Abstract
In freespace optical communications that use both amplitude and phase data modulation (for example, in quadrature amplitude modulation (QAM)), the data are typically recovered by mixing a Gaussian local oscillator with a received Gaussian data beam. However, atmospheric turbulence can induce power coupling from the transmitted Gaussian mode to higherorder modes, resulting in a significantly degraded mixing efficiency and system performance. Here, we use a pilotassisted selfcoherent detection approach to overcome this problem. Specifically, we transmit both a Gaussian data beam and a frequencyoffset Gaussian pilot tone beam such that both beams experience similar turbulence and modal coupling. Subsequently, a photodetector mixes all corresponding pairs of the beams’ modes. During mixing, a conjugate of the turbulenceinduced modal coupling is generated and compensates the modal coupling experienced by the data, and thus the corresponding modes of the pilot and data mix efficiently. We demonstrate a 12 Gbit s^{−1} 16QAM polarizationmultiplexed freespace optical link that is resistant to turbulence.
Main
Compared with radio, freespace optical (FSO) communications have gained substantial interest due to their higher data capacity and lower probability of interception^{1,2,3}. Often, an amplitudeonlymodulated Gaussian data beam (for example, as in pulseamplitude modulation (PAM)) is transmitted and recovered^{2}; since data are encoded as distinct amplitude levels, the data constellation points of PAM lie on a onedimensional line in the twodimensional inphase (I) and quadrature (Q) constellation^{4}. Alternatively, FSO systems can benefit from simultaneously recovering the data beam’s amplitude and phase to enable complex modulation formats^{5,6} such as quadrature amplitude modulation (QAM)^{7}. Since data are encoded as distinct vectors, QAM I/Q constellation points can be arranged in a twodimensional array^{4}. In comparison with PAM of the same number of constellation points (that is, modulation order) and average power per bit, QAM is generally less demanding in terms of the optical signaltonoise ratio (OSNR) of the transmitted data due to its larger Euclidean distance in the twodimensional I/Q constellation^{4}. This advantage tends to be more pronounced as the modulation order increases^{4}. In addition, phase recovery can enable various digital signal processing (DSP) functions^{8} that might benefit future FSO systems^{6,9} (for example, compensation for hybrid fibre/FSO systems^{6} and adaptive probabilistic shaped modulations^{9}).
Intensity modulation/direct detection (IM/DD) FSO links typically receive amplitudeencoded data by directly detecting the beam’s intensity levels, yet phase information is not readily recovered^{2,5,10}. Alternatively, FSO systems can recover both amplitude and phase by using coherent detection, which mixes the data beam with a receiver Gaussian local oscillator (LO) beam^{5,9,11}. However, atmospheric turbulence generally limits coherent detection because it induces power coupling of the data beam from the Gaussian mode to other Laguerre–Gaussian (LG) spatial modes^{12,13,14}. Such turbulenceinduced modal coupling can significantly degrade the data–LO mixing efficiency due to ‘mode mismatch’ between the LO and data beams^{12,13,15,16}. Without turbulence, the photodetector (PD) efficiently mixes the data and the LO since they typically occupy the same singleGaussian mode^{17}, and hence are ‘mode matched’ in their spatial distributions^{18,19}. With turbulence, however, significant power of the data beam can be coupled into higherorder LG modes and degrade the mixing efficiency by >20 dB (refs. ^{12,13,15}) since data power coupled to orthogonal higherorder modes does not efficiently mix with the Gaussian LO^{15,20}.
To enable amplitude and phase recovery in turbulent links, various modalcoupling mitigation approaches have been demonstrated^{21,22,23,24,25}. One technique uses adaptive optics to couple the data power back into the Gaussian mode by measuring the distortion using a wavefront sensor and applying a DSPcalculated conjugate phase to the beam by a wavefront corrector^{21}. Another technique uses multimode digital coherent combining^{22,23,24,25}, wherein much of the data power in higherorder modes is captured by either a multimode fibre^{22,23,25} or an array of singlemode fibre (SMF) apertures^{24}. Subsequently, the power from each of the multiple modes is recovered by a separate coherent detector and combined using DSP^{22,23,24,25}. The performance depends on the number of recovered modes, and the complexity of the detection system tends to increase with the number of detected modes^{22,23,25}. Since turbulence may induce coupling to a large number of modes, a laudable goal towards achieving simultaneous amplitude and phase recovery would be to automatically compensate for such power coupling without additional data processing and do so in a single element that efficiently scales to recover all captured modes.
In this article, we experimentally demonstrate the nearerrorfree transmission of a 12 Gbit s^{−1} 16QAM polarizationmultiplexed (PolM) FSO link that is resilient to turbulenceinduced LG modal power coupling for 200 random turbulence realizations. The amplitude and phase of the transmitted QAM data are retrieved using a pilotassisted selfcoherent detector. We transmit a Gaussian pilot beam with a frequency offset from the Gaussian data beam such that both beams experience similar turbulenceinduced LG modal coupling. Subsequently, a single freespacecoupled PD mixes the received multimode data beam with the multimode pilot beam in ‘selfcoherent’ detection^{26}. During mixing, a conjugate of the turbulenceinduced modal coupling of the pilot beam is automatically generated and used to compensate for the modal coupling in the data beam. Specifically, each data–pilot LG modal pair efficiently mixes and contributes to the intermediate frequency (IF) signal. Since the data and pilot experience similar modal coupling, our approach can simultaneously mix and recover nearly all of the captured data modes using a single PD. Experimental results for the turbulence strength (that is, ratio of the beam size to the Fried parameter) 2w_{0}/r_{0} ≈ 5.5 show an average mixing loss of ~3.3 dB.
Results
Concept of pilotassisted selfcoherent detection using optoelectronic mixing
In an FSO link, a fundamental Gaussian beam (that is, LG_{0,0}(x,y)) carrying a data channel (denoted as S(t,f) with the carrier frequency f) is transmitted through a turbulent atmosphere. Owing to a random spatial and temporal refractive index distribution, the turbulence effects can induce a transverse, spatially dependent wavefront distortion to the Gaussian beam^{27}. Moreover, since such distortion induces modal power coupling, the electrical field of the data beam (E_{data}) at the receiver aperture can be expressed as a superposition of LG modes^{12,28}:
where LG_{l,p}(x,y) represents the electrical field of the LG mode^{17} with an azimuthal index l and a radial index p; \({a}_{l,p}=\int\int U\left(x,y\right){{\rm{LG}}}_{l,p}^{* }\left(x,y\right){\rm{d}}x{\rm{d}}y\) is the complex coefficient of the corresponding LG_{l,p} component in the wavefront, * denotes the conjugate of the modal electrical field, and the portion of the optical power coupled to the LG_{l,p} mode is a_{l,p}^{2}; and \(U(x,y)=\sum _{l}\sum _{p}{a}_{l,p}{{\rm{LG}}}_{l,p}\left(x,y\right)\) represents the turbulenceinduced LG modal coupling. Ideally, the complex weights a_{l,p} for all modal components tend to satisfy \(\sum _{l}\sum _{p}{\left{a}_{l,p}\right}^{2}\cong 1\) if the receiver aperture can collect almost the entire beam^{28}.
A turbulent IM/DD FSO link (that is, S(t,f) is amplitudeonly encoded) may suffer from turbulenceinduced modalcoupling loss if an SMFcoupled PD is used because higherorder modes are not efficiently captured by the SMF^{13}. For a freespacecoupled PD, however, an IM/DD FSO link may not be significantly affected by modal coupling if the receiver aperture can collect most of the distorted beam^{29}. This freespacecoupled PD can utilize the detected optical intensity (that is, S(t,f)^{2}) to recover the amplitudeencoded data, but the beam’s phase information is not readily recoverable.
As shown in Fig. 1a, coherentdetection FSO links can recover both the amplitude and phase of the data although they suffer from performance degradation caused by turbulenceinduced modal coupling. Here, the transmitted data S(t,f) contain both amplitude and phaseencoded data (for example, 16QAM data). By way of a simple illustrative example, the continuouswave LO at the receiver in a singlePD heterodyne coherent detector has an optical frequency offset Δf from the data carrier (denoted as C(f − Δf)) and is a Gaussian beam (that is, C(f − Δf)·LG_{0,0}(x,y)). The squarelaw mixing in the PD of the coherent receiver results in a photocurrent^{26,30}
where Re[·] is the real part of a complex element; I is the generated photocurrent; C(f − Δf)^{2} and S(t,f)^{2} are the direct current (d.c.) and the signal–signal beating interference (SSBI) photocurrent, respectively; and 2Re[S(t,f)C*(f − Δf)] generates the desired signal–LO beating (SLB) photocurrent. However, the Gaussianmode LO does not mix efficiently with the multipleLGmode data beam due to the mode mismatch between their LG spectra, which is expressed as^{15}
where orthogonality amongst the LG modes ensures that \(\int\int {{\rm{LG}}}_{0,0}\left(x,y\right){{\rm{LG}}}_{0,0}^{* }\left(x,y\right){\rm{d}}x{\rm{d}}y=1\) and \(\int\int {{\rm{LG}}}_{l,p}\left(x,y\right){{\rm{LG}}}_{0,0}^{* }\left(x,y\right){\rm{d}}x{\rm{d}}y=0\), given that l ≠ 0 or p ≠ 0. Equation (3) shows that only the portion of the transmitted power that remains LG_{0,0} after turbulence can be efficiently mixed with the LO and utilized for recovering the QAM data. Such modalcoupling loss can result in severe degradation of the mixing IF power and thus the recovered data quality^{20}. We note that this mixingefficiency degradation in coherent detection can occur for a PD that is: (1) freespacecoupled due to orthogonality between the higherorder modes and the Gaussian LO^{15,20} and (2) SMFcoupled due to power in the higherorder modes not being efficiently coupled into the fibre^{13}.
Figure 1b illustrates the simultaneous recovery of the amplitude and phase of QAM data by utilizing pilotassisted selfcoherent detection, which automatically compensates for the turbulenceinduced modal coupling. In addition to the Gaussian data beam, we transmit a coaxial Gaussian beam carrying a continuouswave pilot tone with a frequency offset Δf, producing a frequency gap between the pilot and data beams of roughly the channel bandwidth (B) to avoid SSBI. The electrical fields of the data and pilot beams are likely to experience similar turbulenceinduced distortion and modal coupling due to their frequency difference being orders of magnitude smaller than their carrier frequencies^{27}. This similar distortion produces automatic ‘mode matching’ between the beams, such that the electric field of the pilot tone is^{31}:
Importantly, a turbulenceinduced LGcoupling conjugate U* is automatically generated from the pilot to compensate for the modal coupling experienced by the distorted data beam, and the total generated photocurrent is:
where S(t,f)C*(f − Δf) generates the desired signal–pilot beating (SPB) photocurrent at an IF of Δf. The modal coupling is (ideally) corrected in an automatic fashion and the mixing efficiency is:
where each LG_{l,p} component of the data beam is efficiently mixed with the corresponding LG_{l,p} component of the pilot beam. Consequently, almost all the captured optical power carried by higherorder LG spatial modes can contribute to the IF signal and can be automatically recovered using a single squarelaw freespace PD. The recovered QAM data can thus exhibit resilience against modalcoupling loss due to the efficient mixing between the data and pilot beams.
We note that the pilotassisted selfcoherent approach shares some similarities with both IM/DD and coherent detection: (1) similar to IM/DD, our approach does not use a receiverbased LO; and (2) similar to coherent detection, our approach recovers the amplitude and phase by mixing an ‘LOlike’ transmittergenerated pilot with the data beam and is often called ‘selfcoherent detection’^{32,33}. Notably, the pilot in our selfcoherent system would experience similar FSO channel loss as the data beam, which may be noteworthy in longerdistance FSO links, whereas the LO in coherent detection would not^{6}.
Generally, the OSNR needed to achieve a desired bit error rate (BER) depends on both the modulation formats and the detection approaches^{4,32,34}. When comparing our selfcoherent detection with heterodyne coherent detection for amplitude and phaseencoded data, the transmitted power of selfcoherent detection is shared between the pilot and data beams, resulting in selfcoherent detection being more OSNRdemanding compared with coherent detection (without turbulence effects)^{32}. For example, to achieve a given BER for the same QAM order, our selfcoherent approach is likely to require an OSNR of around 3 dB higher when the carrier (that is, pilot)tosignal power ratio (CSPR) is ~1 compared with heterodyne coherent detection^{32,33}. When comparing our amplitudeandphase selfcoherent approach with amplitudeonly IM/DD, the OSNR advantage of selfcoherent QAM over IM/DD PAM (with the same modulation order) becomes more pronounced as the modulation order increases (for example, conventionally regarded to be many decibels for ≥16QAM)^{4,32,33,34}.
In longerdistance FSO links, the required optical power per bit for a desired BER can be a limiting factor^{10,16}. Since the transmitted power is shared between the pilot and data beams, selfcoherent detection will probably have a lower signaltonoise ratio (SNR) compared with freespacecoupled IM/DD with the same received optical power and receiver thermal noise. Moreover, the SNR advantage of QAM over PAM diminishes as the modulation order decreases^{4}. Consequently, IM/DD may have a better BER performance than pilotassisted selfcoherent detection for low modulation orders, such as 2PAM^{16,34}. We also note that IM/DD may have a better performance than selfcoherent detection under lower SNR conditions even at higher modulation formats^{4,32,34}.
Moreover, since atmospheric turbulence tends not to induce significant depolarization effects^{35}, our pilotassisted system should be compatible with PolM techniques by transmitting pilot–data pairs on each orthogonal polarization. Experimental results for a PolM system are shown later in Figs. 4 and 5.
Our approach transmits a pilot along with the data, and the pilot serves to help probe the turbulence and create a conjugate of the distortion from modal coupling. In optical communications, we note that pilotassisted techniques have been demonstrated to probe a channel’s signature and apply a conjugate of that signature to help mitigate various channel impairments, including crossphase modulation^{36} and laser phase noise^{37}. More specifically, it has been shown via simulation that turbulenceinduced modal crosstalk can be reduced by mixing a pilot beam and datacarrying LG beams in a modedivisionmultiplexed FSO link^{38}. In that approach, the pilot acquires the turbulence signature, is split into multiple copies at the receiver, and generates a conjugate of the turbulence for each of the LG data beams in separate PDs.
Experimental setup of freespace optical communications with emulated turbulence
We experimentally demonstrate pilotassisted selfcoherent detection in a 12 Gbit s^{−1} PolM 16QAM 1mlong FSO link with emulated turbulence. Figure 2 shows the experimental setup (see the Methods section for more details). The strengths (that is, the ratio of the beam size 2w_{0} to the Fried parameter r_{0}) of the weaker and stronger turbulence effects are 2w_{0}/r_{0} ≈ 2.2 and 5.5, respectively.
We emulate atmospheric turbulence effects using a single rotatable phase plate. Generally, turbulence effects can be more accurately emulated using multiple phase plates^{27}. To address our emulation accuracy, we simulate the optical and electrical mixing power loss using single and multiple random phase screen (RPS) models; the simulation results show similar loss distributions and trends for both 1RPS and 5RPS models (see Supplementary Figs. 1 and 2 for more details).
Characterization of optical and electrical mixing power loss
We measure the turbulenceinduced optical power loss and electrical mixing power loss of the pilotassisted selfcoherent detector for each polarization at 1,000 random realizations of the emulated turbulence. For both X and Y polarizations, Fig. 3a shows that stronger turbulence induces <2 dB of optical power loss for selfcoherent detection since the freespacecoupled PD can capture most of the power; we note that freespacecoupled IM/DD systems are likely to have similar captured power loss. As shown in Fig. 3b, the selfcoherent detector has an electrical mixing power loss of <3 dB and <6 dB for 99% weaker and 90% stronger turbulence realizations among 1,000 random turbulence realizations, respectively. The relatively low mixing power loss for selfcoherent detection is due to efficient mixing of the pilot and data beams, which is likely to recover almost all the data power from the captured modes.
As discussed, turbulenceinduced modal coupling can result in significant power loss for ‘modeselective’ SMFcoupled IM/DD or coherent detectors. Figure 3a shows that the optical power loss for SMFcoupled systems ranges from ~2 to ~22 dB and from ~7 to ~30 dB under ~2.2 and ~5.5 turbulence strengths, respectively. Among the 1,000 emulated turbulence realizations, Fig. 3b shows that the coherent detector can suffer from a mixing power loss of ~28 dB for 99% and 90% of weaker and stronger turbulence, respectively. This mixing loss is due to the SMFcoupled detector not efficiently capturing the power coupled to higherorder modes^{13}.
To help further validate our experimental results, we simulate the selfcoherent system using 1RPS (see Supplementary Equations (1)–(6) for simulation details). As shown in Fig. 3c, the simulation results indicate that selfcoherent detection suffers <4 dB of average optical and electrical mixing power loss as the turbulence strength 2w_{0}/r_{0} is increased from ~1 to ~7. Moreover, the plotted experimental results are generally in agreement with the simulation.
Turbulenceresilient 12 Gbit s^{−1} 16QAM PolM freespace optical transmission
We demonstrate 12 Gbit s^{−1} PolM FSO transmission under emulated turbulence effects, with each polarization carrying 1.5 Gbaud 16QAM data. The transmitted total optical power per polarization (including pilot and data beams) is ~7 dBm. The transmitted CSPR values are ~1.1 and ~1 for X and Y polarizations, respectively. Figure 4 shows the recovered 16QAM constellations using the selfcoherent detector under example realizations of the weaker and stronger turbulence. We measure the turbulenceinduced LG spectra for l and p indices of −5 to +5 and 0 to 10, respectively. The complex wavefront is measured using offaxis holography (see Methods)^{39}.
With no turbulence effects, Fig. 4a shows that the pilotassisted selfcoherent detector can achieve a nearerrorfree performance and recover an error vector magnitude (EVM) of ~8% for the 16QAM data. Under one random realization of weaker turbulence, the measured LG spectrum of Fig. 4b shows that the data power is mainly coupled to the neighbouring LG modes. Under two different random realizations of stronger turbulence, Fig. 4c,d show that turbulence effects can induce a power loss of >25 dB and that power can be coupled to a large number of LG modes. The performance of the selfcoherent detector is not severely affected by these turbulence effects and the 16QAM data can be recovered with EVM values from ~8% to ~10% for both realizations. This turbulence resiliency is due to the automatic modalcoupling compensation by the pilot–data mixing, enabling almost all captured LG modes to be efficiently recovered.
To elucidate the effects of turbulenceinduced modal coupling on coherent detection, we also show the recovered 16QAM data for an SMFcoupled heterodyne coherent detector in Fig. 4; the recovered data quality degrades for both polarizations, from EVM values of ~7.5% without turbulence (Fig. 4a) to >16% for stronger turbulence (Fig. 4c,d). This degradation is due to data power coupled to higherorder modes that is not efficiently captured by the SMF^{13}.
We also measure the electrical spectra for the selfcoherent and coherent detectors under these example turbulence realizations. Compared with the case of no turbulence, there is a ~3 dB and ~18 dB SNR degradation of the IF signal measured for the selfcoherent and coherent detectors, respectively, under the turbulence realizations of Fig. 4 (see Supplementary Fig. 3 for more details).
Figure 5 shows measured BER values for the pilotassisted selfcoherent detector under 200 random realizations of weaker and stronger turbulence. Results show that the selfcoherent detector can achieve BER values below the 7% forward error correction limit for all realizations. Since turbulence can cause strong modalcouplinginduced power loss, the performance of the coherent detector can degrade and does not achieve the 7% forward error correction limit for some realizations.
We further characterize the performance of the selfcoherent detector by measuring the BER as a function of the transmitted power. We find power penalties of ~3 dB for both polarizations under one realization of the stronger turbulence (see Supplementary Fig. 4).
Enhancing spectral efficiency using Kramers–Kronig detection
In our selfcoherent approach, a frequency gap between the pilot and data beams is needed to avoid SSBI. This gap is roughly equal to the data bandwidth, such that our spectrum is around 2× the data bandwidth. However, this frequency gap can be reduced to increase the spectral efficiency using SSBI mitigation techniques^{40,41} such as Kramers–Kronig (KK) detection^{6,41}. Therefore, we demonstrate a reduction of the data–pilot gap to ~0.1 GHz (IF ≈ 0.9 GHz) using KK detection (see Supplementary Fig. 5 for more details); the recovered 16QAM data exhibit EVM values of <12% for both polarizations under example realizations of weaker and stronger turbulence. Using KK detection, the spectral efficiency of the pilotassisted approach could be increased by roughly 2×. Importantly, the KK scheme typically utilizes a stronger pilot than the nonKK approach. Hence, it is typically less power efficient than the nonKK pilotassisted approach^{41}, resulting in a tradeoff between power efficiency and spectral efficiency.
Discussion
The following issues are interesting to consider:

(i)
Our 1.5 GHz baud rate is limited by the ~3.5 GHz bandwidth of the PD. However, freespacecoupled PDs with a bandwidth of ~49 GHz have been reported^{42}, making >100 Gbit s^{−1} possible.

(ii)
We use LG modes to analyse modal coupling. However, we could utilize other bases (for example, Hermite–Gaussian^{22}). Importantly, we do not need to specify a priori the basis used because our approach is ‘automatic’ and the pilot and data can be described in different bases.

(iii)
We note that differentialphaseshiftkeyed (DPSK) systems are also referred to as ‘selfcoherent’^{43,44}. In DPSK systems: (1) data are typically encoded in the optical phase difference between neighbouring symbols; (2) the received data beam is split into two copies of which one is delayed; (3) these copies are coherently combined using a Mach–Zehnder interferometer; and (4) both Mach–Zehnder interferometer output branches are detected by two PDs simultaneously to recover the differentialencoded data^{43}. Different from our pilotassisted approach, almost all the captured optical power in DPSK systems contains data^{44}. However, to recover the amplitude and phase of QAM data, differential systems typically utilize a more complex receiver than that of the pilotassisted approach^{43,45}. Interestingly, it might be possible to use multimode mixing as described in this paper to achieve automatic turbulence resiliency in a differential, highorder QAM system.

(iv)
A beam diverges with the link distance. Consequently, both the data and pilot beams can suffer from truncation by a limitedsize receiver aperture causing power loss for longerdistance links^{46}. Moreover, truncation can cause power coupling to higherorder modes^{46}. These higherorder modes tend to be automatically mixed by the pilotassisted selfcoherent detection since the pilot and data beams experience similar truncation effects.

(v)
We use a freespacecoupled PD. Can our approach use fibrecoupled PDs? One possibility might be to use a multimode fibrecoupled PD^{10} such that many modes are captured and then impinge on the PD.

(vi)
Although FSO propagation is dependent on a beam’s carrier frequency, it is likely that beam divergence and turbulenceinduced spatial distortions are similar for the pilot and data beams. This is because their typical frequency difference (<1 nm) is substantially smaller than their carrier frequencies (~1.55 μm)^{27,47}.
This paper has described the concept and experimental/simulation results of pilotassisted selfcoherent links to automatically mitigate modal coupling for recovering the amplitude and phase of data. However, there are important questions for further study as to limits and dependencies of our approach, including: (1) the frequency dependence of spatial distortions and (2) its effectiveness as a function of distance, divergence, turbulence strength, signal bandwidth and signalcarrier frequency separation.
Methods
Experimental details of freespace optical communications in emulated turbulence
As shown in Fig. 2, we transmit a pair of datacarrying and pilot Gaussian beams on both X and Y polarizations. A 6 Gbit s^{−1} 16QAM data channel at a wavelength of λ_{1} ≈ 1.55 μm is generated, amplified using an erbiumdoped fibre amplifier (EDFA) and equally split into two copies. One copy is delayed using a >15 m SMF to decorrelate the data channels and two independent data channels are individually combined with another pilot tone at a wavelength of λ_{2} (with a frequency offset of ~2.6 GHz from λ_{1}, Δλ ≈ 0.02 nm). The polarizations of the signals and pilots are adjusted and subsequently combined using a polarization beam combiner to transmit PolM 16QAM signals. The total optical power including the pilot and data beams is ~7 dBm for each of the polarizations. The optical signal is coupled to free space using an optical collimator (Gaussian beam size of diameter 2w_{0} ≈ 2.2 mm), is distorted using a rotatable turbulence emulator (see the section ‘Experimental emulation of atmospheric turbulence effects’) and then propagates in free space for ~1 m. In this demonstration, we emulate different strengths of atmospheric turbulence using two separate turbulence emulators with different Fried parameters r_{0} of 1.0 mm and 0.4 mm. The emulated turbulence distortion for the transmitted Gaussian beam is characterized by the ratio of the beam size to the Fried parameter^{27}, and these are 2w_{0}/r_{0} ≈ 2.2 and 5.5 for the two emulators.
At the receiver, we demultiplex one polarization at a time using a halfwave plate cascaded with a polarizer. The receiver has an aperture diameter of ~10 mm. We measure the spatial amplitude and phase profiles of the turbulencedistorted beam and calculate its LG decomposition using offaxis holography^{39} (see the section ‘Offaxis holography for complex wavefront measurement’). After polarization demultiplexing, the distorted beam is equally split into two copies that are sent to the pilotassisted selfcoherent detector and a singlePD LObased heterodyne coherent detector.
In the pilotassisted selfcoherent detector, the entire spatial profiles of the distorted data and pilot beams are focused into a freespacecoupled InGaAs PD (3 dB bandwidth <3.5 GHz) using an aspheric lens with a focal length of 16 mm and a numerical aperture of ~0.79. The coupling efficiency of the received Gaussian beam, defined as the ratio of the optical power detected by the PD over the total received optical power by the receiver aperture (without turbulence effects), is measured to be >92%. The generated photocurrent is recorded using a realtime digital oscilloscope and the I–Q information of the data channel is subsequently retrieved using offline DSP algorithms (see the section ‘Digital signal processing for retrieving the I–Q information at the receiver’). The Nyquistshaped 16QAM data channel has a symbol rate of 1.5 GHz with a rolloff factor of 0.1, expanding the data’s spectrum to ~1.7 GHz. To avoid SSBI effects, we set the IF (that is, the difference between the pilot and data beams’ carrier frequencies) at Δf ≈ 2.6 GHz, which includes a frequency gap of ~1.8 GHz between the pilot and data beams. Thus, the total transmitted pilotassisted signal spectrum is ~3.5 GHz, which is roughly twice that of the data spectrum (see Supplementary Fig. 3 for more details).
At the singlePD LObased heterodyne coherent detector (the pilot λ_{2} is turned off), we set the same IF value as the pilotassisted selfcoherent receiver. The distorted Gaussian beam is coupled into an SMF via a collimator (aperture diameter ≈ 3.5 mm), amplified using an EDFA, and mixed with an LO (at the same wavelength λ_{2} as the pilot) at the SMFcoupled PD. The received optical signal is amplified by the EDFA to meet the power sensitivity requirement of the SMFcoupled PD. The electrical signal is subsequently recorded using a realtime digital oscilloscope and processed to retrieve the data channel’s I–Q information using the same offline DSP algorithms as the pilotassisted selfcoherent detector. Note that we measure the optical power loss and electrical mixing power loss of this detector (shown in Fig. 3) without using the EDFA inside this receiver. The mixing power loss is measured at the IF of ~2.6 GHz in the electrical domain.
To evaluate the effectiveness of the pilotassisted selfcoherent detector for various turbulence scenarios, we measure the BER values of the 16QAM data channels carried by both polarizations over 200 random turbulence realizations. To measure turbulenceinduced modalpowercoupling effects on an SMFcoupled coherent detector, we also measure the BER performance for the LObased heterodyne coherent detector over 200 random turbulence realizations. Note that we measure the BER performance for one polarization at a time due to the limitations of our measurement setup. Therefore, the BER values for X and Y polarizations with the same realization label may correspond to different turbulence realizations and are difficult to be compared directly.
Experimental emulation of atmospheric turbulence effects
We experimentally emulate the turbulenceinduced distortion by utilizing glass plates (Lexitek), the refractive index distributions of which are fabricated to emulate Kolmogorov power spectrum statistics^{20,27}. Two rotatable glass plates are used separately in the experiment with different Fried parameters (r_{0}) of 1.0 mm (weaker turbulence effects) and 0.4 mm (stronger turbulence effects). Different ‘random’ turbulence realizations are implemented by rotating the single glass plate to different orientations. The diameter of the transmitted Gaussian beam is 2w_{0} ≈ 2.2 mm. The datacarrying Gaussian beams are distorted by the glass plate and then propagate in free space for a distance of ~1 m before reaching the receiver. The strength of the turbulence distortion is given by the ratio of the beam diameter to the Fried parameter^{27}, that is, 2w_{0}/r_{0}. For a proofofconcept demonstration, we investigate the performance of the pilotassisted selfcoherent detector at two different turbulence strengths (2w_{0}/r_{0} ≈ 2.2 and 5.5). Under even stronger turbulence effects, the selfcoherent FSO systems may suffer from strong beamwandering effects and subsequent optical power loss^{48}. A beam pointing and tracking system can be used to compensate for these beamwandering effects^{49}.
In this demonstration, we use a single phase plate to emulate the turbulence distortions for this ~1 m FSO link. However, a multiplephaseplate emulation can generally provide a higher accuracy for emulating the volume atmospheric turbulence effects^{27}. To illustrate the validity of our emulation method, we simulate 1RPS and 5RPS turbulence effects; similar trends for turbulenceinduced system degradations were found (see Supplementary Figs. 1 and 2 for more details). We note that our turbulence emulation provides an approximation of the Gaussian beam’s propagation in a turbulent medium and may not fully reflect the effects of real atmospheric turbulence. To further enhance the accuracy of turbulence emulation, some advanced modelling or emulation methods could potentially be applied^{27,50}.
Offaxis holography for complex wavefront measurement
We use offaxis holography to measure the complex wavefront (that is, the amplitude and phase) of the distorted Gaussian beam and its corresponding LG spectrum. An offaxis reference Gaussian beam (beam diameter ~7 mm) on the same wavelength as the distorted pilot Gaussian beam is incident on the infrared camera with a tilted angle. We record the offaxis interferogram and apply digital image processing to extract the complex wavefront (see Supplementary Fig. 6 for more details). The datacarrying beam is turned off when we measure the complex wavefront of the turbulencedistorted pilot beam.
After the complex wavefront of the distorted Gaussian beam is obtained, we decompose it into a twodimensional LG modal spectrum in which the two indices l and p range from −5 to +5 and from 0 to 10, respectively, as expressed in equation (7)^{28}:
where E_{rec}(x,y) and LG_{l,p}(x,y) are the measured complex field of the distorted Gaussian beam and the theoretical complex field of an LG_{l,p} mode, respectively. The ratio of optical power coupling to the LG_{l,p} mode is given by a_{l,p}^{2}.
Digital signal processing for retrieving the I–Q information at the receiver
The detected electrical signal is sampled using a realtime oscilloscope (20 GHz bandwidth and 50 gigasamples per second sampling rate) and recorded for offline DSP. The recorded signals from the pilotassisted selfcoherent detector and the singlePD LO heterodyne coherent detector are processed using the same DSP procedures. Each signal is filtered using a rootraisedcosine finite impulse response filter with a rolloff factor of 0.1, and the filtered signal is subsequently equalized using a constant modulus algorithm. After equalization with the constant modulus algorithm, carrier frequency offset estimation and carrier phase recovery are sequentially performed to reduce the frequency and phase difference between the signal and the LO (or pilot). Finally, the EVM and BER values of the demodulated signal are calculated to evaluate the quality of the data transmission. The EVM of the detected signal is calculated using equation (8) as follows^{7}:
where the x_{i} and \(\widehat{{x}_{i}}\) represent the transmitted and recovered data symbols, respectively, and N is the total number of detected symbols. In this demonstration, ~180,000 symbols are collected to calculate the EVM and BER values of the 16QAM data signals.
Data availability
All data, theory detail, simulation detail that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
All relevant computing codes that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
This work is supported by the Vannevar Bush Faculty Fellowship sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research (N000141612813); US Office of Naval Research through a MURI award (N000142012558); Defense Security Cooperation Agency (DSCA 4441006051); Airbus Institute for Engineering Research; Naval Information Warfare Center Pacific (N6600120C4704); and the US Office of Naval Research (N000141712443). R.Z. and H.Z. acknowledge the support of a Qualcomm Innovation Fellowship; R.W.B. acknowledges the Canada Research Chairs Program and Natural Sciences and Engineering Research Council of Canada.
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All the authors contributed to the interpretation of the results and writing of the article; R.Z., N.H., H.Z. and A.E.W. conceived the idea; R.Z., N.H., H.Z., K.Z., Haoqian Song, Z.Z. and A.E.W. designed the experiments; R.Z., N.H., H.Z., X.S., Haoqian Song and A.M. conducted the experimental measurements; N.H., Hao Song. and A.M. carried out the numerical simulations; H.Z., K.Z. and X.S. performed the digital signal processing; Y.Z. and R.W.B. helped with the offaxis holography; Haoqian Song, K.P., Hao Song, A.M., Z.Z., C.L., K.M., A.A., B.L. and M.T. contributed to the data interpretation, presentation and visualization; B.L., R.W.B., M.T. and A.E.W. provided the technical support for data analysis and results interpretation. The project was supervised by A.E.W.
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Zhang, R., Hu, N., Zhou, H. et al. Turbulenceresilient pilotassisted selfcoherent freespace optical communications using automatic optoelectronic mixing of many modes. Nat. Photon. 15, 743–750 (2021). https://doi.org/10.1038/s4156602100877w
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