Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis

IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 11, NO. 9, SEPTEMBER 1999

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Parametric Gain in Multiwavelength Systems: A New Approach to Noise Enhancement Analysis G. Bosco, A. Carena, Member, IEEE, V. Curri, R. Gaudino, Member, IEEE, P. Poggiolini, Member, IEEE, and S. Benedetto, Fellow, IEEE

Abstract—In this paper, we introduce a new formalism to study noise enhancement induced by parametric gain in wavelengthdivision-multiplexed (WDM) systems. We also analyze the phenomenon impact in a typical long-haul, dispersion-compensated link, and show its peculiar relevance in WDM systems.

equation (NLSE) [1], we get

(2)

Index Terms—Optical fiber communications, fiber nonlinearities, long-haul transmission, system analysis, WDM systems.

I. INTRODUCTION

I

N LONG-HAUL WDM optical communication systems, the impact of fiber Kerr nonlinearity on system performance can be detrimental: four-wave mixing (FWM) crosstalk, crossphase modulation (XPM) pulse distortion, and nonlinear noise enhancement induced by parametric gain (PG) are the main causes. Whereas FWM, XPM and single channel PG have been extensively studied [1]–[4], PG impact on WDM systems was not yet investigated in detail. The goal of the present work is the analysis of PG-induced nonlinear noise enhancement in a WDM environment. When one or more signals propagate along the fiber, the interaction of Kerr nonlinearity and quadratic dispersion yields parametric phenomena. In particular, the optical carriers can act as pumps and generate spectral regions where small signals experience gain. In long-haul amplified links, signals propagate together with amplified spontaneous emission (ASE) noise added by in-line EDFA’s. Consequently, optical power can be transferred from signal to noise, thus potentially degrading system performance. II. PARAMETRIC GAIN (PG) ANALYSIS

In multichannel systems, the propagating transversal electric field may be written as [1]

where we have neglected all FWM terms, i.e., all terms oscillating around the beat angular frequencies is the nonlinearity coefficient in W km [1], accounts is the th term of the Taylor series for fiber loss and around the angular expansion of the propagation constant We analyze a fiber span whose parameters do frequency not change along the propagation coordinate and study the continuous-wave (CW) optical propagation of a comb of carriers together with ASE noise. The signal around each carrier can be written as (3) and are respectively the power and the phase where is the ASE noise (the perturbing of the th carrier, is the group signal) around the th carrier, velocity, and (4) is the nonlinear phase-shift. We assume that (2), which is exact for narrow-band signals only, is valid for signals such as (3); this assumption will be verified a posteriori through comparisons with numerical simulations. Substituting (3) into (2), and following the same steps outlined in [4], we get

(1) is the number of channels, is the modal where and are the modal amdistribution function, and plitude and the propagation constant for the angular frequency and for the th carrier. Substituting (1) into the fiber wave equation, i.e., into the well-known nonlinear Schr¨odinger Manuscript received February 12, 1999; revised June 4, 1999. The authors are with the Dipartimento di Elettronica, Politecnico di Torino, 10129 Turin, Italy. Publisher Item Identifier S 1041-1135(99)06859-7.

(5) , as done where we have neglected all terms of the order of is the spectral separation between the th carrier in [4]. and the reference angular frequency , set at the center of the carrier spectral comb. “ ” means conjugation.

1041–1135/99$10.00  1999 IEEE

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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 11, NO. 9, SEPTEMBER 1999

(a)

Fig. 1. ASE noise spectral gain at the output of the nonamplified, single span four-channel WDM system.

By multiplying both sides of the th equation in (5) by , and adding and subtracting the complex conjugate, a system of differential equations in the unknowns: and is obtained. and are the in-phase and quadrature components of the th noise with and respect to the th CW signal phase [5], respectively. have different system impacts, because generates the component after the photodetection, while generates the component only. Taking the Fourier transform of both sides of the resulting equations yields: (b) Fig. 2. Single-sideband ASE noise spectral gain around the optical carrier (ch. 1), separated into the (a) in-phase and (b) quadrature components, for a number of channels ranging from = 14. Results refer to the nonamplified, single span system.

N

(6) where a solution vector

If we group all components defined as

in

where “ ” means transposed, the general solution can be put [4] in the form of a transfer matrix (7) The PG transfer matrix is evaluated numerically by solving (6) using a standard Runge–Kutta method. Then the ASE noise gain, i.e., the PG-induced noise enhancement, can be easily evaluated following as done in [4]. III. NUMERICAL RESULTS AND SYSTEM APPLICATIONS We first analyze a 50-km optical link where four high-power 100 GHz-spaced carriers (40 mW per carrier) are launched

into a fiber ps , W km and dB/km , together with a spectrally flat ASE noise, modeled as a white Gaussian noise. The ASE noise power spectrum at the output of the four-channel system is shown in Fig. 1. As previously stated, the graph shows that PG strongly enhances ASE noise around the optical carrier. The ASE growth can be detrimental since it cannot be filtered out at the receiver, being close to the carriers. Moreover, signal is depleted due to the transfer of power to the noise. Fig. 2 shows the spectrum at the output of a system having the same characteristics, for 1, 2, and 4 transmitted carriers. Results refer to the spectral area around channel 1 and are shown separately for the in-phase [Fig. 2(a)] and quadrature [Fig. 2(b)] components. Due to their symmetry around each carrier, only single-sideband spectra have been drawn. The two components show a completely different behavior. While the in-phase component [Fig. 2(a)] has a 0-dB gain for and it is only marginally affected by the number of channels , the quadrature component [Fig. 2(b)] shows a strong gain around the origin that increases with . We refer to the quadrature noise extra gain as to the “multichannel PG effect.” To our knowledge, this effect was never investigated before. We also developed an analytical theory taking into account the birefringent dual-polarization nature of the fiber, as we did

BOSCO et al.: PARAMETRIC GAIN IN MULTIWAVELENGTH SYSTEMS

PARAMETER

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TABLE I MULTISPAN SYSTEM

OF THE

Fig. 4. Single-sideband ASE noise spectral gain around the optical carrier (ch. 1), separated into the in-phase and quadrature components for a multispan, four channel system. System parameters are described in Table I. The curves obtained from time-domain simulation and for the single channel case are reported for comparison.

in Section II. We found out that the multichannel PG effect , with the power per channel and with increases for the number of channels , while it does not depend on the dispersion value. Moreover, the phenomenon “bandwidth,” i.e., the spectral region where the gain is relevant, increases when increasing the power per channel and the number of channels, and when decreasing the channel separation and the dispersion value. IV. DISCUSSION Fig. 3. Single-sideband ASE noise spectral gain around the optical carrier (ch. 1), separated into the in-phase and quadrature components for a multispan, 2-channel system. System parameters are described in Table I. The curves obtained from time-domain simulation and for the single channel case are reported for comparison.

for the single-channel case [6]. This theory has shown that the multichannel PG effect is weakly dependent on polarization and is just slightly reduced when the signals are launched on orthogonal polarizations. This is a main difference with respect to the FWM. We have also studied a 3000-km long-haul amplified system where two and four 100-GHz-spaced carriers are transmitted over a dispersion-compensated link. System parameters are reported in Table I, while the output power ASE noise spectra, separated into the in-phase and quadrature components, are shown in Fig. 3 for the two-channel system, and in Fig. 4 for the four-channel system. Influence of noise reshaping induced by optical filtering has not been considered. In both cases, the single channel gain is shown for comparison, together with the results obtained using the OptSim [7] timedomain simulator, which solves the NLSE using the timedomain split-step method. The good agreement between the analysis and simulation confirms the validity of our approach. Residual differences are induced by problems in extraction of phase references and by the simplifying assumptions made

AND

CONCLUSION

We have presented a formalism that allows to study the PG noise enhancement in WDM systems. The analysis permits a better understanding of a still not well known phenomenon, i.e., the nonlinear interaction of several CW optical carriers with ASE noise. Moreover, we have shown that the PGinduced noise enhancement grows together with the number of channels, potentially becoming a limiting factor in WDM systems. REFERENCES [1] G. P. Agrawal, Nonlinear Fiber Optics. Boston, MA: Academic, 1995. [2] K. Kikuchi, “Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers,” IEEE Photon. Technol. Lett., vol. 5, pp. 221–223, Feb. 1993. [3] R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of the noise in fibers with dispersion and its impact on optically amplified systems,” IEEE Photon. Technol. Lett., vol. 9, pp. 392–394, Mar. 1997. [4] A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett., vol. 9, pp. 535–537, Apr. 1997. [5] L. W. Couch II, Digital and Analog Communication Systems, 4th ed. New York: Macmillan, 1993, pp. 244–245. [6] A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol., vol. 16, pp. 1149–1157, July 1998. [7] A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization related effects in fiber,” IEEE J. Select. Areas Commun., vol. 15, pp. 751–765, May 1997.