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Abstract
Owing to the random birefringence of optical fibers, the recovery of the state of polarization (SOP) is urgently needed, especially in the nonlinear spectrum division multiplexing transmissions. Based on the variance of the polarization power ratio among symbols as the cost function, we propose a novel algorithm for the blind SOP recovery of eigenvalue communications. In the single eigenvalue transmissions with phase-shift keying or 16-ary amplitude and phase-shift keying constellations, at least 25.3 dB polarization extinction ratio can be achieved by using a block length of 30, even under 7 dB OSNR condition. It also shows that the proposed algorithm can be employed in multi-eigenvalue NFDM transmissions and full-spectrum modulated NFDM system. In the experiment, our proposed algorithm performs the same as the training symbol based method in back-to-back and less than 3000 km fiber link conditions; a maximum performance gain of 1.6 dB was obtained in ultra-long-haul condition (4300 km). It also shows that the impact of the polarization mode dispersion of a single-mode fiber on the algorithm is negligible.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid maturation of coherent optical fiber communications, the linear effects in optical fibers, such as dispersion, can be effortlessly addressed by digital signal processing (DSP) algorithms [1]. For decades, nonlinear effects, notably the Kerr effect, have been the most limiting factors in long-haul optical communications [2], and are termed channel noise together with amplifier spontaneous emission (ASE). Consequently, nonlinear frequency division multiplexing (NFDM) was developed to counteract the impact of the Kerr effect [3,4]. Based on the nonlinear Schrödinger equation, the nonlinear Fourier transformation (NFT) provides an opportunity to solve the nonlinear Kerr effect as part of the channel characteristics, which may transform a time-domain signal into a nonlinear frequency domain for linear transmission [4]. The nonlinear spectrum λ consists of discrete and continuous spectra. The discrete spectrum contains a series of discrete points on the upper plane of the nonlinear domain, also known as eigenvalues [5–7]. In the continuous spectrum, λ is real and located on the real axis [8,9]. Both spectra can be used for communication. NFDM relies fundamentally on the interplay between dispersion and nonlinearity and is sensitive to the emission optical power. Typically, the estimation power for an eigenvalue transmission (+4 dBm) [10] is much higher than that of continuous spectrum ones (-4 dBm) [9], indicating that using an eigenvalue communication system can obtain an OSNR gain of up to 8 dB in long-haul transmissions [11]. In contrast, unlike linear spectrum transmissions, the signal noise in the eigenvalue transmissions does not exhibit a Gaussian distribution, making amplitude and phase-shift keying (APSK) constellations more suitable for eigenvalue communications, instead of quadrature amplitude modulation (QAM) [12]. Although high-order modulation formats such as 64-APSK have been experimentally realized [10], limited by the OSNR after long-haul fiber link, the quadrature phase shift keying (QPSK) and 16-APSK constellations are generally used [13–15].
In addition to high-order modulation formats, the transmission bit rate can be easily doubled using the dual-polarization (DP) technique, which has been widely adopted in linear spectrum optical transmissions. When light is transmitted through a standard single-mode fiber (SSMF), random birefringence causes a random rotation of the state of polarization (RSOP) of the light, resulting in crosstalk between the two polarized signals. This can be resolved by using state of polarization (SOP) recovery algorithms, such as the constant modulus algorithm (CMA) and decision-directed least mean square (DD-LMS) algorithm in linear spectrum optical transmissions [16]. The first DP-NFDM transmission was reported by S. Gaiarin in Technical University of Denmark. However, despite theories for DP-NFDM being proposed and employed both simultaneously and experimentally [3,5,17–19], insufficient SOP recovery algorithms have been established for NFDM transmission. One brute-force method is manually aligning the signal polarization using a polarization controller (PC) at the receiver to substitute the SOP recovery procedure in the DSP process [7,18]. However, the time-varying property of the RSOP in the SSMF is not considered in this approach. A more practical method is to use the training symbols (TS) [5,6,9]. However, it comes with spectral efficiency costs and the performance of this approach degrades after long-haul transmission. The defects of the TS-based method can be eliminated using blind SOP recovery techniques. In the continuous spectrum NFDM systems, a blind NFDM polarization equalization scheme has been proposed with linear approximation [20], which ignores the impact of nonlinear terms on continuous spectrum signals and shows that the Kalman filter equalizer is efficient. However, this method is not applicable to eigenvalue communications.
Based on the variance of the polarization power ratio (PPR) among symbols as the cost function, a blind SOP recovery algorithm is presented in this paper. By using the single eigenvalue transmissions with phase-shift keying (PSK) or two-ring APSK constellations, the principle of the SOP recovery algorithm is introduced, and extended to the multi-eigenvalue condition. It also shows that the proposed algorithm is adaptable for the full spectrum modulated NFDM transmission. We then experimentally measured the performance of the proposed algorithm under back-to-back (B2B) and long-haul transmission conditions. The impact of the polarization mode dispersion (PMD) was also investigated.
2. Principle of the NFT theory
The propagation of the DP signal along an ideal, noiseless, and nonlinear fiber can be modeled by the Manakov equation [21]
As a nonlinear partial differential equation (PDE), the Manakov equation can be solved using the inverse scattering transform (IST). The so-called extended Zakharov-Shabat (Z-S) problem bring Eq. (6) as a closed-form solution to Eq. (3) [22]:
Assuming that the signal power is distributed in the time interval of $[{{T_1},{T_2}} ]$, the scattering data can be defined as
The scattering data are three time-invariant parameters that can be utilized to reconstruct the signal ${\mathbf q}(t )$. While the optical signal propagates through an ideal fiber, the evolution of the scattering coefficients along the transmission follows the following rules:
3. Blind SOP recovery algorithms
3.1 Algorithm for the PSK constellations
A blind SOP recovery algorithm can be established based on the power ratio property of each pulsed symbol between polarizations. The intensity envelope of the eigenvalues is shown in Fig. 1 (a) for the case in which the SOP is unrotated and the PSK constellations are used (e.g., QPSK). The intensity envelopes of each pulse share the same sech-like shape on the two orthogonal polarizations, and the phase of each pulse is the same as its modulated signal $\angle ({{b_{\lambda ,j}}} ), \textrm{j} \in \{{X,Y} \}$. However, the RSOP of an SSMF link generates random crosstalk between the two orthogonal polarizations for each received symbol. The intensity envelope of the SOP-rotated signal is shown in Fig. 1(b). Crosstalk between polarizations arises within each symbol, resulting in the optical power being converted from one polarization to another. Thus, we employ the PPR to indicate the power ratio between the two polarizations as
where ${P_{k,j}}$ is the optical power of the k-th symbol on the j-th ($\textrm{j} \in \{{X,Y} \}$) polarization. $PP{R_k} = 0dB $ indicates that the optical pulse has equivalent optical power on the two polarizations, whereas $PP{R_k} > 0dB $ or $PP{R_k} < 0dB $ indicates that the X-polarization power of the pulse is larger or smaller than that of the Y-polarization, respectively.
Fig. 1. Intensity envelope of the eigenvalues using QPSK constellation with (a) unrotated and (b) rotated SOP; (c)$PPR$ of 4096 symbols under the 13 dB OSNR and QPSK input condition, with rotated and unrotated SOP.
Figure 1(c) shows the $PPR$ of 4096 symbols under 13 dB OSNR and QPSK input condition, with rotated and unrotated SOP. Each data point indicates the $PPR$ result for a DP symbol as the outcome of a time-domain pulse. For unrotated SOP, the pulses have equivalent optical power in both polarizations; this leads to the $PPR$ results distributed around 0 dB. When SOP is rotated, the inter polarization crosstalk leads to the data points spread out. Thus, the separation of $PPR$ among symbols may be utilized to represent the crosstalk level. We can utilize the variance of $PPR$ among $N$ symbols (block length) to describe the separation as:
The Jones vectors and matrix are commonly used for the description of polarization. The SOP recovery matrix is written as [24,25]
Fig. 2. (a) The flow process for establishing the Jones RSOP model. (b) $\sigma _{PPR}^2$ varying over $\alpha $ and $\delta $, searching under the 13 dB OSNR condition with QPSK constellation.
Consequently, the last issue is to develop a simple algorithm to determine the global minimum cost function in the Jones space. The simplest way to derive the minimum cost function is to traverse the Jones space with a fine grid, as shown in Fig. 2(b). However, this is accompanied by intolerable computational complexities. Most fast 2D minimum searching methods require an exact derivative expression, represented by the gradient descent algorithm [26], which is difficult to calculate for our proposed cost function. An alternative 2-D minimum searching algorithm without a derivative is the 2D-golden section search (GSS) [27], which is one of the quickest minimum searching algorithms without derivative expression. Theoretically, for a noiseless signal, by using the 2D-GSS algorithm, the uncertain range can be less than 0.1% on both axis after only 16 iterations, requiring 49 time calculations on the cost function, and the ideal PER can be 64.79 dB. To achieve such accuracy, more than 1.8 million time calculations on the cost function is required by using the grid search method.
3.2 Algorithm for the two-ring APSK constellations
The intensity envelope of the DP-eigenvalues for two-ring constellations (e.g., 16QPSK) is shown in Fig. 3. In this case, depending on whether the symbol is located on the inner or outer ring of the constellation, three possible outcomes exist, ${P_X} = {P_Y}$, ${P_X} > {P_Y}$ and ${P_X} < {P_Y}$, which correspond to $|{{b_{\lambda ,X}}} |= |{{b_{\lambda ,Y}}} |$, $|{{b_{\lambda ,X}}} |> |{{b_{\lambda ,Y}}} |$ and $|{{b_{\lambda ,X}}} |< |{{b_{\lambda ,Y}}} |$, respectively. Thus, in contrast to the PSK constellation condition, the $PPR$ parameter can converge on three possible values, + 6 dB, 0 dB and -6 dB, as shown in Fig. 3. The converged value depends on the radius ratio of the two-ring constellation. If we assume that the variance of $PPR$ is still unitized as the cost function, the $\sigma _{PPR}^2$ for each of the three scenarios should be calculated individually and averaged. These scenarios can be distinguished with threshold of $th ={\pm} 3{\kern 1pt} \;dB $. Thus, the variance outcomes for the three scenarios are:
Fig. 3. Intensity envelope of DP-eigenvalues for the 16-APSK constellation with unrotated SOP (outer) and $PPR$ of 4096 symbols under 13 dB OSNR, with rotated and unrotated SOP (inner).
Similar to the PSK condition, the cost function is scanned over the 2-D Jones space using a 16-APSK constellation and 13 dB OSNR, with $N = 30$, as shown in Fig. 4. There is a single global minimum in the Jones space close to zero, indicating that the polarization crosstalk can be compensated for. However, in this case, because the $PPR$ data points may be categorized into incorrect $\overline {\sigma _{PPR}^2}$ calculation conditions due to the hard decision thresholds, several local minima may be observed during the cost function scanning of the Jones space. If the 2D-GSS is directly unitized across the entire Jones space, it may converge to one of the local minima. Consequently, a preliminary search is required before the GSS search. A coarse grid (e.g., $15 \times 15$) can be used first to find the approximate value of the $({\alpha ,\delta } )$ parameters. Then, an accurate Jones matrix can be found using the 2-D GSS approach. Note that during the 2D-GSS accurate search, the valid range for $\alpha $ can be compressed, but the range for $\delta $ should remain [0, $2\pi $].
Fig. 4. $\sigma _{PPR}^2$ varying for $\alpha $ and $\delta $ searching under the 13 dB OSNR condition with the 16-APSK constellation.
3.3 Extensive situations
In short, while a single eigenvalue is used to employ PSK constellations, the $PPRs$ converge to a single intensity level, which is located around 0 dB, and its variance $\sigma _{PPR}^2$ can be utilized directly as the cost function during the SOP recovery. A complex modulation format leads to more $PPR$ convergence levels. Using a single eigenvalue employing a two-ring APSK constellation, three converged $PPR$ levels can be monitored, and the average of variance of each convergence level $\overline {\sigma _{PPR}^2}$ can be used as the cost function. Generally, the algorithm can provide good performance as long as the $PPRs$ converge to a few intensity levels.
There are other conditions in which our proposed algorithm can be utilized, such as multiple eigenvalue communication with $\lambda $ sharing the same real part using PSK constellations which could use the $\sigma _{PPR}^2$ algorithm, or two eigenvalue transmission with $\lambda $ sharing the same imaginary part using PSK constellations, which could use the $\overline {\sigma _{PPR}^2}$ algorithm. Under these conditions, although the signal have complex temporal shape, and the transfer function $H(\lambda )= \exp ({ - 4j{\lambda^2}z} )$ may cause a walk off between eigenvalues, their $PPR$ convergence remains constant.
The suggested RSOP recovery algorithm is also applicable to NFDM transmissions modulated on both discrete and continuous spectrum. The full spectrum NFDM signal is modulated and demodulated within each burst time. During each burst time, multiple (up to 256) subcarriers of continuous spectrum signal and several (normally 1∼3) discrete spectrum signal are transmitted. Among them, the $PPR$ feature of the discrete spectrum component follows the character introduced by the manuscript. The continuous spectrum component of each burst is comprised of many nonlinear frequencies modulating random intensity and phase sequences, resulting in an irregular and zero-mean characteristic. Thus, the continuous spectrum component has identical impact with the gaussian noise during the RSOP recovery procedure. Consequently, one can apply the RSOP recovery algorithm of discrete spectrum NFDM directly on the full spectrum modulated signal.
4. Transmission scheme
The performance of our proposed SOP recovery algorithm is verified experimentally, with the TS-based SOP recovery method as a reference. Eight X-polarization symbols and eight Y-polarization symbols were interleaved and deposited before each 4096 symbols, as reported in the literature [5].
Figure 5 shows the experimental construction and DSP flow of the DP-eigenvalue communication system. In the Tx DSP, after signal generation, the serial signal is mapped to the QPSK or 16-APSK constellations. The 16-APSK constellation is geometrically shaped based on the channel response by transmitting a regularly distributed APSK constellation through the link [10]. During the DP-INFT process, a 1.5 GBaud modulated signal is generated with the time duration ${T_0} = 353ps$. After denormalization, the signal is down-sampled to 93 GSa/s, which is the sampling rate of our arbitrary waveform generator (AWG, 93 GSa/s). At the transmitter, to avoid the influence of frequency offset and phase noise, a 1550 nm fiber laser with the linewidth of less than 100 Hz is used as the light source and the local oscillator (LO). A DP in-phase and quadrature-phase (DP-IQ) modulator is then used to modulate the electrical signal generated from the AWG on the light. The long-haul transmission link is demonstrated with a loop structure that consists of four 75 km SSMF and four erbium doped fiber amplifier (EDFAs), with a noise figure (NF) of 4.2 dB, while an additional EDFA and a variable optical attenuator (VOA) are used to adjust the optical power in the loop. Two 100 GHz optical bandpass filters are used to filter out the out-of-band ASE noise. The loop is driven by two program-controlled acousto-optic modulators (AOMs) with zero frequency shifting. AOM1 has an insertion loss of 5.5 dB, while AOM2 has an insertion loss of 3 dB. A polarization scrambler is used to rotate the SOP with a speed of 250 rad/s. Finally, the signal is detected by a 40 GHz integrated coherent receiver (ICR) and sampled by a digital storage oscilloscope (DSO) with sampling rate of 80 GSa/s. In the Rx DSP, the signal is first up-sampled to 192 GSa/s (128 samples per signal). A low pass filter is used to filter out the out-of-band noise. The bit synchronization (SYNC) procedure can locate the time slices of each symbol and ensure that a single pulse is calculated each time. The TS-based or our proposed blind SOP recovery methods are then used to recover the SOP rotation. After signal power normalization and DP-INFT, the BER can be finally calculated based on the support vector machine (SVM) approach [28].
5. Results and discussions
As shown in Eqs. (10) and (13), the cost function is the variance of $PPR$ among $N$ signals, with $N$ determines the precision of the SOP recovery against ASE noise and computational complexity. Consequently, the block length parameter should be optimized. We introduce the polarization extinction ratio (PER) to indicate the SOP recovery accuracy as
Fig. 6. $|{PER} |$ versus block length and OSNR using (a) QPSK constellation and (b) 16-APSK constellation.
The RSOP recovery capacity of the proposed algorithm for discrete spectrum modulation of $\lambda \textrm{ = }0.25\textrm{j}$ with QPSK constellation, $\lambda = [0.25j,0.5j,0.75j]$ with 8-PSK constellation using the $\sigma _{PPR}^2$ method and $\lambda \textrm{ = }0.25\textrm{j}$ with 16-APSK constellation, $\lambda = [ - 0.15 + 0.25j,0.15 + 0.25j]$ with 8-PSK constellation using the $\overline {\sigma _{PPR}^2}$ method, with or without continuous spectrum modulation under 18 dB or 11 dB OSNR conditions are shown in Fig. 7. The continuous spectrum signal has 128 subcarriers using 16-QAM constellation. The modulation share the same time duration ${T_0} = 353ps$, while the optical power for continuous spectrum component is 0.73 dBm. Under conditions of high OSNR, the adoption of a continuous spectrum component has similar effect on the RSOP recovery performance using methods $\sigma _{PPR}^2$ and $\overline {\sigma _{PPR}^2}$. However, both the influence of signal noise and continuous spectrum component would cause data points to be incorrectly categorized by the hard decision thresholds as the $\overline {\sigma _{PPR}^2}$ calculation circumstances, hence making the $\overline {\sigma _{PPR}^2}$ approach more vulnerable to OSNR distortion.
Fig. 7. $|{PER} |$ versus block length and modulation formats under: (a) 18 dB OSNR and (b) 11 dB OSNR condition.
The back-to-back (B2B) performance of the blind SOP recovery algorithm and the TS-based algorithm using QPSK and 16-APSK constellations modulated on the $\lambda \textrm{ = }0.25\textrm{j}$ eigenvalue are experimentally compared in Fig. 8(a) and 8(b), respectively, with $N = 30$. The Q-factor was derived from the BER through $Q_{\textrm{dB}}^2 = 20{\log _{10}}\left[ {\sqrt 2 {{erfc }^{ - 1}}({BER } )} \right]$. In both experiments, the Q-factor curves of the two polarization states and SOP recovery methods are approximately consistent, indicating that identical performance can be achieved with the blind SOP recovery and TS-based methods, regardless of the received OSNR. At the soft-decision forward error correction (SD-FEC) limit, the OSNR tolerance for the QPSK and 16-APSK constellations are 9.7 dB and 9.1 dB, respectively.
Fig. 8. B2B Q-factor performance vs. OSNR of the blind SOP recovery algorithm and the TS-based algorithm using (a) QPSK constellation and (b) 16-APSK constellation.
The Q-factor performance of the DP-eigenvalue communication versus transmission distance is shown in Fig. 9. The transmission distance for the QPSK link reached approximately 4500 km, whereas that for the 16-APSK link reached 2000 km at the SD-FEC limit. Similar to the B2B condition, identical performance based on the blind SOP recovery algorithm and the TS-based algorithm can be obtained using a 16-APSK constellation within 3000 km of transmission. However, for the QPSK constellation, using the blind SOP recovery method yields a better performance, which can be attributed to the transmission impairments of the TS, which is generated according to literature [5] as follows: $q_X^1$, $q_Y^2$, $q_X^3$, $q_Y^4$, …The cross-polarization modulation (XPolM) factor is considered an important element in the DP-eigenvalue generation procedure, as shown in Eq. (1). Nonetheless, the generation procedure while eliminating the X-polarization signal or Y-polarization signal for each symbol destroys the eigenvalue feature of the TS. More specifically, there should be a Y-polarized signal $q_Y^1$ corresponding to the X-polarized signal $q_X^1$ during the same time slot, so that it can transmit linearly on the nonlinear spectrum. Owing to the absence of X-polarization or Y-polarization signal, the Kerr effect cannot compensate for the dispersion distortion during propagation. Consequently, after 3000 km of SSMF transmission, the TS symbol is distorted by the dispersion of the fiber, which degrades the SOP recovery performance based on TS, while our proposed blind SOP recovery algorithm retains outstanding performance. Consequently, for the 4500 km transmission, a performance gain of 1.6 dB can be obtained around the SD-FEC threshold. Even if TS distortion can be avoided by generating a TS using the single-polarization INFT algorithm, the intensity of the single-polarization TS would be remarkably higher than that of the communication symbols, which are generated by the DP-INFT algorithm, and the communication signal would be impacted by the quantization noise of the AWG. Hence, adopting the blind SOP recovery algorithm rather than the TS-based method would result in a superior performance.
Fig. 9. Long-haul transmission Q-factor performance vs. transmission distance of the blind SOP recovery algorithm and the TS-based algorithm using (a) QPSK constellation and (b) 16-APSK constellation.
The proposed SOP recovery algorithm is based on a rotation angle search in Jones space, during which the PMD effect is not considered. However, as a major polarization impact, the influence of PMD should be studied. PMD is the probability description of differential group delay (DGD), whose probability density function (PDF) $p(\tau )$ follows a Maxwellian distribution [29]. This can be simulated using $\textrm{T}(\tau )= diag({{e^{j\omega \tau /2}},{e^{ - j\omega \tau /2}}} )$ after the RSOP distortion. The DGD value can be set with the probability of $6.5 \times {10^{ - 8}}$, which is the standard outage probability of 2 s/year [30]. Figure 10 shows the impact of PMD on the DP-eigenvalue communication system based on a blind SOP recovery algorithm, where the transmission link only provides a random RSOP and a fiber length-related DGD. The received OSNR for the QPSK and 16-APSK signals were 10 and 12 dB, respectively. We showed that the Q-factor degrades only under extremely large PMD and long virtual-distance conditions. Given that the typical PMD of an SSMF is 0.04 ${{ps} / {\sqrt {km} }}$, the impact of the PMD is negligible.
Fig. 10. Q-factor performance vs. the virtual fiber length and PMD intensity in simulations using (a) QPSK constellation and (b) 16-APSK constellation.
6. Conclusions
A blind SOP recovery algorithm for the discrete spectrum NFDM system or full-spectrum modulated NFDM system was proposed and experimentally verified. For the multi-eigenvalue transmission, the PER can reach more than 30 dB in the simulations, with or without continuous spectrum signal. For the single eigenvalue transmission, compared with the TS-based algorithms, the blind SOP recovery method has an identical performance in B2B and for less than 3000 km conditions. Under ultra-long-haul conditions, a performance gain of 1.6 dB can be obtained around the SD-FEC limit. The results also show that the impact of the PMD of an SSMF is negligible. The proposed algorithm can provide a good performance as long as $PPRs$ converge to a few intensity levels, such as the conditions mentioned in Section 3. However, it cannot deal with these conditions well if the $PPRs$ converge to a large number of intensity levels; this remains a direction for future research. Even so, as far as we know, this is the only blind SOP recovery algorithm for eigenvalue transmissions.
Funding
National Natural Science Foundation of China (61931010, 62225110).
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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