An analytical expression for the variance of nonlinear phase noise for a quasi-linear system using the midpoint optical phase conjugation (OPC) is obtained. It is shown that the the system with OPC and dispersion inversion (DI) can exactly cancel the nonlinear phase noise up to the first order in nonlinear coefficient if the amplifier and the end point of the system are equidistant from the OPC. It is found that the nonlinear phase noise variance of the midpoint phase-conjugated optical transmission system with DI is smaller than that of the system without DI.
©2007 Optical Society of America
The addition of the amplified spontaneous emission (ASE) on the signal field leads to energy fluctuations of the signal which is translated into phase fluctuations by fiber nonlinearity  leading to bit errors in phase-shift keying (PSK) or differential phase-shift keying (DPSK) transmission systems. This ASE induced nonlinear phase noise (Gordon-Mollenauer phase noise) has drawn significant attention -. The analytical expression for the phase noise variance in the dispersion managed soliton systems were first obtained in Ref. 3. The experimental results of Ref.  show that the nonlinear phase noise can be suppressed using midpoint optical phase conjugation (OPC). In this paper, a theoretical description for such a noise reduction is provided and analytical expressions for the variance of ASE-induced phase noise due to selfphase modulation (SPM) are obtained. In Ref. , it is shown that the nonlinear phase noise can be reduced significantly by a small number of in-line phase conjugators for soliton systems with constant dispersion. Recently, Ref.  also shows the reduction in nonlinear phase noise using optical phase conjugation for dispersion managed soliton systems. In this paper, it is shown that for quasi-linear systems, the variance can be reduced roughly by factors of four and nine if the OPC is located at Ltot/2 and 2Ltot/3, respectively. This is consistent with previously published results obtained -, for other propagation regimes. Further reduction in the variance of nonlinear phase noise is possible if the accumulated dispersion of the link after OPC is inverted with respect to the link before OPC. Previously, a similar dispersion map without OPC has been shown to be effective in reducing the intra-channel nonlinear impairments .
Traditional OPC can undo the effects of fiber nonlinearity only if the fiber link is symmetric in dispersion and power profiles with respect to the location of the OPC . However, it is hard to achieve a symmetric power profile in practice. Instead, Refs. - proposed to place the OPC appropriately within a dispersion map to reduce the intra-channel nonlinear effects. The intra-channel nonlinear impairments can also be suppressed by OPC combined with symmetric dispersion profile , scaled translational symmetry -, or symmetric fiber link . In this paper, it is shown that to suppress the ASE-induced nonlinear phase noise, it is not essential to have the fiber link that is symmetric in power profile with respect to the midpoint OPC, but it is required that the power and dispersion profiles of the fiber link before the OPC should be identical to the link after the OPC, which holds true in many practical systems. Further suppression of the nonlinear phase noise is possible if the links before and after the OPC are identical except for the sign of the dispersion.
In this paper, we use the first order perturbation approach or linearization approach to study the ASE-induced nonlinear phase. In the past, the first order perturbation approach has been used in many applications -. For example, in Ref. , the linearization approach was used to calculate the bit error rates in dispersion managed soliton systems. In Ref. , such an approach was used to prove the existence of breathing solitons in transmission lines with periodic amplifications and periodic dispersion compensation, and in Ref. , it was used to calculate the Shannon’s channel capacity for quasi-linear systems. In this paper, we first calculate the nonlinear phase change using the first order perturbation theory in the absence of ASE for the fixed energy of a pulse. ASE leads to the energy fluctuations and the variance of the resulting phase fluctuations due to nonlinear propagation in fiber is calculated. The use of first order perturbation theory relies on the assumption that nonlinearity is small, i.e. the nonlinear length is much longer than the dispersion length. At higher launch powers, the nonlinear phase shift is comparable to the dispersive phase shift and a higher order theory is required for the accurate description of the optical field evolution . Therefore, the results of this paper is valid only at relatively low launch powers that are typically used in quasi-linear systems.
2. Nonlinear phase noise
The optical field envelope in a periodically amplified transmission system is governed by the nonlinear Schrodinger equation in the lossless form
where β2(z) is the dispersion profile, γ is the nonlinear coefficient, w(z)=∫z 0α(s)ds, α(z) is the fiber loss/amplifier gain profile. The field can be expanded into a series
where uj denotes the jth-order correction. The zeroth order field u 0 corresponds to the solution of the linear part of Eq. (1). Without loss of generality, let us consider an amplifier located at z=0−. The total field after the amplifier can be written as
where x(t,0) ≡ x 0(t) is the input signal field and n(t,0) ≡ n 0(t) is the noise field due to the amplifier. We assume that the noise field n 0(t) is much smaller than the signal filed x 0(t).Consider a dispersion compensated fiber system H of length Lopc and accumulative dispersion profile S(z)=∫z 0β2(x)dx with S(Lopc)=0. The system H consists of several amplifiers and dispersion managed fibers as shown in Fig. 1. Let L be the amplifier spacing. Following the first order perturbation theory -, total field at the end of the system H up to the first order in γ can be written as
denotes the first order perturbation, ⊗ denotes convolution, u 0(t, z) is the zeroth order solution,
with u(t,0)=u 0(t ,0) and uj(t,0)=0,j >0 and m٭(t) is the linear impulse response of the fiber,
Eq. (4) can be written in a matrix form as
where U H in and U H out are vectors corresponding to u(t,0) and u(t,Lopc), respectively,V(S(z),U H in) is a vector corresponding to |u 0(t, z)|2 u 0(t, z) and M(S) is a matrix corresponding to the kernel m(t, z). For simplicity, let us assume that the input signal field x 0(t) is real. The zeroth order field can be decomposed into signal field and noise field as
The nonlinear phase change at the end of the system H is defined as
where X(t)=Xr(t)+ iXi(t), X=Y,n 0. In the above approximation, the second and higher order terms in γ are ignored. To understand the evolution of nonlinear phase noise, let us consider a part of integrand of Eq. (5):
The first term on the right hand side of Eq. (12) is deterministic, fourth and fifth terms are second order in n(t) and we ignore them. Therefore, ASE-induced nonlinear phase noise originates mainly from the second and third terms in Eq. (12). Using Eqs. (9), (5) and (11), the ASE-induced nonlinear phase shift corresponding to the second term in Eq. (12) is given by
Similarly, the nonlinear phase shift corresponding to the third term in Eq. (12) is
Total ASE-induced nonlinear phase change at the end of the system H excluding the first, fourth and fifth terms of Eq. (12) is given by
As shown in Fig.1, the output of the system H passes through an OPC. The output of OPC is simply the complex-conjugate of its input UH out and is given by
The output of OPC passes through a system G as shown in Fig. 1. Therefore, the signal input
to G is
The output of G can be written as
In Eq.(22), the first and second integrals originate from the fiber nonlinearities of systems G and H, respectively.
2.1. System without dispersion inversion
Let us first consider the scheme shown in Fig. 1(a) in which the systems G and H are identical,i.e,
To prove the reduction in nonlinear phase noise using OPC, we substitute Eq. (9) in Eq. (24) and calculate the ASE-induced nonlinear phase change at the end of the system G (corresponding the second term in Eq. (12)) as
In Eq. (16), there are four noise terms in the curly bracket whereas in Eq.(25), there are only two noise terms which are same as the second and third term in the curly bracket of Eq.(16). The absence of terms containing the in-phase noise component n0r in Eq.(25) indicates that the contribution to the phase noise originating from n0r is eliminated using the OPC. Since n0r and n0i are statistically independent and convolution with mr and mi does not change this property, it follows that the OPC provides a partial reduction of the variance of nonlinear phase noise. A similar analysis for the nonlinear phase noise originating from the third term in Eq. (12) also shows the partial reduction.
2.2. System with dispersion inversion
Thus we see that the nonlinear phase noise due to an amplifier located at z=0 can be exactly cancelled up to the first order in γ if the dispersion profile after the OPC is inverted as shown in Fig. 2(b). For a system without dispersion inversion (DI), the nonlinear phase noise due to the in-phase noise component n0r of an amplifier at z=0 can be eliminated using OPC (Eq. (25)) whereas for a system with DI, nonlinear phase noise due to both n0r and n0i are eliminated. The nonlinear phase noise due to an amplifier located at z=nL, z < Lopc becomes zero at (Na − n)L and it builds up from (Na − n)L to the end of the system. Refs. - have shown that intrachannel and inter-channel nonlinear impairments can be significantly reduced using such a dispersion map. For the practical implementation of the DI after OPC, conventional dispersion managed link consisting of standard single mode fibers (SMF) and reverse dispersion fibers (RDF) can not be used because of large difference in effective areas. However, a combination of a non-zero dispersion shifted fiber such as Truewave and a negative dispersion fiber  can be used. In Ref. , fabrication of a dispersion managed fiber consisting of short spans of alternating sections of positive and negative dispersion fibers is reported. For this dispersion managed fiber, the dispersion and dispersion slopes of the adjacent sections are nearly equal in magnitude and opposite in sign and effective areas are nearly equal . The dispersion inversion condition given by Eq. (26) can be easily achieved using the dispersion managed fiber of Ref. . In contrast, for a system without DI (Fig. 1(a)), the effective areas of positive and negative dispersion fiber need not be equal and reverse dispersion fiber (RDF) can be used to compensate the dispersion of a standard single mode fiber.
The systems H or G could be represented by several spans of a dispersion managed fiber consisting of two equal segments with the dispersion of the first segment being anomalous whereas that of the second segment is equal in magnitude but opposite in sign, as shown in Fig.1. From the above derivation, we see that the noise source and the end point of the system should be equidistant from the OPC to cancel the nonlinear phase noise. However, the above condition is not fulfilled for all the amplifiers and clearly, the noise added by the amplifiers located after the OPC can not be compensated and therefore, only partial compensation of nonlinear phase noise is achieved.
3. Variance of nonlinear phase noise
Calculation of the variance of nonlinear phase noise for the system without DI (Fig. 1(a)) directly from Eq. (25) is quite cumbersome. Instead, we extend the approach of Ref.  to include OPC. Two degrees of freedom of the noise field are of importance . One of the noise modes is in phase with the signal and produces an energy shift while the other is in quadrature and produces a linear phase shift. When the noise bandwidth is equal to the signal bandwidth, the contributions from the other noise modes becomes less significant . The analytical expression for the nonlinear phase variance obtained in Ref.  is valid when the degree of freedom is two. Here, the approach of Ref. is extended to include the OPC.
We consider a Gaussian pulse incident on the fiber link shown in Fig. 1(a). The dispersion managed fiber between amplifiers consists of a fiber with anomalous dispersion followed by a fiber with normal dispersion of the same length so that the average dispersion within an amplifier spacing is zero. Let the number of amplifiers in the link be Na-1 and L be the amplifier spacing.Consider an amplifier located at mL, m<=Na/2. Let the signal field at mL be
After complex conjugating Eq. (32) and transmitting the signal in the system G, we find that the signal field at the output of G is
The in-phase component of the noise field added by the amplifier at mL changes the signal energy by δE which is translated into phase fluctuations by the self-phase modulation (SPM).Therefore, change in nonlinear phase due to the noise added by the amplifier at mL up to the first order in γ is
where h r(t)=Re[h(t)]. For the standard configuration without midpoint OPC, the corresponding nonlinear phase change is given by 
Comparing Eqs. (40) and (41), we see that for the system without midpoint OPC, the ASE-induced nonlinear phase noise builds up from mL (m<N a/2) to the end of the transmission line Na L while for the system with midpoint OPC, it builds up only from (Na − m)L to NaL. The amplifier noise added after the OPC is not compensated. Therefore, we have
where ρ is the ASE power spectral density per polarization. For the standard configuration without OPC, we have 
Thus, we see that the variance of nonlinear phase noise is reduced by a factor of 4. For the system with midpoint OPC and DI (Fig. 1(b)), a similar analysis shows that the standard deviation of nonlinear phase noise is same as that for the system without DI (Fig. 1a) up to the first order in γ . If we include the small higher order correction , variance of nonlinear phase noise in a system without DI is found to be slightly higher than the system with DI.
For simplicity, we have assumed that average dispersion within an amplifier spacing is zero and there is no pre/post dispersion compensation. Because of these assumptions, an amplifier and a dispersion managed fiber can be considered as a unit cell and the phase variance due to each of these unit cells becomes identical leading to simple analytical expressions such as Eqs. (43) and (44). In a practically relevant case of non-zero average dispersion and non-zero pre-compensation, the contributions from each of these unit cells could be unequal and a modified analysis is required.
3.1. Optimal location of OPC
So far we have assumed that the OPC is at the midpoint of a transmission system. Now we wish to find the optimal location of OPC. We assume that the systems H and G of Fig. 1a are identical except that the transmission distances in each of these systems are unequal. Let rNaL be the length of the system H where Na − 1 is the total number of amplifiers in a combined system (H and G), L be the amplifier spacing and rNa is an integer.
Proceeding as before, the nonlinear phase change at the end of the combined system due to an amplifier located at mL (m<=rNa) is given by
where f (r,t) is given by Eq. (34). When m>rNa, the nonlinear phase change is (Na−m)γhr( t)δE/Teff. Therefore, the variance of the peak nonlinear phase noise due to all the amplifiers can be written as
The optimal location of OPC can be obtained by differentiating Eq. (46) with respect to r which gives
Comparing the above result with that corresponding to the standard configuration (Eq. (44)), we see that the variance is reduced by a factor of 9 which is consistent with previously published results obtained -, for other propagation regimes.
4. Numerical simulations
To validate the analytical model, numerical simulations of the nonlinear Schrodinger (NLS) equation is carried out with the following parameters: nonlinear coefficient = 2.43 W−1km−1, bit rate=40 Gb/s, wavelength=1.55 μm, fiber lossα =0.2 dB/km, spacing between amplifiers=80 km, nsp=3, launched peak power=3 mW and computational bandwidth=2.4 THz. The dispersion management is achieved by using a 40 Km long anomalous dispersion fiber followed by a normal dispersion fiber of the same length and same absolute dispersion. Pre- /post-compensation fibers are not used. Total transmission distance=800 Km. A Gaussian pulse with full width half-maximum (FWHM) of 12.5 ps is launched to the fiber link. White Gaussian noise with a power spectral density per polarization as given by
is added at each amplifier location. At the end of the transmission line, an ideal optical bandpass filter with a bandwidth of 75 GHz is inserted. The Monte-Carlo simulations of the nonlinear Schrodinger equation is carried out using the split-step Fourier algorithm with 2000 realizations and the variance of the peak phase is computed. Figure 3 shows the variance of the linear and nonlinear phase noise of a single pulse for three different configurations: (i) standard configuration with no OPC and no dispersion inversion (DI) (ii) midpoint OPC only (Fig. 1(a)) and (iii) midpoint OPC and DI (Fig. 1(b)). The horizontal axis in Fig. 3 is the absolute dispersion (|β 2|) of the first or the second fiber segment within an amplifier spacing. The discrepancy between analytical and numerical results are due to the following reasons: (i) ignoring the third order terms in γ in Eq.(43) and (ii) ignoring the second order perturbation term in γ for the optical field in Eq. (32).
In a single channel system, the nonlinear phase change occurs due to the following effects: (i) self-phase modulation (SPM), (ii) intra-channel cross phase modulation (IXPM),(iii) intra-channel four wave mixing (IFWM), (iv) ASE-induced SPM, (v) ASE-induced IXPM, and (vi) ASE-induced IFWM. The phase changes due to SPM and IXPM are deterministic in phase-modulated systems and they do not lead to performance degradation. However, the ASE-induced phase noise due to SPM, IXPM and IFWM could degrade the transmission performance. The phase change due to IFWM (without ASE) depends on the bit-pattern and could lead to nonlinear inter-symbol interference. As the dispersion increases, the ASE-induced phase noise due to intra-channel cross phase modulation (IXPM) could become important , . Therefore, the numerical simulations have been carried out using a psuedo random bit sequence consisting of 4096 bits at a bit rate of 40 Gb/s. To separate the SPM, IXPM and IFWM from the ASE-induced nonlinear phase noise, we first turn off the amplifier noise by setting spontaneous noise factor nsp=0 and obtain the variances σ2 10 for a bit ZERO and σ2 11 for a bit ONE at the end of the link. Next we turn on the amplifier noise and obtain the variance σ2 2j, j=0, 1. The variance of linear and ASE-induced nonlinear phase noise (which includes SPM, IXPM and IFWM) is given by σ2 j=σ2 2j −σ2 1j and is plotted in Fig. 4 for a bit ZER0. The results for bit ONE is very similar since the noise statistics for a bit ZERO and a bit ONE are the same.Comparing Figs. 3 and 4 for a system without OPC, we see that the variance has increased only a little due to ASE-IXPM interaction while according to Ref. , the ASE-induced phase variance due to IXPM is significant. The discrepancy could be due to the fact that the length of the transmission fiber with anomalous dispersion is 100 Km in Ref.  whereas in our simulation it is only 40 Km. In other words, in our example, when the dispersion coefficient β 2=-10 ps2/km, a given bit interacts with at most three or four neighboring bits on both sides while in Ref. , it interacts with 14 or more neighboring bits on both sides, thereby enhancing the IXPM effect. From Fig. 4, we see that the nonlinear phase variance in a system with OPC and without DI is slightly higher than the system with OPC and with DI. This is because the higher order correction to the variance given by Eq. (43) for the system with DI is larger than that for the system without DI.
An analytical expression for the variance of ASE-induced phase noise due to SPM for the quasilinear systems that use OPC is derived. Our analysis pertains to systems that have simplified dispersion maps with zero pre- and post- compensation and zero average dispersion between amplifiers. The results show that the variance can be reduced roughly by a factor of 4 for the systems with midpoint OPC. The nonlinear phase noise due to an amplifier can be exactly cancelled for systems with DI that lie within the validity of first order perturbation theory if the amplifier and the end point of the system are equidistant from the OPC. The variance of the nonlinear phase noise for a system without DI is slightly higher than the system with DI.
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