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Abstract
An orbital angular momentum division multiplexing free space optical communication (OAM-DM FSO) system with channel coding can compensate atmospheric channel fading and improve system performance. An OAM-DM FSO system based on hybrid channel coding is proposed in this paper. The coding gain is improved by taking into account mode dependent channel fading difference caused by atmospheric turbulence. Simulation results show that compared with single channel coding, the coding gain is increased by 1.85 dB under Cn2=1E-14, BER=1E-5 with non-uniform LDPC code (0.7 code rate) for an OAM1/OAM3 multiplexing system. In addition, for four OAM modes (+1,+3,+5,+7) multiplexing systems, the coding gain is increased by more than 3.8 dB under Cn2=1E-14 and BER=1E-5.
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1. Introduction
Free space optical communication (FSO) has advantages such as high bandwidth, inherent security, and ease of deployment [1]. Therefore, it has potential applications in urban local area networks [1], air/space/ground networks [2,3], and 5G fronthaul networks [4]. To improve the transmission capacity of FSO systems, spatial division multiplexing (SDM) FSO systems have received a lot of attention recently [5,6]. SDM takes the spatial orthogonal mode group of light field (Laguerre Gaussian mode group or orbital angular momentum mode (OAM) group) as the multiplexing dimension, and improves the transmission capacity through parallel transmission of multiple spatial mode channels [7]. At present, SDM-FSO systems based on OAM mode group have been widely investigated [8–10]. The reason is that the generation and demultiplexing of OAM modes only require phase only optical component.
OAM-DM FSO systems increase the system capacity by N times by transmitting N OAM mode channels in parallel. However, the signal fading caused by atmospheric channels (including geometric loss caused by beam broadening, intensity scintillation caused by atmospheric turbulence (AT), etc.) severely limits the system performance. To compensate for the impact of atmospheric channel fading, OAM-DM FSO systems based on channel coding (turbo code [11], LDPC code [12,13], space-time coding [14]) have received a lot of attention. For example, Ivan B. Djordjevic et al. from the University of Arizona have conducted extensive theoretical and experimental research on OAM-DM FSO systems based on LDPC codes [10,15,16]. The results show that using LDPC code in AT channels can suppress the influence of turbulence and improve system performance. However, most of the current work on channel coding OAM-DM FSO systems involve separate encoding and decoding of each OAM mode channel. This single channel coding method does not take into account the differences of channel fading caused by AT on different OAM beams. In fact, for the structure of different OAM beams is different, the impact of atmospheric turbulence on the beam is directly related to beam structure, so the AT induced fading felt by different OAM beams is different. In addition, OAM-DM FSO system also suffers from mode dependent loss during mode multiplexing/demultiplexing.
To compensate for the mode dependent fading difference induced by AT, using bit interleaved LDPC coding between mode channels is a solution, but it is not the best method for the different roles of different code element in decoding are not considered. In fact, for multi-channel transmission systems with channel fading differences, hybrid channel coding combined with optimal code element allocation scheme can bring higher coding gain. For example, in FSO/Radio Frequency(RF) cooperative system, hybrid channel coding improves link availability and achieves more than two orders of magnitude improvement in bit error rate (BER) [17–21]. This paper proposes an OAM-DM FSO system based on hybrid channel coding. After encoding at the transmitting end, the code element is allocated to two different OAM mode channels for parallel transmission according to the optimized code element allocation method. At the receiving end, two channels of signal are combined according to corresponding allocation rules and decoded uniformly. Simulation results show that for OAM$_1$/OAM$_3$ multiplexing system, compared with single channel coding, the coding gain is increased by 1.85dB under C$_n^2$=1E-14, BER=1E-5 with non-uniform LDPC code (0.7 code rate). In addition, for four OAM modes (+1,+3,+5,+7) multiplexing systems, the coding gain is increased by more than 3.8dB under C$_n^2$=1E-14, BER=1E-5.
2. Scheme
The proposed scheme of OAM-DM FSO system based on hybrid channel coding is shown in Fig. 1. The information bits are first coded by LDPC encoder, and then the code elements are allocated to two channels. Then, after pulse position modulation (PPM) and phase plate for OAM mode excitation, two channels of information are loaded onto OAM$_1$ and OAM$_3$ mode fields respectively. After the beam combiner, two OAM mode fields are multiplexed together and pass through the atmospheric turbulence channel. At the receiving end, after passing through the beam splitter, two beams are demultiplexed and coupled to single mode fibers (SMFs). Subsequently, they are connected to the detectors through optical filters. Finally, values of log likelihood ratio (LLR) are calculated for the received electrical signals and combined for decoding. Other link parameters are shown in Table 1.
Fig. 1. Scheme of OAM-DM FSO system based on hybrid channel coding. BS: beam splitter, M: mirror, PD: photodetector.
2.1 AT induced channel fading and crosstalk for different OAM modes
For multiple OAM beams in the OAM-DM FSO system, the AT induced channel fading felt by different OAM beam is different. In this paper, turbulence screens are utilized to simulate AT and analyze channel fading differences among different OAM modes. The turbulence screen is generated by the Fourier transform method, and the basic idea is to filter the Gaussian random noise using the atmospheric phase spectrum according to the atmospheric turbulence theory. The turbulence screen is calculated by the following Equations.
where $\sigma ^2 (k_x, k_y)$ is the variance of the phase spectrum and $C$ is a matrix consisting of complex Gaussian random numbers. $r_0$ is the atmospheric coherence length, which indicates the turbulence intensity and can be approximated by the simple expression as Eq. (3). where $C_n^2$ is the refractive index structure constant, $k$ is the wave number and $d$ is the link distance.For our proposed OAM-DM FSO system (Fig. 1). Considering the link parameters shown in Table 1. One turbulence screen simulating 200 m AT is placed in the middle of the link. Using the Monte Carlo method, the distribution of AT induced fading coefficient for OAM$_1$/OAM$_3$ multiplexing system is shown in Fig. 2 under different AT level. For OAM$_1$ or OAM$_3$ mode field, after passing through AT, the spot is distorted. The fading of the OAM$_1$ or OAM$_3$ mode channel is defined as the ratio of the power of OAM$_1$/OAM$_3$ mode field in the distorted spot to the transmitting OAM$_1$/OAM$_3$ mode power, and crosstalk is defined as the ratio of the power of OAM$_1$ or OAM$_3$ mode field in the distorted spot to the transmitting OAM$_3$ or OAM$_1$ mode power. Results show that the fading of OAM$_3$ mode channel is greater than that of OAM$_1$ mode channel, thus there is a mode dependent channel fading difference. As turbulence intensity increases, this difference increases. In addition, as turbulence intensity increases, both channel fading and inter-mode crosstalk become severe. When the turbulence intensity reaches C$_n^2$ = 5E-14, crosstalk is obvious.
Fig. 2. The distribution of turbulence fading coefficient and crosstalk under C$_n^2$ = 1E-15, 1E-14, 2E-14 and 5E-14.
Table 1. Link parameters
2.2 Hybrid channel coding and 4PPM modulation
Due to the fading difference of AT acting on different OAM beams, hybrid channel coding method with optimized bit allocation to improve the reliability of OAM-DM system (Fig. 3) is proposed. The information bits are firstly hybrid coded by uniform or non-uniform LDPC encoder. In this paper, four different types of LDPC codes are constructed, including (i) uniform LDPC codes with code length of 16000 and code rate of 0.5; (ii) uniform LDPC code with code length of 16000 and code rate of 0.7; (iii) non-uniform LDPC code with code length of 15998 and code rate of 0.5 (the degree distribution of variable nodes is given by Eq. (4)), as well as (iv) non-uniform LDPC code with code length of 16000 and code rate of 0.7 (the degree distribution of variable nodes is given by Eq (5)).
After encoding and bit allocation, two channels of information are loaded onto OAM$_1$ and OAM$_3$ mode fields after 4PPM modulation respectively (Fig. 3). At the receiver side, the Max-Max based low-complexity soft information calculation method is utilized for 4PPM modulation system (Fig. 4) [22,23]. Finally, the LLR values are combined for decoding. Using hybrid channel coding especially non-uniform hybrid coding has following two advantages: 1) both channels could collaboratively compensate the shortcomings of each other and thereby, improve the performance of the system as a whole, 2) Non-uniform LDPC codes can provide unequal error protection thus to having a good overall performance.
Fig. 4. Principle of Max-Max based low-complexity LLR calculation method corresponding to 4PPM modulation.
2.3 System model
A theoretical model of the OAM-DM FSO system is given. At the receiver, detectors receive signal light field and background radiation light field. Taking the OAM$_1$ mode channel as an example, the received light field can be expressed as:
Signal terms including components transmitted through its own mode channel and crosstalk components from other mode channels, and the total signal terms are expressed in the following equations for OAM$_1$ and OAM$_3$ mode channel respectively,
Assuming that thermal noise and background radiation noise are the main noise, ignoring signal related noise terms. The relevant noise terms are given by Eq. (10). where $\sigma _{b\times b}^2$ is background light and background light beat noise as Eq. (11), $\sigma _{th}^2$ is thermal noise as Eq. (12), $\sigma _{shot}^2$ is shot noise as Eq. (13). $I_{b\times b}$ is the DC term caused by background radiation light. here, $B_e$ is the electronic bandwidth of the system, $K$ is the Boltzmann constant, $T$ is the temperature in Kelvin, and $R_L$ is the photodetector load resistance. Values of these system parameters are shown in Table 2. In addition, the noise terms for OAM$_3$ mode channel are the same. In summary, the signal terms $($8, 9$)$ and noise terms $($10-14$)$ form the theoretical model of OAM-DM FSO system. The average signal-to-noise ratio $($SNR$)$ of the received signal can be calculated according to Eq. (15).Table 2. System parameters
3. Results analysis
Next, the performance of OAM-DM FSO system is analyzed using the Monte Carlo method. 1000 realizations of AT-induced fading and crosstalk coefficients were calculated for each turbulence intensity. System parameters are utilized according to Table 2. The bit allocation scheme followed hybrid channel coding is determined by comparing the system performance under various allocation methods, and the principle is to choose the method with the best performance. For example, under C$_n^2$=1E-15, for non-uniform LDPC codes with code rate of 0.5, Fig. 5 compares the BER performance of the following six different bit allocation methods: (1) All variable nodes with degree distribution of 3 and as many variable nodes with degree distribution of 6 as possible are allocated to OAM$_1$ mode channel. The remaining variable nodes are allocated to OAM$_3$ mode channel. (2) All variable nodes with degree distribution of 4, 6, 7 and some variable nodes with degree distribution of 15 are allocated to OAM$_1$ mode channel. The remaining variable nodes are allocated to OAM$_3$ mode channel. (3) All variable nodes with degree distribution of 3 and some variable nodes with degree distribution of 4 are allocated to OAM$_1$ mode channel. The remaining variable nodes are allocated to OAM$_3$ mode channel. (4) Some variable nodes with degree distribution of 4 and all variable nodes with degree distribution of 6, 7, 15, 16 are allocated to OAM$_1$ mode channel. The remaining variable nodes are allocated to OAM$_3$ mode channel. (5) Some variable nodes with degree distribution of 3 and all variable nodes with degree distribution of 4, 6 are allocated to OAM$_1$ mode channel. The remaining variable nodes are allocated to OAM$_3$ mode channel. (6) Parity interleaved allocation: assign odd numbered nodes to the OAM$_1$ mode channel, and even numbered nodes to the OAM$_3$ mode channel.
Fig. 5. BER performance using different bit allocation method for non-uniform LDPC code with 0.5 code rate under C$_n^2$=1E-15.
From Fig. 5, it can be seen that OAM-DM FSO system with hybrid channel coding performs best when using the 4th bit allocation method. Therefore, for non-uniform LDPC codes, this bit allocation method is selected for subsequent simulations. In addition, for uniform codes, allocating as many information bits as possible to OAM$_1$ mode channel, and allocating the rest to OAM$_3$ mode channel is the chosen solution.
Firstly, for OAM$_1$/OAM$_3$ DM-FSO system with uniform LDPC codes, the BER performance with code rates of 0.5 and 0.7 under the AT level of C$_n^2$=0, C$_n^2$=1e-15 and C$_n^2$=1e-14 are shown in Fig. 6. The simulation results show that compared with single channel coding, the hybrid channel coding OAM-DM FSO system has better performance. As AT level intensifies, the advantages become more apparent. The reason is that the stronger the turbulence level, the greater the difference in channel fading, and therefore the greater the role of hybrid channel coding. Assuming the coding gain is defined as the average SNR penalty improvement at BER=1E-5. In the case of C$_n^2$=1E-14 and BER=1E-5, the coding gain of hybrid channel coding corresponding to 0.5 and 0.7 code rates is 1.19dB and 1.6dB respectively. Therefore, LDPC code with 0.7 code rate have higher coding gain.
Fig. 6. BER curves with uniform LDPC codes and code rates of 0.5 (left), 0.7 (right) for OAM$_1$ mode and OAM$_3$ mode multiplexing system under the conditions of C$_n^2$=0, C$_n^2$=1E-15 and C$_n^2$=1E-14.
When using non-uniform LDPC code, the BER curves with code rates of 0.5 and 0.7 under C$_n^2$=0, C$_n^2$=1E-15, C$_n^2$=1E-14 are shown in Fig. 7. In the case of C$_n^2$=1E-14 and BER=1E-5, the coding gain of hybrid channel coding corresponding to 0.5 and 0.7 code rates is 1.36dB and 1.85dB respectively. Compared with uniform LDPC, non-uniform LDPC shows higher coding gain. When the electronic bandwidth increases to 20GHz, the coding gain of hybrid channel coding compared with single channel coding is more significant than that of 2GHz (Fig. 8).
Fig. 7. BER curves with non-uniform LDPC codes and code rates of 0.5 (left), 0.7 (right) under C$_n^2$=0, C$_n^2$=1E-15 and C$_n^2$=1E-14 for OAM$_1$ mode and OAM$_3$ mode multiplexing system.
Fig. 8. BER curves with non-uniform LDPC codes and $B_e=20GHz$ for OAM$_1$ mode and OAM$_3$ mode multiplexing system under C$_n^2$=1E-15 and C$_n^2$=1E-14.
In addition, the performance of OAM-DM FSO system based on hybrid channel coding was investigated when three OAM modes and four OAM modes are multiplexed for transmission (Fig. 9). Non-uniform LDPC codes with 0.5 code rates are utilized to analyze the BER performance under C$_n^2$=1E-14. Compared with OAM$_1$/OAM$_3$ two modes multiplexing system, hybrid channel coding based three modes or four modes multiplexing system has greater coding gain under the same turbulence level. The coding gain is increased by more than 3.8dB compared with single channel coding under C$_n^2$=1E-14, BER=1E-5. The reason is that the fading of the OAM$_5$ or OAM$_7$ mode channel is greater. Therefore, the difference in channel fading is greater, and the effect of hybrid channel coding is more pronounced. This coding method also can be extended to other modulation formats, such as OOK (Fig. 10). The coding performance may vary depending on the different characteristics of the modulation format.
Fig. 9. BER curves with non-uniform LDPC codes under C$_n^2$=1E-14 for two OAM modes, three OAM modes and four OAM modes multiplexing systems, respectively.
Fig. 10. BER curves of non-uniform LDPC codes with code rates of 0.5 under C$_n^2$=1E-14 for OAM$_1$ mode and OAM$_3$ mode multiplexing system with OOK modulation.
4. Conclusion
In order to suppress the impact of atmospheric channel fading on OAM-DM FSO systems, a coding method based on hybrid channel coding is proposed. Compared to single channel coding, the proposed scheme takes into account mode dependent channel fading differences, resulting in higher coding gain.
Funding
National Natural Science Foundation of China (62271084, U22B2009).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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