Effect of an external periodic potential on pairs of dissipative solitons

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1 PHYSICAL REVIEW A, 35 Eec o an exernal periodic poenial on pairs o dissipaive solions Wonkeun Chang, Nail Akhmediev, and Sean Wabniz Opical Sciences Group, Research School o Physics and Engineering, The Ausralian Naional Universiy, Canberra, Ausralian Capial Terriory, Ausralia Diparimeno di Eleronica per l Auomazione, Universiy o Brescia, Via Branze 3, 53 Brescia, Ialy Received 4 May ; published 7 July We sudy dissipaive solion pair soluions o he complex cubic-quinic Ginzburg-Landau equaion wih periodic phase modulaion erm. The exernal modulaion changes he solion-pair ime separaion as well as heir phase dierence, wihou desroying he exising wo-solion bound sae. Transiions beween dieren sable orms o he pair occur in he orm o biurcaions. Quie remarkably, wo ypes o bound saes may coexis, which leads o hyseresis loops when he modulaion deph is varied. DOI:.3/PhysRevA..35 PACS number s : 4.65.Tg, 4.55., 4.65.Re I. INTRODUCTION The maser-equaion approach describing passively modelocked laser sysems has been iniiaed by Haus. Inone orm or anoher, such approach leads o he cubic-quinic complex Ginzburg-Landau equaion CGLE 5. One o he signiican advanages o using his echnique is ha he properies o he Ginzburg-Landau equaion and o is soluions have been sudied o grea exen in he physics lieraure 6. The caviy-synchronous phase or ampliude modulaion echnique 7, ransorms passively mode-locked opical oscillaors ino acively mode-locked lasers. Mixing passive and acive mode-locking in he same device resuls in a new class o opical oscillaors capable o generaing shor pulses. Clearly, pulses emied by such ype o lasers should exhibi some peculiar properies, depending upon he adjusmen o he many relevan parameers. In paricular, in mixed passively and acively mode-locked lasers, he ineracions among neighboring pulses will be dieren wih respec o he case o purely passive or acive lasers. The descripion o such ineracions naurally raises he quesion o how o ake ino accoun in he maser equaion he presence o exernal modulaion. The naural soluion here would be ha o adding a ime-periodic erm ino he CGLE equaion,. In his work, we sudy sable pulse soluions o he CGLE which has been modiied in such a way. In paricular, we consider pulse pairs generaed by passively mode-locked lasers, and we sudy he eec on he CGLE pulse pair o an exernal ime-periodic phase modulaion. Experimenal resuls on his subjec were recenly repored in Re.. Here, we provide a deeper look ino his problem. Solion pairs in dissipaive sysems were sudied exensively boh heoreically 5 and experimenally 6,7. On he oher hand, he inluence o periodic phase modulaion on solion pairs and heir sabiliy has been irs sudied in Re.. Phase modulaion is clearly an imporan mechanism or he acive mode-locking o lasers. Indeed, pulses ou o laser sysems can be combined ino pairs even wihou he presence o exernal modulaion. Ican also happen ha he phase modulaion isel can induce pulse pair ormaion. Experimenally, solion pairs in acively mode-locked lasers have been observed in 7. Wedono consider his laer case in he presen work. Insead, we assume here ha solion pairs already exis in he passively mode-locked sysem. Their genesis requires a special adjusmen o he laser sysem parameers. We suppose ha his case is reached, and hen apply he exernal phase modulaion o he properly adjused passive laser sysem. The presence o he addiional modulaion may be expeced o improve he sabiliy o pulse-o-pulse ime separaion, or example. Our aim is o provide a general sudy ino wha happens o he bound CGLE solion pairs under he inluence o periodic phase modulaion. In oher words, in his work we resric ourselves o sudy he inluence o phase modulaion on already exising pulse pairs. II. MODEL To model he laser sysem, we use he complex cubicquinic Ginzburg-Landau equaion wih erms corresponding o acive mode-locking in addiion o he usual passive mode-locking erms i z + D i + i + i + i 4 + cos. In absence o he modulaion erm, he parameers and noaions in Eq. are he same as in Re..,z is he normalized envelope o he ield, z is he disance raveled in he caviy, and is he ime rame moving a he group velociy. D accouns or he dispersion, and akes he value D, depending on wheher he group velociy dispersion GVD is anomalous or normal, respecively, is he linear loss i negaive, i conrols he gain dispersion i posiive. and are he cubic and quinic nonlinear gain coeiciens, respecively, and accouns or he quinic nonlineariy. The addiional modulaion-erm cos produces wo more parameers, namely, and. In he presen paper, we assume ha is real. In he res o he CGLE, we use he same parameers as in Re.. These parameers are shown in Fig.. Noe ha we use hose parameers ha are locaed in he middle o he region o exisence o solion pairs denoed as SP, o make sure ha solion pairs do exis and are sable wihou he presence o he modulaion erm. An example o solion pair 5-47// / The American Physical Sociey

2 CHANG, AKHMEDIEV, AND WABNITZ PHYSICAL REVIEW A, 35. SP D ε.4 δ. µ.5 ν.75 β.5 ε.4 S D, δ -. µ -.5, ν β x specra 5 FIG.. Region o exisence o sable saionary solion pairs denoed as SP. This region is enirely locaed wihin he region o sable single-solions S. In he presen paper, we use he parameers o CGLE in he middle o he region SP marked wih a cross. Namely,.4 and.5. This igure is reproduced rom Re.. obained numerically or he SP poin is shown in Fig. a. Solion-pair separaion in his case depends enirely on he parameers o he passive laser sysem. Once ha we have numerically deermined he ime separaion beween he wo solions, we can choose he parameer in such a way ha he period o he periodic poenial ha i creaes exacly corresponds o he solion separaion. i.e., / i, where i is he iniial pair separaion which is purely deermined by he passive mode locking. Clearly his choice provides a minimal disurbance o he solion pair when he periodic poenial is inroduced. This case is shown by he solid line in Fig. c. The disance beween he wo minima o he cosine uncion is he same as he solion ime separaion. The dashed and he doed curves in Fig. c correspond o values ha are greaer or smaller han. Clearly, he disance beween he minima o he cosine uncion or hese values o is eiher narrower or wider han he sable solion separaion. In Fig. b, we also show he specrum or he unperurbed solion pair. The specrum is asymmeric, and i corresponds o a pair wih / phase dierence beween he solions. III. SOLITON PAIRS UNDER THE INFLUENCE OF PHASE MODULATION Nex we consider he eec o adjusing he phasemodulaion ampliude-parameer. When, he periodic poenial vanishes, and he pair separaion corresponds o he one which is given by passive mode locking, i.e., by he res o he parameers o he CGLE. Wih he increasing o, we ound ha he sable pulse separaion also changes. The resuling changes depend upon he paricular value o, asi is shown in Fig. 3. Namely, Fig. 3 a shows he variaion o when is greaer han, while Fig. 3 b shows he variaion o when is smaller han. All curves sar rom he same poin, which corresponds o he iniial ime-separaion i, and cos( ) FIG.. a Solion pair soluion a D,.4,.5,.,.5, and.75 when no exernal modulaion is applied. Is specrum shown in b demonsraes he asymmery o he pair ha has / phase dierence beween he solions. The pair has separaion i.3. c Exernal modulaion applied o he pair is shown in a. Three dieren modulaion requencies are shown here wih he roughs o he modulaion coinciding solid line, narrower dashed line, and wider doed line wih respec o he spacing beween he pair. converge o a new separaion ha is deined by he value o. More precisely, / as is increased. This convergence is aser or he curves in Fig. 3 b, since in his case he exernal modulaion pushes solions urher away rom each oher, and he CGLE ineracions beween he pair become weaker as he pair separaion is increased. Quie remarkably, he presence o a periodic phase modulaion no only inluences he ime separaion bu also he phase dierence beween he solions. Namely, when is increased rom, he phase dierence beween he pair becomes eiher or. This phase dierence converges o or, depending upon he value o. When is above he criical value i.e.,, he phase dierence converges o, while i converges o when. The new resuling ypes o solion pairs are shown in Fig. 4. This phenomenon can be undersood i we will ake ino accoun ha in he ormer case he periodic poenial compresses he wo pulses while in he laer case, i decompresses he pair. The presence o such an exernal orce pushes he solion pair ino he nex saionary posiion. An ineresing and unexpeced observaion is ha he parameers o he solion pair under he inluence o he periodic modulaion are no unique. Le us sar wih he case 35-

3 EFFECT OF AN EXTERNAL PERIODIC POTENTIAL ON phase (π rad) PHYSICAL REVIEW A, 35 (d) 3 (e) () phase (π rad) specra 5 specra where he period o he poenial coincides wih he solionseparaion i. The values o he separaion and phase-dierence vs in his case are shown in Fig. 5. As we can see, he curves are no single valued. When we increase, he ixed poin o he dynamical sysem may dier rom ha obained when we decrease. This eaure o he ixed poins is represened by he arrows on he diagram o Fig. 5. Thus, here is a hyseresis in he curves ha describe he pulse pair separaion and phase dierence, and wo ypes o solion pairs may exis simulaneously. We can also see in Fig. 5 ha he phase-dierence approaches he value when reaches some paricular value. Whe his limi is reached, he phase dierence says consan upon urher increases o. However, he separaion coninues o change. On he conrary, when is decreased, he phase dierence and separaion jump o heir second values a a lower hreshold. This ransiion is shown in Fig. 5 by he verical dashed lines wih arrows. Similar ransiions or he parameers o he solion pairs occur a oher values o. They are no shown in Fig. 3 as he ransiions happen on a much smaller scale. Magniied 6 7 FIG. 3. The pair-separaion agains he modulaion-deph is shown or a and b. The res o he parameers o Eq. are he same as in Fig FIG. 4. Solion-pair a, d proile, b, e phase, and c, specra when.. Three le-hand side LHS panels are or 3/ and hree righ-hand side RHS panels are or /. Noe ha he phase-dierence is zero when and when. porions o he curves near small -s are shown in Fig. 6. The laer correspond o he recangular boxes in Fig. 3 surrounded by he dashed lines. The curves o Fig. 6 reer o he cases when is eiher slighly above or slighly below he value, respecively. In he ormer case, he ime separaion and he phase dierence are decreasing when is increasing. In he laer case, each o hese parameers is increasing. A small, he solion separaion says almos consan, while he phase dierence beween he solions moves gradually o eiher he or he values, depending upon. When one o hese wo values or he phase dierence is reached, a biurcaion occurs. Above he biurcaion FIG. 5. a The pair-separaion and b he phase-dierence vs modulaion-deph or he case. Bisabiliy is observed in he inerval.5.5. The soluion ollows he branch designaed by an arrow when is varied in he posiive or in he negaive direcions. 35-3

4 CHANG, AKHMEDIEV, AND WABNITZ poin, he phase dierence says consan, while he solion separaion changes unil i is adjused o he value ixed by he periodic poenial. The dierence beween he wo values o ha correspond o he biurcaions inerval o bisabiliy is he highes when is equal o he criical-value. This can be undersood as he inluence o he poenial on he solion pair is minimal when. IV. PHASE-DIFFERENCE TRANSFORMATIONS Anoher ineresing eaure o he solion-pair dynamics is revealed a values o ha are small even in he scale o Fig. 6. Namely, in each o Figs. 6 a and 6 c, here is a jump in he separaion beween he iniial separaion i, denoed wih he cross on he verical axis, and he acual branch o soluions. This jump is due o he phase dierence beween he solions ha is / a zero. This phase dierence resuls in asymmeric solion pairs ha possess nonzero velociy 4,7. The direcion o moion o he pair is deined by wheher he above phase dierence is posiive or negaive, respecively. However, when a periodic poenial wih suicien ampliude is applied, i prevens he solion pair rom moving. Thus, when he parameer grows saring rom zero, he solion pair has o readjus is phase dierence o he new condiions. The readjusmen occurs a very small values o. Indeed, small may no preven he pair rom moving across he periodic poenial. However, he velociy o such a moion becomes a uncion o. There is a hreshold value o above which he solion-pair moion sops compleely. As a maer o ac, here are wo regimes o evoluion or he solion pairs in he periodic poenial. A small s, he (d) FIG. 6. a, c The pair-separaion and b, d he phasedierence vs modulaion-deph when wo LHS panels or wo RHS panels. The wo plos a he op are magniicaions o pars o he curves in Fig. 3 wihin he recangles bounded by dashed lines. Bisabiliy is observed when.4.7 panels a and b and.6.76 panels c and d. The arrows show he branches o he curves ha are ollowed when is varied in he posiive or negaive direcions. sin φ cos φ PHYSICAL REVIEW A, sin φ.6..4 walls o he periodic poenial canno capure he solion pair. Solions can move across he periodic poenial and converge o an oscillaing moving sae, being disurbed periodically. The phase dierence beween he wo solions in he pair remains close o /. This means ha he pair is ormed mainly by he passive mode-locking par o he CGLE. The phase rajecory o he soluion on he ineracion plane in his case is shown in Fig. 7 a. The deiniion o ineracion plane is given in Re. and should be clear rom he axes labels in Fig. 7. Due o he moion o he pair across he periodic poenial, he rajecory on he ineracion plane has a double spiraling eaure. This means ha he moion conains wo requencies. One o he requencies is deined by he inernal oscillaions o he solion molecule. Namely, he wo solions bound ogeher oscillae relaive o he common cener o mass. The second lower requency is deined by he moion o he pair across he periodic poenial. I is deined by he period o poenial and velociy o he pair. One complee roaion around a poin locaed a.46,.33 in Fig. 7 a corresponds o his moion across one period o he poenial, while smaller loops o he rajecory correspond o he inernal oscillaions. These inernal oscillaions evenually decay, and he rajecory converges o a limi cycle around he same poin.46,.33. The convergence occurs over a scale o more han a hundred periods. Hence, he double spiral represens only a small par o he rajecory. The resuling limi cycle is shown in Fig. 7 a wih a hick line. This complicaed moion o he solion pair can be seen in Fig.. The sripes ha correspond o each solion show acceleraion and deceleraion o he solion pair ha are synchronized wih he moion in he periodic poenial. Sill exising and more requen oscillaions show insead he inernal pulsaions o he pair. When grows larger han a cerain hreshold, he periodic poenial conines he pair compleely, so ha he solion pair canno move across i. The phase rajecory in his case is shown in Fig. 7 b. The periodic moion o he pair across he poenial is sopped, and he rajecory converges ino a. cos φ FIG. 7. Color online Trajecories on he ineracion plane Re. showing convergence o he solion pair o a sable sae when 7/6. a A small values o, he pair jumps over he periodic poenial causing double spiral on convergence. This converges o a limi cycle shown wih a hick dashed line. b A larger values o he pair converges direcly o he minima o he periodic poenial wihou jumping beween he periods. Thus, convergence o a ixed poin occurs as a single spiral. 35-4

EFFECT OF AN EXTERNAL PERIODIC POTENTIAL ON z 5 4 3 FIG.. Color online Moion o he dissipaive solion pair across he periodic poenial.

5 EFFECT OF AN EXTERNAL PERIODIC POTENTIAL ON z FIG.. Color online Moion o he dissipaive solion pair across he periodic poenial. This case corresponds o he rajecory on he ineracion plane shown in Fig. 7 a. The sinusoid below he plo shows he poenial. Solion pair experiences inernal decaying oscillaions as well as periodic acceleraions and deceleraions when moving across he poenial. PHYSICAL REVIEW A, 35 sable ixed poin along a simple single spiral. The ixed poin here corresponds o a pair wih he phase dierence beween solions which is neiher / nor or. The phase dierence shis o or when increases. The plos presened in Fig. 6 are relaed o hese ixed poins. A ew experimenal papers on he eec o synchronous phase modulaion on pulse pairs rom mode-locked iber lasers have recenly appeared in he lieraure 7,. In boh Res. 7,, he period o he imposed phase modulaion i.e., ps and ns was much higher han he observed solion-pair separaion 4 ps and 5 ps, respecively. Thereore i appears ha in hose experimens, a dieren regime was considered wih respec o he siuaion ha we have analyzed in our work. Indeed, in he presen paper, we provide a deailed invesigaion o he dynamics o solion-pair separaion in he complemenary siuaion o a phasemodulaion period which is close o he pulse-pair separaion as i is deined by he properies o he passive mode-locked caviy. In his case, we have shown ha he synchronizaion beween he phase modulaion and he solion pairs exhibi a complex dynamics, involving solion-pair biurcaions. We believe ha exploiing he mechanism o such solion ineracions in hybrid mode-locked lasers would allow or a more accurae conrol o he requency and sabiliy o he generaed pulse rains. V. CONCLUSIONS We sudied he inluence o a periodic poenial on he pair o dissipaive solions when he wo neares minima o he poenial are close o he separaion beween he solions. The pair separaion approaches / and he phase dierence beween he pair become eiher or, depending upon he modulaion requency as he modulaion deph is increased. These ransiions rom one ype o solion pair o anoher comprise hyseresis loops. For a very small value o he modulaion deph, he periodic poenial is no srong enough o conine he moving pairs. This causes periodic acceleraion and deceleraion o he solion pairs when moving across he poenial. ACKNOWLEDGMENTS N.A. and W.C. graeully acknowledge he suppor o he Ausralian Research Council Discovery Projec No. DP534. H. A. Haus, IEEE J. Quanum Elecron., A. Komarov, H. Leblond, and F. Sanchez, Phys. Rev. E 7, 564 R 5. 3 W. H. Renninger, A. Chong, and F. W. Wise, Phys. Rev. A 77, J. N. Kuz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundi, and W. H. Knox, J. Op. Soc. Am. B 4, J. D. Moores, Op. Commun. 6, I. Aranson and L. Kramer, Rev. Mod. Phys. 74,. 7 W.-W. Hsiang, C.-Y. Lin, and Y. Lai, Op. Le. 3, W.-W. Hsiang, C.-Y. Lin, M. Tien, and Y. Lai, Op. Le. 3, J. D. Moores, Op. Le. 6, 7. J. J. O Neil, J. N. Kuz, and B. Sandsede, IEEE J. Quanum Elecron. 3, 4. N. D. Nguyen and L. N. Binh, Op. Commun., ;, 34. H. R. Brand and R. J. Deissler, Phys. Rev. Le. 63,. 3 B. A. Malomed, Phys. Rev. A 44, N. N. Akhmediev, A. Ankiewicz, and J. M. Soo-Crespo, J. Op. Soc. Am. B 5, D. Turaev, A. G. Vladimirov, and S. Zelik, Phys. Rev. E 75, 456 R 7. 6 J. M. Soo-Crespo, N. Akhmediev, Ph. Grelu, and F. Belhache, Op. Le., Ph. Grelu and N. Akhmediev, Op. Express, S. Wabniz, Elecron. Le., 7 3. N. N. Akhmediev, A. Ankiewicz, and J. M. Soo-Crespo, Phys. Rev. Le. 7, J. M. Soo-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, Phys. Rev. E 75,

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still. Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in