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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 18 — Sep. 6, 2004
  • pp: 4282–4296
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Intra-channel nonlinearity compensation with scaled translational symmetry

Haiqing Wei and David V. Plant  »View Author Affiliations


Optics Express, Vol. 12, Issue 18, pp. 4282-4296 (2004)
http://dx.doi.org/10.1364/OPEX.12.004282


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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

Group-velocity dispersion and optical nonlinearity are the major limiting factors in high-speed long-distance fiber-optic transmissions [1

1. A. H. Gnauck and R. M. Jopson, “Dispersion compensation for optical fiber systems,” in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

, 2

2. F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

]. Dispersion-compensating fibers (DCFs) have been developed to offset the dispersion effects of transmission fibers over a wide frequency band. The most advanced DCFs are even capable of slope-matching compensation, namely, compensating the dispersion and the dispersion slope of the transmission fiber simultaneously [3

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

, 4

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

]. By contrast, it proves more difficult to compensate the nonlinear effects of optical fibers because of the lack of materials with negative nonlinearity and high group-velocity dispersion simultaneously [5

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]

]. Optical phase conjugation (OPC) in the middle of a transmission line may compensate the nonlinear effects between fibers on the two sides of the phase conjugator [6

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]

], especially when the two sides are configured into a mirror [7

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

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

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).

] or translational [9

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

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

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]

] symmetry in a scaled sense, although the benefit of OPC may still be appreciable in the absence of such scaled symmetry [12

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

]. However, wide-band optical phase conjugation exchanges the channel wavelengths, so to complicate the design and operation of wavelength-division multiplexed (WDM) networks. Also, the performance and reliability of prototype conjugators are not yet sufficient for field deployment. Fortunately, it has been found that ordinary fibers could compensate each other for the intra-channel Kerr nonlinear effects without the help of OPC. The intra-channel nonlinear effects, namely, nonlinear interactions among optical pulses within the same wavelength channel, are the dominating nonlinearities in systems with high modulation speeds of 40 Gb/s and above [13

13. R.-J. Essiambre, G. Raybon, and B. Mikkelson, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

], where the nonlinear interactions among different wavelength channels become less-limiting factors. As a result of the short pulse width and high data rate, optical pulses within one channel are quickly dispersed and overlap significantly so to interact through the Kerr nonlinearity. In the past a few years, intra-channel nonlinearities have been extensively investigated by several research groups [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]

, 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

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

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

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

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

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

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

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]

]. A method has been identified for suppressing the intra-channel nonlinearity-induced jitters in pulse amplitude and timing, using Raman-pumped transmission lines manifesting a lossless or mirror-symmetric map of signal power [16

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

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]

]. However, there is a problem with such mirror-symmetric power map. Namely, the loss of pump power makes it difficult to maintain a constant gain in a long transmission fiber. Consequently, the significant deviation of signal power profile from a desired mirror-symmetric map degrades the result of intra-channel nonlinear compensation using mirror symmetry [23

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]

]. By contrast, we shall demonstrate here that two fiber spans in a scaled translational symmetry [9

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).

, 11

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]

] could cancel out their intra-channel nonlinear effects to a large extent without resorting to OPC, and a significant reduction of intra-channel nonlinear effects may be achieved in a multi-span system with scaled translationally symmetric spans suitably arranged. The results shall be derived analytically from the nonlinear Schrödinger equation and verified by numerical simulations using commercial software.

2. Basics of nonlinear wave propagation in fibers

The eigenvalue solution of Maxwell’s equations in a single-mode fiber determines its transverse model function and propagation constant β(ω) as a function of the optical frequency ω[24

24. J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995), Chapter 4.

, 25

25. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic Press, San Diego, 1995), Chapter 2.

]. 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,

Ezt=Aztexp[izβ0(ς)iω0t],
(1)

with β0(z)=defβ(ω0,z), is often employed to represent an optical signal, which may be of a single time-division multiplexed channel or a superposition of multiple WDM channels. The evolution of the envelope A(z,t) in an optical fiber of length L is governed by the nonlinear Schrödinger equation (NLSE) [9

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).

, 25

25. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic Press, San Diego, 1995), Chapter 2.

],

Aztz+k=2+ik1βk(z)k!(t)2Azt+α(z)2Azt=
iγ(z)Azt2Azt+i[gztAzt2]Azt,
(2)

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,

βk(z)=def12β0(z)k[β2ωz]ωkω=ω0,k2,
(3)

are the 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

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).

]. 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 = dvg1/, namely, the derivative of the inverse of group-velocity with respect to the optical wavelength, and dispersion slope for S = dD/ [1

1. A. H. Gnauck and R. M. Jopson, “Dispersion compensation for optical fiber systems,” in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

]. 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.

Had there been no nonlinearity, namely γ(z) = g(z, t) ≡ 0, Eq. (2) would reduce to,

Aztz+k=2+ik1βk(z)k!(t)kAzt+α(z)2Azt=0,
(4)

which could be solved analytically using, for example, the method of Fourier transform. Let F denote the linear operator of Fourier transform, a signal A(z, t) in the time domain can be represented equivalently in the frequency domain by Ã(z,ω)=defF[A(z,t)]. Through a linear fiber, a signal 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 2z 1, where the transfer function H(z 1,z 2, ω) is defined as,

H(z1,z2,ω)=defexp[ik=2+ωkk!z1z2βk(z)dz12z1z2α(z)dz].
(5)

Pz1z2=defF1H(z1,z2,ω)F.
(6)

Namely, P(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,

h(z1,z2,t)=defF1[H(z1,z2,ω)],
(7)

P(z 1,z 2) may also be represented as,

P(z1,z2)=h(z1,z2,t).
(8)

That is, the action of P(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 1z 2, also known as propagators, form a semigroup [26

26. K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer-Verlag, New York, 2000).

] for the linear evolution governed by Eq. (4).

A0(z2,t)=P(z1,z2)A(z1,t),
(9)
A1(z2,t)=z1z2P(z1,z2){iγ(z)A0(z,t)2A0(z,t)
+i[g(z,t)A0(z,t)2]A0(z,t)}dz,
(10)

where 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

Within one wavelength channel, it is only necessary to consider the Kerr nonlinearity, while the Raman effect may be neglected. The translational symmetry [9

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).

, 11

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]

] requires that the corresponding fiber segments have the same sign for the loss/gain coefficients but opposite second-and higher-order dispersions, which are naturally satisfied conditions in conventional fiber transmission systems, where, for example, a transmission fiber may be paired with a DCF as symmetric counterparts. The scaled translational symmetry further requires that the fiber parameters should be scaled in proportion and the signal amplitudes should be adjusted to satisfy [9

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).

, 11

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]

],

α(z)α(z)=β2(z)β2(z)=β3(z)β3(z)=γ(z)A(z,t)2γ(z)A(z,t)2=zz=1R,
(11)

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

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).

, 11

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]

], it is without loss of generality to consider two spans that are in translational symmetry with the ratio R = 1. The cascade of such two spans would constitute a transmission line stretching from z = 0 to z = 2L, with the fiber parameters satisfying,

α(z)α(z+L)=β2(z)β2(z+L)=β3(z)β3(z+L)=γ(z)A(z,t)2γ(z+L)A(z+L,t)2=1,
(12)

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.

Fig. 1. The signal power and dispersion maps for a cascade of two fiber spans in scaled translational symmetry with scaling ratio R = 1. Top: the variation of signal power along the propagation distance. Bottom: the dispersion map, namely, the variation of accumulated dispersion along the propagation distance.

The amplitude envelope of a single channel may be represented by a sum of optical pulses, namely, A(z,t) = Σk uk (z,t), where uk (z,t) denotes the pulse in the kth bit slot and centered at time t = kT, with kZ and T > 0 being the bit duration. The following NLSE describes the propagation and nonlinear interactions among the pulses [13

13. R.-J. Essiambre, G. Raybon, and B. Mikkelson, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

],

ukz+iβ2(z)22ukt2+α(z)2uk=(z)mnumunum+nk*,kZ,
(13)

where the right-hand side keeps only those nonlinear products that satisfy the phase-matching condition. The nonlinear mixing terms with either m = k or n = k contribute to self-phase modulation and intra-channel cross-phase modulation (XPM), while the rest with both mk and n = k are responsible for intra-channel four-wave mixing (FWM) [13

13. R.-J. Essiambre, G. Raybon, and B. Mikkelson, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

]. 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 uk , ∀ kZ 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 ϕk = ϕ 0 + kϕ, ∀kZ, with ϕ 0,∆ϕ ∈ [0,2π),because Eq. (13) is invariant under the multiplication of phase factors exp(k ) to uk , ∀ kZ. The linear dependence of ϕk 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.

Were there no nonlinearity in the fibers, the signal propagation would by fully described by the dispersive transfer function,

H(z1,z2,ω)=exp[i2b2(z1,z2)ω212z1z2α(z)dz],
(14)

with z 1,z 2 ∈ [0,2L], and,

b2(z1,z2)=defz1z2β2(z)dz,
(15)

or equivalently the corresponding impulse response,

h(z1,z2,t)=1b2(z1,z2)exp[it22b2(z1,z2)12z1z2α(z)dz],
(16)

which is calculated from F-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:

vk(z,t)=P(0,z)uk(0,t),
(17)
vk(z,t)=im0znP(s,z)[γ(s)vm(s,t)vn(s,t)vm+nk*(s,t)]ds,
(18)

kZ, where uk (z,t) ≈ vk (z,t) is the zeroth-order approximation which neglects the fiber nonlinearity completely, whereas the result of first-order perturbation uk (z,t) ≈ vk (z,t)+vk (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), 0z+L α(s)ds = 0z α(s)ds, γ(z + L) = γ(z), ∀ z ∈ [0,L], and vk (2L,t) = vk (L,t) = vk (0,t) = uk (0,t), which is real-valued by assumption, ∀ kZ. It further follows that h(0,z + L,t) = h *(0,z,t) and h(z + L,2L,t) = h *(z,2L,t), hence,

P(0,z+L)=P*(0,z)=defh*(0,z,t),
(19)
P(z+L,2L)=P*(z,2L)=defh*(z,2L,t),
(20)

z ∈ [0,L]. Consequently, the pulses at z and z+L are complex conjugate,namely,vk (z+L,t) = vk*(z,t), ∀ kZ, ∀ z ∈ [0,L]. At the end z = 2L, a typical term of nonlinear mixing reads,

02LP(z,2L)[γ(z)vm(z,t)vn(z,t)vm+nk*(z,t)]dz
=0LP(z,2L)[γ(z)vm(z,t)vn(z,t)vm+nk*(z,t)]dz+
0LP(z+L,2L)[γ(z)vm(z+L,t)vn(z+L,t)vm+nk*(z+L,t)]dz
=0LP(z,2L)[γ(z)vm(z,t)vn(z,t)vm+nk*(z,t)]dz+
0LP*(z,2L)[γ(z)vm*(z,t)vn*(z,t)vm+nk(z,t)]dz,
(21)

which is therefore real-valued. It follows immediately that the first-order nonlinear perturbation vk (2L,t) is purely imaginary-valued, which is in quadrature phase with respect to the zeroth-order approximation vk (2L,t) = uk (0,t), ∀ kZ. 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 - ½b 2(0,L), so that the input pulses to the second span at z = L are pre-chirped by ½b 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 vk(z + L,t) = vk* (z,t), ∀ k ∈ [0,L],∀kZ, are still valid, so are the above argument and the conclusion that vk and vk are real- and imaginary-valued respectively when brought chirp-free.

Mathematically, that vk and vk are in quadrature phase implies |uk |2= |vk + vk |2 = |vk |2 + |vk |2, where |vk |2 is quadratic, or of the second-order, in terms of the Kerr nonlinear coefficient, ∀ kZ. 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

13. R.-J. Essiambre, G. Raybon, and B. Mikkelson, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

, 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

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

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

]. The quadrature-phased nonlinear products due to intra-channel FWM lead to the generation of “ghost” pulses in the “ZERO”-slots [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]

,18

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

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

] and the addition of noise power to the “ONE”-bits. As second-order nonlinear perturbations, these effects are less detrimental. Secondly, it eliminates pulse timing jitter due to intra-channel XPM up to the first-order nonlinear perturbation. Using the moment method [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

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]

], the time of arrival for the center of the kth pulse may be calculated as,

tk=tuk2dtuk2dttvk2dtvk2dt=kT,
(22)

4. Optimal setups of fiber-optic transmission lines

For DCFs having absolute dispersion values much higher than the transmission fiber, it is suitable to coil the DCF into a lumped dispersion-compensating module (DCM) and integrate the module with a multi-stage optical amplifier at each repeater site. Two fiber spans in scaled translational symmetry for intra-channel nonlinearity compensation should have oppositely ordered transmission fibers and DCFs. As shown in Fig. 2, one span has a piece of transmission fiber from A to B, in which the signal power decreases exponentially, and an optical repeater at the end, in which one stage of a multi-stage optical amplifier boosts the signal power up to a suitable level and feeds the signal into a lumped DCM, where the signal power also decreases exponentially along the length of the DCF from B to C, finally the signal power is boosted by another stage of the optical amplifier. The other span has the same transmission fiber and the same DCM, with the signal power in the DCF from C to D tracing the same decreasing curve. However, this span has the DCM placed before the transmission fiber. Ironically, the efforts of improving the so-called figure-of-merit [1

1. A. H. Gnauck and R. M. Jopson, “Dispersion compensation for optical fiber systems,” in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

, 4

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

] by DCF manufacturers have already rendered the loss coefficients of DCFs too low to comply with the scaling rules of Eq. (11). To benefit from nonlinearity compensation enabled by scaled translational symmetries, DCFs, at least parts of them carrying high signal power, may be intentionally made more lossy during manufacturing or by means of special packaging to introduce bending losses. As illustrated in Fig. 2, the DCFs from B to C and from C to D are arranged in scaled translational symmetry to the transmission fibers from D to E and from A to B respectively, such that the transmission fiber from A to B is compensated by the DCF from C to D, and the DCF from B to C compensates the transmission fiber from D to E, for the most detrimental effects of jittering in pulse amplitude and timing due to intra-channel FWM and XPM. In practice, the DCMs from B to D and the multi-stage optical amplifiers may be integrated into one signal repeater, and the same super-span from A to E may be repeated many times to reach a long-distance, with the resulting transmission line enjoying the effective suppression of intra-channel nonlinear impairments. In case distributive Raman pumping in the transmission fibers [30

30. K. Rottwitt and A. J. Stentz, “Raman amplification in lightwave communication systems,” in Optical Fiber Telecommunications IVA: Components, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

, 31

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]

, 32

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

, 33

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

, 34

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.

] is employed to repeat the signal power, the DCFs may also be Raman pumped [35

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

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.

] or erbium-doped for distributive amplification [37

37. E. Desurvire, Erbium-Doped Fiber Amplifiers: Principles and Applications (John Wiley & Sons, New York, 1994).

] to have similar (scaled) power profiles as that in the transmission fibers for optimal nonlinearity compensation.

Fig. 2. The signal power and dispersion maps for a cascade of two fiber spans in scaled translational symmetry with lumped dispersion compensators. Top: the variation of signal power along the propagation distance. Bottom: the dispersion map, namely, the variation of accumulated dispersion along the propagation distance.

Fig. 3. A transmission line consists of 6 pairs of fiber spans, with the first span in each pair having 50 km SMF followed by 50 km RDF then 15.74 dB EDFA gain, and the second span having 39.35 km RDF followed by 39.35 km SMF then 20 dB EDFA gain.

5. Simulation results and discussions

Numerical simulations using commercial software are carried out to support our theoretical analysis and verify the effectiveness of our method of suppressing intra-channel nonlinearity using scaled translational symmetry. In one test system, as depicted in Fig. 3, the transmission line consists of 6 pairs of compensating fiber spans totaling a transmission distance of 1072.2 km. The first span in each pair has 50 km SMF followed by 50 km RDF then an erbium-doped fiber amplifier (EDFA) with gain 15.74 dB, the second span has 39.35 km RDF followed by 39.35 km SMF then an EDFA with gain 20 dB. The other test system consists of the same number of spans with the same span lengths, which are constructed using the same fibers and EDFAs as the first system except that the second span in each span-pair has the 39.35-km SMF placed before the 39.35-km RDF, as shown in Fig. 4. The EDFA noise figure is 4 dB. The SMF has loss α = 0.2 dB/km, dispersion D = 16 + δD ps/nm/km, and dispersion slope S = 0.055 ps/nm2/km, effective modal area A eff = 80 μm2, while the RDF has α = 0.2 dB/km, D = -16 ps/nm/km, S = -0.055 ps/nm2/km, and A eff = 30 μm2. 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 m2/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

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

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

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

] due to the uncompensated and quadrature-phased components of intra-channel FWM. When the amplifier noise is turned back on, as shown in Fig. 6, the received signals become slightly more noisy, but the suppression of nonlinear distortions is still remarkable when there is scaled translational symmetry. Then 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.

Fig. 4. A transmission line consists of 6 pairs of fiber spans, with the first span in each pair having 50 km SMF followed by 50 km RDF then 15.74 dB EDFA gain, and the second span having 39.35 km SMF followed by 39.35 km RDF then 20 dB EDFA gain.
Fig. 5. The transmission results with δD = 0 and amplifier noise turned off to signify the nonlinear effects. Left: received optical eye diagram of the scaled translationally symmetric setup in Fig. 3. Right: received optical eye diagram of the setup in Fig. 4 without scaled translational symmetry.
Fig. 6. The transmission results with δD = 0 and amplifier noise turned on. Left: received optical eye diagram of the scaled translationally symmetric setup in Fig. 3. Right: received optical eye diagram of the setup in Fig. 4 without scaled translational symmetry.

6. Conclusion

In conclusion, we have demonstrated through analytical derivation and numerical simulations that two fiber spans in a scaled translational symmetry could cancel out their intra-channel nonlinear effects to a large extent. And a significant reduction of intra-channel nonlinear effects may be achieved in a long-distance transmission line consisting of multiple pairs of translationally symmetric spans. We have also discussed a method of packaging dispersion-compensating fibers to optimally compensate the nonlinear effects of transmission fibers and to minimize the signal power loss at the same time.

Fig. 7. The transmission results with δD = 0.2 ps/nm/km and amplifier noise turned on. Left: received optical eye diagram of the scaled translationally symmetric setup in Fig. 3. Right: received optical eye diagram of the setup in Fig. 4 without scaled translational symmetry.
Fig. 8. The transmission results with δD = 0.6 ps/nm/km and amplifier noise turned on. Left: received optical eye diagram of the scaled translationally symmetric setup in Fig. 3. Right: received optical eye diagram of the setup in Fig. 4 without scaled translational symmetry.

Acknowledgments

This work was supported by the Natural Sciences and Engineering Research Council (NSERC) and industrial partners, through the Agile All-Photonic Networks (AAPN) Research Network.

References and links

1.

A. H. Gnauck and R. M. Jopson, “Dispersion compensation for optical fiber systems,” in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

2.

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “Fiber nonlinearities and their impact on transmission systems,” in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).

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. 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.

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. 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]

13.

R.-J. Essiambre, G. Raybon, and B. Mikkelson, “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

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]

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]

24.

J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995), Chapter 4.

25.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic Press, San Diego, 1995), Chapter 2.

26.

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer-Verlag, New York, 2000).

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.

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 Optical Fiber Telecommunications IVA: Components, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).

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]

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, Erbium-Doped Fiber Amplifiers: Principles and Applications (John Wiley & Sons, New York, 1994).

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

  1. A. H. Gnauck and R. M. Jopson, �??Dispersion compensation for optical fiber systems,�?? in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).
  2. F. Forghieri, R. W. Tkach and A. R. Chraplyvy, �??Fiber nonlinearities and their impact on transmission systems,�?? in Optical Fiber Telecommunications III A, I. P. Kaminow and T. L. Koch, eds. (Academic Press, San Diego, 1997).
  3. V. Srikant, �??Broadband dispersion and dispersion slope compensation in high bit rate and ultra long haul systems,�?? OFC 2001, 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. 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.
  9. 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).
  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), <a href ="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1938">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1938</a> [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]
  13. R.-J. Essiambre, G. Raybon, and B. Mikkelson, �??Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,�?? in Optical Fiber Telecommunications IV B: Systems and Impairments, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).
  14. P. V. Mamyshev and N. A. Mamysheva, �??Pulse-overlapped dispersion-managed data transmission and intrachannel 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]
  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]
  24. J. A. Buck, Fundamentals of Optical Fibers (Wiley, New York, 1995), Chapter 4.
  25. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic Press, San Diego, 1995), Chapter 2.
  26. K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer-Verlag, New York, 2000).
  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. 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 Optical Fiber Telecommunications IV A: Components, I. P. Kaminow and T. Li, eds. (Academic Press, San Diego, 2002).
  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]
  32. J.-C. Bouteiller, K. Brar, and C. Headley, �??Quasi-constant signal power transmission,�?? European Conference on Optical Communication 2002, 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, 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.
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