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Abstract
Ultrahigh-order optical orbital angular momentum (OAM) states of the identification over ±270 orders are implemented by annular phase grating (APG) and Gaussian beams with different wavelengths. Particularly, the far-field diffraction intensity patterns feature the spiral stripes instead of Hermitian-Gaussian (HG)-like fringes. It’s worth noting that the spiral stripes present uniform distribution, thus the order of OAM states can be intuitively acquired. More specifically, the OAM states can be confirmed from the total amount and rotating direction of the spiral stripes. Compared with traditional methods, the propose scheme contributes to the perfect-distributed and sharper spiral stripes. Moreover, it also makes an easier observation of the patterns in the CCD camera with limited imaging targets. In our experimental setup, the optical filter is removed and the APG parameters are not strictly required. Therefore, the propose optical transmission system is equipped with the advantages of efficiency, robustness and low cost, which paves a promising way for the communication capacity enhancement.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Orbital angular momentum (OAM) beams are shining the next generation of optical communication [1–6] systems because of their effective spectrum utilization, excellent security and high transmission rate [7]. Vortex beams with azimuthal phase exp(ilφ) carry OAM of lh per photon. Here, φ is azimuthal angle, l is the topological charge (TC) or the order of OAM states and h is the reduced Planck’s constant [8–10]. Theoretically, the value of l is assumed to be unlimited, and it induces an infinite number of attainable OAM states. Particularly, the aforementioned characterization of high-order vortex beams enables its widely application in various fields, such as optical imaging [11], quantum information processing [12,13], optical micro-manipulation [14,15] and super-resolution microscopy [16]. Among the these continuously OAM applications of development, it is essential and vital to realize the high-order TC measurement.
Up to now, many different schemes for detecting OAM states have been proposed. Generally, the most popular approach is to observe the interference stripes by interfering a vortex beam with its own mirror image or a reference wave. Then this method determines the TC with the help of the relationship between the interference fringes and the topological number, such as the scheme based on Mach-Zehnder interferometer [17–20]. However, the interfering condition demands strictly of the system and limits the measurement of the OAM states. At the same time, the measurement methods for the amplitude diffraction comprise different slits and apertures [21–25], optical transformations [26,27], special diffraction gratings [28,29] and rotating Doppler effects [30]. Nevertheless, most of these experiments require the use of specialized optical components or a certain number of optical elements with precise alignment. On the basis of various unfavorable factors, the current ability for these approaches to measure high-order OAM states is limited. To overcome the drawbacks of these techniques, Vaity et al. proposed to use a tilted lens to detect OAM states of vortex beams [31]. This solution is capable of detecting both the value and sign of the TC up to ±14. Furthermore, Hermitian-Gaussian (HG) and Laguerre-Gaussian (LG) transitions have been established to measure OAM states, and a mode converter consisting of a pair of cylindrical lenses has been used [32–34]. These methods greatly improve the detection range of OAM states with the robustness and low cost, however, they are not highly adaptable or switchable.
Commendably, Dai et al. reported that a gradually-changing-period amplitude grating can determine the OAM states by observing the HG-like diffraction intensity patterns [35]. The computer usage for flexible operations makes this approach manageable and simple. Unfortunately, its energy efficiency is relatively lower because of the amplitude grating. Fu et al. then developed a phase-diffractive optical element to improve measure efficiency [36]. In the same year, Li et al. proposed a specially designed gradually-changing-period phase grating to improve the detection sensitivity of OAM states [37]. Wang et al. offered another method based on annular gratings to achieve a TC measurement up to ±25 later [38]. Li et al. proposed to determine the topological charges of perfect vortices using the phase shift method, which can determine the TC generated by perfect vortices and other types of vortices in the Fourier plane of the SLM [39]. In addition, Liu et al. proposed a deep learning method to accurately identify OAM patterns with fractional topological charges. The minimum recognition interval between adjacent patterns is reduced to 0.01, which is practical for the next generation of deep learning-based OAM optical communication [40].
Recently, various shapes of period phase gratings including radial phase grating [41], hyperbolic phase grating (HPG) [42] and spiral spoke grating [43] were proposed for high-order OAM states detection, and the measurable TC is up to the high orders of ±120, ±100 and ±160 respectively. These identification methods are essentially the same, and both values and signs of TC correspond to the numbers and tilt directions of the stripes in the HG-like intensity patterns, respectively. Notably, these methods are switchable, reconfigurable and scalable, because they can be implemented by a liquid-crystal spatial light modulator (SLM) and it has the potential for the further improvement of the detection range of OAM states. However, for better recognition, these methods essentially need to be adjusted by the optical filters and the off-axis grating parameters, which are relatively complicated and cumbersome. In addition, for high-order OAM states, stripes are presented non-uniform shapes due to the defects of the HG-like diffraction intensity patterns, which characterize wider at the edges and narrower in the middle. With the increment of OAM orders, the stripes become more and more blurred. Besides, the useful stripes patterns may overflow the imaging region of a general CCD camera. Therefore, the aforementioned reasons may lead to difficulty about ultrahigh-order OAM states measurement.
In this paper, we propose a novel scheme for measuring ultrahigh-order OAM states with an annular phase grating (APG). Meanwhile, a Gaussian beam of different wavelengths is used as auxiliary beams for the LG beam, as shown in Fig. 1. Both numerical analysis and experimental results demonstrate the propose scheme is an efficient and simple. Compared with predecessors, our scheme simplifies the operation by removing the adjustments of the grating parameters and the optical filter. Moreover, this scheme changes the inhomogeneity of the previous HG-like intensity patterns to obtain highly uniformed and distinct spiral stripes, which makes their reception and identification easier. The TC detection can be achieved higher than ±270 through this method, and this scheme is useful for detecting the multiplexed OAM states.
2. Theory model and simulation analysis
The LG beam is commonly used as a target for detection because it carries well-defined OAM. Here, we choose the LG beam to detect the OAM states to demonstrate the feasibility of the scheme. The phase function of the APG can be described as:
where the APG transmission changes along the radial direction r with a period of a. The optical field of the LG beams can be described as:The far-field diffraction intensity is obtained by Eq. (1) and Eq. (2), and it can be expressed as:
The following simulation demonstrates how to measure the high-order OAM states of vortex beams. The measurement schematic for TC of vortex beams with an APG and a Gaussian beam is illustrated in Fig. 1. It is well known that phase singularities exist in the vortex beam. However, it appears multiple phase singularities with the help of Gaussian beams, which correspond to their TC values. Meanwhile, the central part of the APG has the ability to wrap it into a regular stripe. In the figure, a Gaussian beam is used as an auxiliary beam combined with vortex beams to illuminate on or near the center of the APG. In the detection plane, a different result from the previous methods [35–41] was obtained, spiral and highly uniform stripes appeared in the far-field diffraction pattern. Compared with the previous classical scheme [40], the shape of stripes in the propose scheme is circular and the light intensity distribution becomes uniform, which provides a basis for being able to detect larger TCs. Conveniently, we can also identify OAM states by the numbers and rotating directions of dark or light stripes. In addition, both this paper and Ref. [37] use APG, but we use APG center and its phase period asymptotically changes faster, i.e., another unique property of APG is exploited. At the same time, our system is much easier to set up. Most importantly, we have the same advantages that have been compared with the above literature. The right half of the figure shows the number of spiral stripes with l = ±10. The positive sign indicates left rotation while the negative sign indicates right rotation and the TC value corresponds to the number of stripes. In addition, the lower part of the figure records the process from the generation of the vortex beam to the formation of the spiral stripes.
The simulation model can be built based on Eq. (3). The APG can be coded and projected from the phase function Eq. (1), where the grating period a has no strict parameter constraint. However, when the APG period is too large, the phase gradient speed is too slow, resulting in the detected beam not being wrapped. When the APG period is too small, the detected beam will show a large halo and the OAM mode cannot be measured. Therefore, an APG period that is too large or too small will not detect the OAM modes, which is the same characteristic as other grating properties. For simplicity, a is set to 0.0004 mm in our whole task and is feasible around this period. Figures 2(a)-(c) are the simulation results for l = ±20, l = ±100, and l = ±160, with D = 0, respectively, where D is the gap between the position of the incident beam and the grating center. As with the identification method of APG in Fig. 1. Figure 2(a) shows the rotating direction to obtain the left and right rotations of stripes. The number of both light and dark stripes is 20, which is an advantage in this scheme. The direction is determined by the signs of TC, where the positive sign represent the left rotation and the opposite represent the right rotation. There is no error recognition through the distinction of the dark or light stripes.
Fig. 2. (a)-(c) The simulation results for l = ±20, l = ±100, and l = ±160, where the left rotation represents a positive sign and the opposite represents a negative sign. Figures 2(b1) and 2(c1) The enlarged upper right parts for l = +100, and l = +160, respectively.
It can be visually observed that nearly 100% diffraction efficiency is achieved by APG, because almost all of the energy are concentrated in the diffraction ring and no diffraction levels of other orders appear. Meanwhile, the far-field diffraction intensity pattern exhibits a spiral and uniform distribution, where the pattern shape is different from conventional HG-like stripes. It can be seen that the spiral stripes with uniform distribution become clearer and more compact, which are less demanded on the receiving surface. Figure 3(a) shows the simulation of the light intensity distribution for l = -20, and the features of the pattern are more transparently represented and its distribution is perfect. Therefore, the advantage of this scheme is that the stripes take up a smaller useful area of the pattern, making a low-cost reception possible and reducing the complexity of pattern recognition as well. Figures 2(b1) and 2(c1) show the correspondence of the enlarged upper right parts of the simulation results for l = +100, and l = +160, respectively. One more point is that the area and thickness occupied by the stripes in the pattern are affected by the period a. Moreover, when the period a is constant, the halo of the inner ring in the pattern becomes more and more visible as the number of stripes increases. Normally, higher-order OAM states lead to the difficulty to distinguish them in other schemes, and this issue can be overcome in a significant level due to the advantage of the stripes obtained in the proposed scheme.
Fig. 3. (a) The simulation of the light intensity distribution for l = -20. The upper figure is a three-dimensional view with the horizontal and vertical axes in pixels, and the spiral stripes are shown as a top view in the lower figure. (b) The simulation results for the offset positions of l = +20, are Dx = 0.3 mm, 0.6 mm, and 0.9 mm, respectively. (c) Dx = -0.3 mm, -0.6 mm, and -0.9 mm, respectively. (d) Dy = 0.3 mm, 0.6 mm, and 0.9 mm, respectively. (e) Dy = -0.3 mm, -0.6 mm, and -0.9 mm, respectively. (f) Dx = ±0.6 mm, and Dy = ±0.6 mm, respectively. Dx is positive indicates that the position of the beam illuminated to the center of the grating (D = 0) is shifted to the right, and is negative to the left. Similarly, a positive Dy indicates an upward shift, and is negative to a downward shift.
Figures 3(b)-(f) show that the beam offset from the grating center in each direction with different distances for l = +20, respectively. We have selected three different distances of 0.3 mm, 0.6 mm, and 0.9 mm in each direction under discussion, respectively. For example, Fig. 3(b) shows the results of the simulation where the incident point is shifted to the right by three different distances from the APG center. Similarly, Fig. 3(f) shows the simulation results obtained by shifting the incident point to the different off-axis directions. It is observed that the best results are obtained when the beam illuminates the grating center. Moreover, when the beam illuminates the vicinity of the APG center, it still can produce spiral stripes but only changes the light intensity distribution of the pattern. Although the distribution has been changed, the number and direction of the spiral stripes can still be clearly observed as long as the point of the beam incidence is not shifted too much. At the same time, this also shows the stability of the proposed scheme.
In addition, it can be noticed that the light intensity distribution of the stripes is regularly changing with the point of incidence of the light beam in Fig. 3. For example, the light intensity on the right half of the pattern in Fig. 3(b) become weaker if the incident point is shifted to the right. At the same time, the light intensity of the left half of the pattern in Fig. 3(c) becomes weaker with the reverse offset of the incident point, and we have a similar situation for Figs. 3(d)-(f). Thus, the regularity shows that the light intensity of the pattern becomes indistinct in the corresponding direction when the incident point is shifted away from the center of the APG. It is worth noting that the regularity also demonstrates the high symmetry of the APG, which contributes to a better tuning of the optical path. The simulation results prove the advantages of the scheme and can highlight its features as well. Furthermore, the stripes become more uniformly and easy to be received, and can be more clearly observed and identified compared with the previous classical schemes. Therefore, the propose method can easily and efficiently measure the high-order OAM states. Based on the results of numerical simulations, the following experiments were conducted to verify the feasibility of the proposed approach.
3. Experimental setup and results
The experimental setup for the generation and detection of LG beams is illustrated in Fig. 4. A He-Ne laser1 with a center wavelength 1550 nm is employed as the light source to produce a zero-order Gaussian beam. The light beam travels through a beam expander consisting of lenses L1 and L2 with a roughly 2 mm beam size. The dimensions are consistent with the simulated setup and adaptable to the liquid crystal SLM. Since the SLM is sensitive to the polarization state of a light beam, the polarization direction can be adjusted by a polarization beam splitter to obtain a P-polarized Gaussian beam. The beam illuminates the SLM1 loaded with the phase hologram to generate the LG beam. Meanwhile, laser2 emits a Gaussian beam with a center wavelength 1570 nm as an auxiliary beam. The two beams are combined by a beam splitter (BS) and used as a detection object. None interference occurs due to their different frequencies, because the presence of interference can have a significant impact on the results. Finally, the detection object illuminates the SLM2 loaded with the APG in Fig. 1, and the grating is a gray-scale picture which is known to all. The APG we use is generated by the computer and loaded onto the SLM, where the schematic diagram of the APG is shown in Fig. 1.
Fig. 4. Experimental setup for measuring multiplexed OAM states. laser1: ${\lambda _1} = 1550$ nm; laser2: ${\lambda _1} = 1570$ nm; PBS1, PBS2: polarization beam splitter; BS: non-polarized beam splitter; SLM1, SLM2: spatial light modulator; L1, L2, L3: plano-convex lens; CCD: charge-coupled device.
In our experiments, the APG period a is set to 0.0004 mm and the gap D between the grating centers and the LG beam is set to 0. For the purpose of imaging the far-field diffraction pattern on the picture plane of the CCD camera, a plano-convex lens L3 with a focal distance of 200 mm is utilized. Since the laser has a certain optical power and the CCD has a limited receiving surface, it needs to be received by lens L3. The resolution of the SLM (PLUTO-TELCO-013-C) is 1920 × 1080 and the pixel pitch is 8.0 µm. The resolution of the CCD (Digital CamIR1550 463125) is 1296 × 964 and the pixel pitch is 3.75 µm. In Fig. 4, the experimental system removes the frequently used optical filter in previous classical schemes [35–41], because LG beams of other orders have no effect on our experimental results or observations. Meanwhile, they will be helpful for detecting the ultrahigh-order OAM states. The reason is that the stripes produced by the other spatial frequencies will make them to be measured more coherent and well-defined. Therefore, the system in Fig. 4 simplifies the experimental procedure and is more compact when measuring the high-order OAM states.
Figures 5(a)-(c) show the experimental results for the TC detection of the LG beams with l = ±20, l = ±100, and l = ±160, respectively. The original experimental image has been binaried, and the subsequent processing is also similar. The number of stripes and their rotating directions corresponding to the TC values and sign can be clearly distinguished. Figures 5(b1) and 5(c1) show the enlarged upper right part of l = +100, and l = +160, respectively. Consistent with the simulation results, the light intensity distribution and the thickness distribution of the spiral stripes in the Fig. 5 are especially uniform. Figure 6(a) shows the three-dimensional light intensity distribution of the original experimental plot for l = -20, which is in general agreement with the simulation results. The light intensity distribution is not perfect due to the lack of the proper adjustment of the optical path. However, we can refine it by the regularity found in Fig. 3. In addition, the gaps among the spiral stripes are extraordinarily homogeneous. Based on this advantage and the circumference of the circle formed by the spiral stripes, the TC of the high-order OAM can be calculated experiment directly by the image processing algorithms. From above experiment, the effect of our scheme to measure the high-order OAM states is obvious. To further demonstrate the advantages of the proposed scheme, the detection of ultrahigh-order OAM states is shown in the following.
Fig. 5. (a)-(c) The experimental results for l = ±20, l = ±100, and l = ±160, respectively, where the left rotation represents a positive sign and the opposite represents a negative sign. Figures 5(b1) and 5(c1) The enlarged upper right part of l = +100, and l = +160, respectively.
Fig. 6. (a) The light intensity distribution of the original experimental plot for l = -20. The upper part is the three-dimensional view with the horizontal and the vertical axes as pixels. The spiral stripes in the lower part are its top view. (b) and (c) The experimental results for l = ±220 and l = -270, respectively, where the left rotation represents a positive sign and the opposite represents a negative sign. Figures 6(b1), 6(b2) and 6(c1) The enlarged upper right part of l = +220, l = -220, and l = -270, respectively.
Figures 6(b) and 6(c) show the experimental results for l = ±220 and l = -270, respectively. The numbers and rotating directions of the spiral stripes with uniformly distributed can be clearly observed, which corresponding to the TC values and signs of OAM. The enlarged upper right part for l = +220, and l = -220 are shown in Figs. 6(b1) and 6(b2), respectively. Figure 6(c1) shows the enlarged upper right part for l = -270. From this figure, when the TC value increases, the stripes in the pattern become narrower, but the identification of the ultrahigh-order OAM states is not affected. The above experimental results would support our theoretical analysis and show an extraordinary method to detect ultrahigh-order OAM states.
In addition, due to the limitation of the light source power and the imperfect adjustment of the optical path, this scheme only measures the OAM states up to l = 270, but this is not the peak for this scheme. However, we can combine image processing algorithms and deep learning methods to provide several addressing options. Firstly, the number of indistinct stripes is calculated by an image processing algorithm with the uniformity of the resulting intervals of stripes, and the TC of OAM is directly identified. Secondly, the width of the stripes and the gap among them correspond to each an OAM state, which means an ultrahigh-order OAM can be identified directly and quickly by this relation. Finally, the magnitude of the tilt in the direction of the stripes, which is an OAM state, is also different and it can be observed, and provides us with one of the solutions. The measured TC of OAM is up to ±270, and the higher-order OAM states can be determined more directly and quickly with these solutions.
Meanwhile, when the perfection of the generated LG beams or devices is taken into account, the detection range of this scheme can be substantially greater with tolerable distortion. It is worth mentioning that if a multiplexed LG beam is used to illuminate the APG, the stripes in the pattern may be related to the two OAM states of the multiplexed LG beam. Therefore, it can be helpful to use this scheme in this paper to detect the superimposed LG beams, which have been effective in our subsequent experiments.
4. Conclusions
In conclusion, we proposed a flexible, efficient and simple scheme to measure ultrahigh-order OAM states which reach over ±270-order with APG. Meanwhile, a Gaussian beam with different wavelengths is used as auxiliary beams for the LG beam. The advantages of our experiments are that we can eliminate the steps to adjust various parameters or devices, thus the operation process is simplified and the robustness of the experimental system is improved. It is worth noting that the inhomogeneity of the previous HG-like intensity patterns is broken, which converts to a uniformly distributed spiral stripe. As the stripes become more compact and sharper, they provide more space for the CCD camera to collect patterns of the high-order OAM states, which can save cost. Moreover, for the high-uniformity of spiral stripes in the pattern, the TC of OAM can be obtained directly and quickly by image processing algorithms and other deep learning methods, which will improve the recognition efficiency.
More promisingly, the indistinct stripes obtained from the ultrahigh-order OAM states can be calculated directly, through the above-mentioned methods, which will be more efficient and practical. Our research demonstrates that the proposed approach has excellent performance and applicability for analyzing the ultrahigh-order OAM states. In addition, it can contribute to an effective solution for measuring multiplexed OAM states. The proposed scheme will be useful for many applications based on OAM.
Funding
National Natural Science Foundation of China (62105373, 62171487); Knowledge Innovation Program of Wuhan-Shuguang Project (2022010801020408); Key Technology R&D Program of Hubei Province (2020BBB097); Fundamental Research Funds for Central Universities of the Central South University (CZZ22001).
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.
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