1. Introduction

Standard optical fibers exhibit a nonlinear response that introduces a power-dependent phase retardation into the passing wave [1], causing a multitude of effects like self-phase modulation (SPM), cross-phase modulation (XPM) and cross-polarization modulation (XPolM). Undesired phase modulations from nonlinear impairments get converted to amplitude fluctuations during propagation [2], further deteriorating the signal quality. The greater the instantaneous power variation of the optical waveform, the greater the nonlinear impairments.

Modulation formats using only phase and polarization manipulation help reduce the nonlinear impact and increase propagation distance. Polarization Shift (PS)–Quadrature Phase Shift Keying (QPSK) proved to be a good candidate for ultra long-haul transmission allowing greater launch power and increased reach [3,4]. With the advent of optical coherent receivers, polarization multiplexing (PM) rapidly gained interest, as it allows modulation on a 4 dimensional (4D) space and an increase in the binary transmission rate. As an example, a recent publication analyses legacy formats of on-off keying (OOK) and binary phase shift keying (2PSK) when multiplexed on 4 states of polarization (SOP) as a novel modulation format employing PM [5]. Full dual polarization IQ modulators allow the study and use of novel 4D signaling schemes [4]. Formats of higher dimensions were suggested lately, as in [6] where the authors propose a signal constellation design algorithm generating multidimensional signaling schemes that increase the number of dimensions above 4D, where spatial multiplexing in few mode fibers is employed to increase the number of degrees of freedom.

PM offers drawbacks, amongst which the inherent loss of 3dB signal power per polarization to maintain the total optical power. Some formats like Polarization Shift Keying (PolSK) rely solely on modulating the State of Polarization (SOP) to imprint information. Such format benefits from the constant power property, but lack bits per symbol from the smaller set size N_SOP of possible SOPs, log₂(N_SOP).

Higher order modulation formats are widely employed as a way to increase the data throughput by allowing a larger number of different amplitudes and phases. However, such formats inherently exhibit larger power variations from symbol to symbol, accentuating nonlinear effects, especially when polarization multiplexed is used [7]. Higher order modulation formats that do not exhibit larger power variation are of great interest in order to meet the growing need of reach and data throughput.

In this paper, we present a new modulation format for single mode fibers relying on a mixture of PolSK and PSK. We entitled this format 8PolSK-QPSK. The format provides 5 bits of information per symbol where each orthogonal polarization exhibits an 8QAM constellation like Dual Polarization (DP)–8QAM format, itself providing 6 bits per symbol. One of the main characteristics differentiating the 2 formats is the waveform independence on orthogonal states: instead of having completely independent 8QAM formats on each polarization as is the case for DP-8QAM, the proposed 8PolSK-QPSK format couples part of the information on each polarization such that each symbol in the dual polarization symbol family share the property of having equal total optical power. We show both analytically and experimentally that modulating only in polarization and phase allowing a symbol set of equal power attenuates the impact of the optical Kerr effects. To tackle both intra- and inter- channel nonlinear effects, we experimentally transmit 7 channels, all modulated in either DP-8QAM or 8PolSK-QPSK, and study the performance of the central channel. For a fair format comparison, the bit rate per channel is kept constant at 129 Gbit/s. To satisfy this constant bit rate the symbol rate of 8PolSK-QPSK has to be higher than that of DP-8QAM by 20%. Besides this modest baud rate difference, the transmitter architecture and the transmitter implementation penalty are the same for both formats.

This paper is organized as follows. First, the proposed 8PolSK-QPSK modulation format is presented in Section 2 and compared to the DP-8QAM format. The advantage of combining PolSK and QPSK as a modulation scheme are highlighted. We also describe a novel minimum distance detection scheme applicable to any constant power format that is less computationally demanding than the conventional complex minimum distance detection. Section 3 presents an analysis of the nonlinear Manakov-PMD equation [8] and shows how the modified DP-8QAM format can mitigate nonlinear effects. Propagation simulations using both formats demonstrate where and how the 8PolSK-QPSK format suffers less from the Kerr effect. Subsequently, Section 4 presents the experimental test bed used to compare performance of both formats. In this section, we also introduce a way to generate 8PolSK-QPSK format with a generic delay-and-add dual polarization emulator and a controlled single polarization (SP) 8QAM signal. Finally, system performance using both formats is presented in Section 5, where we show performance improvements using the proposed format, including an increased propagation distance of 34%, or 975 km, compared to DP-8QAM, and a reduction of the required OSNR for a BER of 1.4 × 10⁻² of 0.95dB. The paper concludes in Section 6.

2. Proposed 8PolSK-QPSK modulation format

The proposed modulation scheme is very similar to the polarization multiplexed 8QAM format (DP-8QAM). For DP-8QAM, each polarization carries independently an 8QAM modulation format. 8QAM formats can be thought of as a QPSK format (2 bits/symbol) from a symbol family {1, i, −1, −i} on top of a binary (1 bit/symbol) format of symbol family {a, be^iπ/4}, giving a total of 3 bits/symbol/polarization. Of course, both QPSK and binary symbols are independent on each polarization for DP-8QAM. The binary symbol’s amplitude corresponds to the 2 possible rings as depicted in Fig. 1. For a total mean signal power of 1, the total mean power per polarization is 0.5 and for equidistant constellation points, the 2 radii are a = 0.4597 and b = 0.8881. Consequently, for DP-8QAM formats, the total optical power jumps from symbol to symbol to any of the 3 possible power levels: a² + a² = 0.4226, a² + b² = 1, or b² + b² = 1.5774.

 

Fig. 1 Star-8QAM format. In DP-8QAM, the 2 polarizations can both independently be at amplitude a or b. For 8PolSK-QPSK format, that independence is removed.

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In the proposed 8PolSK-QPSK format, the binary amplitude bit is not independent on each polarization. This constitutes the main difference between DP-8QAM and the proposed format. In 8PolSK-QPSK, if the x-pol. shows a symbol in the inner ring (binary symbol a), the simultaneous symbol on the y-pol. has to be a symbol in the outer ring (symbol be^iπ/4), and vice versa. This constraint removes one bit of freedom from DP-8QAM’s 6 bits, providing 5 bits per symbol for the format. As one can compute, enforcing different amplitude on each polarization restricts the total optical power to always be a² + b² = 1 for any given symbol, relieving DP-8QAM from its large 3-level power fluctuation

In the proposed 8PokSK–QPSK format, we only modulate information on the polarization and phase of an optical waveform. The advantage of combining polarization and phase shift keying is found in the analysis of the nonlinear (NL) Schrödinger (NLS) equation. A well- known, simplistic way to study the Kerr nonlinear effects in a single mode fiber is to neglect all impairments but attenuation and Kerr nonlinearity in the NLS equation and study the equation governing the field evolution during propagation. If we call the optical field at z = 0 |u(z = 0,t)〉 = |u_o〉, the field |u(z,t)〉 evolves following |u(z,t)〉 = |u_o〉exp(iϕ_NL(z,t)), where ${\varphi }_{\text{NL}}(\text{z},t)=\gamma (8/9){\int }_0^{\text{z}}〈u_{\text{o}}|u_{\text{o}}〉e^{-\alpha {\text{z}}^{\prime }}\text{d}{\text{z}}^{\prime }$ [2]. The SPM-induced spectral broadening, or the NL impairment, is a consequence of the time dependence of the nonlinear phase ϕ_NL(z,t) [2]. If the temporal variation of the total power 〈u_o|u_o〉 is reduced, the strength of the NL impairments during propagation is also reduced. When modulating in polarization, the total power does not change as PolSK relies on applying a unitary rotation R(t) to the input field |u_CW〉 such that |u_o〉 = R(t)|u_CW〉, where R^†R = I. Additionally, modulating the phase of a continuous wave by applying e^−iϕ(t)|u_CW〉 also maintains the power. The combination of both formats, |u_o〉 = e^−iϕ(t)R(t)|u_CW〉, give a total power of P_o(t), and the NL phase noise ϕ_NL grows as γ(8/9)P_o(t)(1−e^−αz)/α. P_o(t) is the initial launched power waveform, that depends on the pulse shaping at the transmitter. For an intersymbol interference-free pulse shape, P_o(kT) = 1, ∀k. Consequently, formats having a symbol set sharing constant power, as is the case for formats combining polarization and phase shift keying, take advantage of the smaller temporal variation of ϕ_NL to reduce signal deterioration from NL effects.

There is a drawback, however, when using solely polarization and phase as a way to imprint information on a waveform. The 8PolSK-QPSK format is a 4D, constant symbol power modulation format that carries M = 5 bits/symbol and consequently has a symbol set of size 2^M = 32. By defining the dimensions as X̂_REAL, X̂_IMAG, Ŷ_REAL and Ŷ_IMAG, the constraint is written as ${\widehat{\text{X}}}_{\text{REAL}-i}^2+{\widehat{\text{X}}}_{\text{IMAG}-i}^2+{\widehat{\text{Y}}}_{\text{IMAG}-i}^2+{\widehat{\text{Y}}}_{\text{IMAG}-i}^2=1$, for all symbol i in the set. Increasing the symbol set size is equivalent of packing more states in a 4D space, which naturally reduces the minimum squared Euclidean distance (SED) between neighboring states, increasing the required SNR for a target BER [9]. However, by constraining the symbols to satisfy the unit power criterion we lose the independence of all dimensions in the 4D space; all 2^M symbols now have to be sitting on the surface of a 4 dimensional sphere. We end-up with a 4D space of only 3 degrees of freedom. Packing states on a 4D surface normally brings the states closer to each other than when packing them on independent 4Ds. Consequently, the drawback of constant total power formats modulating solely polarization and phase is that the SED is reduced, consequently requiring a larger SNR compared to the unconstrained 4D modulation format of equal symbol set size. However, for the specific cases of 64-states DP–star-8QAM and its proposed constrained 32-states 8PolSK–QPSK, the two formats do share the same smallest SED between closest states: a² + a² = 0.4226.

As the Stokes space represents well the power and state of polarization of symbols, we depict in Fig. 2 the constellations in the Stokes space of (a) the 8PolSK-QPSK format and (b) the DP-8QAM format, both with added white Gaussian noise giving 16 dB of signal to noise ratio (SNR). In Stokes space, the radius of a point represents its power. Figure 2(b) clearly depicts the increased power variation of the DP-8QAM format. For DP-8QAM, one fourth of all possible symbols have radii (r = 0.4226) below the grey sphere of radius 1, half the symbols lie exactly on the sphere and one fourth reside outside (r = 1.5774). For 8PolSK-QPSK format, however, all generated states reside on the sphere. One can observe that the 8 SOP states of DP-8QAM lying on the sphere are exactly the same 8 states for 8PolSK-QPSK. If we distinguish SOPs having the same orientation but of different magnitude, we realize that DP-8QAM has 16 SOPs. For both formats, each SOP can carry 4 different absolute phases. Hence, we can compute that 8PolSK-QPSK supports 8 × 4 = 32 symbols and DP-8QAM, 16 × 4 = 64. When considered as a pure 3 bit/symbol Stokes space constellation, 8PolSK is referred to as “cubic”-PolSK [10,11], clearly pictured in Fig. 2(a).

 

Fig. 2 Stokes space representation of the a) 8PolSK-QPSK and b) DP-8QAM format, both with added noise giving 16 dB of SNR. The axis of both figures are equal. The higher power variations from symbol to symbol is clearly observed for DP-8QAM.

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2.1 Minimum distance detection for 8PolSK-QPSK

The 8PolSK-QPSK format shows a total of 2⁵ = 32 possible symbols. Each symbol can be represented by a Jones vector of the form

The proposed format exhibits constant total power, so 〈S_i|S_i〉 = 1 for any symbol, and the first term 〈R|R〉 is independent of different symbols |S_i〉. The only relevant term becomes the central one. Consequently, instead of computing three terms to obtain Eq. (4), the minimum distance detector’s complexity is alleviated by computing only one term, as in Eq. (3), for the 8PolSK-QPSK format. It is noteworthy to mention that the computation of the minimum distance using Eq. (3) is only valid for formats having a symbol set exhibiting constant power.

A different symbol-to-bits mapping is indeed required for the two formats. For DP-8QAM, we use the optimal bit-to-symbol mapping for 8QAM [12] applied independently to both orthogonal polarizations. Optimal mapping means a bit mapping that yields the smallest average number of bits in error per symbol in error. This mapping uses identical Gray-coded QPSK mapping on each ring of Fig. 1, where one ring is rotated by 45°, with an extra amplitude bit, for a total of 6 bits. For the 8PolSK-QPSK format, the first bit of each symbol is assigned to power, defining if x̂ or ŷ was decided to have more power, equivalent of slicing Fig. 2(a) on the plane A₁ = 0. Then, 2 phase bits on x̂ and on ŷ are Gray coded independently, complying with the optimal mapping scheme found in [12], for a total of 5 bits.

3. Nonlinearity mitigation by modulating with constant power

The idea behind the 8PolSK-QPSK is to reduce the total power variation at the transmitter while keeping a high bit per symbol efficiency. From the wave equation, we know that the nonlinear phase depends on the total optical power in the fiber. The vectorized form of the NLS equation for the modulated optical field, referred to as the Manakov-PMD equation, is described as [1, 8,13,14]

If |u〉 is the sum of modulated envelopes at different frequencies, for instance in the case of a WDM transmission, $|u〉={\sum }_{k=1}^{\text{N}}|u_k(\text{z},t)〉e^{i{\omega }_{k}t}$ and one can show that the nonlinear phase term 〈u|u〉|u〉 of Eq. (5) becomes

The third term in Eq. (6), representing XPolM, can be cast as (∑_m≠nu⃗_m) · σ⃗ =|∑_m≠nu⃗_m|(v̂ · σ⃗) where v̂ =(∑_m≠nu⃗_m)/|∑_m≠nu⃗_m| is a unitary Stokes vector. Put in this form, one can realize that the XPolM term is expressed similarly as the birefringence term of Eq. (5), with a birefringence vector and amplitude −8γ|∑_m≠nu⃗_m|/9. In some literature, XPolM is in fact called nonlinear birefringence [2, 16], as clearly depicted in Eq. (6). As a consequence of XPolM, the state of polarization of the channel of interest is modified by a rotation around the instantaneous sum of the Stokes vectors of the interfering channels, ∑_m≠nu⃗_m, and this nonlinear birefringence term depends on the statistics of the interfering channels’ cumulative power, |∑_m≠nu⃗_m|, and that of their cumulative direction [17, 18]. This XPolM term is the only nonlinear term that is polarization dependent, and XPolM’s impact depends on the SOPs of all other co-propagating channels.

3.1 Nonlinear variance study for 8PolSK-QPSK and DP-8QAM

In the following subsection, we study the variance of each nonlinear term appearing in Eq. (6) for the two modulation formats: the proposed format and the DP-8QAM. We study the central channel of an N-channel system using Eq. (5). The full split-step Fourier method is employed for propagation. We use, in the simulation, parameters that will be used in a subsequent experimental demonstration presented below, namely N = 7 co-propagating channels of identical modulating format and power, launched with a pulse shape of root-raised cosine (RRC) of roll-off factor α = 1/8, at a baud rate of 25.8 Gbaud for 8PolSK-QPSK and 21.5 Gbaud for DP-8QAM. The channels sit in a 50 GHz grid and we study the impact on the central channel. In the simulations, the SOPs of all channels at launch are random. In this simulation, we deliberately neglected ASE noise added by EDFAs, PMD, and birefringence. ASE is neglected because we don’t want to interfere the statistics of the signal with external additive white Gaussian noise, as is shown in [19]. PMD is neglected because we do not want to convolve XPolM induced rotations with frequency dependent rotation from PMD, and finally birefringence is neglected as it simply acts as a global unitary rotation equally applied to the entire waveform. Amplification is lumped at every 80 km and the fiber is standard SMF-28 with β₂ = −21.4 ps²/km and an attenuation coefficient of α_dB/km = 0.2 dB/km. The launch power per channel is set at + 1dBm to trigger significantly some nonlinear impairments. We also use γ = 1.3 W⁻¹km⁻¹.

We are interested in the variance of all nonlinear effects applied to the channel of interest at every distance z, with respect to the mean power of said channel at such distance. As the nonlinear noise is proportional to the signal power [20], one can understand that the NL variance will be proportional to the squared signal power. Consequently, as applied in [21], we normalize the strength of the NL impairments by the squared signal power to alleviate the NL noise from its power dependence. In both our simulations and experiments we set the power per channel to be equal for all channels, allowing normalizing the strength of different NL impairments by the squared power of a single channel. The equations allowing nonlinear monitoring are

As |u〉 = |u(z,t)〉, the nonlinear term 〈u|u〉 of Eq. (5), expended in Eq. (6) as 〈u|u〉_SPM + 〈u|u〉_XPM+〈u|u〉_XPolM, is both time and distance dependent. The variance of those three nonlinear terms is applied over the time variable, yielding the z-dependent variances ${\sigma }_{SPM}^2$, ${\sigma }_{XPM}^2$, and ${\sigma }_{XPolM}^2$. As the solution of the propagation equation of Eq. (5) evolves for every dz, the local temporal variance of the NL processes also evolves with distance. The variance of the three NL processes of Eq. (5), namely namely x₁= 〈u_n|u_n〉, x₂=∑_m≠n3/2〈u_m|u_m〉 and x₃=∑_m≠n½u⃗_m · σ⃗, are depicted in Fig. 3. It is important that the y-axis of Fig. 3 be understood as the normalized variance of x₁, x₂ and x₃. By removing the normalization we obtain the observable nonlinear strength variances, ${\tilde{\sigma }}_{NL}^2={\sigma }_{NL}^2\times {\overline{〈u_n|u_n〉}}^2$ where the subscript NL is either SPM, XPM or XPolM. Of course, ${\overline{〈u_n|u_n〉}}^2$ decays with distance as e^−2αz, where α=α_dB/km / (10log₁₀(e)). The averaging operator X(t) is a statistical ensemble average and time average, where the time averaging operator is ${\text{lim}}_{\text{T}\to \infty }1/\text{T}{\int }_{-\text{T}/2}^{\text{T}/2}\text{X}(\text{t})\text{dt}$ [22]. Monte Carlo simulations of 100 propagation runs were realized to determine the properties in Eqs. (7) through (9) with confidence. Each propagation consists of a PRBS sequence of 2¹² symbols with 8 samples per symbols for each of the 7 channels. The results are presented in Fig. 3.

 

Fig. 3 Temporal variance of SPM, XPM and XPolM, normalized by mean power squared, for 8PolSK-QPSK (solid lines: –––) and DP-8QAM (circled solid lines: –○–) a) Main figure showing all NL strengths for first two spans b) Zoom-in from 0 to 40 km showing SPM and XPolM strengths. c) SPM only, from 3rd to 9th span. d) XPM only between 400 and 650 km

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We observe that the behaviour of the variance of nonlinear processes at launch, when z=0, is much different than that after 40 km and more, and that this is applicable for both modulation formats. The interplay between chromatic dispersion with channel walk-off and nonlinearity greatly modifies the initial statistics. At launch, we obtain a SPM variance ${\sigma }_{SPM}^2$ of 0.138 for 8PolSK-QPSK and 0.26 for DP-8QAM. Those values show that the constant power format exhibits a power variance for SPM that is roughly 1.9 times less than that of DP-8QAM. The same is applicable to XPM: at launch, XPM’s variance ${\sigma }_{XPM}^2$ is roughly 1.86 for 8PolSK-QPSK and 3.52 for DP-8QAM. We deliberately added markers, ⊳, at those nonlinear strengths at z=0. We will come back and relate those values to theory in Section 3.2. As the waveform propagates in the first span, the relative variances ${\sigma }_{SPM}^2$, ${\sigma }_{XPM}^2$ and ${\sigma }_{XPolM}^2$ increase very rapidly until they each reach a value roughly maintained for the rest of the fiber length. After about only 40 km, the interplay of CD with nonlinearity renders respective variances seemingly the same for both formats.

We will show in Section 5 experimental results demonstrating that the constant power 8PolSK-QPSK format performs better than DP-8QAM format. Propagation simulation results in Fig. 3 well explain the reasons. This performance improvement is due to the smaller SPM and XPM variances at launch.

It is well known that fiber dispersion plays two roles in the process of XPM induced intensity interference. The first is translating the nonlinear phase modulation into intensity fluctuations and the second is causing signals to propagate with different velocities [1,23]. It was previously concluded that the nonlinear XPM phase interaction between the signals takes place only in the beginning of the fiber [1,18,23], occurring over a short length in which the walk-off between the signals is small. In that part of the fiber, the phase of the channel of interest gets modulated by neighboring channels, as the walk-off between the signals is minimal and pulses are still well defined and contained in their symbol slot, while the intensity modulation is small, as cumulative CD is almost null. After the first walk-off length, the signals then propagate linearly over the rest of the fiber, where the action of dispersion on the phase modulated signal are continuously translated into intensity fluctuations [23], appearing as a continuous additive Gaussian noise [24]. As the bulk of nonlinear XPM phase shift is induced over the first walk-off length only, if that can be kept small, the overall XPM penalty will be smaller [18]. Consequently, the better performance of 8PolSK-QPSK format is explicable by the fact that it exhibits a much smaller XPM strength at launch, and SPM for that matter, hence suffering less nonlinear penalty. Results in Fig. 3 also concur with those observations from the literature: with a dispersion parameter of β₂ = −21.4 ps²/km and a grid spacing of 50 GHz, the first neighboring channels are delayed at a rate of 1.07 ps/km. With a pulse duration of 38.5 or 46.5 ps, the walk-off length is around 40 km: the required distance before nonlinear variances converge, as we can observe in Fig. 3.

Figures (c) and (d) in Fig. 3 depict the behavior of normalized nonlinear strengths after longer propagation distances by providing a closer look at and, respectively. Although almost identical, we observe that the SPM and XPM variances for DP-8QAM are slightly higher than that of 8PolSK-QPSK. Albeit not the primary argument justifying improved performance of the proposed format, this subtle reduction of and occurring at longer distances would also benefit the 8PolSK-QPSK format, as well as the principal and dominant reduction occurring in the first walk-off length. Finally, we also observe an oscillation of the normalized nonlinear strengths of period equal to the EDFA spacing of 80 km, showing a ramp up of the relative power variances for the first 20 to 25 km after each EDFAs, followed by a slow decay for the rest of the span.

3.2 Variance of SPM, XPM and XPolM at launch

The relative power variance numbers obtained at launch (z=0) describing SPM and XPM can be recovered analytically. In order to analytically compute Eq. (7) for SPM, we need to calculate $\overline{{〈u_n|u_n〉}^2}$. If the launched pulse shape function is g(t) and the polarization multiplexed waveform |u_i〉 of this central channel i is $|u_i(\text{t})〉={\sum }_{p=-\infty }^{\infty }|S_p〉g(t-p\text{T})$ where |S_p〉’s are the possible PM symbol, then $\overline{{〈u_n|u_n〉}^2}$ equals

After keeping only terms of 〈u_m|u_n〉 〈u_k|u_l〉 that have a non-zero statistical ensemble average, we end up with

This equation is valid for the symbol set |S_p〉 of any modulation format, and for any pulse shape g(t), even for a pulse shape that includes CD. The respective terms $X_{kkll}=\overline{〈u_k|u_k〉〈u_l|u_l〉}$, $X_{kllk}=\overline{〈u_k|u_l〉〈u_l|u_k〉}$, and $X_{kkkk}=\overline{{〈u_k|u_k〉}^2}$ are modulation format dependent. If the magnitude of the symbols |S_p〉 are such that the mean total power is unitary, one can show that X_kkll = 1, X_kllk = 0.5 and X_kkkk = 1 for 8PolSK-QPSK, whereas X_kkll = 1, X_kllk = 0.5 and X_kkkk = 7/6 for DP-8QAM. If the pulse shape is root-raised cosine (RRC) of any roll-off factor 0<α<1, the mean power $\overline{〈u_i|u_i〉}$ can be relatively easily analytically calculated to be 1/T. However, the analytical expression for $\overline{{〈u_i|u_i〉}^2}$ is more complicated and it is out of the scope of this paper to derive it. $\overline{{〈u_i|u_i〉}^2}$ is both pulse shape dependent and format dependent. It is easy to obtain the numerical value of Eq. (11) using a mathematical software.

Now that we have an expression for $\overline{{〈u_i|u_i〉}^2}$ and its value, we can obtain the relative nonlinear variance of SPM and XPM from Eqs. (7) and (8) for each format. From the result of ${\sigma }_{SPM}^2$, ${\sigma }_{XPM}^2$ can easily be obtained: for a N-channel co-propagation, ${\sigma }_{XPM}^2=(N-1){\left(3/2\right)}^2{\sigma }_{SPM}^2$. The numerical evaluation of ${\sigma }_{SPM}^2$ for RRC with α=1/8 is 0.13785 and 0.26015 for 8PolSK-QPSK and DP-8QAM respectively. Consequently, with N = 7, ${\sigma }_{XPM}^2$ yields 1.861 and 3.512 for the 2 formats, respectively. Those values, obtained from Eqs. (7) and (8), map exactly to our propagation simulation results at z = 0 in Fig. 3. Small right-pointing triangle markers “⊳” located at those theoretical values of ${\sigma }_{SPM}^2$ and ${\sigma }_{XPM}^2$ for both formats show the match between simulation and theory at launch for ${\sigma }_{SPM}^2$ and ${\sigma }_{XPM}^2$, when no CD nor nonlinearity is applied on the waveform.

A further discussion on the XPM impact on a channel when an increasingly larger counts of copropagating channels are added to the link is of interest. It is known that XPM occurs only if the two pulses from different channels overlap temporally [2]. It is also established that XPM-induced phase shift decreases as the relative group-velocity difference (GVD) increases, where the latter is proportional to the frequency difference between channels [2]. Consequently, only close frequency neighbours to a channel have a significant XPM impact on said channel. Channels added increasingly farther from a reference channel observe increasingly larger GVD, and pulses from the two different channels will traverse at an increasingly different speed, blurring the effect of XPM. As Fig. 3 only shows the evolution of the variance of the three nonlinear elements in Eq. (5) as it is numerically propagated, the reduced effect of XPM impairment from increasingly farther channels is not directly shown in this Fig. As the number of aggressor channels grows, located farther away from a reference channel, their XPM impact on said reference channel would be negligible compared to XPM’s impact from the closest neighbours.

The variance of XPolM’s rotation strength, one the other hand, is random by nature and increases to the same steady-state average, and becomes rapidly agnostic to the modulation format. The rapid settling of the 3 variances ${\sigma }_{SPM}^2$, ${\sigma }_{XPM}^2$ and ${\sigma }_{XPolM}^2$ to an approximately steady-state value concurs with a recent conclusion that nonlinear interferences can be modeled as excess additive Gaussian noise [24]. We show here that the variance of that additive Gaussian noise does tend to a coarse steady state value very rapidly.

For XPolM, the statistics of the variable X=|∑u⃗_m| at launch, namely the mean μ_X and variance ${\sigma }_X^2$, depend on the modulation format, the number of other co-propagating channels and the pulse shape applied to all channels. The nonlinear XPolM strength X is naturally a random process due to both the instantaneous randomness of the data of every other channel affecting one channel and due to walk-off between channels [18]. As |∑u⃗_m| is the norm of a Stokes space vector, the variable represents time varying power. The mean and variance of |∑u⃗_m| over time can easily be numerically obtained. For a N=7 channel WDM propagation where each channel’s power is normalized to 1, with RRC pulse shaped with α=1/8, one can compute that μ_X ≈ 2.412 and ${\sigma }_X^2\approx 1.01$ for 8PolSK-QPSK where μ_X ≈ 2.523 and ${\sigma }_X^2\approx 1.19$ for DP-8QAM. Consequently, ${\sigma }_{XPolM}^2$ in Eq. (9) equals 1.01 for 8PolSK-QPSK and 1.19 for DP-8QAM. We can observe that the XPolM variance obtained in the propagation simulation in Fig. 3, at launch, match exactly those numbers. Again, this match is depicted in Fig. 3 by markers ⊳ added at those theoretical values for XpolM. The variance difference of |∑u⃗_m| for both modulation formats can be explained by the fact that the mean squared power $\overline{{〈u|u〉}^2}$ is larger for DP-8QAM compared to that of 8PolSK-QPSK; $\overline{{〈u|u〉}^2}=1$ for 8PokSK-QPSK and 7/6 for DP-8QAM.

When a large number of channels are launched, all randomly polarized, we can consider that every $u_{\text{i}}^{\text{TOT}}$ in $\sum {\overrightarrow{u}}_m={\left[u_1^{\text{TOT}}{u}_2^{\text{TOT}}{u}_3^{\text{TOT}}\right]}^{\text{T}}$ is a Gaussian variable of zero-mean and variance σ². Consequently, the variable $\left|\sum {\overrightarrow{u}}_m\right|=X=\surd \left(u_1^{{\text{TOT}}_2}+u_2^{{\text{TOT}+}_2}+u_3^{{\text{TOT}}_2}\right)$ is a random variable of Chi-distribution with 3 degrees of freedom. It is known that for a Chi-distribution where the generating Gaussian variables all have zero-mean and a variance of σ², the Chi variable X has a mean of μ_X = σ√2×Γ(2)/Γ(3/2) where Γ is the Gamma function, and a variance of ${\sigma }_X^2=3{\sigma }^2-{\mu }_X^2={\sigma }^2\left[3-2{\left(\mathrm{\Gamma }(2)/\mathrm{\Gamma }\left(3/2\right)\right)}^2\right]\approx 0.4535{\sigma }^2$. One can show numerically that the variance σ² of every $u_{\text{i}}^{\text{TOT}}$ is larger by ≈10% for the DP-8QAM format compared to that of 8PolSK-QPSK. This explains why both the mean and variance of the strength of XPolM |∑u⃗_m| is smaller for 8PolSK-QPSK at launch. Finally, we can conclude that even the impact of XPolM is lessened for 8PolSK-QPSK format during the early stage of propagation.

It is noteworthy to observe in Eq. (6) that the impact of XPolM is 3 times less efficient than that of XPM. Hence the benefit of focusing on reducing the XPM effect over XPolM. The above analysis validates our assumption that having constant power modulation on all wavelengths would reduce nonlinear effects. The experimental test and results presented below confirm our analysis.

4. Experimental test bed

To assess the performance of the two modulation formats, we use the following test bed shown in Fig. 4. In order to use the same transmitter configuration, and consequently the same transmitter implementation penalty, the only parameter modified when switching from one format to the other is the waveform baud rate imprinted in the Digital to Analog Converters (DACs) using a fixed clock rate. The test bed begins with 7 lasers combined to generate the WDM waveform. Six of which are DFBs and the central channel of interest is an ECL with laser linewidth < 100 khz. Lasers are placed on a 50 GH grid. The 6 DFBs, of linewidths smaller than 1 MHz, are used as aggressors acting on the central channel. All channels have equal power. This WDM configuration allows assessing the full nonlinear degradation, where all nonlinear impairments are included. A SP-IQ modulator bulk modulates the laser tones using two DACs clocked at a rate of 32 GSa/s for both formats. To modulate the different formats at a their desired symbol rates, a proper Root Raised Cosine (RRC) matched pulse shaping filter with a roll-off factor of 1/8 is applied. In order to yield the same bit rate of 129 Gbit/s, the 6 bits/symbol DP-8QAM format runs at a baud rate of 21.5 Gbaud, whereas the 5 bits/symbol 8PolSK-QPSK run at 25.75 Gbaud. The reason why this latter symbol rate is not 129/5 = 25.8 GBaud is twofold. First, the DACs’ memory has to be a power of 2 and second, the DAC rate is deliberately fixed for both formats. The difference between a baudrate of 25.8 and 25.75 is assumed negligible. The DAC’s pattern length is set to 2¹⁹. The SP-IQ modulator is driven in the linear regime. Following the IQ modulator is the dual polarization emulator with a delay set at 108 ns. This delay translates to an integer number of symbols at both 21.5 GBaud (2322 Symbols) and 25.75 GBaud (2781 Symbols).

 

Fig. 4 Experiental testbed.

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Then, two 50/100 GHz passive interleavers are used to temporally decorrelate even and odd channels. The WDM waveform is then boosted and attenuated to sweep the launch power before launching in an optical recirculating loop. The latter is composed of 4 spans of 80 km of Corning^® SMF-28 e + LL fiber with dual-stage inline EDFAs of noise figure ≈5.5 dB. A gain flattening filter is inserted after the second span inside the loop. Chromatic dispersion is uncompensated. A noise loading stage follows the recirculating loop for OSNR sweeping followed by a channel selecting filter, isolating the central channel of interest. Finally, a pre-amp and a 0.4 nm filter are preceding the optical coherent receiver. The receiver consists of a dual polarization 90° optical hybrid, 4 balanced photodiodes (U²T BPDV2020R) and a 4 channel real-time scope sampling at 80 GSa/s.

4.1 8PolSK-QPSK signal generation with a dual polarization emulator

Generating a 8PolSK-QPSK format with a SP-IQ modulator and a dual polarization delay-and-add emulator requires some handling of the signal provided to the SP-MZM. The dual polarization emulator consists of splitting the incoming SP waveform on 2 orthogonal branches, delaying one, and recombining on two orthogonal polarizations, as pictured in Fig. 4. For regular Dual Polarization (DP) formats like DP-8QAM, the delay of the DP emulator as no impact on the output format: it will always be DP-8QAM as long as the delay is an integer number of symbols and is not zero. Generating a 8PolSK-QPSK format out of a SP-8QAM format, however, requires extra manipulation on the SP waveform out of the Mach-Zehnder modulator (MZM). The SP waveform generated has to be related to the DP emulator’s delay. As explained in Section 2, the symbol constellation of a single polarization of the 8PolSK-QPSK format is exactly the same as that of the DP-8QAM format, i.e. the 8QAM constellation of Fig. 1. However, the 8PolSK-QPSK signal out of the DP emulator has to exhibit the constant power constraint, not observable before the emulator.

To generate a SP-8QAM waveform that will get converted to 8PolSK-QPSK after the DP emulator having a delay of N = 2781 symbols, a specific repetitive pattern has to be imprinted on the DACs. Let’s mention that for a DAC sampling rate of 32 GSa/s, a DAC memory of 2¹⁹ samples and a symbol rate of 25.75 GBaud, a total of M = 421888 symbols are repeatedly generated by the DACs. Two different symbols streams are needed to properly mimic both 8PolSK-QPSK and DP-8QAM formats. The first and most simple symbol stream is a random QPSK sequence of length M. The second symbols stream is what will determine if the waveform after the DP emulator is a 8PolSK-QPSK or DP-8QAM format. If the second symbol stream is a random sequence also of length M of symbols {a, be^iπ/4} and both streams are multiplied symbol by symbol, then the waveform generated is DP-8QAM. To generate an 8PolSK-QPSK format, the second symbol stream is constructed as follows. First, we generate a random sequence of length N from symbol family {S₀=a, S₁=beiπ/4}. We call this sequence S. Then, a sequence of length 2N is generated by concatenating the inverse of S to itself, generating the sequence S’ = [S S̄]. If S’(k)=S₀, S’(N+k)=S₁ and vice versa. This new sequence S’ is then replicated and truncated after M symbols are generated. By multiplying this symbol stream with the QPSK stream, we will generate the desired 8PolSK-QPSK format after the DP emulator of DP delay N. One can prove that the generated dual polarization signal |u(n)〉 = (u_x(n), u_y(n)) = (S’(n), S’(n + N))^T will have the desired constant power 〈u(n)|u(n)〉 = 1 for every symbol n. Of course, this tedious way of generating this constant power signal is avoided with a full Dual Polarization-Dual Parallel MZM.

5. Results

In this section we present the performance of both 8PolSK-QPSK and DP-8QAM modulation formats. We assume BER free operation with 29% coding overhead (OH) for hard decision forward error correction (HD-FEC) with a BER threshold of 1.4 × 10⁻², providing slightly more coding overhead than the 25% used in references [25, 26]. First, we present in Fig. 5 the maximum reach achievable for a BER of 1.4 × 10⁻² as a function of the launch power per channel. As the launch power jumps were relatively large by 2dB, the resolution and hence the optimum powers of Fig. 5 are coarse. At such granularity the optimum launch power appears to be the same for both formats at −1 dBm. However, by looking carefully at the trend of the curves, we realize that the optimum launch power for DP-8QAM would sit slightly below −1 dBm had we used a smaller launch power step size. On the other hand, the 8PolSK-QPSK format shows an apparent true optimum launch power at −1 dBm. We can observe, after interpolation of the DP-8QAM curve, that 8PolSK-QPSK allows a slightly greater optimum launch power, by around 0.4 dB. Secondly and most importantly, Fig. 5 shows that the 8PolSK-QPSK format allows a much greater propagation distance compared to DP-8QAM, 3800 km compared to 2825km, translating to an additional 975 km, or 34%.

 

Fig. 5 Launch Power per channel against maximum reach [km] for a BER = 1.4 × 10⁻². The optimum launch power is −1dBm for 8PolSK-QPSK and around −1.4 dBm for DP-8QAM. 8PolSK-QPSK format propagates 34% more than DP-8QAM, or 975 km more.

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In the following Fig. 6, we show the experimental BER as a function of the OSNR in 0.1 nm for both formats in back-to-back, along with their theoretical values. The theoretical curves in Fig. 6 show a 0.6 dB difference between 8PolSK-QPSK and DP-8QAM at a BER of 1.4 × 10⁻², benefitting 8PolSK-QPSK. Experimentally, we obtain a difference of 0.5 dB. The offset from theory for both formats, mainly due to the transmitter implementation penalty, is rather large at 4.37 dB. Our claim that both formats have the same implementation penalty is well proven here. The inset (a) in Fig. 6 shows the theoretical OSNR (in 0.1 nm) difference between DP-8QAM and 8PolSK-QPSK as a function of the BER, for BERs varying between 2.5 × 10⁻³ and 2.5 × 10⁻². The independent variable, BER, is put in the y-axis to be consistent with the main outermost axes. This inset demonstrates that as the BER increases, the 8PolSK-QPSK format requires an increasingly smaller OSNR with respect to the DP-8QAM. This property is beneficial for the proposed format as improving FEC schemes increase BER thresholds; a recent FEC scheme can bring a BER of 2.4 × 10⁻² down to 10⁻¹⁵ [27].

 

Fig. 6 BER against OSNR in 0.1 nm for both 8PolSK-QPSK and DP-8QAM formats. Inset a): Theoretical OSNR (in 0.1 nm) difference between formats as a function of BER threshold.

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The theoretical BER against SNR function for DP-8QAM format can be approximated by BER(SNR) = 2/3 erfc(√(3 SNR/14)), in the BER range of 10⁻² to 10⁻⁴, with a maximal deviation of ± 0.05 dB [24]. Unfortunately, to the author’s knowledge, there is no such theoretical curve in the literature for 8PolSK-QPSK. By using a model of

The next Fig. 7 depicts the required OSNR (ROSNR) in 0.1 nm to obtain a BER of 1.4 × 10⁻² after propagation of 1920 km as a function of the launch power, for both formats. We can see that in the fully linear propagation regime, at −5 dBm, the 8PolSK-QPSK format requires 0.5 dB less OSNR compared to DP-8QAM format, as predicted by results in Fig. 6. As the launch power increases, the strength of nonlinear impairments accordingly increase and Fig. 7 exhibits the better nonlinear tolerance of 8PolSK-QPSK. The smaller nonlinear impairments received by using the proposed format is expressed by the clear decoupling of the 2 curves with growing launch power. At the optimal launch power of −1dBm where the simultaneous impact of linear and nonlinear effects are at minimum, the OSNR requirement is 0.95 dB less for the proposed format. At + 1dBm of launch power, where nonlinearity clearly supersedes linear impairments, 8PolSK-QPSK outperform DP-QAM by 2.1 dB of ROSNR.

 

Fig. 7 Required OSNR in 0.1 nm [dB] for BER = 1.4 × 10⁻² after 1920 km.

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As shown in Fig. 3 using simulations, the better performance of 8PolSK-QPSK after 1920 km for increasing launch powers comes almost solely from the first 40 km or so, where the proposed format receives significantly less NL impairments. As the launch power increases, the NL noise applied to both formats also naturally increases, but even more NL noise is applied to the DP-8QAM format within the first 40 km; a supplemental impairment still observable after longer distances like 1920 km. Figure 7 demonstrate very well both the linear and nonlinear advantages of the proposed format.

Finally, as a last experimental result, we present in Fig. 8 BER against Distance curves for the two formats at their optimum launch power of −1 dBm. We can see that at any fixed distance, the BER is always lower when the 8PolSK-QPSK format is employed compared to DP-8QAM. Another curve was added to the 2 BER against Distance curves of Fig. 8, showing the ratio of the BERs of the 2 formats as a function of distance, expressed as BER_DP-8QAM(z) ÷ BER_8PolSK-QPSK(z) × 10⁻⁶ and expressed in log-scale. An offset of 10⁻⁶ was deliberately applied to the ratio to have it fit at the bottom of the y-axis while using the same y-scale. This curve shows that the ratio of the BERs of the 2 formats is essentially constant with distance. The fact that the BER equally increases with distance for the 2 formats concurs with the simulations observations of Fig. 3. It was shown in Fig. 3 that the variance of the nonlinear strength quickly saturates to the same value after the second span for both formats. Consequently the nonlinear noise applied to both formats can be assumed equal after a few hundred kilometers. Both NL and linear noise equally increase with distance, explaining the experimental even growth of the BER for both formats.

 

Fig. 8 BER against Distance [km] for both 8PolSK-QPSK and DP-8QAM. Aslo shown in inset (b) is the reach difference of the two formats as a function of BER in linear scale.

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Another informative curve was added to Fig. 8 in inset (b), showing the excessive reach of 8PolSK-QPSK over DP-8QAM at a specific BER, calculated for different BERs. As an example, at a BER of 1.4 × 10⁻², DP-8QAM’s maximum reach is marked by “A” and 8PolSK-QPSK’s maximum reach is marked by “B”. B–A is the excessive distance when 8PolSK-QPSK is employed compared to when DP-8QAM is used for a target BER threshold of 1.4 × 10⁻². This reach difference is shown on the left-hand side of Fig. 8, with the same x-axis but now representing excessive distance instead of absolute distance. Starting at a BER threshold BER_th of 3 × 10⁻³, the added reach R₊ when using 8PolSK-QPSK over DP-8QAM can be well approximated by a linear relation with BER_th: R₊ ≈375 + 43475 × BER_th. The inset (b) replicates the reach difference curve with BER in linear scale and put on the abscissa. This would further benefit the proposed format as a large effort is recently put in increasing FEC thresholds for optical communications [27]. With such supplemental reach monotonically increasing with BER threshold, we can predict a excessive reach by more than 1400 km at the BER threshold of 2.4 × 10⁻²; a threshold currently applicable for soft-decision (SD) FEC [27,28]. The excessive FEC overhead of 29% used in our experiments would be sufficiently large for such SD-FEC with extra space for protocol overhead.

6. Conclusion

We have presented a novel modulation format obtained by constraining the well-known DP-8QAM format to exhibit constant symbol power. The proposed format, providing 5 bits of information per symbol and entitled 8PolSK-QPSK, presents a passive way to mitigate nonlinear impairments by significantly reducing the optical power variation from symbol to symbol in the first 40 km of propagation. Using both theoretical explanations, simulations and experimental validations, we confirmed the better nonlinear tolerance of the proposed format. We showed a back-to-back decrease of the required OSNR for a BER of 1.4 × 10⁻² by 0.5 dB for 8PolSK-QPSK, showing an improved ASE noise tolerance for 8PolSK-QPSK at such BER. After a distance of 1920 km with a launch power of −1 dBm/ch with 6 other co-propagating channels, the proposed format exhibits a ROSNR reduction of 0.95 dB. We also demonstrated that at the optimum launch power of −1dBm/ch, the proposed format extends the propagation reach by 975 km, or by 34%, for a BER threshold of 1.4 × 10⁻², from 2825 km to 3800 km. We also presented a relation between supplemental reach provided by the format against increasing BER threshold. Finally, to bolster the analysis of the format, we presented a detailed study, both analytically and via simulations, of a comparison of the strengths of the three main nonlinear impairments occurring during propagation, namely SPM, XPM and XPolM, where we showed significantly reduced SPM and XPM strengths by using the 8PolSK-QPSK format.