1. Introduction

To meet the ever increasing network capacity demand while facilitating the continuous reduction of cost-per-bit, the next-generation optical networks are required to maximize the data rate per wavelength beyond 1 Tbps by employing extremely high-speed high-order modulation formats [1], such as 120-Gbaud dual-polarization (DP) probabilistically-shaped (PS) 64 quadrature amplitude modulation (QAM) [2]. Such advanced formats are highly vulnerable to the analog impairments inherent in the optical and electrical components. This poses increasing challenges on the design and implementation of the next-generation optical transceivers. Further tightening the specifications and tolerances of the analog components themselves is no more cost-effective in many cases. In-service optical monitoring followed by adaptive digital compensation is a promising solution for keeping the analog imperfections within the tight penalty limits over wavelength and temperature during the product lifetime [3–6]. Moreover, such monitors and actuators are indispensable to cope with the hardware resource diversification in the emerging open and disaggregated optical transport networks.

One of the most important analog impairments limiting the performance of optical transponders is the IQ imbalance (IQI) [7,8], that is, amplitude, phase, and/or timing skews between the in-phase (I) and quadrature (Q) tributaries in the optical IQ modulator (IQM). For extremely high-order signal generation, it is necessary to compensate the IQI down to its slight frequency-dependent variations, namely, the frequency-dependent IQI (FD-IQI) [9]. Thus far, several IQI monitoring techniques have been proposed based on either direct detection (DD) or coherent detection. The transmitter IQI estimation using DSP at the far-end coherent receiver is a typical approach for in-service IQI monitoring and compensation [3–6,10]. However this approach often suffers from receiver-side IQI and transmission impairments, such as the polarization rotation and carrier frequency offset (CFO). Some sophisticated algorithms are needed for separating the impairments at the transmitter, the receiver, and the channel [6]. In addition, the approach requires a long feedback path for transmitter calibration. Meanwhile, several DD-based IQI monitors utilizing monitor photodetectors (PDs) have also been proposed as low-cost alternatives to the coherent detection-based monitors mainly for in-field calibration of pluggable optics [11–17]. These monitors generally employ a specific pilot tone and/or on-off encoding to decouple the I and Q components (and the X and Y polarization channels in the polarization-multiplexing systems) with DD. This limits their application in in-service scenarios. Moreover, many of the conventional DD-based and coherent-based monitors pertain to the frequency-independent IQI models. Recently, a DD-based FD-IQI monitor via phase retrieval (PR), the so called single-pixel optical modulation analyzer (SP-OMA), has been proposed and demonstrated [18]. The individual impulse responses of the IQ channels including their phase components are reconstructed computationally via the pilot-aided widely-linear (WL) PR technique from the phase-less measurements. There are few constraints on the design of the pilot signal for the SP-OMA. It can exploit the standard preamble sequence or even the information-bearing signals if this low-complexity monitor is integrated with the transmitter. Thus it has potential to be an affordable in-service monitoring solution for FD-IQI. In addition, as it relies on DD, SP-OMA is immune to the receiver-side IQI and CFO.

In this work, we extend the concept of the SP-OMA to the dual-polarization IQ modulator (DP-IQM), most commonly employed in the recent coherent optical transponders. In the case of the DP-IQM, the SP-OMA is required to estimate the four different impulse responses, namely the $I_X, Q_X, I_Y$, and $Q_Y$ channels, simultaneously from a monitor PD output. The resulting PR problem has distributed magnitude constraints and the conventional PR algorithms are not applicable in a straightforward manner. To overcome this, we propose a novel pilot-aided WL-PR algorithm by introducing the alternating minimization (AM) procedure. The validity and convergence property of the PR algorithm are investigated via numerical simulation. The feasibility of the proposed DP-SP-OMA is further demonstrated experimentally with a 63.25-Gbaud DP-16QAM signal.

The remainder of the paper is organized as follows. In Section 2, we describe the details of the proposed PR technique for a DP-IQM. To show the feasibility of the proposed approach, some numerical and experimental results are provided in Sections 3 and 4, respectively. Finally, Section 5 concludes this study.

A part of this work has been presented at the Optical Fiber Communication Conference 2021 [19]. In this work, we will provide the detail of the proposed PR algorithm and analyze its performance via numerical simulation. In addition, almost all the experimental results have been refined from [19] by updating the sampling method; interleaved sampling was employed in [19], whereas this work relies on sequential sampling and processes more data.

2. Optical modulation analyzer via phase retrieval for a dual-polarization optical IQ modulator

Figure 1 shows schematic of the proposed SP-OMA for a DP-IQM. The SP-OMA comprises a single (broadband) monitor PD, an analog-to-digital converter (ADC) at the symbol rate, and a digital signal processing unit for PR. Based on the intensity-only measurement at the modulator output and the transmitted symbol information, the SP-OMA computationally recovers the lost phase information and estimates the FD-IQI, or equivalently the frequency/impulse responses of the individual IQ tributaries, including their phase components. The transmitted signal can be a random sequence and with an arbitrary modulation format. This feature facilitates the in-service monitoring from the standard preamble/pilot sequences, such as the overhead bits in 400ZR [20], without sending a dedicated pilot tone or sequence additionally. If the SP-OMA is integrated with the transmitter and allowed to access the transmitter DSP circuit directly, the information-bearing symbols can also be a pilot sequence.

 

Fig. 1. Schematic architecture of the proposed single-pixel optical modulation analyzer for a dual-polarization IQ modulator.

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Actually, the schematic and hardware architecture of this SP-OMA are the same as those of the conventional SP-OMA for a single-polarization IQM [18]. However the phase recovery problem for a DP-IQM can no longer be cast into the standard form of the generalized PR problem [21] and the conventional pilot-aided PR-based channel estimation algorithm [18] is not applicable. Therefore, in this study, we extend the algorithm by introducing an alternating minimization procedure so that it can simultaneously monitor the four different channels in a DP-IQM, i.e., the $I_X, Q_X, I_Y$, and $Q_Y$ channels. Hereafter, for convenience, SP-SP-OMA denotes the SP-OMA for a single-polarization IQM and DP-SP-OMA is the SP-OMA for a DP-IQM.

As in the previous work [18], we consider FD-IQI as a WL filter in the baseband domain [22,23]. The IQ-distorted modulator output field $E_{X}(t)$ in the X-polarization branch is given by

The IQ-distorted modulator output in the Y-polarization channel can be represented in the same way: $E_{Y}(t)\propto \frac {E_0}{\sqrt {2}} \left \{ g_+(t)\ast y(t) + g_-(t)\ast y^*(t)\right \}$, where $y(t)$ and $\{g_+(t), g_-(t)\}$ are the transmitted signal and the WL filter pair for the FD-IQI in the Y-polarization channel. With a polarization misalignment at an arbitrary angle $\phi$, the monitor PD input field is given by

Note that, any other (linear) polarization impairments inside the DP-IQM, such as the differential group delay (DGD) and the polarization-dependent loss (PDL), are involved in $\{h_+(t), h_-(t), g_+(t), g_-(t)\}$ in our model. For instance, the DGD is represented as a timing mismatch between the peaks in $h_+(t)$ and $g_+(t)$. By assuming that the bandwidth of the monitor PD is sufficiently large and in the absence of noise, we have the monitor PD output signal,

In this work, we define $|\boldsymbol {x}|$, $|\boldsymbol {x}|^2$, and $\sqrt {\boldsymbol {x}}$ as element-wise operators for an input vector $\boldsymbol {x}$ with a slight abuse of notations. In (6), $\boldsymbol {h}:=[\boldsymbol {h}_+^{\rm T}\ \boldsymbol {h}_{-}^{\rm T}]^{\rm T}$ and $\boldsymbol {g}:=[\boldsymbol {g}_+^{\rm T}\ \boldsymbol {g}_{-}^{\rm T}]^{\rm T}$ are the composite channel vectors. The $N\times 2L$ matrix $\boldsymbol {X}$ and $\boldsymbol {Y}$ are composed of the corresponding pilot symbols. The structure of the matrices depends on the pilot frame structure and the sampling method. With sequential sampling, i.e., $\boldsymbol {r}:=[r_1, \ldots, r_N]^{\rm T}$, $\boldsymbol {X}$ and $\boldsymbol {Y}$ become block-wise Toeplitz matrices,

The task of the DP-SP-OMA is to estimate $\{\boldsymbol {h}, \boldsymbol {g}\}$, including their phase components, from the intensity-only samples $\boldsymbol {r}$ and the known feature matrices $\boldsymbol {X}$ and $\boldsymbol {Y}$, i.e.,

In the single-polarization case, i.e., $\boldsymbol {g}=\boldsymbol {0}$, the estimation problem falls into the standard form of (generalized) phase retrieval [21] arising in many areas such as crystallography and diffraction imaging. One can rely on the sophisticated algorithms developed in those areas [24,25] for solving the problem. For instance, a robust algorithm based on the alternating direction method of multipliers (ADMM) [26] has been adopted for the SP-SP-OMA. However, in the dual-polarization case, the modulus constraint in (8) is distributed and the generic PR algorithms are not applicable straightforwardly. Thus, we introduce a simple alternating minimization (AM) procedure to make the PR algorithms applicable to such a distributed PR problem.

In the presence of noise, one may obtain the maximum likelihood estimate of $\{\boldsymbol {h}, \boldsymbol {g}\}$ by minimizing the nonlinear cost function

Regarding the proposed algorithm, there are three issues to be discussed: its global convergence property/correctness, its initialization, and its intrinsic ambiguities. Actually, despite empirical success in many non-convex problems, there are no theoretical guarantees regarding global convergence of AM for most cases (see [29], for example). Moreover, in the proposed algorithm, each AM step involves the non-convex optimization steps, making the analysis difficult. Therefore, we will investigate the global convergence/correctness of the proposed approach numerically and experimentally in the following sections. Meanwhile, most iterative PR algorithms work well only if the initialization is “close” to the solution [25]. Fortunately, this will not be a problem in the SP-OMA. We are interested in a small error from the ideal pulse-shaping filter in general, e.g., $\boldsymbol {h}_+=[1, 0,\ldots,0]^{\rm T}$ and $\boldsymbol {h}_{-} = \boldsymbol {0}$, and the ideal filter response can be a good initial guess. Another issue is the intrinsic ambiguities: PR problems suffer from unavoidable ambiguities of global phase shift and conjugate-flipping: $|\boldsymbol {X}\boldsymbol {h}|=|\boldsymbol {X}(e^{j\alpha }\boldsymbol {h})|$ and $|\boldsymbol {X}\boldsymbol {h}|=|\boldsymbol {X}^{*}\boldsymbol {h}^{*}|$ (and the same for $\boldsymbol {g}$). Fortunately, the global phase is not an issue in the FD-IQI identification in most cases. Further, because $\|\boldsymbol {h}_+\|_2^2 >> \|\boldsymbol {h}_{-}\|_2^2$ typically, the conjugate-flipping can be easily detected and removed (see [18] for details). In addition, such ambiguities in a polarization channel do not impact another polarization channel: $\sqrt {\boldsymbol {r}-|\boldsymbol {X}\boldsymbol {h}_{t+1}|^2}$ in (11) is insensitive to the phase/conjugate ambiguities of $\boldsymbol {h}_{t+1}$.

Algorithm 1. Phase retrieval-based channel estimation algorithm for DP-SP-OMA

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It is worth mentioning that, although we assume the modulator is operated in the linear region of its transfer function in this work, it is possible to extend the concept of the SP-OMA to nonlinear system identification. See [30] for more details.

3. Numerical results

As it is generally difficult to guarantee the global convergence of a hybrid iterative algorithm for nonlinear programming analytically, we investigate the correctness and convergence property of the DP-SP-OMA with Algorithm 1 numerically.

In the simulation, the DP-SP-OMA input signal is generated based on the complex baseband equivalents of (1)–(4). Two independent 16QAM signal streams $\{\boldsymbol {x}_n\}$ and $\{\boldsymbol {y}_n\}$ are generated, and each of them is distorted by a different WL filter $\boldsymbol {h}$ or $\boldsymbol {g}$. On the X-polarization channel, the WL filter $\boldsymbol {h} = [\boldsymbol {h}^{\rm T}_+, \boldsymbol {h}^{\rm T}_- ]^{\rm T}$ is randomly generated based on (2):

The WL filter on the Y-polarization channel, i.e., $\boldsymbol {g}$, is randomly generated in the same manner as $\boldsymbol {h}$.

The distorted 16QAM streams are polarization-multiplexed as in (3), where the polarization angle $\theta$ is uniformly distributed over $[0, 2\pi )$. The polarization-multiplexed signal is transmitted over a complex-valued additive white Gaussian noise channel and detected by an ideal square-low detector as in (4). The DP-SP-OMA estimates the WL filter responses $\boldsymbol {h}$ and $\boldsymbol {g}$ from the detector output, thus a phase-less measurement, via Algorithm 1 with $\rho = 0.3$ and $\boldsymbol {h}_0=\boldsymbol {g}_0=[\boldsymbol {l}^{\rm T}_0, 0, \ldots, 0]^{\rm {T}}$.

We evaluate the polarization-wise normalized mean-square error (NMSE) performance:

Figures 2 and 3 are contour plots of the a) average and b) absolute difference of the polarization-wise NMSEs, i.e., $\frac {\Delta _X + \Delta _Y}{2}$ and $|\Delta _X - \Delta _Y|$, for different numbers of AM iterations ($T$) and ADMM iterations per AM iteration (K). The received SNRs are 12 and 20 dB for Figs. 2 and 3, respectively. The channel memory length $L=11$ and the pilot signal is composed of 4,400 DP-16QAM symbols in both cases. Each data point is an average of 1000 independent trials. As can be observed in Figs. 2(a) and 3(a), the NMSE performance does not improve simply by increasing the iteration counts $K$ and $T$. Typically, ADMMs do not converge monotonically for the first few iterations. Further, a large ADMM iteration count per AM iteration ($K$) leads to a bias between the polarization channels due to over-fitting during the AM process, as in Figs. 2(b) and 3(b). The bias cannot be removed just by increasing the number of AM iterations $T$. The optimum set of $K$ and $T$ may depend on the noise level and the characteristic of the WL filter. As shown in Figs. 2 and 3, for a practical SNR range of 12 to 20 dB, $K=2$ to $4$ with $T\geq 15$ provide unbiased estimations with the NMSE < −20 dB stably. Hereafter, we set $K=3$ unless otherwise noted.

 

Fig. 2. a) Average and b) absolute difference of polarization-wise NMSEs with $K=1, \ldots, 10$, and $T=4, \ldots, 24$ at SNR = 12 dB.

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Fig. 3. a) Average and b) absolute difference of polarization-wise NMSEs with $K=1, \ldots, 10$, and $T=4, \ldots, 24$ at SNR = 20 dB.

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Figure 4 shows the average of NMSEs over the polarization channels versus SNR with different pilot lengths. Here, $L=11$ and $T=25$. The result is an average of 3000 trials. As in the figure, the NMSE performance improves monotonically as SNR increases; the improvement is almost proportional to SNR in a practical SNR range of 10 to 20 dB. The performance can also be improved monotonically by increasing the pilot length $N$; the improvement is independent of SNR. Although the DP-SP-OMA relies on nonlinear programming, the result indicates its simplicity in application.

 

Fig. 4. Average of polarization-wise NMSEs versus SNR for different pilot lengths $N$.

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In Fig. 5, we further assess the impact of the pilot length on the NMSE performance. The probability of PR success depends on the number of intensity measurements per unknown, namely, the sample complexity (SC) [21]. In the DP-SP-OMA, the pilot length $N$ corresponds to the number of measurements, and the number of unknowns is proportional to the filter length $L$. Thus the SC is given by $\frac {N}{4L}$. Figure 5 represents the average of NMSEs across the polarization channels versus the SC for different filter lengths. Here, $T=25$ and SNR is 20 dB. As in the figure, Algorithm 1 converges for SC > 10 and achieves a NMSE < −20 dB independently of the problem size $L$. The NMSE performance further improves monotonically as the SC increases up to approximately SC = 200 for SNR = 20 dB. The NMSE performance is slightly better for larger $L$. The DP-SP-OMA relies on the random projections $\boldsymbol {X}$ and $\boldsymbol {Y}$, and a larger problem size facilitates stable phase recovery owing to the diversity effect.

 

Fig. 5. Average of NMSEs over the polarization channels versus sample complexity for different filter lengths $L = 7, \ldots, 41$.

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In practice, a long pilot sequence may affect the system data rate and the DSP cost. Fortunately, the state of an optical modulator does not change so rapidly in practice, and the pilot overhead for the semi-real-time SP-OMA has less impact on the data rate. For instance, the actual pilot lengths are 2,800 to 16,400 symbols for SC = 100 in Fig. 5. Such overhead in every few seconds is almost negligible in beyond 100G systems. As for the DSP cost, the allowable DSP complexity largely depends on the implementation. For reference, we use MATLAB R2021a on a laptop computer with 16 GB of RAM and a 1.8 GHz Intel Core i7-10610U CPU in the simulation and experiment. The CPU time for Algorithm 1 with $T = 25$, $K=4$, $L=11$, and SC = 100 is 0.24 seconds on average. We believe the complexity is feasible in many implementation scenarios. In addition, it is worth mentioning that a long continuous preamble sequence is not necessary to achieve the required SC. One can send short pilot frames intermittently and concatenate the received frames at the receiver to achieve the SC. In fact, an interleaved sampling technique was employed in our preliminary work [19].

Finally, the NMSE performance versus the image rejection ratio (IRR) is plotted in Fig. 6. Here, SNR = 20 dB, $L = 11$, $T = 25$, and SC is 100 (4,400 symbols). The IRR is defined by ${\rm IRR}(\boldsymbol {h}) = \frac {\|\boldsymbol {h}_-\|_2^2}{\|\boldsymbol {h}_+\|_2^2}$ for a WL filter $\boldsymbol {h}$ and represents the degree of IQ distortion. As in the figure, the IRR varies more than 20 dB among the trials. However, the DP-SP-IQM successfully achieves per-sample, per-polarization NMSEs below −25 dB for all samples almost independently of the IQI level. The slight performance degradation in the high IRR region, IRR > - 10 dB, is mostly due to the initialization step. $\boldsymbol {h}_0=\boldsymbol {g}_0=[\boldsymbol {l}^{\rm T}_0, \ldots, 0]^{\rm T}$ is no more effective in such a high IRR region. Some sophisticated initialization techniques, such as spectral initialization [29], are needed in this region.

 

Fig. 6. NMSE versus image rejection ratio (IRR) per sample.

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These numerical results validate the proposed PR-based monitoring approach for the DP-IQM. In a practical range of SNR and an IQI-level with some practical pilot lengths, the proposed PR algorithm with AM enables the simultaneous and stable estimation of the FD-IQI in polarization-multiplexing signals from the intensity-only measurement.

4. Experimental results

We further investigate the feasibility of the DP-SP-OMA experimentally with a 63.25-Gbaud DP-16QAM signal.

Figure 7 shows the experimental setup. The optical transmitter comprised a 1550.92-nm laser with a 100-kHz linewidth, a 4-channel 92-GSa/s arbitrary waveform generator (AWG), and a DP LiNO₃ Mach-Zehnder IQ modulator (DP-IQM). The modulation format was 63.25-Gbaud DP-16QAM. To emulate the FD-IQI, the 16QAM signal on each polarization was digitally pre-distorted by appending 11-tap WL filters. The modulator output was amplified using an erbium-doped fiber amplifier (EDFA) and input to the DP-SP-OMA.

 

Fig. 7. Experimental setup for DP-SP-OMA with a 63.25-Gbaud DP-16QAM signal.

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The DP-SP-OMA comprised a 70-GHz PD and a 160-GSa/s digital storage oscilloscope (DSO) with offline processing. The optical input power to the DP-SP-OMA was set to +10 dBm; it directly sampled the PD output by the DSO without transimpedance amplifiers. In the offline processing, the received signal was down-sampled to 63.25 GSa/s and then the FD-IQIs on the X- and Y-polarization channels, or equivalently the WL filters $\boldsymbol {h}$ and $\boldsymbol {g}$, were estimated via Algorithm 1 with $\rho = 0.3, K=4$, and $T=25$. The FD-IQI estimates comprised not only the digital pre-distortion filter response, but also the inherent FD-IQI of DP-IQM. The inherent FD-IQI term was mitigated by polarization-wise WL equalizers. Their weights were obtained via the DP-SP-OMA without appending any digital pre-distortion filter.

For reference, the EDFA output was also detected by an optical coherent receiver. The coherent receiver consisted of a standard analog coherent optics with polarization diversity configuration and a 4-channel 80-GSa/s DSO. The linewidth of the local oscillator laser was 100 kHz. For simplicity, the input polarization state was manually aligned with that of the coherent receiver by a polarization controller (PC). Note that, there was no PC in front of the DP-SP-OMA and the input polarization angle could rotate randomly measurement by measurement. In the offline processing at the coherent receiver, first, the CFO was compensated based on short, periodic pilot frames. Then, we mitigated the FD-IQI inherent in the transmitter and the receiver via polarization-wise WL equalizers; the intradyne coherent receivers themselves suffer from FD-IQI.

First, to show the validity of the proposed single-pixel approach intuitively, we tested the DP-SP-OMA for four “eye-friendly” IQ distortion patterns. In this demonstration, we equalized the coherent receiver outputs based on the estimates from the DP-SP-OMA: the successful recovery indicates that the FD-IQI on the X- and Y-polarization channels are properly and simultaneously estimated including their phase from the intensity-only measurement via phase retrieval. Figure 8 depicts the pre-distorted, received, and equalized constellation maps. The first row shows the constellation diagrams of the IQ pre-distorted driving signals. The second is the received signal at the coherent receiver after the CFO and inherent IQI compensation. As can be observed, the IQI patterns in the first row were faithfully generated in the optical domain. The equalizer outputs are shown in the third and fourth rows. In the third row, the equalizer weights were derived from the DP-SP-OMA output based on the polarization-wise WL minimum mean-square error (MMSE) criterion [31]. The WL-MMSE equalizer was also employed in the fourth row, but the equalizer weights were obtained based on the coherent detection. As in the figure, the IQI was successfully mitigated in all four cases via the WL-MMSE equalizer based on the DP-SP-OMA. Moreover, the equalization performance was similar between the equalizers with the DP-SP-OMA and the OMA based on coherent detection. The error-vector magnitude (EVM) performance before and after the IQI compensation is summarized in Table 1; each EVM value is the average over the polarization channels in each IQI pattern. In all four cases, the EVM improved from approximately 20% to 14% after equalization. The EVM performance in the experiment was sub-optimal owing to the residual FD-IQI and cross-talk between the polarization channels at the coherent receiver, which were not known to the DP-SP-OMA.

 

Fig. 8. Pre-distorted, received, and equalized constellation maps of a 63.25-Gbaud DP-16QAM signal for four different IQ imbalance (IQI) patterns in the experiment.

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Table 1. EVM performance for four different IQI patterns

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Next, to reveal the accuracy and robustness of the DP-SP-OMA more quantitatively, we evaluated the NMSE between the digital pre-distortion filter response and the DP-SP-OMA output by randomly changing the pre-distortion filter response. For the X-polarization channel, the WL filer coefficients were generated in the same manner as described in Section 3: Eq. (13) with $L=11$, $\boldsymbol {h}_{I/Q}\sim$ $\mathcal {N}_c(\boldsymbol {l}_{0},$ $0.005\boldsymbol {I}_{L\times L})$, $\epsilon _{X}\sim \mathcal {N}(0,(\frac {1}{25})^2)$, and $\theta _{X}\sim \mathcal {N}(0,(\frac {2\pi }{25})^2)$. Meanwhile, we induced a more severe imbalance for the Y-polarization channel: $L=11$, $\boldsymbol {g}_{I/Q}\sim$ $\mathcal {N}_c(\boldsymbol {l}_{0},$ $0.02\boldsymbol {I}_{L\times L})$, $\epsilon _{Y}\sim \mathcal {N}(0,(\frac {1}{25})^2)$, and $\theta _{Y}\sim \mathcal {N}(0,(\frac {2\pi }{25})^2)$.

Figure 9 shows the per-polarization NMSE $\Delta _X$ and $\Delta _Y$ averaged over 90 independent trials against the sample complexity, i.e., $\frac {N}{4L}$. The DP-SP-OMA converged quickly and achieved the NMSE < −20 dB for SC > 16 for both polarization channels. The performance gap between the polarization channels was almost negligible despite of the more severe IQI in the Y-polarization. For SC > 100, the NMSE was below −26 dB. Examples of the estimated impulse/frequency responses are shown in Figs. 10 and 11. Note that, there exists the NMSE error floor at approximately −28 dB in Fig. 9. The error floor might be caused by the residual transmitter-side IQI; the pre-distortion filter for the IQI emulation could not be implemented perfectly due to the IQI inherent in the DP-IQM module.

 

Fig. 9. NMSE performance versus sample complexity in the experiment.

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Fig. 10. Examples of a) impulse and b) frequency response of the X-polarization channel estimated by DP-SP-OMA in the experiment.

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Fig. 11. Examples of a) impulse and b) frequency response of the Y-polarization channel estimated by the DP-SP-OMA in the experiment.

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Figure 12 depicts the NMSE per polarization and sample versus IRR. IRR varied nearly 15 dB among the samples and the Y-polarization channel had an IRR approximately 5 dB higher than that of the X-polarization channel. However, the DP-SP-OMA successfully achieved an NMSE of < −20 dB for all 90 samples for both polarization channels.

 

Fig. 12. NMSE per sample versus IRR in the experiment.

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These experimental results clearly demonstrated the feasibility and practicality of DP-SP-OMA. The low-complexity single-pixel monitoring solution based on PR [18] has been successfully extended to polarization-multiplexed systems.

It is also worth mentioning that, in this experiment, the polarization angle at the DP-SP-OMA input was arbitrary. The processing delay of the WL-PR was, for reference, 0.11 and 0.24 seconds on average for SC = 16 and 100 (4,400 symbols long), respectively, with the same laptop computer in Sec. 3. The fast and stable FD-IQI estimation for all 90 samples indicates that the DP-SP-OMA can be a promising solution for the fast tracking of the transmitter FD-IQI independent of the polarization rotation and the receiver-side impairments.

5. Conclusions

In this study, we have extended the low-complexity monitoring solution for single-polarization optical IQ modulators (IQMs) based on phase retrieval (PR), namely, the single-pixel optical modulation analyzer (SP-OMA), to typical dual-polarization optical IQ modulators (DP-IQMs). A DP-IQM under the frequency-dependent IQ imbalance (FD-IQI) can be characterized by a set of four complex-valued impulse (frequency) responses, i.e., the $I_X, Q_X, I_Y$, and $Q_Y$ channels. A novel distributed widely-linear (WL) PR algorithm has been proposed by introducing an alternating minimization procedure to estimate the four responses simultaneously from a single monitor PD output. The validity and feasibility of the proposed SP-OMA for DP-IQM (DP-SP-OMA) were demonstrated numerically and experimentally. The FD-IQI in a 400-Gbps DP-16QAM transmitter was precisely identified from a monitor PD output without using a coherent receiver or imposing specific pilot tones.