Nonlinear phase noise reduction using digital back propagation and midpoint optical phase conjugation

Saber Rahbarfam; Shiva Kumar

doi:10.1364/OE.27.008968

1. Introduction

The performance of a long haul fiber optic system is mainly limited by nonlinear effects in fibers [1, 2]. The nonlinear penalty in fiber optic systems can be divided into two types. (i) Deterministic (although random symbol pattern dependent) nonlinear impairments such as self-phase modulation (SPM), intra-channel cross-phase modulation (IXPM) and inter-channel four wave mixing (IFWM) [3, 4], and inter-channel XPM and FWM [1]. Some authors call this nonlinear interference noise (NLIN), although strictly speaking, this is not noise, just a distortion. (ii) Stochastic nonlinear impairments such as nonlinear phase noise (NLPN) or also known as Gordon-Mollenauer phase noise [5]. Noise due to an inline amplifier changes the amplitude of the signal pulse randomly. Due to Kerr effect, this amplitude change translates into random phase change which is known as nonlinear phase noise (NLPN). In principle, deterministic nonlinear impairments can be fully compensated for using the digital back propagation (DBP) [6–8], but it would be hard to compensate for NLPN due to its stochastic nature. Gordon and Mollenauer analyzed this problem for the first time and derived an analytical expression for the NLPN variance [5]. They pointed out that only two degrees of freedom (DOFs) of the noise modes are sufficient to describe the ASE noise added to the signal and the higher order noise modes play much less significant role if the optical bandwidth is not too large. These noise modes have the same form as the signal pulse. One of the noise modes is in phase with the signal and the other in quadrature. The in-phase component of the noise changes the amplitude of the signal pulse and, hence, leads to energy change while the quadrature component leads to a linear phase shift. The energy change is translated into an additional phase shift due to fiber nonlinearity [5]. Afterwards, taking into account higher order noise modes, Mecozzi in [9] showed that if we use matched filter with its bandwidth equal to the signals bandwidth, two DOFs suffice to model the NLPN; however, filters with larger bandwidth will require additional DOFs. Ho [10] and Mecozzi [11] developed analytical expressions for the probability density function (PDF) of NLPN. Authors in [5] and [9–11] have ignored the impact of dispersion, and when the fiber dispersion is included, in [12, 13], Kumar has found that the variance of NLPN decreases significantly. In [14], Minzioni et al. found that as the pulse distortion due to fiber dispersion becomes significant, the penalty produced by the NLPN becomes negligible as compared to that due to deterministic nonlinear impairments. However, in [14], DBP was absent. In the presence of DBP, deterministic nonlinear impairments are fully compensated for and hence, one may expect that NLPN would become important for systems with DBP. An extensive review on NLPN can be found in [15].

Many efforts have been made to analyze and compensate for NLPN in different scenarios, some of which are as follows. Effect of NLPN on the fiber-optic communication systems using M-ary PSK signals is studied in [16]. In [17], Xu et al. developed a novel Viterbi-type adaptive maximum likelihood sequence detection algorithm considering both laser phase noise and NLPN in coherent fiber-optic communication systems. An analytical formula for estimating the variance of NLPN in coherent orthogonal frequency division multiplexing (OFDM) systems considering SPM, XPM and FWM has been developed in [18]. An algorithm using soft-decision error-control code along with the time correlation of the impairments due to nonlinearities has been proposed in [19] to mitigate NLPN in a dual-polarization 16 QAM WDM system. In [20], Wang et al. developed a machine-learning based algorithm to mitigate NLPN in an M-ary PSK-based coherent optical system.

In this paper, we analyze the NLPN in various system configurations. The amount of NLPN depends on the splitting ratio of DBP between transmitter and receiver. We compare various system configurations by keeping the fiber optic system with the receiver-based DBP as our reference system. We have developed simple analytical expressions for the variance of NLPN for various system configurations with a few approximations. The main purpose of this analytical study is not to derive a rigorous expression for NLPN variance, but instead to develop a simple measure of the NLPN variance that helps to better design a fiber optic system. Our analytical results show that the variance of NLPN decreases by a factor of 4 if the DBP is equally split between transmitter and receiver. The numerical simulation of single channel, 120 span standard single mode fiber link with the split-DBP provides an improvement in Q-factor by 1.44 dB as compared to the reference system. Using a Gaussian noise model, Lavery et al. have shown in [21] that the split-nonlinearity compensation (NLC) provides 1.5 dB improvement in signal-to-noise ratio over the receiver-sided DBP. However, this improvement was not attributed to the reduction of nonlinear phase noise. In [22] Semrau et al. demonstrated that the optimum splitting ratio of NLC between transmitter and receiver is strongly dependent on whether the transceiver noise or ASE noise beating dominates.

Typically, nonlinear impairments are compensated for either in optical domain such as midpoint OPC [23, 24] or in digital domain such as DBP. In this paper, we combine the mid-point OPC and DBP to mitigate NLPN. In [25], Kumar and Liu showed that the mid-point OPC compensates partially for NLPN in addition to compensation of nonlinear distortions such as XPM and FWM. Mid-point OPC may be interpreted as a temporal mirror which provides a phase-conjugated image at the receiver of the temporal object placed at the transmitter (i.e. signal waveform at the transmitter output). The OPC also provides the images at the receiver of NLPN sources located at inline amplifiers. If we combine the mid-point OPC with the receiver-based DBP, we find that the variance of NLPN decreases by a factor of 4 as compared to the system with receiver-based DBP, but without the midpoint OPC. Next, using our analytical model, we found that the optimum splitting ratio of DBP between transmitter and receiver in the presence of midpoint OPC is 25%:75%, i.e. if 25% of DBP is done at the transmitter and the rest at receiver (along with midpoint OPC), our analytical results show that the variance of NLPN decreases by a factor of 16 as compared to the reference. Our numerical simulation results show that this system configuration provides a Q-factor improvement of 2.6 dB for a single channel as compared to the reference system.

Fig. 1 Close-up representation of a fiber-optic communication link. TF= Transmission fiber.

Download Full Size | PDF

2. Mathematical analysis of the NLPN

Consider the fiber optic system shown in Fig. 1. Let the signal pulse launched to this system be

q (0, t) = A f (t),

where A² is the signal energy, and

f (t)

is the normalized pulse shape, i.e.,

\int | f (t) |^{2} = 1

. For simplicity, we ignore fiber dispersion. In the presence of dispersion, the variance obtained here has to be multiplied by a factor that depends on the dispersion map [12, 13]. The signal at the output of the first amplifier is

q (L^{+}, t) = A f (t) e^{j γ | f (t) |^{2} L_{e f f} A^{2}} + n_{1} (t),

where γ is the fiber nonlinear coefficient,

n_{1} (t)

is the noise field envelope due to the first amplifier, L is the fiber length,

L_{e f f} = [1 - \exp (- α L)] / α

, and α is the fiber loss coefficient.

Gordon and Molleneuer assumed that the noise has the same form as the signal and two degrees of freedom (DOFs) of noise are of importance [5]. Following their approach, the noise may be written as

n_{1} (t) = δ A_{1} f (t) e^{j θ_{n_{1}} (t)} .

Here $δ A_{1}$ and $θ_{n_{1}}$ are the two DOFs of noise which should be sufficient, provided the matched filters are used at the receiver [9]. Substituting Eq. (3) in Eq. (2), we obtain

q (L^{+}, t) = (A + δ A_{1}) f (t) e^{j γ | f (t) |^{2} L_{e f f} A^{2} + j θ_{n_{1}} (t)},

From Eq. (4), we see that $δ A_{1}$ represents the amplitude change due to amplifier noise and $θ_{n_{1}} (t)$ represents the linear phase noise. After propagating through the second span, the optical field envelope is

q (2 L^{-}, t) = (A + δ A_{1}) f (t) e^{- α L / 2 + j θ_{n_{1}} (t)} e^{j γ | f (t) |^{2} [A^{2} L_{e f f} + {(A + δ A_{1})}^{2} L_{e f f}]} .

In Eq. (5), the first and second terms in the square bracket, represent the nonlinear phase change due to the first and second fiber spans, respectively. We assume that $δ A_{1} < < A$ , and approximate ${(A + δ A_{1})}^{2}$ as $A^{2} + 2 A δ A_{1}$ . Under this approximation, Eq. (5) may be rewritten as

q (2 L^{-}, t) ≅ (A + δ A_{1}) f (t) e^{- α L / 2 + j θ_{n_{1}} (t)} e^{j γ | f (t) |^{2} L_{e f f} [2 A^{2} + 2 A δ A_{1}]} .

2.1. Scheme 1: full DBP at the receiver (standard configuration)

Fig. 2 shows the fiber optic systems consisting of N spans of fibers and inline amplifiers, and a full DBP at the receiver. We refer to this scheme as the standard configuration. Using the procedure outlined in Sec. 2, the optical field envelope at the output of the fiber optic link is

q (N L^{+}, t) = (A + \sum_{m = 1}^{N} δ A_{m}) f (t) e^{j θ_{n} (t) + j K (t) [N A^{2} + 2 A \sum_{m = 1}^{N} (N - m) δ A_{m}]},

where,

θ_{n} (t) = \sum_{i = 1}^{N} θ_{n_{i}} (t),

K (t) = γ | f (t) |^{2} L_{e f f} .

In Eq. (7), the first term in the square bracket represents the deterministic nonlinear phase shift due to SPM, whereas the second term represents the stochastic nonlinear phase change due to SPM-ASE interaction. In principle, DBP can mitigate the deterministic nonlinear phase shift. However, the second term is stochastic in nature, and it is hard to compensate for it. The main focus of this paper is to compare the variance of this stochastic nonlinear phase change in various system configurations.

Fig. 2 Scheme 1: Full DBP at the receiver. The standard configuration. TF stands for transmission fiber, and VF stands for the virtual fiber.

Download Full Size | PDF

Let the input of the DBP be the output of the fiber optic link, i.e., $q_{b} (0, t) = q (N L^{+}, t)$ . Here, the subscript “b” denotes the signal in back propagation. After passing through an attenuator and first span of virtual fiber, the signal is

q_{b} (L^{+}, t) = (A + \sum_{m = 1}^{N} δ A_{m}) f (t) e^{j θ_{n} (t) + j K (t) [(N - 1) A^{2} + 2 A \sum_{m = 1}^{N} (N - m - 1) δ A_{m}]} .

We assume that the loss and nonlinear coefficient of the virtual fiber are equal in magnitude, but opposite in sign of those of the transmission fiber.

From the first term in the square bracket of Eq. (10), it may be noted that the first virtual fiber compensated for the deterministic nonlinear phase change due to the last span of the transmission fiber, and the second term in the square bracket of Eq. (10) shows that the multiplication factor of the stochastic nonlinear phase change is modified as compared to that at the output of the fiber optic link (see Eq. (7)). Proceeding in this way, the signal at the output of DBP is

q_{b} (N L^{+}, t) = (A + \sum_{m = 1}^{N} δ A_{m}) f (t) e^{j θ_{n} (t) - j 2 A K (t) (\sum_{m = 1}^{N} m δ A_{m})} .

Comparing Eqs. (7) and (11), we note that the deterministic nonlinear phase change ( $K (t) N A^{2}$ ) is removed by the DBP. At the end of the fiber optic link, NLPN due to the first inline amplifier ( $2 A K (t) (N - 1) δ A_{1}$ in Eq. (7)) has the highest contribution, whereas it has the least contribution after the DBP ( $2 A K (t) δ A_{1}$ , in the phase term in Eq. (11)).

The NLPN after the DBP is

δ ϕ_{N L_{1}} = - 2 A K (t) (\sum_{m = 1}^{N} M (m) δ A_{m}) .

where

M (m) = m

is the multiplication factor.

M (m)

is a measure of the impact of the noise due to

m^{t h}

amplifier. In this case, the last inline amplifier has the highest impact since NLPN due to the last amplifier propagates through N spans of virtual fibers.

Assuming that the noises of inline amplifiers are statistically independent, the variance can be added leading to the following expression for the total variance of NLPN in the presence of DBP

σ_{N L_{1}}^{2} = 〈 δ ϕ_{N L_{1}}^{2} 〉 = {(2 A K (t))}^{2} 〈 δ A^{2} 〉 (\sum_{m = 1}^{N} m^{2}),

where

〈 δ A^{2} 〉 = 〈 δ A_{m}^{2} 〉

. We have assumed that all the amplifiers have identical gains and noise figures.

For a more rigorous derivation, a factor that depends on the dispersion map has to be included in Eq. (13) [13]. However, in this paper we are mainly interested in the scaling of the phase variation with distance and omit that term. Simplifying Eq. (13), for large N, we find

σ_{N L_{1}}^{2} \propto {(2 A K (t))}^{2} N^{3} .

For WDM or OFDM systems, stochastic nonlinear phase change due to XPM [26], and FWM [18], become important. However, the scaling of the variance with the number of spans would remain the same in the presence of XPM or FWM and hence, we ignore those terms in this paper.

Fig. 3 Scheme 2: DBP split between transmitter and receiver. TF = transmission fiber, VF = virtual fiber.

Download Full Size | PDF

2.2. Scheme 2: symmetric DBP split between transmitter and receiver (split-DBP)

Fig. 3 shows the scheme in which the DBP at the transmitter compensates for the propagation impairments of the first half of the fiber optic link, while the DBP at the receiver compensates for the other half. This scheme is referred as split-NLC [21]. Proceeding as before, the signal at the output of the DBP at the receiver is

q_{b} (N L^{+}, t) = (A + \sum_{m = 1}^{N} δ A_{m}) f (t) e^{j δ ϕ_{N L_{2}} (t) + j θ_{n} (t)},

where the NLPN is

δ ϕ_{N L_{2}} (t) = 2 A K (t) \sum_{m = 1}^{N} M (m) δ A_{m},

where the multiplication factor is

M (m) = \frac{N}{2} - m .

Consider the subsystem consisting of the second half of the fiber optic link and the DBP at the receiver. This subsystem is the same as the standard configurations except that the number of spans in this subsystem is $N / 2$ . Hence, we expect that the NLPN due to the $N^{t h}$ amplifier ( $δ A_{N}$ ) is the strongest. However, its strength (i.e. absolute of multiplication factor) is only $N / 2$ (see Eq. (17)), since the length of this subsystem is $N L / 2$ , whereas the strength of the NLPN due to $N^{t h}$ amplifier is N for the case of full DBP at the receiver (see Eq. (14)).

The variance of NLPN can be found using Eq. (16) as before,

σ_{N L_{2}}^{2} \propto {(2 A K (t))}^{2} (\frac{N^{3}}{4}) .

Comparing Eqs. (18) and (14), we find that the variance of NLPN decreases by a factor of 4 for the case of the split-DBP, as compared to standard configuration.

Fig. 4 Scheme 3: Full DBP at the receiver with OPC set-up. OPC = optical phase conjugation, DPC = digital phase conjugation, TF = transmission fiber.

Download Full Size | PDF

2.3. Scheme 3: full DBP at the receiver with mid-point OPC

Fig. 4 shows the system with mid-point OPC and DBP at the receiver. The DBP consists of $N / 2$ spans of virtual fibers and attenuators followed by digital phase conjugation (DPC), and $N / 2$ spans of virtual fibers and attenuators. The output of the fiber optic link with mid-point OPC is

q (N L^{+}, t) = (A + \sum_{m = 1}^{N} δ A_{m}) f (t) e^{j θ_{n} (t) + j 2 A K (t) [\sum_{m = 1}^{\frac{N}{2}} m δ A_{m} + \sum_{m = \frac{N}{2} + 1}^{N} (N - m) δ A_{m}]} .

It may be noted that the deterministic nonlinear phase change due to SPM is absent in Eq. (19) since the fiber spans following the OPC compensate for SPM due to fiber spans prior to OPC. However, the OPC cannot fully compensate for NLPN; it could provide a partial compensation of NLPN [25], since the NLPN due to amplifiers after the OPC cannot be compensated for. We note that in the presence of dispersion, the OPC cannot compensate fully for even deterministic nonlinear impairments in systems with lumped amplifications due to asymmetric power profile about the conjugation point. The first and second terms in the square bracket in Eq. (19) represent the NLPN due to the amplifiers prior to OPC and after OPC, respectively. The NLPN due to amplifiers after the OPC are unaffected by OPC and hence, the second term in the square bracket in Eq. (19) is the same as the corresponding terms in Eq. (7). Multiplication factor m in the first term can be explained as follows. Consider the $m^{t h}$ amplifier ( $m \leq N / 2$ ). The OPC may be interpreted as a temporal mirror located at $N L / 2$ (See Fig. 5). The NLPN generated by the $m^{t h}$ amplifier is located at a distance of $(N / 2 - m) L$ from the mirror, and it has the temporal image at $(N - m) L$ . This temporal image propagates a distance of $[N - (N - m)] L = m L$ leading to a nonlinear phase shift of $K (t) m L {(A + δ A_{m})}^{2}$ .

Fig. 5 Noise source located at mL has an image at point (N − m)L, from which the distance to the end of the line is mL.

Download Full Size | PDF

Fig. 6 Scheme 4: Fiber optic system with a mid-point OPC and asymmetric DBP. OPC = optical phase conjugation, DPC = digital phase conjugation, TF = transmission fiber.

Download Full Size | PDF

After passing through the DBP, the signal is

q_{b} (N L^{+}, t) = (A + \sum_{m = 1}^{N} δ A_{m}) f (t) e^{j θ_{n} (t) + δ ϕ_{N L_{3}} (t)},

where,

δ ϕ_{N L_{3}} = 2 A K (t) [\sum_{m = 1}^{N} M (m) δ A_{m}],

where

M (m) = {\begin{array}{l} - m & if 1 \leq m \leq \frac{N}{2} \\ m - N . & if \frac{N}{2} + 1 \leq m \leq N \end{array}

The sign of the multiplication factor is reversed after the DBP (compare Eq. (22) and the terms in the square bracket of Eq. (19)). Using Eq. (21), NLPN variance is calculated as

σ_{N L_{3}}^{2} \propto {(2 A K (t))}^{2} (\frac{N^{3}}{4}) .

Comparing Eqs. (14) and (23), we find that the variance decreases by a factor of 4 using the mid-point OPC as compared to the standard configuration. Hence, the NLPN reduction factor is the same for mid-point OPC scheme with the receiver DBP (scheme 3) and split DBP (scheme 2).

2.4. Scheme 4: split DBP at transmitter with midpoint OPC

In the absence of midpoint OPC, it can be shown that the optimum splitting ratio of DBP between transmitter and receiver is 50%-50% (i.e. scheme 2). In this section, we find the optimum splitting ratio analytically in the presence of midpoint OPC. Fig. 6 shows the system configuration. Let the number of virtual fiber spans of DBP on the transmitter side be XN where $0 \leq X \leq 1 / 2$ . We first find the the NLPN variance as a function of X, and then by setting its derivative with respect to X to zero, we find the optimum splitting ratio. Nonlinear impairments of $N / 2$ transmission fiber spans after the OPC is compensated for using $N / 2$ virtual fiber spans prior to the DPC and $(1 / 2 - X) N$ spans of virtual fibers after the DPC compensate for rest of the transmission fiber and virtual fiber spans in the transmitter. The output of the fiber optic link with split DBP and mid-point OPC is

q (N L^{+}, t) = (A + \sum_{m = 1}^{N} δ A_{m}) f (t) e^{j θ_{n} (t) + j K (t) A^{2} X N + j 2 A K (t) [\sum_{m = 1}^{\frac{N}{2}} m δ A_{m} + \sum_{m = \frac{N}{2} + 1}^{N} (N - m) δ A_{m}]} .

The second term in the exponent in Eq. (24) represents the deterministic phase change due to the DBP at the transmitter. The system configuration between transmitter and the receiver, which includes N TFs and amplifiers, and a mid-point OPC, resembles the corresponding subsystem in scheme 3. Therefore, the terms inside the square bracket in Eqs. (19) and (24) are exactly the same. However, since the DBP at the receiver of scheme 4 is different from that of scheme 3, the output of the DBP will be different. It is found to be

q_{b} (N L^{+}, t) = (A + \sum_{m = 1}^{N} δ A_{m}) f (t) e^{j θ_{n} (t) + δ ϕ_{N L_{4}} (t)},

where,

δ ϕ_{N L_{4}} = 2 A K (t) [\sum_{m = 1}^{N} M (m) δ A_{m}],

where

M (m) = {\begin{array}{l} N X - m & if 1 \leq m \leq \frac{N}{2} \\ m - N (1 - X) . & if \frac{N}{2} + 1 \leq m \leq N \end{array}

Using Eq. (26), the variance of the NLPN is calculated as

σ_{N L_{4}}^{2} (X) = {(2 A K (t))}^{2} 〈 δ A^{2} 〉 [\sum_{m = 1}^{N} M^{2} (m)],

where

〈 δ A^{2} 〉

represents the variance of

δ A

, and we have

〈 δ A^{2} 〉 = 〈 δ A_{i}^{2} 〉, i = 1, 2, \dots, N .

Taking the derivative of $σ_{N L_{4}}^{2}$ in Eq. (28) with respect to X, and setting it to be zero, we find the optimum splitting ratio X to be $1 / 4$ . Therefore, transmitter side DBP consists of $N / 4$ spans of virtual fibers and attenuators.

Using Eq. (26) with $X = 1 / 4$ , we find the variance of NLPN as

σ_{N L_{4}}^{2} \propto {(2 A K (t))}^{2} (\frac{N^{3}}{16}) .

Comparing Eq. (30) with Eq. (14), we find that the variance decreases by a factor of 16 as compared to the standard configuration.

In this paper, we have fixed the location of OPC at the mid-span and optimized the splitting ratio of DBP. In the case of soliton systems, in [28], McKinstrie et al. have shown that the optimum location of OPC is two thirds of system length. In [14], Minzioni et al. showed that when the OPC is located at the middle of system length, the performance is better than that of the system when the OPC is located at two-thirds of the system length. However, the DBP is absent in [14]. Although the OPC at two thirds of system length cannot fully compensate for deterministic nonlinear impairments, to compensate for NLPN, it may be better than midpoint OPC. Hence, when DBP is present, the OPC at two thirds of system length may outperform the midpoint OPC. However, we defer the simultaneous optimization of location of OPC and splitting ratio to a future investigation.

Fig. 7 Multiplication factor profile. Multiplication factor M(m) is the strength of NLPN originated at the point mL. N = 120.

Download Full Size | PDF

2.5. Multiplication factor

Fig. 7 compares the absolute value of multiplication factor (or strength) of the NLPN as a function of inline amplifier index for various system configurations. From Fig. 7, we see that for split-DBP (scheme 2), the inline amplifiers located close to the middle of the link do not contribute much to the NLPN; however, for full DBP with mid-point OPC (scheme 3), the highest contribution is due to those amplifiers. In both scheme 2 and scheme 3, the peak multiplication factor drops by a factor of 2 as compared to the case of the reference (scheme 1).

The multiplication factor profile for scheme 4 has two peaks located at $N / 2$ and N. Depending on the splitting ratio of DBP between transmitter and receiver, the peak locations and peak values can differ. For example, if the DBP is equally split between transmitter and receiver, the peak location is at N and peak value is $N / 2$ . When the splitting ratio of DBP is optimum, we find that the peak multiplication factor is $N / 4$ which is the lowest of all the schemes (see Fig. 7).

3. Numerical simulations

In this section we carry out the numerical simulations to compare the performance of the system configurations discussed in Sec. 2. The simulations include both dual polarization (DP) single channel and DP-WDM with 5 channel transmission systems, and we study the performance in terms of Q-factor. The Q-factor is calculated using [2],

Q = 20 l o g_{10} (\sqrt{2} e r f c i n v [2. B E R]),

where the BER is the bit error rate computed by the error counting. We use the following parameters throughout the paper: symbol rate= 28 GBaud, amplifier noise figure, NF= 4.77 dB, channel spacing= 50 GHz, and amplifier spacing is 80km. The transmission fiber is the standard single mode fiber (SSMF) at 1550 nm with the following parameters: dispersion coefficient (β₂)=

- 21 p s^{2} / k m

, loss coefficient (α)=

0.2 d B / k m

, nonlinear coefficient (γ)=

1.1 W^{- 1} k m^{- 1}

. QAM-16 has been used as the modulation format for all simulations. Raised cosine pulses with the roll off factor= 0.2 are used. Number of symbols simulated= 16384 in each polarization. Simulation bandwidths are 123.2 GHz and 308 GHz, respectively, for single channel and WDM systems. The fiber optic system as well as DBP are modeled by Manakov equations [27]. The split-step Fourier scheme [1, 2], is used to solve Manakov equations. The step size (for the fiber optic link and DBP) is so chosen that the peak nonlinear phase accumulated in each step is 0.04 rad. The number of samples per symbol are 8 and 20, respectively, for single channel and WDM systems. We compute the mean BER of 15 statistical runs and use it in Eq. (31) to calculate the Q-factor.

Fig. 8 Q-factor performance comparison of the four schemes, for single channel systems.

Download Full Size | PDF

Fig. 9 Maximum achievable reach for single channel systems. The forward error correction (FEC) limit is 4.7 × 10⁻³, (i.e. Q = 8.29 dB) [29].

Download Full Size | PDF

3.1. Single channel systems

Fig. 8 compares the system performance of the four schemes. For all the four schemes, the Q-factor drops at higher launch powers. This is because, although deterministic (and symbol-pattern dependent) nonlinear impairments are compensated for by the DBP, stochastic nonlinear impairments (i.e. NLPN) cannot be fully compensated for. Hence, at higher launch power, NLPN is one of the dominant impairments. For the standard configuration, the optimum launch power is 6 dBm, which shifts to 8 dBm for the case of split-DBP. Also, the optimum Q-factor improves by 1.44 dB for the case of split-DBP as compared to the standard configuration, consistent with the theoretical model of Secs. 2.1 and 2.2. In [21], Lavery et al. also showed 1.5 dB improvemnt in SNR using the split-DBP. However, the improvement was not attributed to reduction in stochastic nonlinear phase shift. Fig. 8 shows that by combining the split-DBP with mid-point OPC (scheme 4), the Q-factor improves by 2.6 dB as compared to standard configuration. The simple model of Sec. 2.4 predicts an improvement of ∼6 dB for scheme 4 (as compared to standard configuration) at higher launch powers when the NLPN is the dominant impairment, but the simulations shows only 4.78 dB improvement at a launch power of 10 dBm. This discrepancy can be explained as follows. The simple model in Secs. 2.3 and 2.4 ignores fiber dispersion. In the absence of dispersion, the fiber span located at $(N / 2 + m) L$ , compensates exactly for the nonlinear distortion due to the fiber span located at $(N / 2 - m) L$ . However, in the presence of dispersion, this compensation is not exact due to fiber loss, i.e., power symmetry with respect to OPC location is broken. Hence, the temporal image at $(N - m) L$ of a NLPN source at mL (see Sec. 2.3) is not exact, which leads to higher NLPN variance than that predicted by Eq. (30). If the distributed Raman amplifiers are used instead of EDFAs, power-symmetry with respect to OPC location can be maintained and we should expect a higher optimum Q-factor for schemes 3 and 4 in this scenario. This would be the subject of future investigation.

Fig. 9 shows the Q-factor as a function of achievable reach. For each reach, the launch power is optimized to get the highest Q-factor which is shown in Fig. 9. At the FEC limit of $4.7 \times 10^{- 3}$ , (i.e. $Q = 8.29 d B$ ) [29], the maximum achievable reach is limited to 10400 Km for the standard configuration, which is increased to 12640 Km and 14640 Km, using the split-DBP (Scheme 2) and split-DBP with OPC (Scheme 4), respectively. Scheme 2 has a slightly better performance than scheme 3 and difference in Q-factor between scheme 2 and scheme 3 become smaller at longer distances. We also found that the optimum launch power for scheme 2 and scheme 3 become roughly the same at longer distances (>14,300 km). Please note that the FEC limit that is introduced in [29], is derived using staircase codes that are a class of hard-decision codes suitable for high-speed optical communications.

Fig. 10 Q-factor performance comparison of the four schemes, for WDM system with 5 channels.

Download Full Size | PDF

Fig. 11 Maximum achievable reach for 5 channel WDM systems. The forward error correction (FEC) limit is 4.7 × 10⁻³, (i.e. Q = 8.29 dB) [29].

Download Full Size | PDF

3.2. Five-channel WDM systems

To mitigate nonlinear impairments of a WDM system, we use the full field DBP approach of Ref. [8]. The central channel is demultiplexed after the DBP using an ideal bandpass filter of full bandwidth 31.5 GHz. Fig. 10 shows the Q-factor as a function of launch power for various system configurations. WDM results have the same trend as that of single-channel systems. The split-DBP (scheme 2) and split-DBP with mid-point OPC (scheme 4) provide Q-factor improvements of 1.55 dB and 2 dB, respectively, as compared to the standard configuration. Fig. 11 shows the maximum achievable reach for various system configurations. The maximum achievable reach for standard configuration is now reduced to 9040 Km as compared to the single channel system owing to NLPN arising from XPM-ASE and FWM-ASE interactions. However, the split-DBP (Scheme 2) and split-DBP with OPC (Scheme 4) enhances the reach by 31% and 39%, respectively, as compared to the standard configuration.

Table 1. Reach decrease in WDM systems compared to single channel case.

View Table

Table 1 compares the FEC limited transmission distances of single channel and WDM systems. As can be seen, the schemes that employ mid-point OPC (Schemes 3 and 4) have suffered more inter-channel nonlinear penalties in WDM systems than the schemes that do not have OPC. The reason is that the schemes that employ mid-point OPC are not able to mitigate NLPN as effectively as predicted by the model due to the power asymmetry caused by EDFA-amplified links. If distributed Raman amplifiers are used, these schemes should provide significant performance benefits as predicted by the model (i.e. reduction in NLPN by a factor of 16 for scheme 4).

Although we have analyzed WDM systems, the results are also applicable to superchannel systems in which there exists NLPN due to ASE-XPM and ASE-FWM interactions. Maher et al. [30] and Galdino et al. [31] have shown the significant increase in reach when nonlinearity compensation is applied jointly to all received channels. The proposed scheme 4 is expected to further improve the performance of the superchannel systems investigated in [30, 31].

4. Conclusion

The nonlinear impairments due to a long haul fiber optic link can be reduced to some extent using the DBP at the receiver. Although the DBP can compensate for deterministic (but dependent on random symbol pattern) nonlinear impairments, it cannot compensate for stochastic nonlinear impairments such as NLPN arising due to SPM-ASE, XPM-ASE, and FWM-ASE interactions. In this paper, we have first analyzed the evolution of NLPN in various system configurations and derived simple analytical expressions for the variance of NLPN. When the DBP is equally split between transmitter and receiver, we found that the variance of NLPN by a factor of 4 as compared to the standard configuration of the fiber optic link with full DBP at the receiver. Numerical simulations of 28 GBaud, 16-QAM system with 120 fiber spans confirmed that the split-DBP provides $\sim 1.44 d B$ impairment in Q-factor as compared to the standard configuration. To further improve the transmission performance, we introduced the optical phase conjugation (OPC) at the midpoint of the link. Our analytical results showed that by combining the split-DBP with midpoint OPC, the variance of NLPN can be reduced by a factor of 16. We found that 25% of DBP at the transmitter and the rest of DBP at the receiver to be optimum in the presence of midpoint OPC. Numerical simulations showed that the scheme with split-DBP and midpoint OPC outperformed the standard configuration by 2.6 dB in Q-factor.

References

1. G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

2. S. Kumar and M. J. Deen, Fiber Optic Communications: Fundamentals and Applications (John Wiley & Sons, 2014). [CrossRef]

3. R.-J. Essiambre, G. Raybon, and B. Mikkelsen, “Pseudo-Linear Transmission Of High-Speed TDM Signals: 40 And 160 Gb/s,” in Optical Fiber Telecommunications IV-B, 4th ed., I. P. Kaminow and T. Li, eds. (Academic, 2002). [CrossRef]

4. S. Kumar, J. C. Mauro, S. Raghavan, and D. Q. Chowdhury, “Intrachannel nonlinear penalties in dispersion-managed transmission systems,” IEEE J. Sel. Top. Quantum Electron. 8(3), 626–631 (2002). [CrossRef]

5. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990). [CrossRef] [PubMed]

6. R.-J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in 2005 European Conference on Optical Communication (IEEE, 2005), pp. 191–192.

7. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

8. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, Fatih Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008). [CrossRef] [PubMed]

9. A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994). [CrossRef]

10. K. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B: Opt. Phys. 20(9), 1875–1879 (2003). [CrossRef]

11. A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004). [CrossRef] [PubMed]

12. S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005). [CrossRef]

13. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009). [CrossRef]

14. P. Minzioni, V. Pusino, I. Cristiani, L. Marazzi, M. Martinelli, and V. Degiorgio, “Study of the Gordon-Mollenauer effect and of the optical-phase-conjugation compensation method in phase-modulated optical communication systems,” IEEE Photonics J. 2(3), 284–291 (2010). [CrossRef]

15. A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007). [CrossRef]

16. N. Ekanayake and H. Herath, “Effect of nonlinear phase noise on the performance of M-ary PSK signals in optical fiber links,” J. Lightwave Technol. 31(3), 447–454 (2013). [CrossRef]

17. Z. Xu, P. Y. Kam, and C. Yu, “Adaptive maximum likelihood sequence detection for QPSK coherent optical communication system,” IEEE Photonics Technol. Lett. 26(6), 583–586 (2014). [CrossRef]

18. X. Zhu and S. Kumar, “Nonlinear phase noise in coherent optical OFDM transmission systems,” Opt. Express 18(7), 7347–7360 (2010). [CrossRef] [PubMed]

19. C. Pan, H. Bülow, W. Idler, L. Schmalen, and F. Kschischang, “Optical nonlinear phase noise compensation for 9 × 32 -Gbaud PolDM-16 QAM transmission using a code-aided expectation-maximization algorithm,” J. Lightwave Technol. 33(17), 3679–3686 (2015). [CrossRef]

20. D. Wang, M. Zhang, Z. Li, Y. Cui, J. Liu, Y. Yang, and H. Wang, “Nonlinear decision boundary created by a machine learning-based classifier to mitigate nonlinear phase noise,” in 2015 European Conference on Optical Communication (IEEE, 2015), pp. 1–3.

21. D. Lavery, D. Ives, G. Liga, A. Alvarado, S. J. Savory, and P. Bayvel, “The benefit of split nonlinearity compensation for single-channel optical fiber communications,” IEEE Photonics Technol. Lett. 28(617), 1803–1806 (2016). [CrossRef]

22. D. Semrau, D. Lavery, L. Galdino, R. I. Killey, and P. Bayvel, “The impact of transceiver noise on digital nonlinearity compensation,” J. Lightwave Technol. 36(3), 695–702 (2018). [CrossRef]

23. A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4(2), 52–54 (1979). [CrossRef] [PubMed]

24. S. Watanabe, G. Ishikawa, T. Naito, and T. Chikama, “Generation of optical phase-conjugate waves and compensation for pulse shape distortion in a single-mode fiber,” J. Lightwave Technol. 12(12), 2139–2146(1994). [CrossRef]

25. S. Kumar and L. Liu, “Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems,” Opt. Express 15(5), 2166–2177 (2007). [CrossRef] [PubMed]

26. K. P. Ho, “Cross-Phase Modulation-Induced Nonlinear Phase Noise for Quadriphase-Shift-Keying Signals,” in Impact of Nonlinearities on Fiber Optic Communications, S. Kumar, ed. (Springer-Verlag, 2011). [CrossRef]

27. P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996). [CrossRef]

28. C. McKinstrie, S. Radic, and C. Xie, “Reduction of soliton phase jitter by in-line phase conjugation,” Opt. Lett. 28(17), 1519–1521 (2003). [CrossRef] [PubMed]

29. L. M. Zhang and F. R. Kschischang, “Staircase codes with 6% to 33% overhead,” J. Lightwave Technol. 32(10), 1999–2002 (2014). [CrossRef]

30. R. Maher, D. Lavery, D. Millar, A. Alvarado, K. Parsons, R. Killey, and P. Bayvel, “Reach enhancement of 100% for a DP-64QAM super-channel using MC-DBP,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper Th4D.5.

31. L. Galdino, D. Semrau, D. Lavery, G. Saavedra, C. B. Czegledi, E. Agrell, R. I. Killey, and P. Bayvel, “On the limits of digital back-propagation in the presence of transceiver noise,” Opt. Express 25(4), 4564–4578 (2017). [CrossRef] [PubMed]

	Single Channel Reach	WDM Channel Reach	Decrease
Scheme 1	10,400km	9,000km	1,400km
Scheme 2	12,600km	11,900km	700km
Scheme 3	12,600km	11,100km	1,500km
Scheme 4	14,600km	12,500km	2,100km

Nonlinear phase noise reduction using digital back propagation and midpoint optical phase conjugation

Abstract

1. Introduction

2. Mathematical analysis of the NLPN

2.1. Scheme 1: full DBP at the receiver (standard configuration)

2.2. Scheme 2: symmetric DBP split between transmitter and receiver (split-DBP)

2.3. Scheme 3: full DBP at the receiver with mid-point OPC

2.4. Scheme 4: split DBP at transmitter with midpoint OPC

2.5. Multiplication factor

3. Numerical simulations

3.1. Single channel systems

3.2. Five-channel WDM systems

4. Conclusion

References

Cited By

Figures (11)

Tables (1)

Equations (31)

Optics Express