1. Introduction

The exponentially-growing bandwidth demand in optical networks, which we experience nowadays, is inciting studies aimed at improving the reach and throughput of core fiber-optic communication systems [1–3]. The nonlinear effects, determined mainly by the Kerr nonlinear silica fiber-media response, are widely accepted as culpable for the optical systems’ efficacy degradation at high signal powers [1,4]. To reduce the effect of nonlinearity backlash and improve the optical systems’ data-transmission capacity, many nonlinearity mitigation techniques have been proposed up to date [3,5]. Among the existing methods, the nonlinear Fourier transform (NFT) based optical signal processing and modulation techniques, and, in particular, the nonlinear frequency division multiplexing (NFDM) as the most efficient constituent of the latter group, have been intensively studied over the last decade [6–8].

Within the NFDM systems, the data modulation and transmission take place inside the special nonlinear Fourier (NF) domain, where the nonlinear intermodal cross-talk (due to the Kerr effect) between the effective “nonlinear modes” is virtually absent [7], provided that the signal’s evolution is well approximated by an intergable evolutionary equation. When no propagation effects other than the chromatic dispersion and Kerr nonlinearity are present, the signal’s propagation in NFDM systems is unaffected by the fiber nonlinearity capping the channel capacity of conventional systems at high powers. The idea of using the parameters of NF modes rather than those of “traditional” space-time waveforms for data modulation and transmission dates back to the now well-recognized work of Hasegawa and Nyu [9]. The more efficient NFDM variants based on continuous NF spectrum components and efficient modulation formats, which we consider here, were introduced in Refs. [6,10,11]: we notice that it is precisely the utilization of continuous NF spectrum that opened up the possibility for our reaching relatively high data rates in NFDM optical transmission, see, e.g., Ref. [12, Fig. 1] for direct spectral efficiency comparisons. But the systems based on solitonic (discrete NF) modes are also being actively studied [13–16], and more sophisticated techniques incorporating both discrete and continuous NF spectrum components have been developed as well [17–20], albeit we do not address these here.

 

Fig. 1. Normalized PSD for $b_1$ (a), $b_2$ (b), and cross-density (c), as the function of the total power factor $b$ for different values of inter-polarization power split ratio $\theta$.

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Initially, the proof-of-concept NFDM systems predominantly dealt with the scalar optical channel, i.e., with single-polarization signaling. Within such an approach, we operate with the one-component nonlinear Schrödinger equation describing the signal evolution down the fiber and utilize the respective NFT form [6,10,11]. But up to now, the various variants of NFDM have been successfully generalized to the dual-polarization (DP) case [12,13,19,21–23]. This generalization became possible because the narrowband light evolution down the single-mode fiber for the DP signal can be well approximated by the integrable version of the Manakov equation; see more details in the next Section. The integrable Manakov equation emerges if we disregard amplified spontaneous emission (ASE) noise, losses, and other higher-order perturbations. The integrability means that the NFT operations associated with the Manakov equation are known [24], so we can explicitly construct the respective DP-NFDM systems based on the DP-NFT signal processing.

It is worthwhile to notice that despite NFDM’s revealing some promising features in combating the nonlinearity-induced signal distortions, it has not so far delivered a truly superlative performance [12], compared to other state-of-the-art optical transmission systems [25]. For example, the latter Ref. reported the spectral efficiency results in the vicinity of 12 bits/s/Hz while Ref. [12] only reported estimates that are 3 times lower (per polarization). Among the NFDM difficulties related to the deviation of the actual transmission channel from the idealized integrable models [7,26], we underline the signal-noise interaction due to inline ASE noise that emerges due to the amplification process; some works name the ASE noise as the main NFDM performance degradation factor [5,27,28]. Yet the signal-noise interplay occurring inside the NF domain for NFDM systems has been studied only for particular single-polarization cases [27–31], and there is still a lack of analytical results regarding the noise inside the NF domain for the DP-NFDM, aside from some derivations for a less interesting single-soliton case [32–34]. But, naturally, the DP is the most practically important scenario insofar as we do not want to lose the second polarization’s modulation degrees of freedom. We emphasize that doubling the results for the single-polarization case is generally incorrect for the nonlinear signal processing methods to which the NFT belongs. At the same time, the analytical expressions for the channel law are also useful for, e.g., the estimates on the achievable information rates with a mismatched decoding [35–39], and, generally, can give us the vision on how to develop more efficient transmission systems [28,40]. Thus, our current work is dedicated to deriving the asymptotic continuous channel model for the DP-NFDM systems, which can help improve their efficiency and explain the physics behind their noise-related performance degradation.

Recently, a specific continuous NFDM-type system, the so-called $b$-modulation [41–44], has demonstrated the improved efficiency over previously proposed concepts [12,23,40,45]. In this scheme, we modulate the so-called Jost coefficient $b(\xi )$ [46,47], see Eqs. (6)–(10) below: in contrast to the earlier-proposed scattering coefficient modulation [10], the $b$-modulation allows us to generate the signals having compact support, and the latter helps explicitly control the signal duration and improve system’s efficiency. Therefore, in this paper, we specifically focus on the $b$-modulated DP-NFDM and provide the analytical derivation for the respective continuous (modulation format-independent) channel model in the leading approximation. However, our results can also be readily generalized to the scattering coefficient modulation. Our analytical findings are corroborated by the direct numerical evaluation of the effective NF noise correlation properties for an exemplary $b$-modulated DP-NFDM system. Our results can be seen as the continuation and further extension of single-polarization models from Refs. [28,31,40]. Nonetheless, our findings related to the DP channel are non-trivial and do not boil down to simply applying the single-polarization outcomes to the second polarization.

2. Model, DP-NFT operations, and some NFDM transmission systems details

2.1 Evolutionary equation

We start with writing down the generalized Manakov system, our main model, for the two-component vector electric field envelope $\mathbf {E}=(E_1(z,t),E_2(z,t))$ evolving along the fiber [48] (see also [12,13,21]):

$\mathbb {E}$ is the expectation value, $\delta _{ij}$ is the Kronecker symbol, $||\mathbf {E}||^2$ is the squared vector norm, $T$ is the time in the reference frame co-moving with the envelope, and $l$ is the propagation distance down the fiber. In Eq. (2), $N_{ASE}$ is the noise power spectral density (PSD) per polarization.

There are two different schemes where model Eqs. (1)–(2) are valid. The first one is the ideal distributed Raman amplification case, so the gain-loss term in Eq. (1) is effectively zero. In this case one has $N_{ASE}=\alpha \, L \,h\,\nu \, K_T$, where the factor $K_T \approx 1.13$ at room temperature, $L$ is the total propagation distance, $\nu$ is the carrier frequency, and $h$ is the Plank constant [28].

But this model also applies to the path-averaged approximation of a lumped (erbium-doped fiber amplifiers, EDFA) amplification scheme, where we should substitute the nonlinear coefficient with the path-averaged value $\gamma _{eff}$ [49]:

In our theoretical analysis below, we do not distinguish between the two aforementioned amplification schemes and assume that Eqs. (1)–(2) describe the true light evolution well enough. However, for the numerical verification of the results given in Section 4, we do consider different implementations.

2.2 Normalization and NFT operations for DP case

Using the standard normalization units (also called soliton units), i.e. adopting some characteristic temporal scale $T_s$ for the time normalization, and then using $L_s=2\,T_s^2/|\beta _2|$ for the propagation distance normalization, and $P_s=(8 \gamma T_s^2/(9 |\beta _2|)^{-1}$ for the power units, we can write the Manakov system down in the standard dimensionless form [13,21,24] for the normalized field variable $q_i=E_i/\sqrt {P_s}$, time $t=T/T_s$, and distance $z = l/L_s$:

Neglecting the noise in Eqs. (3), and, thus, dealing with the integrable Manakov system, we now introduce the Jost functions (we omit the $z$-dependence further for brevity), the main quantities in determining the nonlinear eigenmodes. The Jost functions for Manakov equations are the special solutions of the following linear scattering problem [24], written for auxiliary vector- or matrix-valued function $v(t)$:

The Jost functions are defined as four vectors/matrices satisfying the following boundary conditions [13,46] at the leading ($t \to \infty$) and trailing ($t \to -\infty$) ends of our sufficiently localized pulse $[q_1(t),q_2(t)]$:

Note that the Jost functions in the form of three 3-component different vectors are also used [51], but we found it more convenient to present our further derivations in vector-matrix form (5). The relation between these functions is as follows:

The fundamental $3 \times 3$ matrices $\Phi$ and $\Psi$ of system (4) are obtained by horizontal concatenation of the Jost functions:

Since the $M$-operator in (4) is skew-Hermitian, it follows that $\Phi ^{\dagger} \Phi$ and $\Psi ^{\dagger} \Psi$ are $t$-independent for real $\lambda$. Indeed since matrices $\Phi$ and $\Psi$ contain column Jost solutions of system (4) they satisfy the matrix form of this equation where the vector $v$ is substituted by the corresponding matrix. Then taking $\Phi$ as an example one immediately obtains:

Next, from boundary conditions (5), one can deduce that both $\Phi$ and $\Psi$ are also unitary for each $t$. From Eqs. (6), we can see that the two matrices are related: $\Phi = \Psi \,S$, where $S$ is the unimodular scattering matrix [46,51] that is also unitary for real $\lambda$. Its explicit form is:

This matrix depends on the spectral parameter $\lambda$ (and the propagation distance, $z$) but not on time, $t$.

The most general conventional NF spectrum consists of the two-entry row and column vectors of the left, $\rho$, or right, $\bar \rho$, reflection coefficients, appended with the set of discrete eigenvalues $\lambda _k$ with respective norming constants (spectral amplitudes) $C_k$ (also left and right). The full left, $\Sigma$, and right, $\bar \Sigma$, scattering data sets are written as:

However, in what follows we will not be dealing with the discrete (solitonic) NF components as these will be absent by the design in $b$-modulated waveforms, and so that the full NF spectrum is given by just either left or right reflection coefficient.

The following symmetries between the two sets of scattering coefficients exist (for real $\lambda$) [46]:

To reduce the number of degrees of freedom, we shall work with the upper set: $a$ and $b$, as in this case $a$ is just a scalar. For the forward NFT operation, we need to solve Eq. (4) associating the solution $v$ with the Jost vector $\phi$, assuming the initial condition on the signal’s $\mathrm {q}(t)$ trailing end (left boundary) $t=T_0$: $v(T_0)=(1, 0, 0)^T \,\exp (-i \lambda T_0)$. So we solve Eqs. (4) using the numerical scheme described in [21] and restore the Jost coefficients at $t=T_1$ via:

For the completeness of our exposition, we discuss the scalar limit for the DP-NFT. This is obtained by taking the limit $q_2 \to 0$. Then the evolution of the third component is decoupled so that $v_3'=i\lambda v_3$, and the solution is determined by the boundary conditions. In particular, it follows that $\phi _3=\bar {\phi }_{31}=\psi _{31}=\bar {\psi }_3=\psi _{12} = \psi _{32} = 0$. Then from Eq. (6) we obtain that $b_2=\bar {a}_{21}=0$, $\bar {a}_{22}=1$. From symmetry relations (9), it also follows that $\bar {a}_{12}=\bar {b_2}=0$.

The inverse NFT, performed at the NFDM system’s transmitter to generate the time-domain shape of our data-bearing signal, produces the initial DP waveform $\mathbf {q}(t,z=0)=[q_1(t,0),q_2(t,0)]$ from the modulated DP NF spectrum (see the schematics in Refs. [18,40]). Recall that in the case considered, the NF spectrum does not contain discrete solitonic components. Then the resulting signal is launched into the optical fiber. At the receiver, located at some desired distance $z=L$, we apply the direct NFT operation retrieving the modulated NF spectrum and removing the accumulated phase rotation by a single-tap channel equalization according to Eq. (10). However, for our exposition here, we do not need to analyze the explicit form of the inverse DP-NFT. For our numerical analysis, to generate the signal, we used the inverse NFT code from Ref. [21].

2.3 Some notes on the NFDM transmission scheme

First, we notice that the details of the NFDM that we consider here are not different from the standard schemes discussed previously, see, e.g., [40] for the analogous single-polarization system; thus, we give below a very brief description, as it is not crucial for our further analysis. In our study, we address the $b$-modulated NFDM communication as is the recent trend [12,13,21,23]. In this approach, the information is imparted on both components of the $b$-coefficient. However, since according to the third identity in (9) the absolute values of both components of the $b$-vector are upper-bounded by unity, the corresponding squeezing transformation is applied [23,40]. Specifically, the data are encoded onto two loading spectra (one for each polarization), as we typically have for the ordinary polarization division-multiplexing:

Next we apply the exponential mapping from the loading spectra to the $b$-coefficient [12,23]:

This is followed by pre-compensation: $b_j(\lambda,0) = b_{in,j}(\lambda )\,\exp (-4 i\,s\,\lambda ^2 L)$, where $L$ is the propagation distance (in normalized units), and $s \in [0,1]$ is the precompensation factor: $s=0$ corresponds to 100% post-compensation, $s=1$ – to 100% pre-compensation. The most popular choice in NFDM is $s=1/2$ corresponding to the halfway pre-compensation [52] that provides the densest data packing. Then, the spectrum is used for the inverse NFT operation to generate the initial data-bearing two-component waveform at the transmitter.

At the receiver, the operations are performed in reversed order. First, the post-compensation is performed: $b_{out,j}(\lambda ) = b_j(\lambda, L) \exp (-4 i \,(1-s)\,\lambda ^2 \,L)$; next, the loading spectra are restored by inverting Eq. (12). Finally, demodulation is performed for each polarization via the cross-correlation with the corresponding orthogonal wave-carriers $\psi _k(\lambda )$.

In what follows, we do not specialize any particular modulation format and present the general analytical expressions; the particular shape of $b_{1,2}(\lambda )$ is used only for the numerical verification of our results.

3. Perturbation theory

The exact theory of noise in the NFT domain is unavailable, and hence no general channel model exists neither for single nor dual polarization cases. However, when the signal-to-noise ratio is large, we can treat the ASE noise as a perturbation and use the perturbation theory methods to develop an approximate channel model. This was done successfully in a single polarization case for $r$-modulated [28] and $b$-modulated system [40]. Here, we aim to generalize these results for the case of two modulations.

3.1 General perturbative framework for the Manakov system

Perturbation theory for Manakov Eq. (3) was considered in several publications [32,53–55]. However, the notation of the above references was slightly cumbersome, and the authors there concentrated mostly on single soliton perturbation theory whereas our goal here is the perturbative evolution of the vector $b$-coefficient. Therefore, in order to make the presentation self-consistent, we develop here the perturbation theory for the $b$-coefficient along the lines of the similar treatment by Kaup [56] for single polarization.

We start by applying Eq. (4) to the Jost vector $\phi$ (see its definition in the previous section). Differentiating both sides of Eq. (4) w.r.t. $z$ and defining vector $u = \partial \phi /\partial z$, we obtain the following inhomogeneous system of equations:

The solution of (13) can be sought using the method of variation of parameters: $u = \frac {\partial \phi }{\partial z} = \Phi \varepsilon$, where $\varepsilon (t,z)$ is a 3 by 1 column vector to be determined. Substituting this into (13) we obtain:

We are interested in the $z$-evolution of the Jost scattering coefficients $a$ and $b$. Introducing a vertically concatenated $3 \times 1$ vector $c = [a, \, b]$, the first equation in (6) can be written in a compact form as: $\phi = \Psi \,c$. Differentiating this equation w.r.t. $z$ and taking the limit $t \to +\infty$, we obtain:

One can verify that in the scalar limit, the first two equations transform into the scalar perturbation equations given in Kaup [56]. The above results were first obtained by Midrio et al. [53] using a slightly different method. In the perturbative equations for the $b$-coefficient, we need to substitute the r.h.s. of the Manakov Eq. (1) for $\partial q_i/\partial z$ and their complex conjugates. The terms corresponding to the integrable (noiseless) equation must produce the results compatible with the known spatial evolution of the scattering data (10), i.e. $\partial b/\partial z = 4 \,i\,\lambda ^2 \,b$. The noise terms, therefore, provide the perturbative corrections for this evolution.

The final master model to be used in this paper is therefore:

An equation for $a$, which is needed when we consider the modulation of the scattering coefficient $\rho (\lambda )$ itself, can be derived similarly:

We do not use the latter equation in the current work but provide it for completeness. Equations (15) can be conveniently rewritten as:

Again, we have omitted the explicit dependencies of the Jost function components on $z$ and $t$ in the r.h.s. of the expressions above.

3.2 Asymptotic correlation properties of the ASE noise in the NF domain

Equations (15) can be integrated to produce a continuous input-output relation for the vector $b = (b_1, b_2)^T$:

The correlation properties of the effective NF noise $N_j(\lambda )$ can be readily derived from Eq. (18):

Unfortunately, expressions (24) are generally of little practical use, since the Jost functions for the unperturbed problem entering matrices $\mathcal {C}$, $\bar {\mathcal {C}}$, are rarely available in the analytical form. However, at large distances, one can use asymptotic expressions for these coefficients, as was done in [28,40]. The usual method relies on asymptotic expressions first derived by Zakharov and Manakov [57] and can be further elaborated via the steepest descent for Riemann–Hilbert factorization problem, see [40,58]. However, when one is interested only in the leading terms in the large distance asymptotic expansion of the covariance and pseudocovariance matrices, it is possible to use a simplified analysis first applied for the $r$-modulated scalar transmission, see Ref. [28, Supplementary Note III]. In this approach, we assume that the time domain pulse has a finite extent: $q_j(t) \equiv 0$ if $t<T_1$ and $t>T_2$, with arbitrary fields $q_j(t)$, containing no solitonic component in its NFT decomposition, inside the finite interval $[T_1,T_2]$ (as before we omit the $z$-dependence for brevity). Such an assumption complies with the “burst mode” modulation requirement that has to be used for the NFT-based transmission methods to avoid intersymbol interference, as we always process a finite interval. It is also true for the original version of unscaled $b$-modulation by design [41]. It is now convenient to split the integrals in the general expression for the (pseudo) densities $A_j$, $\bar A_j$, $B$, and $\bar B$, into tree subintervals: $t<T_1$, $T_1<t<T_2$, and $t>T_2$. The quantities referring to the first interval, i.e. to the left from the finite $q(t)$ extent, will be marked with the superscript “<”; the quantities for the interior interval, where $q_j(t) \neq 0$, will be indexed with the tilde accent “$\sim$” and the remaining interval to the right from the $q(t)$ extent is marked with “>”. Further, we will need the Fourier transform of the Heaviside unit step function, $H(t)=1$ for $t>0$ and 0 otherwise:

Using (5), we immediately obtain:

We also need opposite end asymptotes for both $\phi$ and $\psi$. These can be obtained via the unitary scattering matrix $S$, Eq. (6):

Next, using $\Psi = \Phi \, S^{\dagger}$, we obtain:

Now we can calculate the partial contributions to the densities. We start from the region to the left from the $q_j(t)$ extent, substituting Eqs. (26), (27) into expressions (19)–(22), we obtained:

A similar analysis applies to the right region, $t>T_2$:

Apart from these terms, there are the contributions from the interval $[T_1,T_2]$: we designate these as $\tilde {A}_j$, $\bar {\tilde {A}}_j$, $\tilde {B}$ and $\bar {\tilde {B}}$. The total values of the densities can be obtained by summing up the individual contributions from left, right, and center subintervals.

The crucial observation first made in Ref. [28], is that the covariance and pseudocovariance matrices (24) contain spatial integral of highly oscillating factors. Therefore, at large distances $z$, the leading order contribution to the covariance matrix comes from the values $\lambda \approx \lambda '$. Hence, the contributions from the intermediate region $[T_1,T_2]$ as well as the principle value terms of the densities only contribute $O(1)$ terms, and the main contribution comes only from the Dirac-delta terms in the boundary regions. For the pseudocovariance, as we have shown above, $\mathcal {C}^\lessgtr =0$, and the contribution of the intermediate region is again of the order $O(1)$.

Therefore, keeping only the leading-order diagonal Dirac-delta terms, we obtain:

Finally, from the unitarity of the $S$ matrix (7) it follows that $|b_i|^2+|\bar {a}_{i1}|^2 +|\bar {a}_{i2}|=1$, $\bar {a}_{11}\,\bar {a}^*_{21}+\bar {a}_{12}\,\bar {a}^*_{22} + b_1\,b_2^*=0$ and $|a|^2+|b_1|^2+b_2|^2 =1$. This allows us to arrive at the following remarkably simple answer:

The above is the main result of our work. When only one polarization is used, $b_2=0$, one recovers the results for the scalar case given by the upper left corners of the covariance and pseudocovariance matrices:

On the other hand, interestingly, the noise in an initially “empty” polarization is squeezed: $B_{22}=1-|b_1(\lambda )|^2/2<1$. The maximum squeezing amount is 50%.

In the general case, we can see that the contribution of the cross-polarization to the noise PSD reduction is half of the self-term. Also, due to the constraint $|b_1|^2+|b_2|^2 \leq 1$, the normalized PSDs $B_{11}$ and $B_{22}$ are both positive as required.

4. Numerical verification

In order to perform the numerical verification of the above result, we used two popular transmission schemes corresponding to ideal distributed Raman amplification and lumped EDFA [1] with the parameters given in Table 1: the fiber parameters there are typical for the standard single mode fiber in the C-band regime [7].

Table 1. The parameters of the single-mode fiber link used for the numerical verification of our results.

View Table

For the signal propagation in the fiber we employed a standard split-step Fourier method [59] with a fixed step size of 0.2 km; for the direct and reciprocal NFT code we used the algorithms of Ref. [21].

4.1 Rectangular spectra

For simplicity, in the first set of simulations we consider the case where the initial components of the $b$ vector are the rectangular spectral shapes of constant real-valued amplitudes $b_1$ and $b_2$, localized in the interval $\xi \in [-2,2]$, which roughly corresponds to the two-sided linear bandwidth $W \approx 4/(\pi T_s) =12$ GHz. Since we must have $|b_1|^2 + |b_2|^2 \leq 1$, it makes sense to use the trigonometric parametrization: $b_1 = b\cos \theta$, $b_2 = b\sin \theta$, where $0 \leq b \leq 1$ is the measure of the total power in two polarizations, and $0 \leq \theta \leq \pi /2$ characterizes the power splitting ratio between them.

We have performed a series of Monte-Carlo runs for the distributed Raman case, and the results are depicted in Fig. 1, where we plot the computed normalized PSDs of the individual components $B_{11}$ and $B_{12}$ as well as the cross-density $\mathrm {Re}(B_{12})$, see Eq. (32), vs. the total power factor $b$ for different power split ratios parameterized by $\theta$. One can readily observe a very good agreement between theory and numerics for the self-PSDs, Fig. 1(a), (b), and slight discrepancy for the cross-correlation, Fig. 1(c). The latter is more pronounced at large input powers, and we attribute it to two factors. First, the absolute value of the leading order contribution to $B_{12}$ is relatively low, and, thus, the next orders of the $B_{12}$-expansion (that we did not account for in the final analytical formula) contribute a visually distinguishable amount. Then, we also have the contribution from the processing noise [31,40], i.e. from the final accuracy of the numerical NFT routines used in the simulations. In the next subsection, we provide a brief description of how one can “subtract” the processing noise contribution and extract the ASE-related PSD.

Observe also that in the case of a single polarization, $\theta =0$, the PSD of the noise in the second polarization is reduced up to the maximum level of half the linear density, Fig. 1(b), as discussed at the end of the previous section and in full correspondence with our analytical predictions.

4.2 NFDM spectra

Next, we analyze the performance of our theoretical model in more realistic NFDM settings. The basic setup of b-modulated NFDM systems for both single a dual polarization was described in Section 2.3 and can be also found in multiple references (see e.g. [12,18,23,40]).

In this more applicable scenario, we note that we ought to extract the ‘’processing noise” from the overall result to obtain the correspondence with the analytics. The procedure was first suggested in [31]. Briefly, the extraction procedure works as follows: we assume that the total signal distortion at the receiver is the sum of ASE and the processing noise that are assumed to be statistically independent zero mean processes. This means that their variances (which are properly normalized nonlinear spectral densities and pseudo-densities studied here) should add up. Therefore to extract the ASE-only contribution to the signal which is the subject of the present theory we have performed a separate set of simulations where at each power level the inline ASE noise was artificially switched off in the propagation model. In this simulation, we randomly generated different sequences of input symbols (see below) by keeping the overall power scaling factor fixed. The resulting densities obtained from this noise-free run were then subtracted from the results of full simulation (at the same level of input power) to provide ASE only contribution.

We used 128 subcarriers with 32-QAM pseudorandom constellation points and a sinc carrier shape for both polarizations: $\psi _n= \mathrm {sinc}(\lambda -n)$. We also used half-way pre-compensation, $s=1/2$. The scaling of the NFT parameters corresponded to lumped EDFA amplification case as discussed in Section 2.1 with the parameters from Table 1. The corresponding results are shown in Fig. 2.

 

Fig. 2. Normalized PSD for: $b_1$ (a), (d); $b_2$ (b), (e), and the real part of the cross-density (c), (f), as the function of the total power of NFDM burst. Top row: full simulation, bottom row: after the subtraction of processing noise.

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One can see that since both $b_1$ and $b_2$ have the same statistical distribution of the input symbols, their average PSDs are identical (we checked that the relative error does not exceed $5 \times 10^{-3}$). Next, one can clearly see from the top row figures that the noise densities and cross densities begin to grow with power at around $-5$ dBm, contrary to the prediction of the theory. This is exactly the manifestation of the processing noise not related to the ASE but rather to the numerical accuracy of the NFT-related DSP, burst tails truncation, insufficient sampling rate, etc. [31,40]. When we extract the processing noise, see the lower row of panels in Fig. 2, the agreement with the theory becomes almost perfect for the PSDs, while the cross-density if almost negligible in all the cases (note that the scale of the panes (c) and (f) vertical axis is reduced compared to other figures).

5. Conclusion

In our paper, we derived a channel model for NFDM b-modulated system in the presence of the inline ASE noise, exploring the previously unattended case of double polarization modulation. The aim of our study was to derive analytically and verify the respective channel model numerically, i.e., to study the effective noise emerging in the NF domain due to the progenitor ASE noise. In fact, our work extends the previously-developed perturbative approach for the derivation of noise statistics onto the DP case. We note that our analytical results refer to the leading order of continuous input-output signal relation but are universal and applicable to any DP $b$-modulation type; the expressions for the reflection coefficient modulations, requiring our considering the $a$-coefficient, can also be obtained using the first equation from the set (14), but the b-modulation concept seems to be more appealing in terms of achievable efficiency.

We want to emphasize the non-trivial findings of our work here. In particular, we demonstrated that the contribution of the second polarization to the diagonal elements of the covariance matrix in the DP case amounts to just half of the signal’s own polarization contribution. It means, in particular, that if we set the coefficient $b$ in one polarization close to 1 and to 0 in the second polarization, it would bring about the appearance of the non-zero noise in the second polarization, while the noise effect in the first polarization will dwindle. Our results were verified through direct Monte-Carlo simulations: it was shown that the components of the covariance matrix for the effective input-dependent noise in the NF domain are in very good agreement with direct numerical results. The deviations of the theory and the numerical data for the off-diagonal covariance matrix elements are attributed to the contributions of the high-order expansion terms and the effects of processing noise; when we extract the latter, the correspondence becomes almost perfect.

This paper follows our previous works [28,31,40] devoted to the channel model derivation for the NFDM systems that use the continuous NF spectrum modulation. However, we note that the case where the solitary modes (the discrete NF spectrum) coexist with the continuous NF spectrum components, as in Refs. [18–20], still requires a separate study: the noise-perturbed evolutionary equations for the NF spectral quantities, given in Ref. [60], have to be used together with the respective Jost functions that account for the coexisting NF spectra, and the analytical expressions for the asymptotics of the latter functions seem to be unavailable at the moment. Finally, specifically for the DP case, the master Manakov model (1) represents only an approximation and does not take into account the effects of polarization mode dispersion (PMD) or polarization-dependent loss. The implications of these effects on DP-NFDM transmission are still poorly studied but we mention that a recent publication [61] does consider this subject and offers an effective scheme for channel equalization in this case.