# Nonlinear Compensation for Digital Coherent Transmission

This is a continuation from the previous tutorial - ** laser oscillation dynamics and oscillation threshold**.

## 1. Introduction

The degree to which fiber impairments are compensated determines the transmission capacity of fiber-optic transmission systems.

Dispersion-compensating fiber (DCF) is commonly used to compensate chromatic dispersion.Wavelength-division multiplexing (WDM) systems suffer from both intra- and interchannel nonlinearities such as cross-phase modulation (XPM) and four-wave mixing (FWM).

These effects can be suppressed using dispersion management. Compensation of nonlinear impairments in fiber has become the next logical step in increasing the capacity of WDM systems.

A few optical nonlinearity compensation schemes have been demonstrated such as lumped compensation of self-phase modulation (SPM) and optical phase conjugation for the compensation of both chromatic dispersion and Kerr nonlinearity in fibers.

Recently, coherent detection opened new venues for long-haul optical communication systems. Among these are the possibility of using higher-order modulation formats, the ability to pack channels more tightly using orthogonal frequency-division multiplexing and orthogonal wavelength-division multiplexing.

Also, it was noticed early on that since coherent detection provided complete information about the electric field, including its intensity, phase, and even polarization, fiber-induced linear impairments such as dispersion could be eliminated using digital signal processing.

Dispersion, being a linear and scalar impairment, can be compensated in a single step, which is commonly referred to as lumped compensation. Subsequently, it was shown that if the dispersion of the fiber was small enough, even the impairments caused by the Kerr nonlinearity, which is often data dependent, could be compensated.

However, in the more general case where both dispersion and nonlinearity have appreciable impact on the signal, these impairments cannot be removed in a single step.

In this tutorial, we describe a universal postcompensation scheme to compensate all deterministic impairments in fiber called digital backward propagation (DBP).

## 2. Digital Backward Propagation (DBP)

### 2.1. How DBP Works

To understand how DBP works, first let us look in detail how the electric field of the signal, \(A(z, t)\), propagates in the forward direction in fiber, which is governed by the nonlinear Schrodinger equation (NLSE)

\[\tag{9.1}\frac{\partial}{\partial{z}}A=(\widehat{D}+\widehat{N})A\]

with the dispersion operator

\[\tag{9.2}\widehat{D}=-\frac{j}{2}\beta_2\frac{\partial^2}{\partial{t^2}}+\frac{1}{6}\beta_3\frac{\partial^3}{\partial{t^3}}-\frac{\alpha}{2}\]

and the nonlinear operator

\[\tag{9.3}\widehat{N}=j\gamma_\text{NL}|A|^2\]

where \(\beta_j\) represent the \(j\)th-order dispersion, \(\alpha\) is the absorption coefficient, \(\gamma\) is the nonlinear parameter, and \(t\) is the retarded time.

In general, the NLSE does not have an analytical solution. To model the nonlinear dispersive propagation, the fiber is divided into small sections in the split-step method (SSM), as shown in Figure 9.1(a).

If each section is short enough, the effect of dispersion and nonlinearity can be calculated independently. In one step, the NLSE is solved ignoring the nonlinear term, and in the next step it is solved ignoring the dispersion term. The dispersion effect can be calculated in the time domain using finite-impulse response (FIR) filtering or in the frequency domain using the split-step Fourier method (SSFM).

The interplay between dispersion and nonlinearity manifests after the first dispersion and nonlinearity computation pair. At the end of the fiber, the signal will be different from the transmitted signal.

If the received signal is detected coherently, preserving both the amplitude and phase, it can be sent into a virtual fiber in the digital domain whose dispersion and nonlinearity are exactly opposite to those of the real transmission fiber. The dispersion and nonlinear effects in the real fiber are then piecewise canceled by the virtual fiber as shown in Figure 9.1(b).

At the end of propagation in the virtual fiber, the signal will be the same as the transmitted signal except with a delay. Since propagation in a fiber with opposite dispersion and nonlinearity is the same as propagating backwards in the real fiber

\[\tag{9.4}\frac{\partial}{\partial{z}}A=(-\widehat{N}-\widehat{D})A\rightarrow\frac{\partial}{\partial(-z)}A=(\widehat{N}+\widehat{D})A\]

this digital nonlinear compensation method is called DBP. Even though physical fiber with negative nonlinearity does not exist, the beauty of digital signal processing is that the virtual fiber can assume any dispersion and nonlinearity.

### 2.2 Experimental Demonstration of DBP

The first DBP experiment was demonstrated in 2008. Three distributed-feedback lasers were used as WDM carriers. The center channel (\(\lambda_C\)) and two adjacent channels (\(\lambda_R\) and \(\lambda_L\)) are binary phase-shift keying (BPSK) modulated using Mach–Zehnder modulators (MZM) driven at 6 GBd by a pattern generator (PG) with a pseudorandom bit sequence (PRBS) of length \(2^{23}-1\).

A symbol rate of 6 GBd was chosen in order to fit three WDM channels within the double-sided analog bandwidth of 24 GHz of the real-time oscilloscope (RTO). Larger analog bandwidth would allow higher symbol rates and more WDM channels.

The WDM channels are tightly spaced at 7 GHz rather than 6 GHz, which was found to give the lowest linear crosstalk due to nonideal waveforms. The adjacent channels have synchronized symbol times (using radio frequency delay) but decorrelated data content compared with the center channel (through an optical delay in the center channel path).

Polarization controllers (PCs) were placed at the appropriate locations to insure that all the channels are copolarized. The launch power in each channel was equalized using optical attenuators.

To demonstrate the feasibility of digital impairment compensation for the WDM environment with low channel count limited by the bandwidth of the RTO, a relatively high launching power was used so significant nonlinear effects accumulate.

A total launching power (\(P_L\) = 6 dBm) into the optical fiber was set using an erbium-doped fiber amplifier (EDFA), calibrated to the insertion losses associated with the recirculating loop components.

The recirculating loop consists of two NZ-DSF spools, with a combined length of 152.82 km. The fiber parameters are \(\alpha\) = 0.2 dB/km, \(\beta_2\) = −4.9 ps\(^2\)/km, and \(\gamma_\text{NL}\) = 1.9/km/W. A phase diversity receiver is used to beat the transmitted optical signal with the local oscillator (LO) tuned to the center channel. The RTO was used to digitize the two signal quadratures at 40 GSamples∕s. DSP was performed offline.

After employing backward propagation, proper orthogonal WDM filtering, phase estimation, and decision threshold \(Q\)-factor calculation for the center channel were performed.

To evaluate the benefit of using SSM/FIR, other possible compensation techniques are considered; chromatic dispersion compensation (CDC) only and lumped compensation which includes CDC followed by a phase shift proportional to the signal power.

The eye diagrams for the various compensation schemes after 760km are shown in Figure 9.2. The effectiveness of DBP can be clearly seen.

### 2.3 Computational Complexity of DBP

Once the feasibility of DBP is demonstrated experimentally, the number of operations, which translates to power consumption, required for DBP becomes the critical issue for practical applications.

The number of operations for a given optical link depends on the number of steps (\(n_s\)), and hence, on the step size (\(h=L/n_s\)). The SSFM accuracy depends fundamentally on the mutual influence of dispersion and nonlinearity within the step length.

Due to the nature of the dispersion and nonlinearity operators, the step size has to be chosen such that

- The nonlinear phase shift is small enough to preserve the accuracy of the dispersion operation
- The optical power fluctuations due to dispersion effects are small enough to preserve the accuracy of the nonlinear operation.

The total optical field of the WDM channels are written as

\[\tag{9.5}A=\boldsymbol{\sum}_mA_m\exp(ik_mz)\]

where \(A_m\) and \(k_m\) are the complexity amplitude and the linear propagation constant of the \(m\)th channel, respectively.

One way to set the upper bound for the step size is to identify the characteristic physical lengths, which correlate the optical field fluctuations with the propagation distance. Four physical lengths are of interest here, namely, the dispersion, \(L_D\), the nonlinear length \(L_\text{nl}\), the walk-off length \(L_\text{wo}\), and the FWM length \(L_\text{fwm}\).

The nonlinear and walk-off lengths can be defined, for a multichannel system, as follows:

\[\tag{9.6}L_\text{nl}=\frac{1}{\gamma{P_T}\frac{2N-1}{N}},\quad{L_\text{wo}}=\frac{1}{2\pi|\beta_2|(N-1)\Delta{f}B}\]

where \(P_T=\sum_m|A_m|^2\) is the total launched power, \(B\) is the symbol rate (effectively the inverse of the pulse width), and \(\Delta{f}\) is the channel spacing.

The nonlinear length has been defined as the length after which an individual channel achieves a 1 radian phase shift due to SPM and XPM. The walk-off length is defined as the distance after which the relative delay of pulses from the edge channels is equal to the pulse width.

The above-mentioned characteristic lengths are well known and widely used to qualitatively describe the optical field behavior through fiber propagation. However, when FWM is considered, the nonlinear and walk-off lengths are not enough to qualitatively identify the range where the fastest field fluctuations take place.

The nonlinear term, including FWM, can be expressed as follows for the \(m\)th channel.

\[\tag{9.7}-i\gamma\left(2\sum_{q\in{I}}|A_q|^2-|A_m|^2\right)A_m-i\gamma\left[\sum_{[rslm]\in{I}}A_rA_sA_l^*\exp(i\delta{k_{rslm}z})\right]\]

with the following conditions: \(l=r+s-m\), \([m, r, s]\in{I}\), and \(r\ne{s}\ne{m}\).

The first condition neglects fast time-oscillating terms (frequency matching). The second condition forces the newly generated waves to lay within the WDM band. Finally, the third condition excludes SPM and XPM terms.

\(\delta{k}_{rslm}\) is the phase mismatch parameter, given by

\[\tag{9.8}\delta{k}_{rslm}=k_r+k_s-k_l-k_m=\frac{1}{2}\beta_2\Delta\omega^2[r^2+s^2-(r+s-m)^2-m^2]\]

In order to identify the fastest z-fluctuations for the \(m\)th channel, let us set \(r=1\) and \(s=N\) corresponding to the indexes of the edge channels. By maximizing Equation 9.8, the expression for the maximum phase-mismatch is given by

\[\tag{9.9}\delta{k}_\text{max}=\frac{1}{4}|\beta_2|(N-1)^2\Delta\omega^2\]

This expression leads to the following definition for the FWM length:

\[\tag{9.10}L_\text{FWM}=\frac{1}{\pi^2|\beta_2|(N-1)^2\Delta{f^2}}\]

This expression represents the length after which the argument of the fastest FWM term is shifted by 1 radian; hence, it can be understood as the distance after which power fluctuations due to FWM start to take place.

The definition of the FWM length assumes that the FWM-induced variations on a given channel are governed by the linear (dispersive) phase mismatch. However, nonlinearity also contributes to the overall phase mismatch through SPM and XPM. This contribution is only relevant in high-power regimes and it is not expected to play a role in the analysis of fiber transmission.

## 3. Reducing DBP Complexity for Dispersion-Unmanaged WDM Transmission

For the experimental demonstration described in Section 2, the dispersion length is \(\approx\) 355 km, where the full bandwidth 24 GHz is taken. Considering the effects of SPM and XPM, and neglecting loss, the nonlinear length is \(\approx\) 282 km, while the FWM length is \(\approx\) 105 km.

Typically, especially for large channel count WDM systems, the FWM length is much shorter than other physical lengths. Yet, FWM can be effectively suppressed with sufficient local dispersion in a well-designed WDM system. Hence, it is possible to reduce the DBP computational load for WDM transmission by ignoring FWM.

In a coherent detection system, a full reconstruction of the optical field can be achieved by beating the received field with a copolarized local oscillator. The reconstructed field will be used as the input for backward propagation in order to compensate the transmission impairments.

Let \(\widehat{E}_m\) be the envelope of optical field of the \(m\)th channel, where \(m\in{I}\), \(I=\{1, 2, … ,N\}\) and \(N\) is the total number of WDM channels. By rewriting the field expression as \(A_m=\widehat{A}_m\exp[im\Delta\omega{t}]\), where \(\Delta\omega=2\pi\Delta{f}\), the expression of the full optical field can be expressed as \(A=\sum_mA_m\).

The total-field backward propagation equation, that is, T-NLSE, is given by Taylor

\[\tag{9.11}-\frac{\partial{A}}{\partial{z}}+\frac{\alpha}{2}A+\frac{i\beta_2}{2}\frac{\partial^2A}{\partial{t^2}}-\frac{\beta_3}{6}\frac{\partial^3A}{\partial{t^3}}-i\gamma|A|^2A=0\]

which is the backward propagation equation corresponding to the forward propagation equation (Eq. 9.4). Equation 9.11 governs the backward propagation of the total field including second- and third-order dispersion, SPM, XPM, and FWM compensation.

Alternatively, the effect of FWM can be omitted in backward propagation by introducing the expression for \(A\) into Equation 9.11, expanding the \(|A|^2\) term and neglecting the so-called FWM terms, that is,

\[\tag{9.12}-\frac{\partial{A_m}}{\partial{z}}+\frac{\alpha}{2}A_m+\frac{i\beta_2}{2}\frac{\partial^2A_m}{\partial{t^2}}-\frac{\beta_3}{6}\frac{\partial^3A_m}{\partial{t^3}}-i\gamma\left(2\sum_{q\in{I}}|A_q|^2-|A_m|^2\right)A_m=0\]

The system of coupled equations (Eq. 9.12) describe the backward propagation of the WDM channels where dispersion, SPM, and XPM are compensated.

Figure 9.3 illustrates simulation results comparing selective compensation of transmission impairments for 12 channels × 25 GBd (16QAM) WDM transmission with a 50 GHz spacing. The fiber dispersion is \(D=4.4\) ps/km/nm, and the total length is 1000 km. As can be seen, the benefits of FWM compensation are not as significant as XPM compensation.

The benefits of selective compensation of fiber nonlinearity using DBP can be appreciated by defining a complexity parameter

\[F=\frac{\text{Number of equations}\times\text{Samples per symbol}}{\text{Step size}}\]

The number of equations for CDC and total field DBP is one while the number equations for SPM and XPM compensation is equal to the number of WDM channels.

Samples per symbol for total field compensation are larger by a factor equal to the channel space \(\Delta{f}\) divided by the Baud rate \(B\). As a result, the overall computational complexity for XPM compensation, as shown in Table 9.1 is at least one order of magnitude smaller than total-field compensation that addresses FWM penalties.

In solving Equation 9.12 using the SSM, the nonlinear part is computed in the time domain, having the following solution:

\[\tag{9.13}A_m(t,z+h)=A_m(t,z)\exp(i\phi_{m,\text{SPM}}+i\phi_{m,\text{XPM}})\]

where

\[\tag{9.14}\phi_{m,\text{SPM}}(t,z+h)=\gamma\int_{z}^{z+h}P_m(t,\widehat{z})e^{\alpha{z}}d\widehat{z}\]

\[\tag{9.15}\phi_{m,\text{XPM}}(t,z+h)=2\gamma\sum_{q\ne{m}}\int_{z}^{z+h}P_q(t,\widehat{z})e^{\alpha{\hat{z}}}d\widehat{z}\]

and \(P_q=|A_q|^2\).

In the conventional SSM, the above-mentioned integrals are approximated by

\[\tag{9.16}\phi_{m,\text{SPM}}(t,z+h)=\gamma{h}_\text{eff}P_m(t,z)\]

\[\tag{9.17}\phi_{m,\text{XPM}}(t,z+h)=2\gamma{h}_\text{eff}\sum_{q\ne{m}}P_q(t,z)\]

where \(h_\text{eff}=[\exp(\alpha{h})-1]/\alpha\) is the effective step size.

Equations 9.16 and 9.17 are valid provided variations of the optical fields due to dispersion take place in a length scale much longer than the step size.

In the case of SPM, such a length scale is given by the intrachannel pulse broadening, that is, \(L_D=1/(\beta_2B^2)\), where \(B\) is the Baud-rate.

For XPM, the dispersive walk-off between channels has to be considered. This delay occurs in a length scale given by \(L_\text{wo}=1/(2\pi\beta_2N\Delta{f}B)\). Typically, \(N\) is large and \(L_\text{wo}\ll{L_D}\), which results in XPM being the limiting effect for the step-size.

To relax the step-size requirements for XPM compensation, the effects of pulse broadening and walk-off must be separated. For that, Equation 9.17 can be rewritten, by including the time delay caused by the dispersive walk-off, as follows:

\[\tag{9.18}\phi_{m,\text{XPM}}(t,z+h)=2\gamma\sum_{q\ne{m}}\int_z^{z+h}P_q(t-d_{mq}\hat{z},\hat{z})e^{\alpha\hat{z}}d\hat{z}\]

where \(d_{mq}=\beta_2(\omega_m-\omega_q)\) is the walk-off parameter.

By Fourier transforming Equation 9.18, the following expression is obtained for the XPM phase shift, now in the frequency domain:

\[\tag{9.19}\phi_{m,\text{XPM}}(\omega,z+h)=2\gamma\sum_{q\ne{m}}\int_z^{z+h}P_q(\omega,\hat{z})e^{-id_{mq}\omega\hat{z}+\alpha\hat{z}}d\hat{z}\]

After some algebra, and by grouping the SPM and XPM contributions, the total nonlinear phase shift is given by the following relations:

\[\tag{9.20}\phi_m(t,z+h)=F^{-1}\left[\sum_{\forall{q}}F(P_q(t,z))W_{mq}(\omega,h)\right]\]

\[\tag{9.21}W_{mq}(\omega,h)=\begin{cases}\gamma{h}_\text{eff},\qquad\quad\;\;\qquad\text{for }q=m\\2\gamma\frac{e^{(\alpha+id_{mq}\omega)h}-1}{\alpha+id_{mq}\omega},\qquad\text{for }q\ne{m}\end{cases}\]

Equation 9.19 is valid provided that the individual power spectra do not change significantly over the step-size, which is now limited by the minimum of the intrachannel dispersion length and the XPM nonlinear length.

Despite the increased complexity of the above-mentioned approach (hereafter advanced-SSM), the walk-off effect factorization substantially increases the step size in WDM systems, leading to remarkable savings in computation.

Figure 9.4 compares the step-size requirements for the XPM compensation using the conventional and the advanced-SSMs introduced in the previous section. Results in Figure 9.4 correspond to the optimum power, whereas vertical markers indicate the optimum step-size.

This value is obtained by cubic interpolation of the simulation results and by choosing the step-size value corresponding to a \(Q\)-factor penalty of 0.25 dB with respect to the plateau value. The advantage of the walk-off factorization is clear in terms of step size, which can be increased by a factor of 17 using the advanced-SSM.

It should be noted that low-pass filtering can be applied to the computation of SPM, and XPM to increase the step size and thus reduce the computational load.

Intrachannel FWM can also be effectively compensated using a perturbation approach, especially for QPSK modulation since multiplication of QPSK symbols remains in the constellation and can be computed using a look-up table.

Despite the optimization methods presented earlier for reducing computational complexity, DBP for WDM transmission is still expected to have computational complexities at least two orders of magnitude higher than that of electronic dispersion compensation.

Further reduction of DBP computational complexity will likely require clever optics to assist DSP. Such is the case of DBP for dispersion-managed transmission.

## 4. DBP for Dispersion-Managed WDM Transmission

For dispersion-managed fiber-optic transmission systems, it can be assumed, without loss of generality, that each fiber span with a length of \(L\) is one period of the dispersion map.

For long-haul fiber-optic transmission, an optimum power exists as a result of the trade-off between optical signal-to-noise ratio (OSNR) and nonlinear impairments.

The total nonlinear phase shift at the optimum power level is on the order of 1 radian. Therefore, for transoceanic fiber transmission systems, which consist of many (>100) amplified spans, the nonlinear effects in each span are weak.

As a result, chromatic dispersion is the dominant factor that determines the evolution of the waveform within each span.

One can analyze the nonlinear behavior of the optical signal in dispersion-managed transmission systems using a perturbation approach. The NLSE governing the propagation of the optical field, \(A_j(z,t)\), in the \(j\)th fiber span can be expressed as

\[\tag{9.22}\frac{\partial{A}_j(z,t)}{\partial{z}}=[D+\epsilon\cdot{N}(|A_j(z,t)|^2)]\cdot{A_j}(z,t)\]

where \(0\lt{z}\lt{L}\) is the propagation distance within each span, \(D\) is the linear operator for dispersion, fiber loss, and amplifier gain, \(N(|A_j(z,t)|^2)\) is the nonlinear operator, \(\epsilon\) (to be set to unity) is a parameter indicating that the nonlinear perturbation is small for the reasons mentioned earlier.

The boundary conditions are

\[\tag{9.23}A_1(0,t)=a(0,t)\]

\[\tag{9.24}A_j(0,t)=A_{j-1}(L,t)\quad\text{for}\quad{j\ge2}\]

where \(a(0, t)\) is the input signal at the beginning of the first span. The solution of Equation 9.22 can be written as

\[\tag{9.25}A_j(z,t)=A_{j,\text{l}}(z,t)+\epsilon\cdot{A}_{j,\text{nl}}(z,t)\]

Substituting Equation 9.25 into Equation 9.22 and expanding the equation in power series of \(\epsilon\) yields

\[\tag{9.26}\frac{\partial{A}_{j,\text{l}}(z,t)}{\partial{z}}-D\cdot{A}_{j,\text{l}}(z,t)+\epsilon\cdot\left[\frac{\partial{A}_{j,\text{nl}}(z,t)}{\partial{z}}-D\cdot{A}_{j,\text{nl}}(z,t)-N(|A_{j,\text{l}}(z,t)|^2)\cdot{A}_{j,\text{l}}(z,t)\right]+O(\epsilon^2)=0\]

Equating to zero the successive terms of the series lead to

\[\tag{9.27}\frac{\partial{A}_{j,\text{l}}(z,t)}{\partial{z}}=D\cdot{A}_{j,\text{l}}(z,t)\]

\[\tag{9.28}\frac{\partial{A}_{j,\text{nl}}(z,t)}{\partial{z}}=D\cdot{A}_{j,\text{nl}}(z,t)+N(|A_{j,\text{l}}(z,t)|^2)\cdot{A}_{j,\text{l}}(z,t)\]

The boundary conditions are

\[\tag{9.29}A_{1,l}(0,t)=a(0,t)\]

\[\tag{9.30}A_{j,l}(0,t)=A_{j-1}(L,t)\quad\text{for}\quad{j\ge2}\]

and

\[\tag{9.31}A_{j,\text{nl}}(0,t)=0\]

First, it is assumed that dispersion is completely compensated in each span. As a result, at the end of the first span'

\[\tag{9.32}A_{1,l}(L,t)=a(0,t)\]

and

\[\tag{9.33}A_2(0,t)=A_1(L,t)=a(0,t)+\epsilon\cdot{A}_{1,\text{nl}}(L,t)\]

where \(A_{1,\text{nl}}(z,t)\) is the solution of Equation 9.28 with \(j=1\).

In the second span,

\[\tag{9.34}A_{2,l}(z,t)=A_{1,l}(z,t)+\epsilon\cdot\bar{A}(z,t)\]

where the first and second terms are solutions to Equation 9.27 with boundary conditions \(A_{2,l}(0,t)=a(0,t)\) and \(A_{2,l}(0,t)=\epsilon\cdot{A}_{1,\text{nl}}(L,t)\), respectively, as a result of the principle of superposition.

At the end of the second span, because of complete dispersion compensation,

\[\tag{9.35}A_{2,l}(L,t)=a(0,t)+\epsilon\cdot{A}_{1,\text{nl}}(L,t)\]

The nonlinear distortion in the second span is governed by Equation 9.28 with \(j=2\). Since

\[\tag{9.36}|A_{2,l}|^2=|A_{1,l}+\epsilon\cdot{\bar{A}}|^2=|A_{1,l}|^2+O(\epsilon)\]

the differential equation and the boundary conditions for \(A_{2,\text{nl}}(z,t)\) and \(A_{1,\text{nl}}(z,t)\) are identical, so

\[\tag{9.37}A_{2,\text{nl}}(L,t)=A_{1,\text{nl}}(L,t)\]

As a result, the optical field at the end of the second span is given by

\[\tag{9.38}A_2(L,t)=A_{2,l}(L,t)+\epsilon\cdot{A}_{2,\text{nl}}(L,t)=a(0,t)+\epsilon\cdot2A_{1,\text{nl}}(L,t)\]

That is, the nonlinear distortion accumulated in 2 spans is approximately the same as the nonlinear distortion accumulated in 1 span with the same dispersion map and ** twice** the nonlinearity.

It follows that, assuming weak nonlinearity and periodic dispersion management, the optical field after \(K\) spans of propagation can be written as

\[\tag{9.39}A_N(L,t)=a(0,t)+\epsilon\cdot{K}A_{1,\text{nl}}(L,t)\]

which is the solution of the NLSE

\[\tag{9.40}\frac{\partial{A}_j(z,t)}{\partial{z}}=[D+\epsilon\cdot{K}N(|A_j(z,t)|^2)]\cdot{A_j}(z,t)\]

The nonlinear term \(N(|A_j(z,t)|^2)\) in Equation 9.22 is proportional to the fiber nonlinear parameter \(\gamma\), so the NLSE describing optical propagation in a fiber span where the nonlinearity is \(K\) times of that in the original fiber but with the same dispersion map can be written as

\[\tag{9.41}\frac{\partial{A}_j(z,t)}{\partial{z}}=[D+\epsilon\cdot{k}\cdot{N}(|A_j(z,t)|^2)]\cdot{A_j}(z,t)\]

The equivalence of 9.40 and 9.41 suggests that DBP for \(K\) spans can be folded into a single span and \(K\) times the nonlinearity. This method is distance-folded DBP.

Assuming that the step size for the split-step implementation of DBP is unchanged, the computational load for the folded DBP can be saved by the folding factor \(K\).

The above-mentioned derivation is based on the assumption that waveform distortion due to fiber nonlinearity and the residual dispersion per span (RDPS) are negligible, and consequently, the nonlinear behavior of the signal repeats itself in every span.

This assumption is not exactly valid since, first, fiber nonlinearity also changes the waveform, and second, dispersion is not perfectly periodic if the RDPS is nonzero or the dispersion slope is not compensated.

In order for the nonlinearity compensation to be accurate, it might be necessary to divide the entire long-haul transmission system into segments of multiple dispersion-managed spans so that the accumulated nonlinear effects and residual dispersion is small in each segment.

Moreover, in order to minimize the error due to residual dispersion, folded DBP should be performed with a boundary condition calculated from lumped dispersion compensation for the first half of the segment.

For a fiber link with \(M\times{K}\) spans, the distance-folded DBP is illustrated in Figure 9.5.

A better way of solving disparities in dispersion characteristics in the amplified spans is the so-called dispersion-folded DBP. The dispersion map of a typical dispersion-managed fiber transmission system with RDPS is illustrated in Figure 9.6.

After the dispersion-managed fiber transmission and coherent detection, conventional DBP can be performed in the backward direction of the fiber propagation. Multiple steps are required for each of the many fiber spans, resulting in a large number of steps.

Dispersion-folded DBP exploits the fact that, under the weakly nonlinear assumption, the optical waveform repeats at locations where accumulated dispersions are identical.

Since the Kerr nonlinear effects are determined by the instantaneous optical field, the nonlinear behavior of the optical signal also repeats at locations of identical accumulated dispersion. Hence, it is possible to fold the DBP according to the accumulated dispersion.

The propagation of the optical field, \(A(z, t)\), is governed by Equation 9.22. The solution of Equation 9.22 can be written as

\[\tag{9.42}A(z,t)=A_l(z,t)+\epsilon\cdot{A}_\text{nl}(z,t)\]

Substituting Equation 9.42 into Equation 9.22, expanding the equation in power series of \(\epsilon\), and equating to zero the successive terms of the series yields

\[\tag{9.43}\frac{\partial{A}_l(z,t)}{\partial{z}}=D\cdot{A_l}(z,t)\]

\[\tag{9.44}\frac{\partial{A}_\text{nl}(z,t)}{\partial{z}}=D\cdot{A}_\text{nl}(z,t)+N(|A_l(z,t)|^2)\cdot{A}_l(z,t)\]

which describes the linear evolution and the nonlinear correction, respectively. It is noted that the nonlinear correction \(A_\text{nl}(z, t)\) is governed by a linear partial differential equation with nonzero forcing, which depends on the linear solution only.

Therefore, as shown in Figure 9.6, the dispersion map can be divided into m divisions as indicated by the horizontal dashed lines. The fiber segments within a division have the same accumulated dispersion.

Based on the principle of superposition, the total nonlinear correction is the sum of nonlinear corrections due to nonzero forcing at each fiber segment. In conventional DBP, the contribution from each fiber segment is computed separately.

However, it is advantageous to calculate the total nonlinear correction as the sum of nonlinear corrections due to nonzero forcing at different accumulated dispersion divisions, each having multiple fiber segments.

This is because, with the exception of different input power levels and effective lengths, the linear component \(A_l(z, t)\) that generates the nonlinear correction and the total dispersion for the generated nonlinear perturbation to reach the end of the transmission are identical for the fiber segments with the same accumulated dispersion.

Therefore, the nonlinear corrections due to these multiple fiber segments with the same accumulated dispersion are identical except a constant and can be calculated all at once using a weighting factor, as described later.

In dispersion-folded DBP, the fiber segments with the same accumulated dispersion (e.g., the thick gray lines) can be folded into one step. For a fiber link with positive RDPS, a lumped dispersion compensator (\(D_\text{lumped}\)) can be used to obtain the optical field (\(A_1\)) in the first dispersion division. Then, dispersion compensation (\(D\)) and nonlinearity compensation (NL) are performed for each of the subsequent dispersion divisions.

To take into account the different power levels and effective lengths of the fiber segments, a weighting factor (\(W_i\)) is used in the nonlinearity compensator of each step. The nonlinear phase shift in the \(i\)th step of dispersion-folded DBP is given by \(\varphi_i=W_i\cdot|\bar{A}_i(t)|^2\), where \(\bar{A}_i(t)\) is the optical field with the power normalized to unity.

The weighting factor is given by \(W_i=\sum_k\gamma\int{P}_{i,k}(z)dz\), where \(P_{i,k}(z)\) is the power level as a function of distance within the \(k\)th fiber segment in the \(i\)th dispersion division. The effect of loss for each fiber segment is taken into account in the calculation of this weighting factor.

Nonlinear compensation for multiple fiber segments with the same accumulated dispersion is performed in a single step in dispersion-folded DBP, resulting in orders-of-magnitude savings in computation.

In the SSM, the linear and nonlinear effects can be decoupled when the step size is small enough. The dispersion within a fiber segment is neglected in a nonlinearity compensation operator.

Meanwhile, the power level, effective length, and nonlinear coefficient of a fiber segment have been taken into account in the calculation of the nonlinear phase shift.

Thus, the DBP can be folded even if the fiber link consists of multiple types of fibers. Note that calculating the weighting factors does not require real-time computation.

To demonstrate the effectiveness of the dispersion-folded DBP, single-channel 6084 km transmission of NRZ-QPSK signal at 10 Gbaud was demonstrated. The experimental setup is shown in Figure 9.7(a).

At the transmitter, carrier from an external-cavity laser is modulated by a QPSK modulator using a \(2^{23}-1\) PRBS. The optical signal is launched into a recirculating loop controlled by two acousto-optic modulators (AOMs). The recirculating loop consists of two types of fibers: 82.6 km SSMF with 0.2 dB/km loss and 70 μm\(^2\) effective area, and 11 km DCF with 0.46 dB/km loss and 20 μm\(^2\) effective area.

By optimizing the performance in the training experiments of EDC, the dispersion of the SSMF and the DCF are determined as 17.06 ps/nm/km and −123.35 ps/nm/km, respectively. The RDPS is 53 ps/nm.

Two EDFAs are used to completely compensate for the loss in the loop. An optical band-pass filter (BPF) is used to suppress the EDFA noise. At the receiver, the signal is mixed with the local oscillator from another external cavity laser in a 90\(^\circ\) hybrid.

The \(I\) and \(Q\) components of the received signal are detected using two photodetectors (PDs). A RTO is used for analog-to-digital conversion and data acquisition at 40 Gsamples/s. The DSP is performed offline with Matlab.

Note that for realistic terrestrial systems, the parameters of the fiber spans may not be available with good accuracy. It is expected that the parameters for dispersion-folded DBP can also be obtained via adaptive optimization.

For long-haul transmission, the DBP step size is usually limited by dispersion. In this experiment, DBP steps with equal dispersion per step were used for simplicity. The \(Q\)-value as a function of the number of steps is shown in Figure 9.7(b).

The required number of steps to approach the maximum \(Q\)-value can be reduced from 1300 to 30 by using the dispersion-folded DBP. The number of multiplications per sample for DBP is reduced by a factor of 43.

Figure 9.7(c) shows the \(Q\)-value as a function of the launching power. With only EDC for the accumulated residual dispersion, the maximum \(Q\)-value is 9.1 dB. With nonlinearity compensation using dispersion-folded DBP, the maximum \(Q\)-value is increased to 10.7 dB.

The performance after the 30-step dispersion-folded DBP is almost the same as that after the 1300-step conventional DBP. There is a trade-off between complexity and performance using either conventional DBP or dispersion-folded DBP.

A \(Q\)-value of 10.2 dB, corresponding to a 1.1 dB improvement in comparison with EDC, can be achieved using 130-step conventional DBP or 5-step dispersion-folded DBP.

## 5. DBP for Polarization-Multiplexed Transmission

In many WDM transmission systems, not all WDM channels have the same state of polarization, such as in the case of polarization multiplexing or polarization interleaving. In those systems, the effects of polarization on the nonlinear interactions have to be taken into account.

Whether the total electric field is polarized or not, its propagation in a fiber can be described by the vectorial form of the NLSE:

\[\tag{9.45}\begin{align}\frac{\partial{A_x}}{\partial{z}}&=ib_x(z)A_x-\frac{\alpha}{2}A_x+\frac{i\beta_2}{2}\frac{\partial^2A_x}{\partial{t^2}}+i\gamma\left(|A_x|^2+\frac{2}{3}|A_y|^2\right)A_x+\frac{i\gamma}{3}A_x^*A_y^2\\\frac{\partial{A_y}}{\partial{z}}&=-ib_y(z)A_y-\frac{\alpha}{2}A_y+\frac{i\beta_2}{2}\frac{\partial^2A_y}{\partial{t^2}}+i\gamma\left(|A_y|^2+\frac{2}{3}|A_x|^2\right)A_y+\frac{i\gamma}{3}A_y^*A_x^2\end{align}\]

where \(A_x\) and \(A_y\) are the two orthogonal polarization components of the electric field, \(b_x\) and \(b_y\) represent linear birefringence of the fiber.

According to Equation 9.45, the strength of the nonlinear processes, SPM, XPM, and FWM depends not only on the relative orientations of different channels, but also on the state of polarizations. For instance, a channel polarized linearly accumulates more nonlinear phase due to SPM than another channel polarized elliptically.

Optical transmission fibers are nominally not birefringent; however, they still exhibit the so-called residual birefringence that randomly scatters the polarization of the electric field in length scales less than 100∼m.

This polarization scattering length is much smaller than the nonlinear interaction length, which is typically tens of kilometers. Since the polarization state of the electric field changes so fast, the resulting nonlinearity is not what is expected from a linearly or circularly polarized field, but an average over the entire Poincaré sphere.

By substituting the polarization-rotating transformation

\[\tag{9.46}\begin{bmatrix}A_x'\\A_y'\end{bmatrix}=\begin{bmatrix}\cos\theta&\sin\theta{e}^{i\phi}\\-\sin\theta{e}^{-i\phi}&\cos\theta\end{bmatrix}\begin{bmatrix}A_x\\A_y\end{bmatrix}\]

into Equation 9.45 and averaging over \(\theta\) and \(\phi\), the nonlinear effects are averaged over the fast polarization changes resulting in the Manakov equation given by Agrawal, Marcuse et al., and Wai et al.,

\[\tag{9.47}\begin{align}\frac{\partial{A_x}}{\partial{z}}&=-\frac{\alpha}{2}A_x+\frac{i\beta_2}{2}\frac{\partial^2A_x}{\partial{t^2}}+\frac{8i\gamma}{9}(|A_x|^2+|A_y|^2)A_x\\\frac{\partial{A_y}}{\partial{z}}&=-\frac{\alpha}{2}A_y+\frac{i\beta_2}{2}\frac{\partial^2A_y}{\partial{t^2}}+\frac{8i\gamma}{9}(|A_y|^2+|A_x|^2)A_y\end{align}\]

The Manakov equation is simpler than the full vectorial NLS. Since the fast polarization rotations are averaged already, it is not necessary to follow these changes in the fiber.

Moreover, since polarization changes are so fast and random, it does not matter anymore what the input polarization is. For instance, if only two copolarized channels propagate through the fiber, it does not matter whether both channels have linear polarization or circular polarization and the accumulated nonlinearity will be the same at the end of the fiber.

However, the strength of the nonlinear interaction still depends on the relative orientations of the channels. If the channels have the same polarization, they interact more strongly than if they have orthogonal polarizations.

The Manakov equation as given in Equation 9.47 assumes that the relative orientations of the polarizations of different channels remain the same throughout the fiber.

This assumption is true as long as the bandwidth of the total field is narrow enough so that polarization-mode dispersion (PMD) can be ignored.

A consequence of the fast and random polarization rotations in the fiber is that, at the receiver, the electric field is rotated with respect to the transmitter. To demultiplex the orthogonal channels properly, this random rotation has to be corrected.

As these random rotations are slow, several electronic polarization demultiplexing methods based on digital signal processing have been devised to track these rotations and correct them.

However, most of these methods are either data-aided or \(Q\)-value directed and, therefore, they rely on high signal-to-noise ratio. Because of the linear and nonlinear impairments, the signal at the receiver end may be significantly distorted.

This may make it difficult to separate the polarization-multiplexed channels using the data-aided or \(Q\)-value directed methods.

However, a closer look at the Manakov equation shows that it is not necessary to know on what polarization basis the data are encoded to implement backward propagation.

This can be observed easily by verifying that the Manakov equation remains the same under the unitary transformation:

\[\tag{9.48}B=\mathbf{U}A\]

where \(\mathbf{U}\) is an arbitrary 2 × 2 unitary polarization-rotation matrix and \(A=[A_x, A_y]^T\). Therefore, DBP can be applied first and the electronic polarization demultiplexing techniques can be used subsequently to demultiplex the polarization channels correctly.

## 6. Future Research

Even though tremendous progress has been made to understand the techniques of optical fiber nonlinearity compensation including DBP – the focus of this chapter – there is still much to be desired before DBP becomes commercially indispensable.

The two main reasons are computational load and reliability. In terms of computational load, DBP is still too intense for dispersion-unmanaged transmission systems. In terms of reliability, DBP for transmission systems with large PMD becomes ineffective.

Future research must address these two main issues. It is possible to track the effect of PMD in real time and incorporate the channel state of PMD into DBP.

In terms of computational load, it might be worthwhile to consider dispersion-folded DBP not only for installed dispersion-managed systems but also for future transmission systems.

It is possible for a dispersion-managed transmission system using dispersion-folded DBP to outperform a dispersion-unmanaged transmission system using electronic dispersion compensation, both of which are within the realm of current DSP capability. This is true especially if the low-loss of dispersion compensation modules can be further improved.

The next tutorial introduces ** acousto-optic devices and applications**.