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Optics Letters

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  • Editor: Alan E. Willner
  • Vol. 37, Iss. 14 — Jul. 15, 2012
  • pp: 2931–2933
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Dispersion-dominated nonlinear fiber-optic channel

Sergei Turitsyn, Mariia Sorokina, and Stanislav Derevyanko  »View Author Affiliations


Optics Letters, Vol. 37, Issue 14, pp. 2931-2933 (2012)
http://dx.doi.org/10.1364/OL.37.002931


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Abstract

We propose to apply a large predispersion (having the same sign as the transmission fiber) to an optical signal before the uncompensated fiber transmission in coherent communication systems. This technique is aimed at simplification of the following digital signal processing of nonlinear impairments. We derive a model describing pulse propagation in the dispersion-dominated nonlinear fiber channel. In the limit of very strong initial predispersion, the nonlinear propagation equations for each Fourier mode become local and decoupled. This paves the way for new techniques to manage fiber nonlinearity.

© 2012 Optical Society of America

The averaged evolution of the two orthogonal polarizations of the field envelope [U1(z,t),U2(z,t)] of the optical field along the fiber is well approximated by the Manakov equations [1

1. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

,5

5. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, IEEE J. Sel. Top. Quantum Electron. 16, 1217 (2010). [CrossRef]

,6

6. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, J. Lightwave Technol. 15, 1735 (1997). [CrossRef]

]:
U1z=α2U1iβ222U1t2+i8γ9J×U1+η1,U2z=α2U2iβ222U2t2+i8γ9J×U2+η2,
(1)
where J=|U2|2+|U1|2 is the nonlinearity averaged over polarization inhomogeneities and β2, γ, η1,2, and α are the group velocity dispersion, nonlinear coefficient, distributed noise and linear loss, respectively [1

1. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

]. We focus here on deterministic nonlinear dynamics and the introduction of a simplified nonlinear propagation model. The impact of the additive noise η1,2 on statistical signal properties will be presented in a separate publication.

These equations are integrable:
An(z,ω)=An(0,ω)exp[169iγπ|β2|f(αz,|β2|zK)I(ω)],
(5)
with I(ω)=|A1(0,ω)|2+|A2(0,ω)|2 and f(x,y)=ex/y(Ei[x/yx]Ei[x/y]), which in the lossless limit takes a simple form f[0,y]=ln(1+y). Here Ei(x) is the exponential integral function.

The general form of the spectrum of an input sequence of M symbols sampled at a Nyquist rate 1/B is given by
A(0,ω)=1Bf˜(ω/B)m=M/2M/2xmexp(imωB),
(7)
where f˜(x) is the dimensionless Fourier transform of a single band-limited wave form f(τ) and B is the bandwidth; for instance, it can be sinc-shaped pulses f(τ)=sin(πτ)/(πτ) [8

8. R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol. 28, 662 (2010). [CrossRef]

]. There are two scales in the spectrum (7). The first one relates to the FWHM of the signal waveform f. For the sinc, the spectrum is of course rectangular inside the band |ω|<πB. The second scale is due to the pattern effect, which is given by the modulation factor in Eq. (7). One can see that the more distant symbols with large values of |m| introduce shorter ω scales Δω|m|=B/|m|. The smallest scale introduced by this factor is due to the boundary symbols |m|=M/2.

On the other hand, the typical scale of fast oscillations in the exponent in Eq. (3) (as well as its scalar counterpart) is ΔωKK1/2. This means that there is a competition of the two oscillating factors in the nonlinear integral determining the spectral dynamics. In order for the saddle-point model [Eq. (5) or (6)] to work at the initial stages of pulse dynamics, the precompensation has to ensure that ΔωKΔωf; i.e., at least the base waveform has longer scale spectral variation than the precompensation. For all pulses adjacent to the central one and such that the inequality ΔωKΔω|m| holds (m is the symbol number counted from the central one), the saddle-point approximation [Eq. (5),(6)] works well and produces the rate of change of the spectrum (ΔωK)2. For more distant pulses, such that the inverse inequality holds, the reciprocal regime occurs: the three-wave beating term is oscillating faster than the dispersion exponential. Such terms are not accounted for correctly by the model [Eq. (5) or (6)], but their contribution to the field equation should naturally be of a higher order than (ΔωK)2 and hence can be neglected considering the central part of the pattern.

To sum up, the saddle-point model works well if the inequality K1/B2 holds. Additionally, when considering a lengthy pattern of symbols, only the symbols up to m±BK contribute to the spectral dynamics—the memory effects from more distant pulses are lost.

Let us now turn to numerical verification of the proposed model {specifically, the lossless scalar case [Eq. (6)]}. Here, without loss of generality the following typical parameters are used: β2=20ps2/km, z=1000km, γP0=0.021km1 (P0 is the signal peak power), and K=2×105ps2. The results of the model [Eq. (6)] are compared with full numerical propagation in the NLSE. We use for illustration a four-level quadrature phase shift keying format (QPSK) [8

8. R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol. 28, 662 (2010). [CrossRef]

] to encode a pseudorandom sequence of symbols. The results are presented in Fig. 1. Note, that although we show here results for sinc-shaped pulses, in our simulations we have also made separate runs (not shown) with Gaussian pulses of comparable width, which also demonstrated good agreement between the analytical results and the numerics. In each figure we also include for comparison the initial symbol amplitudes, |xm| (as black dots) sampled at the Nyquist rate. Because we work with the single ring PSK, all symbols have the same absolute value: |xm|=P0. The bandwidth is B=30GHz (corresponding to a pulse width of 1/B=33ps). Note that an increase of bandwidth B only relaxes the condition of the predispersion parameter K.

Fig. 1. Comparison of the full numerics with the saddle-point model for (a) N=8 symbols, (b) N=32 symbols, and (c) N=64 symbols. For the latter two patterns, only a central fragment is shown. No inline dispersion compensation was used in the link.

The number of symbols contributing to the saddle-point field dynamics is around BK13. One can see that there is no loss of accuracy of the model when the pattern length exceeds that value. Note that including a distributed optical noise into the model is straightforward [1

1. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

].

In conclusion, we have proposed the application of a large predispersion (having the same sign as the transmission fiber) to an optical signal before uncompensated fiber transmission in coherent communication systems in order to simplify the following DSP. In such a nonlinear channel, the spectral dynamics of the signal can be described analytically. In the limit of very strong initial predispersion, the nonlinear propagation equations for each spectral component become local and decoupled. The relative simplicity of the mapping from input to output signal opens the possibility of fast DSP of the nonlinear effects.

We acknowledge support of the European Research Council and the Engineering and Physical Sciences Research Council (EPSRC) project UNLOC.

References

1.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

2.

S. J. Savory, Opt. Express 16, 804 (2008). [CrossRef]

3.

S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, Opt. Express 15, 2120 (2007). [CrossRef]

4.

K. Kikuchi, IEICE Electron. Express 8, 1642 (2011). [CrossRef]

5.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, IEEE J. Sel. Top. Quantum Electron. 16, 1217 (2010). [CrossRef]

6.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, J. Lightwave Technol. 15, 1735 (1997). [CrossRef]

7.

J. P. Gordon and L. F. Mollenauer, Opt. Lett. 151351 (1990). [CrossRef]

8.

R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol. 28, 662 (2010). [CrossRef]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(190.4370) Nonlinear optics : Nonlinear optics, fibers

ToC Category:
Nonlinear Optics

History
Original Manuscript: May 14, 2012
Manuscript Accepted: May 27, 2012
Published: July 12, 2012

Virtual Issues
August 1, 2012 Spotlight on Optics

Citation
Sergei Turitsyn, Mariia Sorokina, and Stanislav Derevyanko, "Dispersion-dominated nonlinear fiber-optic channel," Opt. Lett. 37, 2931-2933 (2012)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-14-2931


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References

  1. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).
  2. S. J. Savory, Opt. Express 16, 804 (2008). [CrossRef]
  3. S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, Opt. Express 15, 2120 (2007). [CrossRef]
  4. K. Kikuchi, IEICE Electron. Express 8, 1642 (2011). [CrossRef]
  5. D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, IEEE J. Sel. Top. Quantum Electron. 16, 1217 (2010). [CrossRef]
  6. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, J. Lightwave Technol. 15, 1735 (1997). [CrossRef]
  7. J. P. Gordon and L. F. Mollenauer, Opt. Lett. 151351 (1990). [CrossRef]
  8. R. J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol. 28, 662 (2010). [CrossRef]

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