## Intra-channel nonlinearity compensation with scaled translational symmetry

Optics Express, Vol. 12, Issue 18, pp. 4282-4296 (2004)

http://dx.doi.org/10.1364/OPEX.12.004282

Acrobat PDF (436 KB)

### Abstract

It is proposed and demonstrated that two fiber spans in a scaled translational symmetry could cancel out their intra-channel nonlinear effects to a large extent without using optical phase conjugation. Significant reduction of intra-channel nonlinear effects may be achieved in a long-distance transmission line consisting of multiple pairs of translationally symmetric spans. The results have been derived analytically from the nonlinear Schrödinger equation and verified by numerical simulations using commercial software.

© 2004 Optical Society of America

## 1. Introduction

5. C. Pare, A. Villeneuve, and P.-A. Belanger, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. **21**, 459–461 (1996). [CrossRef] [PubMed]

6. D. M. Pepper and A. Yariv, “Compensation for phase distortions in nonlinear media by phase conjugation,” Opt. Lett. **5**, 59–60 (1980). [CrossRef] [PubMed]

7. S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol. **14**, 243–248 (1996). [CrossRef]

8. I. Brener, B. Mikkelsen, K. Rottwitt, W. Burkett, G. Raybon, J. B. Stark, K. Parameswaran, M. H. Chou, M. M. Fejer, E. E. Chaban, R. Harel, D. L. Philen, and S. Kosinski, “Cancellation of all Kerr nonlinearities in long fiber spans using a LiNbO3 phase conjugator and Raman amplification,” OFC 2000, paper PD33.

10. M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Cancellation of third-order nonlinear effects in amplified fiber links by dispersion compensation, phase conjugation, and alternating dispersion,” Opt. Lett. **20**, no. 8, 863–865 (1995). [CrossRef] [PubMed]

11. H. Wei and D. V. Plant, “Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation,” Opt. Express **12**, no. 9, 1938–1958 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1938. [CrossRef] [PubMed]

12. A. Chowdhury and R.-J. Essiambre, “Optical phase conjugation and pseudolinear transmission,” Opt. Lett. **29**, no. 10, 1105–1107(2004). [CrossRef] [PubMed]

14. P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett. **24**, 1454–1456 (1999). [CrossRef]

15. 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]

16. A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett. **13**, 445–447 (2001). [CrossRef]

17. J. Martensson, A. Berntson, M. Westlund, A. Danielsson, P. Johannisson, D. Anderson, and M. Lisak, “Timing jitter owing to intrachannel pulse interactions in dispersion-managed transmission systems,” Opt. Lett. **26**, 55–57 (2001). [CrossRef]

18. P. Johannisson, D. Anderson, A. Berntson, and J. Martensson, “Generation and dynamics of ghost pulses in strongly dispersion-managed fiber-optic communication systems,” Opt. Lett. **26**, 1227–1229 (2001). [CrossRef]

19. M. J. Ablowitz and T. Hirooka, “Resonant nonlinear intrachannel interactions in strongly dispersion-managed transmission systems,” Opt. Lett. **25**, 1750–1752 (2000). [CrossRef]

20. M. J. Ablowitz and T. Hirooka, “Intrachannel pulse interactions in dispersion-managed transmission systems: timing shifts,” Opt. Lett. **26**, 1846–1848 (2001). [CrossRef]

21. M. J. Ablowitz and T. Hirooka, “Intrachannel pulse interactions in dispersion-managed transmission systems: energy transfer,” Opt. Lett. **27**, 203–205 (2002). [CrossRef]

22. T. Hirooka and M. J. Ablowitz, “Suppression of intrachannel dispersion-managed pulse interactions by distributed amplification,” IEEE Photon. Technol. Lett. **14**, 316–318 (2002). [CrossRef]

16. A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett. **13**, 445–447 (2001). [CrossRef]

22. T. Hirooka and M. J. Ablowitz, “Suppression of intrachannel dispersion-managed pulse interactions by distributed amplification,” IEEE Photon. Technol. Lett. **14**, 316–318 (2002). [CrossRef]

23. R. Hainberger, T. Hoshita, T. Terahara, and H. Onaka, “Comparison of span configurations of Raman-amplified dispersion-managed fibers,” IEEE Photon. Technol. Lett. **14**, 471–473 (2002). [CrossRef]

11. H. Wei and D. V. Plant, “Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation,” Opt. Express **12**, no. 9, 1938–1958 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1938. [CrossRef] [PubMed]

## 2. Basics of nonlinear wave propagation in fibers

*β*(

*ω*) as a function of the optical frequency

*ω*[24, 25]. When a fiber transmission line is heterogeneous along its length, the propagation constant could also depend on the longitudinal position

*z*in the line, and may be denoted as

*β*(

*z,ω*). The slow-varying envelope form,

*A*(

*z,t*) in an optical fiber of length

*L*is governed by the nonlinear Schrödinger equation (NLSE) [9, 25],

*z*∈ [0,

*L*], in the retarded reference frame with the origin

*z*= 0 moving along the fiber at the signal group-velocity. In the above equation, α(

*z*) is the loss/gain coefficient,

*z*-dependent dispersion coefficients of various orders,

*γ*(

*z*) is the Kerr nonlinear coefficient of the fiber,

*g*(

*z,t*) is the impulse response of the Raman gain spectrum, and ⊗ denotes the convolution operation [9]. Note that all fiber parameters are allowed to be

*z*-dependent, that is, they may vary along the length of the fiber. Because of the definition in terms of derivatives,

*β*

_{2}may be called the second-order dispersion (often simply dispersion in short), while

*β*

_{3}may be called the third-order dispersion, so on and so forth. The engineering community has used the term dispersion for the parameter

*D*=

*dλ*, namely, the derivative of the inverse of group-velocity with respect to the optical wavelength, and dispersion slope for

*S*=

*dD*/

*dλ*[1]. Although

*β*

_{2}and

*D*are directly proportional to each other, the relationship between

*β*

_{3}and

*S*is more complicated. To avoid confusion, this paper adopts the convention that dispersion and second-order dispersion are synonyms for the

*β*

_{2}parameter, while dispersion slope and third-order dispersion refer to the same

*β*

_{3}parameter, and similarly the slope of dispersion slope is the same thing as the fourth-order dispersion

*β*

_{4}.

*A*(

*z, t*) in the time domain can be represented equivalently in the frequency domain by

*Ã*(

*z*

_{1},ω) at

*z*=

*z*

_{1}would be transformed into

*Ã*(

*z*

_{2},

*ω*) =

*H*(

*z*

_{1},

*z*

_{2},

*ω*)

*Ã*(

*z*

_{1},

*ω*) at

*z*

_{2}≤

*z*

_{1}, where the transfer function

*H*(

*z*

_{1},

*z*

_{2},

*ω*) is defined as,

*z*

_{1},

*z*

_{2}) is the concatenation of three linear operations: firstly Fourier transform is applied to convert a temporal signal into a frequency signal, which is then multiplied by the transfer function

*H*(

*z*

_{1},

*z*

_{2},

*ω*), finally the resulted signal is inverse Fourier transformed back into the time domain. In terms of the impulse response,

*z*

_{1},

*z*

_{2}) may also be represented as,

*z*

_{1},

*z*

_{2}) on a time-dependent function is to convolve the function with the impulse response. All linear operators P(

*z*

_{1},

*z*

_{2}) with

*z*

_{1}≤

*z*

_{2}, also known as propagators, form a semigroup [26] for the linear evolution governed by Eq. (4).

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]

*A*(

*z*

_{2},

*t*) as a result of nonlinear propagation of a signal

*A*(

*z*

_{1},

*t*) from

*z*

_{1}to

*z*

_{2}≥

*z*

_{1}, may be approximated using,

*A*(

*z*

_{2},

*t*)≈

*A*

_{0}(

*z*

_{2},

*t*) amounts to the zeroth-order approximation which neglects the fiber nonlinearity completely, whereas the result of first-order approximation

*A*(

*z*

_{2},

*t*)≈

*A*

_{0}(

*z*

_{2},

*t*) +

*A*

_{1}(

*z*

_{2},

*t*) accounts in addition for the lowest-order nonlinear products integrated over the fiber length. The term

*A*

_{1}(·,

*t*) is called the first-order perturbation because it is linearly proportional to the nonlinear coefficients

*γ*(·) and

*g*(·,

*t*).

## 3. Theory of intra-channel nonlinearity compensation using scaled translational symmetry

11. H. Wei and D. V. Plant, “Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation,” Opt. Express **12**, no. 9, 1938–1958 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1938. [CrossRef] [PubMed]

**12**, no. 9, 1938–1958 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1938. [CrossRef] [PubMed]

*z*∈ [0,

*L*] and ∈

*t*∈ (-∞, +∞), where

*α*(

*z*),

*β*

_{2}(

*z*),

*β*

_{3}(

*z*), and

*γ*(

*z*) denote the loss coefficient, second-order dispersion, third-order dispersion, and Kerr nonlinear coefficient respectively for one fiber stretching from

*z*= 0 to

*z*=

*L*> 0, while the primed parameters are for the other fiber stretching from

*z′*= 0 to

*z′*=

*L*/

*R*,

*R*> 0 is the scaling ratio,

*A*(

*z,t*) and

*A′*(

*z′,t*) are the envelopes of optical amplitude in the two fiber segments respectively. Even though the effect of dispersion slope may be neglected within a single wavelength channel, the inclusion of the

*β*

_{3}-parameters in the scaling rules of Eq. (11) ensures that good dispersion and nonlinearity compensation is achieved for each wavelength channel across a wide optical band. When a pair of such fiber segments in scaled translational symmetry are cascaded, and the signal power levels are adjusted in accordance with Eq. (11), it may be analytically proved that both the timing jitter and the amplitude fluctuation due to intra-channel nonlinear interactions among overlapping pulses are compensated up to the first-order perturbation of fiber nonlinearity, namely, up to the linear terms of the nonlinear coefficient. Since the dispersive and nonlinear transmission response is invariant under the scaling of fiber parameters and signal amplitudes as in Eq. (11) [9, 11

**12**, no. 9, 1938–1958 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1938. [CrossRef] [PubMed]

*R*= 1. The cascade of such two spans would constitute a transmission line stretching from

*z*= 0 to

*z*= 2

*L*, with the fiber parameters satisfying,

*z*∈ [0,

*L*] and ∀

*t*∈ (-∞, +∞). The translational symmetry is illustrated in Fig. 1 with plots of signal power and accumulated dispersion along the propagation distance.

*A*(

*z,t*) = Σ

_{k}

*u*

_{k}(

*z,t*), where

*u*

_{k}(

*z,t*) denotes the pulse in the

*k*th bit slot and centered at time

*t*=

*kT*, with

*k*∈

**Z**and

*T*> 0 being the bit duration. The following NLSE describes the propagation and nonlinear interactions among the pulses [13],

*m*=

*k*or

*n*=

*k*contribute to self-phase modulation and intra-channel cross-phase modulation (XPM), while the rest with both

*m*≠

*k*and

*n*=

*k*are responsible for intra-channel four-wave mixing (FWM) [13]. It is assumed that all pulses are initially chirp-free or they may be made so by a dispersion compensator, and when chirp-free the pulses

*u*

_{k}, ∀

*k*∈

**Z**should all be real-valued. This includes the modulation schemes of conventional on-off keying as well as binary phase-shift keying, where the relative phases between adjacent pulses are either 0 or

*π*. It is only slightly more general to allow the pulses being modified by arithmetically progressive phase shifts

*ϕ*

_{k}=

*ϕ*

_{0}+

*k*∆

*ϕ*, ∀

*k*∈

**Z**, with

*ϕ*

_{0},∆

*ϕ*∈ [0,2

*π*),because Eq. (13) is invariant under the multiplication of phase factors exp(

*iϕ*

_{k}) to

*u*

_{k}, ∀

*k*∈

**Z**. The linear dependence of

*ϕ*

_{k}on

*k*is in fact equivalent to a readjustment of the frequency and phase of the optical carrier. The pulses may be return-to-zero (RZ) and nonreturn-to-zero (NRZ) modulated as well, for an NRZ signal train may be viewed the same as a stream of wide RZ pulses with the half-amplitude points (with respect to the peak amplitude) on the rising and falling edges separated by one bit duration.

*z*

_{1},

*z*

_{2}∈ [0,2

*L*], and,

^{-1}[

*H*(

*z*

_{1},

*z*

_{2},

*ω*)] up to a constant phase factor. The impulse response defines a linear propagator P(

*z*

_{1},

*z*

_{2}) as in Eq. (8). In reality, the signal evolution is complicated by the Kerr nonlinear effects. Nevertheless, the nonlinearity within each fiber span may be sufficiently weak to justify the application of the first-order perturbation theory:

*k*∈

**Z**, where

*u*

_{k}(

*z,t*) ≈

*v*

_{k}(

*z,t*) is the zeroth-order approximation which neglects the fiber nonlinearity completely, whereas the result of first-order perturbation

*u*

_{k}(

*z,t*) ≈

*v*

_{k}(

*z,t*)+

*z,t*) accounts in addition for the nonlinear products integrated over the fiber length. For the moment, it may be assumed that both fiber spans are fully dispersion- and loss-compensated to simplify the mathematics. It then follows from the translational symmetry of Eq. (12) that

*b*

_{2}(0,

*z*+

*L*) = -

*b*

_{2}(0,

*z*),

*α*(

*s*)

*ds*=

*α*(

*s*)

*ds*,

*γ*(

*z*+

*L*) =

*γ*(

*z*), ∀

*z*∈ [0,

*L*], and

*v*

_{k}(2

*L,t*) =

*v*

_{k}(

*L,t*) =

*v*

_{k}(0,

*t*) =

*u*

_{k}(0,

*t*), which is real-valued by assumption, ∀

*k*∈

**Z**. It further follows that

*h*(0,

*z*+

*L,t*) =

*h*

^{*}(0,

*z,t*) and

*h*(

*z*+

*L*,2

*L,t*) =

*h*

^{*}(

*z*,2

*L,t*), hence,

*z*∈ [0,

*L*]. Consequently, the pulses at

*z*and

*z*+

*L*are complex conjugate,namely,

*v*

_{k}(

*z*+

*L,t*) =

*z,t*), ∀

*k*∈

**Z**, ∀

*z*∈ [0,

*L*]. At the end

*z*= 2

*L*, a typical term of nonlinear mixing reads,

*L,t*) is purely imaginary-valued, which is in quadrature phase with respect to the zeroth-order approximation

*v*

_{k}(2

*L,t*) =

*u*

_{k}(0,

*t*), ∀

*k*∈

**Z**. When the span dispersion is not fully compensated, namely,

*b*

_{2}(0,

*L*) ≠ 0, the input pulses to the first span at

*z*= 0 should be pre-chirped by an amount of dispersion equal to - ½

*b*

_{2}(0,

*L*), so that the input pulses to the second span at

*z*=

*L*are pre-chirped by ½

*b*

_{2}(0,

*L*) as a consequence. In other words, the input signals to the two spans should be oppositely chirped. Under such condition, the equation

*v*(

_{k}*z*+

*L,t*) =

*z,t*), ∀

*k*∈ [0,

*L*],∀

*k*∈

**Z**, are still valid, so are the above argument and the conclusion that

*v*

_{k}and

*v*

_{k}and

*u*

_{k}|

^{2}= |

*v*

_{k}+

^{2}= |

*v*

_{k}|

^{2}+ |

^{2}, where |

^{2}is quadratic, or of the second-order, in terms of the Kerr nonlinear coefficient, ∀

*k*∈

**Z**. This fact has significant implications to the performance of a transmission line. Firstly, it avoids pulse amplitude fluctuations due to the in-phase beating between signal pulses and nonlinear products of intra-channel FWM, which could seriously degrade the signal quality if not controlled [13, 15

15. 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]

16. A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett. **13**, 445–447 (2001). [CrossRef]

21. M. J. Ablowitz and T. Hirooka, “Intrachannel pulse interactions in dispersion-managed transmission systems: energy transfer,” Opt. Lett. **27**, 203–205 (2002). [CrossRef]

14. P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett. **24**, 1454–1456 (1999). [CrossRef]

18. P. Johannisson, D. Anderson, A. Berntson, and J. Martensson, “Generation and dynamics of ghost pulses in strongly dispersion-managed fiber-optic communication systems,” Opt. Lett. **26**, 1227–1229 (2001). [CrossRef]

19. M. J. Ablowitz and T. Hirooka, “Resonant nonlinear intrachannel interactions in strongly dispersion-managed transmission systems,” Opt. Lett. **25**, 1750–1752 (2000). [CrossRef]

15. 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]

**13**, 445–447 (2001). [CrossRef]

*k*th pulse may be calculated as,

*k*∈

**Z**. In the above calculation, the |

^{2}terms are simply neglected as they represent second-order nonlinear perturbations. It may be noted that our mathematical formulation and derivation are straightforwardly applicable to transmission lines with scaled mirror symmetry for compensating intra-channel nonlinear effects without using OPC, and provide a theoretical framework of intra-channel nonlinearity that is more general than previous discussions [14

14. P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett. **24**, 1454–1456 (1999). [CrossRef]

**12**, 392–394 (2000). [CrossRef]

**13**, 445–447 (2001). [CrossRef]

17. J. Martensson, A. Berntson, M. Westlund, A. Danielsson, P. Johannisson, D. Anderson, and M. Lisak, “Timing jitter owing to intrachannel pulse interactions in dispersion-managed transmission systems,” Opt. Lett. **26**, 55–57 (2001). [CrossRef]

18. P. Johannisson, D. Anderson, A. Berntson, and J. Martensson, “Generation and dynamics of ghost pulses in strongly dispersion-managed fiber-optic communication systems,” Opt. Lett. **26**, 1227–1229 (2001). [CrossRef]

19. M. J. Ablowitz and T. Hirooka, “Resonant nonlinear intrachannel interactions in strongly dispersion-managed transmission systems,” Opt. Lett. **25**, 1750–1752 (2000). [CrossRef]

20. M. J. Ablowitz and T. Hirooka, “Intrachannel pulse interactions in dispersion-managed transmission systems: timing shifts,” Opt. Lett. **26**, 1846–1848 (2001). [CrossRef]

21. M. J. Ablowitz and T. Hirooka, “Intrachannel pulse interactions in dispersion-managed transmission systems: energy transfer,” Opt. Lett. **27**, 203–205 (2002). [CrossRef]

22. T. Hirooka and M. J. Ablowitz, “Suppression of intrachannel dispersion-managed pulse interactions by distributed amplification,” IEEE Photon. Technol. Lett. **14**, 316–318 (2002). [CrossRef]

## 4. Optimal setups of fiber-optic transmission lines

**12**, no. 9, 1938–1958 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1938. [CrossRef] [PubMed]

31. M. Vasilyev, B. Szalabofka, S. Tsuda, J. M. Grochocinski, and A. F. Evans, “Reduction of Raman MPI and noise figure in dispersion-managed fiber,” Electron. Lett. **38**, no. 6, 271–272 (2002). [CrossRef]

34. C. Rasmussen, T. Fjelde, J. Bennike, F. Liu, S. Dey, B. Mikkelsen, P. Mamyshev, P. Serbe, P. van der Wagt, Y. Akasaka, D. Harris, D. Gapontsev, V. Ivshin, and P. Reeves-Hall, “DWDM 40G transmission over trans-Pacific distance (10,000 km) using CSRZ-DPSK, enhanced FEC and all-Raman amplified 100 km UltraWave fiber spans,” OFC 2003, paper PD18.

31. M. Vasilyev, B. Szalabofka, S. Tsuda, J. M. Grochocinski, and A. F. Evans, “Reduction of Raman MPI and noise figure in dispersion-managed fiber,” Electron. Lett. **38**, no. 6, 271–272 (2002). [CrossRef]

34. C. Rasmussen, T. Fjelde, J. Bennike, F. Liu, S. Dey, B. Mikkelsen, P. Mamyshev, P. Serbe, P. van der Wagt, Y. Akasaka, D. Harris, D. Gapontsev, V. Ivshin, and P. Reeves-Hall, “DWDM 40G transmission over trans-Pacific distance (10,000 km) using CSRZ-DPSK, enhanced FEC and all-Raman amplified 100 km UltraWave fiber spans,” OFC 2003, paper PD18.

*D′*= -80 ps/nm/km to compensate 100 km NZDSF with dispersion

*D*= 8 ps/nm/km and loss

*α*= 0.2 dB/km. The first 4 km of the DCF may be made highly lossy by a special treatment in manufacturing or packaging, with a loss coefficient

*α′*= 2 dB/km to form a scaled translational symmetry with respect to the first 40 km NZDSF for optimal nonlinearity compensation. However, the remaining 6 km DCF may ignore the scaling rules and have a much lower nominal loss

*α′*= 0.6 dB/km. The total loss is reduced by 8.4 dB as compared to a DCM that complies strictly with the scaling rules throughout the length of the DCF. Another important parameter of DCFs is the effective modal area, or more directly the nonlinear coefficient. Traditional designs of DCFs have always strived to enlarge the modal area so to reduce the nonlinear effects of DCFs. However, for DCFs used in our method of nonlinearity compensation, there exists an optimal range of modal area which should be neither too large nor too small. According to the scaling rules in Eq. (11), a DCF with a large modal area may require too much signal power to generate sufficient nonlinearity to compensate the nonlinear effects of a transmission fiber, when optical amplifiers may have difficulty to produce that much signal power. On the other hand, when the effective modal area is too small, the scaling rules of Eq. (11) dictate a reduced power level for the optical signal in the DCF, which may be more seriously degraded by optical noise, such as the amplified-spontaneous-emission noise from an amplifier at the end of the DCF.

## 5. Simulation results and discussions

*α*= 0.2 dB/km, dispersion

*D*= 16 +

*δD*ps/nm/km, and dispersion slope

*S*= 0.055 ps/nm

^{2}/km, effective modal area

*A*

_{eff}= 80

*μ*m

^{2}, while the RDF has

*α*= 0.2 dB/km,

*D*= -16 ps/nm/km,

*S*= -0.055 ps/nm

^{2}/km, and

*A*

_{eff}= 30

*μ*m

^{2}. Fiber-based pre- and post-dispersion compensators equalize 11/24 and 13/24 respectively of the total dispersion accumulated in the transmission line. Both the SMF and the RDF have the same nonlinear index of silica

*n*

_{2}= 2.6 × 10

^{-20}m

^{2}/W. The transmitter has four 40 Gb/s WDM channels. The center frequency is 193.1 THz, and the channel spacing is 200 GHz. All four channels are co-polarized and RZ-modulated with 33% duty cycle and peak power of 15 mW for the RZ pulses. The MUX/DEMUX filters are Bessel of the 7th order with 3dB-bandwidth 80 GHz. The electrical filter is third-order Bessel with 3dB-bandwidth 28 GHz. The results of four-channel WDM transmissions have been compared with that of single-channel transmissions, with no clearly visible difference observed, which indicates the dominance of intra-channel nonlinearity and the negligibility of inter-channel nonlinear effects. Several trials with various values for

*δD*have been simulated for each test system. The following figures present the eye diagrams of optical pulses after wavelength DEMUX, in order to signify the nonlinear deformation (timing and amplitude jitters) of optical pulses and the generation of ghost pulses. Fig. 5 shows the received optical pulses of

*δD*= 0 for the two test systems, with the amplifier noise being turned off to signify the nonlinear impairments (right diagram) and the effectiveness of nonlinearity compensation (left diagram). Clearly shown is the suppression of nonlinear impairments by using scaled translational symmetry, and especially visible is the reduction of pulse timing jitter, as seen from the thickness of the rising and falling edges as well as the timing of pulse peaks. In both eye diagrams, there are optical pulses with small but discernable amplitudes above the floor of zero signal power, which could be attributed to ghost-pulse generation [14

**24**, 1454–1456 (1999). [CrossRef]

**26**, 1227–1229 (2001). [CrossRef]

**25**, 1750–1752 (2000). [CrossRef]

*δD*= 0.2 ps/nm/km was set for the two test systems of Fig. 3 and Fig. 4 respectively, in order to showcase that a mirror-symmetric ordering of pairwise translationally symmetric fiber spans is fairly tolerant to the residual dispersions in individual fiber spans. In this setting, each fiber span has 10 or 7.87 ps/nm/km worth of residual dispersion, and the accumulated dispersion totals 107.22 ps/nm/km for the entire transmission line. Importantly, the pre- and post-dispersion compensators are set to compensate 11/24 and 13/24 respectively of the total dispersion, ensuring at least approximately the complex conjugation between the input signals to each pair of spans in scaled translational symmetry. The amplifier noise is also turned on. The transmission results, as shown in Fig. 7, are very similar to that with

*δD*= 0, which demonstrates the dispersion tolerance nicely. In a better optimized design to tolerate higher dispersion mismatch |

*δD*|, either SMFs or RDFs may be slightly elongated or shortened in accordance with the value of

*δD*, such that the same residual dispersion is accumulated in all spans. As an example,

*δD*is set to 0.6 ps/nm/km and each 39.35-km SMF is elongated by 0.385 km, so that all spans have the same residual dispersion of 30 ps/nm/km, and the whole transmission line accumulates 360 ps/nm/km worth of dispersion. The pre- and post-dispersion compensators equalize 360×11/24= 165 and 360×13/24 = 195 ps/nm/km worth of dispersion respectively. The gain of each 15.74-dB EDFA is increased to 15.817 dB in correspondence to the elongation of the 39.35-km SMF. The amplifier noise is still on. The transmission results are shown in Fig. 8.

## 6. Conclusion

## Acknowledgments

## References and links

1. | A. H. Gnauck and R. M. Jopson, “Dispersion compensation for optical fiber systems,” in |

2. | F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” in |

3. | V. Srikant, “Broadband dispersion and dispersion slope compensation in high bit rate and ultra long haul systems,” OFC2001, paper TuH1. |

4. | M. J. Li, “Recent progress in fiber dispersion compensators,” European Conference on Optical Communication 2001, paper Th.M.1.1. |

5. | C. Pare, A. Villeneuve, and P.-A. Belanger, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. |

6. | D. M. Pepper and A. Yariv, “Compensation for phase distortions in nonlinear media by phase conjugation,” Opt. Lett. |

7. | S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol. |

8. | I. Brener, B. Mikkelsen, K. Rottwitt, W. Burkett, G. Raybon, J. B. Stark, K. Parameswaran, M. H. Chou, M. M. Fejer, E. E. Chaban, R. Harel, D. L. Philen, and S. Kosinski, “Cancellation of all Kerr nonlinearities in long fiber spans using a LiNbO3 phase conjugator and Raman amplification,” OFC 2000, paper PD33. |

9. | H. Wei and D. V. Plant, “Fundamental equations of nonlinear fiber optics,” in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE5178, 255–266 (2003). |

10. | M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, “Cancellation of third-order nonlinear effects in amplified fiber links by dispersion compensation, phase conjugation, and alternating dispersion,” Opt. Lett. |

11. | H. Wei and D. V. Plant, “Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation,” Opt. Express |

12. | A. Chowdhury and R.-J. Essiambre, “Optical phase conjugation and pseudolinear transmission,” Opt. Lett. |

13. | R.-J. Essiambre, G. Raybon, and B. Mikkelson, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in |

14. | P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion-managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett. |

15. | A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett. |

16. | A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, “Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,” IEEE Photon. Technol. Lett. |

17. | J. Martensson, A. Berntson, M. Westlund, A. Danielsson, P. Johannisson, D. Anderson, and M. Lisak, “Timing jitter owing to intrachannel pulse interactions in dispersion-managed transmission systems,” Opt. Lett. |

18. | P. Johannisson, D. Anderson, A. Berntson, and J. Martensson, “Generation and dynamics of ghost pulses in strongly dispersion-managed fiber-optic communication systems,” Opt. Lett. |

19. | M. J. Ablowitz and T. Hirooka, “Resonant nonlinear intrachannel interactions in strongly dispersion-managed transmission systems,” Opt. Lett. |

20. | M. J. Ablowitz and T. Hirooka, “Intrachannel pulse interactions in dispersion-managed transmission systems: timing shifts,” Opt. Lett. |

21. | M. J. Ablowitz and T. Hirooka, “Intrachannel pulse interactions in dispersion-managed transmission systems: energy transfer,” Opt. Lett. |

22. | T. Hirooka and M. J. Ablowitz, “Suppression of intrachannel dispersion-managed pulse interactions by distributed amplification,” IEEE Photon. Technol. Lett. |

23. | R. Hainberger, T. Hoshita, T. Terahara, and H. Onaka, “Comparison of span configurations of Raman-amplified dispersion-managed fibers,” IEEE Photon. Technol. Lett. |

24. | J. A. Buck, |

25. | G. P. Agrawal, |

26. | K.-J. Engel and R. Nagel, |

27. | E. E. Narimanov and P. Mitra, “The channel capacity of a fiber optics communication system: perturbation theory,” J. Lightwave Technol. |

28. | S. N. Knudsen and T. Veng, “Large effective area dispersion compensating fiber for cabled compensation of standard single mode fiber,” OFC 2000, paper TuG5. |

29. | K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, “New type of dispersion management transmission line with MDFSD for long-haul 40 Gb/s transmission,” OFC 2002, paper ThGG2. |

30. | K. Rottwitt and A. J. Stentz, “Raman amplification in lightwave communication systems,” in |

31. | M. Vasilyev, B. Szalabofka, S. Tsuda, J. M. Grochocinski, and A. F. Evans, “Reduction of Raman MPI and noise figure in dispersion-managed fiber,” Electron. Lett. |

32. | J.-C Bouteiller, K. Brar, and C. Headley, “Quasi-constant signal power transmission,” European Conference on Optical Communication2002, paper S3.04. |

33. | M. Vasilyev, “Raman-assisted transmission: toward ideal distributed amplification,” OFC 2003, paper WB1. |

34. | C. Rasmussen, T. Fjelde, J. Bennike, F. Liu, S. Dey, B. Mikkelsen, P. Mamyshev, P. Serbe, P. van der Wagt, Y. Akasaka, D. Harris, D. Gapontsev, V. Ivshin, and P. Reeves-Hall, “DWDM 40G transmission over trans-Pacific distance (10,000 km) using CSRZ-DPSK, enhanced FEC and all-Raman amplified 100 km UltraWave fiber spans,” OFC 2003, paper PD18. |

35. | L. Gruner-Nielsen, Y. Qian, B. Palsdottir, P. B. Gaarde, S. Dyrbol, T. Veng, and Y Qian, “Module for simultaneous C + L-band dispersion compensation and Raman amplification,” OFC 2002, paper TuJ6. |

36. | T. Miyamoto, T. Tsuzaki, T. Okuno, M. Kakui, M. Hirano, M. Onishi, and M. Shigematsu, “Raman amplification over 100 nm-bandwidth with dispersion and dispersion slope compensation for conventional single mode fiber,” OFC 2002, paper TuJ7. |

37. | E. Desurvire, |

38. | A. Striegler, A. Wietfeld, and B. Schmauss, “Fiber based compensation of IXPM induced timing jitter,” OFC 2004, paper MF72. |

**OCIS Codes**

(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 16, 2004

Revised Manuscript: August 28, 2004

Published: September 6, 2004

**Citation**

Haiqing Wei and David Plant, "Intra-channel nonlinearity compensation with scaled translational symmetry," Opt. Express **12**, 4282-4296 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-18-4282

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### References

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- H. Wei and D. V. Plant, �??Fundamental equations of nonlinear fiber optics,�?? in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE 5178, 255-266 (2003).
- M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, �??Cancellation of third-order nonlinear effects in amplified fiber links by dispersion compensation, phase conjugation, and alternating dispersion,�?? Opt. Lett. 20, no. 8, 863-865 (1995). [CrossRef] [PubMed]
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- A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and A. H. Gnauck, �??Cancellation of timing and amplitude jitter in symmetric links using highly dispersed pulses,�?? IEEE Photon. Technol. Lett. 13, 445-447 (2001). [CrossRef]
- J. Martensson, A. Berntson, M. Westlund, A. Danielsson, P. Johannisson, D. Anderson, and M. Lisak, �??Timing jitter owing to intrachannel pulse interactions in dispersion-managed transmission systems,�?? Opt. Lett. 26, 55-57 (2001). [CrossRef]
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- 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]
- S. N. Knudsen and T. Veng, �??Large effective area dispersion compensating fiber for cabled compensation of standard single mode fiber,�?? OFC 2000, paper TuG5.
- K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, �??New type of dispersion management transmission line with MDFSD for long-haul 40 Gb/s transmission,�?? OFC 2002, paper ThGG2.
- K. Rottwitt and A. J. Stentz, �??Raman amplification in lightwave communication systems,�?? in Optical Fiber Telecommunications IV A: Components, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).
- M. Vasilyev, B. Szalabofka, S. Tsuda, J. M. Grochocinski, and A. F. Evans, �??Reduction of Raman MPI and noise figure in dispersion-managed fiber,�?? Electron. Lett. 38, no. 6, 271-272 (2002) [CrossRef]
- J.-C. Bouteiller, K. Brar, and C. Headley, �??Quasi-constant signal power transmission,�?? European Conference on Optical Communication 2002, paper S3.04.
- M. Vasilyev, �??Raman-assisted transmission: toward ideal distributed amplification,�?? OFC 2003, paper WB1.
- C. Rasmussen, T. Fjelde, J. Bennike, F. Liu, S. Dey, B. Mikkelsen, P. Mamyshev, P. Serbe, P. van der Wagt, Y. Akasaka, D. Harris, D. Gapontsev, V. Ivshin, P. Reeves-Hall, �??DWDM 40G transmission over trans-Pacific distance (10,000 km) using CSRZ-DPSK, enhanced FEC and all-Raman amplified 100 km UltraWaveTM fiber spans,�?? OFC 2003, paper PD18.
- Gruner-Nielsen, Y. Qian, B. Palsdottir, P. B. Gaarde, S. Dyrbol, T. Veng, and Y. Qian, �??Module for simultaneous C + L-band dispersion compensation and Raman amplification,�?? OFC 2002, paper TuJ6.
- T. Miyamoto, T. Tsuzaki, T. Okuno, M. Kakui, M. Hirano, M. Onishi, and M. Shigematsu, �??Raman amplification over 100 nm-bandwidth with dispersion and dispersion slope compensation for conventional single mode fiber,�?? OFC 2002, paper TuJ7.
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- A. Striegler, A. Wietfeld, and B. Schmauss, �??Fiber based compensation of IXPM induced timing jitter,�?? OFC 2004, paper MF72.

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