The dark-soliton timing jitters caused by fluctuations in either the soliton initial phase angle or the background amplitude when such a soliton propagates in a monomode optical fiber under the influence of the stimulated Raman scattering are investigated and compared with those that exist when the stimulated Raman scattering is not present. In addition, it is demonstrated that in the presence of the stimulated Raman scattering, there exists a distance at which, for the negative soliton initial phase angle, the dark-soliton timing jitter caused by fluctuations in the background amplitude becomes zero

Baboiu, D. -M.

Mihalache, D.

Panoiu, N. -C.

English

University of Central Florida (UCF): STARS (Showcase of Text, Archives, Research & Scholarship)

University of Central Florida STARS Faculty Bibliography 1990s Faculty Bibliography 1-1-1995 Dark-Soliton Timing Jitter Caused By Fluctuations In Initial Pulse-Shape N. -C. Panoiu D. Mihalache D. -M. Baboiu University of Central Florida Find similar works at: https://stars.library.ucf.edu/facultybib1990 University of Central Florida Libraries http://library.ucf.edu This Article is brought to you for free and open access by the Faculty Bibliography at STARS. It has been accepted for inclusion in Faculty Bibliography 1990s by an authorized administrator of STARS. For more information, please contact STARS@ucf.edu. Recommended Citation Panoiu, N. -C.; Mihalache, D.; and Baboiu, D. -M., "Dark-Soliton Timing Jitter Caused By Fluctuations In Initial Pulse-Shape" (1995). Faculty Bibliography 1990s. 1429. https://stars.library.ucf.edu/facultybib1990/1429 PHYSICAL REVIEW A VOLUME 52, NUMBER 5 NOVEMBER 1995Dark-soliton timing jitter caused by fluctuations in initial pulse shape¹C.Panoiu, ' D. Mihalache, ' and D.-M. Baboiu"''Department of Theoretical Physics, Institute of Atomic Physics, P 0 Box. MG-6, Bucharest, RomaniaPhysics Department, New York University, 4 Washington Place, New York, New York 10003Department of Signal Theory and Communications, Polytechnic University of Catalonia, 08034 Barcelona, SpainCenter for Research in Electro Opti-cs and Lasers, University of Central Florida, Suite 400, /2424 Research Parkway,Orlando, Florida 32826(Received 29 March 1995)The dark-soliton timing jitters caused by fluctuations in either the soliton initial phase angle or the back-ground amplitude when such a soliton propagates in a monomode optical fiber under the influence of thestimulated Raman scattering are investigated and compared with those that exist when the stimulated Ramanscattering is not present. In addition, it is demonstrated that in the presence of the stimulated Raman scattering,there exists a distance at which, for the negative soliton initial phase angle, the dark-soliton timing jitter causedby fluctuations in the background amplitude becomes zero.PACS number(s): 42.50.RhThe possibility of propagation of dark solitons in the re-gion of normal (positive) group-velocity dispersion (GVD)in monomode optical fibers has been theoretically predictedby Hasegawa and Tappert [1]and has been successfully dem-onstrated in a number of elegant experiments [2].Since then,the properties of dark-soliton propagation in optical fibershave been thoroughly studied in several works [3—10] andhave been reviewed in two recent articles [11,12].The dark-soliton propagation in real fibers is subject to various kindsof perturbations such as the third-order dispersion, the ab-sorption loss, the stimulated Raman scattering (SRS), and thestochastic perturbations. The stochastic perturbations couldbe classified into two groups [13]: the homogeneous one,when the stochasticity originates in the random perturbationsin the fiber itself, and the inhomogeneous one, when thestochasticity is present in the input pulse to the fiber. In thelatter case, the propagation is entirely deterministic; only theinitial parameters of the pulse are random quantities. Onemajor source of the inhomogeneous stochasticity is the am-plified spontaneous emission (ASE) noise. The ASE noiseproduces a soliton timing jitter (the Gordon-Haus effect) andthe theory of this effect has been given in [14] for brightsolitons and in [15] for dark ones.In this paper we study the dark-soliton timing jitter causedby fluctuations in the soliton initial parameters when such asoliton propagates in a monomode optical fiber in the pres-ence of the SRS. We focus on the inhuence of the SRS ondark-soliton timing jitter because, as it is well known, theSRS has the most deleterious effects for dark-soliton propa-gation in optical fibers.For the normal (positive) GVD the propagation of pico-second optical pulses in monomode optical fibers is gov-erned by the famous nonlinear Schrodinger equationz = P2Z/To and t =(T Z/Us)/—To, where P2 is the GVDcoefficient, To is the pulse width (TFwHM= 1.763Tp, whereFWHM is the full width at half maximum), v is the group-velocity, and Z and T are the physical distance and time,respectively.When we deal with subpicosecond optical pulses, an ac-curate treatment of their propagation calls for the introduc-tion of new terms in Eq. (1). Thus, to take into account theSRS, we have to modify Eq. (1) as follows:(2)where o.= T& ITO and Tz is related to the slope of the Ramangain (TR 6 fs) [1——6].It is well known that Eq. (1) supports the modulationally~ 2stable continuous-wave (cw) background P= pe't' ', p beingthe background amplitude. Because the Raman term tends tozero when t + ~ ~, it does not affect the background ampli-tude p. Now, in order to simplify our further analysis, we~ 2make the transformation P(z, t) =e'~ 'u(z, t). Thus Eq. (2)becomesBR 188 ot( u )i — ——, + (IuI' —p')u=a-uBz 2 Bt Bt (3)u, (z, t) = p(cosPtanhj —isin@), (4)A dark soliton of Eq. (3) without the Raman term(a.=0) is a kink-type soliton that connects two stable back-ground waves with the same amplitude but with differentphases and has the formBP 18$where g is the normalized distance along the fiber, t is thenormalized time, and P is the normalized complex amplitudeof the pulse envelope. The normalized units are well known: Iu, I = p (1 —cos P sech s). (5)where (= rg(t Az), r/= peas@, and —0=psinP. Such asoliton is characterized by its phase angle @ (I @I (vr/2),which describes the darkness of the soliton through thesimple relation1050-2947/95/52(5)/4182(5)/$06. 00 4182 1995 The American Physical Society52 DARK-SOLITON TIMING JITTER CAUSED BY. . . 4183This represents a dip with the depth equal tog = p cos P and the velocity II = psing, which moves on abackground intensity with the amplitude p . When P(&1 wecall this soliton a dark-gray one and whenP= ~ m/2~ e, e(&1, we call it a light-gray one [8]. FromEq. (4) we see that the dark-gray soliton has a small velocityand large amplitude while the light-gray one has a large ve-locity and small amplitude so that it could be considered as asmall excitation on the cw background.For the dark soliton (4) the shift in the inverse groupvelocity is /320/Tp, that is, 1/Vg = 1/Us+ p2A/Tp. Thus thesoliton arrival time T in the absence of the Raman term isgiven by the formula4 p3H, = cos @. (12)situated at the edges of the allowed discrete spectrum of theinverse problem. In the presence of the Raman term Eq. (3)ceases to be a Hamiltonian system, so that the phase angle@ is no longer a constant parameter becoming a z-dependentfunction. We suppose that the Raman term is a small pertur-bation so that we can apply the adiabatic approximation. Inthis approximation the emission of radiation and the creationof new solitons are disregarded. Thus we start from theHamiltonian of the unperturbed system (O.=O), which forthe dark-soliton solution (4) takes the value(zdZ'' ( Z p2psln@plT 0 + z,Jo Vg I Us To / (6)Next, by using Eqs. (3) and (ll), we calculate the derivativeof H with respect to z,where Po represents the soliton initial phase angle. Equation(6) can be put in its dimensionless form asg(~u~ )(=l u* --- —u dt.dZ 2 J ~ Bt I, Bt Bt(13)Zt = + psin(@p)z.UgTp(7)Now, let us suppose that either the soliton initial phaseangle Pp or the background amplitude p exhibits fiuctuationswith standard deviations 6@p and Ap, respectively. As aconsequence, the soliton arrival time t exhibits fluctuations(the so-called soliton timing jitter) given byAt~ pcos(@—o—)zh@p, 3 t = ~singo~zhp. (8)Next we calculate the soliton timing jitter due to Auctua-tions in either the soliton initial phase angle Pp or the back-ground amplitude p in the presence of the SRS. For thispurpose we have to establish the dependence of the phaseangle @ on the normalized propagation distance z. In orderto derive this dependence we use the Hamiltonian approach(see, e.g. , Ref. [17] and the references therein). It is wellknown that Eq. (3), with o.=O and finite boundary condi-tions, is a Hamiltonian dynamical system. This means that itis equivalent to the Hamilton equationsBLE—= lH, uj, = (H, u*),BZwhere the Poisson brackets are defined as('" (8F gG PF PGN/F, G)/= t — dr, (lo)where F and G are functionals of u and u* and the Hamil-tonian H is taken as1 ( BuBu*H= — — +(iud —p ) dt.2) Bt BtIn these formulas the asterisk stands for the complex conju-gate.It is known that in the case of dark solitons the presenceof certain perturbations could lead to the creation of newsolitons with small amplitudes and large velocities. Thesenew solitons correspond to zeros of the scattering matrixFor the dark-soliton solution this equation givesdH, 16dz 15= ——o.p cos @sing,so that the resulting equation describing the evolution of thesoliton phase angle @ with respect to z can be written as[8,9]dP 4= —o.p cos P. (15)sing 1+sin@F(gz, fp, p)= 2 +lnCOS COSSill @pCOS @o1+singp 8—ln ——~p3Z=o,cos@p 15 (16)where Pp= P(O) is the soliton initial phase angle. The ar-rival time T in the presence of the perturbation is(zdZ' Z p2p ( zT = = +~SlllgdZp Vg Ug Tp Jo (17)or in its dimensionless formThis expression shows us that the soliton phase angle P canonly increase with distance under the Raman perturbation, soif the initial phase angle Pp is positive (soliton with positivevelocity), the soliton depth r/ decreases, leading to the soli-ton decay, and if the initial phase angle @p is negative (soli-ton with negative velocity), the soliton, in a first stage, willturn into a black one (soliton with zero velocity) and then itsphase angle continues to grow, leading also to soliton decay.In other words, Eq. (15) shows that the SRS leads to a soli-ton self-frequency shift and soliton decay. These effects havebeen observed experimentally in [3] and explained analyti-cally in [18] in the small-amplitude limit.By integrating Eq. (15) with respect to z we obtain thephase angle P= P(z, Po) as the solution of the implicit equa-tion4184 ¹C.PANOIU, D. MIHALACHE, AND D.-M. BABOIU 52Zt = + p sin@(z')dz'UgTp ) p (18) 45—(a)When either the soliton initial phase angle Po or the back-ground amplitude p Auctuates, it is obvious from Eq. (18)that the soliton arrival time t fluctuates, that is, a solitontiming jitter arises.In the case of soliton timing jitter caused by fluctuationsin the soliton initial phase angle Po, we have, from Eq. (18)30—fz 8At&= p cos(P) dz' b, Po.Jp 0 (19)Taking into account that from Eq. (16) we haveoIQ COS PoI@p COS @p(2o)10—we obtainpAt&= 3 cos P(z')dz'rCOS p p152 3 [sing(z) —singo]b, /II.4rrp cos p (21)I I I j I I I I j I I I I j I I I I j I I I I j I I I I j I I I I j I I I j I I I I j I I I I j I I I I0 5 10 'l5 20 25 30 35 40 45 50 5550—We note that in deriving this result we made use of Eq. (15).In order to compare the timing jitter At@ in the presence ofthe SRS with the timing jitter At@ in its absence we considerthe difference between themfz5t~ b, t~ 3—[cos —P(Z') —cos Po]dZ' AItIo.C OS p II(22)Bearing in mind that the phase angle P is an increasingfunction of z, we deduce from Eq. (22) that, for positivevalues of /II, the timing jitter b t& is less than the timingjitter At@, while for negative values of po there is a thresh-old value of the propagation distance z th so that for distancesunder this threshold At@)At& and for distances above thisthreshold At&(At&. This is illustrated in Figs. 1(a) andl(b). In Figs. 1(a) and 1(b) the soliton initial phase angle isgo = m./6 and /II= —m/9, respectively. The other parametersare p=l and o.=0 (dashed lines) or o.=0.1 (solid lines).From Fig. 1(b) we evaluate z~~h=45.In the case of soliton timing jitter caused by fluctuationsin the background amplitude p, from Eq. (18) we haveIz rzAt = sin@(z')dz'+ p cos(@) dz' Ap. (23)Jo Jo ~pTaking into account that from Eq. (16) we have35—3025:20—0 I I I I j I I I I j I I I I j I I I I j I I I I j I I I I j I I I I j I I I I j I I I I j I I I I j I I I I j0 5 'l0 15 20 25 30 35 40 45 50 55FIG. 1. Normalized soliton timing jitter caused by fiuctuationsin the soliton initial phase angle. Here p= 1, o.=0 (dashed lines),and o.=0.1 (solid lines). (a) Po= vr/6 and (b) Po= —7r/9.Now we compare the soliton timing jitter with and with-out SRS. The analysis depends on the sign of the solitoninitial phase angle Po.In the case @o)0 we haveAt =sin(ztIo)zb PoIQ 4Bp 5= —op zcos P,after simple calculations Eq. (23) becomesfzAt = 3zsin@(z) —2 sing(z')dz' Ap.P Jp(24)(25)so that( I'z&t = 3zsin@(z) —2 sin@(z')dz' &p,Jo (27)52 DARK-SOLITON TIMING JITTER CAUSED BY. . . 4185(a) as In the case Po(0 we consider the function f(z) defined50 f(z) =3zsin@(z) —2 sin@(z')dz'.Jo (29)Q 40~ 3025:20—15:10—5:0 ! I I I i I I I I [ I I I I i I I I I I I I I I i I I I I i I I I I ] I I I I i I I I I i I I I I i I I I I t0 5 'IO 15 20 25 30 35 40 45 50 55The derivative of this function with respect to z isf'(z) = sin@z)+3zcos@(z)@'(z), where the prime means dif-ferentiation with respect to z. From this expression it can beseen that f '(0) (0, so that, by taking into account thatf(0) = 0, we conclude that for small values of z the functionf(z) is negative. In turn, for large values of z the functionf(z) becomes positive. So there exists a value zo at whichf(zII) =0, that is, At (zo) =0.Now let us compare the jitters A t and A t . For0&z&zo we have( f.At At = —2 I sing(z')dz' —zsin@(z)I~o+z(sinIt!o —sin@) Ap, (30)(b)so that Atp(Atp For z~zo we have60—~ 50-40—fgA t At = 3z—sin@(z) —2 sing(z ')dz'+ zsin@o A p.Jo(31)From this expression it can be inferred that there exists athreshold propagation distance z h~ zo so that A t A twhen z~zth and At t when z~z,h. This behavior isillustrated in Fig. 2(b). From this figure we evaluate zo —27and zPh=40.Finally, we briefly discuss the soliton timing jittersAt&, At&, At, At in the limit cases of dark-gray and light-gray solitons.In the case of dark-gray solitons (@o(&1) the formulas (8)become0 I I I I f I I I I i I I I I f I I I I J I I I I i I I I I i I I I I i I I I I f I I I I ( I I I I i I I I I0 5 'IO 15 20 25 30 35 40 45 50 55( 2$At@= p' I — zA@o At = I @olzAp.o@o oFIG. 2. Normalized soliton timing jitter caused by fluctuationsin the background amplitude. Here p= 1, o =0 (dashed lines), andrr=0. 1 (solid lines). (a) go= 7r/6 and (b) Po= —7r/4.In order to And the expressions for soliton timing jitters inthe presence of SRS we first solve Eq. (15) in the limit@,@o(&I. After simple calculations we obtain4o+ tanh4Gp15At At = zl sin@(z) —sin@o](+2 zsin@(z) —I sin@(z')dz' I Ap.r30(28)It!otanh4c7 p15(33)By introducing this expression in Eqs. (21) and (25) we ob-tainThis formula and the fact that @ is an increasing function ofz show us that in this case A t ~ A t . This is illustrated inFig. 2(a). and8At~=At~ —a.Pop z—A@ +Oo(o).(34)4186 ¹C.PANOIU, D. MIHALACHE, AND D.-M. BABOIU85t = b, t + sgn(@o) —o.p z 6p+ O(o ), (35)where sgn(go) =1 for @o)0 and sgn(Po) =—1 for go&0.We study the light-gray soliton case ( Po = ~ vr/2M Eo, Ep && 1 ) with positive velocity ( Po = 7r/2 —eo,so&&1). The analysis of the other case is similar. From theformulas (8) we deduceb, t~=peozA@o, At = 1 ——zAp.2) (36)As before, we derive the expression for the small parametere, which corresponds to the soliton phase angle @,I' 86'= Ep 1+ op E'pz (37)and for the timing jitters in the presence of the SRS8ht~=At~ —op —eoz Ago+0(o- ), (38)8&t, = &t,+ trp'eoz'& p—+ O(tr'). (39)In conclusion, we have studied the dark-soliton timingjitter caused by fluctuations in the soliton initial pulse shape.In our analysis we first supposed that the dark-solitonspropagate unperturbed and we have calculated the solitontiming jitters caused by fluctuations in either the soliton ini-tial phase angle Po, that is, At&, or the background ampli-tude p, that is, At . Then we have calculated the same jit-ters, supposing that the dark solitons propagate under theinfluence of the SRS, that is, At& and At . It has beenestablished that for dark solitons with a positive soliton ini-tial phase angle (@o)0) the soliton timing jitter caused byfluctuations in soliton initial phase angle is smaller in thepresence of SRS than in the unperturbed case, while the soli-ton timing jitter caused by fluctuations in background ampli-tude is larger in the presence of SRS than in the unperturbedcase. Furthermore, for solitons with @o)0 both jitters arealways strictly positive. For dark solitons with a negativesoliton initial phase angle (go&0), it has been establishedthat there exists a threshold propagation distance z,& so thatAt&~At& for 0&z&ztI, and At&~At& for z)ztI, and athreshold propagation distance ztI, so that 5t 5t forO~z&z,„and At At for z)z,„.In addition, it has beendemonstrated that, for solitons with $o & 0, there exists apropagation distance zp at which At becomes zero whileAt& is strictly positive. Finally, we have calculated the soli-ton timing jitters in the two limit cases: dark-gray and light-gray solitons.One of the authors (D.M.) acknowledges the partial sup-port of the European Community through the Human Capitaland Mobility Program and the hospitality of the Supercom-puter Center of Catalonia. D.M. is also grateful to ProfessorL. Torner for helpful discussions.[1]A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).[2] Ph. Emplit, J.P. Hamaide, F. Reynaud, G. Froehly, and A. Bar-thelemy, Opt. Commun. 62, 374 (1987);D. 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Akhmediev and A. Ankiewicz, ibid 47, 3213.(1993); L. Gagnon, J. Opt. Soc. Am. B 10, 469 (1993).[11]A.M. Weiner, in Optical Solitons: Theory and Experiment, ed-ited by J.R. Taylor (Cambridge University Press, Cambridge,1992).[12] Yu.S. Kivshar, IEEE J. Quantum Electron. 28, 250 (1993).[13]J.N. Elgin, Phys. Lett. A 110, 441 (1985); Phys. Rev. A 47,4331 (1993); Opt. Lett. 18, 10 (1993).[14] J.P. Gordon and H.A. Haus, Opt. Lett. 11, 665 (1986).[15] Yu.S. Kivshar, M. Haelterman, Ph. Emplit, and J.P. Hamaide,Opt. Lett. 19, 19 (1994).[16] H.A. Haus and M. Nakazawa, J. Opt. Soc. Am. B 4, 652(1987).[17]Yu.S. Kivshar and B.A. Malomed, Rev. Mod. Phys. 61, 763(1989).[18]Yu.S. Kivshar and V.V. Afanasjev, Opt. Lett. 16, 285 (1991).

Dark-Soliton Timing Jitter Caused By Fluctuations In Initial Pulse-Shape

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Dark-Soliton Timing Jitter Caused By Fluctuations In Initial Pulse-Shape

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