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Abstract
We propose a novel phase recovery scheme designed for coherent space division multiplexing (SDM) systems with independently-operated asynchronous light sources. The proposed scheme is based on the approach of the extended Kalman filter and is referred to as multiple-input multiple-output carrier phase recovery (MIMO-CPR). In the minimum mean squared error (MMSE) sense, it simultaneously and optimally obtains estimates of the multiple phase errors arising from phase-unlocked asynchronous light sources. To ensure the scheme’s application for SDM fibers with a time-varying property, we also describe a modification to incorporate a MIMO equalization scheme and analyze the computational complexity. The performance of the proposed MIMO-CPR scheme is investigated through numerical simulation, which shows that it has a tolerance for the sum linewidth symbol duration product of up to 3.4 × 10−4, 1.0 × 10−4 and 2.2 × 10−5 for QPSK, 16QAM, and 64QAM signals, respectively, if 1-dB optical signal-to-noise ratio (OSNR) penalty is allowable to achieve BER of 10−3. Transmission experimental results using three spatial modes in a 51-km-long few-mode fiber (FMF) also verify the applicability of the MIMO-CPR scheme to carrier-asynchronous coherent SDM-MIMO systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Tremendous traffic growth has driven world-wide research efforts to develop next-generation fiber-optic communication systems using space division multiplexing (SDM) technology [1]. SDM transmission in which independent data streams are delivered to their destinations over spatial modes excited in multi-mode or few-mode fibers (MMFs/FMFs), which is known as mode division multiplexing (MDM), is an appealing approach to dramatically enhance system capacity because of the high potential to aggregate many spatial channels even with standard-cladding fiber. To date, the demonstration of MDM transmission over a length of 26.5 km exploiting up to 45 spatial channels has been reported [2].
One important remaining issue for realizing future-deployable SDM transport systems based on MDM technology is to relax the hardware requirement for scaling the number of spatial channels at the transmitters and the receivers [3,4]. Here, we take an example of an MDM-based coherent SDM system in which the optical signals are spatially-multiplexed over MMFs or FMFs. Such an SDM system requires an increased number of electrical/optical devices at both ends, including modulators, photodetectors and analog-to-digital/digital-to-analog converters (ADCs/DACs), even if parts of them are monolithically integrated. Moreover, for coherent detection, another stringent requirement exists for the light sources which are used as signal lasers and local oscillator (LO) lasers, in terms of frequency stabilization and the high-power inputs distributed into the optical circuits of each spatial channel. Indeed, previous reports of MDM transmission have implicitly assumed the use of synchronous light sources to ensure the condition of feeding light into receivers within their coherence length [3]. Straightforward solutions for light synchronization may include light sharing among spatial channels that are split after boosting by multiple optical amplifiers (especially in the case of using high-count spatial modes [2]), or the introduction of the electrical/optical injection locking technique [5]. Nevertheless, these solutions would still require additional hardware, such as a semiconductor optical amplifier, a master/slave laser, a thermoelectric cooler, and a feedback loop for monitoring, which results in an increased footprint and higher power consumption by the transmitters and receivers.
To alleviate these hardware requirements for MDM-based transmitters and receivers, we propose a novel multiple-input multiple-output digital signal processing (MIMO-DSP) technique, especially designed for coherent SDM systems with phase-unlocked asynchronous light sources. The system is described with a state space model involved with nonlinear functions, hence the proposed technique, referred to as MIMO carrier phase recovery (MIMO-CPR) in the present work, is derived from the approach of the extended Kalman filter (EKF) which optimally estimates and removes the carrier phase error in the minimum mean squared error (MMSE) sense. Moreover, we discuss the simultaneous use of the proposed MIMO-CPR technique and MIMO equalization to deal with a time-varying optical channel in an SDM-MIMO link; we also analyze the computational cost. The carrier phase error estimation peformance is investigated through a numerical simulation of signal transmission with multiple signal formats over a $2 \times 2$ system. The performance is also verified through a three-mode MDM transmission experiment in a back-to-back configuration and over a 51.2-km three-mode FMF with three independent LO lasers.
The rest of this paper is organized as follows. Section 2 formulates the phase synchronization problem observed in carrier-asynchronous coherent SDM systems. Section 3 describes the proposed MIMO-CPR scheme, followed by the modification to incorporate a MIMO equalization scheme and the complexity analysis. In Section 4, we numerically verify the performance of the MIMO-CPR scheme with an investigation of its tolerance against the increased linewidth of laser sources. Section 5 describes the experimental results on phase synchronization among the three independent LO lasers used in three-mode FMF transmission over 51.2 km. Finally, Section 6 concludes the paper with a brief summary.
Notations: $(\cdot )^T$, $(\cdot )^H$, and $(\cdot )^*$ denote the matrix transpose, the matrix complex-conjugate transpose, and the matrix complex-conjugate, respectively. $\| \cdot \|$ denotes the $L2$-norm of a vector, and $\textrm {diag}(\cdot )$ indicates a diagonal matrix with a vector input. $E\left [ \cdot \right ]$ indicates the expectation of a variable.
2. Problem formulation
The advent of the CPR technique in the digital domain as a key functional block in real-time high-speed DSP has significantly contributed to the commercialization of coherent transmission systems. With CPR, the phase errors that arise from the phase difference between the light sources in transmitters and receivers are effectively removed, hence enabling to fully exploit two degrees of freedom in signals (i.e., the in-phase and quadrature components). Although a number of CPR techniques have been proposed including $M$th-power carrier phase estimation [6], blind phase search [7], and EKF-based CPR [8–11], these applications are substantially restricted to the estimation of scalar-valued phase error. This section is devoted to the problem formulation of the phase errors observed in SDM systems with independent light sources by introducing a SDM-MIMO model to deal with the appearance of vector-valued phase errors in such a system.
2.1 Phase errors in a carrier-asynchronous SDM-MIMO system
We begin formulating the problem by considering an SDM-MIMO transmission system as shown in Fig. 1. For simplicity, we omit the configuration for wavelength multiplexing. The system comprises $N_t$ transmitters and $N_r$ receivers, each containing a laser diode (LD) as a light source, which generates a continuous-wave (CW) light of optical frequency $f_0$ for the purpose of optical signal generation or LO application. At the receiver end, a simultaneous reception of multiple optical signals is performed by polarization- and phase-diversity coherent receivers. Each LD has a linewidth of $\Delta f$ as an origin of independently and identically distributed (i.i.d.) Gaussian random phase noise with zero mean and variance $\sigma ^2 = 2 \pi \Delta f T$, where $T$ is a symbol duration of a signal. Even after the optical signals are down-converted by $f_0$ and digitized by ADCs, the signal-LO beat signals is contaminated by phase noises $\phi _p^t(t) (p \in \{1, \ \cdots , \ N_t \})$ and $\phi _q^r(t) (q \in \{1, \ \cdots , \ N_r\})$, which originate from the signal laser and LO laser, respectively [12]. Additionally, due to the existence of a spatial coupling process occurring in propagation over SDM fibers (especially in MMFs/FMFs), the received signal waveforms are expressed by a linear combinations of the $N_t$ signals in the linear transmission regimes. Denoting the phase error component as $\phi _{p,q}^e \triangleq \phi _{p}^t - \phi _{q}^r$, due to the employment of $N_t + N_r$ asynchronous light sources at the system ends, each digitized signal reconstructed in the digital domain contains not only $N_t$ signals but also $N_t N_r$ phase errors, indicating that MIMO signal detection could be properly performed only when these phase errors are accurately estimated and/or removed.
2.2 SDM-MIMO model
Next, we introduce a baseband SDM-MIMO model with a discrete-time representation designed for use in the digital domain. To simplify the notation, we assume an instantaneous coupling process between spatial modes. Denoting $x_i(k)$ and $y_i(k)$ as the transmitted symbol and received symbol on an $i$-th spatial channel at a sample time $k$, respectively, the end-to-end signal propagation in a linear transmission regime is described by a linear MIMO model:
where we define $\boldsymbol{x}(k) \triangleq [x_1(k), \ x_2(k), \ \cdots , \ x_{N_t}(k)]^T$ $\in \mathbb {Z}_j^{N_t \times 1}$, $\boldsymbol{y}(k)$ $\triangleq [y_1(k), \ y_2(k), \ \cdots , \ y_{N_r}(k)]^T$ $\in \mathbb {C}^{N_r \times 1}$, $\boldsymbol{z}(k) \in \mathbb {C}^{N_r \times 1}$, and $\boldsymbol{H}(k) \in \mathbb {C}^{N_r \times N_t}$ as the transmitted symbol vector, received symbol vector, and noise vector of the circularly symmetrical Gaussian with zero mean, channel transfer matrix, respectively. Note that $\mathbb {Z}_j \triangleq \mathbb {Z} + j \mathbb {Z}$ denotes the set of Gaussian integers. We assume that each component of the signals and noise is spatially independent. Thus we have $E[\boldsymbol{x}\boldsymbol{x}^H] = P_x \boldsymbol{I}_{N_t}$ and $E[\boldsymbol{z}\boldsymbol{z}^H] = P_z \boldsymbol{I}_{N_r}$, where $P_x$, $P_z$, and $\boldsymbol{I}_n$ denote the transmitted signal power per spatial channel, the noise power per spatial channel, and the identity matrix of size $n \times n$, respectively. Hereafter we omit the notation of $k$ unless explicitly stated. By adding the phase noise term after detection at the coherent receivers, Eq. (1) can be rewritten asIn Eq. (3), we define $\boldsymbol{F} \triangleq \boldsymbol{H}^H \boldsymbol{H} + \frac {1}{\gamma } \boldsymbol{I}_{N_t}$, where $\gamma$ denotes the signal-to-noise ratio (SNR) of each spatial mode, defined as $\gamma \triangleq P_x/P_z$. Multiplying $\boldsymbol{D}_t^H \boldsymbol{W}_{\textrm {opt}}^H \boldsymbol{D}_r^H$ by both sides of Eq. (2) yields
3. Proposed algorithm: MIMO-CPR for carrier-asynchronous SDM-MIMO reception
In this section, we derive a MIMO-CPR technique for application to SDM-MIMO reception with asynchronous light sources in the transmitters and receivers. Successive estimation of phase noise terms $\phi _i^t$ and $\phi _i^r$ which typically vary in time can be performed optimally in the MMSE sense if we utilize the framework of the Kalman filtering technique. We also provide a practical procedure for combined use of the proposed MIMO-CPR technique and MIMO equalization, followed by the analysis of the computational load.
3.1 Extended Kalman filter based MIMO-CPR
Optimal estimates of the phase noise are recursively obtained by prediction and update of state estimates based on the process and measurement equations. In Sec. 2, we derived the measurement equation, which can be rewritten with a nonlinear function $h(\boldsymbol{\phi })$ as follows:
where we define $\boldsymbol{\phi }$ and $h(\boldsymbol{\phi })$ as $\boldsymbol{\phi } \triangleq [\phi _1^t, \ \dots , \ \phi _{N_t}^t, \ \phi _1^r, \ \cdots , \ \phi _{N_r}^r]^T \in \mathbb {R}^{(N_t + N_r) \times 1}$ and $h(\boldsymbol{\phi }) \triangleq \boldsymbol{D}_t^H \boldsymbol{W}_\textrm {opt}^H \boldsymbol{D}_r^H \boldsymbol{y}$, respectively, and $\tilde {\boldsymbol{z}}$ is the measurement noise which has a covariance matrix of $\boldsymbol{R} \triangleq E[\tilde {\boldsymbol{z}}(k) \tilde {\boldsymbol{z}}^H(k)] \in \mathbb {C}^{N_t \times N_t}$. The $i$-th element of $h(\boldsymbol{\phi })$, denoted as $h_i$, is explicitly expressed asOn the basis of the state space model given by Eqs. (6) and (8), the EKF framework enables us to derive the MIMO-CPR procedure as follows. At a sample time $k$, denoting the a priori state of the phase noise, the a priori prediction error covariace matrix, the a posteriori state of the phase noise, and the a posteriori measurement error covariance matrix as $\hat {\boldsymbol{\phi }}^{-} \in \mathbb {R}^{(N_t + N_r) \times 1}$, $\boldsymbol{P}^{-} \triangleq E[(\boldsymbol{\phi } -\hat {\boldsymbol{\phi }}^{-})(\boldsymbol{\phi } - \hat {\boldsymbol{\phi }}^{-})^H] \in \mathbb {R}^{(N_t + N_r) \times (N_t + N_r)}$, $\hat {\boldsymbol{\phi }} \in \mathbb {R}^{(N_t + N_r) \times 1}$, and $\boldsymbol{P} \triangleq E[(\boldsymbol{\phi } -\hat {\boldsymbol{\phi }})(\boldsymbol{\phi } - \hat {\boldsymbol{\phi }})^H] \in \mathbb {R}^{(N_t + N_r) \times (N_t + N_r)}$, respectively, the prediction equations for $\hat {\boldsymbol{\phi }}^{-}(k)$ and $\boldsymbol{P}^{-}$ are obtained based on Eq. (8) as
Because the measurement equation given by Eq. (6) involves a nonlinear function of $h(\cdot )$, the update equations are derived from the approach of the EKF algorithm. Using Eq. (6), we obtain the following set of the update equations:
It is noteworthy that $\boldsymbol{T}$ is constructed with knowledge of the weight matrix $\boldsymbol{W}$, the observation $\boldsymbol{y}$, and the a priori state of the phase noise $\boldsymbol{\phi }^{-}$. This indicates that the proposed MIMO-CPR scheme is suitable for implementation with successive and recursive computation in cooperation with the widely used adaptive MIMO equalization technique. The summary of the MIMO-DSP flow in which the MIMO-CPR scheme is processed together with MIMO equalization will be provided in Sec. 3.2.
3.2 Modification with MIMO equalizer
An optical channel typically exhibits a time-varying property due to the environmental perturbations of a fiber. One major approach in MIMO-DSP for tracking such an optical channel is to include a MIMO equalization technique that outputs a signal estimate $\hat {\boldsymbol{x}}$ with the update of the weight matrix $\boldsymbol{W}$ at each sample time. Our interest in this section is to derive a MIMO-DSP procedure for the combined use of a MIMO equalizer and the proposed MIMO-CPR technique. Hereafter, to make discussion more practical, we assume the use of the weight matrix $\boldsymbol{W}$ instead of $\boldsymbol{W}_\textrm {opt}$ for outputting signal estimates.
Letting $\boldsymbol{Y}$, $\hat {\boldsymbol{D}}_t$, $\hat {\boldsymbol{D}}_r$, and $\hat {\boldsymbol{\theta }}_r$ be $\boldsymbol{Y} \triangleq \textrm {diag}(\boldsymbol{y}) \in \mathbb {C}^{N_r \times N_r}$, $\hat {\boldsymbol{D}}_t \triangleq \textrm {diag}[e^{j \hat {\phi }_1^t}, \ e^{j \hat {\phi }_2^t}, \ \cdots , \ e^{j\hat {\phi }_{N_t}^t}] \in \mathbb {C}^{N_t \times N_t}$, $\hat {\boldsymbol{D}}_r \triangleq \textrm {diag}[e^{j \hat {\phi }_1^r}, \ e^{j \hat {\phi }_2^r}, \ \cdots , \ e^{j\hat {\phi }_{N_r}^t}] \in \mathbb {C}^{N_r \times N_r}$, and $\hat {\boldsymbol{\theta }}_r \triangleq [e^{j \hat {\phi }_1^r}, \ e^{j \hat {\phi }_2^r}, \ \cdots , \ e^{j \hat {\phi }_{N_r}^r}]^T \in \mathbb {C}^{N_r \times 1}$, respectively, the output form of $\hat {\boldsymbol{x}}$ becomes
To conlude this section, we provide a brief summary of the MIMO-DSP procedure to obtain signal estimates $\hat {\boldsymbol{x}}$ by combining MIMO equalization and MIMO-CPR, which is listed in Table 1. At each sample time $k$, prediction with respect to $\hat {\boldsymbol{\phi }}^{-}(k)$, $\boldsymbol{P}^{-}(k)$, and $\hat {\boldsymbol{x}}^-(k)$ is performed according to Eqs. (9), (10), and (14), respectively. Then, in the update stage of MIMO-CPR, the a posteriori quantities $\hat {\boldsymbol{\phi }}(k)$ and $\boldsymbol{P}(k)$ are obtained via Eqs. (11), (12), (13), and (15). By using the current state estimate $\hat {\boldsymbol{\phi }}(k)$, phase derotation is performed to obtain the signal estimate $\hat {\boldsymbol{x}}(k)$. Finally, $\boldsymbol{W}(k)$ is also updated as in (19).
Table 1. MIMO-DSP procedure for a carrier-asynchronous SDM-MIMO system
3.3 Complexity analysis
Next, we evaluate the computational complexity of the proposed MIMO-CPR technique with a view toward its application in SDM-MIMO transmission. In this work we define the computational complexity as the required number of complex multiplications per sample time. The result of the complexity analysis is represented in the rightmost column of Table 1. It is preferable to discuss the overall complexity for both MIMO equalization and MIMO-CPR, because computation steps #1 and #16 are essential for both schemes. In the evaluation, with the equalizer tap length $L$, we expanded the sizes of matrices $\boldsymbol{W}$ and $\boldsymbol{Y}$ to $L N_r \times N_t$ and $L N_r \times L N_r$, respectively; the purpose was to cover the pulse-broadening effect in the impulse responses, which arises from the existense of the inherent spatial modal dispersion in MDM systems. To simplify the notation in the discussion which follows, we set the spatial multiplexing degree as $N = N_t = N_r$.
Upon first glance at the complexity analysis results, we notice that the majority of the computational effort is governed by the computation steps of matrix multiplications and matrix inversions with complexity $O(N^3)$; interestingly, this is the case in the regime of short-reach MDM transmission, but not for transmission over a longer distance. The reason is that, in the latter case, $L$ ordinally reaches the order of several hundreds, or sometimes thousands, when MDM signals driven at several tens of gigahertz are propagated over a transmission fiber with a spatial modal dispersion coefficient on the order of several tens to hundreds of ps/km. Therefore, the computation in the latter case would be dominant in the steps of #1 and #16 in Table 1 where the complexity becomes approximately $2 LN(N+1)$. Importantly, the similar computation steps, which are associated with the signal estimate outputs and the update of the weight matrix of the MIMO equalizer, are also required in conventional MIMO-DSP with a time-domain MIMO equalizer which has the complexity of $2 LN^2 +1$ [15]. Figure 2 shows the computational complexity with a spatial multiplexing degree $N$ of 6 (corresponding to the MDM polarization division multiplexed (PDM) transmission over 2LP-mode fiber) with variation of the equalizer tap length of the MIMO equailzer in the range of $1 \leq L \leq 1000$. Unintuitive but important consequence of these results is that the introduction of the proposed MIMO-CPR technique does not have a significant impact on the overall complexity of MIMO-DSP under the assumption of practical application in long-haul MDM transmission where $L \gg N$ holds.
Fig. 2. Computational complexity with $N = 6$, as a function of the equalizer tap length $L$, for the cases of conventional MIMO equalization (dotted black line) and the MIMO-CPR combination (solid red line).
4. Numerical analysis
The performance of the proposed MIMO-CPR technique is analyzed with a numerical simulation. In the simulation, signals with the formats of QPSK, 16QAM, and 64QAM driven at $1/T$= 10 GBaud in the $2 \times 2$ system (equivalent to a single-mode PDM system) were transmitted and received in a back-to-back configuration. The system used four lasers with an identical linewidth $\Delta f$: two for optical signal generation at the transmitter, and two as LO lasers at the receiver. We also assumed that the MIMO-CPR performance is characterized by the sum linewidth symbol duration product $\Delta \nu \cdot T \triangleq 2 \Delta f T$ [16]. For each parameter set, a total of $10^6$ symbols per spatial channel were transmitted. The update processing of $\hat {\boldsymbol{\phi }}$ and $\boldsymbol{W}$ in the MIMO equalizer operated with input signals at a rate of 1 sample per symbol and MIMO-CPR schemes started with a data-aided mode at the first half period of a transmission frame, then switched to a decision-directed mode. We also periodically inserted 2%-overhead pilot symbols into the transmission frame to avoid cycle slip events which induces catastrophic burst errors in the subsequent symbols. The channel transfer matrix $\boldsymbol{H}$ was set to an unitary matrix expressed as
Figure 3 shows the actual and estimated values of the phase noise and the constellations before and after performing MIMO-DSP for 16QAM signals that were received under the conditions of optical SNR (OSNR) = 15 dB/0.1 nm and $\Delta \nu \cdot T = 2 \times 10^{-5}$. Note that, in the figure, a phase offset of 0.1$\pi$ was deliberately added to the actual phase error to improve the visual presentation. It is obvious that, although the actual phase errors behaved with a statistically independent property, they were simultaneously estimated with a high accuracy by the proposed CPR scheme. The convergence for phase estimation was also fast, as it was obtained within 10 ns (equivalent to a 100-symbol period). The estimation performance was further investigated by changing the modulation format, OSNR, and $\Delta \nu \cdot T$, and the results are represented in Fig. 4. The MIMO-CPR based phase noise estimation worked regardless of the modulation format, which ensures the feasibility of the scheme for an MDM system using signals with a wide range of bitrates. However, a larger deviation from the theoretical BER curves was observed for signals with a higher-order modulation format. Figure 5 depicts the OSNR penalty, which is defined as the OSNR gap to achieve a BER of $10^{-3}$ with respect to the theoretical BER curve. From the results in the figure, we can conclude that if 1-dB OSNR penalty is allowable, the proposed MIMO-CPR technique is capable of estimating phase errors with a sum linewidth symbol duration product $\Delta \nu \cdot T$ of up to $3.4 \times 10^{-4}$ (equivalent to the laser linewidth $\Delta f \sim$ 1.7 MHz for 10-Gbaud signals), $1.0 \times 10^{-4}$ ($\Delta f \sim$ 500 kHz), and $2.2 \times 10^{-5}$ ($\Delta f \sim$ 110 kHz) for QPSK, 16QAM, and 64QAM signals, respectively. These numerical results for our MIMO-CPR scheme agree well with the tolerance ranges of $\Delta \nu \cdot T$ that have been evaluated for other SISO-structured CPR techniques [7,16], while the CPR performance might be degraded in more realistic fiber transmissions due to the phase noise enhancement which may arise from the existence of the fiber nonlinearity.
Fig. 3. (a-c): Actual and estimated phase noise transitions of (a) $\phi _2^t$, (b) $\phi _1^r$, and (c) $\phi _2^r$ for 2 $\times$ 2 16QAM signals with OSNR = 15 dB/0.1 nm and $\Delta \nu \cdot T = 2 \times 10^{-5}$. (d-g): Constellations of the 2 $\times$ 2 16QAM signals before ((d, e)) and after ((f, g)) MIMO-DSP.
Fig. 4. BER performance of MIMO-CPR with $\Delta \nu \cdot T \in \{2 \times 10^{-5}, \ 4 \times 10^{-5}, \ 8 \times 10^{-5}\}$ for 2 $\times$ 2 QPSK, 16QAM, and 64QAM signals as a function of the OSNR.
Fig. 5. OSNR penalty as a function of the sum linewidth symbol duration product for QPSK (circles), 16QAM (squares), and 64QAM (diamonds) signals.
5. Experimental performance evaluation
This section presents the results of an experimental performance evaluation of the proposed MIMO-CPR technique with asynchronous light sources in back-to-back and 51-km three-mode FMF transmission configurations.
5.1 Experimental setup
The setup is illustrated in Fig. 6. The test signal was generated as a 6-GBd PDM-QPSK signal at 1550.57 nm by an external cavity laser with a linewidth $\Delta f$ of 25 kHz. The transmission frame comprised a 33040-symbol-length pattern containing 2%-overhead used as a training sequence that was used for frame synchronization. The genrated optical signal was then split into three and decorrelated with delays of 566, 1151, and 1768 ns for LP$_\textrm {01}$, LP$_\textrm {11a}$, and LP$_\textrm {11b}$ inputs, respectively. These signals were spatially multiplexed using a mode multiplexer/demultiplexer (MUX/DEMUX) pair. In the back-to-back transmission configuration, the MUX and DEMUX were directly jointed with a connector, while the other configuration had a 51-km long graded-index (GI) profile FMF transmission fiber supporting 2-LP modes [17] inserted between them. The launched power was set to -2 dBm/mode. After transmission and mode-demultiplexing, a noise loading setup was also introduced to set the received OSNR to a desired value.
Our evaluation focused on the construction of a carrier-asynchronous (CA) setup for the receiver only. The reason for this was as follows. Suppose that the light sources in the receiver are all synchronized with $\phi _i^r = \phi ^r$ for all $i$, then Eq. (17) is reformulated as
5.2 Experimental results
Figure 7 shows the BER performance averaged over the spatial channels in the back-to-back configuration with the CS or CA receiver setups. The degradation with respect to the ideal BER curve is considered to have originated mainly from the existence of a MUX/DEMUX-induced mode dependent loss (MDL), whose value was estimated as 4.8 dB through a singular value decomposition based analysis. There also exists a slight discrepancy in the BER curves between the CS and CA setups, which could be explained as the decision error of $\hat {\boldsymbol{x}}$ which comprises of $N_t$ symbols at each sample time. The tolerance for $\Delta \nu \cdot T$ may be improved if the overhead portion of the pilot symbol sequences is increased at the expence of the signal data rate. Nevertheless, the results shown in Fig. 7 confirm the feasibility of the MIMO-CPR technique even when a CA-configured setup was employed in an SDM-MIMO system. We further tested FMF transmission over 51 km with the CA setup. Figure 8 shows the learning curve of MIMO-DSP for LP$_\textrm {01}$ signals as an error magnitude evolution, showing that the required convergence time was around 0.5 $\mu$s which is equivalent to 3000-symbol period of 6-GBaud signals. The obtained impulse responses of all spatial modes (including polarization modes) are displayed in Fig. 9. A clear impulse response was estimated from the obtained weight matrix $\boldsymbol{W}$, indicating that the use of MIMO-CPR enabled us to accurately recover phase errors even for dispersive channels with a mode dispersion coefficient of 37 ps/km. The constellations of the six spatial channels are also shown in Fig. 10, from which we conlcude that the MIMO-CPR technique is applicable for carrier-asynchronous reception in an SDM transmission system.
Fig. 7. Experimental performance evaluation with averaged BERs of the MIMO-CPR scheme in back-to-back three-mode transmission with carrier-asynchronous (CA, squares) and carrier-synchronous (CS, diamonds) configured receiver setups.
6. Conclusions
In this paper, a novel scheme for estimation of multiple phase errors was proposed that is applicable to carrier-asynchronous coherent SDM reception, for the first time to date. With a view toward an application to a time-varying channel in an optical transmission link, we showed that the proposed MIMO-CPR scheme can be processed in conjunction with a MIMO equalization scheme with an acceptable increased computational complexity. We numerically showed that the MIMO-CPR scheme has a tolerance for the sum linewidth symbol duration product $\Delta \nu \cdot T$ of up to $3.4 \times 10^{-4}$, $1.0 \times 10^{-4}$ and $2.2 \times 10^{-5}$ for QPSK, 16QAM, and 64QAM signals, respectively; this performance is comparable to that of conventional other SISO-structured phase recovery schemes. Finally, an experimental evaluation with a three-mode back-to-back configuration and 51-km FMF transmission was also conducted, showing that the scheme is feasible to adopt the scheme in carrier-asynchronous coherent SDM-MIMO systems.
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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