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Abstract
The traditional phase gradient method can effectively measure the orbital angular momentum (OAM) number of the vortex beam. However, its spatial sampling phase aperture span is strictly restricted within only π radian. In this paper, we presented a more flexible extended aperture sampling reception (EASR) method for the radio frequency (RF) applications, which can break through this restriction condition. It could make the reception and measurement methods about the OAM mode number become more complete and versatile. By converting the higher-order OAM mode to a lower-order OAM mode, the spatial phase aperture span between the adjacent receiving sampling points can realize extensions. We have conducted a comprehensive theoretical analysis and summarized the general guidelines of this EASR method in the main body of the paper. Subsequently, we perform the related numerical simulation calculations to receive and measure the OAM mode number of a high-order mode vortex beam using the EASR method. Simulation results and theoretical analysis are in good agreement.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It has been several decades since the orbital angular momentum (OAM) was experimentally demonstrated, and the related researches are progressing rapidly. OAM shows important application potential in many fields such as the micromanipulation, imaging, and communication system [1–3]. In the field of optics and quantum information, the OAM mode vortex beam have led to wide researches covering many research directions [4] such as the optical micromanipulation [5], imaging [6], communication [7], remote sensing and quantum information [8]. For most of these applications, the measurement of OAM mode number is of great importance. There were many corresponding OAM mode measurement methods proposed [4,9–11]. At the same time, since the OAM was introduced to radio field [12–14], it was extensively researched in the field of radar imaging and wireless communication [15–22]. It also shows great potentials in 5G backhaul [17], even the future 6G wireless communication [23], physical layer secure communication [24], and Internet of Things (IoT) applications [16,19]. Although there are many argues about the OAM [25–28], but anyway, as a relatively newly developing thing, OAM vortex beam has a reliable physical background and many unusual electromagnetic properties. Its potential applications might not be only restricted in areas such as wireless communication, imaging, and so on. The researches about the OAM vortex beam are still attractive.
There have been many types of devices used to generate electromagnetic OAM vortex beam [29]. Metasurfaces and array antenna are still the two main ways for generating OAM mode vortex beam [30–38]. For most of the OAM mode vortex beam related applications, the measurement of the basic parameter of OAM mode number $\ell$ turns to be of great significance. The most classic and intuitive OAM mode number measurement method we know is the phase gradient method [14], which got the OAM mode number through measuring the spatial phase difference of the vortex beam between the two adjacent reception sample points. But this method has its own constraints conditions for applications, i.e., mathematical inequality (11) of [14], which restricts the spatial sampling phase aperture span within a $\pi$ radian phase change period of the vortex beam. It defined the relationship between the spatial sampling phase aperture span of the adjacent sampling points and the maximum applicable OAM mode number.
Thus, in this paper, we propose a more flexible extended aperture sampling reception (EASR) method, which can break through the previous mentioned constraints of the traditional phase gradient method. Inherently, this method is mainly based on the Fourier transform (FT) relationship of the space domain and the OAM mode domain [39–42]. When the spatial sampling phase aperture span is extended, a higher-order OAM mode will be converted to the corresponding lower-order OAM mode. Then, through the measurement of the lower-order OAM mode number, we achieve the measurement of the higher-order OAM mode number. The paper is organized as follows. In section 2, it is the theory basis and the relevant analysis. Starting from the ideal spatial vortex phase distribution of the OAM vortex beam, we have summarized several general rules about the method of EASR. In section 3, we adopt numerical simulation to generate a higher-order OAM vortex beam. Then, we measure and analyze its OAM mode number using our proposed EASR method. In addition, we define a variable, judgment error rate (JER), for OAM mode number measurement results evaluation, which can be further used to analyze and evaluate the effectiveness and accuracy of this EASR method under the noise environment. When there are noises, for EASR method, by adding the amount of the sampling points, the OAM mode number measurement accuracy can be guaranteed. Finally, it is the conclusions in Section 4.
2. Theory and analysis
In this section, we elaborate the theoretical basis of the extended aperture sample reception (EASR) method. Based on the Fourier transform theory, we consider a transformation relationship analogy between the vortex beam’s space-mode domain (i.e., phase space domain and OAM mode domain) and the traditional time-frequency signal’s time-frequency domain. According to the similarity theorem [39] of the Fourier transform theory, the phase space domain can be extended through converting the high-order OAM mode to a low-order mode. So, the spatial sampling phase aperture span between the adjacent sampling points becomes larger. This is the core idea of the EASR method. Finally, we use an ideal vortex beam with OAM mode number $\ell = 24$ as an example for the analysis. We also summarize the general guidelines of this method, so that it can work more quickly and accurately. The corresponding analysis results are also given.
2.1 Problems description of OAM mode vortex beam reception and OAM mode number measurement
The general vortex beam transmission and reception model can be briefly described by Fig. 1 [42]. The phase gradient method is achieved by subtracting the phase of the adjacent sampling points [14]. The corresponding constraints of the phase gradient method in [14] is,
where $\beta$ is the adjacent two sample points' radian span, $\left | \ell \right |$ is the absolute value of the OAM mode number, and the numerator is the corresponding maximum spatial sampling phase aperture span of the traditional phase gradient method. As shown in Fig. 2, it is an electric field spatial phase change diagram of an ideal vortex beam, in which $\Delta \varphi = 2\pi /M$ is the spatial sampling physical aperture span between the adjacent sampling points.
Fig. 1. The general vortex beam transmission and reception model. The reception plane is perpendicular to the beam axis, and the center of the reception plane is at the beam axis. The OAM mode number is detected through the reception sampling antennas or sampling points on the reception aperture circumference within the reception plane.
Fig. 2. The electric field spatial phase change diagram of an ideal vortex beam with OAM mode number $\ell = 24$ , which is calculated form the pure OAM factor $e^{j\ell \varphi }$. The red point is the sampling points and their corresponding spatial physical aperture position. There are all M = 26 sample points in the reception plane, $\Delta \varphi = 2\pi /M$ is the corresponding spatial sampling physical aperture span between the adjacent sampling points.
Note that these two concepts, the spatial sampling phase aperture span and the spatial sampling physical aperture span, should be distinguished here. The spatial phase aperture span corresponds to the spatial phase change radian span of the OAM vortex beam, and the spatial physical aperture span corresponds to the actual physical three-dimension azimuthal angle space. For example, in Fig. 2, the corresponding spatial physical aperture span is $\Delta \varphi = 2\pi /M$, and the corresponding spatial phase aperture span is $\Delta \varphi _{phase} = \left | \ell \right |\Delta \varphi = 24\Delta \varphi$, which is larger than $\pi$ radian in this case.
It is obvious that the numerator item in Eq. (1) limits the spatial sampling phase aperture span within the value of $\pi$, and thus it constrains the relationship between the variable $\beta$ and variable $\ell$. However, we believe that this restrictive constraint can be further relaxed, i.e., the sampling phase aperture span can be larger than $\pi$, which could lead to the reception and measurement of the OAM mode number be more flexible. Then, we developed the extended aperture sample reception (EASR) method, which will be elaborated in the next subsection.
2.2 EASR method theory
In the cylindrical coordinate system, for the vortex beam carrying OAM modes, its simplified mathematical expression can be written as follows [42],
Next, in order to explain the principle of the EASR method in more detail and clearly, we draw the schematic diagram as shown in Fig. 3 [40]. The original signal is f1(x), whose signal period is T, and the corresponding sample signal is f2(x), whose sampling period is Ts. Here, we divide the discussions into two situations according to the differences of the spatial sampling phase aperture span, $\Delta \varphi _{phase}$. (i) the $\textrm {Rem} (\Delta \varphi _{phase}, 2\pi ) \in (0,\pi )$; (ii) the $\textrm {Rem} (\Delta \varphi _{phase}, 2\pi ) \in (\pi , 2\pi )$. The function Rem(Y, X) returns the remainder after division of Y by X, and if $\Delta \varphi _{phase}$ is negative or less than $2\pi$, by adding the $2\pi$ phase period, (i) and (ii) can still be satisfied. Situation (i) and (ii) correspond to the Figs. 3(a) and (b) respectively.
Fig. 3. The schematic diagram of the EASR method. (a) f1(x) is the original signal with period T = 4, and f2(x) is the extended sample signal whose sampling period is Ts = 5. (b) f1(x) is the original signal with period T = 4, and f2(x) is the extended sample signal whose sampling period is Ts = 3.
Situation (i) corresponds to the schematic diagram of Fig. 3(a), in which Ts = 5/4 T. Its obtained sample signal f2(x) is an expanded form of the original signal f1(x), which will lead to a corresponding scaling in its spectrum domain. Situation (ii) corresponds to the schematic diagram of Fig. 3(b), in which Ts = 3/4 T. Its obtained sample signal f2(x) is not only an expanded form but also a symmetrical positive-negative flip form about the original signal f1(x), which will lead to a corresponding scaling and positive-negative flip in its spectrum domain. This flip transformation corresponds to a change in the positive-negative sign of the spectral domain value. Similarly, we can develop this principle to measure the OAM mode number of the vortex beam. When extending the spatial sampling phase aperture span in OAM vortex beam reception, its corresponding OAM mode number will be scaled. This is the core idea of the EASR method.
Afterwards, we will conduct a simulation analysis of a theoretical example. But before this, we summarize the general constraints of this EASR method first. According to the most primitive phase gradient method, it has the constraints, i.e., Eq. (1), which is actually the embodiment of the Nyquist sampling theorem. Similarly, the EASR method is also subject to the Nyquist sampling theorem. It is also the first basic principle of using the EASR method. The second basic principle is that when the OAM mode number is converted from a high order to a low order, the integer property of the OAM mode number must be guaranteed, which is determined by the nature of the OAM mode number as an integer in general. Thus, we obtain the following two basic principles.
(i) Subject to the Nyquist sampling theorem.
(ii) Ensure that the number of the OAM mode before and after the converter is an integer.
Based on the two basic principles mentioned above, we have summarized the following specific guidelines about the EASR method. Before this, we first give the related parameters settings. The higher-order OAM mode number is L. We set $\textit {T} = 2\pi$ here, which corresponds to a full vortex phase change period of $2\pi$ in the phase aperture. The sampling interval period of the sampling function is Ts. M sampling points are evenly distributed on the complete $2\pi$ circumferential physical aperture as shown in Fig. 2. In this way, the spatial sampling physical aperture span between the adjacent sampling points is $\Delta \varphi = 2\pi /M$, and the corresponding spatial sampling phase aperture span is $\Delta \varphi _{phase} = \textit {L}\Delta \varphi$. It is obviously that Ts = (L/M)T. Therefore, we give the following three specific principles,
(a) $M \leq 2L-1$;
(b) $L \ne kM$, in which k is an integer;
(c) L is an integer multiple of Rem (L, M), in which function Rem(L, M) returns the remainder after division of L by M.
The principle (a) is the prerequisite for an extended aperture of the EASR method, which leads to the spatial sampling phase aperture span be greater than $\pi$. The specific principle (b) is the concrete embodiment of basic principles (i). Because if k is an integer, the corresponding sample phase data of these sample points will be same. The specific principle (c) is the concrete embodiment of basic principles (ii). It should be noted that in principle (c), if M is bigger than L, we set Rem (L, M) = - Rem (M, L).
2.3 Analysis of theoretical simulation example
Based on the theoretical analysis of the EASR method in the above subsection, in this subsection, we analyze a theoretical simulation calculation example. The higher-order OAM mode number that need to be identified is $\ell = 24$ whose ideal spatial phase distribution is shown in Fig. 2. Different spatial sampling phase aperture spans, corresponding to different M, are adopted to measure and analyze the vortex beam’s OAM mode number through the EASR method. According to the specific judgment principles (a)-(c) summarized before, we list a calculation process table, i.e., Table 1, in which $L'$ is the converted lower-order OAM mode number from the higher-order mode, and L = $\ell =24$. From the correspondence relationship between the higher-order OAM mode number and the lower-order OAM mode, we can determine the higher-order OAM mode number from the measurement results of the converted lower-order OAM mode number. Additionally, in order to apply this method more conveniently, we calculated the relationship diagram between different OAM modes number L and different number of sampling points M, which is shown in Fig. 4 as follows. Noting that the calculated range of L given here is from 2 to 24, and the other situations can also be calculated through the calculation process provided in the paper.
Table 1. OAM Mode Reception Analysis Calculation Process
Fig. 4. The relationship diagram between different OAM modes number L and different number of sampling points M. The circle mark means that these values of sample points are available under the corresponding OAM mode number when applying the EASR method. For example, when the OAM mode number L is 5, the corresponding available sample points number of M are 2, 4, and 6.
The specific calculation process of EASR method is shown as follows. As shown in Fig. 2, for the M sampling points uniformly distributed on the full circumferential aperture, the vortex beam’s spatial phase difference of the adjacent sampling points can be calculated according to the following Eq. (5) and Eq. (6). In this way, for M different sampling points, we can correspondingly get M different results of $L'$,
Fig. 5. Calculated converted lower-order OAM mode number L’ through the EASR method. The circle marked line are the actually calculated results, and the dash lines are obtained by a simply data post processing, i.e., a rounding approximation. By multiplying L’ by the corresponding conversion multiple value in Table 1, we can get the original higher-order OAM mode.
3. Numerical simulation and analysis
In this section, we first generate an OAM mode vortex beam of $\ell =24$ at 5.8 GHz through the numerical calculation model of a uniform circular array (UCA) antenna [13]. Its calculated spatial electric field distributions are shown in Fig. 7. Then we use the EASR method with 2-points sampling to measure and analyze this high-order OAM mode vortex beam, in which the $\Delta {\varphi _m}$ of Eq. (5) is obtained by the spatial sampling phase difference of the two adjacent sampling points. Considering the robustness of 2-points sampling EASR method and the impact of noise interference in the practical applications, we further analyze the reception measurement results of 2-points sampling EASR method in the presence of noise. In order to improve the OAM mode number measurement quality, we proposed an improved 3-points EASR method. In the presence of noise, these two methods are compared together to illustrate the superiority of the later method of 3-points EASR method. All of the simulation processes are realized by MATLAB.
Fig. 7. The amplitude and phase distribution of the vortex beam electric field in the observation plane of the OAM mode number $\ell = 24$. The red points are the sampling points and their corresponding aperture positions. There are all M = 21 sample points in the reception plane, $\Delta \varphi = 2\pi /M$ is the spatial physical aperture span between the two adjacent sampling points. R = 0.15 m is the radius of the reception aperture, where the electric field strength is the maximum. R is also suitable later in this section.
The transmitting antenna model used here is a uniform circular array (UCA) antenna at 5.8 GHz. Its radiation pattern can be calculated as follows [36,43],
where andThen, based on the vortex beam electric field distribution results in Fig. 7, and under an ideal conditions without noise interference, we perform the reception and measurement analysis about the OAM mode number $\ell = 24$ according to EASR method with 2-points sampling through Eq. (5) and Eq. (6). However, in practical applications, it is inevitable to consider the noises. To this end, we add Gaussian white noise to the sampled signal and conduct further analysis with 2-points sampling EASR method. The sampled signals added with Gaussian white noise can be expressed by the following formula,
where $s{p_{noise}}(m)$ is the sampled signals with the noise, $sp(m)$ is the original sampled signals, $n(m)$ is the added Gaussian white noise, and signal noise ratio (SNR) used here is 20 dB. In Fig. 8, It shows the calculation results of 2-points sampling under the conditions of presence and absence of the noise.Fig. 8. OAM modes reception measurement results calculated by EASR method about all the sample points, M = 21, of the full circular aperture under different conditions. (a) OAM modes reception results calculated by EASR method with 2-points sample in the presence or absence of noise, and the OAM modes reception measurement results calculated by EASR method with improved 3-points sample in the presence of noise. (b) The OAM mode reception measurement results corresponding to the results in (a), but only after the data post process by function Y = round (X), which means that Y is obtained by rounding X. The dash line at Round (L’) value of three is the accurate theory value of L’. The bars that pop up represent the wrong detections.
It is obvious that the noise will influence the OAM mode number measurement results. Thus, we adopted the improved 3-points sampling EASR method, which calculates the phase gradient of three sampling points at a time as shown in Fig. 7, while 2-points sampling EASR method calculates two sampling points at a time. Similar to the 2-points sampling EASR method, 3-points sampling EASR method is implemented as follows,
In order to further compare the difference between 2-points sampling and 3-points sampling, we define a variable, judgment error rate (JER), to evaluate. The definition expression of JER is as follows,
in which $Num_{\textrm {wrong}}$ is the amount of OAM mode judgment errors, and $Num_{\textrm {total}}$ is the total amount of the OAM mode judgment repetitions. Here, we repeated a total of $M \times {10^7}\; = 21 \times {10^7}$ times of the full circumference aperture sampling points test, and the variation curves between the SNR and the JER are shown in Fig. 9, where the variation range of SNR is from 0 dB to 35 dB. Compared with 2-point sampling, we can clearly see the advantages of 3-point sampling. Under the same SNR, 3-point sampling has lower JER value. Therefore, it also further explains that for the EASR method, increasing the number of sampling calculation points has a positive effect on improving the accuracy of OAM mode number measurement.Fig. 9. Variation curve of judgment error rate (JER) with signal-to-noise ratio (SNR) in two different cases of 2-points EASR method and 3-points EASR method.
4. Conclusion
To summarize, in the RF applications, the proposed extended aperture sample reception (EASR) method can break through the constraints of $\pi$ radian spatial phase aperture span of the traditional phase gradient method. Thus, by introducing the EASR method, the OAM mode number reception and measurement methods will become more complete, and this method can also be used more widely due to its looser constraints. We have proved the correctness and effectiveness of the proposed EASR method from the theoretical derivation and simulation verification. Because of the process of converting the higher-order OAM mode to the lower-order OAM mode in EASR method, it is more useful and powerful when the measured OAM mode number is high. We remark that based on the reliable physical background and unusual electromagnetic properties of the OAM vortex beam, OAM vortex beam could find its considerable application potential in wireless communication, IoT, and radar detection or imaging, and it may not be limited to this. This EASR method will also further promote the related researches and applications of OAM vortex beam.
Funding
Shaanxi Outstanding Youth Science Foundation (2019JC-15); National Key Research and Development Program of China; Fundamental Research Funds for the Central Universities (JCKY2018203C025).
Disclosures
The authors declare no conflicts of interest.
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