## Abstract

In this paper, we experimentally investigate the performance of crosstalk mitigation for 16-ary quadrature amplitude modulation orthogonal frequency division multiplexing (16QAM-OFDM) signals carrying orbital angular momentum (OAM) multiplexed free-space-optical communication (FSO) links using the pilot assisted Least Square (LS) algorithm. At the demodulating spatial light modulators (SLMs), we launch the distorted phase holograms which have the information of atmospheric turbulence obeying the modified Hill spectrum. And crosstalk can be introduced by these holograms with the experimental verification. The pilot assisted LS algorithm can efficiently improve the quality of system performance, the points of constellations get closer to the reference points and around two orders of magnitude improvement of bit-error rate (BER) is obtained.

© 2017 Optical Society of America

## 1. Introduction

Currently, as a new degree of freedom of optical beams except amplitude, phase, wavelength, and polarization, orbital angular momentum (OAM) has been introduced in the optical communication fields with its great performance to enhance the capacity without occupying extra spectrum bandwidth [1]. OAM modes with a helical phase front of $\mathrm{exp}\left(il\phi \right)$, where $l$ is the topological number, are orthogonal with each other so that they can be separated after propagating coaxially, and the possibilities of $l$ are infinite in principle which means we can multiplex as much OAM modes as we can to improve channel capacity theoretically [2].

Orthogonal frequency division multiplexing (OFDM) has been demonstrated in high spectral efficiency communication systems over the past few years [3]. Recent researches have shown the ultra high spectral efficiency potential of OAM multiplexed systems using OFDM signal in free-space-optical communication (FSO) [4,5].

OFDM transfers the complexity of transmitters and receivers from the analog to the digital domain and has been widely used in broadband communication systems as an effective solution to linear signal distortions such as inter-symbol interference (ISI) in a dispersive channel. However, it is well known that FSO links usually are subject to channel fading induced by atmospheric turbulence. The refractive index inhomogeneities of the atmosphere will distort OAM beams’ distinct helical phase-fronts, causing power of each transmitted OAM mode spreading onto neighboring modes, which will essentially result in inter-mode crosstalk between OAM data channels [6]. It has important theoretical significance and reference value to study the method of mitigating interference of atmospheric turbulence on OAM multiplexed FSO communication. Error-correction coding such as Low-Density-Parity-Check (LDPC) code, Bose-Chaudhuri-Hocquenghem (BCH) code and Reed-Solomon (RS) code are used to restrain the effect of turbulence noise [7–11], for instance, BER is declined 1 to 2 orders of magnitude when the refractive index structure parameter ${C}_{n}^{2}$ is less than $1\times {10}^{-14}{m}^{-2/3}$ by BCH code in FSO communication system, and Peak Signal-to-Noise ratio (PSNR) of the reconstructed image is improved 9 to 20 by RS code in holographic ghost imaging (HGI) system when ${C}_{n}^{2}$ is from $1\times {10}^{-1\text{5}}{m}^{-2/3}$ to $5\times {10}^{-16}{m}^{-2/3}$. All these coding methods can improve the system performance, but only for weak atmospheric turbulence, because error-correction coding treats the crosstalk from other OAM modes as noise. In order to mitigate the interference of crosstalk between different OAM modes, some approaches of adaptive optics compensation by using a polarization orthogonal or wavelength separated Gaussian beam as a probe beam for wavefront sensing and correction are raised [12–16], but most of previous works with impressive performance have their own limitations and the optical setup becomes more complex with the adaptive optics. The other way to mitigate the crosstalk is using digital signal processing (DSP) which applies multiple-user detection (MUD) or channel equalization (CE) algorithms to recover the signal and shifts the complexity of the optical setup to the digital domain [17,18]. The performance of CE is notable, but the reported demonstrations are all performed on single-carrier modulated signals, including QPSK, BPSK, and NRZ etc. And there has been no study on CE algorithms for mitigating the atmospheric turbulence induced crosstalk in OFDM-carrying OAM multiplexed FSO links.

In this paper, we illustrate the implementation of dual OAM modes (OAM + 1 and OAM-1) multiplexed FSO link, each OAM mode carries 16-ary quadrature amplitude modulation orthogonal frequency division multiplexing (16QAM-OFDM) signals and propagates coaxially. We emulate the crosstalk by applying demodulating holograms with random turbulence. At receiver, we apply pilot assisted Least Square (LS) algorithm to mitigate the crosstalk caused by distorted optical vortex in offline processing. The equalization effects on the quality of constellation and BER are studied, which shows pilot assisted LS algorithm can dramatically mitigate the crosstalk and improve system performance.

## 2. Model of atmospheric turbulence and principles of LS algorithm

#### 2.1 Model of atmospheric turbulence

Atmospheric turbulence is stochastic fluctuations of the atmosphere refractive index in space domain and time domain. The typical atmospheric turbulence spectrum models are Kolmogorov spectrum, Von Karman spectrum, Hill spectrum and modified Hill spectrum [19–22]. In inertial region, Von Karman spectrum is almost the same as Kolmogorov spectrum, but it decreases monotonically and sharply in dissipative region, which is consistent with experimental results. In consideration of the mutagenesis at high wave number in optical transmission, Hill spectrum is the most accurate one in both inertial region and dissipative region, but it is not suitable for theoretical analysis since it is based on experimental results. For the modified Hill spectrum which is proposed by Andrews, although it is approximate spectrum of Hill spectrum, it is more suitable for theoretical analysis, and therefore the turbulence that we introduce into experiment obeying this kind of spectrum. The spectrum of atmospheric refractive index fluctuations is as follows,

Based on the modified Hill spectrum, transmission processes of OAM beam propagation through the different intensities of atmospheric turbulence via changing the value of ${C}_{n}^{2}$ are simulated in MATLAB with the multi-phase screen method, and the equation of distorted OAM beam can be shown as,

The phase hologram of the distorted OAM beam after random turbulent phase screens when ${l}_{m}=+1$ is shown as Fig. 1.

In order to introduce the crosstalk between different OAM modes, we apply the distorted phase holograms that have the information of atmospheric turbulence to the demodulating spatial light modulators (SLMs), which use the imperfection of demodulation to simulate the influence of atmospheric turbulence.

#### 2.2 Principles of LS algorithm

The LS algorithm is one of the most basic algorithm of channel estimation with simple structure that can easy to realize. Using LS algorithm does not need to get the self-correlation matrix of sub-channels and the noise variance. Since the transmitted as well as the received data are known at the reference signal positions, the LS algorithm at those places are shown as below,

Represent as matrix,

In order to improve performance of LS algorithm, the correlation between subcarriers is exploited, and the model updated as follows,

where ${H}_{p}=Fh$, and the estimated channel response is, where $Q={\left({F}^{\prime}diag{\left({X}_{p}\right)}^{\prime}diag\left({X}_{p}\right)F\right)}^{-1}$.## 3. Experimental setup

Figure 2 depicts the experimental setup of 16QAM-OFDM carrying dual OAM modes multiplexed FSO link. In the transmitter, two OFDM signals are generated by MATLAB programs. The generation process of each signal is shown as Fig. 2(f). First of all, a ${2}^{21-1}$ pseudo-random binary sequence (PRBS) is separated into two parts as the original bit streams of two channels, after serial-parallel conversion, bit streams are fed to the 16-level quadrature amplitude modulation (16QAM) encoder to map into complex data. For channel estimation and equalization, we insert 4 pilot symbols between 96 OFDM symbols evenly which is shown as Fig. 2(a). And 64 points-Inverse fast Fourier transform (IFFT) is used to generate time domain digital signal. Before IFFT, Hermitian symmetry is done, thus the output of IFFT can be real-valued. Next, ahead of each OFDM symbols, we add the cyclic prefix (CP). Including the pilot symbols, each frame is composed of 100 symbols and a known training sequence is added to the front of it for symbol synchronization. After parallel-serial conversion, we get 16QAM-OFDM signal. Through the arbitrary waveform generator (AWG), two channels digital signals are converted to analog signals simultaneously. And then, we modulate two lasers at 1550 nm by these two electrical signals respectively.

In the OAM multiplexed part of Fig. 2, collimators at the end of each branch couple the light into free-space in the form of Gaussian beams. These beams are converted to different OAM modes using computer-generated fork holograms shown as Figs. 2(b) and 2(c) via SLMs. They are converted into OAM + 1 and OAM-1 respectively. The two OAM beams are spatially multiplexed by beam splitter (BS) and coaxially propagate in free-space. Because this experiment is under laboratory conditions, no distortion is observed after the propagation of each beam. Instead, we use the phase holograms mentioned in section 2.1 at demodulating part shown as Figs. 2(d) and 2(e) after splitting received beam into two copies by BS to introduce the crosstalk between different OAM modes. As a result, in each arm of the demultiplexer, one of distorted OAM modes is converted to a Gaussian beam, which can be coupled into a standard single mode fiber (SSMF).

After demultiplexing, a digital storage oscilloscope (DSO) is used as ADC to convert received two channels analog signals to digital signals in the receiver. For the received digital signal, a series of offline digital signal processing (DSP) is done, as shown in Fig. 2(g), after symbol synchronization, CP removing and FFT, we extract the pilot symbols and the pilot symbols’ channel estimation is done by the LS algorithm in frequency domain. Then, we use pilot symbols’ channel estimation to get channel frequency responses of all OFDM symbols which is needed for channel equalization. At last, signal constellation diagram and bit error rate are obtained.

The transceiver parameters that we use in this system are listed in Table 1.

## 4. Results and discussions

#### 4.1 Crosstalk

As mentioned above, the distorted phase holograms that have the information of different atmospheric turbulent intensities (${C}_{n}^{2}$ varies from $5\times {10}^{-16}{m}^{-2/3}$ to $1\times {10}^{-13}{m}^{-2/3}$) shown in Fig. 3(a) are used to introduce crosstalk into the communication system that we establish. Figures 3(b)-3(d) show the intensity profiles of OAM + 1 and OAM-1 multiplexed beams’ demodulation by the phase holograms of OAM-1 (thus, OAM + 1 beam will be converted to Gaussian beam, and OAM-1 will be converted to OAM-2) with turbulent information. With the increase of ${C}_{n}^{2}$, the influence of turbulence is more severe: the intensity profiles of demodulating multiplexed beam become irregular in Fig. 3(b), and the intensity of optical spot which is converted from OAM + 1 beam is getting weaker as in Fig. 3(c), as for OAM-1, an optical spot appears at the center where the intensity should be zero without turbulence as in Fig. 3(d), which means crosstalk exists when the central spot is coupled into SSMF.

We also measure the power coupled into the collimators (Col 3: channel of OAM + 1 and Col 4: channel of OAM-1) after demodulating OAM + 1 and OAM-1 multiplexed beam by the hologram with turbulence. Note that the power refers to the light power launched into the PIN detector at the receiver side. Figure 4 shows the average normalized received power of each collimator as a function of ${C}_{n}^{2}$ in three different situations: only OAM + 1 is on (blue lines); only OAM-1 is on (green lines); both OAM + 1and OAM-1 are on (red lines). For each value of ${C}_{n}^{2}$, we test 30 kinds of randomly distorted phase holograms generated by MATLAB and compute the average of normalized received power. As shown in Fig. 4, on the one hand, the increasing ${C}_{n}^{2}$ cause a drop in the normalized received power of current channel, on the other, the crosstalk from other channel is getting stronger as ${C}_{n}^{2}$ grows, which leads to the decrease of the signal to crosstalk ratios and poor communication quality. Notice that signal to crosstalk ratios close to 1 (0 dB) when ${C}_{n}^{2}$ greater than $3\times {10}^{-14}{m}^{-2/3}$.

Figure 5 shows the decomposition of OAM beam with turbulence. We use the phase holograms of topological charges $l=-4\sim +4$ with ${C}_{n}^{2}$ from $5\times {10}^{-16}{m}^{-2/3}$ to $1\times {10}^{-13}{m}^{-2/3}$ to demodulate OAM + 1 and OAM-1 respectively. We measure the received power to detect the decomposition of OAM + 1 and OAM-1 with turbulence and calculate the probabilities of each topological charge. It needs to be noted that we also test 30 times for each ${C}_{n}^{2}$ per topological charge. The results show that for OAM + 1 the probability of OAM mode keeping on $l=+1$ fells from 96.00% to 6.80% when ${C}_{n}^{2}$ ranges between $5\times {10}^{-16}{m}^{-2/3}$ and $1\times {10}^{-13}{m}^{-2/3}$, for OAM-1 is brought down from 96.09% to 9.08%, and the leakage to the topological charges out of $l=-4\sim +4$ rise from 0.006% to 59.86% and 0.002% to 53.91% for OAM + 1 and OAM-1 respectively. We can find that the signal power is distributed into other OAM modes due to the atmospheric turbulence, and it becomes more serious along with the raise of ${C}_{n}^{2}$.

#### 4.2 Performance of LS algorithm

We then measure the bit-error rate (BER) performance of 16QAM-OFDM signals over OAM channels. Figure 6 shows the measured BER curves of received optical power for OAM + 1 and OAM-1 with and without pilot assisted LS algorithm under three kinds of ${C}_{n}^{2}$ which represent weak, medium and strong atmospheric turbulence. When ${C}_{n}^{2}=1.5\times {10}^{-15}{m}^{-2/3}$, as shown in Fig. 6(a), we can find BER performance is not much affected as the crosstalk is very weak in this case, and LS algorithm also help to improve the qualities of signals slightly. And when ${C}_{n}^{2}=9.5\times {10}^{-15}{m}^{-2/3}$ and ${C}_{n}^{2}=1\times {10}^{-13}{m}^{-2/3}$, more crosstalk is introduced to the system, compared with the situations without LS algorithm, the sensitivities at the forward error correction (FEC) threshold of $3.8\times {10}^{-3}$ for channel OAM + 1 and OAM-1 are reduced to ~-17 dBm and ~-13.5 dBm respectively with LS algorithm, and BERs decrease obviously as in Figs. 6(b) and 6(c).

Figure 7 presents the received constellations of 16QAM-OFDM signals for the turbulence scenario mentioned above when received optical power is ~-12 dBm. We can find that the constellations are scattered badly under medium and strong atmospheric turbulence, especially under the circumstance of strong atmospheric turbulence. And after equalization by pilot assisted LS algorithm, the points in constellations are more concentrated to the reference points and fairly clean constellations can be obtained. We can see from Fig. 7, the error-vector magnitude (EVM) is reduced visibly: for ${C}_{n}^{2}=1.5\times {10}^{-15}{m}^{-2/3}$, the EVM decline by ~0.7%, and is down ~8.5% and ~11.5% for ${C}_{n}^{2}=9.5\times {10}^{-15}{m}^{-2/3}$ and ${C}_{n}^{2}=1\times {10}^{-13}{m}^{-2/3}$ respectively from the results without LS algorithm.

We also measure the BER performance with and without pilot assisted LS algorithm as a function of ${C}_{n}^{2}$ (${C}_{n}^{2}$ from $5\times {10}^{-16}{m}^{-2/3}$ to $1\times {10}^{-13}{m}^{-2/3}$). The optical power at transmitter is ~13 dBm. The results are shown in Fig. 8. From the lines that only OAM + 1 is on and only OAM-1 is on, the power consumption due to turbulence affect the system’s performance partly. And it is plain to see that the BER performance become worse dramatically from the green lines when crosstalk from the other channel is introduced. But the BER improvements can be got of more than two orders of magnitude under medium turbulence, and roughly one order under strong turbulence, which benefitted from LS algorithm applied in offline DSP.

## 5. Conclusions

We set up a 16QAM-OFDM carrying dual OAM modes multiplexed FSO link and use the distorted phase holograms which have the information of atmospheric turbulence obeying the modified Hill spectrum to simulate the influence of atmospheric turbulence and introduce the crosstalk into system effectively. And we investigate the crosstalk mitigation performance of the pilot assisted LS algorithm. The EVMs of constellations fell ~10% and the BER performance is improved by around two orders of magnitude.

## Funding

National Natural Science Foundation of China (Project No. 61420106011, 61601279, 61601277); Shanghai Science and Technology Development Funds (Project No. 17010500400, 15530500600, 16511104100, 16YF1403900).

## References and links

**1. **J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics **6**(7), 488–496 (2012).

**2. **A. E. Willner, H. Huang, Y. Yan, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics **7**(1), 66–106 (2015).

**3. **W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express **16**(2), 841–859 (2008). [PubMed]

**4. **J. Wang, S. Li, C. Li, L. Zhu, C. Gui, D. Xie, Y. Qiu, Q. Yang, and S. Yu, “Ultra-high 230-bit/s/Hz spectral efficiency using OFDM/OQAM 64-QAM signals over pol-muxed 22 orbital angular momentum (OAM) modes,” in *OFC/NFOEC* (2014), paper W1H. 4.

**5. **J. Wang, J. Liu, S. Li, L. Zhu, C. Li, M. Luo, Q. Yang, and S. Yu, “Experimental Demonstration of Free-Space Optical Communications Using OFDM-QPSK/16QAM-Carrying Fractional Orbital Angular Momentum (OAM) Multiplexing,” in *OFC/NFOEC* (2015), paper M2F.5.

**6. **B. Rodenburg, M. P. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. **37**(17), 3735–3737 (2012). [PubMed]

**7. **I. B. Djordjevic, “Deep-space and near-Earth optical communications by coded orbital angular momentum (OAM) modulation,” Opt. Express **19**(15), 14277–14289 (2011). [PubMed]

**8. **Y. Ren, Y. Zhang, Y. Yue, N. Bozinovic, G. Xie, H. Huang, M. Tur, P. Kristensen, I. B. Djordjevic, S. Ramachandran, and A. E. Willner, “Efficient Crosstalk Mitigation of OAM Based 400-Gbit/s QPSK Data Transmission in 1.1-km Vortex Fiber by Using Soft-Decision LDPC codes,” in *CLEO* (2013), paper CM2G.5.

**9. **Z. J. Zhao, R. Liao, S. D. Lyke, and M. C. Roggemann, “Reed-Solomon coding for free-space optical communications through turbulent atmosphere,” Aerospace Conference, 1–12 (2012).

**10. **S. M. Zhao, B. Wang, L. Y. Gong, W. W. Cheng, X. L. Dong, and B. Y. Zheng, “Improving the Atmosphere Turbulence Tolerance in Holographic Ghost, Imaging System by Channel Coding,” J. Lightwave Technol. **31**(17), 2823–2828 (2013).

**11. **S. M. Zhao, B. Wang, L. Zhou, L. Y. Gong, W. W. Cheng, Y. B. Sheng, and B. Y. Zheng, “Turbulence Mitigation Scheme for Optical Communications using Orbital Angular Momentum Multiplexing Based on Channel Coding and Wavefront Correction,” 376 (2014).

**12. **Y. Ren, G. Xie, H. Huang, C. Bao, Y. Yan, N. Ahmed, M. P. Lavery, B. I. Erkmen, S. Dolinar, M. Tur, M. A. Neifeld, M. J. Padgett, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive optics compensation of multiple orbital angular momentum beams propagating through emulated atmospheric turbulence,” Opt. Lett. **39**(10), 2845–2848 (2014). [PubMed]

**13. **Y. Ren, G. Xie, H. Huang, L. Li, N. Ahmed, Y. Yan, M. P. Lavery, R. Bock, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Turbulence compensation of an orbital angular momentum and polarization-multiplexed link using a data-carrying beacon on a separate wavelength,” Opt. Lett. **40**(10), 2249–2252 (2015). [PubMed]

**14. **G. Xie, Y. Ren, H. Huang, M. P. Lavery, N. Ahmed, Y. Yan, C. Bao, L. Li, Z. Zhao, Y. Cao, M. Willner, M. Tur, S. J. Dolinar, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Phase correction for a distorted orbital angular momentum beam using a Zernike polynomials-based stochastic-parallel-gradient-descent algorithm,” Opt. Lett. **40**(7), 1197–1200 (2015). [PubMed]

**15. **S. Li and J. Wang, “Compensation of a distorted N-fold orbital angular momentum multicasting link using adaptive optics,” Opt. Lett. **41**(7), 1482–1485 (2016). [PubMed]

**16. **S. M. Zhao, J. Leach, L. Y. Gong, J. Ding, and B. Y. Zheng, “Aberration corrections for free-space optical communications in atmosphere turbulence using orbital angular momentum states,” Opt. Express **20**(1), 452–461 (2012). [PubMed]

**17. **L. Zou, L. Wang, S. M. Zhao, and H. W. Chen, “Turbulence mitigation scheme based on multiple-user detection in an orbital-angular-momentum multiplexed system,” Chin. Phys. B **25**(11), 316–323 (2016).

**18. **H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. **39**(15), 4360–4363 (2014). [PubMed]

**19. **V. E. Zakharov, V. S. L’Vov, and G. Falkovich, “Kolmogorov Spectra of Turbulence: I Wave Turbulence,” Springer-Verlag, (1992)

**20. **E. E. Morfiadakis, G. L. Glinou, and M. J. Koulouvari, “The Suitability of the von Karman Spectrum for the Structure of Turbulence in a Complex Terrain Wind Farm,” J. Wind Eng. Ind. Aerodyn. **62**, 237–257 (1996).

**21. **W. B. Miller, L. C. Andrews, and J. C. Ricklin, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A **11**(10), 2719–2726 (1994).

**22. **L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media **7**(2), 229–244 (1997).