## Abstract

An investigation was carried out on the polarization attraction (PA) of a polarization-scrambled 10.7-GBaud NRZ-BPSK signal in a 1-km-long highly nonlinear fiber (HNLF). For the back-to-back case, PA on an ASE-loaded signal yielded a receiver sensitivity penalty of ≈ 14.5 dB at the ITU-T G.975.1.I3 FEC threshold of 3.5 × 10^{−3}, relative to matched-filter reception theory. After long-haul 100-GHz DWDM transmission in a recirculating loop, PA on the output signal was found to achieve approximately the same receiver sensitivity performance, as that of the back-to-back case. From these experiments, it is concluded that the Gordon-Mollenauer effect due to propagation in the HNLF during PA dominates other impairments including those arising from the long-haul 100-GHz DWDM recirculating loop transmission.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The depolarization of a signal’s state-of-polarization (SOP) during long-haul transmission, due to polarization mode dispersion [1–3] and the fiber Kerr nonlinearity [4–6], is a well-understood phenomenon. Nonlinear all-optical signal processing (NOSP) devices and on-chip integrated optical circuits that are highly sensitive to polarization fluctuations may experience performance penalties as a result of transmission depolarization [7–10]. One potentially lossless polarizer under investigation is characterized by the re-polarization of an unpolarized signal when it interacts with a fully-polarized continuous-wave (CW) beam through a Kerr nonlinear cross-polarization process [11–14]. The process of the re-polarization of an unpolarized signal is termed polarization attraction (PA). PA utilizes the ultra-fast Kerr nonlinearity in optical fibers to achieve response times within the femtosecond range, making it more amenable to telecommunication applications than the first reported lossless polarizer that utilized two-beam coupling in photorefractive crystals [15]. The theory presented in [12,13] however, suggests a characteristic time constant associated with PA, on the order of a few microseconds. PA was first experimentally demonstrated with a counter-propagating CW pump beam in a 2-m-long, isotopic, highly nonlinear fiber (HNLF) [14,16]; but in each case required very high-power signal/pump beams (≈ 45 W) to obtain a sufficiently nonlinear interaction. A reduced power requirement (< 1 W) was later demonstrated in a Non-Zero Dispersion-Shifted Fiber by using a longer length of fiber (≈ 20 km) [17]. Following this experiment, two more demonstrations were reported for a relatively shorter fiber span of 6.2 km and similarly reduced power requirements (< 1.1 W) [18,19]. The theory presented in [12,13] derived a general model for fiber-based nonlinear lossless polarizers. Simulations predicted that PA should be feasible in isotropic fibers [11], highly birefringent spun fibers [20], and randomly birefringent fibers [21,22]. Several more realizations of PA have been demonstrated on an optical pulse train [23] and on-off keyed (OOK) signals [24–28]. In [29] a polarization-scrambled NRZ-BPSK signal was stabilized for the first time in a HNLF to demonstrate that PA is feasible for phase-shift keying.

One commonality between all of these PA experiments is that they were conducted in back-to-back configurations, where the scrambled signal inbound to the nonlinear fiber had a large optical signal-to-noise-ratio (OSNR) and the receiver was limited by amplified spontaneous emission (ASE). To the extent of the authors’ knowledge, PA has yet to be tested on a signal suffering transmission impairments, arising due to the interaction of dispersion, nonlinearity, PMD, and ASE.

The growth of ASE in transmission and its impact on system performance has been summarized [30]. The interaction of ASE with nonlinearity leads to the Gordon-Mollenauer (G-M) effect for PSK signals [31], in which ASE-induced amplitude fluctuations are converted to phase fluctuations via self-phase modulation (SPM) and/or cross-phase modulation (XPM) during propagation. It has been reported that the G-M effect can result in a significant BER penalty [32–34]. The nonlinearity in the PA medium is critical for the efficacy of the cross-polarization processes over a short fiber length [12,13], but it may also intensify the G-M effect for a signal corrupted by ASE. This report demonstrates that when the signal accumulates ASE, either during long-haul transmission or artificially in the transmitter, the resulting G-M effect within the PA medium can diminish the efficacy of PA.

For the first time, the demonstration of PA on a polarization-scrambled and single-polarization 10.7-GBaud NRZ-BPSK signal is evaluated with a coherent receiver to determine the robustness of PA to transmission impairments. In the first demonstration, PA is studied on an ASE-loaded 10.7-GBaud NRZ-BPSK signal. The purpose of this preliminary experiment is to study the impact of ASE on PA. The second demonstration utilizes a 100-GHz DWDM signal transmitted > 1,000 km over a dispersion-managed, standard single-mode fiber (SMF-28) recirculating loop. After transmission, the central NRZ-BPSK channel is isolated via an arrayed waveguide grating (AWG), followed by PA. Section 2 gives an overview of the effectiveness of PA on polarization-scrambled signals after fiber transmission. In Section 3, the results of each experiment is presented, and details the penalties associated with OSNR degradation before PA, after recirculating loop transmission, and the results of PA post-transmission. A summary of the results, and the conclusions are found in Section 4.

## 2. Overview of experiments

The block diagram for the first experiment is shown in Fig. 1. The output of a single-channel 10.7-GBaud NRZ-BPSK transmitter was followed by the addition of ASE, injection into the polarization attraction (PA) module [29], and evaluation in a 33-GHz, 40-GS/s coherent receiver. The purpose of this preliminary experiment was to evaluate the coherent reception ofPA for an ASE-loaded polarization-scrambled NRZ-BPSK signal. It is assumed for this report that a minimum possible PA characterized by a DOP > 0.9, is necessary to mitigate penalties for some arbitrary polarization-sensitive NOSP device. It was found that a signal average launch power of 0.32 W into the HNLF was sufficient to achieve a DOP > 0.9, which required an average pump power of 1.4 W [29]. In the second experiment, PA was carried out on the NRZ-BPSK signal, post-transmission in a recirculating loop (RL). Changes to the experimental block diagram in Fig. 2 included upgrading the single-channel transmitter to a multi-format DWDM transmitter, and replacing the ASE-loading module with RL transmission. ASE-loaded back-to-back transmission can be considered to be the best-case transmission scenario, in the absence of dispersion, nonlinearity, and polarization-mode dispersion (PMD).

In the laboratory, a RL transmission system is utilized to emulate straight-line (SL) transmission, due to constraints on transmission equipment. Unlike SL-transmission, the RL is highly periodic, resulting in some impairments unique to a given RL, such as sideband instability. Other serious RL artifacts are governed by polarization effects. The channel's state-of-polarization (SOP) can be periodically aligned with a principal state of polarization (PSP), leading to little distortion upon reception. However, it is also possible for the channel to be periodically launched between the two PSPs, leading to a maximum signal distortion over long-haul distances. It was also found that when the PDL per round-trip (RT) was minimized to ensure that the PMD was the dominant effect, the PMD grew linearly with distance, instead of with the square root of distance, as would be the case for SL-transmission. This was due to the fact the PMD vector for one RT could be parallel (anti-parallel) to the vector for the following RT, resulting in DGD addition (cancellation) [35]. The probability density function for the RL DGD is thus non-Maxwellian, unlike that for SL-transmission. Moreover, when PDL/RT is significant, the RL fiber realization may be such that the signal SOP is periodically aligned with the PDL low-loss axis (high-loss axis) leading to significant enhancement (degradation) of the received signal OSNR, for long-haul transmission [36]. These problems can all be significantly alleviated, rendering RL behavior closer to that of SL-transmission, by employing polarization scrambling to break the periodicity of the loop. In a later section, it will be shown that BER is optimized at a certain scrambling speed for the RL used.

The following section reports all significant experimental results, and is separated into three sections. In Section 3.1, the results of the ASE-loaded PA experiment are presented. Section 3.2 reviews all the experiments performed to characterize and optimize the RL performance without PA. Section 3.3 presents the results of PA post-transmission.

## 3. Experiments and analyses

#### 3.1 ASE-loaded polarization attraction experiment

Figure 3 shows the setup used to test polarization attraction (PA) on a NRZ-BPSK signal with an artificially lowered OSNR. The transmitted signal is a 10.7-GBaud 2^{15}-1 pseudo-random bit sequence (PRBS) NRZ-BPSK signal centered at 1547.715 nm. It also served as the probe in a PA subsystem. The Transmitter consisted of an external cavity laser (ECL) which optically drove an *x-*cut LiNbO_{3} Mach-Zehnder modulator (MZM). The MZM was biased at its transmission null and driven by the PRBS via an RF-amplifier at twice its *V*_{π}-voltage, and followed by the polarization scrambler with a maximum speed of ≈ 12 kHz. The linewidth ofthe ECL was ≈ 65 kHz. After the Transmitter, the signal could be attenuated and re-amplified by the low-noise-figure EDFA (LNF-EDFA) to reduce OSNR. When PA was required, a CW pump laser centered at 1549.92 nm, was injected backward into the HNLF using a circulator, and was phase-modulated (PM) at ≈ 335 MHz in order to suppress SBS. The phase modulator was erroneously represented by a MZM in the testbed schematic in [29]. To ensure an efficacious nonlinear interaction in the HNLF, the signal and the pump were each amplified in a high-power EDFA (HP-EDFA). The wavelength of the pump was not crucial to the efficacy of PA within the C-band, and a pump-probe detuning of ≈ 2 nm was selected as was the case in [29]. The respective launch powers for the signal and pump were approximately 0.32 W and 1.4 W. The 1-km-long standard HNLF exhibits a nonlinear coefficient of *γ* ≈ 11W^{−1}·km^{−1}, a loss coefficient of *α* ≈ 0.28 dB/km, a normal dispersion coefficient of *D* ≈ -0.16 ps·nm^{−1}·km^{−1}, and a PMD parameter (*σ _{T}*/

*L*

^{1/2}) of ≈ 0.02 ps/km

^{1/2}. The loss coefficient includes splice losses to standard SMF pigtails, implying that the actual propagation loss is significantly lower. The HNLF's normal dispersion precludes modulation instability [37]. Depolarization due to the total PMD was minimal due to the relatively larger coherence time of the ECL, and the low Baud-rate [1]. The polarization analyzer monitored the signal SOP and DOP beyond a 0.3-nm optical bandpass filter tuned to the signal wavelength, which was also used to reject FWM terms. A commercially available, undriven MZM was used in lieu of a polarizer, as the MZM incorporates a polarizer at its input port. The NRZ-BPSK signal was evaluated using a 33-GHz, 40-GS/s coherent receiver. During receiver sensitivity measurements, the OSNR was degraded by the first variable optical attenuator (VOA) before the LNF-EDFA. The “ASE-loading” block in Fig. 1 was comprised of this VOA and the LNF-EDFA. A second VOA before the HP-EDFA was used to keep constant, the power into the launch EDFA. A third VOA after a 0.5-nm OBPF and before the Receiver, was used to keep the power constant at 0 dBm.

The coherent receiver is shown in Fig. 4, and is commercially available from Keysight Technologies, Inc. The receiver consists of a polarization-diverse 90°-hybrid, opto-electronic hardware, and a digital storage oscilloscope (DSO), which carries the analog-to-digital converters (ADCs) and the digital signal processing (DSP) algorithms. The polarization-diverse 90°-hybrid circuit decouples the two polarization states (if any) via a polarization-beam splitter (PBS), as well as the in-phase and the quadrature components of the signal (if any), and mixes the resultant four components with a low-linewidth (≈65 kHz) local oscillator (LO), which was always tuned to be within 0.1 GHz relative to the input signalfrequency. The LO field is distributed nearly symmetrically among the two hybrids. For a polarization-scrambled, single-polarized signal, the signal's power may be in either of the *x-* or the *y-*polarization hybrid, or in both during some instants, depending on the SOP of the inbound signal. However, the DSP algorithms determine whether the signal is single- or dual-polarization, based on any significant inter-polarization decorrelation. The decoupled four signals are then each detected in a balanced *pin-*detector, and subsequently amplified to ensure sufficient input power to the corresponding ADC. The effective bandwidth of an analog opto-electronic detection module, which includes a detector and its attendant low-noise amplifier, is ≈ 33 GHz. Each ADC sampled a signal at 40 GS/s, at more than thrice the Baud rate. The effective number of bits for an ADC is approximately 5.5 [38]. The DSO also hosts proprietary DSP algorithms that compensate for hardware imperfections, polarization recovery, carrier recovery, clock extraction, and symbol estimation. Additional algorithms include dispersion and PMD compensation, and adaptive equalization, some of which were used in the transmission experiment, and discussed later in this report. To expedite algorithmic convergence, a *T*-spaced (instead of a fractionally spaced) adaptive equalization (AE) FIR filter was used. The analysis bandwidth was constrained to 12.5 GHz. At least 10^{6} symbols were evaluated for a given BER.

The receiver sensitivity was consistently carried out at the FEC BER threshold. The 10.7-GBaud signal was assumed to have been encoded with ITU-T G.975.1.I3 FEC, which requires a redundancy ratio of ≈ 7%. The FEC is implemented with concatenated BCH(3860, 3824) and BCH(2040, 1930) codes, with the former serving as the outer code, while the latter, as the inner code. Further, the FEC scheme would also require an interleaver (de-interleaver) between the codes at the transmitter (receiver). Iterative decoding could be applied at reception, with the addition of an interleaver, to improve error-correction (EC). The EC capability of this FEC is dependent on the number of iteration times. For this report, an input BER threshold of 3.5 × 10^{-3} is assumed, which would result in an output BER of 10^{-9} with three-times iterative decoding, and a net coding gain of ≈ 6.7 dB.

To carry out the baseline back-to-back (ASE-limited) receiver sensitivity measurement, all components in Fig. 3, between the LNF-EDFA and the 0.5-nm-OBPF, were bypassed. The first VOA was used to attenuate the signal input to the LNF-EDFA, thus degrading the received OSNR, which was measured using a 10%-coupler after the first stage of the LNF-EDFA. The last VOA after the OBPF was used to maintain constant the power at ≈ 0 dBm to the Coherent Receiver. In the Receiver, only AE was additionally engaged, except in certain demonstrations of PA.

The G-M effect due to HNLF propagation could be observed qualitatively by the coherent receiver without PA, and shown in Fig. 5. The VOA before the HP-EDFA in Fig. 3 was set toits minimum, intrinsic attenuation. Receiver AE was enabled for Fig. 5. All constellation diagrams in Fig. 5 were taken at the HNLF output, at varying HNLF launch powers, when the OSNR was 16 dB/0.1 nm. Increasing the launch power into the HNLF by a factor of four, from Fig. 5(a) to 5(d) enhanced the G-M effect, and resulted in severe symbolic distortion.

For Fig. 6, the launch power was held constant at ≈ 0.32 W while the corresponding OSNR was varied, by adjusting the VOA before the LNF-EDFA (Fig. 3). This launch power was found to be optimal for PA [29]. The VOA before the HP-EDFA was used to ensure a constant launch power into the HNLF. Figure 6(a) and 6(c) show the signal before and after the HNLF when the input had the highest available OSNR. Figure 6(b) and 6(d) show the signal before and after the HNLF when the input signal had been ASE-loaded down to an OSNR of ≈ 16 dB/0.1 nm. The differences between Figs. 6(c) and 6(d) are ascribed to the G-M effect. Ho and Kahn [39] have noted that in the presence of a significant nonlinearity, the detected symbols assume crescent like densities, due to the correlation of nonlinear phase rotation with the received intensity. In this case, the decision boundary becomes spiral-like, asopposed to a straight line normal to the real-axis that carries the two symbols, as in Fig. 6(a) for a low launch power.

Figure 7(a) qualitatively demonstrates that, at the ITU-T G.975.1.I3 FEC BER threshold of 3.5 × 10^{−3}, transmitter imperfections were overwhelmed by ASE, as the symbols are uniformly distorted. By contrast, ASE-induced signal corruption is enhanced during PA due to nonlinear propagation in the HNLF. Figure 8 summarizes receiver sensitivity measurements for the baseline and the ASE-loaded polarization attraction cases. The HNLF was bypassed to obtain the baseline receiver sensitivity, which yielded at the FEC BER threshold, a penalty of 4 dB maximum relative to the expected theoretical behavior,

*B*is the resolution bandwidth (12.5 GHz), and

_{o}*R*is the bit-rate (10.7 GBaud). The theory assumes heterodyne coherent matched-filter reception limited by ASE alone. For ASE-loaded PA, the penalty worsened to > 14.5 dB, when the signal launch power was held fixed at 0.32 W, the pump was 1.4 W, and the output DOP was ≈ 90% and constant over the entire data set. The penalty is seen to steadily increase as the OSNR is degraded (attributable to the G-M effect), as the ASE-loaded PA case diverges from the baseline. This behavior contrasts with that of the baseline, which approximately shows no change in penalty as the OSNR is reduced. The role of the CW pump in PA was revealed by turning off the pump, which yielded a BER floor. This was due to the conversion of scrambling-induced polarization fluctuations to amplitude modulation in the undriven MZM (Fig. 3). However, at the selected FEC BER threshold, the penalty was 0.5 dB maximum. To qualitatively illustrate the efficacy of PA, the polarization recovery and AE algorithms aboard the coherent receiver were disabled, while the inbound signal SOP was aligned with the slow axis of the PBS input in Fig. 4. The results are shown in Fig. 9, which demonstrates a symbolically compact constellation that resulted in an error-free BER, although the symbols do exhibit some pattern-dependence. When the CW pump to the HNLF was turned off, and under the same conditions, the symbols became significantly diffuse and more distorted, and the corresponding BER was no longer error-free.

A receiver sensitivity experiment was carried out on HNLF propagation of a polarization-scrambled NRZ-BPSK signal, to decouple PA effects from purely nonlinear propagation in the HNLF. Its launch power was set to 0.32 W, the same as that used for the PA experiment. The CW pump was turned off. The undriven MZM was bypassed, to focus solely on the G-M effect. The results are shown in Fig. 10 and demonstrate an excess penalty of −0.5 dB maximum, for the HNLF propagation, relative to PA. This indicates that the large penalty arising for PA is mostly attributable to the G-M effect, but not to the process of PA itself. For all cases shown, AE was once again engaged. It was noted by Carena et al. [40], that adaptive equalization forces the effective receiver filter behavior towards that for the matched filter.

Gordon and Mollenauer have derived an estimate for the nonlinear phase shift (NLPS) variance, for error-free long-haul transmission [31]. The theory also assumes matched-filter reception. For phase modulation, they “show that there results a minimum [bit] error rate when the signal power produces a total nonlinear phase shift of approximately 1 rad.” The expression may be naively applied to propagation over the 1-km-long HNLF, to verify the large penalty observed for PA. From Fig. 8 or Eq. (1), for ASE-limited, matched-filter reception, a linear OSNR of 7.7 (8.9 dB) is required for an error-free BER of 10^{−9} [39]. If a maximum NLPS of 1 rad is additionally assumed, the nonlinear phase variance is then constrained as follows:

*Q*is identical with SNR. It can be seen that if the NLPS

*φ*

_{NL}changes by some factor, the linear OSNR must change by the square of that factor, to keep the resultant variance at a value of no more than 0.037 rad

^{2}(which was obtained for a 1-rad NLPS and a linear OSNR of 7.7.) The NLPS generated in the HNLF at a launch signal power of 0.32 W was 8/9

*γP*

_{0}

*L*≈ π rad, implying that the OSNR would have to be increased by π

^{2}, which yields a linear OSNR of 76 or 18.8 dB. From Fig. 8, the extrapolated OSNR for “HNLF Propagation” at 10

^{−9}-BER is ≈ 23 dB, which is a little more than 4 dB higher than that predicted by Eq. (2).

Alternatively, it is possible to use the extrapolated OSNR in the above equation, to solve for the NLPS, which turns out to be 5.1 rad. The theory thus overestimates the experimental NLPS of π rad, by a factor of 1.6. Gordon and Mollenauer attempted an application of their theory to a 2.5-Gb/s Frequency-shift-keyed, long-haul transmission experiment [41], which yielded error-free BER at 2,200 km. They found that the NLPS predicted by their theory for this experimental case was off by a factor of 1.5 [31].

It is possible to specialize the derivation of the GM-theory to the HNLF, the details ofwhich has been relegated to the Appendix section. The condition for OSNR estimation based on the G-M theory can be summarized as follows, in general

^{−9}error-free BER, the minimum required OSNR is 156 or 21.93 dB / 0.1 nm. According to Fig. 8, the extrapolated OSNR required for 10

^{−9}error-free BER, is ≈ 23 dB. Equation (3) is thus in error by ≈ 1.1 dB. However, extrapolation may have produced a fortuitous OSNR, leading to an erroneous conclusion for the accuracy of the Eq. (3). The theory can be verified at the FEC BER threshold instead, which yielded an actual BER measurement, unlike the case just examined.

For the FEC target BER of 3.5 × 10^{−3}, matched-filter reception requires an OSNR of just 1.9 dB / 0.1 nm, from Fig. 8 or Eq. (1). Since the total power launched into the HNLF was fixed at 0.32 W irrespective of OSNR, the actual signal power is reduced at the lower OSNRs. At the FEC target BER, an OSNR of 16 dB / 0.1 nm was required in the experiment, which yields an actual signal power of 0.31 W. At this power, the NLPS generated by the HNLF is then lower, at 3.03 rad. To achieve the BER of 3.5 × 10^{−3} for a NLPS of 3.03 rad, the OSNR would have to be at least 28.8 or 14.6 dB. The error relative to the experimental OSNR of 16 dB is then 1.4 dB, slightly higher than the discrepancy at the error-free BER. However, by using the more accurate version of Eq. (3), which is Eq. (38), and solved simultaneously with Eq. (40), both found in the Appendix, the estimated OSNR is found to be 15 dB. Once again, this is an error of 1 dB. Based on this data, “The Gordon Mollenauer Theory” was plotted on Fig. 10, demonstrating a uniform 1-dB penalty relative to “HNLF Propagation”. The G-M theory is for matched-filter reception, limited to 2 degrees of freedom, and is also a good approximation in practice when the filter bandwidth is very close to the signal’s bandwidth. The theory is optimistic for the predicted OSNR. In the analysis, the filter bandwidth is assumed to be the OSA’s resolution bandwidth of 12.48 GHz, which is > 16% larger than the Baud-rate of 10.7 GBaud. Due to a larger filter bandwidth, the contributions due to the higher order noise modes become less negligible, resulting in increased variances of both linear and nonlinear phase noises. This effect is exacerbated for high launch powers, and low dispersion fibers [42,43], such as the HNLF.

It is possible to use the “Exact Theory” for the BER of a BPSK signal suffering from nonlinear phase noise, and is given by

*θ*. When the mean NLPS is no more than ≈ 1.25 rad, the optimal

_{c}*θ*is lower than the NLPS. For a higher mean NLPS, this angle becomes larger than the NLPS. The deviation may be exacerbated for a realistic BPSK signal suffering pattern-dependence and ISI, captured by a practical receiver. It was already found that, at the FEC BER threshold, baseline NRZ-BPSK BER was at least 4 dB worse than that of matched filter reception of Eq. (1). Arguably, this penalty contributes to constellation distortion unexpected by the exact theory of Eq. (4), thus shifting the optimal decision angle. It was found that

_{c}*θ*had to be approximately a factor of 1.275 larger than the mean NLPS, to achieve a good agreement between Eq. (4) and the experiment (Fig. 10). For ideal BPSK, matched-filter reception, and for the same NLPS, the penalty of the experiment relative to the exact theory (which required a

_{c}*θ*≈ 1.04 Φ

_{c}_{NL}) is approximately 7 dB at the FEC BER threshold.

#### 3.2 Transmission of NRZ-BPSK in a recirculating loop

This section gives an overview of the recirculating loop (RL), including the transmitter/receiver setup, channel plan, optimal launch power, and baseline transmission receiver sensitivity performance. The NRZ-BPSK signal used in section 3.1 now serves as the center-channel in a 100-GHz DWDM transmission system. Polarization attraction on a signal output by the RL is not examined until section 3.3.

Figure 11 shows the setup used for the 100-GHz DWDM transmitter, which produced 11 de-correlated, 10.7-GBaud, 2^{15}-1 PRBS NRZ-BPSK, ECL-channels centered around 1547.715 nm, in accordance with ITU-T G.692. A low-noise 10.7-GHz frequency synthesizer was used to drive a 12.5-Gb/s bit-pattern generator (BPG). The BPG was used to generate the baseband 2^{15}-1 PRBS, and to also provide a 10.7-GHz clock. The NRZ-BPSK signals were generated by combining all 11 channels and injecting them into a MZM biased at its transmission null and PRBS-driven at twice the MZM's *V*_{π}-voltage. The NRZ-BPSK channels were subsequently de-correlated, by demultiplexing them in an AWG, propagating them through different lengths of fiber, and remultiplexing them in a second AWG. The fiber differential decorrelation length between neighboring channels was at least 1 m, equivalent to > 50 symbols. As shown in Fig. 11, decorrelation of the channels was implemented interstage in a LNF-EDFA, to maintain a high channel OSNR. The multi-format DWDM transmitter also produced 13 RZ-pulse-train signals by combining the outputs of 13 distributed feed-back (DFB) lasers into a MZM biased at quadrature and driven at its *V*_{π}-voltage by a 10.7 GHz clock supplied by the BPG. Eight of these RZ-pulse-train channels were tuned to the blue end of the 11 NRZ-BPSK channel block, whereas the remaining five were tuned to the red end of that channel block. The wavelengths of the NRZ-BPSK channels were also chosen to fit within the standard ITU-T 100 GHz grid, and the wavelengths of the RZ-pulse-train channels were limited by the availability of DFB lasers. Due to a dearth of suitable components, it was not possible to data-drive the 13 side-channels. The RZ-BPSK 11-channel block was finally combined with the RZ 13 side-channels using a broadband 2 × 2 coupler. The overall spectral flatness was achieved through a combination of channel pre-emphasis and judicious amplification. To ensure the highest possible OSNR prior to launch within the recirculating loop, amplification was carried out wherever possible to compensate for couplers and modulators losses.

Figure 12 shows the setup used for the RL. To obtain a transmission baseline for the center-channel, the setup of Fig. 3 was used, where the RL of Fig. 12 replaced the first VOA before the HNLF-EDFA of Fig. 3. The Transmitter was the DWDM transmitter of Fig. 11. Additionally, all components were bypassed in Fig. 3, between the LNF-EDFA and up to the 0.5 nm filter before the Receiver. The received OSNR was measured before the AWG in Fig. 12. The transmission line within the RL had a total length of 162.7 km, a residual dispersion of 6.1 ps nm^{-1} km^{-1}, and a net PDL of 0.05 dB. Table 1 summarizes the characteristics of the transmission spans. A polarization scrambler was utilized to break the periodicity of the RL, thus mitigating artifacts due to PMD [35] and PDL [44]. The wavelength-selective switch (WSS) was used to correct for gain-tilt and gain ripple due to repeated amplification in the RL, which could enhance nonlinear effects during propagation. However, this correction wasonly carried out once per circulation, and not after each amplification. The channels were thus launched into each span with some gain artifacts, which could have exacerbated transmission impairments over long-haul distances. The 1% taps before each span were calibrated to monitor launch power. Launch powers could be adjusted using programmable VOAs after each EDFA. The effects of scrambling speed on the received center-channel Q (observed on the coherent receiver) are shown in Fig. 13, where the optimal scrambling speed seems to have been at least 10 kHz. These measurements were taken at a distance of > 5,000 km, but were also reproducible at longer distances.

To optimize transmission, the fiber span launch power was varied while the BER was measured approximately about the ITU-T G.975.1.I3 FEC threshold [45]. All BER measurements were taken for the center-channel (1547.715 nm) after 7,321.5 km of transmission. Figure 14 confirms that the optimal launch power was approximately −10.5 dBm per channel, and is the result of a trade-off between OSNR and nonlinear impairments. During these measurements, the DCF input power was held at −10 dBm per channel to ensure quasi-linear transmission. These launch powers were measured using a 1% tap before each of the SSMF and DCF spools shown in Fig. 12, after adjustment for the loss in each tap. The AE filter was re-optimized, and found to be approximately 11-symbols long (Fig. 15). The residual dispersion at 7,321.5 km was ≈ + 275 ps/nm, for the center-channel investigated. Consequently, some SPM would be required to make up for the under-compensation at reception, as the proper combination of SPM and residual anomalous dispersion lead to pulse compression, and lower ISI [46–48].

Figure 16 shows the degradation of BER with distance, when the channel launch power was approximately −10.5 dBm, and AE enabled, and optimized for each distance. At distances longer than 7,321.5 km, the FEC BER threshold was exceeded. Having measured the OSNR for each BER point in Fig. 16, the data could then be compared against the baseline (back-to-back) receiver sensitivity (Fig. 17). The OSNR penalty between the back-to-back and transmission curves at the FEC threshold was approximately 4 dB/0.1 nm.These two curves diverge as the OSNR decreases (distance increases), and transmission impairments accumulate.

The relevant transmission analysis parameters are found in Table 2. Due to the use of an MZM instead of a phase modulator in the transmitter, the resultant NRZ-BPSK is actually quasi-CW in power. The power exhibits brief transitions between symbols encoded with different phases, giving the power the appearance of an inverted RZ-OOK signal. Otherwise, the NRZ-BPSK signal exhibits constant power in each bit-slot, regardless of whether the encoded phase is 0 or π. A lowest bound on the dispersion length [48] may be based on the power of an isolated NRZ-BPSK 0- or π-symbol, assuming a rise/fall-time of 25 ps. The isolated one is the most common pattern in a 2* ^{n}*-1 PRBS, and occurs 2

^{n-}^{3}times; but occurs twice as many times in the NRZ-BPSK signal power, as symbols carrying isolated phases of 0 and π have identical powers.

The basic optical fiber transmission impairments are now reviewed, and will be followed by a discussion of the various applicable impairments. The propagation of the *k-*th channel field vector$|{{\rm A}}_{k}\u3009$in an *N-*channel DWDM environment, is expressed by the Manakov equation [8,49],

*χ*

^{(1)}; whereas the RHS expresses the Kerr or nonlinear susceptibility tensor

*χ*

^{(3)}effects [37]. The second term on the LHS expresses propagation loss. Polarization-mode dispersion (PMD) is represented in the third term, where

**J**is a random 2 × 2 unitary matrix, and Δ

*β*

_{1}, the deterministic birefringence. This term expresses the group-velocity fluctuations of the two orthogonally polarized modes (components) of a given channel $|{{\rm A}}_{k}\u3009$. Dispersion and dispersion-slope are represented by the fourth and fifth terms on the LHS. All bracketed constant coefficients on the LHS are evaluated at the

*k-*th channel's carrier frequency

*ω*.

_{k}The RHS Kerr nonlinearity terms are most efficient over the effective length $\left(1-{e}^{-\alpha L}\right)/\alpha $ of a transmission span (Table 2). Inter-channel FWM is considered negligible for the system investigated, due to the channel spacing and the spans' large dispersion (Table 1) [50]. The first term on the RHS of the Manakov equation is responsible for intra-channel (IC) nonlinear effects, such as SPM, IC-XPM, and IC-FWM. The second and third terms represent inter-channel nonlinear effects. They may be considered to respectively express polarization-independent XPM, and polarization-dependent XPM. The coefficient (1- *δ _{nk}*) is required to ensure the exclusion of SPM, which is already represented by the first term on the RHS of Eq. (5). In contrast to SPM then, XPM is polarization dependent [51]. The second and third terms are identical for copolarized channels; but only the second term survives for orthogonally-polarized signals. Since this report is focused on single-polarization transmission, the XPM for two co-polarized channels will be twice that when the channels are orthogonally polarized. The third term thus causes cross-polarization modulation (XPolM), also termed nonlinear polarization scattering. Neglecting

*χ*

^{(1)}effects higher than first-order in Eq. (5), XPolM may be re-expressed in Stokes space, in terms of the sum of the Stokes vectors of the (

*N-*1) co-propagating channels, in interaction with the

*k-*th channel, and in its time-frame

*τ*[52–54];

*P*is the channel power equivalent to$\u3008{{\rm A}}_{n}|{{\rm A}}_{n}\u3009$ in Eq. (5), and the walk-off parameter

_{n}*d*

_{nk}= β_{1}

_{n}- β_{1}

*The total Stokes vector is expressed by the sum in the above equation. A*

_{k}.*k-*th channel's SOP is thus modulated by the SOPs of the other channels, causing the

*k-*th channel Stokes vector to precess about the total Stokes vector on the Poincaré sphere, but subject to walk-off relative to the

*k-*the channel.

As random processes, it was found that SPM-induced nonlinear phase noise (NLPN) is non-Gaussian in distribution [55]; whereas the NLPN due to XPM is close to being Gaussian for transmission over dispersion-compensated spans. The latter is a consequence of the central limit theorem. XPM-induced NLPN is also independent of both SPM-induced NLPN, and ASE [56].

Repeated amplification over long-haul transmission results in an OSNR degradation due to the accumulation of ASE, which leads to amplitude and phase fluctuations for a given channel. The ASE-induce amplitude fluctuation is converted to a phase fluctuation due to the nonlinearity, resulting in the G-M effect. The ASE-induced amplitude fluctuation in one channel, may be converted to a phase fluctuation in another channel, which is the XPM-mediated G-M effect. Similarly, a channel's own ASE-induced amplitude fluctuation result in the SPM-mediated G-M effect. For both cases, the G-M effect is exacerbated near the launch plane of a span in the recirculating loop, where channel pulses will have been re-compressed to their original pulse-widths after DCF propagation, resulting in the highest channel peak powers. Further, the channels will have also been temporally re-aligned.

The ASE-induced phase fluctuation result in a chirp. Independent of ASE, SPM and XPM also respectively cause intra-channel and inter-channel chirp. The interaction of chirp with first-order dispersion results in a channel carrier frequency fluctuations, and therefore a random group velocity and timing jitter. Moreover, Chirp is also converted to an amplitude fluctuation due to second-order dispersion, which is especially efficient in G.652 fiber due to its high *D*-parameter of ≈17 ps∙nm^{−1}∙km^{−1} (Table 1). In addition to data format, the efficacy of XPM depends on the effective length, the walk-off length, the dispersion length and the relative polarizations of the propagating channels. The walk-off length ${L}_{W}=\Delta t/\left(D+{\scriptscriptstyle \frac{1}{2}}S\Delta \lambda \right)\Delta \lambda $is determined by a channel pulse's rise-time duration$\Delta t$, dispersion and dispersion-slope parameters, and the channel spacing $\Delta \lambda $(Table 2) [57]. From Table 2, it can be seen that *L _{D} > L >> L*

_{eff}. This implies that XPM could be quite efficient over

*L*

_{eff}, as the signal pulses broaden minimally over this length, thus retaining their peak powers. However,

*L*

_{W}<< L_{eff}, which implies that although XPM maybe significant over a given span, it will be highly mitigated. It was indeed found that for 100-GHz DWDM 10-Gb/s NRZ-OOK long-haul transmission over G.652 fiber, the XPM effect was significantly weaker than SPM [58,59]. Moreover, XPM impairments are most severe for a format exhibiting pattern-dependent power such as OOK; but are significantly mitigated for NRZ-BPSK, which is quasi-CW in power. The residual dispersion of 6.1 ps/nm per circulation in the loop could have also alleviated the constructive build-up of XPM NLPN [60]. Lastly, XPM is further mitigated polarization-wise, by a relatively small walk-off length, the span PMD and polarization scrambling, since for two, orthogonally polarized channels, XPM will be half that for the case of two channels, co-polarized. As for XPolM, the combination of a relatively small walk-off length along with PMD and polarization scrambling should have also rendered XPolM a minor degradation for the single-polarization system investigated [61]. Similarly, since

*L*>

_{D}*L >> L*

_{eff}, SPM would be efficient, and would build up with distance. This argument also implies that intra-channel nonlinearities should have been minimal in the experiment due to the relatively large dispersion length. Unlike XPM however, SPM is independent of the walk-off length, and relatively immune to polarization effects. Based on this reasoning, then, transmission over the recirculating loop was dominated by SPM. As the signal OSNR deteriorated with distance, the interaction of SPM with ASE resulted in the G-M effect. SPM also leads to timing jitter, due to its interaction with first-order dispersion, and to intensity noise, due to second-order dispersion.

#### 3.3 Polarization attraction after recirculating loop transmission

In this section, polarization attraction (PA) was employed on the central NRZ-BPSK channel, after transmission in the recirculating loop (RL), i.e.∙the signal was now directed to the PA module. With PA bypassed, Figs. 16-17 demonstrate that transmission distances ≈∙7,000 km are achievable before reaching the selected FEC BER threshold, with an OSNR of ≈∙10 dB / 0.1 nm. However, due to the sensitivity of the G-M effect to ASE, it precluded PA at such long distances. Dispersion compensation could have been carried out algorithmically in the coherent receiver, obviating the need for periodic dispersion compensation. Without periodic dispersion compensation and over long-haul distances greater than 1,000 km however, dispersion severely broadens the signal, lowering its peak power, and making it difficult to achieve the requisite peak power for PA in the HNLF. Consequently, transmission was carried out with periodic dispersion compensation.

To test the efficacy of PA to transmission impairments, all components up to the LNF-EDFA, are replaced by the RL shown in Fig. 12. Moreover, the Transmitter was replaced by the DWDM transmitter of Fig. 11. Figure 18 shows the Poincare sphere traces, when PA was employed. Qualitatively, the two traces are similar. A different polarization analyzer from the one used in [29], was used for this report, due to triggering capabilities. The polarization analyzer used was the Keysight Technologies N7781A. Figure 19 demonstrates the efficacy of PA post-transmission in the RL. As the transmission distance is increased, the PA is seen to lose its efficiency, and to deviate, in terms of the output DOP, from the trend established at the shorter distances due to a degraded receiver OSNR. However, the PA output DOP remained above 90% for transmission distances up to 2,440.5 km, which includes those used in the receiver sensitivity measurement (Fig. 20). With the PA module employed in the receiver chain, eight circulations (1,301.6 km) were necessary to observe BER on the coherent receiver, otherwise the signal was received error-free. The received spectrum prior to demultiplexing and PA is in Fig. 20. Figure 21 shows the receiver sensitivity measurements for post-transmission PA (red circles). The post-transmission PA measurements were taken after 8–15 circulations (1,301.6–2,440.5 km respectively), and all for a PA output DOP > 90%, as seen in Fig. 19. AE was enabled over the entire data set. Each measurement point ofthe post-transmission PA is found to be within 1.5 dB of the ASE-loaded back-to-back PA case, demonstrating that PA impairments dominated those due to RL transmission. At the FEC threshold, the BER performance for ASE-loaded and RL cases were within 1.5 dB. Figure 22 shows constellations captured at the FEC threshold for each experimental case in Fig. 21. Comparing constellations of baseline transmission and ASE-loaded PA in Figs. 22(b) and 22(c), respectively, the transmission degradations in baseline transmission appear to be much less significant than those of ASE-loaded PA. Figure 22(d) suffers from impairments due to long-haul transmission and PA, but still appears most similar to Fig. 22(c), since the nonlinear impairments in the HNLF dominate.

## 4. Summary and conclusions

For the first time, polarization attraction (PA) on a scrambled 10.7-GBaud NRZ-BPSK signal in an HNLF has been evaluated for robustness to transmission impairments. In a back-to-back transmission evaluation, PA was employed on a signal corrupted by ASE only. A receiver sensitivity penalty at the ITU-T G.975.1.I3 FEC BER threshold of 3.5 x 10^{−3} of ≈ 10.5 dB was achieved relative to the baseline performance. The large penalty is ascribed to the interaction of ASE with the self-phase modulation (SPM) of the signal in the HNLF, resulting in the Gordon-Mollenauer (G-M) effect. This penalty is not attributed to the process of PA itself however, which is governed by a four-wave mixing interaction of two counter-propagating signals; but to the G-M effect in the HNLF, which is purely a single signal effect. This was confirmed by turning off the CW pump used in PA, resulting in a similar penalty to that observed for PA. For the long-haul transmission evaluation, PA resulted in a nearly identical BER relative to PA on an ASE-loaded signal. For both cases, receiver sensitivity measurements and qualitative constellation diagrams confirmed that the SPM-induced G-M effect within the HNLF was an overwhelming hindrance to an efficacious PA. The G-M effect is of no consequence to the OOK format, which was used in all previous successful demonstrations of PA, for which the 10^{-9}-BER receiver sensitivity penalty was always < 0.5 dB [12,17,24,25].

Before PA may be considered a viable technique in practical fiber-optic communication systems based on PSK formats, the penalty due to the G-M effect in the PA medium must be mitigated. ASE, which is the source of the G-M effect, is intrinsically statistical; hence, full compensation for the G-M effect is not possible [62]. One potential approach to partially compensate for the G-M effect may be an all-optical phase sensitive amplifier (PSA) employed directly after PA [63]. A PSA may be implemented with one or two CW pumps co-propagating with the signal [64]. This method requires that the CW pump(s) add coherently to the signal, so additional carrier recovery/pump phase-locking [65] is necessary for a scenario when the original light source for the signal is unavailable. A simpler solution is to implement a PSA with a nonlinear optical loop mirror (NOLM) before PA. A modified NOLM setup [66] would mitigate the amplitude fluctuations while adding no additional phase jitter, which would reduce the G-M effect in the HNLF during PA. Another more recent experimental demonstration of phase-sensitive regeneration was performed without a phase-locked loop using Brillouin amplification [67]. The use of coherent reception can enable an algorithmic solution to transmission impairments, specifically the digital back-propagation (DBP) approach. A DBP algorithm may be used in conjunction with all-optical compensation techniques, if necessary. It has been reported that DBP can be efficacious at alleviating nonlinear distortion, and increasing the effective SNR at reception, with proper attention to the temporal and spatial discretization of the reverse nonlinear operator, as well as the back-propagated bandwidth [68].

An alternative PA configuration is the Omnipolarizer [69], in which the signal polarization is recovered without the use of a CW pump. In the Omnipolarizer however, the signal is required to traverse the nonlinear fiber twice, resulting in a larger nonlinear phase shift well in excess of π radians for an ASE-loaded PSK signal, and therefore an enhanced G-M effect beyond that observed for this report. For this reason, the linear configuration used in the reported experiments would be the preferred approach.

For more complex formats such as QPSK and QAM and at the same Baud-rate, the PA performance should be even worse than for the BPSK reported here, due to a reduced inter-symbol distance for the same average power. Although QAM involves amplitude modulation unlike BPSK and QPSK, QAM also makes use of the signal's phase to encode data, which would therefore makes it susceptible to the G-M effect, like the other formats. It was also previously noted that PA is currently not possible for a polarization-multiplexed signal, without polarization diversity [29].

## 5 Appendix: Estimation of the penalty due to the Gordon-Mollenauer effect

#### 5.1 Introduction

According to [31], error-free long-haul transmission is achievable for a NRZ-BPSK signal, so long as the mean nonlinear phase shift (NLPS)$\u3008{\phi}_{\text{NL}}\u3009\approx 1$rad, which ensures the minimum phase variance for the received signal. The analysis was based on the assumption of two degrees-of-freedom (DOF), and matched-filter reception. The goal of this section is to apply the theory to the HNLF, in order to explain the large penalty experienced by the signal during polarization attraction. Although a significant penalty would be expected due to the magnitude of the acquired NLPS in the HNLF, a penalty as large as ≈ 10 dB relative to the baseline NRZ-BPSK would require some additional verification. The following analysis is based on that of [42,43], which is a much more rigorous derivation of the G-M effect.

In general, a signal of bandwidth *B* and symbol duration *T* will have 2*K=* 2*BT* DOFs, according to the sampling theorem [31]. A DOF is represented by one of a complete orthonormal set of *real* functions that span the signal field. A set of sinc-functions can be used as such to reconstruct the signal in the time domain. For the nomenclature used, the highest frequency component in the signal’s spectrum is at *B/*2 Hz, so the signal can be completely described by a set of samples spaced at 1/*B* s, in the time domain. Alternatively, a signal’s field can be described by a total of *K* complex orthonormal basis functions *F _{k}*,

*K*complex orthonormal basis functions,with expansion coefficients given byThe real and imaginary parts of such coefficients are considered to be independent, identically distributed, (i.i.d.) Gaussian random variables. Then the first and second order statistics of these ASE coefficients are given by

*χ*

^{2}-distributed random variable, of two DOFs. When the inner product Eq. (14) exists, it is construed as the ASE PSD per unit-bandwidth, per polarization state. It has units of

*hν*per unit-bandwidth or [m

^{2}kg/s

^{2}] = Joules, conventionally stated as W/Hz. At this juncture, no assumptions are made about the basis functions - other than that a basis function has units of s

^{-1/2}to ensure a dimensionless result for the inner product of any two such functions.

#### 5.2 NRZ-BPSK transmitter

The NRZ-BPSK signal is generated using a single-drive MZM optically driven by a laser at a frequency *v*_{0}. It is biased at *V _{π}* (its transmission null), and driven at 2

*V*by the AC-coupled electrical baseband NRZ-OOK symbol${a}_{n}\in \{0,1\}$, parenthesized in the numerator of the exponent:

_{π}*T.*The rapidly changing carrier$\mathrm{exp}\left(-2i\pi {\nu}_{0}t\right)$is the term which multiplies the SVE to yield the time dependence of the optical field. The above expression may be alternatively expressed in terms of a cosine of a real argument. The amplitude in Eq. (15) accounts for the laser power and MZM propagation losses. The expression may be simplified to

*n-*th instant

*t*is a contraction for (

_{n}*t - nT*), and where

*a*may be considered to be symbolically either a solitary one or a solitary zero with identical probabilities, and represented temporally by the rect-function. This is a good approximation in practice since the baseband pulses are super-Gaussian in nature, with rise-/fall-times ≈

_{n}*T/*4 at 10.7 GBaud, assuming proper component selection. A solitary bit should be adequate without loss of generality, since the MZM output power is constant for BPSK except during relatively brief bit-transitions, irrespective of whether the baseband data is a zero or a one. Then the peak power is independent of the baseband pattern encoding the optical carrier. This expression neglects impulse response effects such as pattern-dependence in the transmitter, which would lead to a pattern-dependent output power, and to a possible penalty upon reception. The expression can be easily redefined in terms of an orthonormal basis function

*F*

_{0}satisfying the orthonormality condition of Eq. (9)

*s*represents the exp-term in Eq. (16). The rect-function can serve as a basis function for the electrical baseband signal, since any cluster of consecutive ones or zeros could be expressed in terms of an expansion of rect-functions.

_{n}*F*

_{0}(

*t*) also represents the zeroth-order approximation of the main lobe of the sinc function, which is a basis function used for reconstruction in Shannon's original sampling theorem. To obviate the need for joint p.d.f.’s in the following analysis, the signal is considered deterministic.

In the experiment, three amplifiers are used in tandem to launch the signal into the HNLF. For its proper operation, the HPA required a pre-amplifier, which was comprised of two amplification stages with a 0.45-nm interstage filter. The output of the pre-amplifier was once again filtered, by 1-nm filter, to suppress low-power, out-of-band ASE. Based on Frii’s amplifier theory, the signal’s noise figure would be expected to be dominated by the pre-amplifier. For this analysis, it will be assumed that the entire amplifier chain, along with its attendant filters, is considered as one amplifier, which imparts gain *G* to the MZM output, and adds ASE within a 0.45-nm bandwidth to the amplified signal, resulting in the amplified signal input to the HNLF,

*amplified*signal is given byThe peak energy incorporates the gain factor, and the temporal quantities incorporate the sole basis function

*F*

_{0}(

*t*). Substituting Eq. (19) into Eq. (20), it is observed that the in-phase ASE component results in amplitude fluctuations, whereas the quadrature ASE component yields a phase shift. Both components however, will contribute to the mean NLPS, and more significantly at relatively low OSNRs. The equation is also expressible in polar form as

#### 5.3 Propagation in the HNLF

Since the HNLF is a standard fiber (although with a highly reduced effective area) with principle states of polarization (PSPs), propagation of the optical field is not governed by the scalar nonlinear Schrodinger equation (NLSE), but by the Manakov equation for the signal's Jones vector **A**,

*U*and

*V*describe the transverse spatial dependence of the electric field

**E**of the optical field. The two vectors possess different eigenvalues resulting in different group velocities, and necessitating the use of a mean propagation constant. The SVE complex coefficients

*A*(

_{x}*z*,

*t*) and

*A*(

_{y}*z*,

*t*) constitute the Jones vector of the optical field, which describes the SOP of the field at (

*z*,

*t*). The Jones vector incorporates

*s*(

_{in}*t*). The vector components are properly normalized to reproduce the average power of

*s*(

_{in}*t*). Using the above expression for the electric field in the derivation, averaging over the transverse spatial dependence, as well as over the complex exponential, results in the Manakov equation after a few manipulations. It is the single-channel version of Eq. (5).

**J**is the birefringence matrix, and is a random unitary 2 x 2 matrix, a function of (

*z*,

*t*), and results, along with the deterministic birefringence Δ

*β*

_{1}, in random group velocities for the two polarization components. In the frequency domain, its Jones eigenvectors represent the fiber's PSPs. Due to a low PMD coefficient of 0.05 ps/km

^{1/2}, and at 10.7 GBaud, the HNLF PMD may be considered negligible. The 1/9-th reduction in the nonlinear coefficient arises due to averaging the SPM-term over the fiber's birefringence fluctuations. SPM is the sole term on the RHS of Eq. (22). The resultant 8/9th-coefficient has to be retained in the equation, even when PMD is deemed negligible [70,71]. Due to the HNLF's large dispersion and dispersion-slope lengths at 10.7 GBaud, terms representing dispersion may also be ignored, leading to the closed-form solution at the output plane of the HNLF

*L*due to the low propagation loss of the HNLF. At the input plane (

*z =*0) of the HNLF, and for a polarization-stable field, the field is assumed to have the relatively

*static*Jones vector

#### 5.4 The Gordon-Mollenauer Theory for HNLF

In the output signal expression extracted from the solution of the Manakov equation for the optical field, the SPM-induced NLPS may be expanded into a series

*F**(

*t*) over the duration of the symbol,

*h*(

*t*) =

*F**(

*T-t*). When the filter’s impulse response assumes the form

*h*(

*t*) =

*s*(

_{out}*T-t*)*, the filter is termed “matched”. The signal-to-noise ratio (SNR) is generally defined as

*E*) to the unilateral ASE power spectral density per polarization,

*ρ.*SNR is identical with the

*Q-*factor defined in [31]. This is a standard metric used for all non-optical communications systems, and is Baud-rate independent [72]. SNR can be converted to OSNR, a quantity both Baud-rate and bandwidth dependent,The matched-filter receiver provides no advantages for the RZ-format relative to the NRZ-format (as might be the case in practice); but the RZ duty cycle can significantly enhance the NLPS, and therefore, the NLPN. The phase of the demodulated signal is just the sum of the phases due to each term involved in the product in Eq. (27):

The first term of Eq. (30) is now considered. The argument is the ratio of two i.i.d. Gaussian random variables, resulting in a Cauchy-distributed random variable. The Cauchy p.d.f. is not defined in terms of the mean and the variance, which makes further progress difficult. However, under the assumption of a high received SNR that would render the in-phase ASE component negligible relative to the signal amplitude, a Taylor series expansion can be then be carried out and truncated to its first-order,

_{NL}= 8/9

*γP*

_{0}

*L*is the NLPS, and can be used to simplify Eq. (32). The second parenthesized term may not be negligible at the selected FEC BER threshold, when SNR can be << 10. To evaluate the variance of the phase Eq. (32), the expected value of the quantity ${\left(\phi -\u3008\phi \u3009\right)}^{2}$is computed, with the result

_{NL}= 8/9

*γP*

_{0}

*L*in Eq. (34) and using Eq. (14), the dependence on the signal energy is revealed to be

*linear*phase noise, is inversely proportional to the signal's energy, whereas the second term, the

*nonlinear*phase noise, is directly proportional to it. The existence of an extremum may be explored through differentiation with respect to the signal energy

*E*

_{0}, yielding a minimum mean NLPS ofafter using Φ

_{NL}= 8/9

*γP*

_{0}

*L*. The minimum variance is then, in terms of SNR [Eq. (28)] or

*Q*, found by substitution of Eq. (36) into Eq. (34)

*Q*, i.e.

*Q*

_{MF},for a given BER. The minimum variance Eq. (37) represents an upper bound for the variance Eq. (35), which is restated as

*Q*

^{−2}may be ignored, resulting in the simplified condition, after a re-arrangement

*Q*

_{MF}is the quantity corresponding to the matched-filter BERAlternatively, Eq. (39) may be re-expressed in terms of the OSNR [Eq. (29)], which is the measurable quantity. Then results the condition on the minimum

*expected*experimental OSNR (referenced to a resolution bandwidth of

*B*

_{o}) required to achieve an experimentally observed BER, for an

*R-*Baud BPSK signal experiencing a mean NLPS$\u3008{\Phi}_{\text{NL}}\u3009$,

^{−1}-term. Given a BER, a more accurate result may be obtained from a substitution of Eq. (40) into Eq. (38), and solving the resultant quadratic equation for

*Q*(or OSNR). A commonly used BER metric in receiver sensitivity experiments is BER

_{EF}= 10

^{−9}, at which reception is considered error-free (EF). Due to the widespread usage of FEC technology, a relatively high BER may be corrected to < 10

^{−9}. For ITU-T G.975.1.I3 FEC BER threshold of 3.5 x 10

^{−3}, the relevant BER to use in Eq. (41) is BER

_{FEC}= 3.5 x 10

^{−3}.

#### 5.5 An exact theory

An exact theory for the BER of a BPSK signal suffering the G-M effect has been reported for a long-haul transmission line, consisting of *N* dispersion-less, PMD-free amplified spans [73,74]. The 1-km HNLF used in the experiment can be considered to be negligible in terms of both dispersion and PMD, for 10.7 GBaud NRZ-BPSK, using a low linewidth laser. It may be assumed that the effective nonlinear phase shift due to propagation over the transmission line is identical to that generated by the HNLF, assuming that the relevant quantities are appropriately scaled.

The matched filter receiver BER for an NRZ-BPSK signal suffering nonlinear phase noise (NLPN) due to the G-M effect may be computed using the Fourier series expansion of a probability density function (p.d.f.). The p.d.f. is found from the joint characteristic function of the normalized NLPN and the phase of amplifier noise (ASE). For a given OSNR, the BER is exacerbated due to the dependence between the NLPN and the phase of ASE [74]. The probability of error or BER, was found to be related to the integral of the p.d.f. of the received phase Φ* _{r}* [74]

*γP*

_{0}

*L*(as in the previous section), the decision boundary would be centered at $\u3008{\Phi}_{\text{NL}}\u3009$, resulting in the decision angle of $\pm {\scriptscriptstyle \frac{\pi}{2}}-\u3008{\Phi}_{\text{NL}}\u3009$for BPSK. In reality, the p.d.f. is not symmetrical in such a manner. Consequently, the decision boundary is expressed instead as $\pm {\scriptscriptstyle \frac{\pi}{2}}-{\theta}_{c}$. The characteristic function coefficients for Φ

*, are given by*

_{r}*I*is the modified Bessel function of the first kind, and

_{n}*n-*th order, with argument

*x*)/

*x*, and $\u3008\Phi \u3009$is the mean

*normalized*NLPN,andis the product of the number of polarizations, and the ratio of the −3-dB-bandwidth (0.45 nm or 56.25 GHz) of the receiver DWDM demultiplexer filter used in the experiment, to the resolution bandwidth (12.48 GHz). This factor is required to account for the cross-modulation of the out-of-band dual-polarized ASE, with the signal. Additionally, the marginal characteristic function of the

*normalized*nonlinear phase noise is given by

*.*

Lastly, the phase angle *θ _{c}* is critical to the resultant BER, and deserves further discussion. Using the exact BER expression Eq. (42), the optimal

*θ*was found to be smaller than the mean NLPS $\u3008{\Phi}_{\text{NL}}\u3009$, when the mean NLPS was smaller than ≈ 1.25 rad. At a low $\u3008{\Phi}_{\text{NL}}\u3009$, the p.d.f. of the received phase Φ

_{c}*spreads towards the positive phase so that the optimal*

_{r}*θ*< $\u3008{\Phi}_{\text{NL}}\u3009$. At a high $\u3008{\Phi}_{\text{NL}}\u3009$, Φ

_{c}*is dominated by NLPN. Because the p.d.f. of NLPN spreads towards the negative phase, the optimal*

_{r}*θ*>$\u3008{\Phi}_{\text{NL}}\u3009$. The latter was the case for the polarization attraction experiment, which required a NLPS of ≈ π, necessitating a ${\theta}_{c}\approx 1.04\u3008{\Phi}_{\text{NL}}\u3009$ [74]. The difference between the low and the high $\u3008{\Phi}_{\text{NL}}\u3009$ cases is ascribed to the interdependence of the NLPN and the phase of ASE.

_{c}#### 5.6 Applicability of the two theories

Both the G-M theory and the exact theory assume matched-filter reception. It is well known that a matched filter is the optimal linear filter that maximizes the output signal SNR in the presence of additive white Gaussian noise. For non-Gaussian noise, this filter may not necessarily maximize SNR. Practical receivers can significantly deviate from matched filter reception, due to pattern-dependence, and ISI, among other impairments that are not generally Gaussian. At a minimum, both theories explored here are deficient in this respect, and bound to be in error when applied to experimental data. However, the assumption of a matched filter simplifies the analyses, since the optimal receiver for non-Gaussian noise is typically nonlinear, and is dependent on the noise statistics that may not be available *a priori*. For non-Gaussian noise, a (projected) orthogonal matched filter may be more appropriate [75]. Further, the G-M theory assumes the received signal phase to be Gaussian in distribution, with no interdependence between the ASE phase and the nonlinear phase noise, both of which are approximations. By contrast, the exact theory demonstrates that neither process is Gaussian in distribution, nor are they independent of each other, and is thus more accurate.

## Funding

United States Department of Defense (DoD) (100000005).

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