## Abstract

For space division multiplexing self-homodyne coherent systems, we propose a novel digital in-service relative time delay (RTD) estimation method without any additional optoelectronic device. Taking advantage of the frequency-domain periodicity of the colored frequency modulation noise, we manage to find the peak with location reflecting the RTD in its autocorrelation function (ACF). The peak to average ratio is further enhanced by leveraging a low-pass differential finite impulse response filter for robust identification. By simulations, the method is validated to be feasible for various linewidths, formats (16QAM, 32QAM and 64QAM), and links up to 80 km. Particularly, it is proved to be inherently compatible with large-linewidth low-cost lasers for the 10-km link. Also, for a low-complexity implementation, we discuss the way to reduce the number of points used to calculate the ACF while maintaining the same dynamic range. Furthermore, we demonstrate a 50-GBaud 16-QAM experiment to investigate its performances. With received optical power varying from -11 dBm to -17 dBm, 2^{16} points are sufficient to provide an estimation accuracy of standard deviation (STD) less than 0.089 ns for the RTD range of [2.6, 491.0 ns]. The STD can be lowered to 0.036 ns by adopting 2^{18} points. Especially, at -11-dBm ROP, the highest performance has been achieved with an accuracy smaller than the symbol period (0.018-ns STD) and a RTD range of [1.5, 491.0 ns].

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The ever-increasing big data services have brought unprecedented growth of global Internet Protocol (IP) traffic, in which the short-reach optical transmissions, such as the datacenter (DC) networks and access networks, dominate the overall data flow. Different from the long-haul fiber optical transmission systems, intensity modulation direct-detection (IMDD) systems are typically preferred due to the cost and power consumption issues in short-reach networks. However, with the Ethernet interface data rate approaching 800G and 1.6T, even for less than 2-km intra DC interconnects, the multi-lane IMDD systems are faced with serious technical challenges [1], such as the dispersion induced power fading, rapidly increasing cost and limited sensitivity, etc. As another option, the coherent technology benefits from coherent gain and the linear complex-valued field detection. Consequently, it has the advantages of high spectral efficiency, high sensitivity and good tolerance to impairments like the chromatic and polarization-mode dispersion. However, due to the requirement of narrow-linewidth lasers and complex digital signal processing (DSP) algorithms, the coherent technology is still regarded too expensive and too power-consuming for these scenes. As a compromise solution, many self-coherent schemes have been proposed with certain tolerance to linewidths and less implementation complexity. Among them, Kramers-Kronig receiver (KKR) and Stokes vector receiver (SVR) are two typical schemes and have gained much attention [2–3]. Unfortunately, in terms of the product of analog to digital converter (ADC) bandwidth and number of ADCs, they are not cost-efficient to achieve the same capacity in comparison to the conventional coherent technology [4]. Besides, there exists a trade-off between high carrier-to-signal-power-ratio (CSPR) detection and low-CSPR transmission in the KKR [5]. And non-standard receiver structure is usually required for the SVR. To resolve these issues, the space division multiplexing (SDM) self-homodyne coherent (SHC) system has been proposed as another “coherent lite” solution yet inherits the advantages of the coherent technology to the utmost extent [6–8]. In such systems, both signal and local oscillator (LO) originate from the same laser at the transmitter side, then severally delivered by different space channels. The single mode fiber (SMF) pair, multicore fiber (MCF), few-mode fiber (FMF) and multimode fiber (MMF) based SDM-SHC systems have been investigated in [6–11]. Compared with the intradyne scheme utilized in conventional coherent detection systems, the phase noise (PN) and frequency offset (FO) are eliminated during self-homodyne coherent detection (SHCD) if the relative path (signal and LO) delay is aligned. Thus, it does not require the frequency offset estimation (FOE) and carrier phase estimation (CPE) algorithm, enabling the use of low-cost distributed feedback (DFB) lasers with large linewidth. Nevertheless, the advantages are based on the premise of the matched optical path of the signal and LO. In presence of large-linewidth lasers, the SDM-SHC systems are quite sensitive to the residual phase noise (RPN) induced by the relative time delay (RTD) between signal and LO paths [8]. It has been suggested that for the 16-QAM modulation format, to keep less than 1-dB optical signal-to-noise ratio (OSNR) penalty, the product of laser linewidth and the relative path mismatch should be kept below 0.18 MHz×m [12]. To minimize the impairment of the RPN, it is necessary to estimate the RTD and control it within a proper range for reliable SDM-based SHCD.

However, due to the imperfection during fabrication and the environmental perturbation, there exist considerable static RTDs and dynamic RTD fluctuations in practical fibers. Even in homogeneous MCFs generally considered with good consistency among cores, the static RTD of 53.7-km MCF can reach up to 21.4 ns [13]. In terms of the dynamic RTD fluctuation, the SMF pair is typically susceptible to the environmental perturbation. With the laboratory temperatures varying from 23 ℃ to 45 ℃, the dynamic RTD fluctuations of a 53.7-km SMF pair can reach up to 2.7 ns [13]. Namely, the two SMFs of a 53.7-km SMF pair can still accumulate considerable dynamic RTD fluctuations, even under the same temperature changes with a relatively small range. More complex and extreme environmental effects should be taken into consideration for practical links and reliable communication systems, like the much larger operating temperature range of a SMF [14]. Consequently, RTD estimation methods with large dynamic range are not only required for calibration stage, but also for in-service applications. For practical operating systems, the dynamic RTD fluctuation can be estimated in a sporadic way and compensated by the variable optical delay line with a large delay range [15]. Besides, by cascading, some integrated devices are also promising for its compensation [16–17].

To estimate the RTD, the cross correlation of 10Gb/s on-off-key (OOK) probes delivered by different space channels has been utilized [13,18–19]. However, its performance is at the cost of an additional subsystem containing high-speed modulators, photodetectors and ADCs. Furthermore, the method is incompatible with SDM-SHC systems because it will interrupt the traffic while transmitting the OOK signals. By superimposing low frequency phase modulation tones on the signal and LO components, the signal-LO interference can be used for in-service RTD monitoring with sub-centimeter resolution [20]. Despite its high resolution, this configuration is only applicable for m-PSK formats. And its limited estimation range of 9 cm is far from enough for a typical SDM-SHC system. In addition, the modulated phase will burden the CPE algorithm in case of an un-aligned RTD.

In this paper, for SDM-SHC systems, we propose a fully digital RTD estimation method with both large dynamic range and high accuracy, in which no additional optoelectronic device or probe signal is needed. Without interrupting the payload traffic, the method can operate for in-service applications. Through extracting the effective part of the frequency modulation (FM) noise from the RPN calculated by the CPE, we take advantage of its frequency-domain periodicity and achieve the in-service RTD estimation by its autocorrelation function (ACF). The revised ACF model in presence of noise is analytically derived and discussed. Furthermore, to ensure a reliable RTD estimation, a low-pass differential (LPD) finite impulse response (FIR) filter is proposed to suppress the high-frequency interference hence obtain a high peak to average ratio (PAR) of the ACF peaks. By simulation, the optimizations or the impacts of the PAR-related factors, like laser linewidth, truncated length of the FM noise points *N* for ACF calculating, window length of the CPE algorithm (and the linked filter order of the LPD-FIR filter), has been investigated. And the feasibility of the method for different linewidths, formats and links up to 80 km is also verified. Besides, the way to reduce the points for ACF calculating has also been discussed. Finally, we conduct an experiment to validate its feasibility for practical systems.

## 2. Principles

As illustrated in Fig. 1, when a light wave passes through a waveguide, a time delay occurs, which is proportional to the product of the group refractive index and the fiber length. The group refractive index is hard to accurately control during fabrication and is sensitive to the environmental perturbations. With the accumulated group refractive index difference along with the fiber, practical SDM-SHC links will suffer from considerable static RTDs and dynamic RTD fluctuations, resulting in the RPN and a serious deterioration of the system performance. Note that the dynamic RTD fluctuations cannot be addressed by calibration prior to system operation, but must be addressed by in-service methods. Interestingly, the RTD will superimpose a frequency domain periodic modulation term on the laser white FM noise (hereafter referred to as the colored FM noise). Taking advantage of such a periodicity, peaks with locations reflecting the RTD can be generated in the corresponding time-domain ACF (hereafter referred to as position peaks). By searching the location of these peaks, RTD can be accurately estimated in operating system. In this section, we firstly discuss the ideal ACF model of this colored FM noise. Then, given that only the estimation result of the RPN is available for an in-service system, we further discuss its revised model under the additive noise from amplifiers and receivers. Finally, in order to enhance the position peaks for better RTD estimation, the way to extract the effective part of the colored FM noise from the estimated RPN will also be discussed.

#### 2.1 Ideal ACF model of the colored FM noise

Generally, in terms of an ideal SDM SHCD when a RTD ${\tau _m}$ is taken into account, and the noise is ignored, the RPN $\varphi_m (t)$ can be expressed as [12]

The term $\varphi (t)$ is the intrinsic phase noise of the DFB laser. Since the 1/f flicker noise is concentrated near the low frequency, it has minor impacts within a relatively short measurement time [21]. Thus, the white frequency noise dominates the frequency modulation (FM) noise in short time, and the intrinsic phase noise of a DFB laser can be approximately described as a Wiener process for simplicity, namely

In Eq. (2), FM noise $\nu (t)$ is described as a zero-mean Gaussian white process. Its power spectral density (PSD) ${S_\nu }(f)$ is proportional to the Lorentzian linewidth $\delta f$ of the laser, and the corresponding ACF $R_v(\tau )$ is a Dirac delta function

where $IFT\{{\cdot} \} $ stands for inverse Fourier transform.Different from the white FM noise of intrinsic phase noise indicated in Eq. (3), the instantaneous frequency ${\nu _m}(t)$ of RPN is a colored noise with a frequency-domain periodic modulation term $2 - 2\cos (2\pi {\tau _m}f)$ on the PSD ${S_{{\nu _m}}}(f)$, as indicated in Eq. (6). As a result, to distinguish it from the FM noise of the instinct phase noise, the instantaneous frequency ${\nu _m}(t)$ of the RPN will be referred as the colored FM noise in the manuscript.

Its ACF ${R_{{\nu _m}}}(\tau )$ can be used to estimate the RTD precisely by the location of Dirac delta functions $\delta (\tau \pm \tau _m)$

#### 2.2 Revised ACF model in the presence of noise

However, note that neither the exact value of the RPN nor the one of its FM noise is available in a practical system. For an RTD beyond tolerance, we assume a CPE algorithm with a moving average window of length *T* is used to mitigate additive noise while estimating the RPN [22]. As a result, the estimated RPN $\tilde{\varphi }_m(t)$ can be approximated as

*T*[24]. The FM noise can be approximated by the differential of estimated RPN $\tilde{\varphi }_m(t)$, namely

Here we assume ${\varphi _n}(t)$ is independent from ${\nu _m}(t)$. Note that ${\nu _m}(t)$ is a zero-mean process, consequently, under impairments of noise, the ACF in Eq. (7) is revised as

*T*and locations reflecting the RTD. The other is an interference term caused by the additive noise. It should be noted that the interference term also contains two peaks locating near ±

*T*, but their widths are much narrower than the one of position peaks.

#### 2.3 Enhancing position peaks by an LPD-FIR filter

Apart from the interference peaks, another impact arising from the additive noise is the PAR decrease of position peaks, which may lead to the failure of peak search algorithms. After applying the CPE algorithm, the colored FM noise will be shaped by the moving average window. In other words, the effective part of such FM noise is mainly concentrated in the passband frequency range [-1/*T*,1/*T*] of the moving average window. For a digital coherent system, the discrete FM noise points are generally approximated by the difference of adjacent RPN points. And the available RPN point ${\tilde{\varphi }_m}[n] = {\tilde{\varphi }_m}(n{T_s})$ is estimated for each symbol at the moment of *nT _{s}* [21], where

*T*stands for the symbol period. However, due to the high-pass characteristic of the difference operation and the low stopband attenuation of the moving average window, the unwanted out-of-band component of the estimated FM noise cannot be effectively suppressed. As illustrated in Fig. 2(a), both the additive noise and the phase estimation error caused by the CPE algorithms, like the phase estimation error induced by finite test phases of blind phase search algorithm (BPS), will be greatly enhanced at high frequency, leading to considerable high-frequency interferences while estimating FM noise. One of its impacts is the interference peaks locating at ±

_{s}*T*in the ACF, as the additive noise induced high-frequency interference has a frequency periodicity of 1/

*T*, which has been discussed in section 2.2 and indicated in Fig. 2(a). Meanwhile, it is worth mentioning that the finite test phases induced high-frequency interference has no periodicity. More importantly, with such high-frequency interferences, it becomes hard to distinguish the position peak since the high-frequency interferences will decrease the PAR of the position peaks.

To resolve these issues, instead of the difference operation, an LPD-FIR filter with strong stopband attenuation can be adopted to suppress the high-frequency interference while extracting the effective part of the filtered colored FM noise. The LPD-FIR filter can be equivalently regarded as the all-pass differentiator ${{d({\cdot} )} / {dt}}$ followed by a low-pass filter with strong stopband attenuation. The pass-band edge frequency and stop-band edge frequency of the LPD-FIR filter are fixed at 9/(10*T*) and 5/(4*T*), respectively. And the filter order of the LPD-FIR filter is chosen as 200 for the CPE window length of 17 in this manuscript. The CPE window length is normalized by the sampling period for digital systems. As the *T* changes, for various passband frequency ranges [-1/T,1/T] of the FM noise, a key implementation issue for LPD-FIR filter is to avoid the deterioration of its stop-band attenuation, especially when extracting a narrow-band FM noise. Inspired by [25], to deal with the deterioration of the stop-band attenuation for narrow-band case, we select a simple but effective criterion by setting the filter order of LPD-FIR filter proportional to *T/T _{s}*. Figure 2(b) and 2(c) show the amplitude-frequency response and phase-frequency response of the LPD-FIR filter dedicated for the case in Fig. 2(a), respectively.

After generating the discrete RPN point of each symbol by CPE algorithm, the discrete colored FM noise point can be extracted by filtering the RPN point with the LDP-FIR filter. For its discrete-timed-version ACF calculating, the colored FM point is firstly truncated with a length of *N*, which corresponds to the FM noise results at each symbol moment of a *N*-point symbol sequence. Subsequently, *N*-point fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) algorithms are adopted [26], namely

We numerically examine the ACF result for 56-GBaud 16 QAM signals with 15-dB signal to noise ratio (SNR) and 1-MHz Lorentzian linewidth. To evaluate the relative intensity of the position peaks and the interference, the PAR is used as a metric [27], which is defined as follow

## 3. Numerical investigation

#### 3.1 Simulation setup

To investigate the performance of our proposed RTD estimation method and to determine its optimization strategy, we carry out numerical simulations of SDM-SHC systems with various linewidths, modulation formats and links, and discuss the key parameters related to the PAR. The simulation models are given in Fig. 4, which are constructed in the co-simulation environment of MATLAB and VPI TransmissionMaker 9.9.

Light originated from the laser is split into two equal parts, one is for signal modulation and the other is used as a LO. The 56-GBaud PDM 16/32/64 QAM digital signal with a roll-off factor of 0.1 is generated in Matlab, then converted to electrical signals by 4 digital-to-analog converters (DACs) with 8-bit resolution and 64 GSa/s sampling rate in VPI. The band limitation of the DAC is emulated by 5-order Bessel filters with 3-dB bandwidth of 32 GHz. Subsequently, the electrical signal is modulated on the optical field via a dual-polarization IQ modulator (DP-IQM) through a driver. Before injecting into the fiber, to ensure -10-dBm ROP for different formats, the launched optical power of the signal is fixed by a variable optical attenuation (VOA). Then, the signal is delivered by a 10/80-km single mode fiber (SMF). Note that erbium doped fiber amplifiers (EDFAs) are only used to compensate extra 70-km fiber loss of 14 dB for the 80-km link, and no EDFA is included in simulation for 10-km links. The EDFA1 is an ideal optical amplifier excluding amplified spontaneous emission (ASE) noise, while the OSNR is managed by the SetOSNR model. As for the LO path, the noise figure (NF) of EDFA2 is set to 4.5 dB. A delay model on the LO path is used to adjust the RTD between signal and LO paths. To avoid the impact of the random birefringence of SMF2 on the coherent detection, the LO is converted into a linearly polarized light with the state of polarization (SOP) aligned to one axis of the polarization beam splitter (PBS) at 45°. And the ROP of the LO is fixed at 11 dBm. Signal and LO are finally mixed in a coherent receiver, in which the bandwidth and responsivity of balanced photodetectors (BPDs) are set to 40 GHz and 0.9 A/W. Before sent to the receiver-side digital signal processing (DSP), the signal is sampled by 4 analog-to-digital converters (ADCs) with 8-bit resolution and 80 GSa/s sampling rate. A standard DSP is used to recover the noisy signal. Compared with the DSP flow in intradyne coherent systems, the FOE algorithm can be removed in our system, but CPE algorithms is kept in presence of an unacceptable RTD. The chromatic dispersion (CD) compensation block is only adopted for 80-km link. To ensure a fast and robust convergence of the 2×2 multiple-input-multiple-output (MIMO) DSP block, training mode is used to initialize the equalizer coefficients, and frame synchronization is adopted to align the received signal and training symbols. After initialization, the equalizer coefficients are then updated by radius decision constant modulus algorithm (RDCMA). The BPS algorithm is used to recover the QAM signal from the RPN and noise. To trade off the performance and complexity, the number of test phases of BPS are optimized to 32, 40 and 60 for 16QAM, 32QAM and 64QAM, respectively. After obtaining the estimated RPN of one polarization, in the RTD estimation flow, the effective part of the colored FM noise is calculated by the LPD-FIR filter. Then, the corresponding ACF is calculated using the discrete samples by FFT and IFFT algorithms. By searching the location of position peaks, the RTD can be accurately estimated. And the peak width is used as a filter criterion to avoid fake recognitions in the peak search algorithm.

#### 3.2 Simulation results and discussions

To evaluate the estimation accuracy of our proposed method, the root mean square error (RMSE) of multiple measurements is chosen as a metric, which is defined as

The $RT{D_{estim.,i}}$ is here the *i-th* estimation result of *M-*times measurements, while the $RT{D_{true}}$ is the true RTD set by the delay model. For each case, 10-times independent tests will be conducted to calculate the RMSE.

In our simulation, the tested RTDs are varied from 1 ns to 128 ns. When adopting the FM noise estimation result of each symbol in 56-GBaud transmissions, at least 14336 points should be selected to ensure the position peaks within the time window (corresponding to the product of *NT _{s}*) of the ACF. Therefore, the discrete FM noise points is initially truncated with length of

*N*= 2

^{14}, while the linewidth, OSNR, are set to 1 MHz, 25 dB, respectively. Besides, the window length

*T/T*of the CPE moving average window is set to 21, which has been normalized by the sampling period for digital systems. We investigate the performance of the proposed method for 56-GBuad 16QAM transmission over a 10-km link. As shown in Fig. 5(a), the RMSE is less than 0.040 ns under our test range of [1, 128 ns]. It is worth noting that the RMSE increases to 0.040 ns for 128-ns RTD while remaining below 0.030 ns for the RTD interval of [1, 64 ns]. This is primarily due to the fact that the corresponding PAR decreases as the RTD increases, as illustrated in Fig. 5(b). Since identifying the position peak would suffer from the risk of failure when the PAR is small, the PAR is here averaged by successful times of 25-times independent simulations. When the RTD increases from 1 ns to 128 ns, the PAR drops by 2.4 dB in case of 25-dB OSNR. To ensure the robustness of peak recognition and the estimation accuracy, the PAR must be improved to a reasonable level, particularly for large RTDs.

_{s}### 3.2.1 Factors related to the PAR

As the PAR reflects the relative intensity of the position peaks and the interference, any factor that improves the peak intensity or suppressed the interference will be beneficial to the PAR. The linewidth, the truncated length of the FM noise points *N* and the window length of the CPE algorithm are three important factors related to the PAR. It should be noted that the LPD-FIR order is set proportional to the window length. In this way, the following LPD-FIR filter can keep a very high stopband attenuation even when extracting a narrow-band FM noise, ensuring the effective suppression of the high frequency interference as the window length changes. We next investigate their impacts on the PAR of our proposed method under 128-ns RTD. As illustrated in Fig. 6(a) and indicated in Eq. (10), the PAR increases with the linewidth. And increasing *N* is an effective approach to improve the PAR.

Besides, as illustrated in Fig. 6(b), according to the relationships between PARs and the window length under conditions of different OSNRs and *N *= 2^{15}, there exists an optimal window length corresponding to the maximum PAR. This is because increasing the window length suppresses the out-of-band high frequency interference, but broadens the position peak and reduces the intensity as indicated in Eq. (10). However, the choice of window length is not solely determined by the PAR. When the RTD is small enough, an overlap between interference peaks and positioning peaks occurs, resulting in an offset between the estimated and true RTD values. In this case, the location of the position peak may not represent the true RTD. With the RTD decreasing, the overlap occurs earlier as the window length increases. A relatively smaller window length is preferred to ensure a larger dynamic range. Consequently, a tradeoff between the PAR and the dynamic range should be taken into consideration when optimizing the window length.

### 3.2.2 Performance and feasibility for various linewidths, formats and link distances

In order to test the performance and the feasibility of our proposed method for various formats, we firstly investigate the required estimation accuracy to keep less than 1-dB OSNR penalty. Similar to [12], only the RTD induced RPN and the additive Gauss white noise are included when investigating the OSNR penalty. As indicated in Fig. 7(a), the OSNR penalty is related to the product of the RTD and the linewidth for a given format, because the variance of the probability distribution of the RPN is proportional to such products [12]. For 16 QAM, 32QAM and 64 QAM, the products should be kept below 0.841 MHz×ns, 0,455 MHz×ns and 0.221 MHz×ns, respectively, to keep less than 1-dB OSNR penalty. Following that, we investigate the estimation accuracies under various linewidths and *N* by 10-km 16-QAM simulation. The PAR should be greater than about 7 dB to ensure the robustness of peak identifications. As a result, only the RMSE results of the linewidths with the PAR greater than 7 dB in Fig. 6(a) are presented for *N *= 2^{14} and *N* = 2^{15}. As shown in Fig. 7(b), the method is feasible for all tested linewidths and the RMSE can meet the accuracy requirement of 16QAM. Larger linewidths are preferred to realize higher accuracy because of their greater PARs. Compared with the maximum RMSE of 0.067 ns for 0.3-MHz linewidth, the maximum RMSE of 4 MHz linewidth is reduced to 0.011 ns. It is worth noting that the several-megahertz linewidth is a typical linewidth value of the DFB laser [28]. In other words, our method is inherently compatible with such low-cost lasers for high-accuracy RTD estimation. Moreover, for linewidths of 1, 2 and 4 MHz, using more points is also helpful to reduce the overall RMSE. Besides, when the BPS algorithm is adopted for 128-ns RTD, the BERs corresponding to the linewidths of 0.3 MHz, 0.5 MHz, 1 MHz, 2 MHz and 4 MHz are 1.73e-3, 1.94e-3, 2.40e-3 and 3.06e-3 and 3.93e-3, respectively.

Subsequently, we exam the feasibility of various different formats over a 10-km transmission distance. The OSNRs of 16 QAM, 32 QAM and 64 QAM are set to 22 dB, 29 dB and 31 dB, respectively, with the corresponding window lengths optimized to 31, 35 and 37. Aside from that, the linewidth and the *N* are 1-MHz and 2^{15}. The BERs of 16 QAM, 32 QAM, and 64 QAM for 128-ns RTD are 1.68e-2, 1.08e-2, and 2.33e-2, respectively, when the BPS algorithm is used. As shown in Fig. 7(c), the maximum RMSEs are 0.061 ns, 0041 ns and 0.080 ns for 16 QAM, 32 QAM and 64 QAM, respectively, indicating that this method is applicable for all tested formats and can meet the required accuracy with less than 1-dB OSNR penalty. Furthermore, in comparison to other formats, the overall RMSE of 32 QAM is smaller due to its larger optimized PARs. We next verify the feasibility of 16-QAM transmissions over a link distance extending from 10 km to 80 km, with 25-dB OSNR. For the unamplified 10-km link and the amplified 80-km link, the window lengths, *N* and the linewidth are consistently set to 25, 2^{15} and 1 MHz. As illustrated in Fig. 7(d), the maximum RMSE is 0.025 ns and 0.036 ns for 10-km link and 80 km-link, indicating our method is effective for both link distances. Besides, there exists a performance degradation when the link distance extends to 80 km. The RMSE is around 0.02 ns for the10-km link, but it drops to around 0.03 ns for the 80-km link. It should be noted that in our simulation, the degradation comes from the EEPN effect instead of the ASE noise of LO path [29–31]. In our simulations, the beat noise induced by the weak received signal and the LO-ASE noise from EDFA2 is negligible when compared to the interaction of signal-ASE noise and LO. And the beat noise of signal-ASE noise and LO-ASE noise is even smaller [29]. No distinct SNR decrease will be introduced when an EDFA with a 4.5-dB NF is used for LO amplification. However, as opposed to the almost unaffected phase noise of LO, the phase noise in the complex signal is dispersed with the waveform during transmission [31]. An extra phase noise associated with an amplitude noise will be induced by the EEPN effect for the 80-km link [31], which degrades the performance of our method. Therefore, for a link with considerable CD, there exists a tradeoff when selecting the laser linewidth, because increasing the linewidth improves the intensity of position peaks but induces more serious EEPN effect.

### 3.2.3 Reducing the FFT size for low-complexity implementations

By enlarging the truncated length of discrete FM noise points N, a larger dynamic RTD estimation range can be covered, and its further increasing helps to improve the PAR. However, for a practical implementation, large FFT size (equal to the N) introduces significant time or space complexity. It should be noted that we adopt the FM noise result of each symbol, and the temporal resolution of the ACF is equal to the symbol period *T _{s}*. Thus, the RTD estimation accuracy approaching symbol period can be distinguished by such high temporal resolution. But such high temporal resolution is actually superfluous because the required estimation accuracy and the above RMSEs results are usually much larger than the symbol period. Due to the lower accuracy and complexity requirements for implementations, adopting the FM noise estimation result of each symbol is unnecessary. Instead, the RPN can be estimated every

*L*symbols, and the LDP-FIR filter is used to filter every-

*L*-symbol RPN results, yielding the every-L-symbol FM noise result from which the ACF can be calculated. As a consequence, the required FFT size and the complexity can be dramatically reduced while the dynamic range of RTD estimation remains the same.

We extend the simulation of Fig. 5, and change the *L* to investigate its impairments on the RMSE for 16QAM with 25-dB OSNR. The CPE window length and the linewidth are 21 and 1 MHz, respectively, which are the same as set in Fig. 5. As illustrated in Fig. 8, with the *L* rising from 1 to 3, similar performance can be attained with a maximum RMSE of roughly 0.04 ns. As the *L* grows to 4 and 5, the maximum RMSEs are 0.047 and 0.055 ns, respectively. All the RMSE can meet the accuracy requirement, but the FFT size has been dramatically reduced from 16384 to 3278. Note that the time window (corresponding to the product of *NLT _{s}*) is kept the same for different

*L*, and the selection of

*L*needs to avoid the PSD aliasing of the filtered FM noise. Besides, in such case, no significant PAR decrease will be induced by increasing the

*L*while maintaining the same time window. To precisely investigate the RTD estimation accuracy in the following experiments, we continue to use the FM noise estimation result of each symbol, which ensures the high temporal resolution of ACF.

## 4. Experimental verification

#### 4.1 Experimental setup

To evaluate the performance of our RTD estimation method in practical systems. We conduct the SDM-SHC experiments. As illustrated in Fig. 9, the experimental setup is similar to the simulation setup, but the simulation models are replaced by real instruments or devices.

Light from the DFB laser (Yenista Optics OSICS-DFB) is split into two equal parts by a polarization maintaining coupler (PMC), one is for signal modulation and the other is used as a LO. A PDM 50-GBaud 16-QAM signal with a roll-off factor of 0.4 is generated in Matlab. Digital signals are transformed into analog signal by a 4-channel arbitrary waveform generator (AWG, Keysight M8196A) with 90 GS/s sampling rate. Subsequently, the baseband signal is modulated on the optical field via a 23-GHz commercial DP-IQM (Fujitsu FTM7977HQA). After modulation, the signal is delivered by a 2.355-km standard single mode fiber (SSMF). For ease of control, instead of delivering the LO by a SSMF with the SOP finally managed by a PC, a dedicated 2.3-km polarization maintain fiber (PMF) is used for the polarization-stable LO delivery. The initial mismatch fiber length of the two paths is estimated and compensated by a dedicated spliced optical fiber patch cord (OFPC). To realize wide range tunable RTD between the signal and the LO path, extra OFPCs with different nominal lengths varying from 0.3 m to 100 m are then placed in the signal link to coarsely adjust the RTD. According to that the group refraction index of the OFPC is about 1.47, the OFPCs can help to realize a tunable RTD range of roughly [1.5, 490.0 ns]. The ROP of received signal is managed by a VOA, while the ROP of the LO is fixed at about 9.5 dBm. Signal and LO are finally hybrid in a commercial integrated coherent receiver (ICR, NeoPhotonics Class 40). The in-phase and quadrature components of the PDM signal are sampled by the 4-channel digital sampling oscilloscope (DSO, LeCory LabMaster 10-36Zi-A) with 80 GSa/s sampling rate. In the Rx DSP flow, different from the simulation, Gram-Schmidt orthogonalization procedure (GSOP) and 4×4 real-valued MIMO are used to mitigate the impairments of in-phase/quadrature (IQ) amplitude imbalance, phase imbalance and time skew. And the RTD estimation flow is the same as the simulation. In addition, the number of test phases in BPS is optimized to 32 to trade off the performance as well as the complexity.

#### 4.2 Experimental results and discussions

Initially, we perform a linear fit on the estimated RTDs and the mismatch fiber length induced by extra OFPCs. The nominal length of the OFPC is used to approximate the mismatch fiber length. For each case, 13 independent tests are conducted to estimate the RTDs. The linear curve fits well with the estimation results, as shown in Fig. 10. However, for all lengths, the estimated absolute errors from the fitting value are always clustered near a non-zero value, indicating drifts between the nominal lengths and the real lengths of OFPCs.

Therefore, instead of using the nominal length, we will adopt the average of the multiple estimated results for each RTD, and the standard deviation (STD) of the 13-times tests will be used to evaluate the intrinsic estimated error of our method, which is defined as follow

Under a tested ROP range of [-11, -17 dBm], the mismatch fiber length is varied from 0.3 m to 100 m. We next study the optimizations of the essential parameters for PAR improvement, because a high PAR is required to ensure the robustness of the peak search and the accuracy of the method. According to the simulation results, for a given linewidth, the truncated length of discrete FM noise points *N* and the window length are two manipulable parameters related to the PAR. For each case, we average the PAR by the successful times of 13 independent tests, with a default window length of 30. As illustrated in Fig. 11(a), the PAR will be improved as the ROP and the *N* increases. Similar to the simulation, the PAR of 100 m is usually the lowest among that of the three mismatch fiber lengths. Consequently, to ensure the robustness over the tested RTD range, the PAR of 100-m mismatch fiber length should be tuned to greater than about 7 dB. For 100 m mismatch fiber length, we subsequently sweep the window length and investigate the corresponding PAR to determine the proper CPE window lengths. As illustrated in Fig. 11 (b), the PAR initially increases with window length, then tends to saturate, and finally decreases slightly. Thus, instead of selecting the one corresponding to the largest PAR, the window lengths are chosen as marked by purple star in Fig. 11(b) to trade off the PAR and the dynamic range. In our experiments, all the PARs under the chosen window lengths are sufficient to achieve a stable peak search.

We finally test the performance under the specified window lengths. The BERs for ROPs of -11 dBm, -14 dBm and -17 dBm for 100-m mismatch fiber length, are 1.07e-3, 2.92e-3 and 1.10e-2 (the SNRs are 17.1 dB, 16.2 dB and 14.8 dB), respectively. It should be noted that such BERs can meet or come close to the BER levels of practical operating systems. To cover the maximum RTD of about 490 ns (corresponding to the 100-m OFPC) and ensure position peaks within the time window of ACF, the truncated length of discrete FM noise points N should be at least 49000 points. As a result, the performance is evaluated for *N*=2^{16}, *N*=2^{17} and *N*=2^{18}. As illustrated in Fig. 12, the ROP of -11dBm always has the least STD, and the corresponding maximum STDs over the RTD range are 0.054 ns, 0.027 ns and 0.018 ns for *N*=2^{16}, *N*=2^{17} and *N*=2^{18}, respectively. These results indicate that for a RTD range of [1.5, 491.0 ns] (the RTD is substituted by the average of multiple estimated values), accuracy of about symbol period can be achieved when the *N* is larger than 2^{17}. As the ROP decrease to -14 dBm and -17 dBm, the STD can still be kept below 0.024 ns for RTD of [2.0, 491.0 ns] and below 0.036 ns for RTD of [2.6, 491.0 ns] by adopting 2^{18} points. The dynamic ranges of -14-dBm ROP and -17-dBm ROP diminish as the interference peaks and position peaks come to overlap when employing larger optimal windows. In this case, the estimated RTD may not reflect the true RTD and the corresponding results are discarded. It is worth mentioning that for all tested ROPs, 2^{16} points are sufficient to ensure STD below 0.089-ns for a RTD range of [2.6, 491.0 ns]. Furthermore, by employing 2^{18} points, the performance can be improved even further. Over the RTD range of [2.6, 491.0 ns], estimation accuracy with the maximum STD of 0.036-ns (less than two symbol periods) can be achieved for all tested ROPs.

## 5. Conclusion

For SDM-SHC systems, we propose a novel digital in-service RTD estimation method without any additional optoelectronic device. Taking advantages of the frequency-domain periodicity of the colored FM noise, temporal position peaks with locations reflecting the RTD can be generated in its ACF. The peak to average ratio is further enhanced by leveraging a low-pass differential finite impulse response filter for robust identification. By optimizing PAR-related parameters, reliable and precise RTD estimation has been achieved for various linewidths, formats (16QAM, 32QAM and 64QAM) and links with distance up to 80 km in the simulation, demonstrating the feasibility of the method. In particular, it is proved to be inherently compatible with the self-homodyne coherent systems employing large-linewidth lasers for the 10-km link, according to the theory and simulation results. Besides, for low-complexity implementation, we discuss the way to reduce points for ACF calculating. Furthermore, we demonstrate a 50-GBaud 16-QAM experiment to investigate its performances in practical systems. With tested ROP varying from -11 dBm to -17 dBm, 2^{16} points are sufficient to provide an estimation accuracy of standard deviation (STD) less than 0.089 ns for the RTD range of [2.6, 491.0 ns]. The STD can be lowered to 0.036 ns by adopting 2^{18} points. Besides, the highest performance with a STD below 0.018 ns (less than symbol period) and a RTD range of [1.5, 491.0 ns] has been realized for -11-dBm ROP. The results reveal that the RTD estimation method has a large dynamic range as well as a high accuracy. Therefore, the proposed in-service RTD estimation scheme is promising for the practical deployment of future SDM-SHC systems.

## Funding

National Key Research and Development Program of China (2018YFB1801205); National Natural Science Foundation of China (61931010).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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