Timing jitter analysis for optical communication systems using ultrashort solitons and dispersion-decreasing fibers

We use adiabatic perturbation theory to calculate the timing jitter induced by fluctuations in solitons amplitude, frequency and position due to amplifiers noise when ultrashort solitons ( ~ ! ps) are propagated in dispersion-decreasing fibers. The result is applied to a high-speed soliton communication system with amplifiers spacing of 50-100 km. We show that the transition from a regime where frequency fluctuations dominate the timing jitter (Gordon-Haus jitter) to the regime where amplitude fluctuations dominate (Raman-induced jitter) occurs for soliton widths of,,5 ps; the precise value is determined by the total distance of transmission. The contribution of third.order dispersion to the timing jitter is included in our analysis. We also provide an upper estimate of the distance where a soliton-control device (e.g., optical filters, modulators) needs to be inserted to control the timing jitter, Eeywords: Amplifier heir; Solitons; Optical fiber communication; Nonlinear optics; Optical fiber dispersion It is well known [ I ] that the main limitation on the bit rate of average-soliton [ 2] communication systems stems from the timing jitter due to amplifier-noise-induced frequency fluctuations (the Gordon-Haas effect [ 3 ] ). Several techniques such as sliding-frequency filters and synchronous modulation have been used for reducing the timing jitter. In fact, when a soliton-control method is used, bit rates are no longer limited by the timing jitter but rather by the relatively long duration (,,, 15 ps) of the average solitons. To overcome this limitation on the bit rate, it has been proposed to use dispersion-decreasing fibers (DDFs) [ 4,5]. Such DDFs allow stable propagation of solitons shorter than those needed in the average-soliton regime if the system is appropriately designed to take into account various higher-order effects [6]. A natural question is how the amplifier noise affects such ultrashort solitons. Previous analyses [ 7-9 ] have shown that the Raman-induced soliton self-frequency shift (SSFS) can increase the timing jitter associated with propagation of short solitons and may even dominate the total jitter for ultrashort solitons. However, these studies considered constant-dispersion fibers which are unlikely to support stable propagation of ultrashort solitons (less than I0 ps) because of fiber loss, higher-order dispersive and nonlinear effects, and a relatively short soliton period (typically 1 km). It has been recently shown that dispersiondecreasing fibers can support solitons as short as ,,, 100 fs [6,10]. In this paper, we calculate the timing jitter of ultrashort solitons induced by amplifier noise in DDFs by includ'~ both the Raman effect and third-order dispersion ('rOD). 0030-40181961512,00 Copyright ~) 1996 Elsevier Science B.V. All rights reserved. PI! 50030-4018 ( 96 ) 0037 5-6 R.-J. Essiambre, G.P. Agrawal/Optics Communications 131 (1996) 274-278 275 The propagation of ultrashort solitons through a DDF is described by a generalized nonlinear Schr6dinger equation [ 11 ], .Ou 02u ~ "8 03u t.$7 + 1⁄2p( z )-~r2 + lul2u = -1⁄2it~u + rsu ~,, +, da,r3, (1) where 8d = f13 / (6Tolf12 (0)l) is the normalized TOD, p ( z ) = [f12 ( z )/f12 (0) [ is the normalized group-velocity dispersion (GVD), a is the fiber loss, and eR = TR/To is the Raman parameter normalized to the soliton characteristic width To. The time coordinate ¢ is normalized to To and the propagation distance z is normalized to the dispersion length Lt) = T 2/1fl2(0) I. To apply the adiabatic perturbation theory (APT) [ 12,13], we rescale u and z in Eq. ( 1 ) to a new amplitude v and a new distance scale 7/defined by z P v =p-I/2u, rl = / P ( Y ) dy. (2) t a r 0 Eq. ( 1 ) then takes the form of a perturbed nonlinear Schr6dinger equation, ( l d p ) O,v, 2 .~dO~3O . Ov 102v a ~ ~ V+¢RV + t (3) ,~--~. + ~-~r 2 + Ivl2vi ~pp + 2p ~ p 0¢ 3. For a DDF with a nearly ideal dispersion profile (p(z) ~ e x p ( a z ) ) and [fl~in[ larger than ~, 0. l ps 2/km, the three terms on the right side of Eq. (3) become small enough to allow for the use of APT. Here [fl~nin[ is the absolute value of the minimum dispersion lfl2(za)] at the fiber end, where za is the normalized amplifier spacing. We assume a fundamental soliton of the form vs(B,q,~b, to;¢) = B s e c h [ B ( ¢ q) ] exp(i~b ito¢), (4) where the parameters B, q, ~b, and to represent the rescaled soliton amplitude, position, phase, and frequency, respectively, and are slowly varying functions of r/. By using the standard APT [ 13], their evolution is governed by dB ( p + I d p ) (5) dto = _ s ~'n B 4 , ( 6 ) dr/ dq 8,1 B2 38_.d.a 2, (7) = w + + d~ p P where the evolution equation of ~b(~/) was omitted because the timing jitter does not depend on the soliton phase when soliton interaction is negligible. According to Eq. (2), the normalized and rescaled solitons differ only by their different amplitudes related by A = Bp -t/2. Using Eq. (2) and assuming a loss-matched dispersion profile, p(z ) e x p ( a z ), Eqs. (5) (7) can be solved to yield A ( z ) = A0 e x p ( a z / 2 ) , (8) (9) 8 4 W( Z ) = ~ ' r R A 0 Z l ( Z ) -ttoo, q(z) =qGtl(Z)WO+qR(Z)A4 +q~OD (z)A2 +q~x3D(z)t°2 +q¢I(z)A~ +q¢2(z)A4w°+q°' (10) where Ao, too, and qo are the initial values of the soliton amplitude, frequency and position, respectively, and 276 R-J. Essiambre. G.P, Agrawal/Optics Communications 131 (1996) 274-278 qGH(Z) = -Z l (Z) , qR(Z) = "~5'rR [Zi(Z) -Z2(Z)] , q oD( Z ) = SdZ, q oD( Z ) = 38dZ. qcl(Z ) = 192 8d 1"2 48 8d~'r [ Z -ZI(Z ) ] 225 ~2 [Z--2ZI(Z)+Z2(Z)] , qC2(Z) = IS---~ Zl (Z) = 1 -exp(--aZ) Z2(Z ) = 1 -exp(--2aZ ) ( 1 1 ) a ' 2a " Here, the parameters zl (z) and z2( z ) have been introduced for convenience, qGH (Z) is the term leading to the Gordon-Haus effect, qr (z) governs the soliton displacement due to the Raman effect, the two terms q~D(z ) and q~'oD(Z) are associated with the direct effect of TeD on the soliton position, while the last two terms qcl (z) and q ~ ( z ) represent cross-coupling effects of TeD and the Raman effect. Full numerical simulations of Eq. ( 1 ) for various combinations of parameters have shown that Eq. (I0) gives the evolution of the soliton position with an accuracy better than 1% for nearly all the values of the parameters used in this paper. The displacement 8q(za) of a soliton at the end of one DDF segment resulting from variations 8A0, 8oso. and 8q0 in the soliton amplitude, mean frequency, and position at the fiber input is obtained by differentiating Eq. (l 0) and is given by 8q(za) = qGa(Za) &o0 + 4qR( za)A~SAo + 2q'~oD(Za)A08A0 + qc2 A48oso + 8qc (z.) A7, Ao + 8qo. (12) The root-mean-square (RMS) displacement o',~, obtained by summing the individual displacements over a chain of N amplifiers composing the link, is given by ( 2~,s~2As' 2 (2N4q, q~oDA4+4.,s2-6 0"2q = ~N3q~H + ~N4qGHqc2A~ + T~'" '/C2~0}0"co + .~N'qRA0 , 3 ,2.2 " ' + qc, 0 + N,,:,, (13) +~N6qrqc , A~ ° + l N ('¢TODJ ~'0 + f iN qTooqcl A,7 2 -14'~ 2 ', where we have replaced the summation over the N sections by an integration and assumed that the noise of different amplifiers are uncorrelated [3]. The quantities o',~, o',0, and o'q are delined by using o'1, = (8P2) I/2. where or,,, is the RMS deviation of the variable P. The expression above for the timing jitter o-q includes the contributions of the Raman effect and 'roD. An expression similar to Eq. (13) has been derived in Ref. [9] for the case of constant-dispersion fibers but without considering the qc terms which become important for femtosecond solitons. Moreover, its application is limited by the difficulty to fulfill the average-soliton condition for ultrashort solitons in constant-dispersion fibers. Fluctuations of the soliton frequency and position at the output of an amplifier due to the noise added during the amplification process has been derived in Ref. [ 14] and generalized to the soliton amplitude and phase in Ref. [ 13]. The three fluctuations needed here are given by 2 I 2 9'/'2 2 o "2 = AoeonspF(G), o'o, = ~o'A, o'2 = l--~ooo" A, (14) where n.~p is the spontaneous emission factor (set to 2), F(G) = G I with the total gain G : exp(az,,) for each amplifier, and eo = 2hos/E, is the ratio of the energy of a photon to the average energy per bit of information. The average energy per bit corresponds to half the soliton energy when the same number of "zeros" and "ones" are transmitted. Eq. (13), with variances of the soliton parameters given by Eqs. (14), represents the total RMS timing jitter of a soliton as short as few hundreds of femtoseconds in dispersion-tailored fibers and is the main result of this paper.