splitting beams can be fine-tuned through an appropriate grating constant value, which is more conducive to the study of the LG beam distribution.
Fig. 5. Output splitting beam patterns for different angles of two control lasers, where $\varphi$ is about (a) 0.16$^{\circ }$, (b) 0.20$^{\circ }$, (c) 0.22$^{\circ }$, (d) 0.24$^{\circ }$, (e) 0.28$^{\circ }$, (f) 0.30$^{\circ }$, (g) 0.32$^{\circ }$, and (h) 0.38$^{\circ }$.
Compared with the LG beams, the beam splitting of the HG beams is more difficult to achieve due to its larger size and the presence of more lobes. However, in the field of optical communication, the beam splitting of the HG beams is also an important technical requirement [10]. Considering the HG beams as the input laser field, the high-order HG beam can be expressed as [35]
Fig. 6. Theoretical map of the normalized simulation of (a) Gaussian (HG$_{00}$), (b) HG$_{01}$, and (c) HG$_{11}$ beams and corresponding normalized splitting results of the beams passing through the splitter. The parameters are $\Omega _{c}=2.0\Gamma _{2}$, $\Delta _{c}=-3.6\Gamma _{2}$ and $\Delta _{i}=2.8\Gamma _{2}$.
Figure 7 illustrates the experimental results of the output profile of the Gaussian (HG$_{00}$), HG$_{01}$, and HG$_{11}$ beams passing through this HOG beam splitter without and with the periodically modulated optical field. The experimental parameters used here are consistent with those of the LG$_{20}$ beam in Figs. 3(g). A clear three-port output can be observed, and each output beam has the same number of lobes as the input beam, which is consistent with the theoretical simulation shown in Fig. 6. By changing the corresponding parameters of this HOG beam splitter, the change of the high-order HG beam is the same as that of the LG beam. In other words, this all-optical device can be predicted to work well with all types of HOG beams, including the high-order Bessel beams in theory, and may overcome the limitations of existing devices for splitting HOG beams.
Fig. 7. Experimental observation of the output profile of (a) Gaussian (HG$_{00}$), (b) HG$_{01}$, and (c) HG$_{11}$ beams without and with the periodically modulated optical field.
4. Conclusions
In conclusion, a novel all-optical HOG beam splitter using a periodic dielectric atomic structure was demonstrated. When a weak HOG beam is incident on the beam splitter, the output multi-beams inherit the excellent properties of the original beam, such as the topological charge of the LG beams and the number of lobes of the HG beams. The splitting ratio range can be finely adjusted from 0 to 4.8 by changing the control laser power, and the intervals of the output beam at all orders can be fine-tuned by varying the grating constant of the beam splitter. In contrast to the traditional solid-state structure beam splitter, the proposed atomic structure beam splitter can regulate the splitting beams using only optical fields, which endows this all-optical device with a high efficiency and fast response speed. In this work, a distinguishable five-port output beam is observed, and output splitters with up to seven and nine ports are achievable [21,22]. The proposed periodic dielectric atomic structure, which splits a HOG beam spatially into different positions and intensities, can be realized in theory as a $1\times N$ beam splitter, making it possible for the increase of channel capacity, multiple particle trapping and tweezing, and fast micromachining [9,15,39,40]. This work is also a first attempt of HOG beams in periodic dielectric atomic structure, lays a foundation for more interesting experiments about HOG beams with such structure in the future, as well as conducive to the construction of an all-optical network.
Funding
National Key Research and Development Program of China (2017YFA0304203); National Natural Science Foundation of China (61875112, 62075121, 91736209); Program for Sanjin Scholars of Shanxi Province; Key Research and Development Program of Shanxi Province for International Cooperation (201803D421034); 1331KSC.
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. Y. Zhang, J. Shen, C. Min, Y. Jin, Y. Jiang, J. Liu, S. Zhu, Y. Sheng, A. Zayats, and X. Yuan, “Nonlinearity-induced multiplexed optical trapping and manipulation with femtosecond vector beams,” Nano Lett. 18(9), 5538–5543 (2018). [CrossRef]
2. Y. Kozawa, D. Matsunaga, and S. Sato, “Superresolution imaging via superoscillation focusing of a radially polarized beam,” Optica 5(2), 86–92 (2018). [CrossRef]
3. D. Ding, W. Zhang, Z. Zhou, S. Shi, G. Xiang, X. Wang, Y. Jiang, B. Shi, and G. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114(5), 050502 (2015). [CrossRef]
4. Z. Xie, S. Gao, T. Lei, S. Feng, Y. Zhang, F. Li, J. Zhang, Z. Li, and X. Yuan, “Integrated (de)multiplexer for orbital angular momentum fiber communication,” Photonics Res. 6(7), 743–749 (2018). [CrossRef]
5. E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3(5), e167 (2014). [CrossRef]
6. E. Galvez, “Gaussian Beams.” (Colgate University, 2014).
7. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
8. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle–orbital angular momentum variables,” Science 329(5992), 662–665 (2010). [CrossRef]
9. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef]
10. B. Rodenburg, “Communicating with Transverse Modes of Light.” (University of Rochester, 2015).
11. Y. Zhou, J. Zhao, Z. Shi, S. M. H. Rafsanjani, M. Mirhosseini, Z. Zhu, A. E. Willner, and R. W. Boyd, “Hermite-Gaussian mode sorter,” Opt. Lett. 43(21), 5263–5266 (2018). [CrossRef]
12. I. Moreno, J. A. Davis, B. M. Pascoguin, M. J. Mitry, and D. M. Cottrell, “Vortex sensing diffraction gratings,” Opt. Lett. 34(19), 2927–2929 (2009). [CrossRef]
13. J. C. T. Lee, S. J. Alexander, S. D. Kevan, S. Roy, and B. J. McMorran, “Laguerre-Gauss and Hermite-Gauss soft x-ray states generated using diffractive optics,” Nat. Photonics 13(3), 205–209 (2019). [CrossRef]
14. K. Dai, C. Gao, L. Zhong, Q. Na, and Q. Wang, “Measuring OAM states of light beams with gradually-changing-period gratings,” Opt. Lett. 40(4), 562–565 (2015). [CrossRef]
15. T. Lei, M. Zhang, Y. Li, P. Jia, G. Liu, X. Xu, Z. Li, C. Min, J. Lin, C. Yu, H. Niu, and X. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light: Sci. Appl. 4(3), e257 (2015). [CrossRef]
16. D. Marco, M. Sánchez-López, A. Cofré, A. Vargas, and I. Moreno, “Optimal triplicator design applied to a geometric phase vortex grating,” Opt. Express 27(10), 14472–14486 (2019). [CrossRef]
17. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). [CrossRef]
18. I. L. Garanovich, S. Longhi, A. A. Sukhorukova, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012). [CrossRef]
19. H. Ling, Y. Li, and M. Xiao, “Electromagnetically induced grating: Homogeneously broadened medium,” Phys. Rev. A 57(2), 1338–1344 (1998). [CrossRef]
20. J. Yuan, C. Wu, L. Wang, G. Chen, and S. Jia, “Observation of diffraction pattern in two-dimensional optically induced atomic lattice,” Opt. Lett. 44(17), 4123–4126 (2019). [CrossRef]
21. J. Yuan, S. Dong, C. Wu, L. Wang, L. Xiao, and S. Jia, “Optically tunable grating in a $V+\Xi$ configuration involving a Rydberg state,” Opt. Express 28(16), 23820–23828 (2020). [CrossRef]
22. J. Yuan, S. Dong, H. Zhang, C. Wu, L. Wang, L. Xiao, and S. Jia, “Efficient all-optical modulator based on a periodic dielectric atomic lattice,” Opt. Express 29(2), 2712–2719 (2021). [CrossRef]
23. Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, “Four-wave mixing dipole soliton in laser-induced atomic gratings,” Phys. Rev. Lett. 106(9), 093904 (2011). [CrossRef]
24. F. Wen, H. Ye, X. Zhang, W. Wang, S. Li, H. Wang, Y. Zhang, and C. Qiu, “Optically induced atomic lattice with tunable near-field and far-field diffraction patterns,” Photonics Res. 5(6), 676–683 (2017). [CrossRef]
25. A. W. Brown and M. Xiao, “All-optical switching and routing based on an electromagnetically induced absorption grating,” Opt. Lett. 30(7), 699–701 (2005). [CrossRef]
26. D. Wang, H. Zhou, M. Guo, J. Zhang, J. Evers, and S. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110(9), 093901 (2013). [CrossRef]
27. Z. Zhang, R. Wang, Y. Zhang, Y. V. Kartashov, F. Li, H. Zhong, H. Guan, K. Gao, and M. Xiao, “Observation of edge solitons in photonic graphene,” Nat. Commun. 11(1), 1902 (2020). [CrossRef]
28. H. Cai, J. Liu, J. Wu, Y. He, S. Zhu, J. Zhang, and D. Wang, “Experimental observation of momentum-space chiral edge currents in room-temperature atoms,” Phys. Rev. Lett. 122(2), 023601 (2019). [CrossRef]
29. Y. He, R. Mao, H. Cai, J.-X. Zhang, Y. Li, L. Yuan, S. Zhu, and D. Wang, “Flat-band localization in Creutz superradiance lattices,” Phys. Rev. Lett. 126(10), 103601 (2021). [CrossRef]
30. Y. Xiao, M. Klein, M. Hohensee, L. Jiang, and D. F. Phillips, “Slow Light Beam Splitter,” Phys. Rev. Lett. 101(4), 043601 (2008). [CrossRef]
31. Y. Zhang, C. Zuo, H. Zheng, C. Li, Z. Nie, J. Song, H. Chang, and M. Xiao, “Controlled spatial beam splitter using four-wave-mixing images,” Phys. Rev. A 80(5), 055804 (2009). [CrossRef]
32. K. K. Park, T. M. Zhao, J. C. Lee, Y. T. Chough, and Y. H. Kim, “Coherent and dynamic beam splitting based on light storage in cold atoms,” Sci. Rep. 6(1), 34279 (2016). [CrossRef]
33. T. Jeong, Y. T. Chough, and H. S. Moon, “Light manipulation via spontaneous four-wave mixing in a warm double-Λ-type atomic ensemble,” Opt. Express 28(24), 36611–36619 (2020). [CrossRef]
34. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009). [CrossRef]
35. F. Pampaloni and J. Enderlein, “Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer,” arXiv:physics/0410021.
36. S. N. Alperin, R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41(21), 5019–5022 (2016). [CrossRef]
37. A. Chopinaud, M. Jacquey, B. V. de Lesegno, and L. Pruvost, “High helicity vortex conversion in a rubidium vapor,” Phys. Rev. A 97(6), 063806 (2018). [CrossRef]
38. R. F. Offer, A. Daffurn, E. Riis, P. F. Griffin, and A. S. Arnold, “Gouy phase-matched angular and radial mode conversion in four-wave mixing,” Phys. Rev. A 103(2), L021502 (2021). [CrossRef]
39. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24(9), 608–610 (1999). [CrossRef]
40. T. Omatsu, K. Chujo, K. Miyamoto, M. Okida, K. Nakamura, N. Aoki, and R. Morita, “Metal microneedle fabrication using twisted light with spin,” Opt. Express 18(17), 17967–17973 (2010). [CrossRef]
