Dual polarization nonlinear Fourier transform-based optical communication system
Simone Gaiarin Auro Michele Perego Edson Porto da Silva Francesco Da Ros Darko Zibar

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ADCanalog-to-digital converter
AIRachievable information rate
AKNSAblowitz-Kaup-Newell-Segur
AWGarbitrary waveform generator
AWGNadditive white Gaussian noise
B2Bback-to-back
BERbit error rate
BPDbalanced photodetector
BPSblind phase search
DACdigital-to-analog converter
DBPdigital back-propagation
DCdirect current
DFTdiscrete Fourier transform
DP-NFDMdual polarization nonlinear frequency division multiplexing
DSOdigital storage oscilloscope
DSPdigital signal processing
DTDarboux transformation
ENOBeffective number of bits
EDFAerbium-doped fiber amplifier
FECforward error correction
FFTfast Fourier transform
FLfiber laser
FWHMfull-width half-maximum
FWMfour-wave mixing
GLMGelfand-Levitan-Marchenko
HD-FEChard-decision forward error correction
Iin-phase
IFFTinverse fast Fourier transform
INFTinverse nonlinear Fourier transform
ISIinter-symbol interference
ISTinverse scattering transform
IVPinitial value problem
LOlocal oscillator
LPAlossless path-averaged
MSManakov system
MAPmaximum a posteriori
MLmaximum likelihood
MZMMach-Zehnder modulator
MZSPManakov-Zakharov-Shabat spectral problem
NFDMnonlinear frequency division multiplexing
NFTnonlinear Fourier transform
NISnonlinear inverse synthesis
NLSEnonlinear Schrödinger equation
NMSEnormalized mean squared error
OBPFoptical band pass filter
ODEordinary differential equation
OFDMorthogonal frequency-division multiplexing
OSNRoptical signal-to-noise ratio
PAPRpeak-to-average power ratio
PCpolarization controller
PDEpartial differential equation
PMDpolarization mode dispersion
PSKphase shift keying
Qquadrature
QAMquadrature amplitude modulation
QPSKquadrature phase shift keying
SMFsingle mode fiber
SNRsignal-to-noise ratio
SPMself-phase modulation
SSFMsplit step Fourier method
XPMcross-phase modulation
ZSPZakharov-Shabat spectral problem













                         
                         
            
                           
                           
              

                      
               
                


                 
                       


                                    
                                
                          
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    1,*]S. Gaiarin2,*]A. M. Perego1]E. P. da Silva1]F. Da Ros1]D. Zibar[1]DTU Fotonik, Technical University of Denmark, Lyngby, 2800 Denmark[2]Aston Institute of Photonic Technologies, Aston University, Aston Express Way, Birmingham, B4 7ET UK[*]Corresponding authors: simga@fotonik.dtu.dk, peregoa@aston.ac.ukCompiled December 30, 2023
    
    New services and applications are causing an exponential increase in internet traffic.  In a few years, current fiber optic communication system infrastructure will not be able to meet this demand because fiber nonlinearity dramatically limits the information transmission rate. Eigenvalue communication could potentially overcome these limitations. It relies on a mathematical technique called "nonlinear Fourier transform (NFT)" to exploit the "hidden" linearity of the nonlinear Schrödinger equation as the master model for signal propagation in an optical fiber. We present here the theoretical tools describing the NFT for the Manakov system and report on experimental transmission results for dual polarization in fiber optic eigenvalue communications. A transmission of up to 373.5 km with bit error rate less than the hard-decision forward error correction threshold has been achieved. Our results demonstrate that dual-polarization NFT can work in practice and enable an increased spectral efficiency in NFT-based communication systems, which are currently based on single polarization channels.
    
    displaycopyrighttrueDual polarization nonlinear Fourier transform-based optical communication system
        [
        December 30, 2023
    ================================================================================
    
    fancyshortarticle
    
    § INTRODUCTION
    
    Fiber optics telecommunication is the currently established  backbone infrastructure for most of the information flow across the world <cit.>. However, the demand for an always increasing transmission rate, which for the existing channels is necessarily associated with an increment of the launched signal power  to minimize the OSNR degradation, has been predicted to be asymptotically limited by the distortion induced by the optical fiber nonlinearity <cit.>. 
    It is a well-known fact that light propagation in fiber optics is governed by the NLSE <cit.> where the nonlinearity arises due to the Kerr effect.
    Nonlinearity is a problem for transmitting information with the currently used modulation formats in fiber optics communications.
    Indeed, as the power is increased, the signal is more distorted by the nonlinear cross-talk, thus limiting the capability of the receiver in recovering the transmitted information. It is therefore necessary to mitigate the nonlinear effects to compensate for the distortions and to provide novel approaches for the communication over the nonlinear fiber-optic channel.
    Two main paths have been followed up to now to counteract this problem: the first approach consists in mitigating the nonlinear effects through a wealth of techniques such as for instance optical phase-conjugation <cit.> or digital back-propagation <cit.>;
    the second path, more ambitiously, aims at encoding information into the eigenmodes of the nonlinear channel, whose evolution is linear upon spatial propagation.
    This second approach, originally called Eigenvalue communication, has been proposed by Hasegawa and Nyu <cit.> and it is now, with various modifications,  growing as a new paradigm in optical communications <cit.>.
    
    This method exploits the exact integrability of the NLSE through the IST <cit.> as the master evolution equation of the electric field propagating in SMF.
    Integrability of the NLSE was demonstrated by Zakharov and Shabat back in 1972 <cit.>, who found an associated spectral problem, related to a set of ordinary linear differential equations.
    Following this approach, it is possible to identify the eigenvalues, that can be considered the analogous of the frequencies in the classical Fourier transform; and the so called scattering coefficients: complex amplitudes associated to the eigenvalues.
    The application of the IST to fiber optics communications allows the use of various and flexible modulation formats <cit.>.
    Due to integrability, in the lossless and noiseless limit, nonlinearity is not a detrimental factor anymore, but on the contrary, it is a constitutive element of the transmission system itself.
    The parallelism between the linear Fourier transform method used to solve linear initial value problems and the IST used to solve nonlinear ones <cit.>, has driven some authors to rename the IST as NFT<cit.>, which is the name currently used in the engineering communities (see <cit.> for a recent review including historical details).
    The nonlinear Fourier spectrum of a signal consists of a set of eigenvalues and the respective associated scattering coefficients.
    The eigenvalues belong either to a so-called discrete spectrum or to a continuous spectrum; the first describes the solitonic components of the signal, while the second is associated with dispersive waves and reduces to the classical Fourier spectrum in the limit of low power.
    
    Communications channels based on both discrete or/and continuous spectrum modulations have been extensively studied and experimentally demonstrated up to now for the scalar (single polarization) NLSE (see e.g. <cit.> to cite just a few).
    
    A series of key challenges that need to be met in order to bring NFT-based communication to exit the labs and operate in real-world infrastructures, has been described recently <cit.>. One of those challenges consists indeed of endowing the Eigenvalue communication approach with polarization division multiplexing, which allows information to be encoded on both orthogonal polarization components supported by SMF.
    The description of the light propagation, accounting for its polarization dynamics can, under specific conditions that apply to modern communications fiber link, be described by the Manakov equations <cit.>.
    In a milestone paper of nonlinear science, Manakov showed that those equations can be solved analytically by the IST<cit.>. Detailed investigations of the solutions of the Manakov equations especially concerning soliton and multisoliton dynamics in presence of noise and PMD in optical communications, as well as their connection with optical rogue waves formation, are present in the literature <cit.>.
    
    
    To the best of our knowledge the NFT dual polarization problem has never been tackled at the level to demonstrate a working communication system
    and only very preliminary theoretical works are present in the literature on this topic <cit.>.
    
    In this article, we present the mathematical framework underlying the dual polarization NFT and we show an extension of our recent results on the first experimental demonstration of a  DP-NFDM fiber optics communication system <cit.>. We have transmitted up to 373.5 km at the HD-FECBER threshold of 3.8×10^-3, with information encoded in the QPSK modulated scattering coefficients associated with two eigenvalues belonging to the Manakov system discrete spectrum, for both orthogonal polarization components supported by a SMF.
    
    The structure of the paper is the following: in section 2 we will first define the NFT for the dual polarization case and describe the mathematical tools -the DT- needed to generate the waveforms associated with a desired nonlinear spectrum for both field polarizations. In section 3 we will discuss the details of a DP-NFDM system. Finally, in section 4 we present a detailed account of the experimental transmission results, followed by a discussion of the results and conclusions in section 5.
    
    
    
    
    § MATHEMATICAL FRAMEWORK
    
    
     §.§ Channel model
    
    
    The evolution of the slowly varying complex-valued envelopes of the electric field propagating in a SMF exhibiting random birefringence and whose dispersion and nonlinear lengths are much larger than the birefringence correlation length, is described by the averaged Manakov equations <cit.>
    
        {[1]   =- i2[1]+i8/9(|[1]|^2+|[2]|^2)[1]
        [2]   =- i2[2]+i8/9(|[1]|^2+|[2]|^2)[2]
          .
    
    where  and  represent the time and space coordinates,  are the amplitudes of the two electric field polarizations,  is the dispersion coefficient and  is the nonlinearity coefficient.
    
    In order to remove any dependency from a specific channel, it is common to work with the normalized version of (<ref>).
    The normalized MS <cit.> is obtained by performing the change of variable
    
    
        [j] = [j]√(P),  = T_0,  = -ℒ
    
    with P = ||/(8/9 T_0^2), ℒ = 2 T_0^2 /|| and T_0 is a free normalization
    parameter, leading to
    
    
    
        {
            i[1]z   = [1]t+2(|[1]|^2+|[1]|^2)[1]
        
            i[2]z   = [2]t+2(|[1]|^2+|[2]|^2)[2]
          .
    
    where z and t represent the normalized space and time variables respectively. In this study we have considered the anomalous dispersion regime (<0), since it is the one that supports solitons and corresponds to the regime of currently deployed SMF.
    
    In realistic systems the field amplitude is attenuated upon spatial propagation at a rate α/2, where α is the attenuation coefficient of the fiber. This breaks the integrability of (<ref>). However it is possible to suitably redefine the fields E_1,2→ E_1,2e^-(α/2) in such a way that they obey a lossless equation with an effective nonlinearity coefficient 
        γ_eff=γ(1-e^-α L)/(α L)
    
    where L is the length of one optical fiber span.
    The evolution equation with the modified nonlinear term can be considered the leading approximation of the lossy system when we account for the periodic signal boosts due to the EDFA.
    This is the so-called LPA model <cit.>, which is in general valid when the amplifiers spacing is smaller than the soliton period, and it has been used across all the present study.
    
    
    
     §.§ Direct NFT
    
    In order to
    compute the NFT of a signal [1,2](t) it is first necessary to associate to the
    MS(<ref>) a so-called spectral problem. This is known for the case of the NLSE as the ZSP, while for the Manakov equations we can call it the MZSP. The MZSP is defined by the following system of linear ordinary differential equations
    
    
        t=(λ𝐀+𝐁)
    
    being
    
    
        𝐀 = [ -i  0  0;  0  i  0;  0  0  i ] 𝐁 = [      0    q_1    q_2; -q_1^*      0      0; -q_2^*      0      0 ]
    
    where  is a solution and  is a spectral parameter.
    
    Assuming the vanishing boundary conditions for the signal, i.e., |[1,2](t)| 0 for t |∞|, it is
    possible to find a set of canonical solutions to (<ref>) called Jost solutions defined as <cit.>:
    
    
    
        →[ 1; 0; 0 ]e^-iλ t;  →[ 0 0; 1 0; 0 1 ]e^iλ t   t→ -∞
    
      
        →[ 0 0; 1 0; 0 1 ]e^iλ t;  →[ 1; 0; 0 ]e^-iλ t   t→ +∞ .
    {, }
    and {, } are two bases for the eigenspace associate to λ. One can
    write  and  as a linear combination of the basis vectors {, } as
    
    
    
        =+ 
           =+
    
    with coefficients a(λ), b(λ), a̅(λ) and b̅(λ), where a(λ) is a scalar,
    a̅(λ) is a 2×2 matrix, b(λ) is a two entries column vector and b̅(λ) is a two entries row
    vector.
    These coefficients are called .
    From the knowledge of the scattering coefficients, it is possible to reconstruct the signal [1,2](t) uniquely.
    
    
    Analogously to the case of the NLSE <cit.> we can define the NFT continuous and discrete spectral amplitudes for the MS as:
    
        =^-1      ∈
        [i]   =[][i][i]^-1   [1,... n]   ∈∖
    
    and [i]=da()/d|_=[i]∀[1,....n]∈∖ such
    that [i]=0.
    Although these spectral amplitudes are commonly used, it is more convenient to work
    directly with the  and  <cit.>; hence, when throughout the whole manuscript we will refer to the nonlinear spectrum, we will implicitly mean the eigenvalues and the associated scattering coefficients.
    The scattering coefficients are time independent and their spatial evolution is given by <cit.>:
    
        a(,z)   =a(, 0)    a̅(,z)   =a̅(,0) 
        
        b(,z)   =b(,0)e^-4i^2z   b̅(,z)   =b̅(,0)e^4i^2z.
    
    In order to not overburden the notation we will drop the explicit space dependence as we did in the beginning of this section.
    The fact that the  are time invariant, allows computing them at an arbitrary instant of time. For example, using (<ref>) and the boundary Jost solutions, they can be computed at t = +∞. At this instant  is known. Moreover, it is possible to propagate  from t = -∞, where it is known, to t = +∞ by integrating (<ref>). Given the particular structure of the Jost solutions it results that the  are given by:
    
    
    
        =lim_t → +∞[[1][1]^-1]
        [1]   =lim_t → +∞[[2] [2,1]^-1]
        [2]   =lim_t → +∞[[3] [3,2]^-1]
    
    and using (<ref>) and (<ref>) gives
    
        =lim_t → +∞[[1]e^iλ t]
           =[ [1]; [2] ] =
              lim_t → +∞[[ [2]; [3] ]e^-iλ t].
    
    It should be noted that, compared to the NLSE case, there is an additional scattering coefficient [2] that can be used to encode information, potentially doubling the system transmission rate.
    
    
    
     §.§ Inverse NFT
    
    
    The INFT is the mathematical procedure that allows constructing a time domain  waveform starting from a given nonlinear spectrum. In our work, we have performed the INFT at the transmitter by using an algorithm based on the DT <cit.>.
    The DT is a natural candidate to build time domain signals, especially when the information is encoded only in the discrete nonlinear spectrum. The method consists in adding iteratively discrete eigenvalues to the nonlinear spectrum while simultaneously updating the signal in time domain. The INFT based on the DT for eigenvalue communications was proposed in <cit.>.
    In our work we have used the DT for the MS derived
    by Wright <cit.>. We summarize here how the DT for the MS works.
    
    
    
    
    Let  be a column vector solution of the MZSP  spectral problem (<ref>) associated with the MS  for the
    signal () and the eigenvalue , then according to <cit.> a new solution of (<ref>), , is given by the following equation:
     
        =(λℐ_3-G_0)
    
    where ℐ_3 is the 3×3 identity matrix, G_0=M_0^-1 with
    
        = [    _1  _2^*  _3^*;    _2 -_1^*     0;    _3     0 -_1^* ]
    
    where the matrix M_0=diag(λ_0, λ_0^*, λ_0^*) and =(_1, _2, _3)^T is a solution of (<ref>) for the seed signal q_j, j=1,2 and a fixed value of = [0]. The DT gives the new signal waveforms in time domain for both polarizations  as a function of the old signals q_j, of the auxiliary solution  and of the new eigenvalue λ_0 we want to add to the nonlinear spectrum:
    
        q̂_j=q_j+2i(λ_0^*-λ_0)u_j^*/1+∑_s=1^2|u_s|^2  (j=1,2)
    
    where u_j=_j+1/_1.
    
    Starting from the "vacuum" solution [j](t)=0, j=1,2, the procedure sketched in  <ref> can be repeated iteratively to generate the dual polarization time domain  signal associated with a nonlinear spectrum containing an arbitrary large number of discrete eigenvalues.
    
    The generic auxiliary solution (k) that satisfies the MZSP for the eigenvalue λ_k reads: (k)=(A(k)e^-iλ_k t, B(k)e^iλ_k t, C(k)e^iλ_k t)^T (being {A(k), B(k), C(k)} some initialization constants). Hence after adding i eigenvalues the auxiliary solutions are modified according to the following chain of matrix multiplications
    
        [ [-3pt]_1^(k); [-3pt]_2^(k); [-3pt]_3^(k) ]=
             (λ_kℐ_3-G_0i-1)...(λ_kℐ_3-G_01)
             [ A^(k)e^-iλ_k t;  B^(k)e^iλ_k t;  C^(k)e^iλ_k t ]
    
    where the G_0i matrices are evaluated as functions of the i-th auxiliary
    solution ^(i) evaluated  after i-1 Darboux transformations, see also the scheme depicted in
     <ref>.
    The initialization constants {A^(k), B^(k), C^(k)}
    have been set respectively equal to {1,-b_1(λ_k),-b_2(λ_k)},  in order to obtain the correct spectrum after performing NFT and INFT in sequence as in the scalar case <cit.>.
    b_1(λ_k) and b_2(λ_k) are the  that we want to associate to the eigenvalue λ_k for the two polarization components respectively.
    
    
    
    
    § DP-NFDM SYSTEM
    
    In this section, the basic structure of a DP-NFDM system using the discrete spectrum will be described. The DSP chain will be introduced first and then the experimental setup.
    
    
    
     §.§ Transmitter and receiver digital signal processing
    
    
    
    
    At the transmitter the data bits are mapped to the  pairs {[1][i], [2][i]} for i=1,2 where the pair of eigenvalues {[1] = i0.3, [2] = i0.6} is used for each symbol. We will refer to these set of coefficients and equivalently to the associated time domain waveform as a DP-NFDM symbol. The  associated with the first eigenvalue can assume values drawn from a QPSK constellation of radius 5 and rotated by π/4 while those associated with the second eigenvalue are drawn from a QPSK constellation of radius 0.14 as shown in  <ref>. This particular structure of the constellations was chosen to reduce the PAPR of the signal at the transmitter, in order to limit the performance losses due to the limited resolution of the DAC and due to the nonlinear characteristic of MZM and electrical amplifiers (See Supplementary Material for a detailed explanation).
    The waveform associated to each DP-NFDM symbol is generated using the DT described in the previous section followed by the denormalization as in (<ref>)  with normalization parameter
    T_0 =47 ps. This choice of T_0 allows fitting the waveform in a time window of 1 ns (1 GBd) with enough time guard band among successive DP-NFDM symbols to satisfy the vanishing boundary conditions required to correctly compute the NFT. The power P_tx of the digital signal thus obtained will later be used to set the power of the corresponding transmitted optical signal.
    
    The channel is assumed to be a link of standard SMF with EDFA
    lumped amplification as in the experiment. In order to take into account the presence of the
    losses, the LPA approximation is used in the normalization and
    denormalization steps of the waveform before computing the NFT and after
    computing the INFT respectively.
    
    
    
    
    At the receiver, the digital signal output by the DSO is first rescaled so that its power is P_tx (the power of the transmitted optical signal). Then an ideal rectangular filter with bandwidth equal to the 99% power bandwidth of the signal is used to filter the out of band noise.
    At this point, cross-correlation-based
    frame synchronization using training sequences is performed in order
    to optimally align the DP-NFDM symbol to the processing window.
    For each DP-NFDM symbol, first the eigenvalues are
    located using the Newton-Raphson search method employing the one-directional trapezoidal method and then the coefficients [1,2][i]
    are computed on the found eigenvalues using the forward-backward trapezoidal  method (see Supplementary Material for more details). The homodyne configuration of the receiver allows not having a frequency offset between the transmitter laser and the coherent receiver LO, but given the non-zero combined linewidth of the two lasers (~ 1kHz) their coherence length is limited to about 90 km. This implies that the received constellations are affected by phase noise when the transmission distance exceeds the coherence length of the laser, causing errors in the detection of the symbols. The phase noise is removed by applying the blind phase search algorithm <cit.> in the NFT domain to each constellation individually.
    Finally, the  are rotated back to remove the phase factor acquired during the transmission ((<ref>)) and the decision on the symbols is taken using a minimum Euclidean distance decisor over the .
    
    
    
    
    
     §.§ Experimental setup
    
    
    The experimental setup and the block diagrams of the DSP are depicted  in  <ref>.
    At the
    transmitter a FL with sub-kHz
    linewidth is modulated using an integrated dual polarization I/Q
    modulator driven by an AWG with 20 GHz analog
    bandwidth and 64 GSa/s. Before uploading it to the AWG, the signal
    generated by the INFT is pre-distorted using the ideal inverse
    transfer function of the MZM (asin(·)). This pre-distortion is
    required in order to have a good trade-off between SNR at the output of
    the MZM and signal distortions caused by its nonlinear transfer function. Nonetheless, given the still high PAPR of the optimized waveform considered (see Supplementary Material), this
    pre-distortion is not optimal and advanced methods can be employed to
    improve further the quality of the transmitted signal <cit.>.
    The channel is a fiber link composed of
    up to 9 spans of SMF fiber with dispersion , nonlinear coefficient , attenuation  and PMD coefficient < 0.1 ps km^-1/2.
    Two different span lengths of L=41.5 and L=83 km were employed.
    Considering these channel parameters, the complex baseband signal generated by the INFT with LPA and denormalized has the following properties: 99% of its power contained within a bandwidth W of 12.7 GHz, a PAPR of 9.49 dB and an average power P_tx of 5.30 dBm and 7.70 dBm for the span lengths L of 41.5 and 83 km, respectively. Given these channel and signal parameters we have that the soliton period, defined as (π/2) L_d, with L_d = (W^||)^-1 the dispersion length <cit.>, is 436 km. Being this  much larger than the typical birefringence correlation length, which is on a scale of few tens of meters <cit.>, guarantees the applicability of the Manakov averaged model.
    
    In order to properly match the transmitted signal to the channel, the gain of the EDFA at the transmitter is tuned in such a way to set the power of the optical signal to P_tx.
    The optical signal is then transmitted through the channel.
    
    At the receiver, the signal is first sent through a 0.9 nm OBPF and then a PC was used to manually align
    the polarization of the signal to the optical front-end. The use of the PC was required to avoid the use of polarization tracking algorithms for the NFT signals, which were not available at the time of the experiment. In the future it could be possible to use modulation independent polarization tracking algorithms, as an example using independent components analysis <cit.>. The signal is then detected by using a standard coherent receiver (33 GHz analog bandwidth, 80 GSa/s), in a homodyne configuration where the transmitter laser is used as LO. The acquired digital signal consisting of 5 blocks of 10^5 DP-NFDM symbols is then fed to the receiver DSP chain described previously.
    
    
    
    
    § EXPERIMENTAL RESULTS
    
    
    
    The system was initially tested in a B2B configuration, where the transmitter output has been directly connected to the receiver, in order to obtain the best performance achievable by the system in the sole presence of the intrinsic transceiver distortions (e.g. transmitter front-end distortions, detectors noise, etc.) and added AWGN as commonly done for linear coherent systems. The OSNR was swept by varying the noise power added to the signal at the receiver input. The adopted metric for measuring the performances allows a direct comparison with standard coherent transmission systems. The OSNR range considered is the region of interest where the system performance is around the HD-FEC threshold.  The measured average BER
    is shown in  <ref>.
    A visible effect is the fact that the BER is not the same for the four different
    constellations, but it is worse for the two constellations associated with
    the eigenvalue with a higher imaginary part.
    This effect can be related to the dependency of the noise variance of both the eigenvalues and the corresponding scattering coefficients on the imaginary part of the eigenvalues themselves <cit.>; an analysis of the noise distribution for the various eigenvalues and scattering coefficients is provided in the Supplementary Material.
    
    In order to demonstrate fiber transmission with the proposed system, a transmission was
    performed over a link of N spans of SMF  with span length L=41.5 and L=83 km. The
    performance in terms of BER as a function of the transmission distance
    is shown in  <ref> (a-b) for the four different constellations. The difference
    in performance in the two eigenvalues appears in this case too. This can also be seen from the constellation
    plots after 373.5 km in  <ref>, where the two constellations associated with [2] are
    sensibly more degraded than those related to [1], which are still well
    defined. Similar performance can instead be seen in the two different
    polarizations of the same eigenvalue.
    
    
    
    
    Finally in  <ref> (c) we compare the average BER for the two span lengths
    used in the test, in order to check the impact of the LPA approximation.
    Worse performances are expected when longer links are used being γ_eff given in (<ref>) farther from  the real γ of the fiber in this case.
    The BER curve for the 41.5 km span contains two outlier points at 124.5 km and 249 km
    that are slightly worse than the general trend of the curve.
    This is believed to be caused by instabilities in the setup when the related experimental traces were acquired, in particular an incorrect alignment of the polarization to the receiver due to a drift in the polarization state of the received signal. Besides these two points, the rest of the 41.5 km curve lies
    under the one for the 83 km spans, confirming that the use of longer
    spans adds a slight degradation in the performance of the system. The
    maximum reach of the system achieved with BER under the HD-FEC threshold
    is 373.5 km using 41.5 km spans and 249 km with spans of 83 km.
    It should be noted that in our experimental setup the PMD effect was not compensated for.
    However, for the transmission lengths and PMD values of the standard SMF employed, the accumulated differential group delay is negligible if compared with the pulse duration <cit.>. The impact of PMD is therefore not expected to have had a major impact on the results shown. New approaches have been developed to compensate for PMD  effects in linear transmission systems <cit.> and a recent work has shown in simulations that for a  DP-NFDM system employing the continuous spectrum, PMD effects could be compensated in the nonlinear domain by using a linear equalizer <cit.>. Similar techniques may be applied to discrete DP-NFDM systems.
    
    
    
    
    § CONCLUSIONS
    
    
    We have demonstrated experimentally, for the first time to the best of our knowledge, an eigenvalue-based optical communication system employing two orthogonal modes of polarization. We encoded 8 bits/DP-NFDM symbol and demonstrated transmission up to 373.5 km. Furthermore, we have shown that a powerful, but rather abstract mathematical technique, the Darboux transformation, can have indeed far-reaching impact in applied nonlinear optics namely in fiber-based telecommunication systems. Our results pave the way towards doubling the information rate of NFT-based fiber optics communication systems. Although more research work needs to be done in this direction, by demonstrating the possibility of using dual polarization NFT channels, we have indeed successfully met one of the key challenges that were explicitly highlighted in a recent review of this research field <cit.> as necessary steps in order to bring eigenvalue communication from a pioneering stage to be a working infrastructure for optical communications in the real-world. Furthermore, the demonstration of the polarization division multiplexing is a significant step forward towards a fair comparison of the NFT-based channels with the currently used linear ones where polarization division multiplexing is an established practice.
    
    
    § FUNDING INFORMATION
    
    This work was supported by the Marie Curie Actions through ICONE Project (no. 608099), the DNRF Research Centre of Excellence, SPOC (no. DNRF123) and the Villum foundation.
    
    
    
    
    
    § ACKNOWLEDGMENTS
    
    
    The authors thank S.K. Turitsyn, M. Kamalian, Y. Prylepskiy, R. T. Jones and J. Diniz for the stimulating discussions, constant encouragement and careful reading of the manuscript and the anonymous reviewers for their constructive criticism and relevant suggestions that improved the quality of the manuscript.
    
    
    
    See Supplement 1 for supporting content.
