Radio Vortex Communication System Using Partial Angular Aperture Receiving Scheme Under Atmospheric Turbulence

For transmission based on long-distance vortex waves, the partial receiving scheme which uses a limited angular aperture receiving and demultiplexing multi-beam can solve the difficulty of the conventional whole beam receiving scheme due to the divergence of the vortex beam. But the atmospheric turbulence is rarely considered in analyzing the stability of the radio vortex (RV) communication system based on the partial angular aperture receiving (PAAR) scheme. Here we first introduce atmospheric turbulence into the RV communication system based on the PAAR scheme. Moreover, in order to compare the effects of turbulence on the PAAR scheme and the whole angular aperture receiving (WAAR) scheme, a new turbulence attenuation degree D model is proposed, which represents the stability of the RV communication system in the atmospheric turbulence environment. Simulation results indicate that the difference of D values between PAAR scheme and WAAR scheme does not exceed the order of 0.01 when the range of refractive index structure constant <inline-formula> <tex-math notation="LaTeX">$C_{n}^{2}$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">${10^{-17}}{m^{-\frac {2}{3}}}-{10^{-12}}{m^{-\frac {2}{3}}}$ </tex-math></inline-formula> and the distance is 90m-120m. When the range of <inline-formula> <tex-math notation="LaTeX">$C_{n}^{2}$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">${10^{-13}}{m^{-\frac {2}{3}}}-{10^{-12}}{m^{-\frac {2}{3}}}$ </tex-math></inline-formula> and the distance is 90m-120m, D value of PAAR scheme is always smaller than that of WAAR scheme. These demonstrations suggest that the RV communication system using PAAR scheme is more stable than that using WAAR scheme in the strong atmospheric turbulence environment.


I. INTRODUCTION
Vortex waves carrying orbital angular momentum (OAM) have infinite orthogonal states in theory, while electromagnetic wave with spin angular momentum (SAM) has only two orthogonal states. Therefore, OAM has attractive potential to significantly increase spectral efficiency and channel capacity for wireless communication [1]. It is verified that OAM multiplexing can achieve great potentials in the radio frequency (RF) wireless communications [2], [3].
We call the wireless communication system based on OAM the radio vortex (RV) communication system. It is well known that the radiation pattern of vortex beams possesses a characteristic [4]: intensity nulls are along the beam axis. Due to the divergence of vortex beams, the further they travel, the larger the radius of the intensity nulls is. Moreover, the larger the OAM mode number l, the more The associate editor coordinating the review of this manuscript and approving it for publication was Muhammad Zubair . severe the divergence [5]. Hence, a large receiving aperture is required to capture the effective power of the transmitting vortex beams in long distance links. To solve this problem, a partial angular aperture receiving (PAAR) scheme has been proposed [6]. This PAAR scheme can be an effective space-saving and cost-effective method for the RV communication system. It is verified that the orthogonality of Laguerre-Gaussian (LG) eigen beams with different integer topological charges is preserved and independent of the aperture sizes and the radial indices [7]. Therefore, we can still utilize orthogonality of LG eigen beams to increase channel capacity of RV communication system based on PAAR scheme in long distance links.
It is verified that angle and angular momentum are linked by a Fourier transformation [8]. A restriction of the angular range within an optical beam profile generates orbital angular momentum sidebands on the transmitted light and the crosstalk occurs [8]- [10]. In recent years, the relationship between aperture size, OAM mode interval and minimum crosstalk has been proved by experiments [6], [11]- [15]. These literatures have proved that crosstalk due to limited aperture can be improved by adopting appropriate partial reception scheme. In [15], the influence of additive white Gaussian noise (AWGN) channel on PAAR scheme was analyzed. The influence of AWGN channel and Rician channel on PAAR scheme and the effect of non-ideal receiving condition on partially receiving aperture were also analyzed. However, it does not analyze the influence of atmospheric turbulence channel on the PAAR scheme. Therefore, it is of immediate significance to study the impact of atmospheric turbulence on the RV communication system based on PAAR scheme.
In this paper the impact of atmospheric turbulence on the RV communication system based on PAAR scheme has been investigated. In order to compare the effects of turbulence on PAAR scheme and whole angular aperture receiving (WAAR) scheme, a novel turbulence attenuation D model is proposed. The contributions of this article are summarized as follows: 1) We use a mask model to represent the angular aperture, and then derive the OAM spiral spectrum of the PAAR scheme.
2) According to the channel capacity formula of [16], a channel capacity model of PAAR scheme based on atmospheric turbulence is proposed.
3) A novel turbulence attenuation D model is proposed. It is necessary to transmit multiple modes simultaneously. However, the number of transmission modes in the scheme of [13] is limited. Therefore, we choose the scheme of [6] to study the influence of atmospheric turbulence on the PAAR scheme. The remainder of this paper is organized as follows. Section II presents the definition of spiral spectrum of the PAAR scheme, capacity model of PAAR scheme based on atmospheric turbulence and turbulence attenuation degree D model. In Section III, the simulation results are analyzed and discussed. Finally, the conclusions are drawn in Section IV.

II. THEORETICAL PRINCIPLE
In order to compare with the WAAR scheme proposed in [17], only the phase distortion of atmospheric turbulence is considered, but the amplitude attenuation caused by atmospheric turbulence is not considered. Fig.1 presents the transmission FIGURE 1. RV communication system using a PAAR scheme in the atmospheric turbulent environment. of RV communication system using a PAAR scheme in the atmospheric turbulent environment. In this paper, we use the WAAR scheme and the PAAR scheme to receive the same OAM state set at the same distance.

A. SPIRAL SPECTRUM OF PAAR SCHEME
The LG beam is used to describe the vortex wave carrying orbital angular momentum. The field distribution of Laguerre-Gauss beam in the source plane (z = 0) is expressed as [18] where r is radial distance, φ is azimuthal angle, z is propagation distance, i is an imaginary unit, n is the order of the Laguerre polynomial L m n (x), n = 0 is generally configured for RV systems, L m n (x) = 1, when n = 0. In this paper, we assume that n = 0. m is the OAM state, whose absolute value describes the number of twists of the helical wavefront. w 0 is the beam waist radius of LG beam at z = 0.
The functional form of the Laguerre-Gauss source mode is well known at any point (z > 0) in free space. One has [18] λ is the Rayleigh distance, λ is the wavelength. = arctan( z z R ) is the Gouy phase, k = 2π λ is wave number. Under the Rytov approximation, when passing through the weak turbulent atmosphere, the beam field received at the receiving aperture at distance z is expressed as [18], [19] , R 1 is the receiving aperture radius. In this paper, we know U ( r R 1 ) = 1. ψ(r, φ, z) is the phase distortion term caused by the atmospheric turbulence. Based on the quadratic approximation [20], ψ(r, φ, z) satisfies e (ψ(r,φ,z)+ψ * (r,φ ,z)) = e .
where r 0 is the spatial coherence radius of the LG beam under the Kolmogorov turbulence flow model. r 0 is expressed as [20] ] 3 5 (1.46C 2 n k 2 z) where = 1 + z R is the curvature parameter of the LG beam at receiver, is the Fresnel ratio of the LG beam at receiver, C 2 n is the refractive index structure constant. The larger C 2 n is, the greater the turbulence strength is [16].
It is well known that the radius of the OAM annular region with the maximum energy strength is denoted by [21] r max (z) = |m| 2 w(z).
r max (z) determines the size of the receiving aperture. Only when the receiving aperture is larger than r max (z), can the complete energy of the beam be received.
For the PAAR scheme, the receiver should be placed at the maximum beam strength, therefore the integral range of the receiving radius of the PAAR scheme is (r 1 , r 2 ). For the WAAR scheme, the receiver is placed at the center of the beam, therefore the integral range of the WAAR scheme is (0, R 1 ). In this article r 2 ≤ R 1 .
The aperture is considered a mask [10]. For the WAAR scheme without angle limitation, the mask is For the WAAR scheme without angle limitation, the received field can be described as E W m,n (r, φ, z)M (φ). For the PAAR scheme with angle limitation, the mask is For the PAAR scheme, since the receiver uses an angular aperture of 2π s , the received field can be described as E W m,n (r, φ, z)M (φ). It can be seen that the partial aperture receiving field is 1 s of the whole aperture receiving field. For the WAAR scheme, according to the reference [22], the field distribution of Eq. (3a) is expanded according to the spiral spectrum harmonics. Therefore, Eq. (3a) can be written as It is well known that the energy of the beam is E = If the field received by the whole aperture is E W m,n (r, φ, z), then the field received by the partial aperture is Therefore, Eq.(9a) can be written as In the PAAR scheme, the spiral spectrum with OAM state l is expressed as In the PAAR scheme, we can derive the spiral spectrum with OAM state l (details are in Appendix) as B. CAPACITY MODEL OF PAAR SCHEME BASED ON ATMOSPHERIC TURBULENT In the absence of atmospheric turbulence, the capacity of the RV communication system using WAAR scheme on the AWGN channel is expressed as [16]: where L is the number of channels. N 0 is the additive white Gaussian noise power, P TX is the transmitted power. erfc(.) is the complementary error function. In the literature [15], [23], the theoretical perfect demodulation performance is independent of the aperture size in the AWGN channel. That is to say, the WAAR scheme has the same capacity as the PAAR scheme when there are the same number of OAM states in the AWGN channel. Next we derive the channel capacity in the presence of atmospheric turbulence.
In PAAR scheme, in order to transmit multiple mutually orthogonal OAM modes simultaneously at the same frequency, it is necessary to select a specific OAM modal set. It can be known from the literature [6] that the OAM state set must satisfy The OAM state set transmitted by the transmitter is assumed as B. In order to better compare the performance of WAAR and PAAR scheme in atmospheric turbulence environments, the transmission mode set of WAAR scheme is the same as that of PAAR scheme. Under the influence of turbulence, the OAM multiplexing beam will deviate from the center of the beam, causing crosstalk between adjacent modes [24]. We choose L = 2k +1 symmetrically distributed channels, B = {l 1 − ks, . . . , l 1 , . . . , l 1 + ks}. Therefore, the crosstalk matrix of RV system with PAAR scheme can be expressed as Eq. (14), shown at the bottom of the next page. where P PAAR l 1 −qs (l 1 − ps, z), −k ≤ p ≤ k, −k ≤ q ≤ k is the power weight of the spiral harmonic component with OAM state l 1 − qs, when the vortex wave with the OAM state l 1 −ps is transmitted in atmosphere environments. When p = q, P PAAR l 1 −qs (l 1 − ps, z) represents that the normalized wave power is spread from the state l 1 − ps into the state l 1 − qs. The p-th row of matrix P PAAR indicates that the normalized wave power of the p-th channel is spread into the vortex channels which are included in the entire set B. The q-th column of the matrix P PAAR includes the desired wave power of q-th vortex channel and the spread wave power from other vortex channels which are included in the set B. Therefore, an expression of the signal-to-interference-and-noise ratio (SINR)can be obtained from each column of crosstalk matrix P PAAR of RV system based on PAAR scheme. The expression is where γ PAAR q is the SINR of the q-th OAM state. When the RV communication systems apply the Quadrature Phase Shift Keying (QPSK) modulation, the bit error rate of OAM channels is derived as with [25] p PAAR where p PAAR q is the bit error rate of the q-th OAM state. According to the bit error rate, the capacity of the q-th vortex channel can be expressed as Therefore, the capacity of the RV communication system based on atmospheric turbulence is which is the row sum of the capacity matrix C PAAR L in Eq.(17b).

C. TURBULENCE ATTENUATION DEGREE MODEL
We express the channel capacity of the WAAR scheme proposed in [17] as C WAAR . In order to compare the turbulence effects on the WAAR scheme and the PAAR scheme, a novel turbulence attenuation degree D model is proposed, which represents the stability of the RV communication system in the atmospheric turbulence environment. The absolute value of channel capacity difference between the RV communication system in turbulent environment (C 2 n = 0.5 × 10 −17 m − 2 3 − 0.5 × 10 −11 m − 2 3 ) and the RV communication system in weak turbulence environment (C 2 n = 0.5 × 10 −17 m − 2 3 ) is defined as the numerator of turbulence attenuation degree D. The channel capacity of the RV communication system in the weak turbulent environment (C 2 n = 0.5 × 10 −17 m − 2 3 ) is defined as the denominator of turbulence attenuation degree D. The channel capacity of RV communication system (including WAAR scheme and PAAR scheme) If the refractive index structure constant C 2 n is given, the spatial coherence radius of the LG beam under the C 2 n can be expressed as r 0 (C 2 n ), substituting C 2 n in Eq.(4a). The channel capacity of RV communication system in the weak turbulent environment (C 2 n = 0.5 × 10 −17 m − 2 3 ) is C C 2 n =0.5×10 −17 . We use the spatial coherence radius of the LG beam at the C 2 n = 0.5 × 10 −17 m − 2 3 and the channel capacity of RV communication system in Eq. (19), to express C C 2 n =0.5×10 −17 The channel capacity of the RV communication system in the turbulent environment is C C 2 n , C C 2 n = C(r 0 (C 2 n )).

VOLUME 8, 2020
Hence, the expression of turbulence attenuation degree D is The larger the value of D, the greater the influence of turbulence.

III. SIMULATION RESULTS AND DISCUSSIONS
Generally speaking, the mode selection of vortex beam in PAAR scheme is limited by different factors, including receiver aperture size and circular arc s. Given a fixed receiver aperture size, a larger OAM mode value may result in a larger beam size at the receiver, which may decrease the recovered power. Hence, special attention should be paid to the selection of OAM mode set. In this section, a partial aperture receiving scheme in the atmospheric turbulent environment is simulated. We set the propagation distance to 100m. In [26], vortex phase properties of OAM keep well after long-distance transmission, which were experimentally demonstrated. Hence, as long as the OAM receiving antenna is improved and the OAM modes of PAAR scheme satisfies Eq.(13), it can provide ideal orthogonality for a set of regular OAM modes after long-distance transmission. The system simulation parameter settings are shown in Table 1. In the absence of atmospheric turbulence, the transmission channel is a Gaussian channel, which is considered to be an ideal situation. Fig.2(a) shows the channel capacity comparison between the PAAR scheme (s = 2, 4) and the WAAR scheme (s = 1) under ideal conditions. We can observe that the PAAR scheme has the same capacity as the WAAR scheme when the transmission distance achieves 100 meters. This is because of the fact that theoretical perfect demodulation performance is independent of the aperture size. Fig.2(b) shows the channel capacity comparison between the PAAR as the non-ideal condition. With an identical number of OAM states, the WAAR scheme has more capacity than the PAAR scheme when the transmission distance achieves 100 meters. For instance, when the L is 13, the capacity of s = 1 in the WAAR scheme is 4.062 bits/L-channels, while that of s = 2(s = 4) in the PAAR scheme is 2.8 bits/L-channels (1.251 bits/L-channels). When the transmission distance achieves over 100 meters, the PAAR scheme still has the same capacity as the WAAR scheme under ideal conditions and the WAAR scheme still has more capacity than the PAAR scheme under a non-ideal condition.    3 ) on the power weight of the PAAR scheme. It is verified that the presence of atmospheric turbulence causes crosstalk between OAM states. We see that the power weight of the PAAR scheme with turbulence decreases with the increase of l. The power weight of the PAAR scheme without turbulence initially maintains a fixed value of 0.0625. When l increases to 33, the power weight of the PAAR scheme without turbulence decreases with the increase of l, mainly because of the divergence caused by excessive OAM state l. We also observe that in the case of the identical OAM state, the PAAR scheme with turbulence has much less power weight than PAAR scheme without turbulence. Taking l = 13 as an example, the power weight of the PAAR scheme without turbulence is 0.0625, and the power weight with turbulence is 0.0428.
The capacity of RV communication system using PAAR scheme (s=4) and RV communication system using WAAR scheme are depicted as functions of C 2 n in Fig.4. The typical value of C 2 n is in the range of 10 −17 m − 2 3 − 10 −12 m − 2 3 . It can be seen from the Fig.4 that in the case of the weak and medium turbulent environment, the channel capacity of the PAAR scheme (L = 5, 9, 13) and the WAAR scheme (L = 5, 9, 13) are fixed. However, when C 2 n is 1.256 × 10 −13 m − 2 3 , the channel capacity of the PAAR scheme and the WAAR scheme decrease rapidly with the increase of C 2 n . With an identical L, the PAAR scheme has much less capacity than the WAAR scheme when the C 2 n increases. For example, when the C 2 n is 3.155 × 10 −13 m − 2 3 , the capacity of 5 vortex channels in the WAAR scheme is 3.553 bits/L-channels, while the capacity of 5 vortex channels in the PAAR scheme is 0.8743 bits/L-channels.
In order to determine the practical feasibility of PAAR scheme, we need to study from the turbulence attenuation degree D. The turbulence attenuation degree D based on PAAR scheme (L = 5, 9, 13) and based on WAAR scheme (L = 5, 9, 13) are simulated as functions of refractive index structure constant C 2 n (see Fig.5(a)). We see that the turbulence attenuation degree D values of PAAR scheme are nearly same as that of WAAR scheme at different distances, suggesting that PAAR scheme can be used in the practical RV wireless communication system transmission. We see that D values of the PAAR scheme and WAAR scheme increase rapidly with the increase of C 2 n . We also observe that the difference of the turbulence attenuation degree D values between PAAR scheme and WAAR scheme does not exceed the order of 0.01 with C 2 n of the range of 10 −17 m − 2 3 − 10 −12 m − 2 3 . For example, when C 2 n is 1.991 × 10 −12 m − 2 3 , z = 100m, the turbulence attenuation degree D in the PAAR scheme (in the WAAR scheme) is 0.7874 (0.8225). The turbulence attenuation degree D values based on PAAR scheme (L = 5, 9, 13) and based on WAAR scheme (L = 5, 9,13) are depicted as functions of distance in Fig.5(b). We see that in the case of the weak and medium turbulent environment (10 −17 m − 2 3 − 10 −14 m − 2 3 ), the turbulence attenuation degree D values in the PAAR scheme are always larger than that the WAAR scheme (except L = 5) with identical distance. We also observe that in the case of the strong turbulent environment (C 2 n = 1.0 × 10 −12 m − 2 3 ), the turbulence attenuation degree D values in the PAAR scheme are always less than that the WAAR scheme with identical distance. D values of the PAAR scheme and WAAR scheme increase rapidly with the increase of distance. The turbulence attenuation degree D curves of WAAR scheme rise in a broken line because of the limitation of the receiving aperture. Fig.6 depicts the influence of signal-to-noise ratio(SNR) on the capacity of RV communication system using PAAR scheme (L = 5, 9, 17) and RV communication system using WAAR scheme (L = 5, 9, 17) in turbulent environment. With an identical L, the channel capacity of the PAAR scheme and the WAAR scheme increase with the increase of SNR. When SNR is less than 36dB, the WAAR scheme has more capacity than the PAAR scheme. With the increase of SNR, the PAAR scheme achieves the same capacity as the WAAR scheme. This is because with the increase of SNR, the crosstalk caused by turbulence is negligible. Fig.7(a) shows the effect of atmospheric turbulent refractive index constant C 2 n and SNR on the turbulence attenuation VOLUME 8, 2020 degree D of the PAAR scheme. Fig.7(b) depicts the effect of atmospheric turbulent refractive index constant C 2 n and SNR on the turbulence attenuation degree D of the WAAR scheme. We can observe that the value of the turbulence attenuation degree D is very small in the weak and medium turbulence environment and the turbulence attenuation degree D increases with the increase of C 2 n in the strong turbulence environment (Fig.7 (a), Fig.7 (b)). We also observe that the turbulence attenuation degree D of the PAAR scheme increases first and then decreases with the increase of SNR ( Fig.7 (a), Fig.7 (b)). This means that there is the value of SNR that maximizes the value of turbulence attenuation degree D. We define the turbulence attenuation degree D, which represents the stability of the  The effect of atmospheric turbulent refractive index constant C 2 n and SNR on (a) the RV communication system capacity using the PAAR scheme, (b) the RV communication system capacity using the WAAR scheme.
RV communication system in the atmospheric turbulence environment. We know that the greater the value of turbulence attenuation degree D, the greater the turbulence impact on the RV communication system. In order to ensure the stability of RV communication system, this SNR should be avoided. Compared with. Fig.7 (a) and Fig.7(b), we find that the turbulence attenuation degree D of PAAR scheme is always smaller than that of WAAR scheme under the same conditions. It is confirmed that in the same strong turbulence environment, the RV communication system using PAAR scheme is more stable than that using WAAR scheme.

IV. CONCLUSION
In this article, the influence of atmospheric turbulence on the stability of RV communication system based on PAAR scheme is investigated. The spiral spectrum of PAAR scheme is first derived. Then the capacity model of PAAR scheme based on atmospheric turbulent is presented. Finally, we propose the turbulence attenuation degree D, which represents the stability of the RV communication system in the atmospheric turbulence environment. Theoretical analysis and numerical results are presented. First, the analysis and numerical results show that in the case of high SNR, RV communication system based on PAAR scheme has a large channel capacity. Second, the turbulence attenuation degree D of RV communication system using PAAR scheme is studied. By comparing the turbulence attenuation degree D of RV communication system using PAAR scheme with that of RV communication system using WAAR scheme, it is found that the difference of the turbulence attenuation degree D values between PAAR scheme and WAAR scheme does not exceed the order of 0.01 with C 2 n of the range of 10 −17 m − 2 3 − 10 −12 m − 2 3 . Consequently, we prove that the RV communication system using PAAR scheme is more stable than that using WAAR scheme in the strong atmospheric turbulence environment. When the range of C 2 n is 10 −13 m − 2 3 −10 −12 m − 2 3 and the distance is 90m-120m, D values of PAAR scheme is always smaller than that of WAAR scheme.
To summarize, we can draw a conclusion in this article that from the perspective of turbulence attenuation degree D, the partial aperture receiving scheme can replace the whole aperture receiving scheme in the environment of strong turbulence. All the results in this article are based on the assumption that only the phase distortion of atmospheric turbulence is considered, but the amplitude attenuation caused by atmospheric turbulence is not considered. If we consider the amplitude attenuation caused by atmospheric turbulence, what will happen remains to be further studied, but we believe that even if we consider the amplitude attenuation caused by atmospheric turbulence, the PAAR scheme can provide an opportunity for RV communication system to increase the transmission distance.

APPENDIX
In this appendix, C PAAR l (m, z) and C initial are derived. VOLUME 8, 2020 The α PAAR l (r, z) 2  The electric field of (1) at z = 0 is expressed as with |β m (r, 0)| 2 = 2π (