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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 5 — Mar. 5, 2007
  • pp: 2166–2177
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Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems

Shiva Kumar and Ling Liu  »View Author Affiliations


Optics Express, Vol. 15, Issue 5, pp. 2166-2177 (2007)
http://dx.doi.org/10.1364/OE.15.002166


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Abstract

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

1. Introduction

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 [25

25. R. Holzlohner, V.S. Grigoryan, C.R. Menyuk, and W.L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002). [CrossRef]

]-[28

28. A. Vanucci, P. Serena, and A. Bononi, “The RP method: a new tool for the iterative solution of the nonlinear Schrodinger equation,” J. Lightwave Technol. 20, 1102–1112 (2002). [CrossRef]

]. For example, in Ref. [25

25. R. Holzlohner, V.S. Grigoryan, C.R. Menyuk, and W.L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002). [CrossRef]

], the linearization approach was used to calculate the bit error rates in dispersion managed soliton systems. In Ref. [26

26. I. Gabitov and S. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996). [CrossRef]

], 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. [27

27. E.E. Narimanov and P. Mitra, “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol. 20, 530–537 (2002). [CrossRef]

], 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 [29

29. S. Kumar and D. Yang, “Second order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23, 2073 (2005). [CrossRef]

]. 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

iuzβ2(z)22ut2=γexp[w(z)]u2u,
(1)

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

u(t,z)=u0(t,z)+γu1(t,z)+γ2u2(t,z)+...,
(2)

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

u(t,0)=x(t,0)+n(t,0)
(3)

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 [26

26. I. Gabitov and S. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996). [CrossRef]

]-[29

29. S. Kumar and D. Yang, “Second order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23, 2073 (2005). [CrossRef]

], total field at the end of the system H up to the first order in γ can be written as

u(t,Lopc)=u(t,0)iγY(t),
(4)

where

Y(t)=0Lopcexp[w(z)]{[u0(t,z)2u0(t,z)]m(t,z)}dz
(5)

denotes the first order perturbation, ⊗ denotes convolution, u 0(t, z) is the zeroth order solution,

u0(t,z)=m٭(t,z)u0(t,0),
(6)

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,

m(t,z)=exp[it22S(z)]2πiS(z)
(7)

Eq. (4) can be written in a matrix form as

UoutH=UinHiγ0Lopcdzexp[w(z)]M(S)V(S,UinH),
(8)

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

u0(t,z)=x(t,z)+n(t,z),
(9)
x(t,z)=x(t,0)m(t,z),n(t,z)=n(t,0)m(t,z)
(10)

The nonlinear phase change at the end of the system H is defined as

δϕoutH(t)=γtan1[Yr(t)x0(t)+n0r(t)+γYi(t)],
γYr(t)x0(t)
(11)

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):

u02u0m=[x2x+2x2n+x2n+n2(2x+n)+n2x]m.
(12)

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

δϕ1H(t)2γx0(t)0Lopcdzexp[w(z)]{[(x2nr)mr][(x2nr)mi]},
(13)
nr(t,z)=n0r(t)mr(t,z)+n0i(t)mi(t,z),
(14)
ni(t,z)=n0i(t)mr(t,z)n0r(t)mi(t,z),
(15)

where X=Xr+iXi, X=m,n. Substituting Eqs. (14) and (15) in Eq. (13), we obtain

δϕ1H(t)2γx0(t)0Lopcdzexp[w(z)]{[x2(n0rmr+n0imi)]mr[x2(n0imrn0rmi)]mi},
(16)

Similarly, the nonlinear phase shift corresponding to the third term in Eq. (12) is

δϕ2H(t)γx0(t)0Lopcdzexp[w(z)]{[(xr2xi2)nr+2xrxini]mr[2xrxinr(xr2xi2)ni]mi},
(17)

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

δϕoutH(t)=δϕ1H(t)+δϕ2H(t).
(18)

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

(UoutH)=(UinH)+iγ0Lopcdzexp[w(z)]M(S)V(S,UinH).
(19)

The output of OPC passes through a system G as shown in Fig. 1. Therefore, the signal input

Fig. 1. System Schematic. 1a. OPC without dispersion inversion (DI) 1b. OPC with DI.

to G is

UinG=(UoutH)=(UinH)+O(γ).
(20)

The output of G can be written as

UoutG=UinGiγLopcLtotdzexp[w(z)]M(S)V(S,UinG)dz,
(21)

where Ltot is the total transmission distance. Using Eqs. (19) and (20) in Eq.(21), we obtain

UoutG=(UinH)iγ0Lopcdzexp[w(z)][M(S)V(S,UinH)]iγLopcLtotdzexp[w(z)][M(S)V(S,(UinH))+O(γ)]dz.
(22)

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,

S(z+Lopc)=S(z).
(23)

Here, Lopc = Ltot/2. The corresponding accumulated dispersion profile is shown in Fig. 2(a).Using Eq. (23) in Eq. (22) and retaining only the terms that are first order in γ, we obtain

UoutG=(UinH)+iγ0Lopcdzexp[w(z)][M(S)V(S,UinH)M(S)V(S,(UinH))]dz.
(24)

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

δϕ1G(t)4γx0(t)0Lopcdzexp[w(z)]{(x2n0imi)mr(x2n0imr)mi},
(25)

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

Next, let us consider the scheme shown in Fig. 1(b) in which the system G is identical to system H except for the sign of S(z)[21

21. P. Kaewplung, T. Angkaew, and K. Kikuchi, “Simultaneous suppression of third order dispersion and side band instability in single channel optical fiber transmission by midway optical phase conjugation employing higher order dispersion management,” J. Lightwave Technol. 21, 1465–1473 (2003). [CrossRef]

]-[24

24. X. Tang and Z. Wu, “Reduction of intrachannel nonlinearity using optical phase conjugation,” IEEE Photon. Technol. Lett. 17, 1863–1865 (2005). [CrossRef]

] as shown in Fig. 1(b), i.e.,

S(z+Lopc)=S(z).
(26)

The corresponding accumulated dispersion map is shown in Fig. 2b. Using Eq. (26) in Eq. (22) and retaining only the first order terms, we obtain

UoutG=(UinH)+iγ0Lopcdzexp[w(z)][M(S)V(S,UinH)M(S)V(S,(UinH))]dz.
(27)
Fig. 2. Accumulated dispersion profile. 2a. without dispersion inversion corresponding to Fig. 1(a), 2b. with dispersion inversion corresponding to Fig. 1(b). The dotted line shows the location of OPC. The dispersion coefficient β2 of the first section is -10 ps2/km.

Using Eqs.(26),(5)-(7), we have

M(S)=M(S),
(28)
VSUinH=VS(UinH)
(29)

Using Eqs. (28) and (29) in Eq. (27), we find that the the ASE-induced nonlinear phase change at the output of G up to the first order in γ is

δϕoutG(t)=0.
(30)

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. [10

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

] to include OPC. Two degrees of freedom of the noise field are of importance [1

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

]. 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 [1

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

]. The analytical expression for the nonlinear phase variance obtained in Ref. [10

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

] is valid when the degree of freedom is two. Here, the approach of Ref.[10

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

] is extended to include the OPC.

x0(t)=ETeffexp[t22T02],
(31)

where T 0 is the half-width at 1/e- intensity point, Teff=√πT 0 and E is the pulse energy. Following the first order perturbation approach [33

33. A. Mecozzi, C.B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000). [CrossRef]

],[10

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

], the signal field just before the OPC is given by

us(LNa2,t)=x0(t)[1+iγEg(mL,t)],
(32)

where

g(z,t)=T02TeffzNaL2f(r,t)dr,
(33)
f(r,t)=exp[w(r)Δ(r)t2]T04+3S2(r)+2iT02S(r),
(34)
Δ(r)=T02iS(r)T02[T02+i3S(r)].
(35)

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

us(NaL,t)=x0(t)[1iγEg(mL,t)+iγEg(0,t)].
(36)

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

δϕm(NaL,t)=γδERe[g(0,t)g(mL,t)].
(37)
g(mL,t)=(Na2m)h(t)Teff,
=g(0,t)mh(t)Teff,
(38)

where

h(t)=T020Lexp[w(r)Δ(r)t2]drT04+3S2(r)+2iT02S(r),
(39)

Using Eq. (38), Eq. (37) reduces to

δϕm(NaL,t)=mγhr(t)δETeff,m<=Na2
(40)

where h r(t)=Re[h(t)]. For the standard configuration without midpoint OPC, the corresponding nonlinear phase change is given by [10

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

]

δϕmstd(NaL,t)=(Nam)γhr(t)δETeff.
(41)

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 (Nam)L to NaL. The amplifier noise added after the OPC is not compensated. Therefore, we have

δϕm(NaL,t)=mγhr(t)δETeff,m<=Na2
=(Nam)γhr(t)δETeff.m>Na2
(42)

Using Eq. (42) and proceeding as in Ref. [10

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

], the variance of the peak nonlinear phase noise due to all the amplifiers can be written as

δϕ2=2γ2ρEhr2(0)Teff2[m=1Na2m2+m=Na2+1Na1(Nam)2],
γ2ρEhr2(0)Na36Teff2,
(43)

where ρ is the ASE power spectral density per polarization. For the standard configuration without OPC, we have [10

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

]

δϕ2std2γ2ρEhr2(0)Na33Teff2
(44)

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 [10

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

], 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

δϕm(NaL,t)=γδET02TeffRe[mLrNaLf(r,t)dr+(r+1)NaNaLf(r,t)dr],
=γδEhr(r)[m+(12r)]Teff,
(45)

where f (r,t) is given by Eq. (34). When m>rNa, the nonlinear phase change is (Nam)γhr( tE/Teff. Therefore, the variance of the peak nonlinear phase noise due to all the amplifiers can be written as

δϕ2=2γ2ρEhr2(0)Teff2[m=1rNa[m+(12r)]2+m=rNa+1Na1(Nam)2],
2γ2ρEhr2(0)Na3(1+6r36r2)3Teff2,
(46)

The optimal location of OPC can be obtained by differentiating Eq. (46) with respect to r which gives

ropt=23,
(47)
δϕ2min=2γ2ρEhr2(0)Na327Teff2.
(48)

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 [14

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

]-[15

15. D. Boivin, G. -K. Chang, J. R. Barry, and M. Hanna, “Reduction of Gordon-Mollenauer phase noise in dispersion-managed systems using in-line spectral inversion,” J. Opt. Soc. Am. B 23, 2019–2023 (2006). [CrossRef]

],[9

9. K. P. Ho, “Mid-Span Compensation of Nonlinear Phase Noise,” Optics Comm. 245, 391–398 (2005). [CrossRef]

] 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

ρ=nsphf(G1),
(49)

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).

Fig. 3. Variance of linear and nonlinear phase noise for a single pulse as a function of the absolute dispersion.

Fig. 4. Variance of linear and nonlinear phase noise of a bit ’0’ using a bit pattern as a function of the absolute dispersion. The parameters are same as that of Fig. 3.

5. Conclusions

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

15.

D. Boivin, G. -K. Chang, J. R. Barry, and M. Hanna, “Reduction of Gordon-Mollenauer phase noise in dispersion-managed systems using in-line spectral inversion,” J. Opt. Soc. Am. B 23, 2019–2023 (2006). [CrossRef]

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A. Chowdhury and R.J. Essiambre, “Optical phase conjugation and psuedo linear transmission,” Opt.Lett. 29, 1105–1107 (2004). [CrossRef] [PubMed]

20.

P. Minzioni, F. Alberti, and A. Schiffini, “Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation,” IEEE Photon. Technol. Lett. 16, 813–815 (2004). [CrossRef]

21.

P. Kaewplung, T. Angkaew, and K. Kikuchi, “Simultaneous suppression of third order dispersion and side band instability in single channel optical fiber transmission by midway optical phase conjugation employing higher order dispersion management,” J. Lightwave Technol. 21, 1465–1473 (2003). [CrossRef]

22.

H. Wei and D.V. Plant, “Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation,” Opt. Express 12, 1938–1958 (2004). [CrossRef] [PubMed]

23.

H. Wei and D.V. Plant, “Intra-channel nonlinearity compensation with scaled translational symmetry,” Opt. Express , 12, 4282–4296 (2004). [CrossRef] [PubMed]

24.

X. Tang and Z. Wu, “Reduction of intrachannel nonlinearity using optical phase conjugation,” IEEE Photon. Technol. Lett. 17, 1863–1865 (2005). [CrossRef]

25.

R. Holzlohner, V.S. Grigoryan, C.R. Menyuk, and W.L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002). [CrossRef]

26.

I. Gabitov and S. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996). [CrossRef]

27.

E.E. Narimanov and P. Mitra, “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol. 20, 530–537 (2002). [CrossRef]

28.

A. Vanucci, P. Serena, and A. Bononi, “The RP method: a new tool for the iterative solution of the nonlinear Schrodinger equation,” J. Lightwave Technol. 20, 1102–1112 (2002). [CrossRef]

29.

S. Kumar and D. Yang, “Second order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightwave Technol. 23, 2073 (2005). [CrossRef]

30.

I. Tomkos, D. Chowdhury, J. Conradi, J. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, R.S. Vodhanel, and C.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan area networks,” IEEE J. Sel. Top. Quantum Electron. 7, 439–460 (2001). [CrossRef]

31.

V. Bhagavatula, G. Berkey, D. Chowdhury, A. Evans, and M. Li, “Novel fibers for dispersion-managed high-bitrate systems,” Optical Fiber Conference, TuD2, 21–22, 1998.

32.

S. Kumar and A.F. Evans, “Collision-induced impairments in in dispersion managed fiber systems” in ”Massive WDM and TDM solitons systems,” A. Hasegawa, Ed. (Kluwer Academic, Dordrecht, 2000), pp. 351–364.

33.

A. Mecozzi, C.B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. 12, 392–394 (2000). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(190.3270) Nonlinear optics : Kerr effect
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.5040) Nonlinear optics : Phase conjugation

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 28, 2006
Revised Manuscript: November 2, 2006
Manuscript Accepted: February 21, 2007
Published: March 5, 2007

Citation
Shiva Kumar and Ling Liu, "Reduction of nonlinear phase noise using optical phase conjugation in quasi-linear optical transmission systems," Opt. Express 15, 2166-2177 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2166


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References

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  15. D. Boivin, G. -K. Chang, J. R. Barry, and M. Hanna, "Reduction of Gordon-Mollenauer phase noise in dispersionmanaged systems using in-line spectral inversion," J. Opt. Soc. Am. B 23, 2019-2023 (2006). [CrossRef]
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  19. A. Chowdhury, and R. J. Essiambre, "Optical phase conjugation and psuedo linear transmission," Opt.Lett. 29, 1105-1107 (2004). [CrossRef] [PubMed]
  20. P. Minzioni, F. Alberti, and A. Schiffini, "Experimental demonstration of nonlinearity and dispersion compensation in an embedded link by optical phase conjugation," IEEE Photon. Technol. Lett. 16, 813-815 (2004). [CrossRef]
  21. P. Kaewplung, T. Angkaew, and K. Kikuchi, "Simultaneous suppression of third order dispersion and side band instability in single channel optical fiber transmission by midway optical phase conjugation employing higher order dispersion management," J. Lightwave Technol. 21, 1465-1473 (2003). [CrossRef]
  22. H. Wei and D. V. Plant, "Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation," Opt. Express 12, 1938-1958 (2004). [CrossRef] [PubMed]
  23. H. Wei and D. V. Plant, "Intra-channel nonlinearity compensation with scaled translational symmetry," Opt. Express,  12, 4282-4296 (2004). [CrossRef] [PubMed]
  24. X. Tang and Z. Wu, "Reduction of intrachannel nonlinearity using optical phase conjugation," IEEE Photon. Technol. Lett. 17, 1863-1865 (2005). [CrossRef]
  25. R. Holzlohner, V. S. Grigoryan, C. R. Menyuk, and W. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightwave Technol. 20, 389-400 (2002). [CrossRef]
  26. I. Gabitov and S. Turitsyn, "Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation," Opt. Lett. 21, 327-329 (1996). [CrossRef]
  27. E. E. Narimanov and P. Mitra, "The channel capacity of a fiber optics communication system: perturbation theory," J. Lightwave Technol. 20, 530-537 (2002). [CrossRef]
  28. A. Vanucci, P. Serena, A. Bononi, "The RP method: a new tool for the iterative solution of the nonlinear Schrodinger equation," J. Lightwave Technol. 20, 1102-1112 (2002). [CrossRef]
  29. S. Kumar and D. Yang, "Second order theory for self-phase modulation and cross-phase modulation in optical fibers," J. Lightwave Technol. 23, 2073 (2005). [CrossRef]
  30. I. Tomkos, D. Chowdhury, J. Conradi, J. Culverhouse, K. Ennser, C. Giroux, B. Hallock, T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, R. S. Vodhanel, and C.-C. Wang, "Demonstration of negative dispersion fibers for DWDM metropolitan area networks," IEEE J. Sel. Top. Quantum Electron. 7, 439-460 (2001). [CrossRef]
  31. V. Bhagavatula, G. Berkey, D. Chowdhury, A. Evans, and M. Li, "Novel fibers for dispersion-managed high-bitrate systems," Optical Fiber Conference, TuD2, 21-22, 1998.
  32. S. Kumar and A. F. Evans, "Collision-induced impairments in in dispersion managed fiber systems" in "Massive WDM and TDM solitons systems," A. Hasegawa, ed., (Kluwer Academic, Dordrecht, 2000), pp. 351-364.
  33. A. Mecozzi, C. B. Clausen, and M. Shtaif, "Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission," IEEE Photon. Technol. Lett. 12, 392-394 (2000). [CrossRef]

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