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Nonlinear distortion experienced by signals during their propagation through optical fibers strongly limits the throughput of optical communication systems. Recently, a strong research focus has been dedicated to nonlinearity mitigation and compensation techniques. At the same time, a more disruptive approach, the nonlinear Fourier transform (NFT), aims at designing signaling schemes more suited to the nonlinear fiber channel. In a short period, impressive results have been reported by modulating either the continuous spectrum or the discrete spectrum. Additionally, very recent works further introduced the opportunity to modulate both spectra for single polarization transmission. Here, we extend the joint modulation scheme to dual-polarization transmission by introducing the framework to construct a dual-polarization optical signal with the desired continuous and discrete spectra. After a brief analysis of the numerical algorithms used to implement the proposed scheme, the first experimental demonstration of dual-polarization joint nonlinear frequency division multiplexing (NFDM) modulation is reported for up to 3200 km of low-loss transmission fiber. The proposed dual-polarization joint modulation schemes enables to exploit all the degrees of freedom for modulation (both polarizations and both spectra) provided by a single-mode fiber (SMF).
Dual-polarization NFDM transmission with
continuous and discrete spectral modulation
F. Da Ros, S. Civelli, S. Gaiarin, E.P. da Silva, N. De Renzis, M. Secondini, and D. Zibar
Abstract—Nonlinear distortion experienced by signals dur-
ing their propagation through optical fibers strongly limits
the throughput of optical communication systems. Recently,
a strong research focus has been dedicated to nonlinearity
mitigation and compensation techniques. At the same time,
a more disruptive approach, the nonlinear Fourier transform
(NFT), aims at designing signaling schemes more suited to
the nonlinear fiber channel. In a short period, impressive
results have been reported by modulating either the continuous
spectrum or the discrete spectrum. Additionally, very recent
works further introduced the opportunity to modulate both
spectra for single polarization transmission. Here, we extend
the joint modulation scheme to dual-polarization transmission
by introducing the framework to construct a dual-polarization
optical signal with the desired continuous and discrete spectra.
After a brief analysis of the numerical algorithms used to
implement the proposed scheme, the first experimental demon-
stration of dual-polarization joint nonlinear frequency division
multiplexing (NFDM) modulation is reported for up to 3200 km
of low-loss transmission fiber. The proposed dual-polarization
joint modulation schemes enables to exploit all the degrees of
freedom for modulation (both polarizations and both spectra)
provided by a single-mode fiber (SMF).
Index Terms—nonlinear frequency division multiplexing,
Raman amplification, nonlinear Fourier transform, inverse
scattering transform
I. Introduction
OPTICAL communication systems have experienced
an impressive growth over the past few decades with
ever increasing transmission rates. Such a growth has been
the results of key enabling technologies that have allowed
to counteract the several physical effects hindering the
transmission. As linear effects such as loss and dispersion
can be dealt with in the telecommunication C-band, one of
the current limiting factor preventing to further enhance
the throughput of SMFs is the impact of Kerr nonlinearity.
Whereas a significant research effort has been devoted
to counteract nonlinear distortion experienced by the
Manuscript received October 18th, 2018;
F. Da Ros, S. Gaiarin, N. De Renzis, and D. Zibar are
with the Department of Photonics Engineering, Technical Uni-
versity of Denmark, Kongens Lyngby, 2800 Denmark, e-mail:
S. Civelli and M. Secondini are with the TeCIP
Institute, Scuola Superiore Sant’Anna, Pisa, Italy, email:
E.P. da Silva was with the Department of Photonics Engineering,
Technical University of Denmark, Kongens Lyngby, 2800 Denmark.
He is now with the Department of Electrical Engineering of the
Federal University of Campina Grande (UFCG), Paraba, Brazil,
F. Da Ros, S. Civelli and S. Gaiarin equally contributed to this
signals during propagation, with solutions presented both
in the optical and digital domain [1], no clear practical
solution has yet been devised. Over the past few years,
a theoretical approach, which had led to soliton-based
communication, has been rediscovered [2], [3]. Soliton-
based communication was developed in the 1980’s [4]–[6],
and, in particular, eigenvalue based communication was
first attempted in [7], before the advent of wavelength-
division multiplexing (WDM), and it was quickly aban-
doned due to its low spectral efficiency, challenges with
fiber loss, noise and soliton-soliton interaction and the
lack of coherent transceivers enabling access to the full
electrical field. Recently, however, a generalization of
the mathematical theory behind soliton communication,
i.e., the inverse scattering transform (IST), has been
exploited to devise a new approach to transmit over single-
mode fibers (SMFs). The IST, re-branded as nonlinear
Fourier transform (NFT) within the optical communica-
tion community, provides a transformation that enables
to effectively linearize the nonlinear Schr¨odinger equation
(NLSE) describing the optical wave propagation through a
SMF. By using such a transformation, the impact of group
velocity dispersion and Kerr nonlinearity can be construc-
tively taken into account to design a novel signaling system
that aims at not being limited by signal-signal nonlinear
interaction. Furthermore, this transformation may provide
more flexible modulation techniques compared to stan-
dard coherent approaches, as it associates two spectral
quantities to one time-domain waveform: a continuous
spectrum corresponding to dispersive waves and a discrete
spectrum representing the solitonic solutions [2]. Since
the NFT theory strictly requires loss-less, noise-less, and
dispersion-slope free transmission, a significant research
effort has been devoted into addressing these requirements
both numerically, e.g., by using the lossless path-averaged
(LPA) approximation [8]–[10], and experimentally, e.g.,
using distributed Raman amplification [11]–[13]. This
has yield several impressive demonstrations by encoding
information in either the continuous [14] (up to 3.3 Gb/s
over 7344 km), or the discrete spectrum [15] (4 Gb/s
over 640 km) [16] (4 Gb/s over 1600 km) [13] (4 Gb/s
over 4900 km) and [17] (1.5 Gb/s over 1800 km), finally
leading to a recent report of a joint continuous and discrete
spectral modulation [18], [19] (up to 64 Gb/s over 976 km).
In parallel, the use of SMFs for transmission provides an
additional degree of freedom to increase the transmission
throughput by exploiting two orthogonal field polariza-
tions. The NFT for the NLSE has therefore been extended
to the Manakov system (MS) [20], where two signal
polarizations are transmitted under the assumption that
the state-of-polarization varies fast enough to be averaged
over the nonlinear and dispersion lengths, and polarization
mode dispersion can be neglected. The theory has been
applied to numerical and experimental demonstrations of
dual-polarization NFT-based transmission using either the
continuous [21]–[23] or the discrete spectrum [11], [24].
A dual-polarization transmission where both continuous
and discrete spectra are jointly modulated would therefore
represent the complete system where all the degrees of
freedom for modulation provided by a SMF are exploited.
In this work, we therefore demonstrate joint dual-
polarization nonlinear frequency division multiplexing
(NFDM) modulation for the first time. First we dis-
cuss the framework to perform a joint inverse nonlinear
Fourier transform (INFT) operation, i.e., the operation
of constructing a dual-polarization time-domain waveform
with a desired dual-polarization continuous and discrete
spectrum. The steps required are described in details
and the numerical algorithms that can be employed to
implement the INFT are briefly discussed and numerically
characterized. By applying such INFT, a dual-polarization
joint NFDM system is experimentally characterized. A
transmission distance of up to 3200 km is demonstrated
by jointly modulating the dual-polarization signal with a
10-GBd quadrature phase-shift keying (QPSK)-modulated
continuous spectrum and a 2-eigenvalue 125-MBd QPSK
discrete spectrum, for a total net line rate of 8.4 Gb/s (af-
ter forward error correction (FEC) overhead subtraction).
The rate is mainly limited by the wide guard intervals
required to fulfill the NFT vanishing boundary conditions
at the receiver, i.e., after the dispersion-induced signal
The paper is organized as follows: first in Section II the
theoretical framework for joint NFT and INFT operation
is provided. The discussion on the transmitter and receiver
digital signal processing (DSP) algorithms chosen for
implementing the dual-polarization joint NFDM system
and the achievable digital back-to-back performance follow
in Section III. The experimental setup used as testbed is
described in Section IV together with the characterization
of the optical back-to-back performance. The transmission
results are reported and discussed in Section V and the
conclusions drawn in Section VI.
II. Dual-polarization joint continuous and discrete
spectral modulation
In this Section, the framework for dual-polarization joint
NFDM modulation is described. First the channel model
will be briefly outlined in Section II-A, followed by the
direct NFT in Section II-B. Finally Section II-C describes
in details one of the key contributions of this work,
i.e., the joint INFT using the Darboux transform (DT).
Throughout this section, bold symbol characters indicate
vector or matrices, while their components are indicated
with subscripts (without bold): e.g., φ= (φ1,...,φN)and
v= (v1,...,vN).
A. Channel model
Let us consider a SMF exhibiting random birefringence,
and whose dispersion and nonlinear lengths are much
longer than the birefringence correlation length [20]. Under
those conditions, the averaged MS describes the evolution
in the fiber of the two complex-envelope polarization
components of a signal Ej=Ej(τ, ),j= 1,2[25],
∂ℓ =jβ2
∂τ 2+j8γ
∂ℓ =jβ2
∂τ 2+j8γ
where τis the retarded time, the distance, β2the
group velocity dispersion (GVD), and γthe Kerr nonlinear
coefficient of the fiber.
The normalized MS [26] for the anomalous dispersion
regime (β2<0) is
∂z =2q1
∂t2+ 2 |q1|2+|q2|2q1
∂z =2q2
∂t2+ 2 |q1|2+|q2|2q2
where tis the normalized retarded time, and zthe
normalized distance. The equation is derived from (1)
through the change of variables
qj(t) = Ej(τ)
P, t =τ
, z =
with P=|β2|/(8
9γT 2
0),L= 2T2
0/|β2|, and T0is the free
normalization parameter.
B. Direct NFT
The direct NFT computes the nonlinear spectrum of the
time-domain dual-polarization signal q(t) = (q1(t), q2(t)).
The spectrum is composed of a continuous (dispersive)
part and a finite number of discrete components, which
correspond to the solitonic components. The nonlinear
spectrum is defined through the Zakharov-Shabat (Z-S)
problem [20], [22], [26]
where Lis a 3×3operator that depends on the signal
q(t), and is given by
∂t jq1(t)jq2(t)
∂t 0
2(t) 0 j
The canonical solutions of (4) are the solutions φ,¯
ψ, and ¯
ψdefined by the boundary conditions [22], [26]
as t −∞
z}| {
φ(t, λ;q)
e t,¯
φ(t, λ;q)
0 0
1 0
0 1
e t
(solve GLME)
canonical sol.
NIS mapper
Joint INFT
st Sf Q
λ Q
λ qt
Step 1 Step 2 Step 3
Continuous Discrete
Fig. 1. Schematic diagram of dual-polarization joint NFDM time-domain signal generation through a joint INFT operation. The information
is first separately encoded into discrete and continuous spectrum, through b-modulation and NIS operation (via the signal Fourier transform
S(f)), respectively. Then the three steps of the joint INFT follow to obtain the time-domain waveform q(t).
ψ(t, λ;q)
0 0
1 0
0 1
e t,¯
ψ(t, λ;q)
e t
|{z }
as t+.(7)
The canonical solutions form two pairs of bases of the
same subspace, consequently there exist some coeffi-
cients—known as scattering data—a(λ)C, and b(λ)
C2×1such that
φ(t, λ) = ψ(t, λ )b(λ) + ¯
ψ(t, λ)a(λ).(8)
Such scattering data can be obtained as
a(λ) = lim
t+φ1(t, λ)e+j λt ,(9a)
b1(λ) = lim
t+φ2(t, λ)ej λt ,(9b)
b2(λ) = lim
t+φ3(t, λ)ej λt ,(9c)
and are related to each other by
|a(λ)|2+|b1(λ)|2+|b2(λ)|2= 1.(10)
The continuous spectrum of the time domain signal q(t)
is defined as
Qc(λ) = b(λ)/a(λ), λ R.(11)
The discrete spectrum is made of a finite number NDS of
eigenvalues λiC+such that a(λi) = 0. Each eigenvalue
has an associated spectral amplitude (also referred to as
norming constant) defined as
Qd(λi) = b(λi)/a(λi), λiC+.(12)
with a(λi) = da(λ)
Several methods to perform the NFT numerically are
available for the single polarization scenario [2], [3], [27]
and for the dual polarization case [21], [22], [24]. In
this work, the trapezoidal discretization method was
used for computing both the continuous and the discrete
spectrum [28].
C. Inverse NFT
The INFT is the operation to generate a time domain
signal from a given nonlinear spectrum. Several methods
exist to numerically perform the INFT for the scalar (sin-
gle polarization) NLSE [2], [3], [9], [18], [29]. Additionally,
the theory can be extended to the MS [26] and a few
techniques reported leading to recent numerical analysis
and experimental demonstrations: the DT for the MS [24]
for a multi-soliton signal (no continuous spectrum), the
inverse Ablowitz-Ladik method for a signal with only
continuous spectrum [21], and the Nystr¨om conjugate
gradient (NCG) method—based on the solution of the
Gelfand-Levitan-Marchenko equation (GLME)—that can
be applied to a full spectrum [22], [30]. However, when
the overall energy of the signal, and so of the nonlinear
spectrum, is too high, the latter method may diverge
[22]. This issue becomes relevant in optical communication
when discrete eigenvalues are used for modulation [22]. In
this section, we describe an alternative method to generate
a time domain signal from a given continuous and discrete
nonlinear spectrum. Extending the concepts of [18], [28]
to the MS, the method uses an INFT algorithm (e.g.,
the NCG in this work) to obtain the time-domain signal
corresponding to a pre-modified continuous spectrum, and
then adds the discrete eigenvalues using the DT [24],
[31]. Using this approach, the aforementioned issues of
NCG can be relaxed as the energy of the input to the
NCG is decreased by the large amount carried by the
discrete spectrum, therefore shifting the energy barrier to
higher energy levels. A choice of initialization parameters
is further provided to ensure that the obtained signal
has the desired nonlinear spectrum, by following the
approach in [24], [28]. Whereas the algorithm for single
polarization in [18] is supported by the mathematical
framework demonstrated in [28], a rigorous mathematical
framework is beyond the scope of this work. The numerical
accuracy of the scheme is rather confirmed a posteriori
with numerical simulations (see Section III) and relies on
following the single-polarization framework of [18], [28]
similarly to the approach of [24]. Similarly to the single-
polarization case, the three steps are illustrated below.
Assume that we want to digitally compute the time
domain signal q(t)corresponding to the continuous spec-
trum Qc(λ)and the discrete spectrum {λi,b(λi)}NDS
i=1 ,
adopting b-modulation on the discrete part as will be
discussed in Section III. The choice of b-modulation is
not a strict requirement for the scheme presented in the
following, and the approach can be easily extended to
mapping techniques other than b-modulation.
The algorithm consists of three steps (illustrated in Fig. 1):
1) Use the INFT to compute the time domain signal
q(t)corresponding to the pre-modified continuous
Qc(λ) = Qc(λ)
and with empty discrete spectrum. This can be
achieved by solving the GLME equation, for example
using the NCG method. [22].
2) For each eigenvalue λifor i= 1,...,NDS , obtain the
solution ν(t, λi)of the eigenvalue problem L(e
λivwith boundary conditions
ν1(T, λi) = 1
ν2(T, λi) = b1(λi)
ν3(T, λi) = b2(λi)
where e
q(t) = 0 for t /[T , T ]. The solution ν(t, λi)
can be obtained as
ν(t, λi) = φ(t, λi)
φ1(T, λi)ψ(t, λi)ψ(2)(T , λi)1b(λi),
where φ(t, λi)and ψ(t, λi)are the canonical so-
lutions of L(e
q)v=λiv(see Section II-B) and
ψ(2)(t, λi)is the 2×2matrix made of the second
and the third rows of ψ(t, λi), i.e.,
ψ(2)(t, λi) = ψ21 ψ22
ψ31 ψ32.(16)
Note that Eq. (15) is a solution of the eigen-
value problem because it is a linear combination
of solutions. Furthermore, it verifies the boundary
conditions Eq. (14). The canonical solutions φ(t, λi)
and ψ(t, λi)can be found with standard methods
used for the NFT (see Section II-B). Additionally, if
modulation techniques other than b-modulation are
chosen, a different mapping can be easily derived
starting from Eq. (15).
3) Execute the DT for the MS, as in [24], [31], with in-
put parameters e
i=1 , and {ν(t, λi)}NDS
i=1 to
iteratively add the discrete spectrum {λi,b(λi)}NDS
to the spectrum of e
q(t). The ν(t, λi)are the generic
auxiliary solutions. The solution q(t)obtained in
this manner has continuous spectrum Qc(λ)and
discrete spectrum {λi,b(λi)}NDS
i=1 .
This scheme can be easily verified numerically by comput-
ing the direct NFT of the generated time-domain signal
and comparing the resulting spectra with the desired ones.
Transmitter DSP chain
Bit mapper
Continuous Discrete
PRBS generation
Bit mapper
NIS mapper
IQ-modulator predistortion
PRBS generation
Receiver DSP chain
Continuous Discrete
Signal low-pass filtering
Frame synchronization
Signal amplitude rescaling
Phase estimation Equalization
Decision & BER
CFO compensation
NIS demapper
Equalization Phase estimation
Decision & BER
Add guardbands
Pulse shaper
Inverse channel
transfer func.
Fig. 2. (a) Transmitter and (b) receiver digital signal processing
chains, highlighting the key operations performed on the digital
III. Digital signal processing
The DSP chains implemented at the transmitter and re-
ceiver to properly encode the data into a digital waveform
(transmitter) and extract it back (receiver) are shown in
Fig. 2 (a) and (b), respectively.
A pseudo-random bit sequence is generated at the trans-
mitter side to be encoded in both the continuous and dis-
crete spectra. For the continuous spectrum, after mapping
the bits into 10-GBd QPSK symbols (16 samples/symbol),
guard intervals of 64 symbols are added for each 16-
symbol burst, leading to an overall NFDM symbol (burst)
length of 8 ns. Such guard intervals ensure that no inter-
burst interference takes place after the dispersion-induced
pulse broadening at the maximum transmission distance
considered in this work, i.e., 3200 km. Such long guard
intervals could be decreased by adding dispersion pre-
compensation at the transmitter side. This was avoided
for the experimental investigation since dispersion pre-
compensation would yield digital waveforms with different
peak-to-average power ratio (PAPR) for each transmis-
sion distance, making it more challenging to compare
performance at different distances. The symbols are pulse-
shaped with a raised cosine filter (roll-off of 1) and the NIS
is used to obtain the continuous spectrum as detailed in
Fig. 1: the Fourier transform of the pulse-shaped time-
domain signal is directly mapped into the continuous
spectrum Qc(λ)[10].
The data bits to be encoded onto the discrete spectrum
are also mapped to QPSK symbols. Then, the symbols
are associated to the b(λi)(b-modulation) corresponding
to the two purely-imaginary eigenvalues, λi {0.3j, 0.6j}
that have been chosen. The free time normalization param-
eter defined in (3) is chosen to be T0= 244 ps. The eigen-
values themselves are not modulated, i.e., in each symbol-
slot, both eigenvalues are transmitted. Additionally, sev-
eral recent works clearly showed that modulating directly
the b(λi)(b-modulation) of the discrete spectrum leads
to a lower correlation and thus better performance than
modulating Qd(λi)(Qd(λ)-modulation). In this work we
therefore focus on b-modulation for the discrete spectrum
[12], [32]. The radii of the two QPSK constellations have
been set to 52and 0.052, for the b(λi)associated to
λ1and λ2, respectively. This choice leads to a temporal
separation of the components of the time-domain signal
associated to the NFT coefficients b(λi)corresponding to
different eigenvalues. The separation is such that the dis-
crete spectral components (at the transmitter output) are
placed in time within the guard intervals of the continuous-
spectrum burst. The one corresponding to b(λ1)can be
seen in Fig. 3(a) after the burst encoded in the continuous
spectrum, whereas the waveform corresponding to b(λ2)is
located on the opposite end of the symbol slot. This choice
was made to avoid additional signal-dependent implemen-
tation penalties due to high PAPR at the transmitter side,
as well as to limit the time-frequency product of the multi-
soliton signal [33]. Nevertheless, continuous and discrete
time-components do interact during fiber propagation.
An INFT operation is then performed as described in
Section II-C to generate a time-domain waveform with
the desired continuous and discrete spectra. After proper
denormalization, the waveform shown in Fig. 3(a) is ob-
tained (signal power of -9.2 dBm with the fiber parameters
of Section IV). The figure clearly shows the two discrete
(solitonic) components with the continuous (dispersive)
components in between. Finally, the waveforms are pre-
distorted to account for the nonlinear transfer function of
the IQ modulator by applying an arcsin(·)function. Such
a digital waveform can then be encoded onto an optical
carrier using a standard IQ modulator, after digital-to-
analog conversion, as will be described in Section IV.
The net line rate of the generated signal is 8.4 Gb/s,
taking into account the 80%-guard intervals applied and
the 7%-hard-decision forward error correction (HD-FEC)
overhead [34], [35].
At the receiver side, the digital waveforms are then
processed by the DSP highlighted in Fig 2(b). First,
carrier frequency offset (CFO) compensation is performed
to remove any frequency shift due to frequency mistmatch
between signal and local oscillator (LO), followed by signal
amplitude rescaling, low-pass filtering at twice the 20-
dB signal bandwidth, and frame synchronization. The
direct NFT described in Section II-B is then applied
to recover the continuous and discrete spectra from the
time-domain waveform. The inverse transfer function of
the channel exp42zis first applied to the continuous
part of the spectrum, followed by NIS demapping with
the opposite transformation applied at the transmitter
side: first Qc(λ)is mapped into the Fourier spectrum,
then a inverse Fourier transform is used to recover the
time-domain waveform [10]. The guard intervals are sub-
sequently removed and blind radius-directed equalization
is performed followed by phase estimation using digital
phase-lock loop. Finally, decisions on the symbols are
taken and the bit error rate (BER) is counted. The
steps for the demodulation of the discrete spectrum
consists of first phase recovery using blind phase search
(BPS) independently on each constellation bj(λi), followed
by NFT-domain equalization [11], [12]. As the chosen
eigenvalues are purely imaginary, BPS inherently applies
the ideal inverse channel transfer function, which consists
of a constant phase rotation. After BPS, NFT-domain
equalization reduces the noise on the bj(λi)by exploiting
the correlation between the received eigenvalues and
the spectral amplitudes [11], [12]. This equalizer enables
to partially compensate for the rotation and re-scaling
experienced by b1(λi)and b2(λi)due to the displacement
of the eigenvalues. After equalization, decisions are taken
and BER counting is performed.
The DSP chains and numerical algorithm have been first
benchmarked in a digital back-to-back scenario where the
digital waveforms before the IQ-modulator predistortion
are fed directly into the receiver DSP chain. This analysis
allows to ignore the impact of practical equipment lim-
itations, such as analog-to-digital converter (ADC) and
digital-to-analog converter (DAC) resolution, and electri-
cal/optical noise sources, to focus on the numerical algo-
rithms. The resulting performance is shown in Fig. 3(b),
as a function of the energy in the continuous spectrum,
comparing joint and continuous-only modulation. For the
joint modulation (top figure in Fig. 3(b)), the energy in the
discrete spectrum is kept constant to fulfill the duration-
amplitude relation [2]. The signal quality is evaluated by
calculating the error vector magnitude (EVM) as the BER
values are too low for reliable error counting [36].
In the case of joint modulation, as the energy in the
continuous spectrum increases, the performance of the
continuous spectrum improves (EVM decreases) with an
optimum at approx. 0.18 pJ (-9.2 dBm of launch power).
When the energy in the continuous spectrum approaches
zero, the discrete spectrum is dominant and worsens the
accuracy of the numerical algorithms for the continuous
part. In the case of continuous-only modulation (bottom
figure in Fig. 3(b)), the performance do not degrade as
the energy decreases as for the joint-modulation, thus
ruling out numerical errors of the NCG alone.
Beyond the optimum energy for the joint-modulation,
the performance worsens rather rapidly above -9.0 dBm.
A similar worsening of the performance is indeed re-
flected when no discrete spectrum is present. Note that,
considering b-modulation also for the continuous spec-
trum (instead of modulating directly Qc(λ)) may provide
further improvement [27] even though using it in the
context of joint spectral modulation may present some
challenges [37]. Fig. 3(b) shows also the impact of the
energy in the continuous spectrum on the quality of
the discrete spectrum. As the energy in the continuous
spectrum is increased, the limited precision of the nu-
merical algorithms yields a loss of orthogonality between
continuous and discrete spectrum, thus decreasing the
performance of the latter when the continuous components
at high energy overlap with the solitons. The impact of the
0 2 4 6 8 10 12 14 16
Power (a.u.)
Time (ns)
Error vector magnitude (dB)
Launch power (dBm)
0.0 0.2 0.4 0.6 0.8
Energy continuous spectrum (pJ)
-10.0 -9.1 -8.3 -7.6 -7.0
Joint modulation
Continuous modulation
Fig. 3. (a) Time-domain waveforms (-9.2-dBm launch power,
2 NFDM symbols, X and Y polarizations) showing the discrete
(solitonic) components with the continuous components in between
and (b) digital back-to-back performance: EVM as a function of the
energy in the continuous spectrum for joint modulation (top) and
continuous-only modulation (bottom).
time-overlap is expected to worsen the performance during
transmission over a non-ideal (lossy and noisy) channel,
as discussed in [38]. Furthermore, the numerical precision
of the INFT is also expected to contribute to the overall
worsening of the performance of the discrete spectrum.
Regardless of these limitations and the consequent
energy balance between continuous and discrete spectrum,
the BER that can be estimated from the EVM is well be-
low 1×104, even for the highest power values considered.
IV. Experimental transmission setup
The experimental setup is shown in Fig. 4. The pre-
distorted digital waveforms generated as in Fig. 2(a) are
loaded into a 4-channel 64-GSamples/s arbitrary wave-
form generator (AWG) driving the IQ modulator, which
encodes the dual-polarization NFDM signal into an optical
carrier generated by a low-linewidth (1 kHz) fiber laser.
The same laser is used as LO at the receiver side.
The transmission link consists of a recirculating trans-
mission loop based on four 50-km transmission spans with
distributed Raman amplification applied to each span
as in [11]. Backward pumping combined with low-loss
large effective area fiber (SCUBA fiber) enables to achieve
maximum power variations of approximately 3 dB across
the full 200-km loop length. The power profile measured
by optical time domain reflectometry is shown in inset (a)
of Fig. 4. Loss, dispersion, and nonlinear coefficient of the
transmission fiber are 0.155 dB/km, 22 ps/nm/km, and
0.6 /W/km, respectively. These values have been used for
the (I)NFT (de)normalization as discussed in Section II-A.
In addiction to the transmission fiber, the loop consists of
acusto-optic modulators (AOMs) used as optical switches,
an optical band pass filter (OBPF) (0.5-nm bandwidth)
which suppresses out-of-band amplified spontaneous emis-
sion (ASE) noise, an isolator (ISO), and an erbium-doped
fiber amplifier (EDFA) which compensates for the power
loss of all these components.
After the chosen number of recirculation turns, the
signal is received with a pre-amplified coherent receiver
using four balanced photodetectors (BPDs) and a 80-
GSamples/s digital storage oscilloscope (DSO) acting
as analog-to-digital converter. For simplicity, the signal
polarization is manually aligned at the receiver input
with a polarization controller (PC). However, demulti-
plexing schemes based on training sequences have already
been reported [23]. After analog-to-digital conversion, the
waveforms are processed offline by the DSP discussed
in Section III and the performance are evaluated by
bit error counting performed on more than 106bits,
ensuring a reliable BER above 105. Only BER values
from direct error counting are reported for experimental
measurements. In the following the transmission reach
is evaluated considering the HD-FEC threshold (BER
of 3.8×103) [34], [35]. Remark that the frequency-
offset estimation discussed in Section III is necessary due
to the frequency shift introduced by the AOMs which
results in self-heterodyne detection rather than homodyne.
An example of constellation diagrams for continuous and
discrete spectrum is shown in inset (b) of Fig. 4 after
2800-km transmission, illustrating the high quality of the
received signals.
Before discussing the transmission results in Section V,
the signal performance are evaluated in back-to-back
configuration, i.e., connecting the receiver directly at
the output of the IQ modulator. The results are shown
in Fig. 5, distinguishing between the performance of
continuous and discrete spectral components as well as
showing the total BER. The optical signal-to-noise ratio
(OSNR) at the output of the transmitter was approx.
33.8 dB for all the launch powers considered.
For these back-to-back results, the total BER is domi-
nated by the BER of the continuous spectrum as more bits
are encoded in the continuous spectrum (64 bits/NFDM
symbol) compared to the discrete spectrum (8 bits/NFDM
Continuous Discrete
Transmitter Receiver
Fiber laser
< 1 kHz)
64 GSa/s
Transmission link
90°- Hybrid
80 GSa/s
0 50 100 150 200
power (dB)
Distance (km)
50 km
Fig. 4. Experimental setup for the transmission of the jointly modulated signal in a recirculating transmission loop. Insets: (a) power profile
measured over a loop recirculation and (b) constellation diagrams after 2800-km transmission.
-10.0 -9.5 -9.0 -8.5 -8.0 -7.5 -7.0
Launch power (dBm)
Fig. 5. Optical back-to-back BER performance from direct error
counting as a function of the launch signal power.
symbol). Therefore, a higher error probability can be
tolerated on the discrete spectrum before it starts affecting
the total BER. As discussed for the numerical results of
Fig. 3, the BER of the continuous spectrum improves with
its increased energy, reaches an optimum and worsens due
to numerical instabilities. Note that the optimum power is
the same for the digital back-to-back (see Fig. 3(b)). The
lack of variations in the optimum power hints that the
dominant limitation is currently related to the numerical
algorithms, whereas the impact of electrical/optical noise
at transmitter and receiver, as well as the AWG resolution
are rather negligible. The BER on the discrete spectrum is
also consistent with Fig. 3(b), as errors are only detected
at the highest launch power considered, -7 dBm. At such
a power level, the estimated BER in digital back-to-back
was estimated to almost the same value, showing that a
negligible penalty is introduced by the optical-frontends
(at both transmitter and receiver) also for the discrete
V. Transmission performance
After having evaluated the system performance for
both digital and optical back-to-back, the transmission
0 2 4 6 8
Power (a.u.)
Time (ns)
0 2 4 6 8
Power (a.u.)
Time (ns)
0 2 4 6 8
Power (a.u.)
Time (ns)
0 2 4 6 8
Power (a.u.)
Time (ns)
0 2 4 6 8
Power (a.u.)
Time (ns)
0 2 4 6 8
Power (a.u.)
Time (ns)
!! " #$!! "
$!!! " $%!! "
e) f)
Fig. 6. Examples of time-domain waveforms showing one 8-ns NFDM
symbols (including guard intervals) at a fixed launch power of -
9.2 dBm: (a) digital back-to-back, (b) optical back-to-back and after
(c) 400-km, (d) 1200-km, (e) 2000-km, (f) and 2800-km transmission.
The bit patter is not the same for the same for the different
performance is reported in this section. Fig. 6 shows
the evolution along the fiber of one NFDM symbol at
the optimum launch power of -9.2 dBm. The waveforms
clearly show the interaction in time between the discrete
and continuous spectral components, whereas the guard
interval size is more than sufficient to guarantee the
vanishing boundary conditions required by the NFT also
at the longest transmission distances. The guard interval
size could actually be reduced by pre-dispersing the wave-
forms at the transmitter side by half of the transmission
length, i.e., by applying the inverse of the channel transfer
function [39]. Additionally, by tailoring the guard intervals
to the desired transmission distance, the transmission rate
can be maximized. The total BER results as a function
of the launch power for different transmission distances
are shown in Fig. 7(a). The curves show an optimum
launch power (minimum BER) consistent with the digital
and optical back-to-back performance, i.e., 9.2dBm.
The BER values after a 400-km transmission are actually
0 400 800 1200 1600 2000 2400 2800 3200
Distance (km)
BER total
BER continuous
BER discrete
-9.5 -9.0 -8.5 -8.0 -7.5 -7.0
BER total
Launch power (dBm)
3200 km
2800 km
2000 km
1200 km
400 km
Fig. 7. (a) Total BER performance as a function of the launch
power for different transmission distances, and (b) total BER and
contributions from continuous and discrete spectrum as a function of
the transmission distance for the optimum launch power of -9.2 dBm
(shaded area in (a)).
in close agreement with the optical back-to-back ones
in Figs. 3(b). These results further confirm that the
dominant performance limitation is currently linked to
the numerical precision in performing joint INFT and
NFT at high power values. By increasing the numerical
precision of the different steps (Section II) using improved
numerical algorithms, we believe the performance could be
significantly improved, potentially shifting the optimum
launch power to higher values [40].
The BER as a function of the transmission distance is
shown in Fig. 7(b), for a fixed -9.2-dBm launch power. The
figure highlights that BER below the HD-FEC threshold
can be achieved for up to 3200-km transmission. Beyond
2800 km, the dominant contribution to the total BER
comes from the discrete spectral components and in
particular from the largest eigenvalue λ2= 0.6j. Improved
modulation schemes for the discrete spectral components,
such as the differential modulation proposed in [41] or
soliton detection based on matched filter [18], are thus
expected to increase the transmission reach. Furthermore,
as far as the continuous spectrum is concerned using
improved detection strategies, may also provide additional
performance gain [42], [43]. Finally, the net transmission
rate may be increased by reducing the guard intervals, as
mentioned above, and pre-dispersing the signal by half of
the transmission length at the transmitter side [19], [39],
VI. Conclusion
We have introduced a framework for dual-polarization
NFDM systems which allows encoding data on both
continuous and discrete spectral components. The steps
to perform the joint INFT at the transmitter side are
described and numerically implemented, evaluating its
performance first in a transmission-free scenario without
(digital back-to-back) and with the optical front-ends
(optical back-to-back) considered. The dual-polarization
joint NFDM system has then been experimentally demon-
strated in a transmission scenario using distributed Raman
amplification. A transmission reach of 3200 km is achieved
for a 8.4-Gb/s net rate NFDM signal, mainly limited
by the numerical implementation of the joint INFT and
NFT, which will need to be further improved. This work
demonstrates the use of all the degrees of freedom available
for NFDM-based transmission over SMFs.
This work is supported by the European Research
Council through the ERC-CoG FRECOM project (grant
agreement no. 771878) and the National Council for
Scientific and Technological Development (CNPq), Brazil,
grant 432214/2018-6. We thank OFS fitel Denmark for
providing the SCUBA fiber used in the experiment and
the anonymous reviewers for their constructive feedback.
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... [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][14][15][16], and more sophisticated techniques incorporating both discrete and continuous NF spectrum components have been developed as well [17][18][19][20], albeit we do not address these here. ...
... 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][22][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. ...
... 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][19][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. ...
Full-text available
We consider optical transmission systems based on the nonlinear frequency division multiplexing (NFDM) concept, i.e., the systems employing the nonlinear Fourier transform (NFT) for signal processing and data modulation. Our work specifically addresses the double-polarization (DP) NFDM setup that utilizes the so-called b-modulation, the most efficient NFDM method proposed up-to-date. We extend the previously-developed analytical approach based on the adiabatic perturbation theory for the continuous nonlinear Fourier spectrum (b-coefficient) onto the DP case to obtain the leading order of continuous input-output signal relation, i.e., the asymptotic channel model, for an arbitrary b-modulated DP-NFDM optical communication system. Our main result is in deriving the relatively simple analytical expressions for the power spectral density of the components of effective conditionally Gaussian input-dependent noise emerging inside the nonlinear Fourier domain. We also demonstrate that our analytical expressions are in remarkable agreement with direct numerical results if one extracts the "processing noise" arising due to the imprecision of numerical NFT operations.
... More recently, several groups have studied variations of this approach [13,16,31,[34][35][36]. It is important to note that there are other modulation techniques that use the continuous spectrum alone [37] and some recent work have also studied the combination of those two techniques, potentially increasing spectral efficiency as in [38,39]. In [39], experiments with dual polarization showed that the use of NFT only achieved maxiummu launch power of −9.2 dBm over 3200 km for a system with a total transmission rate of 8.4 GBb/s. ...
... It is important to note that there are other modulation techniques that use the continuous spectrum alone [37] and some recent work have also studied the combination of those two techniques, potentially increasing spectral efficiency as in [38,39]. In [39], experiments with dual polarization showed that the use of NFT only achieved maxiummu launch power of −9.2 dBm over 3200 km for a system with a total transmission rate of 8.4 GBb/s. However, these recent studies indicate that data encoding using the NFT approach is still not competitive for systems operating in the anomalous dispersion regime when compared with conventional encoding method similarly to what we have showed in [27] for a system operating in the normal dispersion regime. ...
... However, this system with the proposed dispersion decreasing fibers only improves the system performance by 2 dB over a propagation distance of 1280 km when compared to the optimized path-averaged nonlinear parameter approximation that can be used with conventional fibers [49]. We have not considered the impact of dual-polarization multiplexing nor polarization-mode dispersion effects in this system, which have been studied in [36,39,[50][51][52][53][54]. The 2-eigenvalue QPSK modulation format has a relatively small SE, which is about 0.12 bits/s/Hz, but it can in principle be increased by adding more eigenvalues (higher-order soliton) and/or by using a more complex quadrature amplitude modulation (QAM) applied to the spectral function of each eigenvalue. ...
Full-text available
We study the robustness of a nonlinear frequency-division multiplexing (NFDM) system, based on the Zakharov-Shabat spectral problem (ZSSP), that is comprised of two independent quadrature phase-shift keyed (QPSK) channels modulated in the discrete spectrum associated with two distinct eigenvalues. Among the many fiber impairments that may limit this system, we focus on determining the limits due to third-order dispersion, the Raman effect, amplified spontaneous emission (ASE) noise from erbium-doped fiber amplifiers (EDFAs), and fiber losses with lumped gain from EDFAs. We examine the impact of these impairments on a 1600-km system by analyzing the Q-factor calculated from the error vector magnitude (EVM) of the received symbols. We found that the maximum launch power due to these impairments is: 13 dBm due to third-order dispersion, 11 dBm due to the Raman effect, 3 dBm due to fiber losses with lumped gain, and 2 dBm due to these three impairments combined with ASE noise. The maximum launch power due to all these impairments combined is comparable to that of a conventional wavelength-division multiplexing (WDM) system, even though WDM systems can operate over a much larger bandwidth and, consequently, have a much higher data throughput when compared with NFDM systems. We find that fiber losses in practical fiber transmission systems with lumped gain from EDFAs is the most stringent limiting factor in the performance of this NFDM system.
... Vahid Aref proposed the first full-spectrum (FS) system with the joint modulation of DS and CS together [19], which increases data rates and SE by using all degrees of freedom offered in the nonlinear spectrum. Though promising results about data transmission encoding the information on both DS and CS have been reported recently [19]- [23], data rates and SE need to be further improved by designing and optimizing CS and DS. ...
Full-text available
Nonlinear frequency division multiplexing (NFDM) system is an optional candidate to overcome the fiber nonlinearity limit. A full-spectrum modulated NFDM system, modulating data on combined continuous spectrum (CS) and discrete spectrum (DS) together, was proposed in recent years to improve the data rates and spectral efficiency (SE) by exploiting all the degrees of freedom offered in the nonlinear spectrum. However, the selection of discrete eigenvalues greatly affects the performance of both CS and DS. Designing appropriate eigenvalues of DS is an important issue to ensure the high SE and excellent performance of the system. In this paper, we discussed the selcection principle of eigenvalues and analyzed it from multiple perspectives, 11 eigenvalues with 64-quadrature amplitude modulation (64QAM) are selected for DS. Besides optimizing the eigenvalues at the transmitter, the linear minimum mean-square estimate (LMMSE) method was used at the receiver to furtherly improve the performance of DS. Through the numerical simulation, a 113 Gb/s (SE of 2.8 bits/s/Hz) full-spectrum modulated NFDM system was set up and transmitted 1120 km distance, where the Q-factors of both CS and DS are above the hard-decision forward error correction (HD-FEC) threshold. The results provide a way to design an efficiently full-spectrum modulated NFDM system.
Nonlinear frequency division multiplexing (NFDM) is a novel optical communication technique that can achieve nonlinear free transmission. However, current design of NFDM is analogous to orthogonal frequency division multiplexing (OFDM), where sinc function is utilized as subcarriers, which may not be optimal for nonlinear spectrums. In this paper, we propose an auto-encoder (AE) assisted subcarrier optimization scheme for dual-polarized (DP) NFDM systems. Numerical verifications show that our scheme can improve the Q-factor by 1.54 dB and 0.62 dB compared to sinc subcarrier and linear minimum mean square error (LMMSE) equalization, respectively, in a 960 km transmission scenario. We also analyze the characteristics of the optimized subcarriers and discuss how they enhance the performance. Furthermore, we demonstrate the robustness of the optimized subcarriers to different modulation formats, transmission distances and bandwidth. Our work provides a new idea in subcarrier design for NFDM.
Nonlinear frequency division multiplexing (NFDM) systems, especially the eigenvalue communications have the potential to overcome the nonlinear Shannon capacity limit. However, the baud rate of eigenvalue communications is typically restricted to a few GBaud, making it challenging to mitigate laser frequency impairments such as the phase noise and frequency offset (FO) using digital signal processing (DSP) algorithms in intradyne detections (IDs). Therefore, we introduce the polarization division multiplexing-self-homodyne detection (PDM-SHD) into the NFDM link, which could overcome the impact of phase noise and FO by transmitting a pilot carrier originating from the transmitter laser to the receiver through the orthogonal polarization state of signal. To separate the signal from the carrier at the receiver, a carrier to signal power ratio (CSPR) unrestricted adaptive polarization controlling strategy is proposed and implemented by exploiting the optical intensity fluctuation of the light in a particular polarization rather than its direct optical power as the feedback. Optical injection locking (OIL) is used subsequently to amplify optical power of pilot carrier and mitigate the impact of signal-signal beat interference (SSBI). Additionally, the effects of cross-polarization modulation (XPolM) and modulation instability (MI) in long haul transmission are explored and inhibited. The results show that the tolerable FO range is about 3.5 GHz, which is 17 times larger than the ID one. When 16-amplitude phase shift keying (APSK) or 64-APSK constellations are used, identical Q-factor performance can be obtained by using distributed feedback (DFB, ∼10 MHz) laser, external cavity laser (ECL, ∼100kHz), or fiber laser (FL, ∼100 Hz), respectively, which demonstrates that our proposed PDM-SHD eigenvalue communication structure is insensitive to the laser linewidth. Under the impact of cycle slip, the Q-factor difference of 16-APSK signal between the ECL-ID system and ECL-SHD system can be up to 8.73 dB after 1500 km transmission.
Carrier frequency offset (CFO) estimation is very important for the optical fiber communications and has been studied widely in linear coherent systems, while only a few works have been reported for nonlinear Fourier transform (NFT) based systems. In continuous spectrum (CS) modulation nonlinear frequency division multiplexing (CS-NFDM) systems, frequency offset (FO) has a great influence on its performance, requiring an improved frequency offset estimation (FOE) method. We found that the oversampling rate R 0 adopted in NFDM to ensure the accuracy of the NFT and inverse NFT (INFT) calculations, would cause the estimation accuracy of the traditional FFT-FOE method to decrease by R 0 times. Moreover, CS-NFDM signals with higher baud rate require more subcarriers and then result in an oversampling factor greater than 16. This makes the traditional FFT-FOE method be ineffective to use the common training sequence (TS) overhead to meet the FOE error requirement of CS-NFDM system. Therefore, a modified FOE method based on FFT assisted by TS and autocorrelation has been proposed. The theoretical analysis and simulation results show that the proposed method is applicable to CS-NFDM system, no matter what modulation format is used. For 512 subcarriers, with a high rate of 70GBaud and the TS length of 8192, the proposed method can obtain a minimum FO estimation error about 0.1 MHz, which is better than the other two typical FFT-FOE and Schmidl & Cox methods. In addition, the proposed method can save at least 87.5% and 50% overhead. Thus, the proposed method has obvious improvement for CS-NFDM system with requiring high oversampling rate.
In recent years, nonlinear Fourier transform (NFT) has been intensely researched as a scheme for communication over the nonlinear optical fiber and characterizing dynamics in nonlinear optical systems. For the accurate computation of b -coefficients on discrete eigenvalues, the bidirectional algorithm has been proposed where b -coefficients are calculated at a cutting point within the temporal window rather than two boundaries. The performance of the bidirectional algorithm depends severely upon the selection criterion of the cutting-point, which has been investigated in nonlinear Schrodinger equation and Korteweg–De Vries equation but not yet in Manakov equation accounting for the dual-polarization signals over optical fiber. In this paper, we propose a cutting-point criterion of b -coefficients for the bidirectional algorithm of Manakov equation and compare it with existing criteria. Numerical results show that, the resulting algorithm has much better accuracy compared with existing criteria, especially for signals containing a large number of eigenvalues with large imaginary parts. The bidirectional algorithm with the proposed criterion is useful for decoding the dual-polarization nonlinear frequency division multiplexing (DP-NFDM) signals for communication and characterization of vector soliton dynamics in Manakov equation described systems.
Nonlinear Fourier transform (NFT)-based nonlinear frequency division multiplexing (NFDM) system has the potential to overcome the limit of fiber nonlinearity. In the NFDM system, the symbol information is modulated onto the nonlinear spectra, which consist of continuous and discrete spectrum. In theory, the dual-polarization (DP) NFDM system with the continuous and discrete spectrum (full spectrum, FS) modulation has the potential to achieve higher spectral efficiency. However, due to the complexity of the NFT and inverse NFT (INFT) numerical algorithms, the DP-FS modulated NFDM system is rarely reported. For the FS-NFDM system, the existing INFT algorithms either only support q-modulation and present poor performance, or have large discrete spectral error as the continuous spectrum energy increases. In this paper, we propose an INFT algorithm design theory. Based the theory, an accurate DP-INFT algorithm is proposed. The algorithm supports not only q-modulation, but also b-modulation with better performance. Employing the proposed INFT algorithm, we demonstrate, for the first time, a 174.5Gbit/s FS modulated NFDM transmission system with 1440km.
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The nonlinear Fourier transform (NFT) has recently gained significant attention in fiber optic communications and other engineering fields. Although several numerical algorithms for computing the NFT have been published, the design of highly accurate low-complexity algorithms remains a challenge. In this paper, we present new fast forward NFT algorithms that achieve accuracies that are orders of magnitudes better than current methods, at comparable run times and even for moderate sampling intervals. The new algorithms are compared to existing solutions in multiple, extensive numerical examples.
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In this work, we study the performance of polarization division multiplexing nonlinear inverse synthesis transmission schemes for fiber-optic communications, expected to have reduced nonlinearity impact. Our technique exploits the integrability of the Manakov equation—the master model for dual-polarization signal propagation in a single mode fiber—and employs nonlinear Fourier transform (NFT) based signal processing. First, we generalize some algorithms for the NFT computation to the two- and multicomponent case. Then, we demonstrate that modulating information on both polarizations doubles the channel information rate with a negligible performance degradation. Moreover, we introduce a novel dual-polarization transmission scheme with reduced complexity which separately processes each polarization component and can also provide a performance improvement in some practical scenarios.
The recently proposed b-modulation method for nonlinear Fourier transform-based fiber-optic transmission offers explicit control over the duration of the generated pulses and therewith solves a longstanding practical problem. The currently used b-modulation however suffers from a fundamental energy barrier. There is a limit to the energy of the pulses, in normalized units, that can be generated. In this paper, we discuss how the energy barrier can be shifted by proper design of the carrier waveform and the modulation alphabet. In an experiment, it is found that the improved b-modulator achieves both a higher Q-factor and a further reach than a comparable conventional b-modulator. Furthermore, it performs significantly better than conventional approaches that modulate the reflection coefficient.
Transmission systems based on the nonlinear Fourier transform (NFT) can potentially address the limitations in transmission reach and throughput set forth by the onset of Kerr-induced nonlinear distortion. Whereas this technique is at a preliminary research stage, a rapid progress has been shown over the past few years leading to experimental demonstrations of dual-polarization systems carrying advanced modulation formats. The lossless transmission required by the NFT to ensure the theoretical validity of the scheme is a fairly strong requirement considering practical transmission links. Here, we address it by using optimized distributed Raman amplification to minimize the power variations to approx. 3 dB over 200 km, thus approaching the lossless transmission requirement. Additionally we experimentally evaluate the improvement provided by equalization schemes applied to the signals in the nonlinear Fourier domain. By combining distributed Raman amplification and nonlinear-Fourier-domain equalization we show transmission reaches for dual-polarization nonlinear frequency division multiplexing (NFDM) systems transmitting both two eigenvalues (8 bit/symbol) up to 2200 km and three eigenvalues (12 bit/symbol) up to more than 600 km at hard-decision (HD) and soft-decision (SD) forward error correction (FEC) threshold, respectively.
Conference Paper
We present a nonlinear Fourier transform algorithm whose accuracy, at a comparable runtime and for moderate step sizes, is orders of magnitude better than that of the classical Boffetta-Osborne method.