Soliton phase jitter control by use of super-gaussian filters

Transcription

1 Optics Communications 250 (2005) Soliton phase jitter control by use of super-gaussian filters Y.J. He, H.Z. Wang * State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan (Sun Yat-Sen) University, Guangzhou , China Received 23 September 2004; received in revised form 28 January 2005; accepted 2 February 2005 Abstract By theoretical calculating, we derive the theoretical results of soliton phase jitter in soliton transmission systems using super-gaussian filters, and we demonstrate that soliton phase jitter can be fully suppressed by super-gaussian filters. By theory and computer simulations, we show that super-gaussian filters are more effective in reducing phase jitter than conventional filters without sacrificing the signal-to-noise ratio. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Solitons; Optical communications; Optical phase jitter; Differential phase-shift keying For amplitude-shift-keyed soliton systems [1] the error-free transmission distance is limited by collision-induced timing-jitter, which depends on the intensity patterns of the bit streams. There has recently been a renewed effort to develop coherent optical communication systems, particularly differential phase-shift keying [2 4] (DPSK), which dose not require a local oscillator to perform decoding. In such systems the essential parameter to be retrieved at the receiving end is the soliton optical phase. The use of DPSK has recently allowed the demonstration of impressive transmission capacities in * Corresponding author. Tel.: ; fax: address: stswhz@zsu.edu.cn (H.Z. Wang). the context of long-haul fiber-optic communication systems [4 9]. In these systems, the error-free distance is mainly limited by random fluctuations of the phase caused by the amplified spontaneous emission (ASE) that occurs in optical amplifiers. Without in-line control, amplitude-to-phase noise conversion occurs through the self-phase modulation (SPM) [10,11] and cross-phase modulation (XPM) [12] effects, producing a phase variance that grows as the cube of the propagation distance [11,13]. However, it has already been [14] shown that in-line synchronous intensity modulation is not efficient and that in-line synchronous phase modulation lowers the phase jitter but does not keep it at an asymptotic level. Since power fluctuations caused by amplifier noise are converted into phase fluctuations by self-phase modulation, /$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi: /j.optcom

2 Y.J. He, H.Z. Wang / Optics Communications 250 (2005) in-line phase conjugation can reduce soliton phase jitter [15]. Nonlinear phase-shift compensation can overcome phase jitter [2,3], because the output phase variance depends on the nonlinear phaseshift coefficient. In-line filtering was also showed to reduce phase jitter efficiently [11,13,15 17] through the damping of amplifiers noise, which result in a linear increase of the phase jitter with distance. The filters are also effective in suppressing the timing jitter [18 26], in reducing the interaction between solitons [27], in overcoming the self-frequency shift induced by Raman scattering [28,29], and in improving the transmission control of dispersion-managed solitons [30,31]. However, the conventional filters usually in the form of lumped Fabry Perot étalons present the drawback whose equivalent distributed transfer function contains higher-order dispersion terms that produce unwanted effects such as a strong asymmetry in the sideband spectrum and an increase in temporal jitter [32]. Butterworth filters introduce additional loss terms resulting from Taylor expansion that negatively affect the soliton stability [23]. Mecozzi [33] proposed the use of higher order filters in overcoming timing jitter induced by ASE because as the order of filter increases a smaller excess gain is needed to achieve a specific damping coefficient. Peral et al. [23] and He et al. [26] has confirmed MecozziÕs prediction using super-gaussian filters. Super-Gaussian filters can be implemented with holographic fiber gratings [23] and its design is performed by means of the inverse scattering technique [19,34]. The transfer function of super-gaussian filters in the form of holographic fiber grating does not introduce any phase distortion and optimizes the jitter reduction [23]. Thus super-gaussian filters in the form of holographic fiber grating are better candidates for soliton transmission control than the previously proposed filters. This solution would permit an all-fiber system that is perfectly compatible with wavelength-division multiplexing [23]. So we predict that super-gaussian filters can more efficiently reduce phase jitter than the conventional filters. In this paper, we derive for the first time an analytical expression for the variance of the phase jitter induced by amplified spontaneous emission noise of a soliton transmission system in the presence of super-gaussian filters in the form of holographic fiber grating. We also perform computer simulations. In standard soliton units, from Ref. [35], we get the simplified propagation equation in the presence of Gaussian and super-gaussian filters is 2n u; ð1þ ou oz ¼ i o 2 u 2 ot þ 2 ijuj2 u þ au Xn g j i o ot j¼1 where n, a positive integer, determines the order of the filter, a and g j excess gain of the amplifiers and filter strength, respectively. The excess gain a is minimized if g j = 0 for j 6¼ n [23]. For n P 2, the filter is super-gaussian. For n = 2, introducing the usual ansatz for the soliton u = Asech- (At q)exp( ixt +ir), from Eq. (1) we obtain a pair of coupled equations for the soliton amplitude A and frequency X and the evolution equation of phase r (we set g 1 = 0 according to g j = 0 for j 6¼ n mentioned above, i.e., we omit the g 1 term): da dz ¼ 2aA 2g 2 7A 4 15 þ 2A2 X 2 þ X 4 A; ð2aþ dx dz ¼ 8 15 g 2 7A 2 þ 5X 2 XA 2 ; ð2bþ dr dz ¼ A2 X 2 : ð2cþ 2 For n = 1, the coupled equations for the soliton amplitude A and frequency X have been stated in Refs. [11,13]. Eqs. (2a) and (2b) have two equilibrium points, say (A a,x a ) and (A b,x b ) with X a 6 X b and A a P A b. The point (A a,x a ) is stable, i.e., for z!1, X! X a = 0 and A! A a = 1 whenever a =7g 2 /15. The point (A b,x b ) is unstable. For the equilibrium point (A a,x a ) = (1,0), introducing the first-order quantities X = X a + d, A = A a + a and r = r + Dr and linearizing Eqs. (2a) (2c) about the steady state, we obtain da dz ¼ g 2a þ F a ðz; tþ; ð3aþ dd dz ¼ g 2d þ F d ðz; tþ; ð3bþ

3 38 Y.J. He, H.Z. Wang / Optics Communications 250 (2005) ddr dz ¼ a þ F rðz; tþ: ð3cþ On the right-hand side of Eqs. (3a) (3c), we have added three delta-correlated noise factors of zero mean and correlation functions to account for the ASE noise added by the in-line amplifiers, which similar to the method of Refs. [22,26]. The statistics of the in-phase and quadrature components of the noise are given by [36,37] hf i ðz; tþf j ðz; t 0 Þi ¼ 1 2 n spn i d i;j dðz z 0 Þdðt t 0 Þ ði; j ¼ a; d; rþ; ð4þ ðg 1Þ ðg 1Þ N a ¼ ; N d ¼ ; N 0 Z a 3N 0 Z a N r ¼ p2 ðg 1Þ 12 þ 1 ; ð5þ 3N 0 Z a where N 0 = P 0 T 0 /hm = A eff Dk 4 /(4p 2 n 2 ht 0 c 2 ) is the ratio of the number of photons to the unit energy (in soliton units), n sp is the spontaneous emission factor, G = exp(cz a ) is the gain of the amplifier, Z a is amplifier spacing, and h is PlanckÕs constant. The normal modes of Eqs. (3a) and (3b) have two eigenvalues (damping constants): c 1 ¼ c 2 ¼ g 2: ð6þ The steady state of soliton transmission systems need Re(c 1 ) = Re(c 2 ) > 0. Because of g 2 > 0, the stable condition c 1 = c 2 > 0 must be satisfied. From Ref. [23], the required excess gain is a n ¼ P n j¼1 g jm j A 2j and damping coefficient is c n ¼ P n j¼1 4ng jm j A 2j, with M j equal to the 2jth derivative of the generating function f ðsþ ¼s= sinðsþ at s =0 [33]. If g j = 0 for j 6¼ n, we get excess gain a n and damping coefficient c n satisfy c n =4na n, here a n = g n M n A 2n, a 1 = 1/3g 1, a 2 = 7/15g 2. When c n is constant, a n µ 1/n, i.e., excess gain decreases with the increase of the order of filter in a specific damping coefficient, which shows high-order super-gaussian filters are more effective than low-order super-gaussian filters in reducing phase jitter. From Eqs. (3a) and (3b), we obtain aðzþ ¼ Z z 0 F a ðzþ expð c 1 zþ dz: ð7þ Combining Eqs. (3c) and (7), by use of standard techniques, we obtain the variance of phase jitter ÆDr 2 (Z)æ with distance. The result (in real units) is hdr 2 ðzþi ¼ Mf ðc 1 ZÞZ 3 þ M where p2 12 þ 1 Z; ð8þ M ¼ 2p2 hc 2 n 2 n sp f ðgþct 0 ; ð9aþ 3k 4 A eff D f ðc 1 ZÞ¼ expð c 1ZÞ þ 1 expð 2c 1ZÞ ; c 2 1 Z2 c 1 Z 2c 1 Z ð9bþ where Z = zl D (L D is dispersion length) and f(c 1 Z) is a jitter-reduction factor, which is the same for phase jitter [11,13] and for temporal position jitter [24,25] and f(g) =F(G)/ln(G), F(G) =(G 1) 2 / [G(lnG)] is the ratio of the soliton peak power at the amplifier output to the peak power of the average soliton. On the right hand of Eq. (8), the first term shows that amplitude-to-phase noise conversion occurs through the self-phase modulation (SPM) [10,11], the last term is a small contribution of direct coupling of the ASE noise on phase. For large values of Z and neglect the last term of Eq. (8), we have f(c 1 Z)=3/(c 1 Z) 2, i.e., hdr 2 ðzþi ¼ 2p 2 hc 2 n 2 n sp f ðgþct 0 Z=ðc 2 1 k4 A eff DÞ, which means that the phase jitter grows only in linear proportion to Z. Without filtering, i.e., g j =0, we have f(c 1 Z) = 1, i.e., ÆDr 2 (Z)æ =2p 2 hc 2 n 2 n sp f(g)c T 0 Z 3 / (3k 4 A eff D), which is agreement with the result shown in Refs. [11,13]. There is no surprise that the phase jitter recovers the cubic growth of unfiltered case with distance, because two damping constants becomes zero. This shows the need for phase jitter control if DPSK is to be successfully implemented. Thus, in-line filtering appears to be an efficient control on soliton optical phase. The main drawback of in-line filtering is that the noise that has exactly the soliton frequency is less attenuated than the soliton itself and grows exponentially with distance, eventually saturating the amplifiers and creating a strong continuum wave [13,18]. This phenomenon leads to a trade-off involving the strength of the filter and limits the achievable length of the communication channel [13]. The use of a sliding-frequency filter whose

4 Y.J. He, H.Z. Wang / Optics Communications 250 (2005) center frequency moves with distance along the transmission line can reduce continuum wave, which leads to solitons transmission in a longer distance [20]. However, the slidingaction of filter increases timing jitter [22,26] and phase jitter [16]. This is the likely that the sliding action produces an extra coupling between amplitude and frequency fluctuations that enhances the timing jitter and phase jitter with respect of the case without sliding. The timing jitter with sliding was measured experimentally by Mollenauer et al. [38]. The phase is evaluated along the propagation by use of the formula given in Ref. [39]: " R # juj 2 r ¼ tan 1 ImðuÞ dt R : ð10þ juj 2 ReðuÞ dt Considering a numerical example given in Ref. [13], we use lumped filters with the equivalent distributed filter strength g 2 = (c 1 = 0.235). Considering a 40-Gbit/s system, we set the pulse width 1.763T 0 = 12 ps. By setting the typical values of other parameters as: k = 1.55 lm, D = 0.25 ps/ (nm km), n sp = 1.5, C = km 1 (0.2 db/km loss), n 2 = cm 2 /W, A eff =50lm 2, we get the dispersion length L D = km. The amplifier spacing Z a = 45 km. The soliton peak power P = 2.8 mw and energy E =38f J. Averaging over 1500 realizations by a slip-step method, we perform the computer simulations. In Fig. 1, we plot the variance of phase jitter with distance theoretically based on Eq. (8) and computer simulation according to Eq. (1) in the presence of super-gaussian filters (n = 2), and the computer simulation [curve (+)] shows a good agreement with theory (solid curve). Dash curve shows the results of theory in the presence of the conventional filters based on Ref. [13]. Dash-dotted curve is the theoretical result without filtering. In Fig. 2, we plot the variance of phase jitter with distance in the presence of sliding-frequency filters with sliding rate 4.5 GHz/Mm (in real unit). Dash curve is the theoretical result in the presence of the conventional sliding-frequency filters based on Ref. [16]. Dotted curve is the simulation result in the presence of the super-gaussain sliding-frequency filters according to Ref. [40]. From Figs. 1 and 2, we realize that super-gaussian filters are superior to the conventional filters in Standard deviation of the phase (rad) Distance (Mm) pffiffiffiffiffiffiffiffiffiffiffi Fig. 1. Standard deviation of the phase hdr 2 i=p versus propagation distance. Theoretical result (dash-dotted curve) unfiltered case, theoretical result (dash curve) with conventional Gaussian filters case (n = 1) without sliding, and theoretical result (solid curve) and numerical simulation (++) with super- Gaussian filters case (n = 2) without sliding. Standard deviation of the phase (rad) Distance (Mm) pffiffiffiffiffiffiffiffiffiffiffi Fig. 2. Standard deviation of the phase hdr 2 i=p versus propagation distance. With sliding and the sliding rate is 4.5 GHz/Mm, theoretical result (dash curve) with conventional Gaussian filters case (n = 1) and numerical simulation (dotted curve) with super-gaussian filters case (n = 2). Other parameters are the same as Fig. 1. reducing phase jitter, and sliding action increases phase jitter. Assuming a perfect phase receiver and a Gaussian probability density function, the standard deviation of the phase that yields a symbol error rate of 10 9 is 0.26 rad.

5 40 Y.J. He, H.Z. Wang / Optics Communications 250 (2005) We have derived the expression for the variance of the phase jitter of an isolated soliton in the presence of super-gaussian filters. Computer simulations have shown a good agreement with the theory. Compared with the conventional filters usually in the form of Fabry Perot étalons, the super-gaussian filters in the form of holographic fiber gratings are better in overcoming phase jitter. Acknowledgments The authors thank the anonymous referee(s)õs insightful comments. This work was supported by the National Natural Science Foundation of China ( ), National 973 (2003CB314901) Project of China, National 863 (2003AA311022) Project of China, and Natural Science Foundation of Guangdong Province of China. References [1] L.F. Mollenauer, J.P. Gordon, P.V. Mamyshev, in: I.P. Kaminow, T.L. Koch (Eds.), Optical Fiber Telecommunications IIIA, 373, Academic Press, San Diego, CA, [2] X. Liu, X. Wei, R.E. Slusher, C.J. Mckinstrie, Opt. Lett. 27 (2002) [3] C. Xu, X. Liu, Opt. Lett. 27 (2002) [4] A.H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, A. Agarwal, S. Banerjee, D. Grosz, S. Hunsche, A. Kung, A. Marhelyuk, D. Maywar, M. Movassaghi, X. Liu, C. Wei, D.M. Gill, in: Optical Fiber Communications ConferenceOSA Technical Digest Series, vol. 70, Optical Society of America, Washington, DC, 2002, post deadline paper FC2. [5] A.H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, E. Burrows, Photon. Technol. Lett. 15 (2003) 467. [6] P.S. Cho, V.S. Grigoryan, Y.A. Godin, A. Salamon, Y. Achiam, Photon. Technol. Lett. 15 (2003) 473. [7] K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, S. Kuroda, T. Mizuochi, Optical Fiber Conference, 2003, paper ThE2. [8] J.-X. Cai, D.G. Foursa, C.R. Davidson, Y. Cai, G. Domagala, H. Li, L. Liu, W.W. Patterson, A.N. Pilipetskii, M. Nissov, N.S. Bergano, Optical Fiber Conference, 2003, paper PD22. [9] C. Xu, X. Liu, L.F. Mollenauer, X. Wei, Photon. Technol. Lett. 15 (2003) 617. [10] J. Gordon, L.F. Mollenauer, Opt. Lett. 15 (1990) [11] E. Iannone, F. Matera, A. Mecozzi, M. Settembre, Nonlinear Optical Communication Networks, Wiley Interscience, New York, 1998, p. 155 and 163. [12] C.J. Mckinstrie, C. Xie, C. Xu, Opt. Lett. 28 (2003) 604. [13] M. Hanna, H. Porte, W.T. Rhodes, J.-P. Goedgebuer, Opt. Lett. 24 (1999) 732. [14] O. Leclerc, E. Desurvire, Opt. Lett. 23 (1998) [15] C.J. Mckinstrie, S. Radic, C. Xie, Opt. Lett. 28 (2003) [16] M. Hanna, D. Boivin, P.-A. Lacourt, J.-P. Goedgebuer, Opt. Commun. 231 (2004) 181. [17] D. Boivin, M. Hanna, P.-A. Lacourt, J.-P. Goedgebuer, Opt. Lett. 29 (2004) 688. [18] A. Mecozzi, J.D. Moores, H.A. Haus, Y. Lai, Opt. Lett. 16 (1991) [19] Y. Kodama, A. Hasegawa, Opt. Lett. 17 (1992) 31. [20] L.F. Mollenauer, J.P. Gordon, S.G. Evangelides, Opt. Lett. 17 (1992) [21] L. Mollenauer, E. Lichtman, M.J. Neubelt, G.T. Harvey, Electron. Lett. 29 (1993) 910. [22] A. Mecozzi, M. Midrio, M. Romagnoli, Opt. Lett. 21 (1996) 402. [23] E. Peral, J. Capmany, J. Marti, Opt. Lett. 21 (1996) [24] H. Toda, K. Mino, Y. Kodama, A. Hasegawa, L. Fellow, P.A. Andrekson, J. Lightwave Technol. 17 (1999) [25] M.F.S. Ferreira, S.C.V. Latas, J. Lightwave Technol. 19 (2001) 332. [26] Y.J. He, G.S. Zhou, X.J. Shi, G. Wang, W.R. Xue, Opt. Eng. 43 (2004) 489. [27] Y. Kodama, S. Wabnitz, Opt. Lett. 19 (1994) 162. [28] M.F. Ferreira, Opt. Commun. 107 (1994) 365. [29] P. Tchofo Dinda, K. Nakkeeran, A. Labruyère, Opt. Lett. 27 (2002) 382. [30] M. Nakazawa, K. Suzuki, E. Yamada, H. Kubota, Y. Kimura, M. Takaya, Electron. Lett. 29 (1993) 729. [31] A. Tonello, A. Capobianco, S. Wabnitz, S. Turitsyn, Opt. Commun. 175 (2000) 103. [32] P.K.A. Wai, C.R. Menyuk, Y.C. Lee, H.H. Chen, Opt. Lett. 11 (1986) 464. [33] A. Mecozzi, Opt. Lett. 20 (1995) [34] E. Peral, J. Capmany, J. Marti, Electron. Lett. 32 (1996) 918. [35] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, Boston, MA, [36] H.A. Haus, J. Opt. Soc. Am. B 8 (1991) 112. [37] J.P. Gordon, H.A. Haus, Opt. Lett. 11 (1986) 665. [38] L.F. Mollenauer, P.V. Mamyshev, M.J. Neubelt, Opt. Lett. 19 (1994) 704. [39] K.J. Blow, N.J. Doran, S.J.D. Phoenix, Opt. Commun. 88 (1992) 137. [40] P.V. Mamyshev, L.F. Mollenauer, Opt. Lett. 15 (1994) 2083.