Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series

Jacklyn D. Reis; António L. Teixeira

doi:10.1364/OE.18.008660

1. Introduction

Coherent Optical transmission has been widely investigated as a mean of increasing the transmission capacity in current fiber-optic communication systems and enabling the deployment of future 100GbE. Over the past few years, multi-level modulation formats such as phase-shift keying (m-ary PSK) [1,2] and quadrature amplitude modulation (m-ary QAM) [3] along with high-speed digital coherent receiver and possibly polarization multiplexing have been successfully demonstrated and gained attention for the possibility of transmitting 100Gb/s per channel and performing electronic equalization of physical impairments [4,5]. Using WDM based network or single-channel sub-carrier multiplexing such as orthogonal frequency-division multiplexing (OFDM) enables increasing even further the capacity up to 1Tb/s per channel for instance [6–8]. On the other hand, when WDM transmission is employed, inter-channel nonlinear crosstalk such as cross-phase modulation (XPM) and four-wave mixing (FWM) impairs the transmission if the input transmitter parameters are not properly optimized. The scenario worsens when the already installed 10Gb/s-IMDD (intensity modulated direct detection) channels being a very effective source of nonlinear crosstalk limits the performance of the coherent receiver in recovering the noisy multi-level modulated channels [9,10]. Therefore, estimating effectively the signal degradation of multi-level modulation formats by fiber nonlinearities requires developing analytical expressions based on closed-form transfer function. Volterra Series Transfer Function (VSTF) method, for instance is a powerful mathematical tool that enables to calculate the several contributions of linear and nonlinear distortion with good accuracy compared to the well-known Split-Step Fourier (SSF). VSTF has been reported in [11] in which the authors showed that 3rd and 5th orders Volterra kernels are sufficient to match SSF in IMDD fiber-optic communication systems. Also, the authors in [12] showed how VSTF can optimize the transmission performance and design of IMDD-WDM systems. On the other hand, one complete analysis of how VSTF can be applied to coherent optical WDM system is still missing. Recently, electronic nonlinear compensation using Volterra series expansion has been reported in [13,14], and the authors in [15] compensate FWM crosstalk between the sub-carrier of coherent optical QPSK-OFDM system.

In this paper, we show, to our knowledge, for the first time that Volterra series method can be applied and very effective in evaluating the performance using vector analysis of multi-level modulation formats WDM transmission employing digital coherent receiver. Firstly, we revise and validate the 3rd order VSTF solution through numerical simulations compared to the SSF method on the transmission of three high-order modulation formats: quadrature phase-shift keying (QPSK), 8-phase-shift keying (8PSK), and square 16-quadrature amplitude modulation (16QAM). In the WDM network, we considered 16 channels with 25Gbaud spaced at 50GHz. Then, we also analyze the impact of nonlinear distortion on those signals by lower data rate IMDD channels co-propagating on the network. In this scenario, we compare the different contributions of nonlinear distortion and showed the effect on the corresponding signal constellation in the complex plane.

2. Analysis of nonlinear effects using Volterra series

Volterra Series Transfer Function was first proposed in [11] as a solution to the nonlinear Schrödinger (NLS) equation defined in the frequency domain. The method was already extensively tested and compared to the Split-Step Fourier (SSF) method mainly in IMDD (intensity modulation direct detection) WDM systems for analysis and optimization of nonlinear effects such as in [12] and [16]. In addition, the solution was already mathematically related to the first and second orders perturbation theory (3rd and 5th orders VSTF solution respectively) [17], which has been used for channel capacity estimation [18] and mitigation of intra-channel distortion [19]. In this paper, however, we will show that VSTF can also be applied for modeling and characterizing the nonlinear effects on CO-WDM systems. Its main advantage relies on the capability to represent the fiber linear and nonlinear effects as a transfer function. As such, one can estimate separately one specific nonlinear effect. Therefore, the estimates can be used either to optimize the transmitter parameters or to design compensation strategies in order to improve the transmission performance.

The solution retained up to the third order Volterra kernel expressed in Eq. (1) [11] is the relationship between the input and output optical signal defined in the frequency through transfer functions representing the linear and nonlinear propagation effects.

\begin{array}{l} A (ω, z) \approx H_{1} (ω, z) A (ω) + \frac{1}{4 π^{2}} \iint H_{3} (ω_{1}, ω_{2}, ω - ω_{1} + ω_{2}, z) A (ω_{1}) \\ A^{*} (ω_{2}) A (ω - ω_{1} + ω_{2}) d ω_{1} d ω_{2} \end{array}

where z is the fiber length,

A (ω) = A (ω, 0)

is the Fourier transform of the input complex envelope of the optical signal

A (t, 0)

, the operator (.)* stands for the complex conjugate,

H_{1} (ω, z)

is the linear transfer function (1st order Volterra kernel) while

H_{3} (ω_{1}, ω_{2}, ω_{3}, z)

is the nonlinear transfer function (3rd order Volterra kernel), defined in Eq. (2) and Eq. (3), respectively.

H_{1} (ω, z) = e^{G_{1} (ω_{1})}

H_{3} (ω_{1}, ω_{2}, ω_{3}, z) = G_{3} (ω_{1}, ω_{2}, ω_{3}) \frac{e^{(G_{1} (ω_{1}) + G_{1}^{*} (ω_{2}) + G_{1} (ω_{3})) z} - e^{(G_{1} (ω_{1} - ω_{2} + ω_{3}) z)}}{G_{1} (ω_{1}) + G_{1}^{*} (ω_{2}) + G_{1} (ω_{3}) - G_{1} (ω_{1} - ω_{2} + ω_{3})}

They are functions of $ω_{1}$ and $ω_{2}$ , frequencies that should scan where the optical signal is defined while $ω_{3}$ can simply be replaced by $ω - ω_{1} + ω_{2}$ , $G_{1} (ω)$ (Eq. (4)) and $G_{3} (ω_{1}, ω_{2}, ω_{3})$ (Eq. (5)) are defined as follow.

G_{1} (ω) = - \frac{α}{2} - j \frac{β_{2} ω^{2}}{2} - j \frac{β_{3} ω^{3}}{6}

G_{3} (ω_{1}, ω_{2}, ω_{3}) = - j (γ + a_{1} (ω_{1} - ω_{2} + ω_{3})) - a_{2} (ω_{1} - ω_{2})

where α is the fiber attenuation,

β_{2}

and

β_{3}

are the second and third orders dispersion coefficients, γ is the nonlinear parameter (Kerr effect),

a_{1}

accounts for Self-Steepening while

a_{2}

accounts for stimulated Raman scattering. If the higher order nonlinear effects are neglected (

a_{1} = a_{2} = 0

) the only contribution to nonlinear phase noise will be from the Kerr effect such that

G_{3} (ω_{1}, ω_{2}, ω_{3}) = - j γ

. Therefore, the evaluation of the double integral in Eq. (1) accounts for SPM (intra-channel), XPM and FWM (inter-channel) effects resulting in phase and intensity distortions. Moreover, the added amplified spontaneous emission (ASE) on the signal leading to energy fluctuations is also translated into phase fluctuations by fiber nonlinearity thus decreasing the coherent receiver performance [20]. The frequencies

ω_{1}

and

ω_{2}

play an important rule on the complexity of the method. Very fast calculation of the double integral in Eq. (1) can be accomplished by correctly defining the limits of integration thus avoiding unnecessary calculations.

VSTF can be used either as a propagation model or an analytical mathematical tool in order to analyze nonlinear effects. For instance if one considers the interactions inside the double integral in Eq. (1) of the input modulated signals $A_{i} (ω_{1}) A_{j}^{*} (ω_{2}) A_{k} (ω - ω_{1} + ω_{2})$ , then: $[i = j = k]$ accounts for SPM; $[i = j \neq k o r i \neq j = k]$ accounts for XPM; $[i - j + k]$ accounts for FWM as described in [21]. It is worth emphasizing that the calculation of the nonlinear effects does not consider the pump-probe analysis in which the test channel is approximated by continuous wave signal. Instead, the VSTF enables the calculation of those effects for any type of input signal such as intensity modulated, phase modulated or both.

2.1 Self-phase modulation

The VSTF solution in Eq. (1) can be represented as $A (ω, z) = A_{L I} (ω, z) + A_{N L} (ω, z)$ being the first right term the linear solution and the second right term being the nonlinear solution, which accounts for intra-channel SPM, inter-channel XPM and FWM if $G_{3} (ω_{1}, ω_{2}, ω_{3}) = - j γ$ . If one needs to calculate only the SPM effect at the i^th channel, then $A_{N L} (ω, z) = A_{i}^{S P M} (ω, z)$ is expressed in Eq. (6) bellow.

A_{i}^{S P M} (ω, z) = \frac{1}{4 π^{2}} \iint H_{3} (ω_{1}, ω_{2}, ω - ω_{1} + ω_{2}, z) A_{i} (ω_{1}) A_{i}^{*} (ω_{2}) A_{i} (ω - ω_{1} + ω_{2}) d ω_{1} d ω_{2}

In this case, the frequencies $ω_{1}$ and $ω_{2}$ scan around the optical signal $A_{i} (ω)$ bandwidth.

2.2 Cross-phase modulation

Since XPM is a multi-channel effect, which the phase of a channel is modulated by the neighboring channels and then converted to intensity fluctuations by chromatic dispersion, we first need to represent the input signal as summation accounting for all the WDM channels. This way, the input optical signal can be $A (ω) = \sum_{n = 1}^{N c h} A_{n} (ω - Δ ω_{n})$ where $N c h$ is the total number of channels and $Δ ω_{n}$ is the channel separation. Therefore, the total XPM effect at the i^th channel is in Eq. (7).

\begin{array}{l} A_{i}^{X P M} (ω, z) = \frac{1}{4 π^{2}} \iint H_{3} (ω_{1}, ω_{2}, ω - ω_{1} + ω_{2}, z) {A_{i} (ω_{1}) [\sum_{j \neq i} A_{j}^{*} (ω_{2}) A_{j} (ω - ω_{1} + ω_{2})] + \\ A_{i} (ω - ω_{1} + ω_{2}) [\sum_{j \neq i} A_{j} (ω_{1}) A_{j}^{*} (ω_{2})]} d ω_{1} d ω_{2} \end{array}

Alternatively, this equation can be simplified to a single integration if one considers that the test channel is represented by a continuous wave channel similar how the authors in [16] performed.

2.3 Four-wave mixing

The best approach for calculating the FWM is first to estimate the several channel combinations of the form $(i, j, k)$ and the resulting FWM product will fall on the channel located at $[i - j + k]$ . Thus, the FWM is calculated as in Eq. (8).

A_{i - j + k}^{F W M} (ω, z) = \frac{1}{4 π^{2}} \iint H_{3} (ω_{1}, ω_{2}, ω - ω_{1} + ω_{2}, z) A_{i} (ω_{1}) A_{j}^{*} (ω_{2}) A_{k} (ω - ω_{1} + ω_{2}) d ω_{1} d ω_{2}

2.4 Numerical solution of the VSTF double integral

The complexity of the 3rd VSTF method can be minimized depending on the approach to numerically solve the double integral in Eq. (1). One strategy can be based on a direct sum or more accurately using trapezoidal rule to perform the integration. Since it is usual to work with discrete signals, the right term (nonlinear part) of this equation can be rewritten as follows in Eq. (9):

A_{N L} {[n]}_{z} = \frac{1}{4 π^{2}} \sum_{i} \sum_{j} H_{3} {[ω_{i}, ω_{j}, ω_{n - i + j}]}_{z} A [i] A^{*} [j] A [n - i + j]

where the integer indexes

(n, i, j)

represent the signal sample points at the discrete angular frequencies

(ω_{n}, ω_{i}, ω_{j})

. Assuming the input discrete signal in the time domain has N symbols and

S P

samples per symbol spaced by the sampling period

T s

(sampling frequency

F s

is

1 / T s

) gives a total of

N F F T = S P \cdot N

points in the time/frequency window. In the Frequency domain, those samples are spread on the interval

[- F s / 2, F s / 2]

centered around

N F F T / 2

. We will consider that each channel has the same bandwidth equal to the channel spacing,

Δ f

. So the total WDM signal bandwidth will be

N c h \cdot Δ f

. The next step is to translate the channel bandwidth

Δ f

in the integer number of sample frequency points per channel denoted as

S P c

. This quantity is approximated to

Δ f

divided by the frequency resolution equal to

F s / N F F T = {(T s \cdot N F F T)}^{- 1}

. So,

S P c = ⌈ Δ f \cdot T s \cdot N F F T ⌉

where the operator

⌈ . ⌉

stands for the largest integer greater or equal to. The total frequency points in the WDM signal,

N F

is the sample frequency points per channel

S P c

times the number of channels

N c h

, or

N F = S P c \cdot N c h

. Therefore, we define the integer indexes

(i, j)

such that

i, j = [- N F / 2 : N F / 2 - 1] + N F F T / 2

or

i, j = [- N c h \cdot S P c / 2 : N c h \cdot S P c / 2 - 1] + N F F T / 2

. Let

K = 1, 2, ..., N c h

represents each channel index, then (i)

n = i [1 + S P c \cdot (K - 1) : K \cdot S P c]

solves Eq. (9) only for channel K; (ii)

n = [- N F / 2 : N F / 2 - 1] + N F F T / 2

solves Eq. (9) for all channels K.

Since we perform $n - i + j$ , we need to guarantee that $1 \leq n - i + j \leq N F F T$ . Equivalently, $\max [i, j] + N F = S P c \cdot N c h / 2 - 1 + N F F T / 2 + S P c \cdot N c h \leq N F F T$ arrives at Eq. (10),

N c h \leq ⌊ \frac{N F F T + 2}{3 \cdot Δ f \cdot T s \cdot N F F T} ⌋

where the operator

⌊ . ⌋

denotes the largest integer less than or equal to. This expression shows the maximum allowable number of channels that we can simulate using VSTF based on the size of the FFT (fast Fourier transform) to represent the input signal on the frequency domain and the sampling period. One pseudo code for solving Eq. (9) is as follows:

1. If the condition by Eq. (10) stands, then;
- a. Create the matrixes $[i, j]$ and $[ω_{i}, ω_{j}]$ ;
2. Define the desirable range of n found in (i) or (ii)
3. For each n, evaluate;
- a. $A_{i, j} = \frac{1}{4 π^{2}} H_{3} {[ω_{i}, ω_{j}, ω_{n - i + j}]}_{z} A [i] A^{*} [j] A [n - i + j]$
- b. $A_{N L} {[n]}_{z} = \sum_{i} \sum_{j} A_{i, j}$
4. Calculate the VSTF complete solution: $A {[n]}_{z} = A_{L I} {[n]}_{z} + A_{N L} {[n]}_{z}$

3. CO WDM transmission system

3.1 System scenario

Our CO WDM setup, depicted in Fig. 1 consists in 16 transmitters operating at 25Gbaud (sampled at 3.2THz with 128 samples per symbol) spaced equally by 50GHz and employing three different modulation formats: quadrature phase-shift keying (QPSK, inset (a) of Fig. 1), 8 phase-shift keying (8PSK, inset (b) of Fig. 1) and square 16 quadrature amplitude modulation (16QAM, inset (c) of Fig. 1). In the QPSK transmitter the continuous wave (CW) optical signal (linewidth~100kHz) is phase modulated by 2 NRZ (non-return to zero, 2¹²-1 bit-long sequences resulting 4096 symbols) electrical signals at 25Gb/s through an IQ modulator composed by 2 Mach-Zehnder modulators (MZM) plus one 90° phase shift. Employing an additional phase-modulator (PM) fed by a third NRZ electrical signal 8PSK modulation is obtained. The square 16QAM is based on a modified IQ modulator (enhanced IQ modulator [22], ) where two phase-modulators are included in the conventional IQ modulator. In this case, this modulator is fed with 4x25Gb/s NRZ electrical signals resulting 100Gb/s (25Gbaud) per channel. Besides the transmitter configuration in the insets (a)-(c) of Fig. 1, we also show the corresponding signal constellation in back-to-back after coherent detection. The channels are multiplexed using an AWG (arrayed waveguide grating) modeled as a super-Gaussian optical filter with 42.5GHz bandwidth. After propagating into 100km of standard single-mode fiber (SSMF) whose parameters are summarized in Table 1 , the optical signal is amplified and dispersion compensated. This was required since we want to analyze only fiber nonlinear effects into the corresponding signal constellation. The optical signal is inputted into the optical 90° Hybrid along with the local oscillator matched to the 8th channel (test channel) almost in the center of the spectrum. The output is then detected by two balanced detectors, electrically filtered and sampled at the symbol rate (one sample per symbol) so that the in-phase and quadrature signal components are calculated. Finally, we perform phase synchronization on the digital domain before recovering the received symbol.

Fig. 1 Coherent Optical WDM scenario. (A) QPSK transmitter; (B) 8PSK transmitter; (C) Square 16QAM transmitter.

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Table 1. Optical fiber parameters.

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3.2 Comparative analysis

This section analyzes the accuracy of the 3rd order VSTF method compared to the Split-Step Fourier (SSF) method through Monte Carlo simulations. To guarantee very accurate results the symmetric SSF had step size h restricted to phase changing not exceeding 0.00010 rad. This requirement can be represented by $γ \cdot h \cdot \max {| A (t, z) |}^{2} \leq 0.00010$ rad. To improve the accuracy of the VSTF method we performed the modified version of the VSTF complete solution as the author proposed in [23]. The comparison was performed in terms of vector analysis through the Error Vector Magnitude (EVM), which measures the signal quality (lower EVM means better transmission performance) [24]. The definition of EVM used throughout this paper is as follows: $E V M [%] = 100 \cdot {[\sum_{i = 1}^{N} {| y_{i} - x_{i} |}^{2} / N]}^{1 / 2} / \tilde{P}$ , where N is the number of evaluated symbols, $y_{i}$ is the received symbol, $x_{i}$ is the ideal symbol and $\tilde{P}$ is average constellation power used as the normalization factor. We measured the EVM of the 8th channel (around the center of the optical spectrum) as a function of the CW input power per channel after fully amplitude and dispersion compensated optical link using both VSTF and SSF propagation models. Figure 2 depicts the EVM calculated after propagation for both SSF (blue) and VSTF (red) models for the tree modulation formats. Clearly the 3rd order VSTF method was sufficiently accurate for the ranges considered in comparison to SSF for all the chosen input power range. For input power equals to 2dBm, it is shown in the insets (a)-(c) of Fig. 2 the signal constellation obtained by both models showing that VSTF can estimate both in-phase and quadrature signals components. In addition, the 16QAM EVM curve has an increasing tendency with respect to the power justified by the fact that its amplitude modulation nature induces more nonlinear effect (mostly XPM). On the other hand, QPSK/8PSK having almost constant power induces less nonlinear effect resulting in almost constant EVM for this scenario. In order to analyze the accuracy of the 3rd VSTF on both XPM and FWM nonlinear regime, we have simulated the same scenario using NZDSF fiber whose parameters are summarized in Table 1. Figure 3 shows the EVM for both models and the insets (a)-(c) show the recovered constellation at 2dBm point. We noticed that for input powers higher than 3dBm per channel, the VSTF solution started diverging of SSF indicating that the 3rd VSTF works to input peak powers around 32mW whose value is in agreement to found by the author in [25].

Fig. 2 EVM (B2B = 1%) of VSTF and SSF after 100km-SSMF. (A) 16x50Gb/s-QPSK transmission. (B) 16x75Gb/s-8PSK transmission. (C) 16x100Gb/s-16QAM transmission.

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Fig. 3 EVM (B2B = 1%) of VSTF and SSF after 100km-NZDSF. (A) 16x50Gb/s-QPSK transmission. (B) 16x75Gb/s-8PSK transmission. (C) 16x100Gb/s-16QAM transmission.

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4. Impact of nonlinear effects on hybrid WDM systems

In the previous section we have analyzed, by comparison to SSF simulations the ability of the VSTF on evaluating the transmission performance of CO-WDM systems. On the other hand, the deployment of CO-WDM systems to further increase the transmission capacity up to 100Gb/s should cope with the already installed 10Gb/s IMDD-WDM systems. Recently, several studies have been carried out in order to investigate the nonlinear tolerance of CO-WDM systems under the presence of OOK channels [9,10]. Therefore, this section discusses how VSTF might be used to model the nonlinear effects induced by IMDD channels on the CO-WDM system previously discussed.

The system scenario has the same 16 channels however we substitute, starting at one edged of the optical spectrum channels of CO-WDM by IMDD channels as follows: (i) 16-0; (ii) 14-2; (iii) 12-4; … (viii) 2-14. The 8th and 9th channels, in the center of the spectrum, were kept as reference CO test signals. The transmission was performed using fully compensated SSMF optical links; each IMDD channel is intensity modulated (NRZ-OOK) by 2¹¹-1 PRBS using a MZM at 12.5Gb/s (to guarantee all the channels have the same size) with extinction ration equal to 15dB. All channels were set with input peak power equal to 2.5dBm and the CO ones carrying the same modulation formats (QPSK, 8PSK and 16QAM) at 25Gbaud and minimum EVM in back-to-back (B2B) around 1%.

4.1 QPSK transmission

Figure 4 depicts the EVM of the 8th QPSK-channel after coherent detection considering different contributions of nonlinear distortion: (i) black squares, representing SPM, XPM and FWM; (ii) blue circles, XPM only; (iii) red triangles, SPM only. Clearly, the induced nonlinear distortion from the OOK channels impairs the QPSK transmission performance and it is evident that XPM is the main contribution for the high EVM. The presence of only 2 IMDD channels is sufficient to increase the EVM in 65% although being separated by almost 400GHz from the channel in analysis. It is also shown in the insets (a)-(c) the recoveredsignal constellation and the effect of nonlinear phase noise for 4, 8 and 12 interfering channels respectively.

Fig. 4 EVM (B2B = 1%) as a function of the number of IMDD channels. (A) 4 IMDD channels; (B) 8 IMDD channels; (C) 12 IMDD channels.

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It is worth emphasizing that the SPM on purely phase-modulated signal (QPSK and 8PSK) is very low so that the EVM measures represents more the signal distortion imposed by coherent crosstalk and filtering than SPM distortion. On 16QAM signals, SPM effect is enhanced because the high variation of the signal intensity.

4.2 8PSK transmission

The same simulations were performed considering the 8th channel as 8PSK instead. Similar behavior on the EVM curve as a function of IMDD channels was obtained, as shown in Fig. 5 . On the other hand, the constellations on the insets (a)-(c) indicate that the induced nonlinear phase noise limits even more, compared to QPSK the transmission performance in terms of BER since the decision region is decreased.

Fig. 5 EVM (B2B = 1%) as a function of the number of IMDD channels. (A) 4 IMDD channels; (B) 8 IMDD channels; (C) 12 IMDD channels.

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4.3 16QAM transmission

Figure 6 shows the EVM curve when the test channel carries 16QAM signaling. The insets (a)-(c) of Fig. 6 depicts the received constellation impaired by nonlinear phase noise induced by the OOK channels through XPM as the dominant effect. The effect of phase noise is even stronger on the symbols located at the corner of the constellation because of their high amplitude, which enhances the nonlinear effect. Therefore, quantifying the statistics of XPM on 16QAM signals requires the calculation for the three values of amplitude whereas on QPSK/8PSK one can assume that all the symbols have similar XPM statistics.

Fig. 6 EVM (B2B = 1%) as a function of the number of IMDD channels. (A) 4 IMDD channels; (B) 8 IMDD channels; (C) 12 IMDD channels.

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5. Conclusion

In this paper we demonstrate through error vector magnitude measures that the VSTF method is a powerful mathematical tool that can be applied to evaluate the transmission performance of future coherent optical WDM systems (16x25Gbaud QPSK, 8PSK or 16QAM spaced at 50GHz grid). Its closed-form transfer function for representing analytically the optical fiber allows designing compensation strategies for both linear and nonlinear physical impairments. The main advantage of VSTF is that enables to analyze each contribution of fiber distortion (SPM, XPM and FWM) taking into account the interplay with chromatic dispersion independently of what kind the probe test channel is: continuous wave, phase modulated (e.g. QPSK/8PSK), amplitude modulated (e.g. OOK) or both phase/amplitude modulated (e.g. 16QAM). In addition, on hybrid WDM networks transporting both IMDD channels and digital modulated channels VSTF can be used as an estimation technique so that the induced nonlinear distortion is mitigated.

In our simulations, we considered the input optical signal having single polarization. Therefore adapting the VSTF method for both two states of polarization, therefore enabling transmission employing polarization multiplexing (e.g. PM-QPSK) is one topic to be covered on future studies.

Acknowledgements

The authors acknowledge Dr. Bo Xu and Darlene M. Neves for their valuable contributions. This work was supported by the European Union within the EURO-FOS project, a Network of Excellence funded by the EU 7th ICT-Framework Programme. Jacklyn D. Reis also acknowledges his PhD grant from FCT (SFRH/43941/2008, “Fundação para a Ciência e a Tecnologia”).

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Parameter	SSMF	NZDSF
Attenuation (dB/km)	0.20	0.20
Dispersion (ps/(nm.km))@1550nm	16.50	2.00
Slope (ps²/(nm.km))	0.0656	0.0656
Nonlinear Parameter (W.km)⁻¹	1.35	1.55

Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series

Abstract

1. Introduction

2. Analysis of nonlinear effects using Volterra series

2.1 Self-phase modulation

2.2 Cross-phase modulation

2.3 Four-wave mixing

2.4 Numerical solution of the VSTF double integral

3. CO WDM transmission system

3.1 System scenario

3.2 Comparative analysis

4. Impact of nonlinear effects on hybrid WDM systems

4.1 QPSK transmission

4.2 8PSK transmission

4.3 16QAM transmission

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (6)

Tables (1)

Equations (10)

Optics Express