We deploy kernel-based time-series prediction to suppress the phase noise induced by small deviations from ideal pump counter-phasing in a dual-pump optical phase conjugation system. We show experimentally 1.5-dB SNR improvement for 16-QAM signals at 4opump-phase mismatch

Ali, Abdallah

Boscolo, Sonia

Ellis, Andrew D.

Nguyen, Thanh Tu

Sugizaki, Ryuichi

Sygletos, Stylianos

Takasaka, Shigehiro

Tan, Mingming

Zhang, Tingting

We deploy kernel-based time-series prediction to suppress the phase noise induced by small deviations from ideal pump counter-phasing in a dual-pump optical phase conjugation system. We show experimentally 1.5-dB SNR improvement for 16-QAM signals at 4° pump-phase mismatch

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Kernel-Based Learning-Aided Phase Noise Compensation inDual-Pump Optical Phase Conjugation Coherent SystemT. T. Nguyen1,†, S. Boscolo1,∗, A. A. I. Ali1, M. Tan1, T. Zhang1, S. Takasaka2, R.Sugizaki2, S. Sygletos1, and A. D. Ellis11Aston Institute of Photonic Technologies, Aston University, Birmingham, B4 7ET, United Kingdom2Furukawa Electric Co. Ltd., 6 Yawata-kaigandori, Ichihara, Chiba, 290-8555, Japan†Now with Infinera PA, 7360 Windsor Dr, Allentown, PA 18106, USA. ∗s.a.boscolo@aston.ac.ukAbstract: We deploy kernel-based time-series prediction to suppress the phase noise inducedby small deviations from ideal pump counter-phasing in a dual-pump optical phase conjugationsystem. We show experimentally 1.5-dB SNR improvement for 16-QAM signals at 4◦ pump-phase mismatch. © 2021 The Author(s)1. IntroductionOptical phase conjugation (OPC) has been shown as a promising method to mitigate the distortions caused by chro-matic dispersion and Kerr nonlinearities in high-transmission rate dispersion-unmanaged coherent optical systemsthrough mid-span spectral inversion [1, 2]. However, a major limitation to the performance of fibre-based OPC de-vices relying on four-wave mixing is the transfer of the pump phase modulation that is used to suppress stimulatedBrillouin scattering from the pump to the conjugated signal (idler). Theoretically, this transfer can be fully suppressedby the use of a dual-pump scheme in which the two pumps are modulated using complementary phase patterns [3].However, this approach requires precise adjustment of the phase and amplitude of the modulating signals, whichis difficult to achieve in practice. Therefore, small deviations from the ideal case commonly exist, and require thedevelopment of advanced digital signal processing (DSP) to counteract their side effects.In this paper, we propose for the first time the aid of machine learning (ML) to conventional DSP phase-noise (PN)compensation [4] to correct the deterministic phase distortion caused by deviations from ideal pump counter-ditheringin the OPC of quadrature-amplitude modulation (QAM) signals. We deploy a kernel-based recursive least-squares(KRLS) time-series prediction method [5] in which the phase of the signal after conventional PN compensation isregarded as the result of a time-varying process. The online algorithm then tracks these time variations by consideringdata in windows of fixed size and calculating the updated solution for each window. Contrary to existing compensa-tion methods [6], the proposed approach requires no prior knowledge of the dithering frequencies and features highsensitivity to the residual pump dithering. The latter characteristic makes it particularly suitable for offsetting smallimperfections in the pump-phase modulation scheme of OPC-assisted systems. The technique is verified experimen-tally with a 28Gbaud dual-polarisation 16-QAM signal, and shown to achieve 1.5dB signal-to-noise ratio (SNR)improvement relative to unaided PN compensation at a pump-phase mismatch of 4◦ within an optical back-to-backOPC configuration.2. Experimental SetupThe experimental setup of the dual-polarisation 28-Gbaud 16-QAM system that was used to validate the proposed PNcompensation method is shown in Fig. 1. At the transmitter, ∼ 64000 random 16-QAM symbols were generated foreach polarisation. The data was then up-sampled at 2 samples per symbol and pulse-shaped using a root-raised-cosinefilter with a 0.1 roll-off factor. The signal was loaded into a Keysight arbitrary waveform generator (AWG) (4-channel8-bit digital-to-analogue converter with 56-GSa/s sample rate) and subsequently converted into the optical domainby a commercial multi-format optical transmitter (∼ 100-kHz laser linewidth on 192.4-THz). An erbium-doped fibreamplifier was used to control the launch signal power.WSS WSSHNLFin out1560.1 nm1540.4 nmIQ mod.16-QAMsymbolEDFAOPC900hybrid + PDs + ADCs EDFALODP QAM coherent receiverAWGQAM de-mappingDSP (offline)resamplingTiming recoveryFrequency offset com.Matched filteringAE + downsamplingML based PNCQAM de-mappingECLdBm-1    0    1 Frequency (Ghz)Fig. 1. Experimental setup of the OPC-assisted dual-polarisation 28Gbaud 16-QAM system equipped withML-assisted PN compensation.The signal was conjugated through a polarisation-insensitive dual-ba d OPC with orthogonally polarised pumpsspectrally located at 1540.4nm and 1560.1nm, and with laser linewidth∼ 30kHz. The details of the OPC setup can befound in [2]. One signal band only was used in this experiment. Two radio-frequency (RF) tones at frequencies f1 =60MHz and f2 = 600MHz were generated by the AWG and used to independently phase-modulate the pump lasers viaoptical phase modulators. The amplitudes and phases θi of the RF tones for each phase modulator i = 1,2, were firstadjusted to minimise the transfer of phase modulation from the pumps to the generated idler. The inset of Fig. 1 showsthe RF spectrum of the photo-detected conjugated copy of a continuous-wave laser after the OPC in these calibrationsettings. It features 37.5dB suppression (relative to the carrier peak) of the phase-modulation sidebands transferredfrom the pumps. We refer to this operating condition as ideal pump counter-phasing, which we can express as δθ =(θ1+θ2)−180◦ = 0◦, assuming a fully compensated modulation-index mismatch between the two modulators. Then,we intentionally increased the phase mismatch δθ up to 8◦ by tuning the phase of one of the RF tones. The HNLF was100m long, and the nominal values of the fibre attenuation coefficient, Kerr-nonlinearity coefficient, zero-dispersionwavelength, and dispersion slope were 1.2dBkm−1, 21.4W−1 km−1, 1550nm, and 0.041psnm−2 km−1, respectively.Detection of the conjugated copy of the signal was performed with a typical intra-dyne coherent receiver includinga local oscillator (∼100-kHz linewidth on 193.1THz), a 90◦ optical hybrid, four balanced photo-detectors and a real-time sampling scope (100-GS/s, 23 GHz analogue bandwidth). A standard DSP procedure for data recovery [7,8] wasimplemented offline. The DSP started with re-sampling the digital signal at 2 samples per symbol. Timing recoveryand frequency-offset error correction were performed using a Gardner phase detector (window size of 1024) and aconventional Fourier-transform-based method (window size of 4096), respectively. After matched filtering, an adap-tive equaliser (15-tap butterfly-structure) demultiplexed the dual-polarisation fields and compensated linear effects.The signal was then down-sampled to 1 sample per symbol before being fed into the proposed PN compensationmodule. At the output of this module, the recovered QAM symbols together with the transmitted symbols were usedto measure an effective SNR defined as SNR = Ek[|x[k]|2]/Ek[|ŷ[k]− x[k]|2], where x[k] and ŷ[k] are the respectivetransmitted and received QAM symbols at the time instance k and E is the expectation operator.3. Kernel-Based Learning-Assisted PN Compensation MethodAssuming for simplicity perfect timing recovery, ideal synchronisation and zero frequency-offset, the phase-distortedsignal at the input to the PN compensation unit can be written in the form y[k] = x[k]exp[i(δφ [k] + φm[k])] + ε[k],where ε[k] is the additive white Gaussian noise in the system. The first phase term φ [k] ∼ N (0,2πδνTs) repre-sents the Wiener random laser PN, where δν is the combined spectral linewidth of the system (total linewidth oftransmit, receive and pump lasers) and Ts is the symbol period. The second phase term φm[k] = φm1[k] + φm2[k] ∼(m+δm)[cos(2π f1tk +δθ)+cos(2π f2tk +δθ)]+m[−cos(2π f1tk)−cos(2π f2tk)] represents the deterministic phasedistortion generated by imperfect pump counter-phasing, where m is the modulation index and the modulation-indexmismatch δm represents possibly different modulation indices. Note that in practice, this phase term contains alsononlinear mixing products arising from the nonlinear response of the RF amplifiers. In an OPC-free coherent system,the well-known blind-phase-search (BPS) method [4] is the preferred scheme to compensate for laser-induced PN forthe 16-QAM format because it does not require the use of pilot symbols. However, the effectiveness of this methodmay degrade significantly under the impact of the PN term arising in an OPC system because of its decision-directedoperation. To circumvent this issue, we deploy an additional ML-based compensation stage to aid the BPS algorithm,as depicted in Fig. 2.BPSDithering-induced phase estimation based on Kernel𝑒"#(.)Kerner-assistedInput signal output signal𝛿𝜃 = 0+𝛿𝜃 = 4+PCMC(a) (b)Fig. 2. (a) Block diagram of the proposed PN compensation scheme. (b) Spectral representation of the esti-mated phase noise after the ML unit. PC: Principal components (i.e., the RF frequencies), MC: mixing compo-nents.Hence, within the proposed scheme, the PN-distorted signal is first phase-compensated by the BPS algorithm(using 32 test phase angles and a filter width of 7). We denote here the output signal from the BPS stage as Ŷ 1 .Then the signal’s phase is stripped off and fed into the ML unit whose operational principle relies upon the sliding-window KRLS algorithm presented in [5]. Slightly differently from [5], our implementation of the algorithm seeks theoptimal model vector h that solves the LS problem: minh‖∠{bŶ 1hcD}−∠{Ŷ 1h}‖2, where ∠{.} and b.cD are the angleand direct-decision operators, respectively. The remainder of the implementation follows Algorithm 1 in [5], wherewe use a window size N, regularisation parameter c and Gaussian kernel width w of 26, 0.5 and 100, respectively.The estimated evolution of the phase is then used to compensate for the remaining phase distortion after the BPSblock (Fig. 2(a)). Fig. 2(b) shows examples of the estimated PN after the ML unit in the frequency domain for thepump-phase mismatches δθ = 0◦ and 4◦ and ∼ 36dB optical SNR. It is noteworthy that even in the “ideal” pumpcounter-phasing configuration, there are still several frequency components transferred from the RF tones and theirlinear and nonlinear mixing to the idler, indicating the persistence of some residual dithering. These components canbe detected by the proposed ML-assisted scheme. We also observe that the larger δθ the higher the amplitude of thesecomponents.4. Results and DiscussionTo evaluate the effectiveness of the proposed PN compensation scheme, we set the OPC transceiver to operate atan optical SNR of ∼ 36dB under ideal pump counter-phasing (δθ = 0◦). Then we assess the performance of PNcompensation in terms of effective SNR when varying the pump-phase mismatch δθ . The results are summarised inFig. 3. We observe that the performance of the conventional BPS method degrades significantly under the impact ofimperfect pump counter-phasing: more than 1-dB SNR penalty is observed at a pump-phase mismatch of only 4◦.Conversely, the ML-aided scheme gives around 0.3dB performance benefit over conventional BPS under the bestpump-dithering settings (i.e., at δθ = 0◦), and this is not lost until the pump-phase mismatch is 4◦. At δθ = 4◦,the ML-assisted BPS method outperforms conventional BPS by 1.5dB, with this value increasing to 3dB when δθreaches 8◦.(a) – 16QAM 5dB4dB(b) – 64QAM (**)(b)𝛿𝜃 (degree)(*)(***)0.3 dB1.5 dBKernel assistedBPSGreen: penalty ≤ 0.1 dB;Green & red: penalty ≤ 0.3 dBYellow: penalty > 0.3 dBFig. 3. Performance comparison between KRLS-assisted and conventional PN compensation in terms of ef-fective SNR versus pump-phase mismatch δθ for 16-QAM system.Fig. 4. Limits of operation of the KRLS-assisted PN compensation scheme under different SNR penalty con-straints in the plane of kernel regularisation parameter c and kernel width w for δθ = 0◦, 2◦, 4◦, 6◦, and 8◦.The X marks indicate the parameter set (c,w) used in Fig. 3.Execution of KRLS algorithm requires knowledge of the kernel’s parameters which are not avail-able analytically. In Fig. 4, we shows the working ranges of kernel’s parameter values within differ-ent SNR penalty constraints for different pump-phase mismatch values. For each δθ value we calcu-late the SNR penalty with respect to the maximum effective SNR obtained at that phase mismatch asmax{∀ SNRc,w}−SNRc,w, where SNRc,w is the SNR at a given parameter set (c,w). The results in Fig. 4 highlightthat there is considerable flexibility in the choice of c and w subject to minor performance degradation. This stemsfrom the adaptive operation of the kernel method. We can see in particular that there is a large overlap between theparameter regions bearing a SNR penalty of≤ 0.1 dB (green areas) for the tested δθ values. This means that the sameset (c,w) can be chosen to feed the PN compensation scheme for different pump-phase mismatch levels.5. ConclusionWe have developed a KRLS algorithm-enhanced PN compensation method able to estimate and correct the phasedistortion arising from small imperfections in the pump-modulation scheme of OPC-assisted coherent systems, whichcommonly exist in practice even in optimised OPC setups. By applying the technique to a 16-QAM system, we havedemonstrated 1.5-dB SNR improvement relative to conventional PN compensation at a pump-phase mismatch of 4◦.Owing to its non-parametric nature, the deployed KRLS algorithm is indifferent to the number of dithering frequenciesused, which makes our method potentially applicable to cascaded OPC systems.Acknowledgements: This work was supported by the UK EPSRC - Grants EP/S003436/1, EP/S016171/1 and EP/R035342/1.References1. A. D. Ellis et al., Adv. Opt. Photon. 9, 429–503 (2017).2. M. A. Z. Al-Khateeb et al., Opt. Express 26, 23960–23969(2018).3. M.-C. Ho et al., J. Lightwave Technol. 20, 469–476 (2002).4. T. Pfau et al., J. Lightwave Technol. 27, 989–999 (2009).5. S. Van Vaerenbergh et al., Proc. IEEE ICASSP 2006, pp. V-V.6. R. Elschner et al., Proc. ECOC 2010, P3.17.7. S. J. Savory, Opt. Express 16, 804–817 (2008).8. T. T. Nguyen et al., J. Lightw. Technol. 39, 388–399 (2021).

Kernel-Based Learning-Aided Phase Noise Compensation in Dual-Pump Optical Phase Conjugation Coherent System

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Kernel-based learning-aided phase noise compensation in dual-pump optical phase conjugation coherent system

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