A tutorial on fiber Kerr nonlinearity effect and its compensation in optical communication systems

The advent of silica-based low-cost standard single-mode fibers revolutionized the whole communication industry. The deployment of optical fibers in the networks induces a paradigm shift in the communication technologies used for long-haul information transfer. However, the communication using the optical fibers is affected by several linear and nonlinear effects. The most common linear effects are attenuation and chromatic dispersion, whereas the dominant nonlinear effect is the Kerr effect. The Kerr effect induces a power-dependent nonlinear distortion for the signal propagating in the optical fiber. The detrimental effects of the Kerr nonlinearity limit the capacity of long-haul optical communication systems. Fiber Kerr nonlinearity compensation using digital signal processing (DSP) techniques has been well investigated over several years. In this paper, we provide a comprehensive tutorial, including the fundamental mathematical analysis, on the characteristics of the optical fiber channel, the origin of the Kerr nonlinearity effect, the theory of the pulse propagation in the optical fiber, and the numerical and analytical tools for solving the pulse propagation equation. In addition, we provide a concise review of various DSP techniques for fiber nonlinearity compensation, such as digital back-propagation, Volterra series-based nonlinearity equalization, perturbation theory-based nonlinearity compensation, and phase conjugation. We also carry out numerical simulation and the complexity evaluation of the selected nonlinearity compensation techniques.


Introduction
The modern high capacity core communication network uses optical fibers to transmit information from one point to another Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
as modulated light pulses.The optical fiber is a dielectric cylindrical waveguide made up of low-cost material silica [1].The basic phenomenon responsible for guiding the light pulses inside the optical fiber is total internal reflection [1].The uncladded optical fibers manufactured in the early 1920s were not suitable for information transfer at long distances [1].In the 1950s, the use of the cladding layer was a starting point for the field of fiber optics and led to a significant improvement in the fiber characteristics [1].During the 1960s, the transmission of images through the glass fibers demonstrated the significant development of the field of fiber optics [1,2].The fibers manufactured during that time had a drawback of extreme power loss of ≫ 1000 dB km −1 while the optical light is transmitted.Further research efforts in the 1970s reduced the loss of the silica-based fibers below 20 dB km −1 [1].During the late 1970s, advancements in fiber fabrication technology significantly reduced the fiber loss down to 0.2 dB km −1 in the 1550 nm wavelength range [1].That eventually led to a breakthrough in the field of optical fiber communication systems [1].During the 1990s, the optical fibers were doped with rareearth elements such as Erbium.That led to the development of the Erbium-doped fiber amplifiers (EDFAs) and fiber lasers [1,3].
The invention of EDFAs and cost-effective manufacturing of standard single-mode fibers (SSMFs) with less than 0.2 dB km −1 attenuation marked the beginning of commercially viable fiber-optic communication systems.Since the deployment, the optical communication systems underwent several technical evolutions to fulfil the requirements of highspeed communications.The emergence of bandwidth-hungry applications, such as cloud services and virtual reality, has fueled the global network traffic increase to a large extent [4][5][6][7].Furthermore, human-centered applications like video gaming and the exchange of multimedia content via smartphones are among the most bandwidth-consuming applications.That leads to a strong requirement for an increase in the access network capacity, and consequently, for the core optical network capacities to meet such ever-increasing traffic demands [7].
The deployment of wavelength division multiplexing (WDM) technology, which enables the multiplexing of several optical signals in the same fiber, dramatically increases the fiber capacity [8,9].A few years ago, an optical superchannel system has been introduced for 400 Gb s −1 or 1 Tb s −1 optical transmission systems [3][4][5][6][7][8][9][10][11][12][13][14][15][16].This system effectively splits the given WDM channel into smaller subcarriers separated by smaller guard-bands and transmitted as a single entity along the optical network.The superchannel technique has several benefits over the single-carrier systems [16].It has lower requirements in terms of optical signal-to-noise ratio (OSNR) and analog-to-digital converters/digital-to-analog converters bandwidth [16].In comparison with single-carrier 400 Gb s −1 and 1 Tb s −1 transmission systems, the superchannel systems exhibit better transmission performance.However, the transmission impairments in the optical fiber degrade the performance of the high data-rate optical superchannel systems.For example, the chromatic dispersion (CD) introduces a frequency-dependent phase shift to the signals and acts as a primary limiting factor in provisioning a reliable long-haul optical communication link [17].It is noteworthy that there are various optical and electrical techniques available to combat the adverse effect of the CD in a long-haul optical communication system [17][18][19][20][21].
Another significant impairment that limits the transmission performance of the long-haul optical communication system is the fiber nonlinearity [7].The nonlinearity effects in the optical fiber are due to an electro-optic effect, referred to as the Kerr effect, which arises from the dependence of the optical fiber refractive index on the transmit signal power [2,7].That makes the optical fiber channel different from other transmission media used for the information transfer.In a linear transmission medium, the information signals are usually perturbed by the additive noise, which generally results in channel capacities monotonically increasing with transmit power, thereby a corresponding increase in the SNR.However, the detrimental effects of Kerr-induced signal nonlinear distortions grow at a faster rate than the SNR capacity gain at higher launch powers.That in turn leads the channel capacity to be a nonmonotonic function of the transmit launch power with a maximum value at a particular launch power termed as optimum launch power.The achievable transmission rate decreases rapidly beyond the optimal power point as the launch power increases due to the corresponding increase in the Kerr-induced signal nonlinear distortions [2].
The transmission performance of the single-channel optical communication systems is mainly limited by the intra-channel Kerr nonlinearity effect.The intra-channel nonlinear interactions can be categorized into three types: (a) self-phase modulation (SPM), (b) intra-channel cross-phase modulation (IXPM), and (c) intra-channel four-wave mixing (IFWM).It is important to mention that the SPM, IXPM, and IFWM can be compensated well using digital nonlinearity compensation (NLC) techniques.In WDM superchannel systems, a significant portion of the nonlinear distortion comes from the nonlinear interaction between the channel under consideration and the co-propagating signals in adjacent channels, referred to as inter-channel nonlinearity effects [2].The interchannel nonlinear effects can be classified into three types: (a) XPM, (b) cross-polarization modulation (XPolM), and (c) FWM [22,23].In recent years, the spectral efficiency of the polarization division multiplexed (PDM) optical communication systems have dramatically improved by introducing a promising detection technique referred to as coherent detection.The coherent detection also enables the implementation of the advanced forward error-correction coding techniques and the adaptive digital signal processing (DSP) algorithms to combat time-varying transmission impairments [24].A plethora of NLC techniques based on DSP has been developed in the literature to deal with the Kerr nonlinearity effect.
In this tutorial, we provide a comprehensive description of the characteristics of the optical fiber channel, the theory behind the origin of the Kerr effect, the development of the pulse propagation equation, the numerical and analytical tools used for solving the pulse propagation equation, and various Kerr-induced nonlinearity effects in the optical fiber channel.We also provide a concise review of various state-of-the-art DSP techniques available in the literature to deal with the detrimental effects of the Kerr-induced fiber nonlinearity effects.For the maximum benefit of the readers, we adopt a didactic approach with the help of a very detailed mathematical interpretation for various optical phenomenons, processes, models, and methods covered in this tutorial.
The remainder of this tutorial is organized as follows.Section 2 explains the characteristics of the optical fiber channel.Section 3 describes the nonlinear susceptibility and the Kerr effect.Section 4 provides the physics behind the pulse propagation in optical fiber.Section 5 demonstrates various mathematical tools used to solve the pulse propagation equation.Section 6 details various Kerr-induced fiber nonlinearity effects in the optical fiber.Section 7 outlines various nonlinearity compensation techniques used to combat the Kerr-induced nonlinearity effects in the optical fiber.Section 8 illustrates the numerical simulation results and the computational complexity evaluation of the selected nonlinearity compensation techniques.Finally, section 9 concludes the tutorial.

Characteristics of optical fiber channel
The optical fiber consists of a glass core surrounded by a cladding layer to confine the modulated light inside the core region.The refractive index n 2 of the cladding layer is slightly lower than that of the core index n 1 to facilitate the total internal reflection [1].Such fibers are generally referred to as stepindex fiber.The step-index optical fibers can be categorized by two parameters, namely core-cladding index difference ∆ and the V parameter, given as: and respectively, where k 0 = 2π λ , λ is the wavelength of light, a is the core radius, and NA = n 2 1 − n 2 2 is the numerical aperture.The V parameter determines the number of propagation modes in the optical fiber.For example, if V < 2.405, then the stepindex fiber supports only one mode.Such fibers are termed as single-mode fibers [1].It is important to mention that, in this tutorial, we consider nonlinearity effects in the single-mode fibers since they are used to realize the long-haul high data rate optical communication systems.
Example 1 .The V parameter of a step-index fiber is given by 2.3 at 1550 nm wavelength.Calculate the NA for a core radius of 5 µm.

Fiber losses
While the optical signal propagates through the optical fiber channel, the fiber losses attenuate the signal power as a function of the transmission distance.The transmitted signal power considering the fiber attenuation can be represented as [1]: where P 0 is the power launched at the input of the fiber, α is the fiber attenuation parameter, and L is the transmission distance.
It is worth mentioning that the fiber attenuation parameter is usually expressed in units of dB km −1 , which can be represented by using (4), as follows: ( The attenuation in optical fiber is caused by several mechanisms, including absorption, scattering, and geometric effects [1].The material impurity in the silica core fiber causes the absorption of the light energy.The hydroxyl ion (OH) absorption is one of the main absorptions in the case of glass fibers [1].The OH absorption causes the multiple absorption peaks in the wavelength range from visible to the infrared band [25].The main source of the scattering loss in optical fiber is due to the Rayleigh scattering [25].During the fabrication of the optical fiber, the variation of the refractive index is caused by the microscopic variations of fiber material component density, randomly distributed material defects, and inhomogeneous material structure [25].The scale of this index variation is much smaller than the wavelength of interest.The energy scattering when the propagating light interacts with such small index variation causes the Rayleigh scattering [25].The significance of the Rayleigh scattering in optical fiber is reduced as the wavelength increases.
The physical bending of the fiber is the main source of the geometric effect causing the signal power attenuation.There are two types of bending loss, including macroscopic and microscopic loss.The macroscopic bending loss is produced whenever the optical fiber is subjected to a significant amount of bending beyond a critical value of curvature [1,2].The microscopic bending loss is comparatively weaker and is caused by the strain or stress distributed along the length of the fiber [1,2].
Example 2 .A single-mode optical fiber exhibits an attenuation of 0.5 dB km −1 and 0.22 dB km −1 at wavelengths 1310 nm and 1550 nm, respectively.Assuming that two optical signals, one at 1310 nm with an optical power of 120 µW and the other at 1550 nm with an optical power of 80 µW are transmitted through the given fiber.Calculate the power levels in µW of these two signals at transmission distances: (a) 15 km and (b) 80 km.

Solution.
Since the attenuation values are given in dB km −1 , we need to convert the given power levels from µW to dBm, as follows:

Chromatic dispersion
In general, the response of a medium to the incident electromagnetic wave depends on the optical frequency of ω.This property of the medium is referred to as the CD [1].The CD has two parts: material dispersion and the waveguide dispersion [26].The material dispersion part of CD effect in optical fiber manifests through the frequency dependence of the refractive index n(w).In the case of the short optical pulses used in the optical communication system, the material dispersion plays a critical role since the optical pulse consists of several frequency components.In the presence of material dispersion, the different spectral components associated with the pulse travel at different speeds given by c/n(w), where c is the speed of light.That will induce pulse broadening [1].
In the weakly nonlinear regime, the dispersion-induced pulse broadening is the dominant impairment and causes severe penalties to the optical communication systems.In the presence of strong nonlinearity, the interplay between dispersion and nonlinearity is quite complicated, which results in quantitatively different behaviour for the pulse distortions in the optical fiber channel [1].The effect of dispersion can be quantitatively accounted by expanding the mode-propagation constant β in a Taylor series about the frequency w 0 at which the pulse spectrum is centered as [1]: where From ( 6), β 2 can be represented as: In reality, the envelope of the optical pulse moves at the group velocity, and the parameter β 2 causes the dispersion of the group velocity, which leads to the pulse broadening [1].This effect is called the group-velocity dispersion (GVD), and β 2 is termed as the GVD parameter [1].
In the optical fiber, a part of the optical signal propagates through the cladding, referred to as the dielectric waveguiding, which causes a slight reduction in the effective mode index n(w) of the core.That results in the waveguide dispersion and must be added to the material dispersion [26].In general, the contribution of the waveguide dispersion is relatively small when compared to the material dispersion except near the zero-dispersion wavelength λ 0 .For standard fibers, the effect of the waveguide dispersion is to shift λ 0 slightly towards the longer wavelengths, such as λ 0 ≈ 1.31 µm.It is important to mention that the quantity dispersion parameter D is related to β 2 as [26]: Example 3 .An optical signal of wavelength 1550 nm is transmitted trough a single-mode fiber having a GVD of −25.509 × 10 3 fs 2 m −1 .Calculate the dispersion parameter.
Solution.Using (9), It is worth mentioning that the waveguide dispersion is dependent on the fiber design parameters such as the core radius a, and the core-cladding index difference ∆.The zerodispersion wavelength λ 0 can be shifted to the vicinity of 1.55 µm by using this feature of the waveguide dispersion.It is also worthy of mentioning that the fiber loss is minimum at 1.55 µm wavelength range [26].Such dispersion-shifted fibers are commonly used in communication systems.The nonlinearity in optical fiber exhibits a different response behaviour depending on the sign of the GVD parameter.For example, for wavelengths λ < λ 0 , the fiber exhibits normal dispersion regime where β 2 > 0. In this regime, the low-frequency components travel faster than the highfrequency components of the same optical pulse.The fiber is said to be in an anomalous dispersion regime when β 2 < 0. In silica-based optical fibers, the anomalous dispersion regime occurs when the light wavelength exceeds the zero-dispersion wavelength, i.e. λ > λ 0 .The anomalous dispersion regime is of particular interest in optical communication systems since it supports the existence of solitons through a balance between the dispersive and nonlinear effects [1].
The CD-induced mismatch in the group velocities of the optical pulses at different wavelengths causes a different speed for those pulses in the optical fiber [1].This feature leads to an important effect referred to as the walk-off effect [1].More specifically, when the fast-moving pulse completely walks through the slower moving pulse, the nonlinear interaction between two optical pulses ceases to occur [1].That is governed by a walk-off parameter defined as [1]: where λ 1 and λ 2 are the centre wavelengths of the two pulses and v g = 1 β1 is the group velocity and is calculated using (7).For pulses of width τ , the walk-off length can be defined as [1]:

Nonlinear susceptibility
For intense electromagnetic fields, the response of the optical fiber is nonlinear.The origin of the nonlinear response is due to the anharmonic motion of bound electrons under the influence of the applied electromagnetic field [27].The electric field of the incident light interacts with the electron and makes it oscillate in accordance with Coulomb's law [27].The oscillating charge resembles an antenna and radiates the electromagnetic energy at the same frequency as the incident field with a different phase shift.The dynamics of the displaced electron under the influence of the applied electric field is a fundamental field of study in quantum mechanics [27].
On the other hand, in the classical electron oscillator model, the electron is modelled as a charged cloud surrounding the nucleus, as shown in figure 1(a).Figure 1(b) shows that the electron charge cloud is displaced when an electric field E x is applied [27].According to Newton's law, the equation of motion for the center of the electron charge cloud can be represented as [27]: where m is the electron mass, x(t) is the displacement, and q e is the electron charge.There exists a force of attraction between the nucleus and the electron charge cloud when the electron charge cloud moves away for the equilibrium position [27].For a small displacement x(t), the restoration force can be approximated as [27]: where K is a constant.The negative sign in (13) indicates that the restoration force acts in a direction opposite to the external force [27].This situation is similar to the case of a simple pendulum pushed away from the equilibrium position by an external force [27].The restoration force due to the gravitation pulls back the pendulum to the equilibrium position.The net force acting on the electron can be represented as [27]: Next, by combining ( 12) and ( 14), we obtain [27]: or where w 0 = K/m is the natural frequency of oscillation.Assume that the applied electric field is of the form as given below [27]: The displacement x(t) due to the applied field also change harmonically in the steady state and try a trial solution as [27]: Substituting ( 17) and ( 18) in ( 16), we get [27]: and The dipole moment of an atom is given as [27]: In general [27]: If we define P = Np as the dielectric polarization, ( 22) can be written as [27]: where N is the number of atoms per unit volume.
For the weak incident electromagnetic field, we can relate the dielectric polarization P to the electric field intensity E as [27]: where ε 0 is the electric permittivity of free space and ε 0 χ (1) is the first-order susceptibility or linear susceptibility [27], and: It is important to mention that if the medium is not isotropic, the susceptibility depends on direction as well and ( 24) is modified as [27]: or P = ε 0 χ (1) .E, (27) where χ (1) is a 3 × 3 matrix and • denotes dot product.
If the incident electromagnetic field is intense, the relation between the restoration force and the displacement is nonlinear, and thereby, the electron cloud oscillation is not harmonic [27].In this case, the relation between the dielectric polarization P and the electric dipoles are nonlinear, which can be generalized as [27]: (1) • E + χ (2) : EE + χ (3) . ..EEE + . . ., (28) where χ ( j) (j = 1, 2, . ..) is the jth order susceptibility.χ ( j) is also a tensor of rank j + 1. χ (1) is the first-order susceptibility and is related to the linear refractive index.The secondorder susceptibility χ (2) generates the second-harmonic and the sum-frequency terms [27].However, since SiO 2 is a symmetric molecule, χ (2) vanishes for the silica glasses [27].As a result, the optical fibers do not exhibit the χ (2) -induced nonlinear effects [27].

The Kerr effect
The nonlinearity effects in optical fiber originate from the third-order susceptibility χ (3) .One of the primary sources of the nonlinearity effect in optical fiber is the χ (3) -induced nonlinear refraction, the Kerr effect, a phenomenon referring to the light intensity-dependent refractive index [27].Assume that the electromagnetic field incident on the optical fiber core has only E x and H y components.Then, the tensor equation in (28) can be simplified for a centrally symmetric dielectric material as [27]: where χ xxxx is a component of the fourth-rank tensor χ (3) .Suppose, the incident optical field is a monochromatic wave given as: To find E 3 x , we should first find the real part of E x , i.e.

Re[E
In the absence of the special phase-matching techniques, the third harmonic terms in (32) can be neglected [27].
Let the polarization at frequency w be: Then, From ( 29), we have: where the imaginary parts of the susceptibility are ignored.Substituting ( 31) and ( 32) into (35), collecting the terms that are proportional to exp (−jwt), and comparing it with (34), we obtain: where χ eff is the effective susceptibility that includes both linear and nonlinear susceptibilities [27].
We can represent the electric field density D as [27]: Substituting ( 36) in (37), we can write: In general, we can represent the electric flux density as [27]: where ε r is the relative permittivity.From ( 38) and ( 39), we can represent ε r as: Since the relative permittivity ε r and refractive index n are related by n 2 = ε r , we can write: where n 0 is the linear refractive index and the second term of ( 41) represents the nonlinear contribution to the refractive index.
From (41), we can represent: where In (43), the term n 2 is called the Kerr coefficient [27].For silica based fiber, the typical value of n 2 varies between 1.2 × 10 −20 m 2 W −1 and 3.2 × 10 −20 m 2 W −1 [27].From (42), it is clear that the nonlinear part of the refractive index n is proportional to the optical intensity |E 0 | 2 .This effect is termed as the Kerr effect [27].

Pulse propagation in optical fibers
The wave propagation in dispersive nonlinear media is governed by the fundamental theory of electromagnetic wave propagation underpinned by the Maxwell's equations [1].Using Maxwell's equations, one can easily show that: where µ 0 is the permeability of the free space.
For the Kerr effect-based nonlinearity in optical fiber, the induced dielectric polarization P(r, t) consists of two parts such as [1]: where P L and P NL are the linear and nonlinear parts which are given as: and

Nonlinear Schrödinger equation
From ( 44) and ( 45), we can represent the wave equation as: where P L and P NL are given by ( 46) and ( 47), respectively.To solve the wave equation in (48), we adopt several simplifying assumptions including [1]: (a) The nonlinear part of the dielectric polarization P NL is treated as a small perturbation to the linear part P L ; (b) A scalar approach is adopted, i.e. the optical field is assumed to maintain the polarization along the length of the fiber; and (c) A quasi-monochromatic assumption for the optical field.
By adopting the slowly varying envelop approximation, we can represent the electric field E(r, t) as [1]: J. Opt.23 (2021) 123502 Tutorial where x is the polarization unit vector and w 0 is the center frequency of the optical pulse spectrum.The dielectric polarization components P L and P NL can also be expressed in similar form as [1]: and The linear component P L can be calculated by substituting (50) in ( 46) as [1]: where Ẽ0 (r, w) is the Fourier transform of E 0 (r, w).
The nonlinear component P NL is obtained by substituting (51) in (47) and after some simplifications we obtain: When ( 49) is substituted in (53) and following the analysis given in section 3.1, we can approximate P NL as [1]: where For simplicity, we adopt a frequency-domain analysis to derive the wave equation for the slowly varying amplitude E 0 (r, t).Substituting ( 49)-( 51) in (48), the Fourier transform Ẽ0 (r, w − w 0 ) can be defined as: which is found to satisfy the Helmholtz equation as given below [1]: where k 0 = w c and Equation ( 57) can be solved using the method of separation of variables.Assuming the solution of the form: where Q(z, w) is a slowly varying function of z and β 0 is the wave number.From (57), we can write two equations for F(x, y) and Q(z, w) as: and The dielectric constant ε(w) can be approximated as: where ∆n is a small perturbation given by: where α is the absorption coefficient.
The first-order perturbation theory can be used to solve (60).After solving using the first-order perturbation theory, the value of β can be represented as [1]: where Similarly, the electric field E(r, t) can be written as [1]: where q(z, t) is the slowly varying pulse envelop [1].The Fourier transform Q(z, w − w 0 ) of q(z, t) satisfies ( 61), which can be represented as [1]: The propagation equation for q(z, t) is obtained by taking the inverse Fourier transform of (67).In (67), the exact functional form of the mode-propagation constant β(w) is rarely known and therefore, it is useful to expand β(w) in Taylor series as in (6) [1].Because of the quasi-monochromatic assumption for the optical field, the cubic and higher-order terms in (6) can be neglected [1].Next, substitute (6) in (67) and take the inverse Fourier transform, we obtain: The term ∆β in (68) includes the effect of loss and the nonlinearity [1].Next, evaluate ∆β using ( 63) and ( 65), and substitute in (68), we obtain the nonlinear Schrödinger equation (NLSE) as: where z is the propagation distance, β 2 is the GVD parameter, α is the signal attenuation parameter, and γ is the nonlinearity coefficient given as [1]: where A eff is the effective area of the fiber and is given as: Using (70), one can calculate the peak nonlinear phase shift accumulated over the link as: where P 0 is the peak power and , with L as the fiber span length.

Solution.
The evaluation of A eff requires the use of modal distribution F(x, y) for the fundamental fiber mode [1].It is worth mentioning that the typical value of A eff varies in the range 20 − 100 µm 2 in the 1550 nm region depending on the fiber design [1].As a result, the nonlinearity coefficient γ takes values in the range 1−10 The NLSE in (69) can be simplified by applying the transformation q(z, t) ≜ u(z, t) exp(− α 2 z), where u(z, t) is the normalized field, and the simplified form can be written as [1]: Example 5 .The long-haul optical fiber transmission system has following parameters: attenuation coefficient α = 0.2 dB km −1 , fiber span length L = 80 km, Kerr coefficient n 2 = 2.6 × 10 −20 m 2 W −1 , wavelength λ 0 = 1550 nm, and the peak power P 0 = 2 dBm.For a transmission distance of 1200 km, calculate lower limit on the effective area A eff of the fiber in which the peak nonlinear phase shift φ NL accumulated over the link should be less than 0.5 rad.Ignore β 2 .
Solution.The first step is to convert α from dB km −1 to 1/km as follows: The number of fiber spans is given by: 1200 km 80 km = 15.
The total nonlinear phase shift = 15φ NL .
As per the requirement 15φ NL < 0.5.
15φ NL < 0.5 Therefore, the effective area A eff should be higher than 105.34 µm 2 to have the peak nonlinear phase shift φ NL less than or equal to 0.5 rad.

Manakov equation
For dual-polarization transmission systems, the coupled NLSE (CNLSE) provides an accurate model for the nonlinear pulse propagation in optical fiber [1].It includes the polarization mode dispersion (PMD) effects in the fiber along with the dispersive and nonlinearity effects.In general, similar to NLSE, the CNLSE also needs to be solved numerically [1].However, the different length scales associated with PMD, GVD, and nonlinearity effects make the numerical evaluation cumbersome.The dispersive and nonlinearity effects vary on a length scale from 10 to 100 km, while the birefringence in optical fiber varies on a length scale of 10-100 m [1].Therefore, the step size used in the numerical evaluation of CNLSE must be less than 1 m [1].That increases the computation time.Based on this fact, we adopt an approximation method to solve the CNLSE.It is observed that the birefringence fluctuation changes the state of polarization (SOP) of the optical field on a short length scale that the field covers the entire Poincaré sphere after a few kilometers [1].As a result, the nonlinearity terms in CNLSE can be averaged over the birefringence fluctuations [1].The resultant propagation equation is referred to as the Manakov equation and can be represented as: where u(z, t) = [u x (z, t) u y (z, t)] † and the superscript † represents the transpose [1].It is clear from (74) that the rapid random variations in the SOP of the optical field reduce the effect of the nonlinearity parameter γ by a factor of 8  9 .For WDM systems, the optical field u(z, t) in ( 74) consists of the sum of all the multiplexed channels, which can be represented as: The Manakov equation for the WDM system consists of both intra-and inter-channel nonlinearity effects such as SPM, XPM, and FWM [1].By substituting (75) in (74), we can represent the propagation equation for the ith channel as: where x and y in this context represent the horizontal and vertical polarizations, respectively, u x/y represents the optical field transmitted on x/y polarization tributaries, m, l, and k denote the symbol indices.It is important to note that in (76), the evolution of the optical field envelop of the ith channel along the optical fiber link consists of the CD part arising from the pulse dispersion, the SPM part generating from the nonlinear interaction of the ith channel to itself, XPM term inducing from the nonlinear interaction of the ith channel with one other co-propagating channel, and the FWM term originating from the nonlinear interaction of the ith channel with two or more co-propagating channels.
In (76), the definitions u CD x/y,i , u SPM x/y,i , u XPM x/y,m̸ =i , and u FWM x/y,k̸ =l,m,l̸ =i,m̸ =i can be represented as: where p ′ , q ′ , and r ′ takes the x and y polarization tributaries.
For simplicity, the space and time variables z, t are not shown in equations ( 77)-(80).

Numerical approximation method
The analytical solution of NLSE in ( 73) is not available other than for some exceptional cases [1].Numerical approximations are typically used to solve the propagation equation in (73).The most commonly used numerical method to solve the pulse propagation problem in nonlinear and dispersive media is the split-step Fourier method (SSFM) [1].
5.1.1.The split-step Fourier method.By separating the linear and nonlinear parts, the NLSE in (73) can be written as [1]: where D and N represent the linear and nonlinear operators, respectively [1].
The SSFM is an iterative signal propagation modelling method in which the optical fiber span is divided into small segments, each having a length of h.More specifically, the signal propagation from z to z + h is carried out in two separate steps [1].First, the linear operator D is set to zero, and only the nonlinearity is taken into account.Second, the nonlinear operator N is set to zero, and only the dispersion is taken into account. Mathematically: (84) In (84), the step size h is selected in such a way that the linear and the nonlinear sections in each fiber segment can be modelled as independent operations [1].
Example 6 .Show that the SSFM in ( 84) is accurate to the second-order in h.

Solution.
We can start with the Baker-Hausdorff formula [28] for two non-commutating operators µA and µB as: where [A, B] = AB − BA.By substituting µ = h, A = D, and B = N in (85), we can write: By substituting (86) in (84), we get: By examining (87), we understand that the dominant error term is the commutator term h 2 2 D, N .From this, we can confirm that the SSFM in (84) is accurate to the second-order in h.The accuracy of the SSFM can be improved by writing the symmetric form of exponential operators in (84).This method is referred to as symmetric SSFM.The symmetric SSFM can be represented as follows [1]: In this method, the nonlinearity is included in the middle of the fiber segment rather than at the segment boundaries, as shown in figure 2. It is important to mention that for small values of step size h, the term exp ´z+h  Example 7 .Show that the symmetric SSFM in ( 88) is accurate to the third-order in h.

Solution.
Using the Baker-Hausdorff formula in [28], we can write: ) . (89) By the substitution 89) can be rewritten as: Considering the part of (90) corresponds to the term µ 2 and substituting the expression for C, we obtain: By expanding (91) following the substitution of µ = h, A = D, B = N and considering only the dominant error term by ignoring all other terms (note that this assumption is only to isolate the dominant error term for the mathematical simplicity of explanation), we obtain the term h 3 16 D, D, N .From this, we can conclude that the symmetric SSFM is accurate to the third-order in h, especially for small values of h.
The symmetric SSFM in (88) has some important advantages over the SSFM implementation in (84), including: (a) The integral in the nonlinear section (i.e. the middle exponential) is useful to incorporate the z dependence of N, especially for large step size h.Also, it can be approximated as exp h N for small values of h.(b) The symmetric SSFM is accurate to the third-order in the step size h compared to the SSFM in (84), which is accurate only to the second-order in h.This fact is evident from examples 6 and 7.
The implementation of the symmetric SSFM is relatively straightforward [1].The fiber span is divided into a large number of segments of size h.The optical signal is propagated from segment to segment as formulated in (88).More specifically, the optical field envelop u(z, t) is first propagated for a distance h/2 with only dispersion effect.At the midplane, the optical field is multiplied by the nonlinear term that represents the nonlinearity effect for the whole segment of length h.Finally, the optical field is propagated the remaining distance of length h/2 with only dispersion effect to obtain the optical field u(z + h, t) [1].

Analytical approximation methods
The SSFM-based numerical solution of the NLSE provides an accurate model for the linear and nonlinear signal propagation in the optical fiber; however, its computational complexity is impractically high [1].This drawback of the SSFM motivates the research community to search for approximation methods to solve the NLSE analytically.The most commonly used analytical approximations used for the solution of NLSE are the Volterra series-based and the regular perturbation (RP) seriesbased methods.

Volterra series-based method.
The Volterra seriesbased analysis is a well-established tool used in the nonlinear systems theory and extensively used to solve nonlinear problems in the field of communications engineering [29][30][31][32][33][34][35].The seminal work of Peddanarappagari and Brandt-Pearce in [34] to analytically approximate the solution of NLSE opens the door of further research interests in the approximate solutions of the fiber propagation equation.These research efforts initiate the adaptation of known communication theory results, such as the equalization techniques for nonlinear channels, for use in the nonlinear optical fiber channels [32].
The Volterra series-based approach can be used to approximate the solution of NLSE analytically.In this approach, the relation between the input and the output of the optical fiber can be represented as a series of transfer function kernels, referred to as Volterra series transfer functions (VSTFs) [29][30][31][32][33][34][35].The VSTF provides the relationship between the Fourier transforms of the input to the fiber X(w) and the output at the end of the fiber link Y(w), as [29][30][31][32][33][34][35]: where H n (w 1 , . . ., w n−1 , w) is the nth order VSTF kernel in the frequency-domain.Due to the symmetries in the silica-based optical fiber, the even-ordered Volterra kernels are zero.The Volterra series is similar to the Taylor series; however, the former captures the memory effect of the nonlinear system, and the latter does not.The VSTF in (92) includes the memory effects caused by the fiber CD and the CD-nonlinearity interplay.More specifically, the VSTF kernel labeled H 1 represents the linear or CD-only response of the fiber channel, and the higher-order kernel terms H 2,3,...,n defines the higher-order nonlinear transfer functions, including the CDnonlinearity interplay.It is important to mention that the VSTF kernels up to third-order provide an acceptable level of accuracy for the typical values of peak launch power and the transmission distance used in the optical fiber communication systems.
For a PDM transmission system, the VSTF kernels up to third-order can be represented as [29][30][31][32][33][34][35]: where U x/y (w) ≜ U x/y (z = 0, w) represents the Fourier transform of the field envelop u x/y (z, t) and VSTF kernels can be given as: Substituting ( 94) and ( 95) in (93) and detaching H 1 (z, w) from H 3 (z, w 1 , w 2 , w), we can modify (93) as [29][30][31][32][33][34][35]: where K 3 is defined as: It is understood from (96) that the pulse propagation in SSMF is modelled by the combination of a linear kernel and a third-order nonlinear kernel.The linear kernel H 1 (z, w) models the dispersion and attenuation effect in the optical fiber; whereas, the third-order nonlinear kernel H 3 (z, w 1 , w 2 , w) models the beating of various frequency components present in the input signal.It is important to mention that, for longhaul links, the complexity of the Volterra series-based method increases corresponding to the increase in the number of spans [29][30][31][32][33][34][35].

Perturbation theory-based method.
The RP theorybased approach has been proposed to analytically approximate the solution of NLSE with a reduced complexity when compared to the Volterra series-based approach.The RP-based solution provides a detailed insight into the complex interaction between the Kerr nonlinearity and the accumulated CD in the optical fiber.In this method, the signal field u(z, t) is expanded in a power series of γ as u(z, t) where k ′ is the order of the solution.Then, substituting u(z, t) in ( 73), we get [36]: where γ represents the nonlinearity coefficient, m, n, l indicate the symbol indices, α denotes the attenuation coefficient, and z stands for the transmission distance.We obtain a set of linear differential equations by equating the terms in (98) that have equal powers of γ on both sides of the equal sign.Accordingly, the differential equation for the k ′ th order solution is given as [36]:

Zeroth-order (or linear) solution.
By substituting k ′ = 0 in (99), the differential equation for the zeroth-order (or linear) solution can be written as [37,38]: By solving (100), the zeroth-order solution at z = L can be written as [37,38]: where ⊗ represents the convolution, h exp −jt 2 2β2z with w as the angular frequency and F −1 {.} as the inverse Fourier transform function.

First-order solution.
The differential equation for the first-order solution is obtained by substituting k ′ = 1 in (99) and can be represented as [37,38]: The first-order distortion field at a transmission distance z = L is obtained by solving (102) and assuming an ideal dispersion compensation at z = L, we get [37,38]: The frequency-domain distortion field at z = L is obtained by taking the Fourier transform of ( 103) and the result is given as [37,38]: where F(z, w) is written as [37,38]: The signal input to the optical fiber can be represented as [37,38]: where P 0 is the peak launch power, a k is the data symbol of the kth pulse, g(z, t) is the pulse temporal waveform at transmission distance z, and T is the symbol duration.Therefore, the product |u 0 (z, t)| 2 u 0 (z, t) in (105) can be represented as [37,38]: Substituting ( 107) in (105), we obtain [36]: where G(z, w) = G(0, w) exp j w 2 β2z 2 with G(0, w) is the Fourier transform of g(0, t).
Substituting ( 108) in ( 104), we get first-order perturbation kernel term in frequency-domain as [37,38]: Next, by taking the inverse Fourier transform of (109), the time-domain first-order kernel term can be represented as [37,38]: For Gaussian pulse shape assumption, i.e.G(0, w) = √ 2πτ 2 exp(− w 2 τ 2 2 ), with τ as the pulse width, the triplet product in (110) can be calculated as [37,38]: It is important to mention that with the Gaussian shape assumption for the input pulse shape, the first-order nonlinearity coefficients can be calculated using analytic expressions, which involve the exponential integral function [39].This will explain in detail in section 7.3.Substituting (111) in ( 110) and integrating w.r.t w, we get [37,38]: ) . ( Next, integrating (112) and carrying out some algebraic simplifications, we obtain the first-order distortion field (or firstorder ghost pulse) as [37,38]: where T m , T l , and T n can be represented as mT, lT, and nT, respectively, k = m + n − l, m, n, l are the symbol indices, a m/l/n is the symbol complex amplitude, and τ is the pulse width.
In the PDM optical system, the input signal field to the fiber is a column vector u(z, t) = [u x (z, t) u y (z, t)] † , where x, y are the polarization tributaries, and † is the transpose.The vector field propagation in the optical fiber is governed by the Manakov equation, which is given as [40]: where I is the identity matrix.It is important to note that (114) is the same as the Manakov equation given in (74), in which ( 114) is written in a more compact form by omitting the space and time variables z, t for the sake of simplicity.By solving (114), the zeroth-and first-order solutions can be written as [40]: and Following the steps from (104) to (113), we get the first-order ghost pulse for the PDM transmission system as [37,38]: (117)

Kerr-induced fiber nonlinearity effects
As stated in section 3, the nonlinearity effects in the optical fiber are due to the Kerr effect, which arises from the dependence of the optical fiber refractive index on the transmit signal power [1]. Figure 3 illustrates different types of Kerr nonlinearity effects in the optical fiber.The nonlinearity effects can be divided into two types: signal-signal and signal-amplified spontaneous emission (ASE) noise nonlinear interaction [7].
In single-channel systems, the intra-channel signal-signal nonlinear interactions can be categorized into three types: (a) SPM, (b) IXPM, and (c) IFWM.The SPM results in a phase modulation induced by the refractive index, which increases with the increase in the input power level.This leads to a frequency chirping effect [7,[41][42][43], which interacts with the CD and causes the spectral broadening of the optical pulse.The IXPM is the result of the refractive index change proportional to the intensity of the neighbouring pulses in the same channel [7,[41][42][43].The IXPM yields a timing jitter between the co-propagating pulses, which leads to performance degradation.The IFWM is caused by the nonlinear interaction between two or more pulses of the same channel [7,[41][42][43].That generates echo or ghost pulses in the time domain, and thus, results in interference between the signal pulses of the same channel.It is important to mention that the SPM, IXPM, and IFWM can be compensated well using digital NLC techniques [7,[41][42][43].
In WDM systems, the inter-channel signal-signal nonlinear effects can also be classified into three types: (a) XPM, (b) XPolM, and (c) FWM.The XPM effect is due to the refractive index change proportional to the intensity of the pulse in the co-propagating channel.The XPM induces frequency chirping and pulses overlapping between channels [7,[41][42][43].The XPM effect consists of two parts, the coherent and incoherent XPM, as given in (79).The coherent XPM is the same as the usual XPM effect, which involves the interaction of two WDM channels.The incoherent XPM results in polarization crosstalk, and the cross-talk coefficient is determined by the polarization cross-product u * y/x,m u x/y,m of the interfering channel [7].The XPolM occurs in PDM-WDM transmission systems when the SOP of a transmitted channel depends on the SOP of other co-propagating channels through Kerr effect.That is due to the random propagation of SOP inside the optical fiber caused by PMD [7,[41][42][43] and causes channel cross-talk for PDM-WDM systems [7,[41][42][43].The FWM in the multichannel system is generated by the nonlinear interaction between two or more co-propagating wavelength channels [7,[41][42][43].The FWM results in significant performance degradation due to cross-talk among different wavelength channels [7,[41][42][43].
Another nonlinearity effect is the Kerr-induced signal-ASE noise interaction, referred to as the Gordon-Mollenauer effect [44,45], which leads to the generation of the nonlinear signalnoise interaction (NSNI) [44,45].The modeling of the impact of NSNI is important because it can contribute to answering the questions regarding the fundamental limits of the performance improvement provided by the NLC techniques [44,45].

Fiber nonlinearity compensation techniques
The impairments due to the Kerr effect degrade the transmission performance of optical systems using higher-order modulation formats.Therefore, NLC is a hot research topic to increase fiber capacity.Several digital and optical techniques have been reported in the literature to combat the effects of fiber nonlinearity.Such techniques are applied either at the transmitter side or in the optical link, or at the receiver side.The introduction of coherent detection in optical communications initiates the use of DSP algorithms to compensate for fiber impairments.Usually, DSP algorithms are implemented either at the transmitter side or at the receiver side, or a combination of both like split nonlinearity compensation techniques [7].Implementing DSP algorithms in the optical fiber link requires optical-to-electrical/electrical-to-optical conversions at the locations where the DSP modules are deployed.That increases the signal latency in the optical fiber link and the overall cost of the link.
Digital NLC techniques represent a key technology and a cost-effective approach to increase the data rate, being adopted for the next generation WDM optical transmission systems [7].In this section, we provide a concise review of the most popular NLC techniques in the literature.

Digital back-propagation
The digital back-propagation (DBP) is based on the SSFM, which represents an effective numerical technique to solve the signal propagation equation.The idea of the DBP technique is to digitally model a fictitious fiber with exactly opposite characteristics when compared to the real fiber used for the transmission [7].In this technique, the optical fiber is first divided into small segments, and the signal propagation in each segment is modelled as a concatenation of linear and nonlinear operations [7].There are different DBP implementations available depending on the order of the linear and nonlinear sections in a given fiber segment [7].The most commonly used approach is the application of linear compensation first because nonlinear effects are more important at high input powers, which is the case at the end of the fictitious fiber [7].The receiver side DBP implementation is shown in figure 4, where N s is the number of steps.
Using SSFM, the output of the CD (or linear) compensation section is given by [7]: x/y (z, w) = U x/y (z, w) exp(−jh( where h is the length of each step, w is the frequency variable and z is the transmission distance.Following that, the nonlinear section is carried out in time-domain as [7]: where 0 < φ < 1 is a real-valued optimization parameter.Single-channel DBP (SC-DBP) has been more widely researched because it is typically considered to be more realistic with the current hardware limitations [7].Since a singlewavelength channel is back-propagated, SC-DBP only compensates for intra-channel nonlinearity, e.g.SPM.In WDM superchannel systems, one way of overcoming the interchannel nonlinear distortions like XPM and FWM induced by the co-propagating subchannels is to use a multi channel-DBP (MC-DBP).The MC-DBP back-propagates the entire WDM channel [7].However, the implementation of the MC-DBP is limited to point-to-point links, and its computational complexity is considered impractical due to the need for massively parallel processing computer systems to implement [7].There are a few reduced complexity DBP implementations are available, such as weighted DBP [7] and correlated DBP [7].However, their computational complexity is still considered high.Even though the DBP implementation complexity is high, it is considered a benchmark to measure the effectiveness of other NLC techniques.

Volterra series-based nonlinear equalizer
The VSTF can be effectively used to model the fiber nonlinearity effects [7].In this technique, after modelling the optical channel as a series of VSTFs, the pth order theory proposed in [33] is adopted to design the inverse VSTF (IVSTF) kernels.These IVSTF kernels can be used to compensate for the fiber Kerr nonlinearity and its interplay with CD.One of the features of the Volterra series-based nonlinear equalizer (VNLE) is that the compensation operation can be performed in parallel [7].That reduces the computational complexity when compared to DBP [7].The principle of the parallel implementation of VNLE is depicted in figure 5, where N spans corresponds to the number of fiber spans.
For each polarization, the compensation operation can be divided into two parts; one is linear, and the other is nonlinear.The linear part consists of CD compensation, and the nonlinear part compensates for the nonlinearity effect.In this technique, the nonlinearity compensation for each span can be carried out in parallel.We obtain the output compensated signal by combining the output of the linear section and the output of each nonlinear stage, as shown  in figure 5.In the literature, there are different variants of VNLE have been proposed, such as modified VNLE (MVNLE) and weighted Volterra series nonlinear equalizer (W-VSNE), to improve the compensation performance and reduce the implementation complexity [7].It is worth noting that the VNLE, MVNLE, and W-VSNE are based on the third-order Volterra series approximation.A fifth-order VNLE has also been proposed in the literature to compensate for the Kerr nonlinearity effect; however, it involves much higher implementation complexity when compared to its third-order counterpart [7].
In the PB-NLC technique, the perturbation coefficients are calculated beforehand and saved in a look-up table.The predistortion technique is implemented by calculating the firstorder distortion field u 1,x/y using (120) and subtract it from the symbol of interest a 0,x/y to generate ã0,x/y , as shown in figure 7.

Phase conjugation-based NLC
A few years ago, a digital phase conjugation (DPC) based technique, referred to as the phase-conjugated twin wave (PCTW), was proposed for the mitigation of the first-order nonlinear distortions in PDM optical transmission systems.However, the performance improvement of the PCTW technique comes at the expense of 50% spectral efficiency reduction [50,51].Figure 8 shows the PCTW scheme applied for a coherent optical orthogonal frequency-division multiplexing (CO-OFDM) system.The data and its phase-conjugate are transmitted on the same subcarrier frequency of the x-and y-polarizations.At the receiver side, the coherent superposition of the phase-conjugate pairs is carried out to cancel the first-order nonlinear distortion, as illustrated in figure 8.
A time-domain implementation of the generalized PCTW was reported in [52], and this method is referred to as conjugate data repetition (CDR).In the CDR technique, each timedomain signal datum is followed by its conjugate pairs, and the received signals in the adjacent time slots are coherently superimposed at the receiver.Since the nonlinearity interference coefficients change slowly in a highly dispersive channel, the nonlinear distortions generated by conjugate repetition data can be self-canceled by superimposing [52].In [52], a theoretical explanation based on time-domain perturbation analysis has been provided for the nonlinear distortion cancellation with the CDR technique.It is also shown that the PCTW and the CDR techniques have similar performance, with a limitation of halving the overall capacity of the coherent optical communication system.The PCTW technique for the CO-OFDM system utilizing the Hermitian symmetry has been proposed in [53] at the expense of 50% spectral efficiency loss.
7.4.1.DPC techniques with improved spectral efficiency.In [54], a spectrally efficient DPC technique for nonlinearity compensation in the CO-OFDM system has been proposed.This technique is based on the transmission of the phaseconjugated pilots (PCPs) and the coherent superposition at the receiver.In this scheme, a portion of the OFDM subcarriers is transmitted as PCPs of the other subcarriers.These phase conjugate pairs are used at the receiver to estimate and compensate for the nonlinear distortions introduced by the channel.In this technique, the spectral redundancy can be adjusted up to 50% according to the targeted performance gain.That can be achieved through the proper selection of the number of PCPs in each OFDM band.In [55], a novel technique is proposed based on the joint processing of two pairs of PCTWs, referred to as dual-PCTW, to avoid the loss of spectral efficiency associated with the use of PCTWs.
In [56], a frequency-domain coding technique, termed as phase-conjugated subcarrier coding (PCSC), combined with electronic dispersion pre-compensation (pre-EDC), has been demonstrated for nonlinearity mitigation in CO-OFDM system.This technique extends the idea of the dual-PCTW concept to process the neighboring OFDM subcarriers jointly.The benefit of this nonlinearity mitigation technique comes from the fact that the nonlinear distortions on neighboring OFDM subcarriers are highly correlated [56].The PCSC scheme cancels the first-order distortion field through coherent superposition without losing spectral efficiency.In [57], two DPC approaches have been proposed to overcome the spectral efficiency issue associated with the CDR and PCTW techniques and the pre-EDC requirement of the PCSC technique.These techniques are referred to as the modified-16-quadrature amplitude modulated (QAM)-CDR (MOD-16-QAM-CDR) and MOD-16-QAM-PCTW techniques.
Table 1 demonstrates a list of the state-of-the-art fiber NLC techniques available in the literature to deal with the adverse effect of the Kerr nonlinearity effect.It also shows the type, location, system, and fiber nonlinearity they compensate for, along with the corresponding list of references.

Numerical simulations and discussions
In this section, we carry out numerical simulations of the selected nonlinearity compensation techniques discussed in section 7. The selected techniques are: (a) DBP, (b) VNLE, (c) PB-NLC, (d) PCSC, and (e) MOD-16-QAM-CDR/MOD-16-QAM-PCTW techniques.In the numerical simulation, we   consider both quasi-Nyquist single-carrier and CO-OFDM systems to evaluate the performances of the selected techniques.Specifically, the single-carrier system is used to evaluate the performance of the DBP, VNLE, and PB-NLC techniques.On the other hand, the CO-OFDM system is used to evaluate the performance of the PCSC and MOD-16-QAM-CDR/MOD-16-QAM-PCTW techniques.For more details on the system setup for the quasi-Nyquist singlecarrier system, please refer to [49].Similarly, refer to [57] for details on the CO-OFDM system setup.The general parameters used for the numerical simulation are listed in table 2.
Figure 9 shows the Q-factor plotted as a function of the launch power for the DBP, VNLE, PB-NLC, and EDC techniques for the quasi-Nyquist single-carrier system at a transmission distance of 2800 km.It is observed that the DBP technique improves the Q-factor performance by ∼1.8 dB when compared to the EDC technique.Also, it can be seen from figure 9 that the VNLE and the PB-NLC techniques show similar Q-factor performances [48,49].[48,57].The results also indicate that the MOD-16-QAM-CDR/MOD-16-QAM-PCTW techniques show considerable Q-factor improvement in the nonlinear regime without using pre-EDC when compared to the PCSC technique, which shows negligible performance improvement in this scenario [57].
Table 3 shows the complexity expressions to compute the number of multiplications/symbol for the considered NLC techniques.In table 3, the parameters N span is the number of fiber spans, N steps is the number of steps/span, N FFT is the fast Fourier transform size, P n is the number of significant perturbation coefficients, and M is the modulation cardinality.
Figure 11 shows the plot of the number of real-valued multiplications/symbol for DBP, VNLE, PB-NLC, and EDC techniques as a function of the number of fiber spans N span for a quasi-Nyquist single-carrier system.In figure 11, N FFT = 4096, and N steps = 1.It is observed that the complexity of the VNLE is slightly higher than the complexity of the PB-NLC technique [7,48].
Figure 12 shows the number of real-valued multiplications/symbol for MOD-16-QAM-CDR/MOD-16-QAM-PCTW, PCSC, and EDC techniques as a function of the number of fiber spans N span for a CO-OFDM system.It is evident from figure 12 that the performance benefit of the MOD-16-QAM-CDR/MOD-16-QAM-PCTW techniques comes with an increase in the computational complexity [48,57].It is important to note that the MOD-16-QAM-CDR/MOD-16-QAM-PCTW and the PCSC techniques are comparatively low complexity NLC techniques, and their complexity is independent of the number of fiber spans N span [48,57].
It is important to mention that the NLC techniques implemented in this section do not consider the effects of the PMD and the laser phase noise to clearly understand the ability of the selected NLC techniques to deal with the fiber nonlinearity effect.However, in reality, the performance of the NLC techniques may be impacted by such impairments.Several works in the literature carried out the investigation on the impact of the PMD on the NLC techniques.In [73], an investigation of the impact of PMD on the performance of  DBP has been carried out.The results indicate that the nondeterministic distributed PMD impairs the DBP performance larger than the Kerr-induced signal-ASE stochastic nonlinear effect and manifests as a fundamental limitation to the achievable fiber channel capacity.In [74], the impact of PMD on the performance of the in-line phase-conjugation technique has been studied.The study results indicate that a more close installation of the in-line phase-conjugation devices significantly improves the NLC performance.Similar to PMD, several research works have been conducted in the literature to investigate the impact of laser phase noise on NLC performance.In [75], analytical modeling of the equalization enhanced phase noise (EEPN) has been developed for evaluating the performance of the optical transmission systems.The findings in [75] demonstrate the importance of considering EEPN in the design of the long-haul optical transmission system.In [76], an experimental study on the impact of EEPN on the optical transmission system has been carried out.The results indicate that the blind phase search algorithm used to recover the carrier phase can partially mitigate the effect of EEPN in an optical transmission system.

Conclusion
In this tutorial, we have elucidated the impact of Kerr-induced fiber nonlinearity effects on long-haul coherent optical transmission systems.Using the classical electron oscillator model, we have explained the origin of the nonlinear susceptibility and the Kerr effect in a silica-based optical fiber.We have discussed the mechanism of the optical pulse propagation in the optical fiber medium using the nonlinear Schrödinger equation derived from Maxwell's equations.Then, we have investigated the numerical and analytical methods commonly used to solve the pulse propagation equation, such as the SSFM, the Volterra series-based analysis, and the regular perturbation seriesbased analysis.Following that, we have demonstrated various state-of-the-art fiber NLC techniques available in the literature to deal with the detrimental effects of fiber Kerr nonlinearity.We have also carried out the performance comparison and the complexity evaluation of the selected NLC techniques for quasi-Nyquist single-carrier and CO-OFDM transmission systems.

Figure 1 .
Figure 1.Classical electron oscillator model.(a) in equilibrium and (b) in the presence of an external field.© 2014 John Wiley & Sons, Ltd Reprinted, with permission, from [27].

Example 4 .
For a given single-mode fiber with Kerr coefficient n 2 = 2.6 × 10 −20 m 2 W −1 and the effective area A eff = 80 µm 2 , calculate the nonlinearity coefficient γ at a wavelength of 1550 nm.

Figure 7 .
Figure 7. Principle of operation of the PB-NLC technique.

Figure 8 .
Figure 8. Principle of the PCTW technique.N is the total number of OFDM subcarriers.

Figure 9 .
Figure 9. Q-factor as a function of the launch power for DBP, VNLC, PB-NLC, and EDC techniques for a quasi-Nyquist single-carrier system at a transmission distance of 2800 km.

Figure 10 .
Figure 10.Q-factor as a function of the launch power for MOD-16-QAM-CDR/MOD-16-QAM-PCTW, PCSC, and EDC techniques for a CO-OFDM system at a transmission distance of 2800 km.

Figure 10
Figure10shows the Q-factor performances of the MOD-16-QAM-CDR/MOD-16-QAM-PCTW and PCSC techniques with and without considering the pre-EDC for a CO-OFDM system at a transmission distance of 2800 km.The result of the 4-QAM with EDC having the same spectral efficiency is also included for the comparison.Results indicate that the MOD-16-QAM-CDR/MOD-16-QAM-PCTW techniques improve the Q-factor performance when compared to the EDC and PCSC techniques at the nonlinear regime of the optical fiber[48,57].The results also indicate that the MOD-16-QAM-CDR/MOD-16-QAM-PCTW techniques show considerable Q-factor improvement in the nonlinear regime without using pre-EDC when compared to the PCSC technique, which shows negligible performance improvement in this scenario[57].Table3shows the complexity expressions to compute the number of multiplications/symbol for the considered NLC techniques.In table 3, the parameters N span is the number of fiber spans, N steps is the number of steps/span, N FFT is the fast Fourier transform size, P n is the number of significant perturbation coefficients, and M is the modulation cardinality.Figure11shows the plot of the number of real-valued multiplications/symbol for DBP, VNLE, PB-NLC, and EDC techniques as a function of the number of fiber spans N span for a quasi-Nyquist single-carrier system.In figure11, N FFT = 4096, and N steps = 1.It is observed that the complexity of the VNLE is slightly higher than the complexity of the PB-NLC technique[7,48].Figure12shows the number of real-valued multiplications/symbol for MOD-16-QAM-CDR/MOD-16-QAM-PCTW,

Figure 11 .
Figure 11.The number of real-valued multiplications/symbol for DBP, VNLE, PB-NLC, and EDC techniques as a function of the number of fiber spans Nspan for a quasi-Nyquist single-carrier system.

Figure 12 .
Figure 12.The number of real-valued multiplications/symbol for MOD-16-QAM-CDR/MOD-16-QAM-PCTW, PCSC, and EDC techniques as a function of the number of fiber spans Nspan for a CO-OFDM system.

Table 1 .
State-of-the-art fiber NLC techniques.